1,1,11,11,0.0168518,"\int \frac{x}{\left(-1+x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[x/(-1 + x^2)^(3/4),x]","2 \sqrt[4]{-1+x^2}","2 \sqrt[4]{-1+x^2}",1,"2*(-1 + x^2)^(1/4)","A",1
2,1,12,12,0.0098225,"\int \frac{1+3 x^2}{\sqrt{-1+x+x^3}} \, dx","IntegrateAlgebraic[(1 + 3*x^2)/Sqrt[-1 + x + x^3],x]","2 \sqrt{-1+x+x^3}","2 \sqrt{-1+x+x^3}",1,"2*Sqrt[-1 + x + x^3]","A",1
3,1,12,12,0.2780065,"\int \frac{-1+x^8}{\sqrt{-1+x^4} \left(1-2 x^4+x^8\right)} \, dx","IntegrateAlgebraic[(-1 + x^8)/(Sqrt[-1 + x^4]*(1 - 2*x^4 + x^8)),x]","-\frac{x}{\sqrt{-1+x^4}}","-\frac{x}{\sqrt{-1+x^4}}",1,"-(x/Sqrt[-1 + x^4])","A",1
4,1,13,13,0.0083216,"\int \frac{x}{\sqrt[3]{-1+x^2}} \, dx","IntegrateAlgebraic[x/(-1 + x^2)^(1/3),x]","\frac{3}{4} \left(-1+x^2\right)^{2/3}","\frac{3}{4} \left(-1+x^2\right)^{2/3}",1,"(3*(-1 + x^2)^(2/3))/4","A",1
5,1,13,13,0.0085334,"\int \frac{x}{\sqrt[4]{-1+x^2}} \, dx","IntegrateAlgebraic[x/(-1 + x^2)^(1/4),x]","\frac{2}{3} \left(-1+x^2\right)^{3/4}","\frac{2}{3} \left(-1+x^2\right)^{3/4}",1,"(2*(-1 + x^2)^(3/4))/3","A",1
6,1,13,13,0.0081753,"\int x \sqrt[4]{-1+x^2} \, dx","IntegrateAlgebraic[x*(-1 + x^2)^(1/4),x]","\frac{2}{5} \left(-1+x^2\right)^{5/4}","\frac{2}{5} \left(-1+x^2\right)^{5/4}",1,"(2*(-1 + x^2)^(5/4))/5","A",1
7,1,13,13,0.0074319,"\int x \sqrt[3]{-1+x^2} \, dx","IntegrateAlgebraic[x*(-1 + x^2)^(1/3),x]","\frac{3}{8} \left(-1+x^2\right)^{4/3}","\frac{3}{8} \left(-1+x^2\right)^{4/3}",1,"(3*(-1 + x^2)^(4/3))/8","A",1
8,1,13,13,0.0083107,"\int x \left(-1+x^2\right)^{2/3} \, dx","IntegrateAlgebraic[x*(-1 + x^2)^(2/3),x]","\frac{3}{10} \left(-1+x^2\right)^{5/3}","\frac{3}{10} \left(-1+x^2\right)^{5/3}",1,"(3*(-1 + x^2)^(5/3))/10","A",1
9,1,13,13,0.0087792,"\int x \left(-1+x^2\right)^{3/4} \, dx","IntegrateAlgebraic[x*(-1 + x^2)^(3/4),x]","\frac{2}{7} \left(-1+x^2\right)^{7/4}","\frac{2}{7} \left(-1+x^2\right)^{7/4}",1,"(2*(-1 + x^2)^(7/4))/7","A",1
10,1,13,13,0.0084317,"\int \frac{x}{\sqrt[3]{1+x^2}} \, dx","IntegrateAlgebraic[x/(1 + x^2)^(1/3),x]","\frac{3}{4} \left(1+x^2\right)^{2/3}","\frac{3}{4} \left(1+x^2\right)^{2/3}",1,"(3*(1 + x^2)^(2/3))/4","A",1
11,1,13,13,0.0084694,"\int \frac{x}{\sqrt[4]{1+x^2}} \, dx","IntegrateAlgebraic[x/(1 + x^2)^(1/4),x]","\frac{2}{3} \left(1+x^2\right)^{3/4}","\frac{2}{3} \left(1+x^2\right)^{3/4}",1,"(2*(1 + x^2)^(3/4))/3","A",1
12,1,13,13,0.0072173,"\int x \sqrt[4]{1+x^2} \, dx","IntegrateAlgebraic[x*(1 + x^2)^(1/4),x]","\frac{2}{5} \left(1+x^2\right)^{5/4}","\frac{2}{5} \left(1+x^2\right)^{5/4}",1,"(2*(1 + x^2)^(5/4))/5","A",1
13,1,13,13,0.0075611,"\int x \sqrt[3]{1+x^2} \, dx","IntegrateAlgebraic[x*(1 + x^2)^(1/3),x]","\frac{3}{8} \left(1+x^2\right)^{4/3}","\frac{3}{8} \left(1+x^2\right)^{4/3}",1,"(3*(1 + x^2)^(4/3))/8","A",1
14,1,13,13,0.0078843,"\int x \left(1+x^2\right)^{3/4} \, dx","IntegrateAlgebraic[x*(1 + x^2)^(3/4),x]","\frac{2}{7} \left(1+x^2\right)^{7/4}","\frac{2}{7} \left(1+x^2\right)^{7/4}",1,"(2*(1 + x^2)^(7/4))/7","A",1
15,1,13,13,0.0110869,"\int \frac{x^2}{\sqrt[4]{-1+x^3}} \, dx","IntegrateAlgebraic[x^2/(-1 + x^3)^(1/4),x]","\frac{4}{9} \left(-1+x^3\right)^{3/4}","\frac{4}{9} \left(-1+x^3\right)^{3/4}",1,"(4*(-1 + x^3)^(3/4))/9","A",1
16,1,13,13,0.0096625,"\int x^2 \sqrt[4]{-1+x^3} \, dx","IntegrateAlgebraic[x^2*(-1 + x^3)^(1/4),x]","\frac{4}{15} \left(-1+x^3\right)^{5/4}","\frac{4}{15} \left(-1+x^3\right)^{5/4}",1,"(4*(-1 + x^3)^(5/4))/15","A",1
17,1,13,13,0.0130686,"\int x^2 \left(-1+x^3\right)^{3/4} \, dx","IntegrateAlgebraic[x^2*(-1 + x^3)^(3/4),x]","\frac{4}{21} \left(-1+x^3\right)^{7/4}","\frac{4}{21} \left(-1+x^3\right)^{7/4}",1,"(4*(-1 + x^3)^(7/4))/21","A",1
18,1,13,13,0.0100932,"\int \frac{x^2}{\sqrt{1+x^3}} \, dx","IntegrateAlgebraic[x^2/Sqrt[1 + x^3],x]","\frac{2 \sqrt{1+x^3}}{3}","\frac{2 \sqrt{1+x^3}}{3}",1,"(2*Sqrt[1 + x^3])/3","A",1
19,1,13,13,0.0107344,"\int \frac{x^2}{\sqrt[4]{1+x^3}} \, dx","IntegrateAlgebraic[x^2/(1 + x^3)^(1/4),x]","\frac{4}{9} \left(1+x^3\right)^{3/4}","\frac{4}{9} \left(1+x^3\right)^{3/4}",1,"(4*(1 + x^3)^(3/4))/9","A",1
20,1,13,13,0.0098699,"\int x^2 \sqrt[4]{1+x^3} \, dx","IntegrateAlgebraic[x^2*(1 + x^3)^(1/4),x]","\frac{4}{15} \left(1+x^3\right)^{5/4}","\frac{4}{15} \left(1+x^3\right)^{5/4}",1,"(4*(1 + x^3)^(5/4))/15","A",1
21,1,13,13,0.0101151,"\int x^2 \sqrt[3]{1+x^3} \, dx","IntegrateAlgebraic[x^2*(1 + x^3)^(1/3),x]","\frac{1}{4} \left(1+x^3\right)^{4/3}","\frac{1}{4} \left(1+x^3\right)^{4/3}",1,"(1 + x^3)^(4/3)/4","A",1
22,1,13,13,0.0095051,"\int x^2 \sqrt{1+x^3} \, dx","IntegrateAlgebraic[x^2*Sqrt[1 + x^3],x]","\frac{2}{9} \left(1+x^3\right)^{3/2}","\frac{2}{9} \left(1+x^3\right)^{3/2}",1,"(2*(1 + x^3)^(3/2))/9","A",1
23,1,13,13,0.0109709,"\int x^2 \left(1+x^3\right)^{2/3} \, dx","IntegrateAlgebraic[x^2*(1 + x^3)^(2/3),x]","\frac{1}{5} \left(1+x^3\right)^{5/3}","\frac{1}{5} \left(1+x^3\right)^{5/3}",1,"(1 + x^3)^(5/3)/5","A",1
24,1,13,13,0.0126972,"\int x^2 \left(1+x^3\right)^{3/4} \, dx","IntegrateAlgebraic[x^2*(1 + x^3)^(3/4),x]","\frac{4}{21} \left(1+x^3\right)^{7/4}","\frac{4}{21} \left(1+x^3\right)^{7/4}",1,"(4*(1 + x^3)^(7/4))/21","A",1
25,1,13,13,0.011896,"\int \left(1+3 x^2\right) \sqrt[3]{x+x^3} \, dx","IntegrateAlgebraic[(1 + 3*x^2)*(x + x^3)^(1/3),x]","\frac{3}{4} \left(x+x^3\right)^{4/3}","\frac{3}{4} \left(x+x^3\right)^{4/3}",1,"(3*(x + x^3)^(4/3))/4","A",1
26,1,13,13,0.1393572,"\int \frac{1}{x^2 \left(-1+x^4\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x^2*(-1 + x^4)^(3/4)),x]","\frac{\sqrt[4]{-1+x^4}}{x}","\frac{\sqrt[4]{-1+x^4}}{x}",1,"(-1 + x^4)^(1/4)/x","A",1
27,1,13,13,0.0113812,"\int \frac{x^3}{\sqrt[3]{-1+x^4}} \, dx","IntegrateAlgebraic[x^3/(-1 + x^4)^(1/3),x]","\frac{3}{8} \left(-1+x^4\right)^{2/3}","\frac{3}{8} \left(-1+x^4\right)^{2/3}",1,"(3*(-1 + x^4)^(2/3))/8","A",1
28,1,13,13,0.0104043,"\int x^3 \sqrt[3]{-1+x^4} \, dx","IntegrateAlgebraic[x^3*(-1 + x^4)^(1/3),x]","\frac{3}{16} \left(-1+x^4\right)^{4/3}","\frac{3}{16} \left(-1+x^4\right)^{4/3}",1,"(3*(-1 + x^4)^(4/3))/16","A",1
29,1,13,13,0.0117835,"\int x^3 \left(-1+x^4\right)^{2/3} \, dx","IntegrateAlgebraic[x^3*(-1 + x^4)^(2/3),x]","\frac{3}{20} \left(-1+x^4\right)^{5/3}","\frac{3}{20} \left(-1+x^4\right)^{5/3}",1,"(3*(-1 + x^4)^(5/3))/20","A",1
30,1,13,13,0.0135889,"\int x^3 \left(-1+x^4\right)^{3/4} \, dx","IntegrateAlgebraic[x^3*(-1 + x^4)^(3/4),x]","\frac{1}{7} \left(-1+x^4\right)^{7/4}","\frac{1}{7} \left(-1+x^4\right)^{7/4}",1,"(-1 + x^4)^(7/4)/7","A",1
31,1,13,13,0.536067,"\int \frac{-1+x^4}{x^2 \sqrt{1+x^4}} \, dx","IntegrateAlgebraic[(-1 + x^4)/(x^2*Sqrt[1 + x^4]),x]","\frac{\sqrt{1+x^4}}{x}","\frac{\sqrt{1+x^4}}{x}",1,"Sqrt[1 + x^4]/x","A",1
32,1,13,13,0.0106377,"\int \frac{x^3}{\sqrt[3]{1+x^4}} \, dx","IntegrateAlgebraic[x^3/(1 + x^4)^(1/3),x]","\frac{3}{8} \left(1+x^4\right)^{2/3}","\frac{3}{8} \left(1+x^4\right)^{2/3}",1,"(3*(1 + x^4)^(2/3))/8","A",1
33,1,13,13,0.0099454,"\int x^3 \sqrt[4]{1+x^4} \, dx","IntegrateAlgebraic[x^3*(1 + x^4)^(1/4),x]","\frac{1}{5} \left(1+x^4\right)^{5/4}","\frac{1}{5} \left(1+x^4\right)^{5/4}",1,"(1 + x^4)^(5/4)/5","A",1
34,1,13,13,0.0101623,"\int x^3 \sqrt[3]{1+x^4} \, dx","IntegrateAlgebraic[x^3*(1 + x^4)^(1/3),x]","\frac{3}{16} \left(1+x^4\right)^{4/3}","\frac{3}{16} \left(1+x^4\right)^{4/3}",1,"(3*(1 + x^4)^(4/3))/16","A",1
35,1,13,13,0.0117414,"\int x^3 \left(1+x^4\right)^{2/3} \, dx","IntegrateAlgebraic[x^3*(1 + x^4)^(2/3),x]","\frac{3}{20} \left(1+x^4\right)^{5/3}","\frac{3}{20} \left(1+x^4\right)^{5/3}",1,"(3*(1 + x^4)^(5/3))/20","A",1
36,1,13,13,0.0119142,"\int x^4 \left(-1+x^5\right)^{2/3} \, dx","IntegrateAlgebraic[x^4*(-1 + x^5)^(2/3),x]","\frac{3}{25} \left(-1+x^5\right)^{5/3}","\frac{3}{25} \left(-1+x^5\right)^{5/3}",1,"(3*(-1 + x^5)^(5/3))/25","A",1
37,1,13,13,0.0112147,"\int x^4 \left(1+x^5\right)^{2/3} \, dx","IntegrateAlgebraic[x^4*(1 + x^5)^(2/3),x]","\frac{3}{25} \left(1+x^5\right)^{5/3}","\frac{3}{25} \left(1+x^5\right)^{5/3}",1,"(3*(1 + x^5)^(5/3))/25","A",1
38,1,13,13,0.0113333,"\int \frac{x^5}{\sqrt[3]{-1+x^6}} \, dx","IntegrateAlgebraic[x^5/(-1 + x^6)^(1/3),x]","\frac{1}{4} \left(-1+x^6\right)^{2/3}","\frac{1}{4} \left(-1+x^6\right)^{2/3}",1,"(-1 + x^6)^(2/3)/4","A",1
39,1,13,13,0.0107414,"\int x^5 \sqrt[4]{-1+x^6} \, dx","IntegrateAlgebraic[x^5*(-1 + x^6)^(1/4),x]","\frac{2}{15} \left(-1+x^6\right)^{5/4}","\frac{2}{15} \left(-1+x^6\right)^{5/4}",1,"(2*(-1 + x^6)^(5/4))/15","A",1
40,1,13,13,0.0101109,"\int x^5 \sqrt[3]{-1+x^6} \, dx","IntegrateAlgebraic[x^5*(-1 + x^6)^(1/3),x]","\frac{1}{8} \left(-1+x^6\right)^{4/3}","\frac{1}{8} \left(-1+x^6\right)^{4/3}",1,"(-1 + x^6)^(4/3)/8","A",1
41,1,13,13,0.0103668,"\int x^5 \sqrt{-1+x^6} \, dx","IntegrateAlgebraic[x^5*Sqrt[-1 + x^6],x]","\frac{1}{9} \left(-1+x^6\right)^{3/2}","\frac{1}{9} \left(-1+x^6\right)^{3/2}",1,"(-1 + x^6)^(3/2)/9","A",1
42,1,13,13,0.0136735,"\int x^5 \left(-1+x^6\right)^{3/4} \, dx","IntegrateAlgebraic[x^5*(-1 + x^6)^(3/4),x]","\frac{2}{21} \left(-1+x^6\right)^{7/4}","\frac{2}{21} \left(-1+x^6\right)^{7/4}",1,"(2*(-1 + x^6)^(7/4))/21","A",1
43,1,13,13,1.7052982,"\int \frac{-2+x^6}{x^3 \sqrt{1+x^6}} \, dx","IntegrateAlgebraic[(-2 + x^6)/(x^3*Sqrt[1 + x^6]),x]","\frac{\sqrt{1+x^6}}{x^2}","\frac{\sqrt{1+x^6}}{x^2}",1,"Sqrt[1 + x^6]/x^2","A",1
44,1,13,13,0.0135269,"\int \frac{x^5}{\sqrt[3]{1+x^6}} \, dx","IntegrateAlgebraic[x^5/(1 + x^6)^(1/3),x]","\frac{1}{4} \left(1+x^6\right)^{2/3}","\frac{1}{4} \left(1+x^6\right)^{2/3}",1,"(1 + x^6)^(2/3)/4","A",1
45,1,13,13,0.0104508,"\int x^5 \sqrt[4]{1+x^6} \, dx","IntegrateAlgebraic[x^5*(1 + x^6)^(1/4),x]","\frac{2}{15} \left(1+x^6\right)^{5/4}","\frac{2}{15} \left(1+x^6\right)^{5/4}",1,"(2*(1 + x^6)^(5/4))/15","A",1
46,1,13,13,0.011563,"\int x^5 \sqrt[3]{1+x^6} \, dx","IntegrateAlgebraic[x^5*(1 + x^6)^(1/3),x]","\frac{1}{8} \left(1+x^6\right)^{4/3}","\frac{1}{8} \left(1+x^6\right)^{4/3}",1,"(1 + x^6)^(4/3)/8","A",1
47,1,14,14,2.0972695,"\int \frac{-4+x^3}{x^2 \left(-1+x^3\right)^{3/4}} \, dx","IntegrateAlgebraic[(-4 + x^3)/(x^2*(-1 + x^3)^(3/4)),x]","-\frac{4 \sqrt[4]{-1+x^3}}{x}","-\frac{4 \sqrt[4]{-1+x^3}}{x}",1,"(-4*(-1 + x^3)^(1/4))/x","A",1
48,1,14,14,0.023901,"\int \frac{1}{x \sqrt{1+x^3}} \, dx","IntegrateAlgebraic[1/(x*Sqrt[1 + x^3]),x]","-\frac{2}{3} \tanh ^{-1}\left(\sqrt{1+x^3}\right)","-\frac{2}{3} \tanh ^{-1}\left(\sqrt{1+x^3}\right)",1,"(-2*ArcTanh[Sqrt[1 + x^3]])/3","A",1
49,1,14,14,2.0879612,"\int \frac{4+x^3}{x^2 \left(1+x^3\right)^{3/4}} \, dx","IntegrateAlgebraic[(4 + x^3)/(x^2*(1 + x^3)^(3/4)),x]","-\frac{4 \sqrt[4]{1+x^3}}{x}","-\frac{4 \sqrt[4]{1+x^3}}{x}",1,"(-4*(1 + x^3)^(1/4))/x","A",1
50,1,14,14,0.183496,"\int \frac{\left(2+3 x^2\right) \sqrt[3]{x+x^3}}{1+x^2} \, dx","IntegrateAlgebraic[((2 + 3*x^2)*(x + x^3)^(1/3))/(1 + x^2),x]","\frac{3}{2} x \sqrt[3]{x+x^3}","\frac{3}{2} x \sqrt[3]{x+x^3}",1,"(3*x*(x + x^3)^(1/3))/2","A",1
51,1,14,14,6.1533363,"\int \frac{-2+x}{(-1+x) \sqrt[4]{-x^2+x^3}} \, dx","IntegrateAlgebraic[(-2 + x)/((-1 + x)*(-x^2 + x^3)^(1/4)),x]","\frac{4 x}{\sqrt[4]{(-1+x) x^2}}","\frac{4 x}{\sqrt[4]{(-1+x) x^2}}",1,"(4*x)/((-1 + x)*x^2)^(1/4)","A",1
52,1,14,14,6.438879,"\int \frac{2+x}{(1+x) \sqrt[4]{x^2+x^3}} \, dx","IntegrateAlgebraic[(2 + x)/((1 + x)*(x^2 + x^3)^(1/4)),x]","\frac{4 x}{\sqrt[4]{x^2 (1+x)}}","\frac{4 x}{\sqrt[4]{x^2 (1+x)}}",1,"(4*x)/(x^2*(1 + x))^(1/4)","A",1
53,1,14,14,0.1449587,"\int \frac{1}{x^2 \left(1+x^4\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x^2*(1 + x^4)^(3/4)),x]","-\frac{\sqrt[4]{1+x^4}}{x}","-\frac{\sqrt[4]{1+x^4}}{x}",1,"-((1 + x^4)^(1/4)/x)","A",1
54,1,14,14,0.6100788,"\int \frac{3+x^4}{x^4 \sqrt{1+x^4}} \, dx","IntegrateAlgebraic[(3 + x^4)/(x^4*Sqrt[1 + x^4]),x]","-\frac{\sqrt{1+x^4}}{x^3}","-\frac{\sqrt{1+x^4}}{x^3}",1,"-(Sqrt[1 + x^4]/x^3)","A",1
55,1,14,14,0.0125747,"\int \frac{-1+4 x^3}{\sqrt{-1-2 x+2 x^4}} \, dx","IntegrateAlgebraic[(-1 + 4*x^3)/Sqrt[-1 - 2*x + 2*x^4],x]","\sqrt{-1-2 x+2 x^4}","\sqrt{-1-2 x+2 x^4}",1,"Sqrt[-1 - 2*x + 2*x^4]","A",1
56,1,14,14,0.1863492,"\int \frac{-4+x^5}{x^2 \left(1+x^5\right)^{3/4}} \, dx","IntegrateAlgebraic[(-4 + x^5)/(x^2*(1 + x^5)^(3/4)),x]","\frac{4 \sqrt[4]{1+x^5}}{x}","\frac{4 \sqrt[4]{1+x^5}}{x}",1,"(4*(1 + x^5)^(1/4))/x","A",1
57,1,14,14,0.184229,"\int \frac{4+x^5}{x^2 \left(-1+x^5\right)^{3/4}} \, dx","IntegrateAlgebraic[(4 + x^5)/(x^2*(-1 + x^5)^(3/4)),x]","\frac{4 \sqrt[4]{-1+x^5}}{x}","\frac{4 \sqrt[4]{-1+x^5}}{x}",1,"(4*(-1 + x^5)^(1/4))/x","A",1
58,1,14,14,1.6649928,"\int \frac{2+5 x^3}{\sqrt{1+x^3} \left(1+x^2+x^5\right)} \, dx","IntegrateAlgebraic[(2 + 5*x^3)/(Sqrt[1 + x^3]*(1 + x^2 + x^5)),x]","2 \tan ^{-1}\left(x \sqrt{1+x^3}\right)","2 \tan ^{-1}\left(x \sqrt{1+x^3}\right)",1,"2*ArcTan[x*Sqrt[1 + x^3]]","A",1
59,1,14,14,0.0148182,"\int \frac{1}{x \sqrt{-1+x^6}} \, dx","IntegrateAlgebraic[1/(x*Sqrt[-1 + x^6]),x]","\frac{1}{3} \tan ^{-1}\left(\sqrt{-1+x^6}\right)","\frac{1}{3} \tan ^{-1}\left(\sqrt{-1+x^6}\right)",1,"ArcTan[Sqrt[-1 + x^6]]/3","A",1
60,1,14,14,0.2768998,"\int \frac{-2+x^6}{x^2 \left(1+x^6\right)^{3/4}} \, dx","IntegrateAlgebraic[(-2 + x^6)/(x^2*(1 + x^6)^(3/4)),x]","\frac{2 \sqrt[4]{1+x^6}}{x}","\frac{2 \sqrt[4]{1+x^6}}{x}",1,"(2*(1 + x^6)^(1/4))/x","A",1
61,1,14,14,0.0167485,"\int \frac{1}{x \sqrt{1+x^6}} \, dx","IntegrateAlgebraic[1/(x*Sqrt[1 + x^6]),x]","-\frac{1}{3} \tanh ^{-1}\left(\sqrt{1+x^6}\right)","-\frac{1}{3} \tanh ^{-1}\left(\sqrt{1+x^6}\right)",1,"-1/3*ArcTanh[Sqrt[1 + x^6]]","A",1
62,1,14,14,0.2756614,"\int \frac{2+x^6}{x^2 \left(-1+x^6\right)^{3/4}} \, dx","IntegrateAlgebraic[(2 + x^6)/(x^2*(-1 + x^6)^(3/4)),x]","\frac{2 \sqrt[4]{-1+x^6}}{x}","\frac{2 \sqrt[4]{-1+x^6}}{x}",1,"(2*(-1 + x^6)^(1/4))/x","A",1
63,1,14,14,3.4151331,"\int \frac{-1+2 x^6}{\sqrt{1+x^6} \left(1-x^2+x^6\right)} \, dx","IntegrateAlgebraic[(-1 + 2*x^6)/(Sqrt[1 + x^6]*(1 - x^2 + x^6)),x]","-\tanh ^{-1}\left(\frac{x}{\sqrt{1+x^6}}\right)","-\tanh ^{-1}\left(\frac{x}{\sqrt{1+x^6}}\right)",1,"-ArcTanh[x/Sqrt[1 + x^6]]","A",1
64,1,14,14,3.3833048,"\int \frac{1+2 x^6}{\sqrt{-1+x^6} \left(-1+x^2+x^6\right)} \, dx","IntegrateAlgebraic[(1 + 2*x^6)/(Sqrt[-1 + x^6]*(-1 + x^2 + x^6)),x]","-\tan ^{-1}\left(\frac{x}{\sqrt{-1+x^6}}\right)","-\tan ^{-1}\left(\frac{x}{\sqrt{-1+x^6}}\right)",1,"-ArcTan[x/Sqrt[-1 + x^6]]","A",1
65,1,15,15,0.0145633,"\int \frac{-1+3 x^2}{\sqrt[3]{-x+x^3}} \, dx","IntegrateAlgebraic[(-1 + 3*x^2)/(-x + x^3)^(1/3),x]","\frac{3}{2} \left(-x+x^3\right)^{2/3}","\frac{3}{2} \left(-x+x^3\right)^{2/3}",1,"(3*(-x + x^3)^(2/3))/2","A",1
66,1,15,15,0.0133764,"\int \left(-1+3 x^2\right) \sqrt[3]{-x+x^3} \, dx","IntegrateAlgebraic[(-1 + 3*x^2)*(-x + x^3)^(1/3),x]","\frac{3}{4} \left(-x+x^3\right)^{4/3}","\frac{3}{4} \left(-x+x^3\right)^{4/3}",1,"(3*(-x + x^3)^(4/3))/4","A",1
67,1,15,15,0.2629405,"\int \frac{2+x-x^3}{\sqrt{1+x+x^3} \left(1+x-x^2+x^3\right)} \, dx","IntegrateAlgebraic[(2 + x - x^3)/(Sqrt[1 + x + x^3]*(1 + x - x^2 + x^3)),x]","2 \tanh ^{-1}\left(\frac{x}{\sqrt{1+x+x^3}}\right)","2 \tanh ^{-1}\left(\frac{x}{\sqrt{1+x+x^3}}\right)",1,"2*ArcTanh[x/Sqrt[1 + x + x^3]]","A",1
68,1,15,15,0.2319933,"\int \frac{1+x^4}{\left(1-x^4\right) \sqrt{-1+x^2+x^4}} \, dx","IntegrateAlgebraic[(1 + x^4)/((1 - x^4)*Sqrt[-1 + x^2 + x^4]),x]","\tanh ^{-1}\left(\frac{x}{\sqrt{-1+x^2+x^4}}\right)","\tanh ^{-1}\left(\frac{x}{\sqrt{-1+x^2+x^4}}\right)",1,"ArcTanh[x/Sqrt[-1 + x^2 + x^4]]","A",1
69,1,16,16,0.0949237,"\int \frac{-4+x^3}{x^4 \sqrt[4]{-1+x^3}} \, dx","IntegrateAlgebraic[(-4 + x^3)/(x^4*(-1 + x^3)^(1/4)),x]","-\frac{4 \left(-1+x^3\right)^{3/4}}{3 x^3}","-\frac{4 \left(-1+x^3\right)^{3/4}}{3 x^3}",1,"(-4*(-1 + x^3)^(3/4))/(3*x^3)","A",1
70,1,16,16,0.067739,"\int \frac{\sqrt[3]{-1+x^3}}{x^5} \, dx","IntegrateAlgebraic[(-1 + x^3)^(1/3)/x^5,x]","\frac{\left(-1+x^3\right)^{4/3}}{4 x^4}","\frac{\left(-1+x^3\right)^{4/3}}{4 x^4}",1,"(-1 + x^3)^(4/3)/(4*x^4)","A",1
71,1,16,16,0.0712225,"\int \frac{\left(-1+x^3\right)^{2/3}}{x^6} \, dx","IntegrateAlgebraic[(-1 + x^3)^(2/3)/x^6,x]","\frac{\left(-1+x^3\right)^{5/3}}{5 x^5}","\frac{\left(-1+x^3\right)^{5/3}}{5 x^5}",1,"(-1 + x^3)^(5/3)/(5*x^5)","A",1
72,1,16,16,0.0680712,"\int \frac{\sqrt[3]{1+x^3}}{x^5} \, dx","IntegrateAlgebraic[(1 + x^3)^(1/3)/x^5,x]","-\frac{\left(1+x^3\right)^{4/3}}{4 x^4}","-\frac{\left(1+x^3\right)^{4/3}}{4 x^4}",1,"-1/4*(1 + x^3)^(4/3)/x^4","A",1
73,1,16,16,0.0688011,"\int \frac{\left(1+x^3\right)^{2/3}}{x^6} \, dx","IntegrateAlgebraic[(1 + x^3)^(2/3)/x^6,x]","-\frac{\left(1+x^3\right)^{5/3}}{5 x^5}","-\frac{\left(1+x^3\right)^{5/3}}{5 x^5}",1,"-1/5*(1 + x^3)^(5/3)/x^5","A",1
74,1,16,16,0.0610637,"\int \frac{\left(-2+x^3\right) \left(1+x^3\right)^{3/2}}{x^6} \, dx","IntegrateAlgebraic[((-2 + x^3)*(1 + x^3)^(3/2))/x^6,x]","\frac{2 \left(1+x^3\right)^{5/2}}{5 x^5}","\frac{2 \left(1+x^3\right)^{5/2}}{5 x^5}",1,"(2*(1 + x^3)^(5/2))/(5*x^5)","A",1
75,1,16,16,0.0292074,"\int \frac{4+x^3}{x^4 \sqrt[4]{1+x^3}} \, dx","IntegrateAlgebraic[(4 + x^3)/(x^4*(1 + x^3)^(1/4)),x]","-\frac{4 \left(1+x^3\right)^{3/4}}{3 x^3}","-\frac{4 \left(1+x^3\right)^{3/4}}{3 x^3}",1,"(-4*(1 + x^3)^(3/4))/(3*x^3)","A",1
76,1,16,16,0.1358026,"\int \frac{1}{x^2 \sqrt[3]{x+x^3}} \, dx","IntegrateAlgebraic[1/(x^2*(x + x^3)^(1/3)),x]","-\frac{3 \left(x+x^3\right)^{2/3}}{4 x^2}","-\frac{3 \left(x+x^3\right)^{2/3}}{4 x^2}",1,"(-3*(x + x^3)^(2/3))/(4*x^2)","A",1
77,1,16,16,0.3303375,"\int \frac{\left(1+x^2\right) \left(3+x^2\right)}{x^6 \sqrt[4]{x+x^3}} \, dx","IntegrateAlgebraic[((1 + x^2)*(3 + x^2))/(x^6*(x + x^3)^(1/4)),x]","-\frac{4 \left(x+x^3\right)^{7/4}}{7 x^7}","-\frac{4 \left(x+x^3\right)^{7/4}}{7 x^7}",1,"(-4*(x + x^3)^(7/4))/(7*x^7)","A",1
78,1,16,16,0.1402608,"\int \frac{\sqrt[3]{x+x^3}}{x^4} \, dx","IntegrateAlgebraic[(x + x^3)^(1/3)/x^4,x]","-\frac{3 \left(x+x^3\right)^{4/3}}{8 x^4}","-\frac{3 \left(x+x^3\right)^{4/3}}{8 x^4}",1,"(-3*(x + x^3)^(4/3))/(8*x^4)","A",1
79,1,16,16,0.1182593,"\int \frac{1}{x^4 \sqrt[4]{-1+x^4}} \, dx","IntegrateAlgebraic[1/(x^4*(-1 + x^4)^(1/4)),x]","\frac{\left(-1+x^4\right)^{3/4}}{3 x^3}","\frac{\left(-1+x^4\right)^{3/4}}{3 x^3}",1,"(-1 + x^4)^(3/4)/(3*x^3)","A",1
80,1,16,16,0.1052286,"\int \frac{\left(-1+x^4\right)^{3/4}}{x^8} \, dx","IntegrateAlgebraic[(-1 + x^4)^(3/4)/x^8,x]","\frac{\left(-1+x^4\right)^{7/4}}{7 x^7}","\frac{\left(-1+x^4\right)^{7/4}}{7 x^7}",1,"(-1 + x^4)^(7/4)/(7*x^7)","A",1
81,1,16,16,0.1582711,"\int \frac{-1+x^4}{x^2 \sqrt{x+x^3}} \, dx","IntegrateAlgebraic[(-1 + x^4)/(x^2*Sqrt[x + x^3]),x]","\frac{2 \left(x+x^3\right)^{3/2}}{3 x^3}","\frac{2 \left(x+x^3\right)^{3/2}}{3 x^3}",1,"(2*(x + x^3)^(3/2))/(3*x^3)","A",1
82,1,16,16,0.1166864,"\int \frac{1}{x^4 \sqrt[4]{1+x^4}} \, dx","IntegrateAlgebraic[1/(x^4*(1 + x^4)^(1/4)),x]","-\frac{\left(1+x^4\right)^{3/4}}{3 x^3}","-\frac{\left(1+x^4\right)^{3/4}}{3 x^3}",1,"-1/3*(1 + x^4)^(3/4)/x^3","A",1
83,1,16,16,0.0331105,"\int \frac{\left(-3+x^4\right) \sqrt[3]{1+x^4}}{x^5} \, dx","IntegrateAlgebraic[((-3 + x^4)*(1 + x^4)^(1/3))/x^5,x]","\frac{3 \left(1+x^4\right)^{4/3}}{4 x^4}","\frac{3 \left(1+x^4\right)^{4/3}}{4 x^4}",1,"(3*(1 + x^4)^(4/3))/(4*x^4)","A",1
84,1,16,16,0.1293957,"\int \frac{\left(-1+x^2\right) \left(1+x^2\right) \sqrt{1+x^4}}{x^4} \, dx","IntegrateAlgebraic[((-1 + x^2)*(1 + x^2)*Sqrt[1 + x^4])/x^4,x]","\frac{\left(1+x^4\right)^{3/2}}{3 x^3}","\frac{\left(1+x^4\right)^{3/2}}{3 x^3}",1,"(1 + x^4)^(3/2)/(3*x^3)","A",1
85,1,16,16,0.0827401,"\int \frac{\left(-3+x^4\right) \left(1+x^4\right)^{2/3}}{x^6} \, dx","IntegrateAlgebraic[((-3 + x^4)*(1 + x^4)^(2/3))/x^6,x]","\frac{3 \left(1+x^4\right)^{5/3}}{5 x^5}","\frac{3 \left(1+x^4\right)^{5/3}}{5 x^5}",1,"(3*(1 + x^4)^(5/3))/(5*x^5)","A",1
86,1,16,16,0.112199,"\int \frac{\left(1+x^4\right)^{3/4}}{x^8} \, dx","IntegrateAlgebraic[(1 + x^4)^(3/4)/x^8,x]","-\frac{\left(1+x^4\right)^{7/4}}{7 x^7}","-\frac{\left(1+x^4\right)^{7/4}}{7 x^7}",1,"-1/7*(1 + x^4)^(7/4)/x^7","A",1
87,1,16,16,0.084521,"\int \frac{\left(-1+x^4\right)^{2/3} \left(3+x^4\right)}{x^6} \, dx","IntegrateAlgebraic[((-1 + x^4)^(2/3)*(3 + x^4))/x^6,x]","\frac{3 \left(-1+x^4\right)^{5/3}}{5 x^5}","\frac{3 \left(-1+x^4\right)^{5/3}}{5 x^5}",1,"(3*(-1 + x^4)^(5/3))/(5*x^5)","A",1
88,1,16,16,0.0352393,"\int \frac{\sqrt[3]{1+x^4} \left(3+x^4\right)}{x^9} \, dx","IntegrateAlgebraic[((1 + x^4)^(1/3)*(3 + x^4))/x^9,x]","-\frac{3 \left(1+x^4\right)^{4/3}}{8 x^8}","-\frac{3 \left(1+x^4\right)^{4/3}}{8 x^8}",1,"(-3*(1 + x^4)^(4/3))/(8*x^8)","A",1
89,1,16,16,0.3302539,"\int \frac{1}{x^2 \sqrt{x+x^4}} \, dx","IntegrateAlgebraic[1/(x^2*Sqrt[x + x^4]),x]","-\frac{2 \sqrt{x+x^4}}{3 x^2}","-\frac{2 \sqrt{x+x^4}}{3 x^2}",1,"(-2*Sqrt[x + x^4])/(3*x^2)","A",1
90,1,16,16,0.2301794,"\int \frac{1+x^3}{x^6 \sqrt[4]{x+x^4}} \, dx","IntegrateAlgebraic[(1 + x^3)/(x^6*(x + x^4)^(1/4)),x]","-\frac{4 \left(x+x^4\right)^{7/4}}{21 x^7}","-\frac{4 \left(x+x^4\right)^{7/4}}{21 x^7}",1,"(-4*(x + x^4)^(7/4))/(21*x^7)","A",1
91,1,16,16,0.1875824,"\int \frac{\left(-2+x^3\right) \sqrt[3]{x+x^4}}{\left(1+x^3\right)^2} \, dx","IntegrateAlgebraic[((-2 + x^3)*(x + x^4)^(1/3))/(1 + x^3)^2,x]","-\frac{3 x^2}{2 \left(x+x^4\right)^{2/3}}","-\frac{3 x^2}{2 \left(x+x^4\right)^{2/3}}",1,"(-3*x^2)/(2*(x + x^4)^(2/3))","A",1
92,1,16,16,0.0826547,"\int \frac{\sqrt[4]{x^2+x^4}}{x^2 \left(1+x^2\right)} \, dx","IntegrateAlgebraic[(x^2 + x^4)^(1/4)/(x^2*(1 + x^2)),x]","-\frac{2 \sqrt[4]{x^2+x^4}}{x}","-\frac{2 \sqrt[4]{x^2+x^4}}{x}",1,"(-2*(x^2 + x^4)^(1/4))/x","A",1
93,1,16,16,0.0629725,"\int \frac{\left(-6+x^5\right) \left(-1+x^5\right)^{2/3}}{x^{11}} \, dx","IntegrateAlgebraic[((-6 + x^5)*(-1 + x^5)^(2/3))/x^11,x]","-\frac{3 \left(-1+x^5\right)^{5/3}}{5 x^{10}}","-\frac{3 \left(-1+x^5\right)^{5/3}}{5 x^{10}}",1,"(-3*(-1 + x^5)^(5/3))/(5*x^10)","A",1
94,1,16,16,0.1105325,"\int \frac{\left(-4+x^5\right) \left(1+x^5\right)^{3/4}}{x^8} \, dx","IntegrateAlgebraic[((-4 + x^5)*(1 + x^5)^(3/4))/x^8,x]","\frac{4 \left(1+x^5\right)^{7/4}}{7 x^7}","\frac{4 \left(1+x^5\right)^{7/4}}{7 x^7}",1,"(4*(1 + x^5)^(7/4))/(7*x^7)","A",1
95,1,16,16,0.1114173,"\int \frac{\left(-1+x^5\right)^{3/4} \left(4+x^5\right)}{x^8} \, dx","IntegrateAlgebraic[((-1 + x^5)^(3/4)*(4 + x^5))/x^8,x]","\frac{4 \left(-1+x^5\right)^{7/4}}{7 x^7}","\frac{4 \left(-1+x^5\right)^{7/4}}{7 x^7}",1,"(4*(-1 + x^5)^(7/4))/(7*x^7)","A",1
96,1,16,16,0.0594357,"\int \frac{\left(1+x^5\right)^{2/3} \left(6+x^5\right)}{x^{11}} \, dx","IntegrateAlgebraic[((1 + x^5)^(2/3)*(6 + x^5))/x^11,x]","-\frac{3 \left(1+x^5\right)^{5/3}}{5 x^{10}}","-\frac{3 \left(1+x^5\right)^{5/3}}{5 x^{10}}",1,"(-3*(1 + x^5)^(5/3))/(5*x^10)","A",1
97,1,16,16,0.2142763,"\int \frac{\left(-3+x^4\right) \left(1+x^4\right)}{x^6 \sqrt[4]{x+x^5}} \, dx","IntegrateAlgebraic[((-3 + x^4)*(1 + x^4))/(x^6*(x + x^5)^(1/4)),x]","\frac{4 \left(x+x^5\right)^{7/4}}{7 x^7}","\frac{4 \left(x+x^5\right)^{7/4}}{7 x^7}",1,"(4*(x + x^5)^(7/4))/(7*x^7)","A",1
98,1,16,16,0.0805675,"\int \frac{\left(-1+x^2\right) \sqrt[4]{x^3+x^5}}{x^2 \left(1+x^2\right)} \, dx","IntegrateAlgebraic[((-1 + x^2)*(x^3 + x^5)^(1/4))/(x^2*(1 + x^2)),x]","\frac{4 \sqrt[4]{x^3+x^5}}{x}","\frac{4 \sqrt[4]{x^3+x^5}}{x}",1,"(4*(x^3 + x^5)^(1/4))/x","A",1
99,1,16,16,3.4628524,"\int \frac{-2 b+3 a x^5}{\sqrt{b+a x^5} \left(b+x^2+a x^5\right)} \, dx","IntegrateAlgebraic[(-2*b + 3*a*x^5)/(Sqrt[b + a*x^5]*(b + x^2 + a*x^5)),x]","-2 \tan ^{-1}\left(\frac{x}{\sqrt{b+a x^5}}\right)","-2 \tan ^{-1}\left(\frac{x}{\sqrt{b+a x^5}}\right)",1,"-2*ArcTan[x/Sqrt[b + a*x^5]]","A",1
100,1,16,16,0.1355604,"\int \frac{1}{x^4 \sqrt{-1+x^6}} \, dx","IntegrateAlgebraic[1/(x^4*Sqrt[-1 + x^6]),x]","\frac{\sqrt{-1+x^6}}{3 x^3}","\frac{\sqrt{-1+x^6}}{3 x^3}",1,"Sqrt[-1 + x^6]/(3*x^3)","A",1
101,1,16,16,0.5475671,"\int \frac{\sqrt[3]{-1+x^6}}{x^9} \, dx","IntegrateAlgebraic[(-1 + x^6)^(1/3)/x^9,x]","\frac{\left(-1+x^6\right)^{4/3}}{8 x^8}","\frac{\left(-1+x^6\right)^{4/3}}{8 x^8}",1,"(-1 + x^6)^(4/3)/(8*x^8)","A",1
102,1,16,16,0.1119619,"\int \frac{\sqrt{-1+x^6}}{x^{10}} \, dx","IntegrateAlgebraic[Sqrt[-1 + x^6]/x^10,x]","\frac{\left(-1+x^6\right)^{3/2}}{9 x^9}","\frac{\left(-1+x^6\right)^{3/2}}{9 x^9}",1,"(-1 + x^6)^(3/2)/(9*x^9)","A",1
103,1,16,16,0.1227627,"\int \frac{1}{x^4 \sqrt{1+x^6}} \, dx","IntegrateAlgebraic[1/(x^4*Sqrt[1 + x^6]),x]","-\frac{\sqrt{1+x^6}}{3 x^3}","-\frac{\sqrt{1+x^6}}{3 x^3}",1,"-1/3*Sqrt[1 + x^6]/x^3","A",1
104,1,16,16,0.6688857,"\int \frac{1}{x^5 \sqrt[3]{1+x^6}} \, dx","IntegrateAlgebraic[1/(x^5*(1 + x^6)^(1/3)),x]","-\frac{\left(1+x^6\right)^{2/3}}{4 x^4}","-\frac{\left(1+x^6\right)^{2/3}}{4 x^4}",1,"-1/4*(1 + x^6)^(2/3)/x^4","A",1
105,1,16,16,0.1383992,"\int \frac{-2+x^6}{x^4 \sqrt[4]{1+x^6}} \, dx","IntegrateAlgebraic[(-2 + x^6)/(x^4*(1 + x^6)^(1/4)),x]","\frac{2 \left(1+x^6\right)^{3/4}}{3 x^3}","\frac{2 \left(1+x^6\right)^{3/4}}{3 x^3}",1,"(2*(1 + x^6)^(3/4))/(3*x^3)","A",1
106,1,16,16,0.1151942,"\int \frac{\sqrt{1+x^6}}{x^{10}} \, dx","IntegrateAlgebraic[Sqrt[1 + x^6]/x^10,x]","-\frac{\left(1+x^6\right)^{3/2}}{9 x^9}","-\frac{\left(1+x^6\right)^{3/2}}{9 x^9}",1,"-1/9*(1 + x^6)^(3/2)/x^9","A",1
107,1,16,16,0.1507625,"\int \frac{\left(-2+x^6\right) \left(1+x^6\right)^{3/4}}{x^8} \, dx","IntegrateAlgebraic[((-2 + x^6)*(1 + x^6)^(3/4))/x^8,x]","\frac{2 \left(1+x^6\right)^{7/4}}{7 x^7}","\frac{2 \left(1+x^6\right)^{7/4}}{7 x^7}",1,"(2*(1 + x^6)^(7/4))/(7*x^7)","A",1
108,1,16,16,0.1277089,"\int \frac{\sqrt[3]{-1+x^6} \left(1+x^6\right)}{x^5} \, dx","IntegrateAlgebraic[((-1 + x^6)^(1/3)*(1 + x^6))/x^5,x]","\frac{\left(-1+x^6\right)^{4/3}}{4 x^4}","\frac{\left(-1+x^6\right)^{4/3}}{4 x^4}",1,"(-1 + x^6)^(4/3)/(4*x^4)","A",1
109,1,16,16,0.1400368,"\int \frac{2+x^6}{x^4 \sqrt[4]{-1+x^6}} \, dx","IntegrateAlgebraic[(2 + x^6)/(x^4*(-1 + x^6)^(1/4)),x]","\frac{2 \left(-1+x^6\right)^{3/4}}{3 x^3}","\frac{2 \left(-1+x^6\right)^{3/4}}{3 x^3}",1,"(2*(-1 + x^6)^(3/4))/(3*x^3)","A",1
110,1,16,16,0.1473028,"\int \frac{\left(-1+x^6\right)^{3/4} \left(2+x^6\right)}{x^8} \, dx","IntegrateAlgebraic[((-1 + x^6)^(3/4)*(2 + x^6))/x^8,x]","\frac{2 \left(-1+x^6\right)^{7/4}}{7 x^7}","\frac{2 \left(-1+x^6\right)^{7/4}}{7 x^7}",1,"(2*(-1 + x^6)^(7/4))/(7*x^7)","A",1
111,1,16,16,0.1867294,"\int \frac{-3+2 x^5}{x^3 \sqrt[4]{x+x^6}} \, dx","IntegrateAlgebraic[(-3 + 2*x^5)/(x^3*(x + x^6)^(1/4)),x]","\frac{4 \left(x+x^6\right)^{3/4}}{3 x^3}","\frac{4 \left(x+x^6\right)^{3/4}}{3 x^3}",1,"(4*(x + x^6)^(3/4))/(3*x^3)","A",1
112,1,16,16,0.1955125,"\int \frac{\sqrt[3]{x+x^4} \left(-2-x^3+x^6\right)}{x^6} \, dx","IntegrateAlgebraic[((x + x^4)^(1/3)*(-2 - x^3 + x^6))/x^6,x]","\frac{3 \left(x+x^4\right)^{7/3}}{7 x^7}","\frac{3 \left(x+x^4\right)^{7/3}}{7 x^7}",1,"(3*(x + x^4)^(7/3))/(7*x^7)","A",1
113,1,16,16,15.1655522,"\int \frac{x \left(2+x^6\right)}{\sqrt{-1+x^6} \left(-1-x^4+x^6\right)} \, dx","IntegrateAlgebraic[(x*(2 + x^6))/(Sqrt[-1 + x^6]*(-1 - x^4 + x^6)),x]","-\tanh ^{-1}\left(\frac{x^2}{\sqrt{-1+x^6}}\right)","-\tanh ^{-1}\left(\frac{x^2}{\sqrt{-1+x^6}}\right)",1,"-ArcTanh[x^2/Sqrt[-1 + x^6]]","A",1
114,1,16,16,0.1473056,"\int \frac{-1+x^8}{x^4 \sqrt{-1+x^4}} \, dx","IntegrateAlgebraic[(-1 + x^8)/(x^4*Sqrt[-1 + x^4]),x]","\frac{\left(-1+x^4\right)^{3/2}}{3 x^3}","\frac{\left(-1+x^4\right)^{3/2}}{3 x^3}",1,"(-1 + x^4)^(3/2)/(3*x^3)","A",1
115,1,16,16,2.4183611,"\int \frac{\left(-1+x^8\right) \sqrt{1+x^8}}{x^7} \, dx","IntegrateAlgebraic[((-1 + x^8)*Sqrt[1 + x^8])/x^7,x]","\frac{\left(1+x^8\right)^{3/2}}{6 x^6}","\frac{\left(1+x^8\right)^{3/2}}{6 x^6}",1,"(1 + x^8)^(3/2)/(6*x^6)","A",1
116,1,17,17,1.171164,"\int \frac{2+x+2 x^2}{(-1+2 x) \sqrt{x+x^4}} \, dx","IntegrateAlgebraic[(2 + x + 2*x^2)/((-1 + 2*x)*Sqrt[x + x^4]),x]","2 \tanh ^{-1}\left(\frac{(-2+x) x}{\sqrt{x+x^4}}\right)","2 \tanh ^{-1}\left(\frac{(-2+x) x}{\sqrt{x+x^4}}\right)",1,"2*ArcTanh[((-2 + x)*x)/Sqrt[x + x^4]]","A",1
117,1,17,17,0.1728807,"\int \frac{-1+x^2}{\left(1+x^2\right) \sqrt{1+x^2+x^4}} \, dx","IntegrateAlgebraic[(-1 + x^2)/((1 + x^2)*Sqrt[1 + x^2 + x^4]),x]","-\tan ^{-1}\left(\frac{x}{\sqrt{1+x^2+x^4}}\right)","-\tan ^{-1}\left(\frac{x}{\sqrt{1+x^2+x^4}}\right)",1,"-ArcTan[x/Sqrt[1 + x^2 + x^4]]","A",1
118,1,17,17,0.2410289,"\int \frac{-1+x^4}{\left(1+x^4\right) \sqrt{1+x^2+x^4}} \, dx","IntegrateAlgebraic[(-1 + x^4)/((1 + x^4)*Sqrt[1 + x^2 + x^4]),x]","-\tanh ^{-1}\left(\frac{x}{\sqrt{1+x^2+x^4}}\right)","-\tanh ^{-1}\left(\frac{x}{\sqrt{1+x^2+x^4}}\right)",1,"-ArcTanh[x/Sqrt[1 + x^2 + x^4]]","A",1
119,1,17,17,8.5309335,"\int \frac{-1+2 x^2+2 x^4}{\left(1+2 x^2\right) \sqrt{-1+x^6}} \, dx","IntegrateAlgebraic[(-1 + 2*x^2 + 2*x^4)/((1 + 2*x^2)*Sqrt[-1 + x^6]),x]","\tanh ^{-1}\left(\frac{x \left(-1+x^2\right)}{\sqrt{-1+x^6}}\right)","\tanh ^{-1}\left(\frac{x \left(-1+x^2\right)}{\sqrt{-1+x^6}}\right)",1,"ArcTanh[(x*(-1 + x^2))/Sqrt[-1 + x^6]]","A",1
120,1,18,18,0.0169315,"\int \frac{x^3}{\left(-1+x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[x^3/(-1 + x^2)^(3/4),x]","\frac{2}{5} \sqrt[4]{-1+x^2} \left(4+x^2\right)","\frac{2}{5} \sqrt[4]{-1+x^2} \left(4+x^2\right)",1,"(2*(-1 + x^2)^(1/4)*(4 + x^2))/5","A",1
121,1,18,18,0.0647626,"\int \frac{3+x}{(-1+x)^2 \sqrt[3]{-1+x^2}} \, dx","IntegrateAlgebraic[(3 + x)/((-1 + x)^2*(-1 + x^2)^(1/3)),x]","-\frac{3 \left(-1+x^2\right)^{2/3}}{2 (-1+x)^2}","-\frac{3 \left(-1+x^2\right)^{2/3}}{2 (-1+x)^2}",1,"(-3*(-1 + x^2)^(2/3))/(2*(-1 + x)^2)","A",1
122,1,18,18,0.0169315,"\int \frac{x^3}{\left(1+x^2\right)^{2/3}} \, dx","IntegrateAlgebraic[x^3/(1 + x^2)^(2/3),x]","\frac{3}{8} \left(-3+x^2\right) \sqrt[3]{1+x^2}","\frac{3}{8} \left(-3+x^2\right) \sqrt[3]{1+x^2}",1,"(3*(-3 + x^2)*(1 + x^2)^(1/3))/8","A",1
123,1,18,18,0.1458291,"\int \frac{1}{x^2 \sqrt[3]{-x+x^3}} \, dx","IntegrateAlgebraic[1/(x^2*(-x + x^3)^(1/3)),x]","\frac{3 \left(-x+x^3\right)^{2/3}}{4 x^2}","\frac{3 \left(-x+x^3\right)^{2/3}}{4 x^2}",1,"(3*(-x + x^3)^(2/3))/(4*x^2)","A",1
124,1,18,18,0.1492605,"\int \frac{\sqrt[3]{-x+x^3}}{x^4} \, dx","IntegrateAlgebraic[(-x + x^3)^(1/3)/x^4,x]","\frac{3 \left(-x+x^3\right)^{4/3}}{8 x^4}","\frac{3 \left(-x+x^3\right)^{4/3}}{8 x^4}",1,"(3*(-x + x^3)^(4/3))/(8*x^4)","A",1
125,1,18,18,0.105819,"\int \frac{-1+x^2}{\left(1+x^2\right) \sqrt{x+x^3}} \, dx","IntegrateAlgebraic[(-1 + x^2)/((1 + x^2)*Sqrt[x + x^3]),x]","-\frac{2 \sqrt{x+x^3}}{1+x^2}","-\frac{2 \sqrt{x+x^3}}{1+x^2}",1,"(-2*Sqrt[x + x^3])/(1 + x^2)","A",1
126,1,18,18,0.1157144,"\int \frac{1}{x \sqrt[3]{x^2+x^3}} \, dx","IntegrateAlgebraic[1/(x*(x^2 + x^3)^(1/3)),x]","-\frac{3 \left(x^2+x^3\right)^{2/3}}{2 x^2}","-\frac{3 \left(x^2+x^3\right)^{2/3}}{2 x^2}",1,"(-3*(x^2 + x^3)^(2/3))/(2*x^2)","A",1
127,1,18,18,0.1470403,"\int \frac{2+x}{x^2 \sqrt[4]{x^2+x^3}} \, dx","IntegrateAlgebraic[(2 + x)/(x^2*(x^2 + x^3)^(1/4)),x]","-\frac{4 \left(x^2+x^3\right)^{3/4}}{3 x^3}","-\frac{4 \left(x^2+x^3\right)^{3/4}}{3 x^3}",1,"(-4*(x^2 + x^3)^(3/4))/(3*x^3)","A",1
128,1,18,18,0.0561125,"\int \frac{\left(-2 b+a x^3\right) \sqrt{b+a x^3}}{x^4} \, dx","IntegrateAlgebraic[((-2*b + a*x^3)*Sqrt[b + a*x^3])/x^4,x]","\frac{2 \left(b+a x^3\right)^{3/2}}{3 x^3}","\frac{2 \left(b+a x^3\right)^{3/2}}{3 x^3}",1,"(2*(b + a*x^3)^(3/2))/(3*x^3)","A",1
129,1,18,18,3.3301467,"\int \frac{1+x+x^2}{(-1+x)^2 \sqrt{-1+x^4}} \, dx","IntegrateAlgebraic[(1 + x + x^2)/((-1 + x)^2*Sqrt[-1 + x^4]),x]","-\frac{\sqrt{-1+x^4}}{2 (-1+x)^2}","-\frac{\sqrt{-1+x^4}}{2 (-1+x)^2}",1,"-1/2*Sqrt[-1 + x^4]/(-1 + x)^2","A",1
130,1,18,18,0.1603633,"\int \frac{-1+x^4}{x^2 \sqrt{-x+x^3}} \, dx","IntegrateAlgebraic[(-1 + x^4)/(x^2*Sqrt[-x + x^3]),x]","\frac{2 \left(-x+x^3\right)^{3/2}}{3 x^3}","\frac{2 \left(-x+x^3\right)^{3/2}}{3 x^3}",1,"(2*(-x + x^3)^(3/2))/(3*x^3)","A",1
131,1,18,18,0.3393478,"\int \frac{1}{x^2 \sqrt{-x+x^4}} \, dx","IntegrateAlgebraic[1/(x^2*Sqrt[-x + x^4]),x]","\frac{2 \sqrt{-x+x^4}}{3 x^2}","\frac{2 \sqrt{-x+x^4}}{3 x^2}",1,"(2*Sqrt[-x + x^4])/(3*x^2)","A",1
132,1,18,18,0.2146515,"\int \frac{1}{x^3 \sqrt[4]{-x+x^4}} \, dx","IntegrateAlgebraic[1/(x^3*(-x + x^4)^(1/4)),x]","\frac{4 \left(-x+x^4\right)^{3/4}}{9 x^3}","\frac{4 \left(-x+x^4\right)^{3/4}}{9 x^3}",1,"(4*(-x + x^4)^(3/4))/(9*x^3)","A",1
133,1,18,18,0.2438882,"\int \frac{-1+x^3}{x^6 \sqrt[4]{-x+x^4}} \, dx","IntegrateAlgebraic[(-1 + x^3)/(x^6*(-x + x^4)^(1/4)),x]","\frac{4 \left(-x+x^4\right)^{7/4}}{21 x^7}","\frac{4 \left(-x+x^4\right)^{7/4}}{21 x^7}",1,"(4*(-x + x^4)^(7/4))/(21*x^7)","A",1
134,1,18,18,0.1813298,"\int \frac{\sqrt[4]{-x+x^4}}{x^5} \, dx","IntegrateAlgebraic[(-x + x^4)^(1/4)/x^5,x]","\frac{4 \left(-x+x^4\right)^{5/4}}{15 x^5}","\frac{4 \left(-x+x^4\right)^{5/4}}{15 x^5}",1,"(4*(-x + x^4)^(5/4))/(15*x^5)","A",1
135,1,18,18,0.1879885,"\int \frac{\left(2+x^3\right) \sqrt[3]{-x+x^4}}{\left(-1+x^3\right)^2} \, dx","IntegrateAlgebraic[((2 + x^3)*(-x + x^4)^(1/3))/(-1 + x^3)^2,x]","-\frac{3 x^2}{2 \left(-x+x^4\right)^{2/3}}","-\frac{3 x^2}{2 \left(-x+x^4\right)^{2/3}}",1,"(-3*x^2)/(2*(-x + x^4)^(2/3))","A",1
136,1,18,18,0.3584375,"\int \frac{\sqrt{-x+x^4}}{x^6} \, dx","IntegrateAlgebraic[Sqrt[-x + x^4]/x^6,x]","\frac{2 \left(-x+x^4\right)^{3/2}}{9 x^6}","\frac{2 \left(-x+x^4\right)^{3/2}}{9 x^6}",1,"(2*(-x + x^4)^(3/2))/(9*x^6)","A",1
137,1,18,18,0.2879624,"\int \frac{x}{\sqrt{x+x^4}} \, dx","IntegrateAlgebraic[x/Sqrt[x + x^4],x]","\frac{2}{3} \tanh ^{-1}\left(\frac{x^2}{\sqrt{x+x^4}}\right)","\frac{2}{3} \tanh ^{-1}\left(\frac{x^2}{\sqrt{x+x^4}}\right)",1,"(2*ArcTanh[x^2/Sqrt[x + x^4]])/3","A",1
138,1,18,18,0.2047552,"\int \frac{-2+x^3}{\left(1+x^3\right) \sqrt[3]{x+x^4}} \, dx","IntegrateAlgebraic[(-2 + x^3)/((1 + x^3)*(x + x^4)^(1/3)),x]","-\frac{3 \left(x+x^4\right)^{2/3}}{1+x^3}","-\frac{3 \left(x+x^4\right)^{2/3}}{1+x^3}",1,"(-3*(x + x^4)^(2/3))/(1 + x^3)","A",1
139,1,18,18,0.0682073,"\int \frac{-1+x^2}{x \sqrt[3]{x^2+x^4}} \, dx","IntegrateAlgebraic[(-1 + x^2)/(x*(x^2 + x^4)^(1/3)),x]","\frac{3 \left(x^2+x^4\right)^{2/3}}{2 x^2}","\frac{3 \left(x^2+x^4\right)^{2/3}}{2 x^2}",1,"(3*(x^2 + x^4)^(2/3))/(2*x^2)","A",1
140,1,18,18,0.0793135,"\int \frac{1}{x^2 \sqrt[4]{x^2+x^4}} \, dx","IntegrateAlgebraic[1/(x^2*(x^2 + x^4)^(1/4)),x]","-\frac{2 \left(x^2+x^4\right)^{3/4}}{3 x^3}","-\frac{2 \left(x^2+x^4\right)^{3/4}}{3 x^3}",1,"(-2*(x^2 + x^4)^(3/4))/(3*x^3)","A",1
141,1,18,18,0.0818875,"\int \frac{\sqrt[4]{x^2+x^4}}{x^4} \, dx","IntegrateAlgebraic[(x^2 + x^4)^(1/4)/x^4,x]","-\frac{2 \left(x^2+x^4\right)^{5/4}}{5 x^5}","-\frac{2 \left(x^2+x^4\right)^{5/4}}{5 x^5}",1,"(-2*(x^2 + x^4)^(5/4))/(5*x^5)","A",1
142,1,18,18,0.0658203,"\int \frac{\left(-1+x^2\right) \sqrt[3]{x^2+x^4}}{x^3} \, dx","IntegrateAlgebraic[((-1 + x^2)*(x^2 + x^4)^(1/3))/x^3,x]","\frac{3 \left(x^2+x^4\right)^{4/3}}{4 x^4}","\frac{3 \left(x^2+x^4\right)^{4/3}}{4 x^4}",1,"(3*(x^2 + x^4)^(4/3))/(4*x^4)","A",1
143,1,18,18,0.187266,"\int \frac{1}{x \sqrt[4]{x^3+x^4}} \, dx","IntegrateAlgebraic[1/(x*(x^3 + x^4)^(1/4)),x]","-\frac{4 \left(x^3+x^4\right)^{3/4}}{3 x^3}","-\frac{4 \left(x^3+x^4\right)^{3/4}}{3 x^3}",1,"(-4*(x^3 + x^4)^(3/4))/(3*x^3)","A",1
144,1,18,18,0.1991681,"\int \frac{(1+x) \sqrt[4]{x^3+x^4}}{x^4} \, dx","IntegrateAlgebraic[((1 + x)*(x^3 + x^4)^(1/4))/x^4,x]","-\frac{4 \left(x^3+x^4\right)^{9/4}}{9 x^9}","-\frac{4 \left(x^3+x^4\right)^{9/4}}{9 x^9}",1,"(-4*(x^3 + x^4)^(9/4))/(9*x^9)","A",1
145,1,18,18,0.1563322,"\int \frac{-2-x+2 x^4}{\left(1+x+x^4\right) \sqrt{1+x+x^2+x^4}} \, dx","IntegrateAlgebraic[(-2 - x + 2*x^4)/((1 + x + x^4)*Sqrt[1 + x + x^2 + x^4]),x]","-2 \tanh ^{-1}\left(\frac{x}{\sqrt{1+x+x^2+x^4}}\right)","-2 \tanh ^{-1}\left(\frac{x}{\sqrt{1+x+x^2+x^4}}\right)",1,"-2*ArcTanh[x/Sqrt[1 + x + x^2 + x^4]]","A",1
146,1,18,18,0.0740084,"\int \frac{3+x^4}{x^3 \sqrt[4]{-x+x^5}} \, dx","IntegrateAlgebraic[(3 + x^4)/(x^3*(-x + x^5)^(1/4)),x]","\frac{4 \left(-x+x^5\right)^{3/4}}{3 x^3}","\frac{4 \left(-x+x^5\right)^{3/4}}{3 x^3}",1,"(4*(-x + x^5)^(3/4))/(3*x^3)","A",1
147,1,18,18,0.2320984,"\int \frac{\left(-1+x^4\right) \left(3+x^4\right)}{x^6 \sqrt[4]{-x+x^5}} \, dx","IntegrateAlgebraic[((-1 + x^4)*(3 + x^4))/(x^6*(-x + x^5)^(1/4)),x]","\frac{4 \left(-x+x^5\right)^{7/4}}{7 x^7}","\frac{4 \left(-x+x^5\right)^{7/4}}{7 x^7}",1,"(4*(-x + x^5)^(7/4))/(7*x^7)","A",1
148,1,18,18,0.1240277,"\int \frac{-3+x^4}{\left(1+x^4\right) \sqrt[4]{x+x^5}} \, dx","IntegrateAlgebraic[(-3 + x^4)/((1 + x^4)*(x + x^5)^(1/4)),x]","-\frac{4 \left(x+x^5\right)^{3/4}}{1+x^4}","-\frac{4 \left(x+x^5\right)^{3/4}}{1+x^4}",1,"(-4*(x + x^5)^(3/4))/(1 + x^4)","A",1
149,1,18,18,0.1026682,"\int \frac{\left(-1+2 x^3\right) \sqrt[3]{x^2+x^5}}{x^3} \, dx","IntegrateAlgebraic[((-1 + 2*x^3)*(x^2 + x^5)^(1/3))/x^3,x]","\frac{3 \left(x^2+x^5\right)^{4/3}}{4 x^4}","\frac{3 \left(x^2+x^5\right)^{4/3}}{4 x^4}",1,"(3*(x^2 + x^5)^(4/3))/(4*x^4)","A",1
150,1,18,18,0.0868392,"\int \frac{-1+x^2}{x \sqrt[4]{x^3+x^5}} \, dx","IntegrateAlgebraic[(-1 + x^2)/(x*(x^3 + x^5)^(1/4)),x]","\frac{4 \left(x^3+x^5\right)^{3/4}}{3 x^3}","\frac{4 \left(x^3+x^5\right)^{3/4}}{3 x^3}",1,"(4*(x^3 + x^5)^(3/4))/(3*x^3)","A",1
151,1,18,18,0.093249,"\int \frac{\left(-1+x^4\right) \sqrt[4]{x^3+x^5}}{x^4} \, dx","IntegrateAlgebraic[((-1 + x^4)*(x^3 + x^5)^(1/4))/x^4,x]","\frac{4 \left(x^3+x^5\right)^{9/4}}{9 x^9}","\frac{4 \left(x^3+x^5\right)^{9/4}}{9 x^9}",1,"(4*(x^3 + x^5)^(9/4))/(9*x^9)","A",1
152,1,18,18,0.1102445,"\int \frac{x^2}{\sqrt{-1+x^6}} \, dx","IntegrateAlgebraic[x^2/Sqrt[-1 + x^6],x]","\frac{1}{3} \log \left(x^3+\sqrt{-1+x^6}\right)","\frac{1}{3} \log \left(x^3+\sqrt{-1+x^6}\right)",1,"Log[x^3 + Sqrt[-1 + x^6]]/3","A",1
153,1,18,18,0.1006677,"\int \frac{x^2}{\sqrt{1+x^6}} \, dx","IntegrateAlgebraic[x^2/Sqrt[1 + x^6],x]","\frac{1}{3} \log \left(x^3+\sqrt{1+x^6}\right)","\frac{1}{3} \log \left(x^3+\sqrt{1+x^6}\right)",1,"Log[x^3 + Sqrt[1 + x^6]]/3","A",1
154,1,18,18,0.2780936,"\int \frac{\left(-1+x^5\right) \left(3+2 x^5\right)}{x^6 \sqrt[4]{-x+x^6}} \, dx","IntegrateAlgebraic[((-1 + x^5)*(3 + 2*x^5))/(x^6*(-x + x^6)^(1/4)),x]","\frac{4 \left(-x+x^6\right)^{7/4}}{7 x^7}","\frac{4 \left(-x+x^6\right)^{7/4}}{7 x^7}",1,"(4*(-x + x^6)^(7/4))/(7*x^7)","A",1
155,1,18,18,0.309409,"\int \frac{1}{x^3 \sqrt[3]{x^2+x^6}} \, dx","IntegrateAlgebraic[1/(x^3*(x^2 + x^6)^(1/3)),x]","-\frac{3 \left(x^2+x^6\right)^{2/3}}{8 x^4}","-\frac{3 \left(x^2+x^6\right)^{2/3}}{8 x^4}",1,"(-3*(x^2 + x^6)^(2/3))/(8*x^4)","A",1
156,1,18,18,0.2263666,"\int \frac{-1+x^4}{x^2 \sqrt[4]{x^2+x^6}} \, dx","IntegrateAlgebraic[(-1 + x^4)/(x^2*(x^2 + x^6)^(1/4)),x]","\frac{2 \left(x^2+x^6\right)^{3/4}}{3 x^3}","\frac{2 \left(x^2+x^6\right)^{3/4}}{3 x^3}",1,"(2*(x^2 + x^6)^(3/4))/(3*x^3)","A",1
157,1,18,18,0.2401646,"\int \frac{\left(-1+x^4\right) \sqrt[4]{x^2+x^6}}{x^4} \, dx","IntegrateAlgebraic[((-1 + x^4)*(x^2 + x^6)^(1/4))/x^4,x]","\frac{2 \left(x^2+x^6\right)^{5/4}}{5 x^5}","\frac{2 \left(x^2+x^6\right)^{5/4}}{5 x^5}",1,"(2*(x^2 + x^6)^(5/4))/(5*x^5)","A",1
158,1,18,18,0.0437038,"\int \frac{\sqrt{1+\sqrt{1+x^2}}}{\sqrt{1+x^2}} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[1 + x^2]]/Sqrt[1 + x^2],x]","\frac{2 x}{\sqrt{1+\sqrt{1+x^2}}}","\frac{2 x}{\sqrt{1+\sqrt{1+x^2}}}",1,"(2*x)/Sqrt[1 + Sqrt[1 + x^2]]","A",1
159,1,19,19,0.4540721,"\int \frac{-2-2 x+x^2}{\left(1+x+x^2\right) \sqrt{-1+x^3}} \, dx","IntegrateAlgebraic[(-2 - 2*x + x^2)/((1 + x + x^2)*Sqrt[-1 + x^3]),x]","-\frac{2 \sqrt{-1+x^3}}{1+x+x^2}","-\frac{2 \sqrt{-1+x^3}}{1+x+x^2}",1,"(-2*Sqrt[-1 + x^3])/(1 + x + x^2)","A",1
160,1,19,19,0.2039833,"\int \frac{b+a x^2}{x \sqrt{-b x+a x^3}} \, dx","IntegrateAlgebraic[(b + a*x^2)/(x*Sqrt[-(b*x) + a*x^3]),x]","\frac{2 \sqrt{-b x+a x^3}}{x}","\frac{2 \sqrt{-b x+a x^3}}{x}",1,"(2*Sqrt[-(b*x) + a*x^3])/x","A",1
161,1,19,19,1.2278297,"\int \frac{-1-2 x+2 x^2}{\left(1+2 x^2\right) \sqrt{x+x^4}} \, dx","IntegrateAlgebraic[(-1 - 2*x + 2*x^2)/((1 + 2*x^2)*Sqrt[x + x^4]),x]","\tan ^{-1}\left(\frac{2 \sqrt{x+x^4}}{-1+2 x}\right)","\tan ^{-1}\left(\frac{2 \sqrt{x+x^4}}{-1+2 x}\right)",1,"ArcTan[(2*Sqrt[x + x^4])/(-1 + 2*x)]","A",1
162,1,19,19,0.2460786,"\int \frac{1+x^4}{\left(-1+x^4\right) \sqrt{-1-x^2+x^4}} \, dx","IntegrateAlgebraic[(1 + x^4)/((-1 + x^4)*Sqrt[-1 - x^2 + x^4]),x]","-\tan ^{-1}\left(\frac{x}{\sqrt{-1-x^2+x^4}}\right)","-\tan ^{-1}\left(\frac{x}{\sqrt{-1-x^2+x^4}}\right)",1,"-ArcTan[x/Sqrt[-1 - x^2 + x^4]]","A",1
163,1,19,19,5.4035301,"\int \frac{x}{\sqrt[5]{1-4 x+6 x^2-4 x^3+x^4}} \, dx","IntegrateAlgebraic[x/(1 - 4*x + 6*x^2 - 4*x^3 + x^4)^(1/5),x]","\frac{5 (-1+x) (5+x)}{6 \sqrt[5]{(-1+x)^4}}","\frac{5 (-1+x) (5+x)}{6 \sqrt[5]{(-1+x)^4}}",1,"(5*(-1 + x)*(5 + x))/(6*((-1 + x)^4)^(1/5))","A",1
164,1,19,19,0.4690874,"\int \frac{-2+x^6}{x^4 \sqrt[4]{1+x^4+x^6}} \, dx","IntegrateAlgebraic[(-2 + x^6)/(x^4*(1 + x^4 + x^6)^(1/4)),x]","\frac{2 \left(1+x^4+x^6\right)^{3/4}}{3 x^3}","\frac{2 \left(1+x^4+x^6\right)^{3/4}}{3 x^3}",1,"(2*(1 + x^4 + x^6)^(3/4))/(3*x^3)","A",1
165,1,19,19,0.4029119,"\int \frac{1+x^8}{x^4 \sqrt[4]{-1+x^4+x^8}} \, dx","IntegrateAlgebraic[(1 + x^8)/(x^4*(-1 + x^4 + x^8)^(1/4)),x]","\frac{\left(-1+x^4+x^8\right)^{3/4}}{3 x^3}","\frac{\left(-1+x^4+x^8\right)^{3/4}}{3 x^3}",1,"(-1 + x^4 + x^8)^(3/4)/(3*x^3)","A",1
166,1,19,19,0.3937808,"\int \frac{-1+x^8}{x^4 \sqrt[4]{1+x^4+x^8}} \, dx","IntegrateAlgebraic[(-1 + x^8)/(x^4*(1 + x^4 + x^8)^(1/4)),x]","\frac{\left(1+x^4+x^8\right)^{3/4}}{3 x^3}","\frac{\left(1+x^4+x^8\right)^{3/4}}{3 x^3}",1,"(1 + x^4 + x^8)^(3/4)/(3*x^3)","A",1
167,1,20,20,0.1083081,"\int \frac{1+x^2}{\left(-1+x^2\right) \sqrt{-x+x^3}} \, dx","IntegrateAlgebraic[(1 + x^2)/((-1 + x^2)*Sqrt[-x + x^3]),x]","-\frac{2 \sqrt{-x+x^3}}{-1+x^2}","-\frac{2 \sqrt{-x+x^3}}{-1+x^2}",1,"(-2*Sqrt[-x + x^3])/(-1 + x^2)","A",1
168,1,20,20,0.0893759,"\int \frac{1}{\left(1+x^2\right) \sqrt[3]{x+x^3}} \, dx","IntegrateAlgebraic[1/((1 + x^2)*(x + x^3)^(1/3)),x]","\frac{3 \left(x+x^3\right)^{2/3}}{2 \left(1+x^2\right)}","\frac{3 \left(x+x^3\right)^{2/3}}{2 \left(1+x^2\right)}",1,"(3*(x + x^3)^(2/3))/(2*(1 + x^2))","A",1
169,1,20,20,0.1255706,"\int \frac{1}{x \sqrt[3]{-x^2+x^3}} \, dx","IntegrateAlgebraic[1/(x*(-x^2 + x^3)^(1/3)),x]","\frac{3 \left(-x^2+x^3\right)^{2/3}}{2 x^2}","\frac{3 \left(-x^2+x^3\right)^{2/3}}{2 x^2}",1,"(3*(-x^2 + x^3)^(2/3))/(2*x^2)","A",1
170,1,20,20,0.0183106,"\int \frac{x^2}{\sqrt{-b+a x^3}} \, dx","IntegrateAlgebraic[x^2/Sqrt[-b + a*x^3],x]","\frac{2 \sqrt{-b+a x^3}}{3 a}","\frac{2 \sqrt{-b+a x^3}}{3 a}",1,"(2*Sqrt[-b + a*x^3])/(3*a)","A",1
171,1,20,20,0.0166959,"\int x^2 \sqrt{-b+a x^3} \, dx","IntegrateAlgebraic[x^2*Sqrt[-b + a*x^3],x]","\frac{2 \left(-b+a x^3\right)^{3/2}}{9 a}","\frac{2 \left(-b+a x^3\right)^{3/2}}{9 a}",1,"(2*(-b + a*x^3)^(3/2))/(9*a)","A",1
172,1,20,20,0.0378781,"\int \frac{\sqrt{-b+a x^3} \left(2 b+a x^3\right)}{x^4} \, dx","IntegrateAlgebraic[(Sqrt[-b + a*x^3]*(2*b + a*x^3))/x^4,x]","\frac{2 \left(-b+a x^3\right)^{3/2}}{3 x^3}","\frac{2 \left(-b+a x^3\right)^{3/2}}{3 x^3}",1,"(2*(-b + a*x^3)^(3/2))/(3*x^3)","A",1
173,1,20,20,0.1055949,"\int \frac{1}{x^2 \sqrt[4]{-x^2+x^4}} \, dx","IntegrateAlgebraic[1/(x^2*(-x^2 + x^4)^(1/4)),x]","\frac{2 \left(-x^2+x^4\right)^{3/4}}{3 x^3}","\frac{2 \left(-x^2+x^4\right)^{3/4}}{3 x^3}",1,"(2*(-x^2 + x^4)^(3/4))/(3*x^3)","A",1
174,1,20,20,0.0885778,"\int \frac{\sqrt[4]{-x^2+x^4}}{x^4} \, dx","IntegrateAlgebraic[(-x^2 + x^4)^(1/4)/x^4,x]","\frac{2 \left(-x^2+x^4\right)^{5/4}}{5 x^5}","\frac{2 \left(-x^2+x^4\right)^{5/4}}{5 x^5}",1,"(2*(-x^2 + x^4)^(5/4))/(5*x^5)","A",1
175,1,20,20,0.0715647,"\int \frac{\left(1+x^2\right) \sqrt[3]{-x^2+x^4}}{x^3} \, dx","IntegrateAlgebraic[((1 + x^2)*(-x^2 + x^4)^(1/3))/x^3,x]","\frac{3 \left(-x^2+x^4\right)^{4/3}}{4 x^4}","\frac{3 \left(-x^2+x^4\right)^{4/3}}{4 x^4}",1,"(3*(-x^2 + x^4)^(4/3))/(4*x^4)","A",1
176,1,20,20,0.2020434,"\int \frac{1}{x \sqrt[4]{-x^3+x^4}} \, dx","IntegrateAlgebraic[1/(x*(-x^3 + x^4)^(1/4)),x]","\frac{4 \left(-x^3+x^4\right)^{3/4}}{3 x^3}","\frac{4 \left(-x^3+x^4\right)^{3/4}}{3 x^3}",1,"(4*(-x^3 + x^4)^(3/4))/(3*x^3)","A",1
177,1,20,20,0.1964632,"\int \frac{(-1+x) \sqrt[4]{-x^3+x^4}}{x^4} \, dx","IntegrateAlgebraic[((-1 + x)*(-x^3 + x^4)^(1/4))/x^4,x]","\frac{4 \left(-x^3+x^4\right)^{9/4}}{9 x^9}","\frac{4 \left(-x^3+x^4\right)^{9/4}}{9 x^9}",1,"(4*(-x^3 + x^4)^(9/4))/(9*x^9)","A",1
178,1,20,20,0.1805962,"\int \frac{\sqrt[4]{-x^3+x^4}}{x^3} \, dx","IntegrateAlgebraic[(-x^3 + x^4)^(1/4)/x^3,x]","\frac{4 \left(-x^3+x^4\right)^{5/4}}{5 x^5}","\frac{4 \left(-x^3+x^4\right)^{5/4}}{5 x^5}",1,"(4*(-x^3 + x^4)^(5/4))/(5*x^5)","A",1
179,1,20,20,0.2197925,"\int x \left(1+2 x^2\right) \sqrt{-1+2 x^2+2 x^4} \, dx","IntegrateAlgebraic[x*(1 + 2*x^2)*Sqrt[-1 + 2*x^2 + 2*x^4],x]","\frac{1}{6} \left(-1+2 x^2+2 x^4\right)^{3/2}","\frac{1}{6} \left(-1+2 x^2+2 x^4\right)^{3/2}",1,"(-1 + 2*x^2 + 2*x^4)^(3/2)/6","A",1
180,1,20,20,0.0923244,"\int \frac{\left(1+2 x^3\right) \sqrt[3]{-x^2+x^5}}{x^3} \, dx","IntegrateAlgebraic[((1 + 2*x^3)*(-x^2 + x^5)^(1/3))/x^3,x]","\frac{3 \left(-x^2+x^5\right)^{4/3}}{4 x^4}","\frac{3 \left(-x^2+x^5\right)^{4/3}}{4 x^4}",1,"(3*(-x^2 + x^5)^(4/3))/(4*x^4)","A",1
181,1,20,20,0.1417597,"\int \frac{-1+x^4}{x^2 \sqrt[4]{-x^3+x^5}} \, dx","IntegrateAlgebraic[(-1 + x^4)/(x^2*(-x^3 + x^5)^(1/4)),x]","\frac{4 \left(-x^3+x^5\right)^{7/4}}{7 x^7}","\frac{4 \left(-x^3+x^5\right)^{7/4}}{7 x^7}",1,"(4*(-x^3 + x^5)^(7/4))/(7*x^7)","A",1
182,1,20,20,0.0970425,"\int \frac{\left(-1+x^4\right) \sqrt[4]{-x^3+x^5}}{x^4} \, dx","IntegrateAlgebraic[((-1 + x^4)*(-x^3 + x^5)^(1/4))/x^4,x]","\frac{4 \left(-x^3+x^5\right)^{9/4}}{9 x^9}","\frac{4 \left(-x^3+x^5\right)^{9/4}}{9 x^9}",1,"(4*(-x^3 + x^5)^(9/4))/(9*x^9)","A",1
183,1,20,20,9.8126963,"\int \frac{-1-2 x^2+2 x^4}{x^2 \left(1+x^2\right) \sqrt{1+x^6}} \, dx","IntegrateAlgebraic[(-1 - 2*x^2 + 2*x^4)/(x^2*(1 + x^2)*Sqrt[1 + x^6]),x]","\frac{\sqrt{1+x^6}}{x \left(1+x^2\right)}","\frac{\sqrt{1+x^6}}{x \left(1+x^2\right)}",1,"Sqrt[1 + x^6]/(x*(1 + x^2))","A",1
184,1,20,20,0.3173377,"\int \frac{1}{x^3 \sqrt[3]{-x^2+x^6}} \, dx","IntegrateAlgebraic[1/(x^3*(-x^2 + x^6)^(1/3)),x]","\frac{3 \left(-x^2+x^6\right)^{2/3}}{8 x^4}","\frac{3 \left(-x^2+x^6\right)^{2/3}}{8 x^4}",1,"(3*(-x^2 + x^6)^(2/3))/(8*x^4)","A",1
185,1,20,20,0.2283112,"\int \frac{1+x^4}{x^2 \sqrt[4]{-x^2+x^6}} \, dx","IntegrateAlgebraic[(1 + x^4)/(x^2*(-x^2 + x^6)^(1/4)),x]","\frac{2 \left(-x^2+x^6\right)^{3/4}}{3 x^3}","\frac{2 \left(-x^2+x^6\right)^{3/4}}{3 x^3}",1,"(2*(-x^2 + x^6)^(3/4))/(3*x^3)","A",1
186,1,20,20,0.2298089,"\int \frac{\left(1+x^4\right) \sqrt[4]{-x^2+x^6}}{x^4} \, dx","IntegrateAlgebraic[((1 + x^4)*(-x^2 + x^6)^(1/4))/x^4,x]","\frac{2 \left(-x^2+x^6\right)^{5/4}}{5 x^5}","\frac{2 \left(-x^2+x^6\right)^{5/4}}{5 x^5}",1,"(2*(-x^2 + x^6)^(5/4))/(5*x^5)","A",1
187,1,20,20,0.2941283,"\int \frac{-1+x^8}{x^4 \sqrt[4]{-x^2+x^6}} \, dx","IntegrateAlgebraic[(-1 + x^8)/(x^4*(-x^2 + x^6)^(1/4)),x]","\frac{2 \left(-x^2+x^6\right)^{7/4}}{7 x^7}","\frac{2 \left(-x^2+x^6\right)^{7/4}}{7 x^7}",1,"(2*(-x^2 + x^6)^(7/4))/(7*x^7)","A",1
188,1,20,20,0.4980926,"\int \frac{1+x^8}{\sqrt[4]{1-x^8} \left(-1+x^8\right)} \, dx","IntegrateAlgebraic[(1 + x^8)/((1 - x^8)^(1/4)*(-1 + x^8)),x]","\frac{x \left(1-x^8\right)^{3/4}}{-1+x^8}","\frac{x \left(1-x^8\right)^{3/4}}{-1+x^8}",1,"(x*(1 - x^8)^(3/4))/(-1 + x^8)","A",1
189,1,20,20,0.0448241,"\int \frac{1}{\sqrt{-x^2+x^8}} \, dx","IntegrateAlgebraic[1/Sqrt[-x^2 + x^8],x]","-\frac{1}{3} \tan ^{-1}\left(\frac{x}{\sqrt{-x^2+x^8}}\right)","-\frac{1}{3} \tan ^{-1}\left(\frac{x}{\sqrt{-x^2+x^8}}\right)",1,"-1/3*ArcTan[x/Sqrt[-x^2 + x^8]]","A",1
190,1,20,20,0.0594496,"\int \frac{\sqrt{1+\sqrt{1+x^2}}}{1+x^2} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[1 + x^2]]/(1 + x^2),x]","2 \tan ^{-1}\left(\frac{x}{\sqrt{1+\sqrt{1+x^2}}}\right)","2 \tan ^{-1}\left(\frac{x}{\sqrt{1+\sqrt{1+x^2}}}\right)",1,"2*ArcTan[x/Sqrt[1 + Sqrt[1 + x^2]]]","A",1
191,1,21,21,0.8231856,"\int \frac{-1+x}{(2+x) \sqrt{-1+x^3}} \, dx","IntegrateAlgebraic[(-1 + x)/((2 + x)*Sqrt[-1 + x^3]),x]","-\frac{2}{3} \tan ^{-1}\left(\frac{3 \sqrt{-1+x^3}}{(-1+x)^2}\right)","-\frac{2}{3} \tan ^{-1}\left(\frac{3 \sqrt{-1+x^3}}{(-1+x)^2}\right)",1,"(-2*ArcTan[(3*Sqrt[-1 + x^3])/(-1 + x)^2])/3","A",1
192,1,21,21,1.0040695,"\int \frac{-2-2 x+x^2}{\left(2+x^2\right) \sqrt{-1+x^3}} \, dx","IntegrateAlgebraic[(-2 - 2*x + x^2)/((2 + x^2)*Sqrt[-1 + x^3]),x]","-2 \tanh ^{-1}\left(\frac{\sqrt{-1+x^3}}{1+x+x^2}\right)","-2 \tanh ^{-1}\left(\frac{\sqrt{-1+x^3}}{1+x+x^2}\right)",1,"-2*ArcTanh[Sqrt[-1 + x^3]/(1 + x + x^2)]","A",1
193,1,21,21,1.0125443,"\int \frac{-2-2 x+x^2}{\left(2 x+x^2\right) \sqrt{-1+x^3}} \, dx","IntegrateAlgebraic[(-2 - 2*x + x^2)/((2*x + x^2)*Sqrt[-1 + x^3]),x]","-2 \tan ^{-1}\left(\frac{\sqrt{-1+x^3}}{1+x+x^2}\right)","-2 \tan ^{-1}\left(\frac{\sqrt{-1+x^3}}{1+x+x^2}\right)",1,"-2*ArcTan[Sqrt[-1 + x^3]/(1 + x + x^2)]","A",1
194,1,21,21,1.1899464,"\int \frac{-1+2 x+2 x^2}{(-1+x) x \sqrt{-x+x^4}} \, dx","IntegrateAlgebraic[(-1 + 2*x + 2*x^2)/((-1 + x)*x*Sqrt[-x + x^4]),x]","-\frac{2 \sqrt{-x+x^4}}{(-1+x) x}","-\frac{2 \sqrt{-x+x^4}}{(-1+x) x}",1,"(-2*Sqrt[-x + x^4])/((-1 + x)*x)","A",1
195,1,21,21,1.26147,"\int \frac{-1+2 x+2 x^2}{\left(1+2 x^2\right) \sqrt{-x+x^4}} \, dx","IntegrateAlgebraic[(-1 + 2*x + 2*x^2)/((1 + 2*x^2)*Sqrt[-x + x^4]),x]","\tan ^{-1}\left(\frac{2 \sqrt{-x+x^4}}{1+2 x}\right)","\tan ^{-1}\left(\frac{2 \sqrt{-x+x^4}}{1+2 x}\right)",1,"ArcTan[(2*Sqrt[-x + x^4])/(1 + 2*x)]","A",1
196,1,21,21,1.1510315,"\int \frac{-1-2 x+2 x^2}{(-1+2 x) \sqrt{x+x^4}} \, dx","IntegrateAlgebraic[(-1 - 2*x + 2*x^2)/((-1 + 2*x)*Sqrt[x + x^4]),x]","\tanh ^{-1}\left(\frac{2 \sqrt{x+x^4}}{1+2 x^2}\right)","\tanh ^{-1}\left(\frac{2 \sqrt{x+x^4}}{1+2 x^2}\right)",1,"ArcTanh[(2*Sqrt[x + x^4])/(1 + 2*x^2)]","A",1
197,1,21,21,0.6042291,"\int \frac{-1-2 x+2 x^2}{\left(1-x+x^2\right) \sqrt{x+x^4}} \, dx","IntegrateAlgebraic[(-1 - 2*x + 2*x^2)/((1 - x + x^2)*Sqrt[x + x^4]),x]","-\frac{2 \sqrt{x+x^4}}{1-x+x^2}","-\frac{2 \sqrt{x+x^4}}{1-x+x^2}",1,"(-2*Sqrt[x + x^4])/(1 - x + x^2)","A",1
198,1,21,21,0.1886852,"\int \frac{1}{(1+x) \sqrt[4]{x^3+x^4}} \, dx","IntegrateAlgebraic[1/((1 + x)*(x^3 + x^4)^(1/4)),x]","\frac{4 \left(x^3+x^4\right)^{3/4}}{x^2 (1+x)}","\frac{4 \left(x^3+x^4\right)^{3/4}}{x^2 (1+x)}",1,"(4*(x^3 + x^4)^(3/4))/(x^2*(1 + x))","A",1
199,1,21,21,0.3399623,"\int \frac{1+2 x^4}{\left(-1+2 x^4\right) \sqrt{-1-x^2+2 x^4}} \, dx","IntegrateAlgebraic[(1 + 2*x^4)/((-1 + 2*x^4)*Sqrt[-1 - x^2 + 2*x^4]),x]","-\tan ^{-1}\left(\frac{x}{\sqrt{-1-x^2+2 x^4}}\right)","-\tan ^{-1}\left(\frac{x}{\sqrt{-1-x^2+2 x^4}}\right)",1,"-ArcTan[x/Sqrt[-1 - x^2 + 2*x^4]]","A",1
200,1,21,21,0.5928129,"\int \frac{-1+2 x^4}{\left(1+2 x^2+2 x^4\right) \sqrt{1+3 x^2+2 x^4}} \, dx","IntegrateAlgebraic[(-1 + 2*x^4)/((1 + 2*x^2 + 2*x^4)*Sqrt[1 + 3*x^2 + 2*x^4]),x]","-\tanh ^{-1}\left(\frac{x}{\sqrt{1+3 x^2+2 x^4}}\right)","-\tanh ^{-1}\left(\frac{x}{\sqrt{1+3 x^2+2 x^4}}\right)",1,"-ArcTanh[x/Sqrt[1 + 3*x^2 + 2*x^4]]","A",1
201,1,21,21,0.2579846,"\int \frac{\sqrt[4]{-1+x^4-x^5} \left(-4+x^5\right)}{x^6} \, dx","IntegrateAlgebraic[((-1 + x^4 - x^5)^(1/4)*(-4 + x^5))/x^6,x]","-\frac{4 \left(-1+x^4-x^5\right)^{5/4}}{5 x^5}","-\frac{4 \left(-1+x^4-x^5\right)^{5/4}}{5 x^5}",1,"(-4*(-1 + x^4 - x^5)^(5/4))/(5*x^5)","A",1
202,1,21,21,0.6167592,"\int \frac{\left(1+x^8\right) \sqrt{-1-2 x^4+x^8}}{x^7} \, dx","IntegrateAlgebraic[((1 + x^8)*Sqrt[-1 - 2*x^4 + x^8])/x^7,x]","\frac{\left(-1-2 x^4+x^8\right)^{3/2}}{6 x^6}","\frac{\left(-1-2 x^4+x^8\right)^{3/2}}{6 x^6}",1,"(-1 - 2*x^4 + x^8)^(3/2)/(6*x^6)","A",1
203,1,37,21,0.0472695,"\int \frac{x^2 \left(-2+x^3\right) \sqrt{1+x^3}}{4+12 x^3+13 x^6+4 x^9} \, dx","IntegrateAlgebraic[(x^2*(-2 + x^3)*Sqrt[1 + x^3])/(4 + 12*x^3 + 13*x^6 + 4*x^9),x]","\frac{1}{3} \tan ^{-1}\left(\sqrt{1+x^3}\right)-\frac{1}{3} \tan ^{-1}\left(\frac{1+2 x^3}{\sqrt{1+x^3}}\right)","-\frac{1}{3} \tan ^{-1}\left(\frac{x^3}{2 \left(1+x^3\right)^{3/2}}\right)",1,"ArcTan[Sqrt[1 + x^3]]/3 - ArcTan[(1 + 2*x^3)/Sqrt[1 + x^3]]/3","A",1
204,1,22,22,0.0988801,"\int \frac{1}{\left(-1+x^2\right) \sqrt[3]{-x+x^3}} \, dx","IntegrateAlgebraic[1/((-1 + x^2)*(-x + x^3)^(1/3)),x]","-\frac{3 \left(-x+x^3\right)^{2/3}}{2 \left(-1+x^2\right)}","-\frac{3 \left(-x+x^3\right)^{2/3}}{2 \left(-1+x^2\right)}",1,"(-3*(-x + x^3)^(2/3))/(2*(-1 + x^2))","A",1
205,1,22,22,0.2107885,"\int \frac{1}{\left(-1+x^3\right) \sqrt[4]{-x+x^4}} \, dx","IntegrateAlgebraic[1/((-1 + x^3)*(-x + x^4)^(1/4)),x]","-\frac{4 \left(-x+x^4\right)^{3/4}}{3 \left(-1+x^3\right)}","-\frac{4 \left(-x+x^4\right)^{3/4}}{3 \left(-1+x^3\right)}",1,"(-4*(-x + x^4)^(3/4))/(3*(-1 + x^3))","A",1
206,1,22,22,0.0881849,"\int \frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{b^2+a x^2}} \, dx","IntegrateAlgebraic[Sqrt[b + Sqrt[b^2 + a*x^2]]/Sqrt[b^2 + a*x^2],x]","\frac{2 x}{\sqrt{b+\sqrt{b^2+a x^2}}}","\frac{2 x}{\sqrt{b+\sqrt{b^2+a x^2}}}",1,"(2*x)/Sqrt[b + Sqrt[b^2 + a*x^2]]","A",1
207,1,23,23,0.0219566,"\int \frac{1}{x \sqrt[4]{1+x^2}} \, dx","IntegrateAlgebraic[1/(x*(1 + x^2)^(1/4)),x]","\tan ^{-1}\left(\sqrt[4]{1+x^2}\right)-\tanh ^{-1}\left(\sqrt[4]{1+x^2}\right)","\tan ^{-1}\left(\sqrt[4]{1+x^2}\right)-\tanh ^{-1}\left(\sqrt[4]{1+x^2}\right)",1,"ArcTan[(1 + x^2)^(1/4)] - ArcTanh[(1 + x^2)^(1/4)]","A",1
208,1,39,23,4.4615874,"\int \frac{-5+2 x}{\sqrt[4]{4-4 x+x^2}} \, dx","IntegrateAlgebraic[(-5 + 2*x)/(4 - 4*x + x^2)^(1/4),x]","\frac{2 \left(-3 \sqrt{-2+x}+2 (-2+x)^{3/2}\right) \left((-2+x)^2\right)^{3/4}}{3 (-2+x)^{3/2}}","\frac{2 \left((-2+x)^2\right)^{3/4} (-7+2 x)}{3 (-2+x)}",1,"(2*(-3*Sqrt[-2 + x] + 2*(-2 + x)^(3/2))*((-2 + x)^2)^(3/4))/(3*(-2 + x)^(3/2))","A",1
209,1,20,23,0.0139646,"\int x^5 \sqrt[3]{1+x^3} \, dx","IntegrateAlgebraic[x^5*(1 + x^3)^(1/3),x]","\frac{1}{28} \left(1+x^3\right)^{4/3} \left(-3+4 x^3\right)","\frac{1}{28} \sqrt[3]{1+x^3} \left(-3+x^3+4 x^6\right)",1,"((1 + x^3)^(4/3)*(-3 + 4*x^3))/28","A",1
210,1,18,23,0.1884189,"\int \frac{\left(1+3 x^2\right) \sqrt[3]{-x+x^3}}{x^2} \, dx","IntegrateAlgebraic[((1 + 3*x^2)*(-x + x^3)^(1/3))/x^2,x]","\frac{3 \left(-x+x^3\right)^{4/3}}{2 x^2}","\frac{3 \left(-1+x^2\right) \sqrt[3]{-x+x^3}}{2 x}",1,"(3*(-x + x^3)^(4/3))/(2*x^2)","A",1
211,1,23,23,0.1305653,"\int \frac{1}{(-1+x) \sqrt[3]{-x^2+x^3}} \, dx","IntegrateAlgebraic[1/((-1 + x)*(-x^2 + x^3)^(1/3)),x]","-\frac{3 \left(-x^2+x^3\right)^{2/3}}{(-1+x) x}","-\frac{3 \left(-x^2+x^3\right)^{2/3}}{(-1+x) x}",1,"(-3*(-x^2 + x^3)^(2/3))/((-1 + x)*x)","A",1
212,1,23,23,0.1088097,"\int \frac{1}{x^2 \sqrt[3]{x^2+x^3}} \, dx","IntegrateAlgebraic[1/(x^2*(x^2 + x^3)^(1/3)),x]","\frac{3 (-2+3 x) \left(x^2+x^3\right)^{2/3}}{10 x^3}","\frac{3 (-2+3 x) \left(x^2+x^3\right)^{2/3}}{10 x^3}",1,"(3*(-2 + 3*x)*(x^2 + x^3)^(2/3))/(10*x^3)","A",1
213,1,23,23,0.1617181,"\int \frac{\left(3+2 x^2\right) \sqrt{x+2 x^3}}{\left(1+2 x^2\right)^2} \, dx","IntegrateAlgebraic[((3 + 2*x^2)*Sqrt[x + 2*x^3])/(1 + 2*x^2)^2,x]","\frac{2 x \sqrt{x+2 x^3}}{1+2 x^2}","\frac{2 x \sqrt{x+2 x^3}}{1+2 x^2}",1,"(2*x*Sqrt[x + 2*x^3])/(1 + 2*x^2)","A",1
214,1,23,23,0.1583731,"\int \frac{1}{x^6 \left(-1+x^4\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x^6*(-1 + x^4)^(3/4)),x]","\frac{\sqrt[4]{-1+x^4} \left(1+4 x^4\right)}{5 x^5}","\frac{\sqrt[4]{-1+x^4} \left(1+4 x^4\right)}{5 x^5}",1,"((-1 + x^4)^(1/4)*(1 + 4*x^4))/(5*x^5)","A",1
215,1,23,23,0.1613773,"\int \frac{1}{x^4 \left(1+x^4\right)^{5/4}} \, dx","IntegrateAlgebraic[1/(x^4*(1 + x^4)^(5/4)),x]","\frac{-1-4 x^4}{3 x^3 \sqrt[4]{1+x^4}}","\frac{-1-4 x^4}{3 x^3 \sqrt[4]{1+x^4}}",1,"(-1 - 4*x^4)/(3*x^3*(1 + x^4)^(1/4))","A",1
216,1,23,23,0.155047,"\int \frac{1}{x^6 \left(1+x^4\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x^6*(1 + x^4)^(3/4)),x]","\frac{\sqrt[4]{1+x^4} \left(-1+4 x^4\right)}{5 x^5}","\frac{\sqrt[4]{1+x^4} \left(-1+4 x^4\right)}{5 x^5}",1,"((1 + x^4)^(1/4)*(-1 + 4*x^4))/(5*x^5)","A",1
217,1,23,23,0.1805107,"\int \frac{-1+x^4}{x^6 \left(1+x^4\right)^{3/4}} \, dx","IntegrateAlgebraic[(-1 + x^4)/(x^6*(1 + x^4)^(3/4)),x]","\frac{\left(1-9 x^4\right) \sqrt[4]{1+x^4}}{5 x^5}","\frac{\left(1-9 x^4\right) \sqrt[4]{1+x^4}}{5 x^5}",1,"((1 - 9*x^4)*(1 + x^4)^(1/4))/(5*x^5)","A",1
218,1,23,23,0.3512627,"\int \frac{1+x^2}{\left(1-x^2\right) \sqrt{1+x^4}} \, dx","IntegrateAlgebraic[(1 + x^2)/((1 - x^2)*Sqrt[1 + x^4]),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{\sqrt{2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{\sqrt{2}}",1,"ArcTanh[(Sqrt[2]*x)/Sqrt[1 + x^4]]/Sqrt[2]","A",1
219,1,23,23,1.1719682,"\int \frac{-1+2 x+2 x^2}{(1+2 x) \sqrt{-x+x^4}} \, dx","IntegrateAlgebraic[(-1 + 2*x + 2*x^2)/((1 + 2*x)*Sqrt[-x + x^4]),x]","\tanh ^{-1}\left(\frac{2 \sqrt{-x+x^4}}{1+2 x^2}\right)","\tanh ^{-1}\left(\frac{2 \sqrt{-x+x^4}}{1+2 x^2}\right)",1,"ArcTanh[(2*Sqrt[-x + x^4])/(1 + 2*x^2)]","A",1
220,1,23,23,0.3790634,"\int \frac{1}{x^5 \sqrt{x+x^4}} \, dx","IntegrateAlgebraic[1/(x^5*Sqrt[x + x^4]),x]","\frac{2 \left(-1+2 x^3\right) \sqrt{x+x^4}}{9 x^5}","\frac{2 \left(-1+2 x^3\right) \sqrt{x+x^4}}{9 x^5}",1,"(2*(-1 + 2*x^3)*Sqrt[x + x^4])/(9*x^5)","A",1
221,1,23,23,0.2543999,"\int \frac{-1+x^3}{x^6 \sqrt[4]{x+x^4}} \, dx","IntegrateAlgebraic[(-1 + x^3)/(x^6*(x + x^4)^(1/4)),x]","-\frac{4 \left(-3+11 x^3\right) \left(x+x^4\right)^{3/4}}{63 x^6}","-\frac{4 \left(-3+11 x^3\right) \left(x+x^4\right)^{3/4}}{63 x^6}",1,"(-4*(-3 + 11*x^3)*(x + x^4)^(3/4))/(63*x^6)","A",1
222,1,23,23,0.2594254,"\int \frac{1+2 x^3}{x^6 \sqrt[4]{x+x^4}} \, dx","IntegrateAlgebraic[(1 + 2*x^3)/(x^6*(x + x^4)^(1/4)),x]","-\frac{4 \left(3+10 x^3\right) \left(x+x^4\right)^{3/4}}{63 x^6}","-\frac{4 \left(3+10 x^3\right) \left(x+x^4\right)^{3/4}}{63 x^6}",1,"(-4*(3 + 10*x^3)*(x + x^4)^(3/4))/(63*x^6)","A",1
223,1,23,23,0.0998225,"\int \frac{-1+x^2}{\left(1+x^2\right) \sqrt[3]{x^2+x^4}} \, dx","IntegrateAlgebraic[(-1 + x^2)/((1 + x^2)*(x^2 + x^4)^(1/3)),x]","-\frac{3 \left(x^2+x^4\right)^{2/3}}{x \left(1+x^2\right)}","-\frac{3 \left(x^2+x^4\right)^{2/3}}{x \left(1+x^2\right)}",1,"(-3*(x^2 + x^4)^(2/3))/(x*(1 + x^2))","A",1
224,1,23,23,0.0860479,"\int \frac{1}{\left(1+x^2\right) \sqrt[4]{x^2+x^4}} \, dx","IntegrateAlgebraic[1/((1 + x^2)*(x^2 + x^4)^(1/4)),x]","\frac{2 \left(x^2+x^4\right)^{3/4}}{x \left(1+x^2\right)}","\frac{2 \left(x^2+x^4\right)^{3/4}}{x \left(1+x^2\right)}",1,"(2*(x^2 + x^4)^(3/4))/(x*(1 + x^2))","A",1
225,1,23,23,0.2085758,"\int \frac{1}{(-1+x) \sqrt[4]{-x^3+x^4}} \, dx","IntegrateAlgebraic[1/((-1 + x)*(-x^3 + x^4)^(1/4)),x]","-\frac{4 \left(-x^3+x^4\right)^{3/4}}{(-1+x) x^2}","-\frac{4 \left(-x^3+x^4\right)^{3/4}}{(-1+x) x^2}",1,"(-4*(-x^3 + x^4)^(3/4))/((-1 + x)*x^2)","A",1
226,1,23,23,0.112238,"\int \frac{1}{x \sqrt{x^3+x^4}} \, dx","IntegrateAlgebraic[1/(x*Sqrt[x^3 + x^4]),x]","\frac{2 (-1+2 x) \sqrt{x^3+x^4}}{3 x^3}","\frac{2 (-1+2 x) \sqrt{x^3+x^4}}{3 x^3}",1,"(2*(-1 + 2*x)*Sqrt[x^3 + x^4])/(3*x^3)","A",1
227,1,23,23,0.2004719,"\int \frac{1}{x^2 \sqrt[4]{x^3+x^4}} \, dx","IntegrateAlgebraic[1/(x^2*(x^3 + x^4)^(1/4)),x]","\frac{4 (-3+4 x) \left(x^3+x^4\right)^{3/4}}{21 x^4}","\frac{4 (-3+4 x) \left(x^3+x^4\right)^{3/4}}{21 x^4}",1,"(4*(-3 + 4*x)*(x^3 + x^4)^(3/4))/(21*x^4)","A",1
228,1,23,23,0.0738512,"\int \frac{1+2 x}{\sqrt{3+x^2+2 x^3+x^4}} \, dx","IntegrateAlgebraic[(1 + 2*x)/Sqrt[3 + x^2 + 2*x^3 + x^4],x]","\log \left(x+x^2+\sqrt{3+x^2+2 x^3+x^4}\right)","\log \left(x+x^2+\sqrt{3+x^2+2 x^3+x^4}\right)",1,"Log[x + x^2 + Sqrt[3 + x^2 + 2*x^3 + x^4]]","A",1
229,1,23,23,0.1592697,"\int \frac{-1+2 x^4}{x^8 \sqrt[4]{-1+x^4}} \, dx","IntegrateAlgebraic[(-1 + 2*x^4)/(x^8*(-1 + x^4)^(1/4)),x]","\frac{\left(-1+x^4\right)^{3/4} \left(-3+10 x^4\right)}{21 x^7}","\frac{\left(-1+x^4\right)^{3/4} \left(-3+10 x^4\right)}{21 x^7}",1,"((-1 + x^4)^(3/4)*(-3 + 10*x^4))/(21*x^7)","A",1
230,1,23,23,0.2128534,"\int \frac{1+2 x^4}{x^4 \left(1+x^4\right)^{5/4}} \, dx","IntegrateAlgebraic[(1 + 2*x^4)/(x^4*(1 + x^4)^(5/4)),x]","\frac{-1+2 x^4}{3 x^3 \sqrt[4]{1+x^4}}","\frac{-1+2 x^4}{3 x^3 \sqrt[4]{1+x^4}}",1,"(-1 + 2*x^4)/(3*x^3*(1 + x^4)^(1/4))","A",1
231,1,23,23,0.0525074,"\int \frac{\left(1+4 x^3\right) \left(1+2 x+2 x^4\right)}{\sqrt{x+x^4}} \, dx","IntegrateAlgebraic[((1 + 4*x^3)*(1 + 2*x + 2*x^4))/Sqrt[x + x^4],x]","\frac{2}{3} \sqrt{x+x^4} \left(3+2 x+2 x^4\right)","\frac{2}{3} \sqrt{x+x^4} \left(3+2 x+2 x^4\right)",1,"(2*Sqrt[x + x^4]*(3 + 2*x + 2*x^4))/3","A",1
232,1,23,23,0.1359222,"\int \frac{-1+x^2}{\left(1+x^2\right) \sqrt[4]{x^3+x^5}} \, dx","IntegrateAlgebraic[(-1 + x^2)/((1 + x^2)*(x^3 + x^5)^(1/4)),x]","-\frac{4 \left(x^3+x^5\right)^{3/4}}{x^2 \left(1+x^2\right)}","-\frac{4 \left(x^3+x^5\right)^{3/4}}{x^2 \left(1+x^2\right)}",1,"(-4*(x^3 + x^5)^(3/4))/(x^2*(1 + x^2))","A",1
233,1,23,23,0.1571627,"\int \frac{1}{x^{10} \sqrt{-1+x^6}} \, dx","IntegrateAlgebraic[1/(x^10*Sqrt[-1 + x^6]),x]","\frac{\sqrt{-1+x^6} \left(1+2 x^6\right)}{9 x^9}","\frac{\sqrt{-1+x^6} \left(1+2 x^6\right)}{9 x^9}",1,"(Sqrt[-1 + x^6]*(1 + 2*x^6))/(9*x^9)","A",1
234,1,23,23,0.2014693,"\int \frac{1+x^6}{x^{10} \sqrt{-1+x^6}} \, dx","IntegrateAlgebraic[(1 + x^6)/(x^10*Sqrt[-1 + x^6]),x]","\frac{\sqrt{-1+x^6} \left(1+5 x^6\right)}{9 x^9}","\frac{\sqrt{-1+x^6} \left(1+5 x^6\right)}{9 x^9}",1,"(Sqrt[-1 + x^6]*(1 + 5*x^6))/(9*x^9)","A",1
235,1,23,23,0.2512137,"\int \frac{-1+3 x^4}{\left(1+x^4\right) \sqrt[3]{x^2+x^6}} \, dx","IntegrateAlgebraic[(-1 + 3*x^4)/((1 + x^4)*(x^2 + x^6)^(1/3)),x]","-\frac{3 \left(x^2+x^6\right)^{2/3}}{x \left(1+x^4\right)}","-\frac{3 \left(x^2+x^6\right)^{2/3}}{x \left(1+x^4\right)}",1,"(-3*(x^2 + x^6)^(2/3))/(x*(1 + x^4))","A",1
236,1,23,23,0.2686228,"\int \frac{-1+x^4}{\left(1+x^4\right) \sqrt[4]{x^2+x^6}} \, dx","IntegrateAlgebraic[(-1 + x^4)/((1 + x^4)*(x^2 + x^6)^(1/4)),x]","-\frac{2 \left(x^2+x^6\right)^{3/4}}{x \left(1+x^4\right)}","-\frac{2 \left(x^2+x^6\right)^{3/4}}{x \left(1+x^4\right)}",1,"(-2*(x^2 + x^6)^(3/4))/(x*(1 + x^4))","A",1
237,1,23,23,4.8910154,"\int \frac{x^2 \left(3+2 x^2\right) \left(1+x^2+2 x^6\right)}{\left(1+x^2\right)^2 \sqrt{1+x^2+x^6}} \, dx","IntegrateAlgebraic[(x^2*(3 + 2*x^2)*(1 + x^2 + 2*x^6))/((1 + x^2)^2*Sqrt[1 + x^2 + x^6]),x]","\frac{x^3 \sqrt{1+x^2+x^6}}{1+x^2}","\frac{x^3 \sqrt{1+x^2+x^6}}{1+x^2}",1,"(x^3*Sqrt[1 + x^2 + x^6])/(1 + x^2)","A",1
238,1,24,24,0.0515896,"\int \frac{1}{\sqrt{3-5 x+x^2+x^3}} \, dx","IntegrateAlgebraic[1/Sqrt[3 - 5*x + x^2 + x^3],x]","-\tanh ^{-1}\left(\frac{-2+2 x}{\sqrt{3-5 x+x^2+x^3}}\right)","-\tanh ^{-1}\left(\frac{-2+2 x}{\sqrt{3-5 x+x^2+x^3}}\right)",1,"-ArcTanh[(-2 + 2*x)/Sqrt[3 - 5*x + x^2 + x^3]]","A",1
239,1,24,24,0.0693359,"\int \frac{-1+x}{(1+x) \sqrt{x+x^2+x^3}} \, dx","IntegrateAlgebraic[(-1 + x)/((1 + x)*Sqrt[x + x^2 + x^3]),x]","-2 \tan ^{-1}\left(\frac{\sqrt{x+x^2+x^3}}{1+x+x^2}\right)","-2 \tan ^{-1}\left(\frac{\sqrt{x+x^2+x^3}}{1+x+x^2}\right)",1,"-2*ArcTan[Sqrt[x + x^2 + x^3]/(1 + x + x^2)]","A",1
240,1,24,24,0.0671117,"\int \frac{-1+x^2}{\left(1+x^2\right) \sqrt{x+x^2+x^3}} \, dx","IntegrateAlgebraic[(-1 + x^2)/((1 + x^2)*Sqrt[x + x^2 + x^3]),x]","-2 \tanh ^{-1}\left(\frac{\sqrt{x+x^2+x^3}}{1+x+x^2}\right)","-2 \tanh ^{-1}\left(\frac{\sqrt{x+x^2+x^3}}{1+x+x^2}\right)",1,"-2*ArcTanh[Sqrt[x + x^2 + x^3]/(1 + x + x^2)]","A",1
241,1,24,24,0.3132954,"\int \frac{-1+x^2}{\left(1+x^2\right) \sqrt{1+x^4}} \, dx","IntegrateAlgebraic[(-1 + x^2)/((1 + x^2)*Sqrt[1 + x^4]),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{\sqrt{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{\sqrt{2}}",1,"-(ArcTan[(Sqrt[2]*x)/Sqrt[1 + x^4]]/Sqrt[2])","A",1
242,1,24,24,0.3389004,"\int \frac{1+x^2}{\left(-1+x^2\right) \sqrt{1+x^4}} \, dx","IntegrateAlgebraic[(1 + x^2)/((-1 + x^2)*Sqrt[1 + x^4]),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{\sqrt{2}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{\sqrt{2}}",1,"-(ArcTanh[(Sqrt[2]*x)/Sqrt[1 + x^4]]/Sqrt[2])","A",1
243,1,24,24,5.0351067,"\int \frac{2-3 x^5}{\sqrt{1+x^5} \left(1-a x^2+x^5\right)} \, dx","IntegrateAlgebraic[(2 - 3*x^5)/(Sqrt[1 + x^5]*(1 - a*x^2 + x^5)),x]","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{1+x^5}}\right)}{\sqrt{a}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{1+x^5}}\right)}{\sqrt{a}}",1,"(2*ArcTanh[(Sqrt[a]*x)/Sqrt[1 + x^5]])/Sqrt[a]","A",1
244,1,24,24,5.0329683,"\int \frac{2+3 x^5}{\sqrt{-1+x^5} \left(-1-a x^2+x^5\right)} \, dx","IntegrateAlgebraic[(2 + 3*x^5)/(Sqrt[-1 + x^5]*(-1 - a*x^2 + x^5)),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{-1+x^5}}\right)}{\sqrt{a}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{-1+x^5}}\right)}{\sqrt{a}}",1,"(-2*ArcTanh[(Sqrt[a]*x)/Sqrt[-1 + x^5]])/Sqrt[a]","A",1
245,1,24,24,2.2456505,"\int \frac{2+3 x^5}{\sqrt{-1+x^5} \left(-a-x^2+a x^5\right)} \, dx","IntegrateAlgebraic[(2 + 3*x^5)/(Sqrt[-1 + x^5]*(-a - x^2 + a*x^5)),x]","-\frac{2 \tanh ^{-1}\left(\frac{x}{\sqrt{a} \sqrt{-1+x^5}}\right)}{\sqrt{a}}","-\frac{2 \tanh ^{-1}\left(\frac{x}{\sqrt{a} \sqrt{-1+x^5}}\right)}{\sqrt{a}}",1,"(-2*ArcTanh[x/(Sqrt[a]*Sqrt[-1 + x^5])])/Sqrt[a]","A",1
246,1,24,24,2.1031289,"\int \frac{-2+3 x^5}{\sqrt{1+x^5} \left(a-x^2+a x^5\right)} \, dx","IntegrateAlgebraic[(-2 + 3*x^5)/(Sqrt[1 + x^5]*(a - x^2 + a*x^5)),x]","-\frac{2 \tanh ^{-1}\left(\frac{x}{\sqrt{a} \sqrt{1+x^5}}\right)}{\sqrt{a}}","-\frac{2 \tanh ^{-1}\left(\frac{x}{\sqrt{a} \sqrt{1+x^5}}\right)}{\sqrt{a}}",1,"(-2*ArcTanh[x/(Sqrt[a]*Sqrt[1 + x^5])])/Sqrt[a]","A",1
247,1,24,24,8.4164759,"\int \frac{-1-2 x^2+2 x^4}{\left(1-x^2+x^4\right) \sqrt{1+x^6}} \, dx","IntegrateAlgebraic[(-1 - 2*x^2 + 2*x^4)/((1 - x^2 + x^4)*Sqrt[1 + x^6]),x]","-\frac{x \sqrt{1+x^6}}{1-x^2+x^4}","-\frac{x \sqrt{1+x^6}}{1-x^2+x^4}",1,"-((x*Sqrt[1 + x^6])/(1 - x^2 + x^4))","A",1
248,1,24,24,2.6654691,"\int \frac{-1+4 x^5}{\left(1-a x+x^5\right) \sqrt{x+x^6}} \, dx","IntegrateAlgebraic[(-1 + 4*x^5)/((1 - a*x + x^5)*Sqrt[x + x^6]),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{x+x^6}}\right)}{\sqrt{a}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{x+x^6}}\right)}{\sqrt{a}}",1,"(-2*ArcTanh[(Sqrt[a]*x)/Sqrt[x + x^6]])/Sqrt[a]","A",1
249,1,24,24,2.6550486,"\int \frac{-1+4 x^5}{\left(a-x+a x^5\right) \sqrt{x+x^6}} \, dx","IntegrateAlgebraic[(-1 + 4*x^5)/((a - x + a*x^5)*Sqrt[x + x^6]),x]","-\frac{2 \tanh ^{-1}\left(\frac{x}{\sqrt{a} \sqrt{x+x^6}}\right)}{\sqrt{a}}","-\frac{2 \tanh ^{-1}\left(\frac{x}{\sqrt{a} \sqrt{x+x^6}}\right)}{\sqrt{a}}",1,"(-2*ArcTanh[x/(Sqrt[a]*Sqrt[x + x^6])])/Sqrt[a]","A",1
250,1,20,25,0.0161056,"\int x^3 \left(-1+x^2\right)^{2/3} \, dx","IntegrateAlgebraic[x^3*(-1 + x^2)^(2/3),x]","\frac{3}{80} \left(-1+x^2\right)^{5/3} \left(3+5 x^2\right)","\frac{3}{80} \left(-1+x^2\right)^{2/3} \left(-3-2 x^2+5 x^4\right)",1,"(3*(-1 + x^2)^(5/3)*(3 + 5*x^2))/80","A",1
251,1,20,25,0.0174816,"\int x^3 \left(-1+x^2\right)^{3/4} \, dx","IntegrateAlgebraic[x^3*(-1 + x^2)^(3/4),x]","\frac{2}{77} \left(-1+x^2\right)^{7/4} \left(4+7 x^2\right)","\frac{2}{77} \left(-1+x^2\right)^{3/4} \left(-4-3 x^2+7 x^4\right)",1,"(2*(-1 + x^2)^(7/4)*(4 + 7*x^2))/77","A",1
252,1,25,25,0.0248162,"\int \frac{1}{x \left(1+x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x*(1 + x^2)^(3/4)),x]","-\tan ^{-1}\left(\sqrt[4]{1+x^2}\right)-\tanh ^{-1}\left(\sqrt[4]{1+x^2}\right)","-\tan ^{-1}\left(\sqrt[4]{1+x^2}\right)-\tanh ^{-1}\left(\sqrt[4]{1+x^2}\right)",1,"-ArcTan[(1 + x^2)^(1/4)] - ArcTanh[(1 + x^2)^(1/4)]","A",1
253,1,20,25,0.0168285,"\int x^3 \left(1+x^2\right)^{3/4} \, dx","IntegrateAlgebraic[x^3*(1 + x^2)^(3/4),x]","\frac{2}{77} \left(1+x^2\right)^{7/4} \left(-4+7 x^2\right)","\frac{2}{77} \left(1+x^2\right)^{3/4} \left(-4+3 x^2+7 x^4\right)",1,"(2*(1 + x^2)^(7/4)*(-4 + 7*x^2))/77","A",1
254,1,20,25,0.013386,"\int x^5 \sqrt[3]{-1+x^3} \, dx","IntegrateAlgebraic[x^5*(-1 + x^3)^(1/3),x]","\frac{1}{28} \left(-1+x^3\right)^{4/3} \left(3+4 x^3\right)","\frac{1}{28} \sqrt[3]{-1+x^3} \left(-3-x^3+4 x^6\right)",1,"((-1 + x^3)^(4/3)*(3 + 4*x^3))/28","A",1
255,1,20,25,0.0172654,"\int x^5 \left(-1+x^3\right)^{3/4} \, dx","IntegrateAlgebraic[x^5*(-1 + x^3)^(3/4),x]","\frac{4}{231} \left(-1+x^3\right)^{7/4} \left(4+7 x^3\right)","\frac{4}{231} \left(-1+x^3\right)^{3/4} \left(-4-3 x^3+7 x^6\right)",1,"(4*(-1 + x^3)^(7/4)*(4 + 7*x^3))/231","A",1
256,1,25,25,0.017519,"\int \frac{x^8}{\sqrt[4]{1+x^3}} \, dx","IntegrateAlgebraic[x^8/(1 + x^3)^(1/4),x]","\frac{4}{693} \left(1+x^3\right)^{3/4} \left(32-24 x^3+21 x^6\right)","\frac{4}{693} \left(1+x^3\right)^{3/4} \left(32-24 x^3+21 x^6\right)",1,"(4*(1 + x^3)^(3/4)*(32 - 24*x^3 + 21*x^6))/693","A",1
257,1,20,25,0.0167961,"\int x^5 \left(1+x^3\right)^{2/3} \, dx","IntegrateAlgebraic[x^5*(1 + x^3)^(2/3),x]","\frac{1}{40} \left(1+x^3\right)^{5/3} \left(-3+5 x^3\right)","\frac{1}{40} \left(1+x^3\right)^{2/3} \left(-3+2 x^3+5 x^6\right)",1,"((1 + x^3)^(5/3)*(-3 + 5*x^3))/40","A",1
258,1,25,25,0.1099552,"\int \frac{1}{x^2 \sqrt[3]{-x^2+x^3}} \, dx","IntegrateAlgebraic[1/(x^2*(-x^2 + x^3)^(1/3)),x]","\frac{3 (2+3 x) \left(-x^2+x^3\right)^{2/3}}{10 x^3}","\frac{3 (2+3 x) \left(-x^2+x^3\right)^{2/3}}{10 x^3}",1,"(3*(2 + 3*x)*(-x^2 + x^3)^(2/3))/(10*x^3)","A",1
259,1,20,25,0.0188462,"\int x^7 \left(1+x^4\right)^{2/3} \, dx","IntegrateAlgebraic[x^7*(1 + x^4)^(2/3),x]","\frac{3}{160} \left(1+x^4\right)^{5/3} \left(-3+5 x^4\right)","\frac{3}{160} \left(1+x^4\right)^{2/3} \left(-3+2 x^4+5 x^8\right)",1,"(3*(1 + x^4)^(5/3)*(-3 + 5*x^4))/160","A",1
260,1,25,25,0.3870731,"\int \frac{1}{x^5 \sqrt{-x+x^4}} \, dx","IntegrateAlgebraic[1/(x^5*Sqrt[-x + x^4]),x]","\frac{2 \left(1+2 x^3\right) \sqrt{-x+x^4}}{9 x^5}","\frac{2 \left(1+2 x^3\right) \sqrt{-x+x^4}}{9 x^5}",1,"(2*(1 + 2*x^3)*Sqrt[-x + x^4])/(9*x^5)","A",1
261,1,25,25,0.2686265,"\int \frac{1+x^3}{x^6 \sqrt[4]{-x+x^4}} \, dx","IntegrateAlgebraic[(1 + x^3)/(x^6*(-x + x^4)^(1/4)),x]","\frac{4 \left(3+11 x^3\right) \left(-x+x^4\right)^{3/4}}{63 x^6}","\frac{4 \left(3+11 x^3\right) \left(-x+x^4\right)^{3/4}}{63 x^6}",1,"(4*(3 + 11*x^3)*(-x + x^4)^(3/4))/(63*x^6)","A",1
262,1,25,25,0.092194,"\int \frac{1+x^2}{\left(-1+x^2\right) \sqrt[3]{-x^2+x^4}} \, dx","IntegrateAlgebraic[(1 + x^2)/((-1 + x^2)*(-x^2 + x^4)^(1/3)),x]","-\frac{3 \left(-x^2+x^4\right)^{2/3}}{x \left(-1+x^2\right)}","-\frac{3 \left(-x^2+x^4\right)^{2/3}}{x \left(-1+x^2\right)}",1,"(-3*(-x^2 + x^4)^(2/3))/(x*(-1 + x^2))","A",1
263,1,25,25,0.1098993,"\int \frac{1}{\left(-1+x^2\right) \sqrt[4]{-x^2+x^4}} \, dx","IntegrateAlgebraic[1/((-1 + x^2)*(-x^2 + x^4)^(1/4)),x]","-\frac{2 \left(-x^2+x^4\right)^{3/4}}{x \left(-1+x^2\right)}","-\frac{2 \left(-x^2+x^4\right)^{3/4}}{x \left(-1+x^2\right)}",1,"(-2*(-x^2 + x^4)^(3/4))/(x*(-1 + x^2))","A",1
264,1,25,25,0.1029865,"\int \frac{1}{x^4 \sqrt[4]{x^2+x^4}} \, dx","IntegrateAlgebraic[1/(x^4*(x^2 + x^4)^(1/4)),x]","\frac{2 \left(-3+4 x^2\right) \left(x^2+x^4\right)^{3/4}}{21 x^5}","\frac{2 \left(-3+4 x^2\right) \left(x^2+x^4\right)^{3/4}}{21 x^5}",1,"(2*(-3 + 4*x^2)*(x^2 + x^4)^(3/4))/(21*x^5)","A",1
265,1,25,25,0.0736343,"\int \frac{-1+2 x}{\sqrt{-3+x^2-2 x^3+x^4}} \, dx","IntegrateAlgebraic[(-1 + 2*x)/Sqrt[-3 + x^2 - 2*x^3 + x^4],x]","\log \left(-x+x^2+\sqrt{-3+x^2-2 x^3+x^4}\right)","\log \left(-x+x^2+\sqrt{-3+x^2-2 x^3+x^4}\right)",1,"Log[-x + x^2 + Sqrt[-3 + x^2 - 2*x^3 + x^4]]","A",1
266,1,25,25,0.1098561,"\int \frac{-1+2 x}{\sqrt{4+x^2-2 x^3+x^4}} \, dx","IntegrateAlgebraic[(-1 + 2*x)/Sqrt[4 + x^2 - 2*x^3 + x^4],x]","\log \left(-x+x^2+\sqrt{4+x^2-2 x^3+x^4}\right)","\log \left(-x+x^2+\sqrt{4+x^2-2 x^3+x^4}\right)",1,"Log[-x + x^2 + Sqrt[4 + x^2 - 2*x^3 + x^4]]","A",1
267,1,25,25,0.2164583,"\int \frac{-1+2 x}{\sqrt{13+x^2-2 x^3+x^4}} \, dx","IntegrateAlgebraic[(-1 + 2*x)/Sqrt[13 + x^2 - 2*x^3 + x^4],x]","\log \left(-x+x^2+\sqrt{13+x^2-2 x^3+x^4}\right)","\log \left(-x+x^2+\sqrt{13+x^2-2 x^3+x^4}\right)",1,"Log[-x + x^2 + Sqrt[13 + x^2 - 2*x^3 + x^4]]","A",1
268,1,25,25,0.2095694,"\int \frac{1}{x^2 \sqrt[4]{-x^3+x^4}} \, dx","IntegrateAlgebraic[1/(x^2*(-x^3 + x^4)^(1/4)),x]","\frac{4 (3+4 x) \left(-x^3+x^4\right)^{3/4}}{21 x^4}","\frac{4 (3+4 x) \left(-x^3+x^4\right)^{3/4}}{21 x^4}",1,"(4*(3 + 4*x)*(-x^3 + x^4)^(3/4))/(21*x^4)","A",1
269,1,25,25,0.1659204,"\int \frac{-1+x^4}{x^8 \sqrt[4]{-1+2 x^4}} \, dx","IntegrateAlgebraic[(-1 + x^4)/(x^8*(-1 + 2*x^4)^(1/4)),x]","\frac{\left(-3-x^4\right) \left(-1+2 x^4\right)^{3/4}}{21 x^7}","\frac{\left(-3-x^4\right) \left(-1+2 x^4\right)^{3/4}}{21 x^7}",1,"((-3 - x^4)*(-1 + 2*x^4)^(3/4))/(21*x^7)","A",1
270,1,25,25,0.1333009,"\int \frac{1+x^2}{\left(-1+x^2\right) \sqrt[4]{-x^3+x^5}} \, dx","IntegrateAlgebraic[(1 + x^2)/((-1 + x^2)*(-x^3 + x^5)^(1/4)),x]","-\frac{4 \left(-x^3+x^5\right)^{3/4}}{x^2 \left(-1+x^2\right)}","-\frac{4 \left(-x^3+x^5\right)^{3/4}}{x^2 \left(-1+x^2\right)}",1,"(-4*(-x^3 + x^5)^(3/4))/(x^2*(-1 + x^2))","A",1
271,1,25,25,0.245642,"\int \frac{1+3 x^4}{\left(-1+x^4\right) \sqrt[3]{-x^2+x^6}} \, dx","IntegrateAlgebraic[(1 + 3*x^4)/((-1 + x^4)*(-x^2 + x^6)^(1/3)),x]","-\frac{3 \left(-x^2+x^6\right)^{2/3}}{x \left(-1+x^4\right)}","-\frac{3 \left(-x^2+x^6\right)^{2/3}}{x \left(-1+x^4\right)}",1,"(-3*(-x^2 + x^6)^(2/3))/(x*(-1 + x^4))","A",1
272,1,25,25,0.2607923,"\int \frac{1+x^4}{\left(-1+x^4\right) \sqrt[4]{-x^2+x^6}} \, dx","IntegrateAlgebraic[(1 + x^4)/((-1 + x^4)*(-x^2 + x^6)^(1/4)),x]","-\frac{2 \left(-x^2+x^6\right)^{3/4}}{x \left(-1+x^4\right)}","-\frac{2 \left(-x^2+x^6\right)^{3/4}}{x \left(-1+x^4\right)}",1,"(-2*(-x^2 + x^6)^(3/4))/(x*(-1 + x^4))","A",1
273,1,25,25,0.4047018,"\int \frac{1}{x^7 \sqrt[3]{x^2+x^6}} \, dx","IntegrateAlgebraic[1/(x^7*(x^2 + x^6)^(1/3)),x]","\frac{3 \left(-2+3 x^4\right) \left(x^2+x^6\right)^{2/3}}{40 x^8}","\frac{3 \left(-2+3 x^4\right) \left(x^2+x^6\right)^{2/3}}{40 x^8}",1,"(3*(-2 + 3*x^4)*(x^2 + x^6)^(2/3))/(40*x^8)","A",1
274,1,26,26,0.0821348,"\int \frac{\sqrt[3]{-1+x^3}}{x^8} \, dx","IntegrateAlgebraic[(-1 + x^3)^(1/3)/x^8,x]","\frac{\sqrt[3]{-1+x^3} \left(-4+x^3+3 x^6\right)}{28 x^7}","\frac{\sqrt[3]{-1+x^3} \left(-4+x^3+3 x^6\right)}{28 x^7}",1,"((-1 + x^3)^(1/3)*(-4 + x^3 + 3*x^6))/(28*x^7)","A",1
275,1,26,26,0.1672586,"\int \frac{b+a x^2}{\left(-b+a x^2\right) \sqrt{-b x+a x^3}} \, dx","IntegrateAlgebraic[(b + a*x^2)/((-b + a*x^2)*Sqrt[-(b*x) + a*x^3]),x]","\frac{2 \sqrt{-b x+a x^3}}{b-a x^2}","\frac{2 \sqrt{-b x+a x^3}}{b-a x^2}",1,"(2*Sqrt[-(b*x) + a*x^3])/(b - a*x^2)","A",1
276,1,26,26,0.5686247,"\int \frac{-2 b+a x^3}{\sqrt{b+a x^3} \left(b-c x^2+a x^3\right)} \, dx","IntegrateAlgebraic[(-2*b + a*x^3)/(Sqrt[b + a*x^3]*(b - c*x^2 + a*x^3)),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b+a x^3}}\right)}{\sqrt{c}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b+a x^3}}\right)}{\sqrt{c}}",1,"(-2*ArcTanh[(Sqrt[c]*x)/Sqrt[b + a*x^3]])/Sqrt[c]","A",1
277,1,26,26,0.5696881,"\int \frac{-2 b+a x^3}{\sqrt{b+a x^3} \left(b+c x^2+a x^3\right)} \, dx","IntegrateAlgebraic[(-2*b + a*x^3)/(Sqrt[b + a*x^3]*(b + c*x^2 + a*x^3)),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b+a x^3}}\right)}{\sqrt{c}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{b+a x^3}}\right)}{\sqrt{c}}",1,"(-2*ArcTan[(Sqrt[c]*x)/Sqrt[b + a*x^3]])/Sqrt[c]","A",1
278,1,26,26,12.1095913,"\int \frac{1+x^2}{\left(-1+x^2\right) \left(2+x^2\right) \sqrt{-3+x^4}} \, dx","IntegrateAlgebraic[(1 + x^2)/((-1 + x^2)*(2 + x^2)*Sqrt[-3 + x^4]),x]","\frac{\tan ^{-1}\left(\frac{x \sqrt{-3+x^4}}{\sqrt{2}}\right)}{3 \sqrt{2}}","\frac{\tan ^{-1}\left(\frac{x \sqrt{-3+x^4}}{\sqrt{2}}\right)}{3 \sqrt{2}}",1,"ArcTan[(x*Sqrt[-3 + x^4])/Sqrt[2]]/(3*Sqrt[2])","A",1
279,1,23,26,0.1370146,"\int \frac{\left(-4+x^4\right) \left(1+x^4\right)^{3/4}}{x^{12}} \, dx","IntegrateAlgebraic[((-4 + x^4)*(1 + x^4)^(3/4))/x^12,x]","\frac{\left(28-27 x^4\right) \left(1+x^4\right)^{7/4}}{77 x^{11}}","\frac{\left(1+x^4\right)^{3/4} \left(28+x^4-27 x^8\right)}{77 x^{11}}",1,"((28 - 27*x^4)*(1 + x^4)^(7/4))/(77*x^11)","A",1
280,1,26,26,0.3478129,"\int \frac{\sqrt{-1+x^4} \left(1+x^4\right)}{x^2 \left(-1+x^2+x^4\right)} \, dx","IntegrateAlgebraic[(Sqrt[-1 + x^4]*(1 + x^4))/(x^2*(-1 + x^2 + x^4)),x]","\frac{\sqrt{-1+x^4}}{x}+\tan ^{-1}\left(\frac{x}{\sqrt{-1+x^4}}\right)","\frac{\sqrt{-1+x^4}}{x}+\tan ^{-1}\left(\frac{x}{\sqrt{-1+x^4}}\right)",1,"Sqrt[-1 + x^4]/x + ArcTan[x/Sqrt[-1 + x^4]]","A",1
281,1,26,26,0.4149986,"\int \frac{\left(-1+x^4\right) \left(1+x^2+x^4\right)}{x^4 \sqrt{1+x^4}} \, dx","IntegrateAlgebraic[((-1 + x^4)*(1 + x^2 + x^4))/(x^4*Sqrt[1 + x^4]),x]","\frac{\sqrt{1+x^4} \left(1+3 x^2+x^4\right)}{3 x^3}","\frac{\sqrt{1+x^4} \left(1+3 x^2+x^4\right)}{3 x^3}",1,"(Sqrt[1 + x^4]*(1 + 3*x^2 + x^4))/(3*x^3)","A",1
282,1,26,26,3.7377915,"\int \frac{\left(-4+x^3\right) \left(-1+x^3+x^4\right)}{x^6 \left(-1+x^3\right)^{3/4}} \, dx","IntegrateAlgebraic[((-4 + x^3)*(-1 + x^3 + x^4))/(x^6*(-1 + x^3)^(3/4)),x]","-\frac{4 \sqrt[4]{-1+x^3} \left(-1+x^3+5 x^4\right)}{5 x^5}","-\frac{4 \sqrt[4]{-1+x^3} \left(-1+x^3+5 x^4\right)}{5 x^5}",1,"(-4*(-1 + x^3)^(1/4)*(-1 + x^3 + 5*x^4))/(5*x^5)","A",1
283,1,26,26,0.5027788,"\int \frac{1+3 x^4}{\left(-1-a x+x^4\right) \sqrt{-x+x^5}} \, dx","IntegrateAlgebraic[(1 + 3*x^4)/((-1 - a*x + x^4)*Sqrt[-x + x^5]),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{-x+x^5}}\right)}{\sqrt{a}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{-x+x^5}}\right)}{\sqrt{a}}",1,"(-2*ArcTanh[(Sqrt[a]*x)/Sqrt[-x + x^5]])/Sqrt[a]","A",1
284,1,26,26,1.1525676,"\int \frac{1+3 x^4}{\left(-a-x+a x^4\right) \sqrt{-x+x^5}} \, dx","IntegrateAlgebraic[(1 + 3*x^4)/((-a - x + a*x^4)*Sqrt[-x + x^5]),x]","-\frac{2 \tanh ^{-1}\left(\frac{x}{\sqrt{a} \sqrt{-x+x^5}}\right)}{\sqrt{a}}","-\frac{2 \tanh ^{-1}\left(\frac{x}{\sqrt{a} \sqrt{-x+x^5}}\right)}{\sqrt{a}}",1,"(-2*ArcTanh[x/(Sqrt[a]*Sqrt[-x + x^5])])/Sqrt[a]","A",1
285,1,26,26,2.1936879,"\int \frac{\left(4+x^5\right) \left(-1+x^4+x^5\right)}{x^6 \left(-1+x^5\right)^{3/4}} \, dx","IntegrateAlgebraic[((4 + x^5)*(-1 + x^4 + x^5))/(x^6*(-1 + x^5)^(3/4)),x]","\frac{4 \sqrt[4]{-1+x^5} \left(-1+5 x^4+x^5\right)}{5 x^5}","\frac{4 \sqrt[4]{-1+x^5} \left(-1+5 x^4+x^5\right)}{5 x^5}",1,"(4*(-1 + x^5)^(1/4)*(-1 + 5*x^4 + x^5))/(5*x^5)","A",1
286,1,26,26,1.877221,"\int \frac{\left(-4+x^5\right) \left(1+x^4+x^5\right)}{x^6 \left(1+x^5\right)^{3/4}} \, dx","IntegrateAlgebraic[((-4 + x^5)*(1 + x^4 + x^5))/(x^6*(1 + x^5)^(3/4)),x]","\frac{4 \sqrt[4]{1+x^5} \left(1+5 x^4+x^5\right)}{5 x^5}","\frac{4 \sqrt[4]{1+x^5} \left(1+5 x^4+x^5\right)}{5 x^5}",1,"(4*(1 + x^5)^(1/4)*(1 + 5*x^4 + x^5))/(5*x^5)","A",1
287,1,26,26,0.1251304,"\int \frac{\sqrt{-1+x^6}}{x^{16}} \, dx","IntegrateAlgebraic[Sqrt[-1 + x^6]/x^16,x]","\frac{\sqrt{-1+x^6} \left(-3+x^6+2 x^{12}\right)}{45 x^{15}}","\frac{\sqrt{-1+x^6} \left(-3+x^6+2 x^{12}\right)}{45 x^{15}}",1,"(Sqrt[-1 + x^6]*(-3 + x^6 + 2*x^12))/(45*x^15)","A",1
288,1,26,26,12.1274471,"\int \frac{-1-2 x^2+2 x^4}{\left(1+2 x^4\right) \sqrt{1+x^6}} \, dx","IntegrateAlgebraic[(-1 - 2*x^2 + 2*x^4)/((1 + 2*x^4)*Sqrt[1 + x^6]),x]","-\tan ^{-1}\left(\frac{x \sqrt{1+x^6}}{1-x^2+x^4}\right)","-\tan ^{-1}\left(\frac{x \sqrt{1+x^6}}{1-x^2+x^4}\right)",1,"-ArcTan[(x*Sqrt[1 + x^6])/(1 - x^2 + x^4)]","A",1
289,1,26,26,2.657521,"\int \frac{1+4 x^5}{\left(-1-a x+x^5\right) \sqrt{-x+x^6}} \, dx","IntegrateAlgebraic[(1 + 4*x^5)/((-1 - a*x + x^5)*Sqrt[-x + x^6]),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{-x+x^6}}\right)}{\sqrt{a}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{-x+x^6}}\right)}{\sqrt{a}}",1,"(-2*ArcTanh[(Sqrt[a]*x)/Sqrt[-x + x^6]])/Sqrt[a]","A",1
290,1,26,26,2.6550085,"\int \frac{1+4 x^5}{\left(-a-x+a x^5\right) \sqrt{-x+x^6}} \, dx","IntegrateAlgebraic[(1 + 4*x^5)/((-a - x + a*x^5)*Sqrt[-x + x^6]),x]","-\frac{2 \tanh ^{-1}\left(\frac{x}{\sqrt{a} \sqrt{-x+x^6}}\right)}{\sqrt{a}}","-\frac{2 \tanh ^{-1}\left(\frac{x}{\sqrt{a} \sqrt{-x+x^6}}\right)}{\sqrt{a}}",1,"(-2*ArcTanh[x/(Sqrt[a]*Sqrt[-x + x^6])])/Sqrt[a]","A",1
291,1,26,26,4.1834298,"\int \frac{\left(2+x^6\right) \left(-1-x^4+x^6\right)}{x^6 \left(-1+x^6\right)^{3/4}} \, dx","IntegrateAlgebraic[((2 + x^6)*(-1 - x^4 + x^6))/(x^6*(-1 + x^6)^(3/4)),x]","\frac{2 \sqrt[4]{-1+x^6} \left(-1-5 x^4+x^6\right)}{5 x^5}","\frac{2 \sqrt[4]{-1+x^6} \left(-1-5 x^4+x^6\right)}{5 x^5}",1,"(2*(-1 + x^6)^(1/4)*(-1 - 5*x^4 + x^6))/(5*x^5)","A",1
292,1,26,26,4.2062126,"\int \frac{\left(-2+x^6\right) \left(1-x^4+x^6\right)}{x^6 \left(1+x^6\right)^{3/4}} \, dx","IntegrateAlgebraic[((-2 + x^6)*(1 - x^4 + x^6))/(x^6*(1 + x^6)^(3/4)),x]","\frac{2 \sqrt[4]{1+x^6} \left(1-5 x^4+x^6\right)}{5 x^5}","\frac{2 \sqrt[4]{1+x^6} \left(1-5 x^4+x^6\right)}{5 x^5}",1,"(2*(1 + x^6)^(1/4)*(1 - 5*x^4 + x^6))/(5*x^5)","A",1
293,1,26,26,3.6289136,"\int \frac{\sqrt{-1+x^6} \left(1+2 x^6\right)}{x^2 \left(-1+x^2+x^6\right)} \, dx","IntegrateAlgebraic[(Sqrt[-1 + x^6]*(1 + 2*x^6))/(x^2*(-1 + x^2 + x^6)),x]","\frac{\sqrt{-1+x^6}}{x}+\tan ^{-1}\left(\frac{x}{\sqrt{-1+x^6}}\right)","\frac{\sqrt{-1+x^6}}{x}+\tan ^{-1}\left(\frac{x}{\sqrt{-1+x^6}}\right)",1,"Sqrt[-1 + x^6]/x + ArcTan[x/Sqrt[-1 + x^6]]","A",1
294,1,26,26,0.128401,"\int \frac{\left(-1+x^3-x^5-2 x^7\right)^{2/3} \left(1-x^3+x^5+2 x^7\right) \left(-3+2 x^5+8 x^7\right)}{x^9} \, dx","IntegrateAlgebraic[((-1 + x^3 - x^5 - 2*x^7)^(2/3)*(1 - x^3 + x^5 + 2*x^7)*(-3 + 2*x^5 + 8*x^7))/x^9,x]","\frac{3 \left(-1+x^3-x^5-2 x^7\right)^{8/3}}{8 x^8}","\frac{3 \left(-1+x^3-x^5-2 x^7\right)^{8/3}}{8 x^8}",1,"(3*(-1 + x^3 - x^5 - 2*x^7)^(8/3))/(8*x^8)","A",1
295,1,26,26,16.8172188,"\int \frac{x^4 \left(9+4 x^5\right)}{\sqrt{x+x^6} \left(-a-a x^5+x^9\right)} \, dx","IntegrateAlgebraic[(x^4*(9 + 4*x^5))/(Sqrt[x + x^6]*(-a - a*x^5 + x^9)),x]","-\frac{2 \tanh ^{-1}\left(\frac{x^5}{\sqrt{a} \sqrt{x+x^6}}\right)}{\sqrt{a}}","-\frac{2 \tanh ^{-1}\left(\frac{x^5}{\sqrt{a} \sqrt{x+x^6}}\right)}{\sqrt{a}}",1,"(-2*ArcTanh[x^5/(Sqrt[a]*Sqrt[x + x^6])])/Sqrt[a]","A",1
296,1,26,26,4.5638003,"\int \frac{x^4 \left(9+5 x^4\right)}{\sqrt{x+x^5} \left(-1-x^4+a x^9\right)} \, dx","IntegrateAlgebraic[(x^4*(9 + 5*x^4))/(Sqrt[x + x^5]*(-1 - x^4 + a*x^9)),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{x+x^5}}{\sqrt{a} x^5}\right)}{\sqrt{a}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{x+x^5}}{\sqrt{a} x^5}\right)}{\sqrt{a}}",1,"(-2*ArcTanh[Sqrt[x + x^5]/(Sqrt[a]*x^5)])/Sqrt[a]","A",1
297,1,26,26,9.4803947,"\int \frac{x^4 \left(9+4 x^5\right)}{\sqrt{x+x^6} \left(-1-x^5+a x^9\right)} \, dx","IntegrateAlgebraic[(x^4*(9 + 4*x^5))/(Sqrt[x + x^6]*(-1 - x^5 + a*x^9)),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{x+x^6}}{\sqrt{a} x^5}\right)}{\sqrt{a}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{x+x^6}}{\sqrt{a} x^5}\right)}{\sqrt{a}}",1,"(-2*ArcTanh[Sqrt[x + x^6]/(Sqrt[a]*x^5)])/Sqrt[a]","A",1
298,1,22,27,0.0167383,"\int x^5 \sqrt{1-2 x^3} \, dx","IntegrateAlgebraic[x^5*Sqrt[1 - 2*x^3],x]","\frac{1}{45} \left(-1-3 x^3\right) \left(1-2 x^3\right)^{3/2}","\frac{1}{45} \sqrt{1-2 x^3} \left(-1-x^3+6 x^6\right)",1,"((-1 - 3*x^3)*(1 - 2*x^3)^(3/2))/45","A",1
299,1,27,27,0.2145087,"\int \frac{2+x}{(-1+x) \sqrt{-1+3 x+x^3}} \, dx","IntegrateAlgebraic[(2 + x)/((-1 + x)*Sqrt[-1 + 3*x + x^3]),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{3} x}{\sqrt{-1+3 x+x^3}}\right)}{\sqrt{3}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{3} x}{\sqrt{-1+3 x+x^3}}\right)}{\sqrt{3}}",1,"(-2*ArcTanh[(Sqrt[3]*x)/Sqrt[-1 + 3*x + x^3]])/Sqrt[3]","A",1
300,1,27,27,0.0835225,"\int \frac{-1+2 x}{(1+x) \sqrt{-x-x^2+x^3}} \, dx","IntegrateAlgebraic[(-1 + 2*x)/((1 + x)*Sqrt[-x - x^2 + x^3]),x]","2 \tan ^{-1}\left(\frac{\sqrt{-x-x^2+x^3}}{1+2 x}\right)","2 \tan ^{-1}\left(\frac{\sqrt{-x-x^2+x^3}}{1+2 x}\right)",1,"2*ArcTan[Sqrt[-x - x^2 + x^3]/(1 + 2*x)]","A",1
301,1,27,27,0.0252587,"\int \frac{x^5}{\sqrt{b+a x^3}} \, dx","IntegrateAlgebraic[x^5/Sqrt[b + a*x^3],x]","\frac{2 \left(-2 b+a x^3\right) \sqrt{b+a x^3}}{9 a^2}","\frac{2 \left(-2 b+a x^3\right) \sqrt{b+a x^3}}{9 a^2}",1,"(2*(-2*b + a*x^3)*Sqrt[b + a*x^3])/(9*a^2)","A",1
302,1,27,27,0.1208864,"\int \frac{1}{x^4 \sqrt[4]{-x^2+x^4}} \, dx","IntegrateAlgebraic[1/(x^4*(-x^2 + x^4)^(1/4)),x]","\frac{2 \left(3+4 x^2\right) \left(-x^2+x^4\right)^{3/4}}{21 x^5}","\frac{2 \left(3+4 x^2\right) \left(-x^2+x^4\right)^{3/4}}{21 x^5}",1,"(2*(3 + 4*x^2)*(-x^2 + x^4)^(3/4))/(21*x^5)","A",1
303,1,27,27,0.0315284,"\int \frac{1}{x \sqrt{b+a x^4}} \, dx","IntegrateAlgebraic[1/(x*Sqrt[b + a*x^4]),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{b+a x^4}}{\sqrt{b}}\right)}{2 \sqrt{b}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{b+a x^4}}{\sqrt{b}}\right)}{2 \sqrt{b}}",1,"-1/2*ArcTanh[Sqrt[b + a*x^4]/Sqrt[b]]/Sqrt[b]","A",1
304,1,27,27,0.4186227,"\int \frac{1}{x^7 \sqrt[3]{-x^2+x^6}} \, dx","IntegrateAlgebraic[1/(x^7*(-x^2 + x^6)^(1/3)),x]","\frac{3 \left(2+3 x^4\right) \left(-x^2+x^6\right)^{2/3}}{40 x^8}","\frac{3 \left(2+3 x^4\right) \left(-x^2+x^6\right)^{2/3}}{40 x^8}",1,"(3*(2 + 3*x^4)*(-x^2 + x^6)^(2/3))/(40*x^8)","A",1
305,1,28,28,0.0186617,"\int \frac{\sqrt{-1+x^3}}{x} \, dx","IntegrateAlgebraic[Sqrt[-1 + x^3]/x,x]","\frac{2}{3} \sqrt{-1+x^3}-\frac{2}{3} \tan ^{-1}\left(\sqrt{-1+x^3}\right)","\frac{2}{3} \sqrt{-1+x^3}-\frac{2}{3} \tan ^{-1}\left(\sqrt{-1+x^3}\right)",1,"(2*Sqrt[-1 + x^3])/3 - (2*ArcTan[Sqrt[-1 + x^3]])/3","A",1
306,1,23,28,0.0904137,"\int \frac{\left(-1+x^3\right) \sqrt[3]{1+x^3}}{x^8} \, dx","IntegrateAlgebraic[((-1 + x^3)*(1 + x^3)^(1/3))/x^8,x]","\frac{\left(2-5 x^3\right) \left(1+x^3\right)^{4/3}}{14 x^7}","\frac{\sqrt[3]{1+x^3} \left(2-3 x^3-5 x^6\right)}{14 x^7}",1,"((2 - 5*x^3)*(1 + x^3)^(4/3))/(14*x^7)","A",1
307,1,28,28,0.0257184,"\int \frac{\sqrt{1+x^3}}{x} \, dx","IntegrateAlgebraic[Sqrt[1 + x^3]/x,x]","\frac{2 \sqrt{1+x^3}}{3}-\frac{2}{3} \tanh ^{-1}\left(\sqrt{1+x^3}\right)","\frac{2 \sqrt{1+x^3}}{3}-\frac{2}{3} \tanh ^{-1}\left(\sqrt{1+x^3}\right)",1,"(2*Sqrt[1 + x^3])/3 - (2*ArcTanh[Sqrt[1 + x^3]])/3","A",1
308,1,28,28,0.0931387,"\int \frac{\sqrt[3]{-1+x^3} \left(1+x^3\right)}{x^8} \, dx","IntegrateAlgebraic[((-1 + x^3)^(1/3)*(1 + x^3))/x^8,x]","\frac{\sqrt[3]{-1+x^3} \left(-2-3 x^3+5 x^6\right)}{14 x^7}","\frac{\sqrt[3]{-1+x^3} \left(-2-3 x^3+5 x^6\right)}{14 x^7}",1,"((-1 + x^3)^(1/3)*(-2 - 3*x^3 + 5*x^6))/(14*x^7)","A",1
309,1,28,28,0.0919795,"\int \frac{\left(-1+x^3\right)^{2/3} \left(2+x^3\right)}{x^9} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(2 + x^3))/x^9,x]","\frac{\left(-1+x^3\right)^{2/3} \left(-5-2 x^3+7 x^6\right)}{20 x^8}","\frac{\left(-1+x^3\right)^{2/3} \left(-5-2 x^3+7 x^6\right)}{20 x^8}",1,"((-1 + x^3)^(2/3)*(-5 - 2*x^3 + 7*x^6))/(20*x^8)","A",1
310,1,28,28,0.0940539,"\int \frac{\left(1+x^3\right)^{2/3} \left(2+x^3\right)}{x^9} \, dx","IntegrateAlgebraic[((1 + x^3)^(2/3)*(2 + x^3))/x^9,x]","\frac{\left(1+x^3\right)^{2/3} \left(-5-6 x^3-x^6\right)}{20 x^8}","\frac{\left(1+x^3\right)^{2/3} \left(-5-6 x^3-x^6\right)}{20 x^8}",1,"((1 + x^3)^(2/3)*(-5 - 6*x^3 - x^6))/(20*x^8)","A",1
311,1,28,28,0.2112498,"\int \frac{1}{x^6 \sqrt[3]{x+x^3}} \, dx","IntegrateAlgebraic[1/(x^6*(x + x^3)^(1/3)),x]","-\frac{3 \left(x+x^3\right)^{2/3} \left(5-6 x^2+9 x^4\right)}{80 x^6}","-\frac{3 \left(x+x^3\right)^{2/3} \left(5-6 x^2+9 x^4\right)}{80 x^6}",1,"(-3*(x + x^3)^(2/3)*(5 - 6*x^2 + 9*x^4))/(80*x^6)","A",1
312,1,28,28,0.4148597,"\int \frac{-2-x+2 x^2}{(-1+x) x \sqrt[4]{-x^2+x^3}} \, dx","IntegrateAlgebraic[(-2 - x + 2*x^2)/((-1 + x)*x*(-x^2 + x^3)^(1/4)),x]","\frac{4 (-1+2 x) \left(-x^2+x^3\right)^{3/4}}{(-1+x) x^2}","\frac{4 (-1+2 x) \left(-x^2+x^3\right)^{3/4}}{(-1+x) x^2}",1,"(4*(-1 + 2*x)*(-x^2 + x^3)^(3/4))/((-1 + x)*x^2)","A",1
313,1,28,28,0.1053693,"\int \frac{2+x}{(-1+x) \sqrt{-x-x^2+x^3}} \, dx","IntegrateAlgebraic[(2 + x)/((-1 + x)*Sqrt[-x - x^2 + x^3]),x]","-2 \tan ^{-1}\left(\frac{\sqrt{-x-x^2+x^3}}{(-2+x) x}\right)","-2 \tan ^{-1}\left(\frac{\sqrt{-x-x^2+x^3}}{(-2+x) x}\right)",1,"-2*ArcTan[Sqrt[-x - x^2 + x^3]/((-2 + x)*x)]","A",1
314,1,28,28,0.7029508,"\int \frac{\left(3+x^2\right) \left(1+x^2+x^3\right)}{x^6 \sqrt[4]{x+x^3}} \, dx","IntegrateAlgebraic[((3 + x^2)*(1 + x^2 + x^3))/(x^6*(x + x^3)^(1/4)),x]","-\frac{4 \left(x+x^3\right)^{3/4} \left(3+3 x^2+7 x^3\right)}{21 x^6}","-\frac{4 \left(x+x^3\right)^{3/4} \left(3+3 x^2+7 x^3\right)}{21 x^6}",1,"(-4*(x + x^3)^(3/4)*(3 + 3*x^2 + 7*x^3))/(21*x^6)","A",1
315,1,28,28,0.0925712,"\int \frac{\sqrt[3]{-1+x^3} \left(-1+2 x^3\right)}{x^{11}} \, dx","IntegrateAlgebraic[((-1 + x^3)^(1/3)*(-1 + 2*x^3))/x^11,x]","\frac{\left(-1+x^3\right)^{4/3} \left(-2+4 x^3+3 x^6\right)}{20 x^{10}}","\frac{\left(-1+x^3\right)^{4/3} \left(-2+4 x^3+3 x^6\right)}{20 x^{10}}",1,"((-1 + x^3)^(4/3)*(-2 + 4*x^3 + 3*x^6))/(20*x^10)","A",1
316,1,23,28,0.0905763,"\int \frac{\sqrt[3]{-1+x^3} \left(-1+2 x^3\right)}{x^8} \, dx","IntegrateAlgebraic[((-1 + x^3)^(1/3)*(-1 + 2*x^3))/x^8,x]","\frac{\left(-1+x^3\right)^{4/3} \left(-4+11 x^3\right)}{28 x^7}","\frac{\sqrt[3]{-1+x^3} \left(4-15 x^3+11 x^6\right)}{28 x^7}",1,"((-1 + x^3)^(4/3)*(-4 + 11*x^3))/(28*x^7)","A",1
317,1,28,28,0.6803015,"\int \frac{\left(3+x^2\right) \left(-1-x^2+2 x^3\right)}{x^6 \sqrt[4]{x+x^3}} \, dx","IntegrateAlgebraic[((3 + x^2)*(-1 - x^2 + 2*x^3))/(x^6*(x + x^3)^(1/4)),x]","-\frac{4 \left(x+x^3\right)^{3/4} \left(-3-3 x^2+14 x^3\right)}{21 x^6}","-\frac{4 \left(x+x^3\right)^{3/4} \left(-3-3 x^2+14 x^3\right)}{21 x^6}",1,"(-4*(x + x^3)^(3/4)*(-3 - 3*x^2 + 14*x^3))/(21*x^6)","A",1
318,1,28,28,0.5474874,"\int \frac{2 b+a x^3}{\sqrt{-b+a x^3} \left(-b+x^2+a x^3\right)} \, dx","IntegrateAlgebraic[(2*b + a*x^3)/(Sqrt[-b + a*x^3]*(-b + x^2 + a*x^3)),x]","2 \tan ^{-1}\left(\frac{x \sqrt{-b+a x^3}}{b-a x^3}\right)","2 \tan ^{-1}\left(\frac{x \sqrt{-b+a x^3}}{b-a x^3}\right)",1,"2*ArcTan[(x*Sqrt[-b + a*x^3])/(b - a*x^3)]","A",1
319,1,23,28,0.1383042,"\int \frac{\left(-4+x^4\right) \left(-1+x^4\right)^{3/4}}{x^{12}} \, dx","IntegrateAlgebraic[((-4 + x^4)*(-1 + x^4)^(3/4))/x^12,x]","\frac{\left(-28-5 x^4\right) \left(-1+x^4\right)^{7/4}}{77 x^{11}}","\frac{\left(-1+x^4\right)^{3/4} \left(28-23 x^4-5 x^8\right)}{77 x^{11}}",1,"((-28 - 5*x^4)*(-1 + x^4)^(7/4))/(77*x^11)","A",1
320,1,28,28,0.0216019,"\int \frac{\sqrt{1+x^4}}{x} \, dx","IntegrateAlgebraic[Sqrt[1 + x^4]/x,x]","\frac{\sqrt{1+x^4}}{2}-\frac{1}{2} \tanh ^{-1}\left(\sqrt{1+x^4}\right)","\frac{\sqrt{1+x^4}}{2}-\frac{1}{2} \tanh ^{-1}\left(\sqrt{1+x^4}\right)",1,"Sqrt[1 + x^4]/2 - ArcTanh[Sqrt[1 + x^4]]/2","A",1
321,1,25,28,0.2223845,"\int \frac{\sqrt[4]{-x+x^4}}{x^8} \, dx","IntegrateAlgebraic[(-x + x^4)^(1/4)/x^8,x]","\frac{4 \left(5+4 x^3\right) \left(-x+x^4\right)^{5/4}}{135 x^8}","\frac{4 \sqrt[4]{-x+x^4} \left(-5+x^3+4 x^6\right)}{135 x^7}",1,"(4*(5 + 4*x^3)*(-x + x^4)^(5/4))/(135*x^8)","A",1
322,1,23,28,0.2193793,"\int \frac{\sqrt[4]{x+x^4}}{x^8} \, dx","IntegrateAlgebraic[(x + x^4)^(1/4)/x^8,x]","\frac{4 \left(-5+4 x^3\right) \left(x+x^4\right)^{5/4}}{135 x^8}","\frac{4 \sqrt[4]{x+x^4} \left(-5-x^3+4 x^6\right)}{135 x^7}",1,"(4*(-5 + 4*x^3)*(x + x^4)^(5/4))/(135*x^8)","A",1
323,1,28,28,0.9518113,"\int \frac{2+x^2}{\left(-1+x^2\right) \sqrt{-1-x^2+x^4}} \, dx","IntegrateAlgebraic[(2 + x^2)/((-1 + x^2)*Sqrt[-1 - x^2 + x^4]),x]","-\tan ^{-1}\left(\frac{\sqrt{-1-x^2+x^4}}{x \left(-2+x^2\right)}\right)","-\tan ^{-1}\left(\frac{\sqrt{-1-x^2+x^4}}{x \left(-2+x^2\right)}\right)",1,"-ArcTan[Sqrt[-1 - x^2 + x^4]/(x*(-2 + x^2))]","A",1
324,1,28,28,3.903042,"\int \frac{\left(4+x^3\right) \left(-1-x^3+x^4\right)}{x^6 \left(1+x^3\right)^{3/4}} \, dx","IntegrateAlgebraic[((4 + x^3)*(-1 - x^3 + x^4))/(x^6*(1 + x^3)^(3/4)),x]","-\frac{4 \sqrt[4]{1+x^3} \left(-1-x^3+5 x^4\right)}{5 x^5}","-\frac{4 \sqrt[4]{1+x^3} \left(-1-x^3+5 x^4\right)}{5 x^5}",1,"(-4*(1 + x^3)^(1/4)*(-1 - x^3 + 5*x^4))/(5*x^5)","A",1
325,1,28,28,0.5890495,"\int \frac{\left(4+x^3\right) \left(-1-x^3+x^4\right)}{x^8 \sqrt[4]{1+x^3}} \, dx","IntegrateAlgebraic[((4 + x^3)*(-1 - x^3 + x^4))/(x^8*(1 + x^3)^(1/4)),x]","-\frac{4 \left(1+x^3\right)^{3/4} \left(-3-3 x^3+7 x^4\right)}{21 x^7}","-\frac{4 \left(1+x^3\right)^{3/4} \left(-3-3 x^3+7 x^4\right)}{21 x^7}",1,"(-4*(1 + x^3)^(3/4)*(-3 - 3*x^3 + 7*x^4))/(21*x^7)","A",1
326,1,28,28,0.5857526,"\int \frac{\left(-4+x^3\right) \left(1-x^3+x^4\right)}{x^8 \sqrt[4]{-1+x^3}} \, dx","IntegrateAlgebraic[((-4 + x^3)*(1 - x^3 + x^4))/(x^8*(-1 + x^3)^(1/4)),x]","-\frac{4 \left(-1+x^3\right)^{3/4} \left(3-3 x^3+7 x^4\right)}{21 x^7}","-\frac{4 \left(-1+x^3\right)^{3/4} \left(3-3 x^3+7 x^4\right)}{21 x^7}",1,"(-4*(-1 + x^3)^(3/4)*(3 - 3*x^3 + 7*x^4))/(21*x^7)","A",1
327,1,28,28,0.1020757,"\int \frac{x}{\sqrt{x^3+x^4}} \, dx","IntegrateAlgebraic[x/Sqrt[x^3 + x^4],x]","\log (x)-\log \left(-x-2 x^2+2 \sqrt{x^3+x^4}\right)","\log (x)-\log \left(-x-2 x^2+2 \sqrt{x^3+x^4}\right)",1,"Log[x] - Log[-x - 2*x^2 + 2*Sqrt[x^3 + x^4]]","A",1
328,1,28,28,0.1981469,"\int \frac{1}{x (1+x) \sqrt[4]{x^3+x^4}} \, dx","IntegrateAlgebraic[1/(x*(1 + x)*(x^3 + x^4)^(1/4)),x]","-\frac{4 (1+4 x) \left(x^3+x^4\right)^{3/4}}{3 x^3 (1+x)}","-\frac{4 (1+4 x) \left(x^3+x^4\right)^{3/4}}{3 x^3 (1+x)}",1,"(-4*(1 + 4*x)*(x^3 + x^4)^(3/4))/(3*x^3*(1 + x))","A",1
329,1,28,28,3.7333023,"\int \frac{\left(4+x^3\right) \left(-1-x^3+2 x^4\right)}{x^6 \left(1+x^3\right)^{3/4}} \, dx","IntegrateAlgebraic[((4 + x^3)*(-1 - x^3 + 2*x^4))/(x^6*(1 + x^3)^(3/4)),x]","-\frac{4 \sqrt[4]{1+x^3} \left(-1-x^3+10 x^4\right)}{5 x^5}","-\frac{4 \sqrt[4]{1+x^3} \left(-1-x^3+10 x^4\right)}{5 x^5}",1,"(-4*(1 + x^3)^(1/4)*(-1 - x^3 + 10*x^4))/(5*x^5)","A",1
330,1,28,28,2.5938266,"\int \frac{\left(-3+x^4\right) \left(1-x^3+x^4\right)}{x^6 \sqrt[4]{x+x^5}} \, dx","IntegrateAlgebraic[((-3 + x^4)*(1 - x^3 + x^4))/(x^6*(x + x^5)^(1/4)),x]","\frac{4 \left(3-7 x^3+3 x^4\right) \left(x+x^5\right)^{3/4}}{21 x^6}","\frac{4 \left(3-7 x^3+3 x^4\right) \left(x+x^5\right)^{3/4}}{21 x^6}",1,"(4*(3 - 7*x^3 + 3*x^4)*(x + x^5)^(3/4))/(21*x^6)","A",1
331,1,28,28,0.0210726,"\int \frac{\sqrt{-1+x^6}}{x} \, dx","IntegrateAlgebraic[Sqrt[-1 + x^6]/x,x]","\frac{1}{3} \sqrt{-1+x^6}-\frac{1}{3} \tan ^{-1}\left(\sqrt{-1+x^6}\right)","\frac{1}{3} \sqrt{-1+x^6}-\frac{1}{3} \tan ^{-1}\left(\sqrt{-1+x^6}\right)",1,"Sqrt[-1 + x^6]/3 - ArcTan[Sqrt[-1 + x^6]]/3","A",1
332,1,28,28,0.0355119,"\int \frac{-1+x^6}{x \sqrt{1+x^6}} \, dx","IntegrateAlgebraic[(-1 + x^6)/(x*Sqrt[1 + x^6]),x]","\frac{\sqrt{1+x^6}}{3}+\frac{1}{3} \tanh ^{-1}\left(\sqrt{1+x^6}\right)","\frac{\sqrt{1+x^6}}{3}+\frac{1}{3} \tanh ^{-1}\left(\sqrt{1+x^6}\right)",1,"Sqrt[1 + x^6]/3 + ArcTanh[Sqrt[1 + x^6]]/3","A",1
333,1,28,28,0.0192071,"\int \frac{\sqrt{1+x^6}}{x} \, dx","IntegrateAlgebraic[Sqrt[1 + x^6]/x,x]","\frac{\sqrt{1+x^6}}{3}-\frac{1}{3} \tanh ^{-1}\left(\sqrt{1+x^6}\right)","\frac{\sqrt{1+x^6}}{3}-\frac{1}{3} \tanh ^{-1}\left(\sqrt{1+x^6}\right)",1,"Sqrt[1 + x^6]/3 - ArcTanh[Sqrt[1 + x^6]]/3","A",1
334,1,23,28,0.5276187,"\int \frac{\sqrt[3]{-1+x^6} \left(1+x^6\right)}{x^{15}} \, dx","IntegrateAlgebraic[((-1 + x^6)^(1/3)*(1 + x^6))/x^15,x]","\frac{\left(-1+x^6\right)^{4/3} \left(2+5 x^6\right)}{28 x^{14}}","\frac{\sqrt[3]{-1+x^6} \left(-2-3 x^6+5 x^{12}\right)}{28 x^{14}}",1,"((-1 + x^6)^(4/3)*(2 + 5*x^6))/(28*x^14)","A",1
335,1,28,28,2.6128021,"\int \frac{\left(1-x^3+x^5\right) \left(-3+2 x^5\right)}{x^6 \sqrt[4]{x+x^6}} \, dx","IntegrateAlgebraic[((1 - x^3 + x^5)*(-3 + 2*x^5))/(x^6*(x + x^6)^(1/4)),x]","\frac{4 \left(3-7 x^3+3 x^5\right) \left(x+x^6\right)^{3/4}}{21 x^6}","\frac{4 \left(3-7 x^3+3 x^5\right) \left(x+x^6\right)^{3/4}}{21 x^6}",1,"(4*(3 - 7*x^3 + 3*x^5)*(x + x^6)^(3/4))/(21*x^6)","A",1
336,1,28,28,2.092788,"\int \frac{\left(-2+x^6\right) \left(1-x^4+x^6\right)}{x^8 \sqrt[4]{1+x^6}} \, dx","IntegrateAlgebraic[((-2 + x^6)*(1 - x^4 + x^6))/(x^8*(1 + x^6)^(1/4)),x]","\frac{2 \left(1+x^6\right)^{3/4} \left(3-7 x^4+3 x^6\right)}{21 x^7}","\frac{2 \left(1+x^6\right)^{3/4} \left(3-7 x^4+3 x^6\right)}{21 x^7}",1,"(2*(1 + x^6)^(3/4)*(3 - 7*x^4 + 3*x^6))/(21*x^7)","A",1
337,1,23,28,1.2602034,"\int \frac{\sqrt[3]{-1+x^6} \left(-1+2 x^6\right)}{x^{15}} \, dx","IntegrateAlgebraic[((-1 + x^6)^(1/3)*(-1 + 2*x^6))/x^15,x]","\frac{\left(-1+x^6\right)^{4/3} \left(-4+11 x^6\right)}{56 x^{14}}","\frac{\sqrt[3]{-1+x^6} \left(4-15 x^6+11 x^{12}\right)}{56 x^{14}}",1,"((-1 + x^6)^(4/3)*(-4 + 11*x^6))/(56*x^14)","A",1
338,1,28,28,0.0692393,"\int \frac{x^2 \left(-2+x^3\right) \sqrt{1+x^3}}{1+3 x^3+x^9} \, dx","IntegrateAlgebraic[(x^2*(-2 + x^3)*Sqrt[1 + x^3])/(1 + 3*x^3 + x^9),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{3} x^3}{\left(1+x^3\right)^{3/2}}\right)}{3 \sqrt{3}}","-\frac{2 \tanh ^{-1}\left(\frac{\left(1+x^3\right)^{3/2}}{\sqrt{3} x^3}\right)}{3 \sqrt{3}}",1,"(-2*ArcTanh[(Sqrt[3]*x^3)/(1 + x^3)^(3/2)])/(3*Sqrt[3])","A",1
339,1,28,28,17.568861,"\int \frac{x^4 \left(-9+4 x^5\right)}{\sqrt{-x+x^6} \left(a-a x^5+x^9\right)} \, dx","IntegrateAlgebraic[(x^4*(-9 + 4*x^5))/(Sqrt[-x + x^6]*(a - a*x^5 + x^9)),x]","-\frac{2 \tanh ^{-1}\left(\frac{x^5}{\sqrt{a} \sqrt{-x+x^6}}\right)}{\sqrt{a}}","-\frac{2 \tanh ^{-1}\left(\frac{x^5}{\sqrt{a} \sqrt{-x+x^6}}\right)}{\sqrt{a}}",1,"(-2*ArcTanh[x^5/(Sqrt[a]*Sqrt[-x + x^6])])/Sqrt[a]","A",1
340,1,28,28,4.5417108,"\int \frac{x^4 \left(-9+5 x^4\right)}{\sqrt{-x+x^5} \left(1-x^4+a x^9\right)} \, dx","IntegrateAlgebraic[(x^4*(-9 + 5*x^4))/(Sqrt[-x + x^5]*(1 - x^4 + a*x^9)),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{-x+x^5}}{\sqrt{a} x^5}\right)}{\sqrt{a}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{-x+x^5}}{\sqrt{a} x^5}\right)}{\sqrt{a}}",1,"(-2*ArcTanh[Sqrt[-x + x^5]/(Sqrt[a]*x^5)])/Sqrt[a]","A",1
341,1,28,28,9.5013087,"\int \frac{x^4 \left(-9+4 x^5\right)}{\sqrt{-x+x^6} \left(1-x^5+a x^9\right)} \, dx","IntegrateAlgebraic[(x^4*(-9 + 4*x^5))/(Sqrt[-x + x^6]*(1 - x^5 + a*x^9)),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{-x+x^6}}{\sqrt{a} x^5}\right)}{\sqrt{a}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{-x+x^6}}{\sqrt{a} x^5}\right)}{\sqrt{a}}",1,"(-2*ArcTanh[Sqrt[-x + x^6]/(Sqrt[a]*x^5)])/Sqrt[a]","A",1
342,1,28,28,0.2776636,"\int \frac{1+x^{12}}{x^{16} \sqrt{-1+x^6}} \, dx","IntegrateAlgebraic[(1 + x^12)/(x^16*Sqrt[-1 + x^6]),x]","\frac{\sqrt{-1+x^6} \left(3+4 x^6+23 x^{12}\right)}{45 x^{15}}","\frac{\sqrt{-1+x^6} \left(3+4 x^6+23 x^{12}\right)}{45 x^{15}}",1,"(Sqrt[-1 + x^6]*(3 + 4*x^6 + 23*x^12))/(45*x^15)","A",1
343,1,29,29,0.0544834,"\int \frac{-1+x}{(-3+x) (1+x) \sqrt[4]{-2-2 x+x^2}} \, dx","IntegrateAlgebraic[(-1 + x)/((-3 + x)*(1 + x)*(-2 - 2*x + x^2)^(1/4)),x]","\tan ^{-1}\left(\sqrt[4]{-2-2 x+x^2}\right)-\tanh ^{-1}\left(\sqrt[4]{-2-2 x+x^2}\right)","\tan ^{-1}\left(\sqrt[4]{-2-2 x+x^2}\right)-\tanh ^{-1}\left(\sqrt[4]{-2-2 x+x^2}\right)",1,"ArcTan[(-2 - 2*x + x^2)^(1/4)] - ArcTanh[(-2 - 2*x + x^2)^(1/4)]","A",1
344,1,29,29,0.050096,"\int \frac{1}{(-1+x) \sqrt[4]{2-2 x+x^2}} \, dx","IntegrateAlgebraic[1/((-1 + x)*(2 - 2*x + x^2)^(1/4)),x]","\tan ^{-1}\left(\sqrt[4]{2-2 x+x^2}\right)-\tanh ^{-1}\left(\sqrt[4]{2-2 x+x^2}\right)","\tan ^{-1}\left(\sqrt[4]{2-2 x+x^2}\right)-\tanh ^{-1}\left(\sqrt[4]{2-2 x+x^2}\right)",1,"ArcTan[(2 - 2*x + x^2)^(1/4)] - ArcTanh[(2 - 2*x + x^2)^(1/4)]","A",1
345,1,29,29,0.049745,"\int \frac{1}{(1+x) \sqrt[4]{2+2 x+x^2}} \, dx","IntegrateAlgebraic[1/((1 + x)*(2 + 2*x + x^2)^(1/4)),x]","\tan ^{-1}\left(\sqrt[4]{2+2 x+x^2}\right)-\tanh ^{-1}\left(\sqrt[4]{2+2 x+x^2}\right)","\tan ^{-1}\left(\sqrt[4]{2+2 x+x^2}\right)-\tanh ^{-1}\left(\sqrt[4]{2+2 x+x^2}\right)",1,"ArcTan[(2 + 2*x + x^2)^(1/4)] - ArcTanh[(2 + 2*x + x^2)^(1/4)]","A",1
346,1,29,29,0.0220058,"\int \frac{1}{x \sqrt[4]{1+x^3}} \, dx","IntegrateAlgebraic[1/(x*(1 + x^3)^(1/4)),x]","\frac{2}{3} \tan ^{-1}\left(\sqrt[4]{1+x^3}\right)-\frac{2}{3} \tanh ^{-1}\left(\sqrt[4]{1+x^3}\right)","\frac{2}{3} \tan ^{-1}\left(\sqrt[4]{1+x^3}\right)-\frac{2}{3} \tanh ^{-1}\left(\sqrt[4]{1+x^3}\right)",1,"(2*ArcTan[(1 + x^3)^(1/4)])/3 - (2*ArcTanh[(1 + x^3)^(1/4)])/3","A",1
347,1,29,29,0.025995,"\int \frac{1}{x \sqrt{-b+a x^3}} \, dx","IntegrateAlgebraic[1/(x*Sqrt[-b + a*x^3]),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt{-b+a x^3}}{\sqrt{b}}\right)}{3 \sqrt{b}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{-b+a x^3}}{\sqrt{b}}\right)}{3 \sqrt{b}}",1,"(2*ArcTan[Sqrt[-b + a*x^3]/Sqrt[b]])/(3*Sqrt[b])","A",1
348,1,29,29,0.0268252,"\int \frac{x^5}{\sqrt{-b+a x^3}} \, dx","IntegrateAlgebraic[x^5/Sqrt[-b + a*x^3],x]","\frac{2 \sqrt{-b+a x^3} \left(2 b+a x^3\right)}{9 a^2}","\frac{2 \sqrt{-b+a x^3} \left(2 b+a x^3\right)}{9 a^2}",1,"(2*Sqrt[-b + a*x^3]*(2*b + a*x^3))/(9*a^2)","A",1
349,1,29,29,0.3745434,"\int \frac{1+x}{(-1+x) \sqrt{1-x^2+x^4}} \, dx","IntegrateAlgebraic[(1 + x)/((-1 + x)*Sqrt[1 - x^2 + x^4]),x]","-2 \tanh ^{-1}\left(\frac{x}{1-2 x+x^2+\sqrt{1-x^2+x^4}}\right)","-2 \tanh ^{-1}\left(\frac{x}{1-2 x+x^2+\sqrt{1-x^2+x^4}}\right)",1,"-2*ArcTanh[x/(1 - 2*x + x^2 + Sqrt[1 - x^2 + x^4])]","A",1
350,1,29,29,0.3704693,"\int \frac{\left(-1+x^4\right) \left(1+x^4\right)}{\left(1+x^2+x^4\right)^{5/2}} \, dx","IntegrateAlgebraic[((-1 + x^4)*(1 + x^4))/(1 + x^2 + x^4)^(5/2),x]","-\frac{x \left(3+2 x^2+3 x^4\right)}{3 \left(1+x^2+x^4\right)^{3/2}}","-\frac{x \left(3+2 x^2+3 x^4\right)}{3 \left(1+x^2+x^4\right)^{3/2}}",1,"-1/3*(x*(3 + 2*x^2 + 3*x^4))/(1 + x^2 + x^4)^(3/2)","A",1
351,1,29,29,2.4668392,"\int \frac{x^2 \left(-4+x^3\right)}{\left(-1+x^3\right)^{3/4} \left(1-x^3+x^4\right)} \, dx","IntegrateAlgebraic[(x^2*(-4 + x^3))/((-1 + x^3)^(3/4)*(1 - x^3 + x^4)),x]","2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^3}}\right)-2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^3}}\right)","2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^3}}\right)-2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^3}}\right)",1,"2*ArcTan[x/(-1 + x^3)^(1/4)] - 2*ArcTanh[x/(-1 + x^3)^(1/4)]","A",1
352,1,29,29,0.1475156,"\int \frac{1+2 x}{\sqrt{-4-4 x-3 x^2+2 x^3+x^4}} \, dx","IntegrateAlgebraic[(1 + 2*x)/Sqrt[-4 - 4*x - 3*x^2 + 2*x^3 + x^4],x]","\log \left(-2+x+x^2+\sqrt{-4-4 x-3 x^2+2 x^3+x^4}\right)","\log \left(-2+x+x^2+\sqrt{-4-4 x-3 x^2+2 x^3+x^4}\right)",1,"Log[-2 + x + x^2 + Sqrt[-4 - 4*x - 3*x^2 + 2*x^3 + x^4]]","A",1
353,1,29,29,0.0263882,"\int \frac{1}{x \sqrt{-b+a x^4}} \, dx","IntegrateAlgebraic[1/(x*Sqrt[-b + a*x^4]),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{-b+a x^4}}{\sqrt{b}}\right)}{2 \sqrt{b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{-b+a x^4}}{\sqrt{b}}\right)}{2 \sqrt{b}}",1,"ArcTan[Sqrt[-b + a*x^4]/Sqrt[b]]/(2*Sqrt[b])","A",1
354,1,29,29,1.0654965,"\int \frac{-1+2 x^6}{\left(1+x^6\right) \sqrt{1-2 x^2+x^6}} \, dx","IntegrateAlgebraic[(-1 + 2*x^6)/((1 + x^6)*Sqrt[1 - 2*x^2 + x^6]),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1-2 x^2+x^6}}\right)}{\sqrt{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1-2 x^2+x^6}}\right)}{\sqrt{2}}",1,"-(ArcTan[(Sqrt[2]*x)/Sqrt[1 - 2*x^2 + x^6]]/Sqrt[2])","A",1
355,1,29,29,1.0590631,"\int \frac{1+2 x^6}{\left(-1+x^6\right) \sqrt{-1-2 x^2+x^6}} \, dx","IntegrateAlgebraic[(1 + 2*x^6)/((-1 + x^6)*Sqrt[-1 - 2*x^2 + x^6]),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{-1-2 x^2+x^6}}\right)}{\sqrt{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{-1-2 x^2+x^6}}\right)}{\sqrt{2}}",1,"-(ArcTan[(Sqrt[2]*x)/Sqrt[-1 - 2*x^2 + x^6]]/Sqrt[2])","A",1
356,1,25,30,0.0165993,"\int x^8 \sqrt[3]{-1+x^3} \, dx","IntegrateAlgebraic[x^8*(-1 + x^3)^(1/3),x]","\frac{1}{140} \left(-1+x^3\right)^{4/3} \left(9+12 x^3+14 x^6\right)","\frac{1}{140} \sqrt[3]{-1+x^3} \left(-9-3 x^3-2 x^6+14 x^9\right)",1,"((-1 + x^3)^(4/3)*(9 + 12*x^3 + 14*x^6))/140","A",1
357,1,30,30,0.0168684,"\int x^8 \sqrt[4]{1+x^3} \, dx","IntegrateAlgebraic[x^8*(1 + x^3)^(1/4),x]","\frac{4 \sqrt[4]{1+x^3} \left(32-8 x^3+5 x^6+45 x^9\right)}{1755}","\frac{4 \sqrt[4]{1+x^3} \left(32-8 x^3+5 x^6+45 x^9\right)}{1755}",1,"(4*(1 + x^3)^(1/4)*(32 - 8*x^3 + 5*x^6 + 45*x^9))/1755","A",1
358,1,30,30,0.2153359,"\int \frac{1}{x^6 \sqrt[3]{-x+x^3}} \, dx","IntegrateAlgebraic[1/(x^6*(-x + x^3)^(1/3)),x]","\frac{3 \left(-x+x^3\right)^{2/3} \left(5+6 x^2+9 x^4\right)}{80 x^6}","\frac{3 \left(-x+x^3\right)^{2/3} \left(5+6 x^2+9 x^4\right)}{80 x^6}",1,"(3*(-x + x^3)^(2/3)*(5 + 6*x^2 + 9*x^4))/(80*x^6)","A",1
359,1,30,30,0.1711723,"\int \frac{1}{x^3 \sqrt[3]{-x^2+x^3}} \, dx","IntegrateAlgebraic[1/(x^3*(-x^2 + x^3)^(1/3)),x]","\frac{3 \left(5+6 x+9 x^2\right) \left(-x^2+x^3\right)^{2/3}}{40 x^4}","\frac{3 \left(5+6 x+9 x^2\right) \left(-x^2+x^3\right)^{2/3}}{40 x^4}",1,"(3*(5 + 6*x + 9*x^2)*(-x^2 + x^3)^(2/3))/(40*x^4)","A",1
360,1,30,30,0.7041242,"\int \frac{\left(-3+x^2\right) \left(1-x^2+x^3\right)}{x^6 \sqrt[4]{-x+x^3}} \, dx","IntegrateAlgebraic[((-3 + x^2)*(1 - x^2 + x^3))/(x^6*(-x + x^3)^(1/4)),x]","-\frac{4 \left(-x+x^3\right)^{3/4} \left(3-3 x^2+7 x^3\right)}{21 x^6}","-\frac{4 \left(-x+x^3\right)^{3/4} \left(3-3 x^2+7 x^3\right)}{21 x^6}",1,"(-4*(-x + x^3)^(3/4)*(3 - 3*x^2 + 7*x^3))/(21*x^6)","A",1
361,1,30,30,0.0726854,"\int \frac{1+x^2}{\left(-1+x^2\right) \sqrt{-x-x^2+x^3}} \, dx","IntegrateAlgebraic[(1 + x^2)/((-1 + x^2)*Sqrt[-x - x^2 + x^3]),x]","-2 \tan ^{-1}\left(\frac{\sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)","-2 \tan ^{-1}\left(\frac{\sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)",1,"-2*ArcTan[Sqrt[-x - x^2 + x^3]/(-1 - x + x^2)]","A",1
362,1,30,30,0.2481981,"\int \frac{-1+x^3}{x^3 \left(1+x^3\right) \sqrt[4]{x+x^4}} \, dx","IntegrateAlgebraic[(-1 + x^3)/(x^3*(1 + x^3)*(x + x^4)^(1/4)),x]","\frac{4 \left(1+7 x^3\right) \left(x+x^4\right)^{3/4}}{9 x^3 \left(1+x^3\right)}","\frac{4 \left(1+7 x^3\right) \left(x+x^4\right)^{3/4}}{9 x^3 \left(1+x^3\right)}",1,"(4*(1 + 7*x^3)*(x + x^4)^(3/4))/(9*x^3*(1 + x^3))","A",1
363,1,30,30,0.1114767,"\int \frac{\sqrt[4]{-x^2+x^4}}{x^6} \, dx","IntegrateAlgebraic[(-x^2 + x^4)^(1/4)/x^6,x]","\frac{2 \sqrt[4]{-x^2+x^4} \left(-5+x^2+4 x^4\right)}{45 x^5}","\frac{2 \sqrt[4]{-x^2+x^4} \left(-5+x^2+4 x^4\right)}{45 x^5}",1,"(2*(-x^2 + x^4)^(1/4)*(-5 + x^2 + 4*x^4))/(45*x^5)","A",1
364,1,30,30,0.3705633,"\int \frac{-1+2 x}{\sqrt{1+x-2 x^3+x^4}} \, dx","IntegrateAlgebraic[(-1 + 2*x)/Sqrt[1 + x - 2*x^3 + x^4],x]","-\log \left(1+2 x-2 x^2+2 \sqrt{1+x-2 x^3+x^4}\right)","-\log \left(1+2 x-2 x^2+2 \sqrt{1+x-2 x^3+x^4}\right)",1,"-Log[1 + 2*x - 2*x^2 + 2*Sqrt[1 + x - 2*x^3 + x^4]]","A",1
365,1,30,30,0.2156432,"\int \frac{1+x}{(-1+x) x \sqrt[4]{-x^3+x^4}} \, dx","IntegrateAlgebraic[(1 + x)/((-1 + x)*x*(-x^3 + x^4)^(1/4)),x]","-\frac{4 (-1+7 x) \left(-x^3+x^4\right)^{3/4}}{3 (-1+x) x^3}","-\frac{4 (-1+7 x) \left(-x^3+x^4\right)^{3/4}}{3 (-1+x) x^3}",1,"(-4*(-1 + 7*x)*(-x^3 + x^4)^(3/4))/(3*(-1 + x)*x^3)","A",1
366,1,30,30,2.5923093,"\int \frac{\left(3+x^4\right) \left(-1-x^3+x^4\right)}{x^6 \sqrt[4]{-x+x^5}} \, dx","IntegrateAlgebraic[((3 + x^4)*(-1 - x^3 + x^4))/(x^6*(-x + x^5)^(1/4)),x]","\frac{4 \left(-3-7 x^3+3 x^4\right) \left(-x+x^5\right)^{3/4}}{21 x^6}","\frac{4 \left(-3-7 x^3+3 x^4\right) \left(-x+x^5\right)^{3/4}}{21 x^6}",1,"(4*(-3 - 7*x^3 + 3*x^4)*(-x + x^5)^(3/4))/(21*x^6)","A",1
367,1,30,30,2.5878453,"\int \frac{\left(3+x^4\right) \left(-1+x^3+x^4\right)}{x^6 \sqrt[4]{-x+x^5}} \, dx","IntegrateAlgebraic[((3 + x^4)*(-1 + x^3 + x^4))/(x^6*(-x + x^5)^(1/4)),x]","\frac{4 \left(-3+7 x^3+3 x^4\right) \left(-x+x^5\right)^{3/4}}{21 x^6}","\frac{4 \left(-3+7 x^3+3 x^4\right) \left(-x+x^5\right)^{3/4}}{21 x^6}",1,"(4*(-3 + 7*x^3 + 3*x^4)*(-x + x^5)^(3/4))/(21*x^6)","A",1
368,1,30,30,0.5121047,"\int \frac{-1+3 x^4}{\left(1-a x+x^4\right) \sqrt{x+x^5}} \, dx","IntegrateAlgebraic[(-1 + 3*x^4)/((1 - a*x + x^4)*Sqrt[x + x^5]),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{x+x^5}}{1+x^4}\right)}{\sqrt{a}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{x+x^5}}{1+x^4}\right)}{\sqrt{a}}",1,"(-2*ArcTanh[(Sqrt[a]*Sqrt[x + x^5])/(1 + x^4)])/Sqrt[a]","A",1
369,1,30,30,1.0387097,"\int \frac{-1+3 x^4}{\left(a-x+a x^4\right) \sqrt{x+x^5}} \, dx","IntegrateAlgebraic[(-1 + 3*x^4)/((a - x + a*x^4)*Sqrt[x + x^5]),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{x+x^5}}{\sqrt{a} \left(1+x^4\right)}\right)}{\sqrt{a}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{x+x^5}}{\sqrt{a} \left(1+x^4\right)}\right)}{\sqrt{a}}",1,"(-2*ArcTanh[Sqrt[x + x^5]/(Sqrt[a]*(1 + x^4))])/Sqrt[a]","A",1
370,1,30,30,7.0600215,"\int \frac{\sqrt{-1+x^6} \left(2+x^6\right)}{x^3 \left(-1-x^4+x^6\right)} \, dx","IntegrateAlgebraic[(Sqrt[-1 + x^6]*(2 + x^6))/(x^3*(-1 - x^4 + x^6)),x]","\frac{\sqrt{-1+x^6}}{x^2}-\tanh ^{-1}\left(\frac{x^2}{\sqrt{-1+x^6}}\right)","\frac{\sqrt{-1+x^6}}{x^2}-\tanh ^{-1}\left(\frac{x^2}{\sqrt{-1+x^6}}\right)",1,"Sqrt[-1 + x^6]/x^2 - ArcTanh[x^2/Sqrt[-1 + x^6]]","A",1
371,1,30,30,3.1827352,"\int \frac{\sqrt{1-x^6} \left(1+2 x^6\right)}{x^2 \left(-1-x^2+x^6\right)} \, dx","IntegrateAlgebraic[(Sqrt[1 - x^6]*(1 + 2*x^6))/(x^2*(-1 - x^2 + x^6)),x]","\frac{\sqrt{1-x^6}}{x}+\tan ^{-1}\left(\frac{x}{\sqrt{1-x^6}}\right)","\frac{\sqrt{1-x^6}}{x}+\tan ^{-1}\left(\frac{x}{\sqrt{1-x^6}}\right)",1,"Sqrt[1 - x^6]/x + ArcTan[x/Sqrt[1 - x^6]]","A",1
372,1,31,31,0.0233418,"\int \frac{1}{x^4 \sqrt{-1+x^3}} \, dx","IntegrateAlgebraic[1/(x^4*Sqrt[-1 + x^3]),x]","\frac{\sqrt{-1+x^3}}{3 x^3}+\frac{1}{3} \tan ^{-1}\left(\sqrt{-1+x^3}\right)","\frac{\sqrt{-1+x^3}}{3 x^3}+\frac{1}{3} \tan ^{-1}\left(\sqrt{-1+x^3}\right)",1,"Sqrt[-1 + x^3]/(3*x^3) + ArcTan[Sqrt[-1 + x^3]]/3","A",1
373,1,31,31,1.0541314,"\int \frac{-2-2 x+x^2}{\left(3-x+x^2\right) \sqrt{-1+x^3}} \, dx","IntegrateAlgebraic[(-2 - 2*x + x^2)/((3 - x + x^2)*Sqrt[-1 + x^3]),x]","-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{-1+x^3}}{1+x+x^2}\right)","-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{-1+x^3}}{1+x+x^2}\right)",1,"-(Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[-1 + x^3])/(1 + x + x^2)])","A",1
374,1,31,31,1.0919137,"\int \frac{-2-2 x+x^2}{\left(-1+3 x+x^2\right) \sqrt{-1+x^3}} \, dx","IntegrateAlgebraic[(-2 - 2*x + x^2)/((-1 + 3*x + x^2)*Sqrt[-1 + x^3]),x]","-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{-1+x^3}}{1+x+x^2}\right)","-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{-1+x^3}}{1+x+x^2}\right)",1,"-(Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[-1 + x^3])/(1 + x + x^2)])","A",1
375,1,31,31,0.0243007,"\int \frac{\sqrt{-1+x^3}}{x^4} \, dx","IntegrateAlgebraic[Sqrt[-1 + x^3]/x^4,x]","-\frac{\sqrt{-1+x^3}}{3 x^3}+\frac{1}{3} \tan ^{-1}\left(\sqrt{-1+x^3}\right)","-\frac{\sqrt{-1+x^3}}{3 x^3}+\frac{1}{3} \tan ^{-1}\left(\sqrt{-1+x^3}\right)",1,"-1/3*Sqrt[-1 + x^3]/x^3 + ArcTan[Sqrt[-1 + x^3]]/3","A",1
376,1,31,31,0.0381028,"\int \frac{1}{x^4 \sqrt{1+x^3}} \, dx","IntegrateAlgebraic[1/(x^4*Sqrt[1 + x^3]),x]","-\frac{\sqrt{1+x^3}}{3 x^3}+\frac{1}{3} \tanh ^{-1}\left(\sqrt{1+x^3}\right)","-\frac{\sqrt{1+x^3}}{3 x^3}+\frac{1}{3} \tanh ^{-1}\left(\sqrt{1+x^3}\right)",1,"-1/3*Sqrt[1 + x^3]/x^3 + ArcTanh[Sqrt[1 + x^3]]/3","A",1
377,1,31,31,0.8258152,"\int \frac{1+x}{(-2+x) \sqrt{1+x^3}} \, dx","IntegrateAlgebraic[(1 + x)/((-2 + x)*Sqrt[1 + x^3]),x]","-\frac{2}{3} \tanh ^{-1}\left(\frac{\frac{1}{3}+\frac{2 x}{3}+\frac{x^2}{3}}{\sqrt{1+x^3}}\right)","-\frac{2}{3} \tanh ^{-1}\left(\frac{\frac{1}{3}+\frac{2 x}{3}+\frac{x^2}{3}}{\sqrt{1+x^3}}\right)",1,"(-2*ArcTanh[(1/3 + (2*x)/3 + x^2/3)/Sqrt[1 + x^3]])/3","A",1
378,1,31,31,0.0277139,"\int \frac{\sqrt{1+x^3}}{x^4} \, dx","IntegrateAlgebraic[Sqrt[1 + x^3]/x^4,x]","-\frac{\sqrt{1+x^3}}{3 x^3}-\frac{1}{3} \tanh ^{-1}\left(\sqrt{1+x^3}\right)","-\frac{\sqrt{1+x^3}}{3 x^3}-\frac{1}{3} \tanh ^{-1}\left(\sqrt{1+x^3}\right)",1,"-1/3*Sqrt[1 + x^3]/x^3 - ArcTanh[Sqrt[1 + x^3]]/3","A",1
379,1,31,31,0.9571005,"\int \frac{1+x}{(-1+2 x) \sqrt{x+x^4}} \, dx","IntegrateAlgebraic[(1 + x)/((-1 + 2*x)*Sqrt[x + x^4]),x]","-\frac{2}{3} \tanh ^{-1}\left(\frac{\frac{1}{3}+\frac{2 x}{3}+\frac{x^2}{3}}{\sqrt{x+x^4}}\right)","-\frac{2}{3} \tanh ^{-1}\left(\frac{\frac{1}{3}+\frac{2 x}{3}+\frac{x^2}{3}}{\sqrt{x+x^4}}\right)",1,"(-2*ArcTanh[(1/3 + (2*x)/3 + x^2/3)/Sqrt[x + x^4]])/3","A",1
380,1,31,31,0.0636694,"\int \frac{-1+x}{\sqrt{-5+4 x^2-4 x^3+x^4}} \, dx","IntegrateAlgebraic[(-1 + x)/Sqrt[-5 + 4*x^2 - 4*x^3 + x^4],x]","\frac{1}{2} \log \left(-2 x+x^2+\sqrt{-5+4 x^2-4 x^3+x^4}\right)","\frac{1}{2} \log \left(-2 x+x^2+\sqrt{-5+4 x^2-4 x^3+x^4}\right)",1,"Log[-2*x + x^2 + Sqrt[-5 + 4*x^2 - 4*x^3 + x^4]]/2","A",1
381,1,31,31,0.1278534,"\int \frac{2+x}{\sqrt{13+16 x^2+8 x^3+x^4}} \, dx","IntegrateAlgebraic[(2 + x)/Sqrt[13 + 16*x^2 + 8*x^3 + x^4],x]","\frac{1}{2} \log \left(4 x+x^2+\sqrt{13+16 x^2+8 x^3+x^4}\right)","\frac{1}{2} \log \left(4 x+x^2+\sqrt{13+16 x^2+8 x^3+x^4}\right)",1,"Log[4*x + x^2 + Sqrt[13 + 16*x^2 + 8*x^3 + x^4]]/2","A",1
382,1,31,31,0.1237488,"\int \frac{x}{\sqrt{b+a x^4}} \, dx","IntegrateAlgebraic[x/Sqrt[b + a*x^4],x]","\frac{\log \left(\sqrt{a} x^2+\sqrt{b+a x^4}\right)}{2 \sqrt{a}}","\frac{\log \left(\sqrt{a} x^2+\sqrt{b+a x^4}\right)}{2 \sqrt{a}}",1,"Log[Sqrt[a]*x^2 + Sqrt[b + a*x^4]]/(2*Sqrt[a])","A",1
383,1,31,31,0.0202152,"\int \frac{1}{x^7 \sqrt{-1+x^6}} \, dx","IntegrateAlgebraic[1/(x^7*Sqrt[-1 + x^6]),x]","\frac{\sqrt{-1+x^6}}{6 x^6}+\frac{1}{6} \tan ^{-1}\left(\sqrt{-1+x^6}\right)","\frac{\sqrt{-1+x^6}}{6 x^6}+\frac{1}{6} \tan ^{-1}\left(\sqrt{-1+x^6}\right)",1,"Sqrt[-1 + x^6]/(6*x^6) + ArcTan[Sqrt[-1 + x^6]]/6","A",1
384,1,31,31,0.0234833,"\int \frac{\sqrt{-1+x^6}}{x^7} \, dx","IntegrateAlgebraic[Sqrt[-1 + x^6]/x^7,x]","-\frac{\sqrt{-1+x^6}}{6 x^6}+\frac{1}{6} \tan ^{-1}\left(\sqrt{-1+x^6}\right)","-\frac{\sqrt{-1+x^6}}{6 x^6}+\frac{1}{6} \tan ^{-1}\left(\sqrt{-1+x^6}\right)",1,"-1/6*Sqrt[-1 + x^6]/x^6 + ArcTan[Sqrt[-1 + x^6]]/6","A",1
385,1,31,31,0.039415,"\int \frac{-1+x^6}{x^7 \sqrt{1+x^6}} \, dx","IntegrateAlgebraic[(-1 + x^6)/(x^7*Sqrt[1 + x^6]),x]","\frac{\sqrt{1+x^6}}{6 x^6}-\frac{1}{2} \tanh ^{-1}\left(\sqrt{1+x^6}\right)","\frac{\sqrt{1+x^6}}{6 x^6}-\frac{1}{2} \tanh ^{-1}\left(\sqrt{1+x^6}\right)",1,"Sqrt[1 + x^6]/(6*x^6) - ArcTanh[Sqrt[1 + x^6]]/2","A",1
386,1,31,31,0.0227521,"\int \frac{\sqrt{1+x^6}}{x^7} \, dx","IntegrateAlgebraic[Sqrt[1 + x^6]/x^7,x]","-\frac{\sqrt{1+x^6}}{6 x^6}-\frac{1}{6} \tanh ^{-1}\left(\sqrt{1+x^6}\right)","-\frac{\sqrt{1+x^6}}{6 x^6}-\frac{1}{6} \tanh ^{-1}\left(\sqrt{1+x^6}\right)",1,"-1/6*Sqrt[1 + x^6]/x^6 - ArcTanh[Sqrt[1 + x^6]]/6","A",1
387,1,31,31,7.4630526,"\int \frac{\left(2+x^6\right) \left(1-2 x^6+x^8+x^{12}\right)}{x^{10} \left(-1+x^6\right)^{3/4}} \, dx","IntegrateAlgebraic[((2 + x^6)*(1 - 2*x^6 + x^8 + x^12))/(x^10*(-1 + x^6)^(3/4)),x]","\frac{2 \sqrt[4]{-1+x^6} \left(1-2 x^6+9 x^8+x^{12}\right)}{9 x^9}","\frac{2 \sqrt[4]{-1+x^6} \left(1-2 x^6+9 x^8+x^{12}\right)}{9 x^9}",1,"(2*(-1 + x^6)^(1/4)*(1 - 2*x^6 + 9*x^8 + x^12))/(9*x^9)","A",1
388,1,32,32,1.3359322,"\int \frac{-1+2 x+2 x^2}{\left(1-x+3 x^2\right) \sqrt{-x+x^4}} \, dx","IntegrateAlgebraic[(-1 + 2*x + 2*x^2)/((1 - x + 3*x^2)*Sqrt[-x + x^4]),x]","\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{-x+x^4}}{1+x+x^2}\right)","\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{-x+x^4}}{1+x+x^2}\right)",1,"Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[-x + x^4])/(1 + x + x^2)]","A",1
389,1,32,32,0.2652197,"\int \frac{1+x^3}{x^3 \left(-1+x^3\right) \sqrt[4]{-x+x^4}} \, dx","IntegrateAlgebraic[(1 + x^3)/(x^3*(-1 + x^3)*(-x + x^4)^(1/4)),x]","-\frac{4 \left(-1+7 x^3\right) \left(-x+x^4\right)^{3/4}}{9 x^3 \left(-1+x^3\right)}","-\frac{4 \left(-1+7 x^3\right) \left(-x+x^4\right)^{3/4}}{9 x^3 \left(-1+x^3\right)}",1,"(-4*(-1 + 7*x^3)*(-x + x^4)^(3/4))/(9*x^3*(-1 + x^3))","A",1
390,1,32,32,1.3801315,"\int \frac{-1-2 x+2 x^2}{\left(-1+3 x+x^2\right) \sqrt{x+x^4}} \, dx","IntegrateAlgebraic[(-1 - 2*x + 2*x^2)/((-1 + 3*x + x^2)*Sqrt[x + x^4]),x]","\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{x+x^4}}{1-x+x^2}\right)","\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{x+x^4}}{1-x+x^2}\right)",1,"Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[x + x^4])/(1 - x + x^2)]","A",1
391,1,32,32,0.1065406,"\int \frac{-1+x^2}{x^2 \left(1+x^2\right) \sqrt[4]{x^2+x^4}} \, dx","IntegrateAlgebraic[(-1 + x^2)/(x^2*(1 + x^2)*(x^2 + x^4)^(1/4)),x]","\frac{2 \left(1+7 x^2\right) \left(x^2+x^4\right)^{3/4}}{3 x^3 \left(1+x^2\right)}","\frac{2 \left(1+7 x^2\right) \left(x^2+x^4\right)^{3/4}}{3 x^3 \left(1+x^2\right)}",1,"(2*(1 + 7*x^2)*(x^2 + x^4)^(3/4))/(3*x^3*(1 + x^2))","A",1
392,1,32,32,0.1438818,"\int \frac{-1+2 x}{\sqrt{-8 x+9 x^2-2 x^3+x^4}} \, dx","IntegrateAlgebraic[(-1 + 2*x)/Sqrt[-8*x + 9*x^2 - 2*x^3 + x^4],x]","-\log \left(-4+x-x^2+\sqrt{-8 x+9 x^2-2 x^3+x^4}\right)","-\log \left(-4+x-x^2+\sqrt{-8 x+9 x^2-2 x^3+x^4}\right)",1,"-Log[-4 + x - x^2 + Sqrt[-8*x + 9*x^2 - 2*x^3 + x^4]]","A",1
393,1,33,33,0.0274926,"\int \frac{-a+2 x}{\left(-1+b-a x+x^2\right) \sqrt[4]{b-a x+x^2}} \, dx","IntegrateAlgebraic[(-a + 2*x)/((-1 + b - a*x + x^2)*(b - a*x + x^2)^(1/4)),x]","2 \tan ^{-1}\left(\sqrt[4]{b-a x+x^2}\right)-2 \tanh ^{-1}\left(\sqrt[4]{b-a x+x^2}\right)","2 \tan ^{-1}\left(\sqrt[4]{b-a x+x^2}\right)-2 \tanh ^{-1}\left(\sqrt[4]{b-a x+x^2}\right)",1,"2*ArcTan[(b - a*x + x^2)^(1/4)] - 2*ArcTanh[(b - a*x + x^2)^(1/4)]","A",1
394,1,33,33,0.0942378,"\int \frac{\left(-4+x^3\right) \left(-1+x^3\right)^{2/3}}{x^{12}} \, dx","IntegrateAlgebraic[((-4 + x^3)*(-1 + x^3)^(2/3))/x^12,x]","\frac{\left(-1+x^3\right)^{2/3} \left(160-95 x^3-26 x^6-39 x^9\right)}{440 x^{11}}","\frac{\left(-1+x^3\right)^{2/3} \left(160-95 x^3-26 x^6-39 x^9\right)}{440 x^{11}}",1,"((-1 + x^3)^(2/3)*(160 - 95*x^3 - 26*x^6 - 39*x^9))/(440*x^11)","A",1
395,1,33,33,1.1005633,"\int \frac{-2+2 x+x^2}{\left(-1-3 x+x^2\right) \sqrt{1+x^3}} \, dx","IntegrateAlgebraic[(-2 + 2*x + x^2)/((-1 - 3*x + x^2)*Sqrt[1 + x^3]),x]","-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{1+x^3}}{1-x+x^2}\right)","-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{1+x^3}}{1-x+x^2}\right)",1,"-(Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[1 + x^3])/(1 - x + x^2)])","A",1
396,1,33,33,1.094137,"\int \frac{-2+2 x+x^2}{\left(3-x+2 x^2\right) \sqrt{1+x^3}} \, dx","IntegrateAlgebraic[(-2 + 2*x + x^2)/((3 - x + 2*x^2)*Sqrt[1 + x^3]),x]","-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{1+x^3}}{\sqrt{2} \left(1-x+x^2\right)}\right)","-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{1+x^3}}{\sqrt{2} \left(1-x+x^2\right)}\right)",1,"-(Sqrt[2]*ArcTan[Sqrt[1 + x^3]/(Sqrt[2]*(1 - x + x^2))])","A",1
397,1,33,33,1.0944229,"\int \frac{-2+2 x+x^2}{\left(2-4 x+3 x^2\right) \sqrt{1+x^3}} \, dx","IntegrateAlgebraic[(-2 + 2*x + x^2)/((2 - 4*x + 3*x^2)*Sqrt[1 + x^3]),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{1+x^3}}{\sqrt{3} \left(1-x+x^2\right)}\right)}{\sqrt{3}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{1+x^3}}{\sqrt{3} \left(1-x+x^2\right)}\right)}{\sqrt{3}}",1,"(-2*ArcTanh[Sqrt[1 + x^3]/(Sqrt[3]*(1 - x + x^2))])/Sqrt[3]","A",1
398,1,28,33,0.0943474,"\int \frac{\left(-1+x^3\right) \sqrt[3]{1+x^3}}{x^{11}} \, dx","IntegrateAlgebraic[((-1 + x^3)*(1 + x^3)^(1/3))/x^11,x]","\frac{\left(1+x^3\right)^{4/3} \left(7-16 x^3+12 x^6\right)}{70 x^{10}}","\frac{\sqrt[3]{1+x^3} \left(7-9 x^3-4 x^6+12 x^9\right)}{70 x^{10}}",1,"((1 + x^3)^(4/3)*(7 - 16*x^3 + 12*x^6))/(70*x^10)","A",1
399,1,28,33,0.0966189,"\int \frac{\sqrt[3]{-1+x^3} \left(1+x^3\right)}{x^{11}} \, dx","IntegrateAlgebraic[((-1 + x^3)^(1/3)*(1 + x^3))/x^11,x]","\frac{\left(-1+x^3\right)^{4/3} \left(7+16 x^3+12 x^6\right)}{70 x^{10}}","\frac{\sqrt[3]{-1+x^3} \left(-7-9 x^3+4 x^6+12 x^9\right)}{70 x^{10}}",1,"((-1 + x^3)^(4/3)*(7 + 16*x^3 + 12*x^6))/(70*x^10)","A",1
400,1,57,33,4.9168295,"\int \frac{-1-2 x+x^2+3 x^3}{\sqrt[4]{-1+3 x-3 x^2+x^3}} \, dx","IntegrateAlgebraic[(-1 - 2*x + x^2 + 3*x^3)/(-1 + 3*x - 3*x^2 + x^3)^(1/4),x]","\frac{4 \left(585 \sqrt[4]{-1+x}+1053 (-1+x)^{5/4}+650 (-1+x)^{9/4}+135 (-1+x)^{13/4}\right) \left((-1+x)^3\right)^{3/4}}{585 (-1+x)^{9/4}}","\frac{4 \left((-1+x)^3\right)^{3/4} \left(47+158 x+245 x^2+135 x^3\right)}{585 (-1+x)^2}",1,"(4*(585*(-1 + x)^(1/4) + 1053*(-1 + x)^(5/4) + 650*(-1 + x)^(9/4) + 135*(-1 + x)^(13/4))*((-1 + x)^3)^(3/4))/(585*(-1 + x)^(9/4))","A",1
401,1,57,33,4.9529469,"\int \sqrt[4]{-1+3 x-3 x^2+x^3} \left(-1-2 x+x^2+3 x^3\right) \, dx","IntegrateAlgebraic[(-1 + 3*x - 3*x^2 + x^3)^(1/4)*(-1 - 2*x + x^2 + 3*x^3),x]","\frac{4 \left(627 (-1+x)^{7/4}+3591 (-1+x)^{11/4}+2926 (-1+x)^{15/4}+693 (-1+x)^{19/4}\right) \sqrt[4]{(-1+x)^3}}{4389 (-1+x)^{3/4}}","\frac{4 \sqrt[4]{(-1+x)^3} \left(731-549 x-1029 x^2+154 x^3+693 x^4\right)}{4389}",1,"(4*(627*(-1 + x)^(7/4) + 3591*(-1 + x)^(11/4) + 2926*(-1 + x)^(15/4) + 693*(-1 + x)^(19/4))*((-1 + x)^3)^(1/4))/(4389*(-1 + x)^(3/4))","A",1
402,1,33,33,0.5165684,"\int \frac{\left(-2 b+a x^3\right) \sqrt{b+a x^3}}{x^2 \left(b-x^2+a x^3\right)} \, dx","IntegrateAlgebraic[((-2*b + a*x^3)*Sqrt[b + a*x^3])/(x^2*(b - x^2 + a*x^3)),x]","\frac{2 \sqrt{b+a x^3}}{x}-2 \tanh ^{-1}\left(\frac{x}{\sqrt{b+a x^3}}\right)","\frac{2 \sqrt{b+a x^3}}{x}-2 \tanh ^{-1}\left(\frac{x}{\sqrt{b+a x^3}}\right)",1,"(2*Sqrt[b + a*x^3])/x - 2*ArcTanh[x/Sqrt[b + a*x^3]]","A",1
403,1,33,33,0.1480828,"\int \frac{1}{\sqrt[4]{1+x^4}} \, dx","IntegrateAlgebraic[(1 + x^4)^(-1/4),x]","\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)+\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)","\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)+\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)",1,"ArcTan[x/(1 + x^4)^(1/4)]/2 + ArcTanh[x/(1 + x^4)^(1/4)]/2","A",1
404,1,33,33,1.0622997,"\int \frac{-1-2 x+2 x^2}{\left(1+2 x+4 x^2\right) \sqrt{x+x^4}} \, dx","IntegrateAlgebraic[(-1 - 2*x + 2*x^2)/((1 + 2*x + 4*x^2)*Sqrt[x + x^4]),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} \sqrt{x+x^4}}{1-x+x^2}\right)}{\sqrt{3}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} \sqrt{x+x^4}}{1-x+x^2}\right)}{\sqrt{3}}",1,"(-2*ArcTan[(Sqrt[3]*Sqrt[x + x^4])/(1 - x + x^2)])/Sqrt[3]","A",1
405,1,33,33,0.3361904,"\int \sqrt{x+x^4} \, dx","IntegrateAlgebraic[Sqrt[x + x^4],x]","\frac{1}{3} x \sqrt{x+x^4}+\frac{1}{3} \tanh ^{-1}\left(\frac{x^2}{\sqrt{x+x^4}}\right)","\frac{1}{3} x \sqrt{x+x^4}+\frac{1}{3} \tanh ^{-1}\left(\frac{x^2}{\sqrt{x+x^4}}\right)",1,"(x*Sqrt[x + x^4])/3 + ArcTanh[x^2/Sqrt[x + x^4]]/3","A",1
406,1,33,33,0.1112641,"\int \frac{1}{\sqrt[4]{-x^2+x^4}} \, dx","IntegrateAlgebraic[(-x^2 + x^4)^(-1/4),x]","\tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)","\tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)",1,"ArcTan[x/(-x^2 + x^4)^(1/4)] + ArcTanh[x/(-x^2 + x^4)^(1/4)]","A",1
407,1,33,33,0.1629988,"\int \frac{2 b+a x^2}{\sqrt[4]{b+a x^2} \left(-b-a x^2+x^4\right)} \, dx","IntegrateAlgebraic[(2*b + a*x^2)/((b + a*x^2)^(1/4)*(-b - a*x^2 + x^4)),x]","\tan ^{-1}\left(\frac{\sqrt[4]{b+a x^2}}{x}\right)-\tanh ^{-1}\left(\frac{x}{\sqrt[4]{b+a x^2}}\right)","\tan ^{-1}\left(\frac{\sqrt[4]{b+a x^2}}{x}\right)-\tanh ^{-1}\left(\frac{x}{\sqrt[4]{b+a x^2}}\right)",1,"ArcTan[(b + a*x^2)^(1/4)/x] - ArcTanh[x/(b + a*x^2)^(1/4)]","A",1
408,1,33,33,0.1254687,"\int \frac{-1+2 x}{\sqrt{-2-2 x+3 x^2-2 x^3+x^4}} \, dx","IntegrateAlgebraic[(-1 + 2*x)/Sqrt[-2 - 2*x + 3*x^2 - 2*x^3 + x^4],x]","-\log \left(-1+x-x^2+\sqrt{-2-2 x+3 x^2-2 x^3+x^4}\right)","-\log \left(-1+x-x^2+\sqrt{-2-2 x+3 x^2-2 x^3+x^4}\right)",1,"-Log[-1 + x - x^2 + Sqrt[-2 - 2*x + 3*x^2 - 2*x^3 + x^4]]","A",1
409,1,33,33,0.1226986,"\int \frac{-1+2 x}{\sqrt{-4-4 x+5 x^2-2 x^3+x^4}} \, dx","IntegrateAlgebraic[(-1 + 2*x)/Sqrt[-4 - 4*x + 5*x^2 - 2*x^3 + x^4],x]","-\log \left(-2+x-x^2+\sqrt{-4-4 x+5 x^2-2 x^3+x^4}\right)","-\log \left(-2+x-x^2+\sqrt{-4-4 x+5 x^2-2 x^3+x^4}\right)",1,"-Log[-2 + x - x^2 + Sqrt[-4 - 4*x + 5*x^2 - 2*x^3 + x^4]]","A",1
410,1,33,33,0.2177782,"\int \frac{1}{x^4 \sqrt[4]{x^3+x^4}} \, dx","IntegrateAlgebraic[1/(x^4*(x^3 + x^4)^(1/4)),x]","\frac{4 \left(-77+84 x-96 x^2+128 x^3\right) \left(x^3+x^4\right)^{3/4}}{1155 x^6}","\frac{4 \left(-77+84 x-96 x^2+128 x^3\right) \left(x^3+x^4\right)^{3/4}}{1155 x^6}",1,"(4*(-77 + 84*x - 96*x^2 + 128*x^3)*(x^3 + x^4)^(3/4))/(1155*x^6)","A",1
411,1,33,33,0.1300371,"\int \frac{x}{\sqrt{-b+a x^4}} \, dx","IntegrateAlgebraic[x/Sqrt[-b + a*x^4],x]","\frac{\log \left(\sqrt{a} x^2+\sqrt{-b+a x^4}\right)}{2 \sqrt{a}}","\frac{\log \left(\sqrt{a} x^2+\sqrt{-b+a x^4}\right)}{2 \sqrt{a}}",1,"Log[Sqrt[a]*x^2 + Sqrt[-b + a*x^4]]/(2*Sqrt[a])","A",1
412,1,33,33,0.1069332,"\int \frac{\left(1+x^3\right)^{2/3} \left(1-x^3+2 x^6\right)}{x^{12}} \, dx","IntegrateAlgebraic[((1 + x^3)^(2/3)*(1 - x^3 + 2*x^6))/x^12,x]","\frac{\left(1+x^3\right)^{2/3} \left(-40+45 x^3-142 x^6-227 x^9\right)}{440 x^{11}}","\frac{\left(1+x^3\right)^{2/3} \left(-40+45 x^3-142 x^6-227 x^9\right)}{440 x^{11}}",1,"((1 + x^3)^(2/3)*(-40 + 45*x^3 - 142*x^6 - 227*x^9))/(440*x^11)","A",1
413,1,33,33,0.7638041,"\int \frac{\left(3+x^4\right) \sqrt{x-x^5}}{1-2 x^4-x^6+x^8} \, dx","IntegrateAlgebraic[((3 + x^4)*Sqrt[x - x^5])/(1 - 2*x^4 - x^6 + x^8),x]","\tan ^{-1}\left(\frac{x^2}{\sqrt{x-x^5}}\right)+\tanh ^{-1}\left(\frac{x^2}{\sqrt{x-x^5}}\right)","\tan ^{-1}\left(\frac{x^2}{\sqrt{x-x^5}}\right)+\tanh ^{-1}\left(\frac{x^2}{\sqrt{x-x^5}}\right)",1,"ArcTan[x^2/Sqrt[x - x^5]] + ArcTanh[x^2/Sqrt[x - x^5]]","A",1
414,1,33,33,0.1996291,"\int \frac{-1+x^{16}}{x^8 \sqrt{-1+x^4}} \, dx","IntegrateAlgebraic[(-1 + x^16)/(x^8*Sqrt[-1 + x^4]),x]","\frac{\sqrt{-1+x^4} \left(-3-5 x^4+5 x^8+3 x^{12}\right)}{21 x^7}","\frac{\sqrt{-1+x^4} \left(-3-5 x^4+5 x^8+3 x^{12}\right)}{21 x^7}",1,"(Sqrt[-1 + x^4]*(-3 - 5*x^4 + 5*x^8 + 3*x^12))/(21*x^7)","A",1
415,1,19,33,0.3463872,"\int \frac{(-1+x)^{3/2}+(1+x)^{3/2}}{(-1+x)^{3/2} (1+x)^{3/2}} \, dx","IntegrateAlgebraic[((-1 + x)^(3/2) + (1 + x)^(3/2))/((-1 + x)^(3/2)*(1 + x)^(3/2)),x]","-\frac{2}{\sqrt{-1+x}}-\frac{2}{\sqrt{1+x}}","\frac{2 \sqrt{-1+x}}{1-x}+\frac{2 \sqrt{1+x}}{-1-x}",1,"-2/Sqrt[-1 + x] - 2/Sqrt[1 + x]","A",1
416,1,34,34,0.0836394,"\int \frac{1+x}{(-1+x) \sqrt{x+x^2+x^3}} \, dx","IntegrateAlgebraic[(1 + x)/((-1 + x)*Sqrt[x + x^2 + x^3]),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{3} \sqrt{x+x^2+x^3}}{1+x+x^2}\right)}{\sqrt{3}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{3} \sqrt{x+x^2+x^3}}{1+x+x^2}\right)}{\sqrt{3}}",1,"(-2*ArcTanh[(Sqrt[3]*Sqrt[x + x^2 + x^3])/(1 + x + x^2)])/Sqrt[3]","A",1
417,1,34,34,0.0929932,"\int \frac{-1+x^2}{\left(1-x+x^2\right) \sqrt{x+x^2+x^3}} \, dx","IntegrateAlgebraic[(-1 + x^2)/((1 - x + x^2)*Sqrt[x + x^2 + x^3]),x]","-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{x+x^2+x^3}}{1+x+x^2}\right)","-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{x+x^2+x^3}}{1+x+x^2}\right)",1,"-(Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[x + x^2 + x^3])/(1 + x + x^2)])","A",1
418,1,34,34,0.594062,"\int \frac{-1+x}{(1+x) \sqrt{1+x^4}} \, dx","IntegrateAlgebraic[(-1 + x)/((1 + x)*Sqrt[1 + x^4]),x]","-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{1+2 x+x^2+\sqrt{1+x^4}}\right)","-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{1+2 x+x^2+\sqrt{1+x^4}}\right)",1,"-(Sqrt[2]*ArcTanh[(Sqrt[2]*x)/(1 + 2*x + x^2 + Sqrt[1 + x^4])])","A",1
419,1,34,34,0.6166581,"\int \frac{1+x}{(-1+x) \sqrt{1+x^4}} \, dx","IntegrateAlgebraic[(1 + x)/((-1 + x)*Sqrt[1 + x^4]),x]","-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{1-2 x+x^2+\sqrt{1+x^4}}\right)","-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{1-2 x+x^2+\sqrt{1+x^4}}\right)",1,"-(Sqrt[2]*ArcTanh[(Sqrt[2]*x)/(1 - 2*x + x^2 + Sqrt[1 + x^4])])","A",1
420,1,34,34,0.4425493,"\int \frac{1+x}{(-1+x) \sqrt{1+x^2+x^4}} \, dx","IntegrateAlgebraic[(1 + x)/((-1 + x)*Sqrt[1 + x^2 + x^4]),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{3} \sqrt{1+x^2+x^4}}{1+x+x^2}\right)}{\sqrt{3}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{3} \sqrt{1+x^2+x^4}}{1+x+x^2}\right)}{\sqrt{3}}",1,"(-2*ArcTanh[(Sqrt[3]*Sqrt[1 + x^2 + x^4])/(1 + x + x^2)])/Sqrt[3]","A",1
421,1,34,34,0.1350968,"\int \frac{1+x^2}{\left(-1+x^2\right) \sqrt{1-x-x^2+x^3+x^4}} \, dx","IntegrateAlgebraic[(1 + x^2)/((-1 + x^2)*Sqrt[1 - x - x^2 + x^3 + x^4]),x]","2 \tanh ^{-1}\left(\frac{x}{-1+x^2-\sqrt{1-x-x^2+x^3+x^4}}\right)","2 \tanh ^{-1}\left(\frac{x}{-1+x^2-\sqrt{1-x-x^2+x^3+x^4}}\right)",1,"2*ArcTanh[x/(-1 + x^2 - Sqrt[1 - x - x^2 + x^3 + x^4])]","A",1
422,1,34,34,0.1307755,"\int \frac{x}{\sqrt{11-11 x-3 x^2+2 x^3+x^4}} \, dx","IntegrateAlgebraic[x/Sqrt[11 - 11*x - 3*x^2 + 2*x^3 + x^4],x]","\frac{2}{3} \tanh ^{-1}\left(\frac{-2+x+x^2}{\sqrt{11-11 x-3 x^2+2 x^3+x^4}}\right)","\frac{2}{3} \tanh ^{-1}\left(\frac{-2+x+x^2}{\sqrt{11-11 x-3 x^2+2 x^3+x^4}}\right)",1,"(2*ArcTanh[(-2 + x + x^2)/Sqrt[11 - 11*x - 3*x^2 + 2*x^3 + x^4]])/3","A",1
423,1,34,34,0.1606986,"\int \frac{1+x}{\sqrt{2-5 x-3 x^2+2 x^3+x^4}} \, dx","IntegrateAlgebraic[(1 + x)/Sqrt[2 - 5*x - 3*x^2 + 2*x^3 + x^4],x]","\frac{2}{3} \tanh ^{-1}\left(\frac{-2+x+x^2}{\sqrt{2-5 x-3 x^2+2 x^3+x^4}}\right)","\frac{2}{3} \tanh ^{-1}\left(\frac{-2+x+x^2}{\sqrt{2-5 x-3 x^2+2 x^3+x^4}}\right)",1,"(2*ArcTanh[(-2 + x + x^2)/Sqrt[2 - 5*x - 3*x^2 + 2*x^3 + x^4]])/3","A",1
424,1,34,34,0.1256271,"\int \frac{x}{\sqrt{-3+3 x-3 x^2+2 x^3+x^4}} \, dx","IntegrateAlgebraic[x/Sqrt[-3 + 3*x - 3*x^2 + 2*x^3 + x^4],x]","\frac{2}{3} \tanh ^{-1}\left(\frac{-2+x+x^2}{\sqrt{-3+3 x-3 x^2+2 x^3+x^4}}\right)","\frac{2}{3} \tanh ^{-1}\left(\frac{-2+x+x^2}{\sqrt{-3+3 x-3 x^2+2 x^3+x^4}}\right)",1,"(2*ArcTanh[(-2 + x + x^2)/Sqrt[-3 + 3*x - 3*x^2 + 2*x^3 + x^4]])/3","A",1
425,1,34,34,6.992844,"\int \frac{\sqrt{1-x^6} \left(2+x^6\right)}{x^3 \left(-1+x^4+x^6\right)} \, dx","IntegrateAlgebraic[(Sqrt[1 - x^6]*(2 + x^6))/(x^3*(-1 + x^4 + x^6)),x]","\frac{\sqrt{1-x^6}}{x^2}-\tanh ^{-1}\left(\frac{x^2}{\sqrt{1-x^6}}\right)","\frac{\sqrt{1-x^6}}{x^2}-\tanh ^{-1}\left(\frac{x^2}{\sqrt{1-x^6}}\right)",1,"Sqrt[1 - x^6]/x^2 - ArcTanh[x^2/Sqrt[1 - x^6]]","A",1
426,1,35,35,0.1656892,"\int \frac{\sqrt{x}}{\left(-2+x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[Sqrt[x]/(-2 + x^2)^(3/4),x]","-\tan ^{-1}\left(\frac{\sqrt{x}}{\sqrt[4]{-2+x^2}}\right)+\tanh ^{-1}\left(\frac{\sqrt{x}}{\sqrt[4]{-2+x^2}}\right)","-\tan ^{-1}\left(\frac{\sqrt{x}}{\sqrt[4]{-2+x^2}}\right)+\tanh ^{-1}\left(\frac{\sqrt{x}}{\sqrt[4]{-2+x^2}}\right)",1,"-ArcTan[Sqrt[x]/(-2 + x^2)^(1/4)] + ArcTanh[Sqrt[x]/(-2 + x^2)^(1/4)]","A",1
427,1,35,35,0.0261956,"\int \frac{a+x}{\left(-1+2 b+2 a x+x^2\right) \sqrt[4]{2 b+2 a x+x^2}} \, dx","IntegrateAlgebraic[(a + x)/((-1 + 2*b + 2*a*x + x^2)*(2*b + 2*a*x + x^2)^(1/4)),x]","\tan ^{-1}\left(\sqrt[4]{2 b+2 a x+x^2}\right)-\tanh ^{-1}\left(\sqrt[4]{2 b+2 a x+x^2}\right)","\tan ^{-1}\left(\sqrt[4]{2 b+2 a x+x^2}\right)-\tanh ^{-1}\left(\sqrt[4]{2 b+2 a x+x^2}\right)",1,"ArcTan[(2*b + 2*a*x + x^2)^(1/4)] - ArcTanh[(2*b + 2*a*x + x^2)^(1/4)]","A",1
428,1,35,35,0.1091176,"\int \frac{1}{\sqrt{c+b x+a x^2}} \, dx","IntegrateAlgebraic[1/Sqrt[c + b*x + a*x^2],x]","-\frac{\log \left(b+2 a x-2 \sqrt{a} \sqrt{c+b x+a x^2}\right)}{\sqrt{a}}","-\frac{\log \left(b+2 a x-2 \sqrt{a} \sqrt{c+b x+a x^2}\right)}{\sqrt{a}}",1,"-(Log[b + 2*a*x - 2*Sqrt[a]*Sqrt[c + b*x + a*x^2]]/Sqrt[a])","A",1
429,1,35,35,0.0194452,"\int x^{11} \sqrt[3]{-1+x^3} \, dx","IntegrateAlgebraic[x^11*(-1 + x^3)^(1/3),x]","\frac{\sqrt[3]{-1+x^3} \left(-81-27 x^3-18 x^6-14 x^9+140 x^{12}\right)}{1820}","\frac{\sqrt[3]{-1+x^3} \left(-81-27 x^3-18 x^6-14 x^9+140 x^{12}\right)}{1820}",1,"((-1 + x^3)^(1/3)*(-81 - 27*x^3 - 18*x^6 - 14*x^9 + 140*x^12))/1820","A",1
430,1,35,35,0.2723968,"\int \frac{-1-2 x+x^2}{\left(1+2 x+3 x^2\right) \sqrt{-x+x^3}} \, dx","IntegrateAlgebraic[(-1 - 2*x + x^2)/((1 + 2*x + 3*x^2)*Sqrt[-x + x^3]),x]","-\frac{2 \tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{3}}+\frac{x}{\sqrt{3}}}{\sqrt{-x+x^3}}\right)}{\sqrt{3}}","-\frac{2 \tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{3}}+\frac{x}{\sqrt{3}}}{\sqrt{-x+x^3}}\right)}{\sqrt{3}}",1,"(-2*ArcTanh[(-(1/Sqrt[3]) + x/Sqrt[3])/Sqrt[-x + x^3]])/Sqrt[3]","A",1
431,1,35,35,0.3442094,"\int \frac{\left(-1+x^2\right) \sqrt{-1-4 x-5 x^2-4 x^3-x^4}}{\left(1+x+x^2\right) \left(1+3 x+x^2\right)^2} \, dx","IntegrateAlgebraic[((-1 + x^2)*Sqrt[-1 - 4*x - 5*x^2 - 4*x^3 - x^4])/((1 + x + x^2)*(1 + 3*x + x^2)^2),x]","\frac{\sqrt{-1-4 x-5 x^2-4 x^3-x^4}}{1+3 x+x^2}","\frac{\sqrt{-1-4 x-5 x^2-4 x^3-x^4}}{1+3 x+x^2}",1,"Sqrt[-1 - 4*x - 5*x^2 - 4*x^3 - x^4]/(1 + 3*x + x^2)","A",1
432,1,37,35,0.0564953,"\int \frac{-1+x^4}{x^3 \sqrt{1+x^4}} \, dx","IntegrateAlgebraic[(-1 + x^4)/(x^3*Sqrt[1 + x^4]),x]","\frac{\sqrt{1+x^4}}{2 x^2}-\frac{1}{2} \log \left(-x^2+\sqrt{1+x^4}\right)","\frac{\sqrt{1+x^4}}{2 x^2}+\frac{1}{2} \log \left(x^2+\sqrt{1+x^4}\right)",1,"Sqrt[1 + x^4]/(2*x^2) - Log[-x^2 + Sqrt[1 + x^4]]/2","A",1
433,1,35,35,0.3336839,"\int \frac{-1+x^3}{x^6 \left(1+x^3\right) \sqrt[4]{x+x^4}} \, dx","IntegrateAlgebraic[(-1 + x^3)/(x^6*(1 + x^3)*(x + x^4)^(1/4)),x]","-\frac{4 \left(x+x^4\right)^{3/4} \left(-1+5 x^3+20 x^6\right)}{21 x^6 \left(1+x^3\right)}","-\frac{4 \left(x+x^4\right)^{3/4} \left(-1+5 x^3+20 x^6\right)}{21 x^6 \left(1+x^3\right)}",1,"(-4*(x + x^4)^(3/4)*(-1 + 5*x^3 + 20*x^6))/(21*x^6*(1 + x^3))","A",1
434,1,35,35,0.3421165,"\int \frac{\sqrt{x+x^4}}{x^3} \, dx","IntegrateAlgebraic[Sqrt[x + x^4]/x^3,x]","-\frac{2 \sqrt{x+x^4}}{3 x^2}+\frac{2}{3} \tanh ^{-1}\left(\frac{x^2}{\sqrt{x+x^4}}\right)","-\frac{2 \sqrt{x+x^4}}{3 x^2}+\frac{2}{3} \tanh ^{-1}\left(\frac{x^2}{\sqrt{x+x^4}}\right)",1,"(-2*Sqrt[x + x^4])/(3*x^2) + (2*ArcTanh[x^2/Sqrt[x + x^4]])/3","A",1
435,1,35,35,0.1576808,"\int \frac{1}{x^8 \sqrt[4]{x^2+x^4}} \, dx","IntegrateAlgebraic[1/(x^8*(x^2 + x^4)^(1/4)),x]","\frac{2 \left(x^2+x^4\right)^{3/4} \left(-77+84 x^2-96 x^4+128 x^6\right)}{1155 x^9}","\frac{2 \left(x^2+x^4\right)^{3/4} \left(-77+84 x^2-96 x^4+128 x^6\right)}{1155 x^9}",1,"(2*(x^2 + x^4)^(3/4)*(-77 + 84*x^2 - 96*x^4 + 128*x^6))/(1155*x^9)","A",1
436,1,35,35,0.2316237,"\int \frac{1-2 x}{\sqrt{5+5 x-4 x^2-2 x^3+x^4}} \, dx","IntegrateAlgebraic[(1 - 2*x)/Sqrt[5 + 5*x - 4*x^2 - 2*x^3 + x^4],x]","\log \left(5+2 x-2 x^2+2 \sqrt{5+5 x-4 x^2-2 x^3+x^4}\right)","\log \left(5+2 x-2 x^2+2 \sqrt{5+5 x-4 x^2-2 x^3+x^4}\right)",1,"Log[5 + 2*x - 2*x^2 + 2*Sqrt[5 + 5*x - 4*x^2 - 2*x^3 + x^4]]","A",1
437,1,35,35,0.2349538,"\int \frac{1}{x^4 \sqrt[4]{-x^3+x^4}} \, dx","IntegrateAlgebraic[1/(x^4*(-x^3 + x^4)^(1/4)),x]","\frac{4 \left(77+84 x+96 x^2+128 x^3\right) \left(-x^3+x^4\right)^{3/4}}{1155 x^6}","\frac{4 \left(77+84 x+96 x^2+128 x^3\right) \left(-x^3+x^4\right)^{3/4}}{1155 x^6}",1,"(4*(77 + 84*x + 96*x^2 + 128*x^3)*(-x^3 + x^4)^(3/4))/(1155*x^6)","A",1
438,1,35,35,0.1467485,"\int \frac{1+2 x}{\sqrt{-4-3 x-2 x^2+2 x^3+x^4}} \, dx","IntegrateAlgebraic[(1 + 2*x)/Sqrt[-4 - 3*x - 2*x^2 + 2*x^3 + x^4],x]","\log \left(-3+2 x+2 x^2+2 \sqrt{-4-3 x-2 x^2+2 x^3+x^4}\right)","\log \left(-3+2 x+2 x^2+2 \sqrt{-4-3 x-2 x^2+2 x^3+x^4}\right)",1,"Log[-3 + 2*x + 2*x^2 + 2*Sqrt[-4 - 3*x - 2*x^2 + 2*x^3 + x^4]]","A",1
439,1,35,35,0.5978919,"\int \frac{4 b+a x^3}{\sqrt[4]{b+a x^3} \left(-b-a x^3+x^4\right)} \, dx","IntegrateAlgebraic[(4*b + a*x^3)/((b + a*x^3)^(1/4)*(-b - a*x^3 + x^4)),x]","2 \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^3}}{x}\right)-2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{b+a x^3}}\right)","2 \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^3}}{x}\right)-2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{b+a x^3}}\right)",1,"2*ArcTan[(b + a*x^3)^(1/4)/x] - 2*ArcTanh[x/(b + a*x^3)^(1/4)]","A",1
440,1,35,35,0.1580048,"\int \frac{x^8}{\sqrt{-1+x^6}} \, dx","IntegrateAlgebraic[x^8/Sqrt[-1 + x^6],x]","\frac{1}{6} x^3 \sqrt{-1+x^6}+\frac{1}{6} \log \left(x^3+\sqrt{-1+x^6}\right)","\frac{1}{6} x^3 \sqrt{-1+x^6}+\frac{1}{6} \log \left(x^3+\sqrt{-1+x^6}\right)",1,"(x^3*Sqrt[-1 + x^6])/6 + Log[x^3 + Sqrt[-1 + x^6]]/6","A",1
441,1,35,35,0.118852,"\int \frac{\sqrt{-1+x^6}}{x^4} \, dx","IntegrateAlgebraic[Sqrt[-1 + x^6]/x^4,x]","-\frac{\sqrt{-1+x^6}}{3 x^3}+\frac{1}{3} \log \left(x^3+\sqrt{-1+x^6}\right)","-\frac{\sqrt{-1+x^6}}{3 x^3}+\frac{1}{3} \log \left(x^3+\sqrt{-1+x^6}\right)",1,"-1/3*Sqrt[-1 + x^6]/x^3 + Log[x^3 + Sqrt[-1 + x^6]]/3","A",1
442,1,35,35,0.1055262,"\int x^2 \sqrt{-1+x^6} \, dx","IntegrateAlgebraic[x^2*Sqrt[-1 + x^6],x]","\frac{1}{6} x^3 \sqrt{-1+x^6}-\frac{1}{6} \log \left(x^3+\sqrt{-1+x^6}\right)","\frac{1}{6} x^3 \sqrt{-1+x^6}-\frac{1}{6} \log \left(x^3+\sqrt{-1+x^6}\right)",1,"(x^3*Sqrt[-1 + x^6])/6 - Log[x^3 + Sqrt[-1 + x^6]]/6","A",1
443,1,35,35,0.1414611,"\int \frac{x^8}{\sqrt{1+x^6}} \, dx","IntegrateAlgebraic[x^8/Sqrt[1 + x^6],x]","\frac{1}{6} x^3 \sqrt{1+x^6}-\frac{1}{6} \log \left(x^3+\sqrt{1+x^6}\right)","\frac{1}{6} x^3 \sqrt{1+x^6}-\frac{1}{6} \log \left(x^3+\sqrt{1+x^6}\right)",1,"(x^3*Sqrt[1 + x^6])/6 - Log[x^3 + Sqrt[1 + x^6]]/6","A",1
444,1,35,35,0.1191616,"\int \frac{\sqrt{1+x^6}}{x^4} \, dx","IntegrateAlgebraic[Sqrt[1 + x^6]/x^4,x]","-\frac{\sqrt{1+x^6}}{3 x^3}+\frac{1}{3} \log \left(x^3+\sqrt{1+x^6}\right)","-\frac{\sqrt{1+x^6}}{3 x^3}+\frac{1}{3} \log \left(x^3+\sqrt{1+x^6}\right)",1,"-1/3*Sqrt[1 + x^6]/x^3 + Log[x^3 + Sqrt[1 + x^6]]/3","A",1
445,1,35,35,0.4162432,"\int \frac{1+x^6}{x^6 \left(1+x^3\right) \sqrt[4]{x+x^4}} \, dx","IntegrateAlgebraic[(1 + x^6)/(x^6*(1 + x^3)*(x + x^4)^(1/4)),x]","\frac{4 \left(x+x^4\right)^{3/4} \left(-3+8 x^3+53 x^6\right)}{63 x^6 \left(1+x^3\right)}","\frac{4 \left(x+x^4\right)^{3/4} \left(-3+8 x^3+53 x^6\right)}{63 x^6 \left(1+x^3\right)}",1,"(4*(x + x^4)^(3/4)*(-3 + 8*x^3 + 53*x^6))/(63*x^6*(1 + x^3))","A",1
446,1,35,35,0.2540587,"\int x \sqrt{x+x^6} \, dx","IntegrateAlgebraic[x*Sqrt[x + x^6],x]","\frac{1}{5} x^2 \sqrt{x+x^6}+\frac{1}{5} \tanh ^{-1}\left(\frac{x^3}{\sqrt{x+x^6}}\right)","\frac{1}{5} x^2 \sqrt{x+x^6}+\frac{1}{5} \tanh ^{-1}\left(\frac{x^3}{\sqrt{x+x^6}}\right)",1,"(x^2*Sqrt[x + x^6])/5 + ArcTanh[x^3/Sqrt[x + x^6]]/5","A",1
447,1,37,35,4.8748154,"\int \frac{-2 b c+a c x^6}{\sqrt[4]{b+a x^6} \left(b-c^4 x^4+a x^6\right)} \, dx","IntegrateAlgebraic[(-2*b*c + a*c*x^6)/((b + a*x^6)^(1/4)*(b - c^4*x^4 + a*x^6)),x]","\tan ^{-1}\left(\frac{\sqrt[4]{b+a x^6}}{c x}\right)-\tanh ^{-1}\left(\frac{c x}{\sqrt[4]{b+a x^6}}\right)","-\tan ^{-1}\left(\frac{c x}{\sqrt[4]{b+a x^6}}\right)-\tanh ^{-1}\left(\frac{c x}{\sqrt[4]{b+a x^6}}\right)",1,"ArcTan[(b + a*x^6)^(1/4)/(c*x)] - ArcTanh[(c*x)/(b + a*x^6)^(1/4)]","A",1
448,1,35,35,0.0474179,"\int \frac{1}{\sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[1/Sqrt[x + Sqrt[1 + x^2]],x]","-\frac{1}{3 \left(x+\sqrt{1+x^2}\right)^{3/2}}+\sqrt{x+\sqrt{1+x^2}}","-\frac{1}{3 \left(x+\sqrt{1+x^2}\right)^{3/2}}+\sqrt{x+\sqrt{1+x^2}}",1,"-1/3*1/(x + Sqrt[1 + x^2])^(3/2) + Sqrt[x + Sqrt[1 + x^2]]","A",1
449,1,36,36,0.024464,"\int \frac{\sqrt[4]{1+x^2}}{x} \, dx","IntegrateAlgebraic[(1 + x^2)^(1/4)/x,x]","2 \sqrt[4]{1+x^2}-\tan ^{-1}\left(\sqrt[4]{1+x^2}\right)-\tanh ^{-1}\left(\sqrt[4]{1+x^2}\right)","2 \sqrt[4]{1+x^2}-\tan ^{-1}\left(\sqrt[4]{1+x^2}\right)-\tanh ^{-1}\left(\sqrt[4]{1+x^2}\right)",1,"2*(1 + x^2)^(1/4) - ArcTan[(1 + x^2)^(1/4)] - ArcTanh[(1 + x^2)^(1/4)]","A",1
450,1,36,36,0.0392498,"\int \frac{\sqrt{-1+x^3}}{x^7} \, dx","IntegrateAlgebraic[Sqrt[-1 + x^3]/x^7,x]","\frac{\left(-2+x^3\right) \sqrt{-1+x^3}}{12 x^6}+\frac{1}{12} \tan ^{-1}\left(\sqrt{-1+x^3}\right)","\frac{\left(-2+x^3\right) \sqrt{-1+x^3}}{12 x^6}+\frac{1}{12} \tan ^{-1}\left(\sqrt{-1+x^3}\right)",1,"((-2 + x^3)*Sqrt[-1 + x^3])/(12*x^6) + ArcTan[Sqrt[-1 + x^3]]/12","A",1
451,1,36,36,0.2965738,"\int \frac{-b+a x^2}{\left(b+c x+a x^2\right) \sqrt{b x+a x^3}} \, dx","IntegrateAlgebraic[(-b + a*x^2)/((b + c*x + a*x^2)*Sqrt[b*x + a*x^3]),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{b x+a x^3}}{b+a x^2}\right)}{\sqrt{c}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{b x+a x^3}}{b+a x^2}\right)}{\sqrt{c}}",1,"(-2*ArcTan[(Sqrt[c]*Sqrt[b*x + a*x^3])/(b + a*x^2)])/Sqrt[c]","A",1
452,1,36,36,0.1370064,"\int \frac{-1+x}{\sqrt{-2-x+6 x^2-4 x^3+x^4}} \, dx","IntegrateAlgebraic[(-1 + x)/Sqrt[-2 - x + 6*x^2 - 4*x^3 + x^4],x]","\frac{2}{3} \tanh ^{-1}\left(\frac{1-2 x+x^2}{\sqrt{-2-x+6 x^2-4 x^3+x^4}}\right)","\frac{2}{3} \tanh ^{-1}\left(\frac{1-2 x+x^2}{\sqrt{-2-x+6 x^2-4 x^3+x^4}}\right)",1,"(2*ArcTanh[(1 - 2*x + x^2)/Sqrt[-2 - x + 6*x^2 - 4*x^3 + x^4]])/3","A",1
453,1,36,36,0.0289126,"\int \frac{\sqrt{-1+x^6}}{x^{13}} \, dx","IntegrateAlgebraic[Sqrt[-1 + x^6]/x^13,x]","\frac{\left(-2+x^6\right) \sqrt{-1+x^6}}{24 x^{12}}+\frac{1}{24} \tan ^{-1}\left(\sqrt{-1+x^6}\right)","\frac{\left(-2+x^6\right) \sqrt{-1+x^6}}{24 x^{12}}+\frac{1}{24} \tan ^{-1}\left(\sqrt{-1+x^6}\right)",1,"((-2 + x^6)*Sqrt[-1 + x^6])/(24*x^12) + ArcTan[Sqrt[-1 + x^6]]/24","A",1
454,1,36,36,13.6929274,"\int \frac{-1-2 x^2+2 x^4}{\left(2-3 x^2+x^4\right) \sqrt{1+x^6}} \, dx","IntegrateAlgebraic[(-1 - 2*x^2 + 2*x^4)/((2 - 3*x^2 + x^4)*Sqrt[1 + x^6]),x]","-\frac{\tanh ^{-1}\left(\frac{x \sqrt{1+x^6}}{\sqrt{2} \left(1-x^2+x^4\right)}\right)}{\sqrt{2}}","-\frac{\tanh ^{-1}\left(\frac{x \sqrt{1+x^6}}{\sqrt{2} \left(1-x^2+x^4\right)}\right)}{\sqrt{2}}",1,"-(ArcTanh[(x*Sqrt[1 + x^6])/(Sqrt[2]*(1 - x^2 + x^4))]/Sqrt[2])","A",1
455,1,37,37,0.1367335,"\int \frac{-1+k x}{(1+k x) \sqrt{(1-x) x \left(1-k^2 x\right)}} \, dx","IntegrateAlgebraic[(-1 + k*x)/((1 + k*x)*Sqrt[(1 - x)*x*(1 - k^2*x)]),x]","-\frac{2 \tan ^{-1}\left(\frac{(1+k) x}{\sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}\right)}{1+k}","-\frac{2 \tan ^{-1}\left(\frac{(1+k) x}{\sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}\right)}{1+k}",1,"(-2*ArcTan[((1 + k)*x)/Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3]])/(1 + k)","A",1
456,1,37,37,0.1317431,"\int \frac{1+k x}{(-1+k x) \sqrt{(1-x) x \left(1-k^2 x\right)}} \, dx","IntegrateAlgebraic[(1 + k*x)/((-1 + k*x)*Sqrt[(1 - x)*x*(1 - k^2*x)]),x]","-\frac{2 \tan ^{-1}\left(\frac{(-1+k) x}{\sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}\right)}{-1+k}","-\frac{2 \tan ^{-1}\left(\frac{(-1+k) x}{\sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}\right)}{-1+k}",1,"(-2*ArcTan[((-1 + k)*x)/Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3]])/(-1 + k)","A",1
457,1,37,37,0.2568603,"\int \frac{\left(-1+x^2\right) \sqrt{1+x^4}}{x^2 \left(1+x^2\right)} \, dx","IntegrateAlgebraic[((-1 + x^2)*Sqrt[1 + x^4])/(x^2*(1 + x^2)),x]","\frac{\sqrt{1+x^4}}{x}+\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)","\frac{\sqrt{1+x^4}}{x}+\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)",1,"Sqrt[1 + x^4]/x + Sqrt[2]*ArcTan[(Sqrt[2]*x)/Sqrt[1 + x^4]]","A",1
458,1,37,37,0.3518884,"\int \frac{1+x^3}{x^6 \left(-1+x^3\right) \sqrt[4]{-x+x^4}} \, dx","IntegrateAlgebraic[(1 + x^3)/(x^6*(-1 + x^3)*(-x + x^4)^(1/4)),x]","-\frac{4 \left(-x+x^4\right)^{3/4} \left(-1-5 x^3+20 x^6\right)}{21 x^6 \left(-1+x^3\right)}","-\frac{4 \left(-x+x^4\right)^{3/4} \left(-1-5 x^3+20 x^6\right)}{21 x^6 \left(-1+x^3\right)}",1,"(-4*(-x + x^4)^(3/4)*(-1 - 5*x^3 + 20*x^6))/(21*x^6*(-1 + x^3))","A",1
459,1,37,37,0.3478909,"\int \sqrt{-x+x^4} \, dx","IntegrateAlgebraic[Sqrt[-x + x^4],x]","\frac{1}{3} x \sqrt{-x+x^4}-\frac{1}{3} \tanh ^{-1}\left(\frac{x^2}{\sqrt{-x+x^4}}\right)","\frac{1}{3} x \sqrt{-x+x^4}-\frac{1}{3} \tanh ^{-1}\left(\frac{x^2}{\sqrt{-x+x^4}}\right)",1,"(x*Sqrt[-x + x^4])/3 - ArcTanh[x^2/Sqrt[-x + x^4]]/3","A",1
460,1,37,37,0.1811361,"\int \frac{1}{x^8 \sqrt[4]{-x^2+x^4}} \, dx","IntegrateAlgebraic[1/(x^8*(-x^2 + x^4)^(1/4)),x]","\frac{2 \left(-x^2+x^4\right)^{3/4} \left(77+84 x^2+96 x^4+128 x^6\right)}{1155 x^9}","\frac{2 \left(-x^2+x^4\right)^{3/4} \left(77+84 x^2+96 x^4+128 x^6\right)}{1155 x^9}",1,"(2*(-x^2 + x^4)^(3/4)*(77 + 84*x^2 + 96*x^4 + 128*x^6))/(1155*x^9)","A",1
461,1,37,37,0.154793,"\int \frac{-1+x}{\sqrt{-1+4 x+2 x^2-4 x^3+x^4}} \, dx","IntegrateAlgebraic[(-1 + x)/Sqrt[-1 + 4*x + 2*x^2 - 4*x^3 + x^4],x]","-\frac{1}{2} \log \left(1+2 x-x^2+\sqrt{-1+4 x+2 x^2-4 x^3+x^4}\right)","-\frac{1}{2} \log \left(1+2 x-x^2+\sqrt{-1+4 x+2 x^2-4 x^3+x^4}\right)",1,"-1/2*Log[1 + 2*x - x^2 + Sqrt[-1 + 4*x + 2*x^2 - 4*x^3 + x^4]]","A",1
462,1,37,37,0.2111977,"\int \frac{\sqrt[4]{-x^3+x^4}}{(-1+x) x} \, dx","IntegrateAlgebraic[(-x^3 + x^4)^(1/4)/((-1 + x)*x),x]","-2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)+2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)","-2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)+2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)",1,"-2*ArcTan[x/(-x^3 + x^4)^(1/4)] + 2*ArcTanh[x/(-x^3 + x^4)^(1/4)]","A",1
463,1,41,37,0.131458,"\int \frac{-2+x^3}{x \sqrt{-1+x^6}} \, dx","IntegrateAlgebraic[(-2 + x^3)/(x*Sqrt[-1 + x^6]),x]","\frac{4}{3} \tan ^{-1}\left(x^3-\sqrt{-1+x^6}\right)-\frac{1}{3} \log \left(-x^3+\sqrt{-1+x^6}\right)","-\frac{4}{3} \tan ^{-1}\left(x^3+\sqrt{-1+x^6}\right)+\frac{1}{3} \log \left(x^3+\sqrt{-1+x^6}\right)",1,"(4*ArcTan[x^3 - Sqrt[-1 + x^6]])/3 - Log[-x^3 + Sqrt[-1 + x^6]]/3","A",1
464,1,41,37,0.1114467,"\int \frac{1+x^3}{x \sqrt{-1+x^6}} \, dx","IntegrateAlgebraic[(1 + x^3)/(x*Sqrt[-1 + x^6]),x]","-\frac{2}{3} \tan ^{-1}\left(x^3-\sqrt{-1+x^6}\right)-\frac{1}{3} \log \left(-x^3+\sqrt{-1+x^6}\right)","\frac{2}{3} \tan ^{-1}\left(x^3+\sqrt{-1+x^6}\right)+\frac{1}{3} \log \left(x^3+\sqrt{-1+x^6}\right)",1,"(-2*ArcTan[x^3 - Sqrt[-1 + x^6]])/3 - Log[-x^3 + Sqrt[-1 + x^6]]/3","A",1
465,1,41,37,0.1364474,"\int \frac{-1+2 x^3}{x \sqrt{-1+x^6}} \, dx","IntegrateAlgebraic[(-1 + 2*x^3)/(x*Sqrt[-1 + x^6]),x]","\frac{2}{3} \tan ^{-1}\left(x^3-\sqrt{-1+x^6}\right)-\frac{2}{3} \log \left(-x^3+\sqrt{-1+x^6}\right)","-\frac{2}{3} \tan ^{-1}\left(x^3+\sqrt{-1+x^6}\right)+\frac{2}{3} \log \left(x^3+\sqrt{-1+x^6}\right)",1,"(2*ArcTan[x^3 - Sqrt[-1 + x^6]])/3 - (2*Log[-x^3 + Sqrt[-1 + x^6]])/3","A",1
466,1,41,37,0.1313204,"\int \frac{1+2 x^3}{x \sqrt{-1+x^6}} \, dx","IntegrateAlgebraic[(1 + 2*x^3)/(x*Sqrt[-1 + x^6]),x]","-\frac{2}{3} \tan ^{-1}\left(x^3-\sqrt{-1+x^6}\right)-\frac{2}{3} \log \left(-x^3+\sqrt{-1+x^6}\right)","\frac{2}{3} \tan ^{-1}\left(x^3+\sqrt{-1+x^6}\right)+\frac{2}{3} \log \left(x^3+\sqrt{-1+x^6}\right)",1,"(-2*ArcTan[x^3 - Sqrt[-1 + x^6]])/3 - (2*Log[-x^3 + Sqrt[-1 + x^6]])/3","A",1
467,1,41,37,0.1337971,"\int \frac{1+4 x^3}{x \sqrt{-1+x^6}} \, dx","IntegrateAlgebraic[(1 + 4*x^3)/(x*Sqrt[-1 + x^6]),x]","-\frac{2}{3} \tan ^{-1}\left(x^3-\sqrt{-1+x^6}\right)-\frac{4}{3} \log \left(-x^3+\sqrt{-1+x^6}\right)","\frac{2}{3} \tan ^{-1}\left(x^3+\sqrt{-1+x^6}\right)+\frac{4}{3} \log \left(x^3+\sqrt{-1+x^6}\right)",1,"(-2*ArcTan[x^3 - Sqrt[-1 + x^6]])/3 - (4*Log[-x^3 + Sqrt[-1 + x^6]])/3","A",1
468,1,37,37,0.4280519,"\int \frac{1+x^6}{x^6 \left(-1+x^3\right) \sqrt[4]{-x+x^4}} \, dx","IntegrateAlgebraic[(1 + x^6)/(x^6*(-1 + x^3)*(-x + x^4)^(1/4)),x]","-\frac{4 \left(-x+x^4\right)^{3/4} \left(-3-8 x^3+53 x^6\right)}{63 x^6 \left(-1+x^3\right)}","-\frac{4 \left(-x+x^4\right)^{3/4} \left(-3-8 x^3+53 x^6\right)}{63 x^6 \left(-1+x^3\right)}",1,"(-4*(-x + x^4)^(3/4)*(-3 - 8*x^3 + 53*x^6))/(63*x^6*(-1 + x^3))","A",1
469,1,37,37,0.8484453,"\int \frac{x \left(8 b+5 a x^3\right)}{\sqrt[4]{b+a x^3} \left(-b-a x^3+x^8\right)} \, dx","IntegrateAlgebraic[(x*(8*b + 5*a*x^3))/((b + a*x^3)^(1/4)*(-b - a*x^3 + x^8)),x]","2 \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^3}}{x^2}\right)-2 \tanh ^{-1}\left(\frac{x^2}{\sqrt[4]{b+a x^3}}\right)","2 \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^3}}{x^2}\right)-2 \tanh ^{-1}\left(\frac{x^2}{\sqrt[4]{b+a x^3}}\right)",1,"2*ArcTan[(b + a*x^3)^(1/4)/x^2] - 2*ArcTanh[x^2/(b + a*x^3)^(1/4)]","A",1
470,1,37,37,0.0437254,"\int \sqrt{x+\sqrt{1+x^2}} \, dx","IntegrateAlgebraic[Sqrt[x + Sqrt[1 + x^2]],x]","-\frac{1}{\sqrt{x+\sqrt{1+x^2}}}+\frac{1}{3} \left(x+\sqrt{1+x^2}\right)^{3/2}","-\frac{1}{\sqrt{x+\sqrt{1+x^2}}}+\frac{1}{3} \left(x+\sqrt{1+x^2}\right)^{3/2}",1,"-(1/Sqrt[x + Sqrt[1 + x^2]]) + (x + Sqrt[1 + x^2])^(3/2)/3","A",1
471,1,38,38,0.1680707,"\int \frac{a b-x^2}{\sqrt{x (-a+x) (-b+x)} \left(a b-(a+b+d) x+x^2\right)} \, dx","IntegrateAlgebraic[(a*b - x^2)/(Sqrt[x*(-a + x)*(-b + x)]*(a*b - (a + b + d)*x + x^2)),x]","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{a b x+(-a-b) x^2+x^3}}\right)}{\sqrt{d}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{a b x+(-a-b) x^2+x^3}}\right)}{\sqrt{d}}",1,"(2*ArcTanh[(Sqrt[d]*x)/Sqrt[a*b*x + (-a - b)*x^2 + x^3]])/Sqrt[d]","A",1
472,1,38,38,0.1854352,"\int \frac{a b-x^2}{\sqrt{x (-a+x) (-b+x)} \left(a b d-(1+a d+b d) x+d x^2\right)} \, dx","IntegrateAlgebraic[(a*b - x^2)/(Sqrt[x*(-a + x)*(-b + x)]*(a*b*d - (1 + a*d + b*d)*x + d*x^2)),x]","\frac{2 \tanh ^{-1}\left(\frac{x}{\sqrt{d} \sqrt{a b x+(-a-b) x^2+x^3}}\right)}{\sqrt{d}}","\frac{2 \tanh ^{-1}\left(\frac{x}{\sqrt{d} \sqrt{a b x+(-a-b) x^2+x^3}}\right)}{\sqrt{d}}",1,"(2*ArcTanh[x/(Sqrt[d]*Sqrt[a*b*x + (-a - b)*x^2 + x^3])])/Sqrt[d]","A",1
473,1,38,38,1.8619477,"\int \frac{-1+k x^2}{\left(1+k x^2\right) \sqrt{\left(1-x^2\right) \left(1-k^2 x^2\right)}} \, dx","IntegrateAlgebraic[(-1 + k*x^2)/((1 + k*x^2)*Sqrt[(1 - x^2)*(1 - k^2*x^2)]),x]","\frac{\tan ^{-1}\left(\frac{(1+k) x}{\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{-1-k}","\frac{\tan ^{-1}\left(\frac{(1+k) x}{\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{-1-k}",1,"ArcTan[((1 + k)*x)/Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4]]/(-1 - k)","A",1
474,1,38,38,1.5853621,"\int \frac{1+k x^2}{\left(-1+k x^2\right) \sqrt{\left(1-x^2\right) \left(1-k^2 x^2\right)}} \, dx","IntegrateAlgebraic[(1 + k*x^2)/((-1 + k*x^2)*Sqrt[(1 - x^2)*(1 - k^2*x^2)]),x]","\frac{\tan ^{-1}\left(\frac{(-1+k) x}{\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{1-k}","\frac{\tan ^{-1}\left(\frac{(-1+k) x}{\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{1-k}",1,"ArcTan[((-1 + k)*x)/Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4]]/(1 - k)","A",1
475,1,38,38,0.1651194,"\int \frac{1-2 k^2 x+k^2 x^2}{x \sqrt{(1-x) x \left(1-k^2 x\right)} \left(-1+k^2 x\right)} \, dx","IntegrateAlgebraic[(1 - 2*k^2*x + k^2*x^2)/(x*Sqrt[(1 - x)*x*(1 - k^2*x)]*(-1 + k^2*x)),x]","-\frac{2 \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{x \left(-1+k^2 x\right)}","-\frac{2 \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{x \left(-1+k^2 x\right)}",1,"(-2*Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3])/(x*(-1 + k^2*x))","A",1
476,1,38,38,0.0278413,"\int \frac{1}{x^7 \sqrt{-1+x^3}} \, dx","IntegrateAlgebraic[1/(x^7*Sqrt[-1 + x^3]),x]","\frac{\sqrt{-1+x^3} \left(2+3 x^3\right)}{12 x^6}+\frac{1}{4} \tan ^{-1}\left(\sqrt{-1+x^3}\right)","\frac{\sqrt{-1+x^3} \left(2+3 x^3\right)}{12 x^6}+\frac{1}{4} \tan ^{-1}\left(\sqrt{-1+x^3}\right)",1,"(Sqrt[-1 + x^3]*(2 + 3*x^3))/(12*x^6) + ArcTan[Sqrt[-1 + x^3]]/4","A",1
477,1,38,38,0.0441462,"\int \frac{1}{x^7 \sqrt{1+x^3}} \, dx","IntegrateAlgebraic[1/(x^7*Sqrt[1 + x^3]),x]","\frac{\sqrt{1+x^3} \left(-2+3 x^3\right)}{12 x^6}-\frac{1}{4} \tanh ^{-1}\left(\sqrt{1+x^3}\right)","\frac{\sqrt{1+x^3} \left(-2+3 x^3\right)}{12 x^6}-\frac{1}{4} \tanh ^{-1}\left(\sqrt{1+x^3}\right)",1,"(Sqrt[1 + x^3]*(-2 + 3*x^3))/(12*x^6) - ArcTanh[Sqrt[1 + x^3]]/4","A",1
478,1,38,38,0.0583045,"\int \frac{\sqrt{1+x^3}}{x^7} \, dx","IntegrateAlgebraic[Sqrt[1 + x^3]/x^7,x]","\frac{\left(-2-x^3\right) \sqrt{1+x^3}}{12 x^6}+\frac{1}{12} \tanh ^{-1}\left(\sqrt{1+x^3}\right)","\frac{\left(-2-x^3\right) \sqrt{1+x^3}}{12 x^6}+\frac{1}{12} \tanh ^{-1}\left(\sqrt{1+x^3}\right)",1,"((-2 - x^3)*Sqrt[1 + x^3])/(12*x^6) + ArcTanh[Sqrt[1 + x^3]]/12","A",1
479,1,38,38,0.0366254,"\int \frac{1+x^3}{x^7 \sqrt{-1+x^3}} \, dx","IntegrateAlgebraic[(1 + x^3)/(x^7*Sqrt[-1 + x^3]),x]","\frac{\sqrt{-1+x^3} \left(2+7 x^3\right)}{12 x^6}+\frac{7}{12} \tan ^{-1}\left(\sqrt{-1+x^3}\right)","\frac{\sqrt{-1+x^3} \left(2+7 x^3\right)}{12 x^6}+\frac{7}{12} \tan ^{-1}\left(\sqrt{-1+x^3}\right)",1,"(Sqrt[-1 + x^3]*(2 + 7*x^3))/(12*x^6) + (7*ArcTan[Sqrt[-1 + x^3]])/12","A",1
480,1,38,38,0.5286829,"\int \frac{\sqrt{-1+x^3} \left(2+x^3\right)}{x^2 \left(-2-4 x^2+2 x^3\right)} \, dx","IntegrateAlgebraic[(Sqrt[-1 + x^3]*(2 + x^3))/(x^2*(-2 - 4*x^2 + 2*x^3)),x]","\frac{\sqrt{-1+x^3}}{x}-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{-1+x^3}}\right)","\frac{\sqrt{-1+x^3}}{x}-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{-1+x^3}}\right)",1,"Sqrt[-1 + x^3]/x - Sqrt[2]*ArcTanh[(Sqrt[2]*x)/Sqrt[-1 + x^3]]","A",1
481,1,38,38,0.0265267,"\int x^5 \sqrt{b+a x^3} \, dx","IntegrateAlgebraic[x^5*Sqrt[b + a*x^3],x]","\frac{2 \sqrt{b+a x^3} \left(-2 b^2+a b x^3+3 a^2 x^6\right)}{45 a^2}","\frac{2 \sqrt{b+a x^3} \left(-2 b^2+a b x^3+3 a^2 x^6\right)}{45 a^2}",1,"(2*Sqrt[b + a*x^3]*(-2*b^2 + a*b*x^3 + 3*a^2*x^6))/(45*a^2)","A",1
482,1,28,38,0.269136,"\int \frac{\left(-3+x^4\right) \sqrt[3]{1+x^4} \left(1+x^3+x^4\right)}{x^8} \, dx","IntegrateAlgebraic[((-3 + x^4)*(1 + x^4)^(1/3)*(1 + x^3 + x^4))/x^8,x]","\frac{3 \left(1+x^4\right)^{4/3} \left(4+7 x^3+4 x^4\right)}{28 x^7}","\frac{3 \sqrt[3]{1+x^4} \left(4+7 x^3+8 x^4+7 x^7+4 x^8\right)}{28 x^7}",1,"(3*(1 + x^4)^(4/3)*(4 + 7*x^3 + 4*x^4))/(28*x^7)","A",1
483,1,28,38,0.2554068,"\int \frac{\left(-3+x^4\right) \left(1+x^4\right)^{2/3} \left(1+x^3+x^4\right)}{x^9} \, dx","IntegrateAlgebraic[((-3 + x^4)*(1 + x^4)^(2/3)*(1 + x^3 + x^4))/x^9,x]","\frac{3 \left(1+x^4\right)^{5/3} \left(5+8 x^3+5 x^4\right)}{40 x^8}","\frac{3 \left(1+x^4\right)^{2/3} \left(5+8 x^3+10 x^4+8 x^7+5 x^8\right)}{40 x^8}",1,"(3*(1 + x^4)^(5/3)*(5 + 8*x^3 + 5*x^4))/(40*x^8)","A",1
484,1,28,38,0.2573175,"\int \frac{\left(-1+x^4\right)^{2/3} \left(3+x^4\right) \left(-2-x^3+2 x^4\right)}{x^9} \, dx","IntegrateAlgebraic[((-1 + x^4)^(2/3)*(3 + x^4)*(-2 - x^3 + 2*x^4))/x^9,x]","\frac{3 \left(-1+x^4\right)^{5/3} \left(-5-4 x^3+5 x^4\right)}{20 x^8}","\frac{3 \left(-1+x^4\right)^{2/3} \left(5+4 x^3-10 x^4-4 x^7+5 x^8\right)}{20 x^8}",1,"(3*(-1 + x^4)^(5/3)*(-5 - 4*x^3 + 5*x^4))/(20*x^8)","A",1
485,1,28,38,2.5767252,"\int \frac{\left(-1+x^5\right)^{3/4} \left(4+x^5\right) \left(-1-x^4+x^5\right)}{x^{12}} \, dx","IntegrateAlgebraic[((-1 + x^5)^(3/4)*(4 + x^5)*(-1 - x^4 + x^5))/x^12,x]","\frac{4 \left(-1+x^5\right)^{7/4} \left(-7-11 x^4+7 x^5\right)}{77 x^{11}}","\frac{4 \left(-1+x^5\right)^{3/4} \left(7+11 x^4-14 x^5-11 x^9+7 x^{10}\right)}{77 x^{11}}",1,"(4*(-1 + x^5)^(7/4)*(-7 - 11*x^4 + 7*x^5))/(77*x^11)","A",1
486,1,28,38,0.8416005,"\int \frac{\left(1+x^5\right)^{2/3} \left(1+x^3+x^5\right) \left(-3+2 x^5\right)}{x^9} \, dx","IntegrateAlgebraic[((1 + x^5)^(2/3)*(1 + x^3 + x^5)*(-3 + 2*x^5))/x^9,x]","\frac{3 \left(1+x^5\right)^{5/3} \left(5+8 x^3+5 x^5\right)}{40 x^8}","\frac{3 \left(1+x^5\right)^{2/3} \left(5+8 x^3+10 x^5+8 x^8+5 x^{10}\right)}{40 x^8}",1,"(3*(1 + x^5)^(5/3)*(5 + 8*x^3 + 5*x^5))/(40*x^8)","A",1
487,1,28,38,0.8271652,"\int \frac{\left(-1+x^5\right)^{2/3} \left(-1+x^3+x^5\right) \left(3+2 x^5\right)}{x^9} \, dx","IntegrateAlgebraic[((-1 + x^5)^(2/3)*(-1 + x^3 + x^5)*(3 + 2*x^5))/x^9,x]","\frac{3 \left(-1+x^5\right)^{5/3} \left(-5+8 x^3+5 x^5\right)}{40 x^8}","\frac{3 \left(-1+x^5\right)^{2/3} \left(5-8 x^3-10 x^5+8 x^8+5 x^{10}\right)}{40 x^8}",1,"(3*(-1 + x^5)^(5/3)*(-5 + 8*x^3 + 5*x^5))/(40*x^8)","A",1
488,1,28,38,2.5971395,"\int \frac{\left(-4+x^5\right) \left(1+x^5\right)^{3/4} \left(2-x^4+2 x^5\right)}{x^{12}} \, dx","IntegrateAlgebraic[((-4 + x^5)*(1 + x^5)^(3/4)*(2 - x^4 + 2*x^5))/x^12,x]","\frac{4 \left(1+x^5\right)^{7/4} \left(14-11 x^4+14 x^5\right)}{77 x^{11}}","\frac{4 \left(1+x^5\right)^{3/4} \left(14-11 x^4+28 x^5-11 x^9+14 x^{10}\right)}{77 x^{11}}",1,"(4*(1 + x^5)^(7/4)*(14 - 11*x^4 + 14*x^5))/(77*x^11)","A",1
489,1,28,38,0.950274,"\int \frac{\left(1+x^5\right)^{2/3} \left(-3+2 x^5\right) \left(4+3 x^3+4 x^5\right)}{x^9} \, dx","IntegrateAlgebraic[((1 + x^5)^(2/3)*(-3 + 2*x^5)*(4 + 3*x^3 + 4*x^5))/x^9,x]","\frac{3 \left(1+x^5\right)^{5/3} \left(5+6 x^3+5 x^5\right)}{10 x^8}","\frac{3 \left(1+x^5\right)^{2/3} \left(5+6 x^3+10 x^5+6 x^8+5 x^{10}\right)}{10 x^8}",1,"(3*(1 + x^5)^(5/3)*(5 + 6*x^3 + 5*x^5))/(10*x^8)","A",1
490,1,38,38,0.0250104,"\int \frac{1}{x^{13} \sqrt{-1+x^6}} \, dx","IntegrateAlgebraic[1/(x^13*Sqrt[-1 + x^6]),x]","\frac{\sqrt{-1+x^6} \left(2+3 x^6\right)}{24 x^{12}}+\frac{1}{8} \tan ^{-1}\left(\sqrt{-1+x^6}\right)","\frac{\sqrt{-1+x^6} \left(2+3 x^6\right)}{24 x^{12}}+\frac{1}{8} \tan ^{-1}\left(\sqrt{-1+x^6}\right)",1,"(Sqrt[-1 + x^6]*(2 + 3*x^6))/(24*x^12) + ArcTan[Sqrt[-1 + x^6]]/8","A",1
491,1,38,38,0.0530435,"\int \frac{-1+x^6}{x^{13} \sqrt{1+x^6}} \, dx","IntegrateAlgebraic[(-1 + x^6)/(x^13*Sqrt[1 + x^6]),x]","\frac{\left(2-7 x^6\right) \sqrt{1+x^6}}{24 x^{12}}+\frac{7}{24} \tanh ^{-1}\left(\sqrt{1+x^6}\right)","\frac{\left(2-7 x^6\right) \sqrt{1+x^6}}{24 x^{12}}+\frac{7}{24} \tanh ^{-1}\left(\sqrt{1+x^6}\right)",1,"((2 - 7*x^6)*Sqrt[1 + x^6])/(24*x^12) + (7*ArcTanh[Sqrt[1 + x^6]])/24","A",1
492,1,38,38,0.050402,"\int \frac{\sqrt{1+x^6}}{x^{13}} \, dx","IntegrateAlgebraic[Sqrt[1 + x^6]/x^13,x]","\frac{\left(-2-x^6\right) \sqrt{1+x^6}}{24 x^{12}}+\frac{1}{24} \tanh ^{-1}\left(\sqrt{1+x^6}\right)","\frac{\left(-2-x^6\right) \sqrt{1+x^6}}{24 x^{12}}+\frac{1}{24} \tanh ^{-1}\left(\sqrt{1+x^6}\right)",1,"((-2 - x^6)*Sqrt[1 + x^6])/(24*x^12) + ArcTanh[Sqrt[1 + x^6]]/24","A",1
493,1,38,38,0.0392855,"\int \frac{1+x^6}{x^{13} \sqrt{-1+x^6}} \, dx","IntegrateAlgebraic[(1 + x^6)/(x^13*Sqrt[-1 + x^6]),x]","\frac{\sqrt{-1+x^6} \left(2+7 x^6\right)}{24 x^{12}}+\frac{7}{24} \tan ^{-1}\left(\sqrt{-1+x^6}\right)","\frac{\sqrt{-1+x^6} \left(2+7 x^6\right)}{24 x^{12}}+\frac{7}{24} \tan ^{-1}\left(\sqrt{-1+x^6}\right)",1,"(Sqrt[-1 + x^6]*(2 + 7*x^6))/(24*x^12) + (7*ArcTan[Sqrt[-1 + x^6]])/24","A",1
494,1,28,38,0.6950533,"\int \frac{\sqrt[3]{-1+x^6} \left(1+x^6\right) \left(-1+x^3+x^6\right)}{x^8} \, dx","IntegrateAlgebraic[((-1 + x^6)^(1/3)*(1 + x^6)*(-1 + x^3 + x^6))/x^8,x]","\frac{\left(-1+x^6\right)^{4/3} \left(-4+7 x^3+4 x^6\right)}{28 x^7}","\frac{\sqrt[3]{-1+x^6} \left(4-7 x^3-8 x^6+7 x^9+4 x^{12}\right)}{28 x^7}",1,"((-1 + x^6)^(4/3)*(-4 + 7*x^3 + 4*x^6))/(28*x^7)","A",1
495,1,28,38,2.6812858,"\int \frac{\left(-1+x^6\right)^{3/4} \left(2+x^6\right) \left(-1-x^4+x^6\right)}{x^{12}} \, dx","IntegrateAlgebraic[((-1 + x^6)^(3/4)*(2 + x^6)*(-1 - x^4 + x^6))/x^12,x]","\frac{2 \left(-1+x^6\right)^{7/4} \left(-7-11 x^4+7 x^6\right)}{77 x^{11}}","\frac{2 \left(-1+x^6\right)^{3/4} \left(7+11 x^4-14 x^6-11 x^{10}+7 x^{12}\right)}{77 x^{11}}",1,"(2*(-1 + x^6)^(7/4)*(-7 - 11*x^4 + 7*x^6))/(77*x^11)","A",1
496,1,28,38,2.3290118,"\int \frac{\left(-2+x^6\right) \left(1+x^6\right)^{3/4} \left(1-x^4+x^6\right)}{x^{12}} \, dx","IntegrateAlgebraic[((-2 + x^6)*(1 + x^6)^(3/4)*(1 - x^4 + x^6))/x^12,x]","\frac{2 \left(1+x^6\right)^{7/4} \left(7-11 x^4+7 x^6\right)}{77 x^{11}}","\frac{2 \left(1+x^6\right)^{3/4} \left(7-11 x^4+14 x^6-11 x^{10}+7 x^{12}\right)}{77 x^{11}}",1,"(2*(1 + x^6)^(7/4)*(7 - 11*x^4 + 7*x^6))/(77*x^11)","A",1
497,1,38,38,0.1483459,"\int \frac{-2 x+3 x^2}{\sqrt{5-4 x^2+4 x^3+x^4-2 x^5+x^6}} \, dx","IntegrateAlgebraic[(-2*x + 3*x^2)/Sqrt[5 - 4*x^2 + 4*x^3 + x^4 - 2*x^5 + x^6],x]","\log \left(2-x^2+x^3+\sqrt{5-4 x^2+4 x^3+x^4-2 x^5+x^6}\right)","\log \left(2-x^2+x^3+\sqrt{5-4 x^2+4 x^3+x^4-2 x^5+x^6}\right)",1,"Log[2 - x^2 + x^3 + Sqrt[5 - 4*x^2 + 4*x^3 + x^4 - 2*x^5 + x^6]]","A",1
498,1,38,38,0.0370791,"\int \frac{-2+5 x^6}{x \sqrt{-1+x^6} \left(2+x^6\right)} \, dx","IntegrateAlgebraic[(-2 + 5*x^6)/(x*Sqrt[-1 + x^6]*(2 + x^6)),x]","-\frac{1}{3} \tan ^{-1}\left(\sqrt{-1+x^6}\right)+\frac{2 \tan ^{-1}\left(\frac{\sqrt{-1+x^6}}{\sqrt{3}}\right)}{\sqrt{3}}","-\frac{1}{3} \tan ^{-1}\left(\sqrt{-1+x^6}\right)+\frac{2 \tan ^{-1}\left(\frac{\sqrt{-1+x^6}}{\sqrt{3}}\right)}{\sqrt{3}}",1,"-1/3*ArcTan[Sqrt[-1 + x^6]] + (2*ArcTan[Sqrt[-1 + x^6]/Sqrt[3]])/Sqrt[3]","A",1
499,1,38,38,0.0388554,"\int \frac{-1+10 x^6}{x \sqrt{-1+x^6} \left(-1+4 x^6\right)} \, dx","IntegrateAlgebraic[(-1 + 10*x^6)/(x*Sqrt[-1 + x^6]*(-1 + 4*x^6)),x]","\frac{1}{3} \tan ^{-1}\left(\sqrt{-1+x^6}\right)+\frac{\tan ^{-1}\left(\frac{2 \sqrt{-1+x^6}}{\sqrt{3}}\right)}{\sqrt{3}}","\frac{1}{3} \tan ^{-1}\left(\sqrt{-1+x^6}\right)+\frac{\tan ^{-1}\left(\frac{2 \sqrt{-1+x^6}}{\sqrt{3}}\right)}{\sqrt{3}}",1,"ArcTan[Sqrt[-1 + x^6]]/3 + ArcTan[(2*Sqrt[-1 + x^6])/Sqrt[3]]/Sqrt[3]","A",1
500,1,39,39,14.5994235,"\int \frac{2+x+x^2}{x^2 \left(1+x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[(2 + x + x^2)/(x^2*(1 + x^2)^(3/4)),x]","-\frac{2 \sqrt[4]{1+x^2}}{x}-\tan ^{-1}\left(\sqrt[4]{1+x^2}\right)-\tanh ^{-1}\left(\sqrt[4]{1+x^2}\right)","-\frac{2 \sqrt[4]{1+x^2}}{x}-\tan ^{-1}\left(\sqrt[4]{1+x^2}\right)-\tanh ^{-1}\left(\sqrt[4]{1+x^2}\right)",1,"(-2*(1 + x^2)^(1/4))/x - ArcTan[(1 + x^2)^(1/4)] - ArcTanh[(1 + x^2)^(1/4)]","A",1
501,1,39,39,0.0979019,"\int \frac{-3+2 x}{\sqrt[4]{-x+x^2} \left(1-x+x^3\right)} \, dx","IntegrateAlgebraic[(-3 + 2*x)/((-x + x^2)^(1/4)*(1 - x + x^3)),x]","2 \tan ^{-1}\left(\frac{\sqrt[4]{-x+x^2}}{x}\right)-2 \tanh ^{-1}\left(\frac{\left(-x+x^2\right)^{3/4}}{-1+x}\right)","2 \tan ^{-1}\left(\frac{\sqrt[4]{-x+x^2}}{x}\right)-2 \tanh ^{-1}\left(\frac{\left(-x+x^2\right)^{3/4}}{-1+x}\right)",1,"2*ArcTan[(-x + x^2)^(1/4)/x] - 2*ArcTanh[(-x + x^2)^(3/4)/(-1 + x)]","A",1
502,1,39,39,0.2483801,"\int \frac{\left(-1+x^2\right) \sqrt{x+x^3}}{\left(1+x^2\right) \left(1+x+x^2\right)^2} \, dx","IntegrateAlgebraic[((-1 + x^2)*Sqrt[x + x^3])/((1 + x^2)*(1 + x + x^2)^2),x]","\frac{\sqrt{x+x^3}}{1+x+x^2}-\tan ^{-1}\left(\frac{\sqrt{x+x^3}}{1+x^2}\right)","\frac{\sqrt{x+x^3}}{1+x+x^2}-\tan ^{-1}\left(\frac{\sqrt{x+x^3}}{1+x^2}\right)",1,"Sqrt[x + x^3]/(1 + x + x^2) - ArcTan[Sqrt[x + x^3]/(1 + x^2)]","A",1
503,1,39,39,0.3545291,"\int \frac{\left(-2+x^3\right) \sqrt{2+x^2+2 x^3}}{\left(1+x^3\right) \left(1+x^2+x^3\right)} \, dx","IntegrateAlgebraic[((-2 + x^3)*Sqrt[2 + x^2 + 2*x^3])/((1 + x^3)*(1 + x^2 + x^3)),x]","-2 \tan ^{-1}\left(\frac{x}{\sqrt{2+x^2+2 x^3}}\right)-2 \tanh ^{-1}\left(\frac{x}{\sqrt{2+x^2+2 x^3}}\right)","-2 \tan ^{-1}\left(\frac{x}{\sqrt{2+x^2+2 x^3}}\right)-2 \tanh ^{-1}\left(\frac{x}{\sqrt{2+x^2+2 x^3}}\right)",1,"-2*ArcTan[x/Sqrt[2 + x^2 + 2*x^3]] - 2*ArcTanh[x/Sqrt[2 + x^2 + 2*x^3]]","A",1
504,1,39,39,0.3062253,"\int \frac{b+a x^2}{\left(-b+c x+a x^2\right) \sqrt{-b x+a x^3}} \, dx","IntegrateAlgebraic[(b + a*x^2)/((-b + c*x + a*x^2)*Sqrt[-(b*x) + a*x^3]),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{-b x+a x^3}}{-b+a x^2}\right)}{\sqrt{c}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{-b x+a x^3}}{-b+a x^2}\right)}{\sqrt{c}}",1,"(-2*ArcTan[(Sqrt[c]*Sqrt[-(b*x) + a*x^3])/(-b + a*x^2)])/Sqrt[c]","A",1
505,1,39,39,0.1395673,"\int x \sqrt{-1+x^4} \, dx","IntegrateAlgebraic[x*Sqrt[-1 + x^4],x]","\frac{1}{4} x^2 \sqrt{-1+x^4}-\frac{1}{2} \tanh ^{-1}\left(\frac{\sqrt{-1+x^4}}{1+x^2}\right)","\frac{1}{4} x^2 \sqrt{-1+x^4}-\frac{1}{2} \tanh ^{-1}\left(\frac{\sqrt{-1+x^4}}{1+x^2}\right)",1,"(x^2*Sqrt[-1 + x^4])/4 - ArcTanh[Sqrt[-1 + x^4]/(1 + x^2)]/2","A",1
506,1,39,39,0.3403486,"\int \frac{\sqrt{-x+x^4}}{x^3} \, dx","IntegrateAlgebraic[Sqrt[-x + x^4]/x^3,x]","-\frac{2 \sqrt{-x+x^4}}{3 x^2}+\frac{2}{3} \tanh ^{-1}\left(\frac{x^2}{\sqrt{-x+x^4}}\right)","-\frac{2 \sqrt{-x+x^4}}{3 x^2}+\frac{2}{3} \tanh ^{-1}\left(\frac{x^2}{\sqrt{-x+x^4}}\right)",1,"(-2*Sqrt[-x + x^4])/(3*x^2) + (2*ArcTanh[x^2/Sqrt[-x + x^4]])/3","A",1
507,1,39,39,0.388945,"\int x^3 \sqrt{x+x^4} \, dx","IntegrateAlgebraic[x^3*Sqrt[x + x^4],x]","\frac{1}{12} \sqrt{x+x^4} \left(x+2 x^4\right)-\frac{1}{12} \tanh ^{-1}\left(\frac{x^2}{\sqrt{x+x^4}}\right)","\frac{1}{12} \sqrt{x+x^4} \left(x+2 x^4\right)-\frac{1}{12} \tanh ^{-1}\left(\frac{x^2}{\sqrt{x+x^4}}\right)",1,"(Sqrt[x + x^4]*(x + 2*x^4))/12 - ArcTanh[x^2/Sqrt[x + x^4]]/12","A",1
508,1,39,39,0.1660639,"\int \frac{1+x}{\sqrt{16+18 x+13 x^2+4 x^3+x^4}} \, dx","IntegrateAlgebraic[(1 + x)/Sqrt[16 + 18*x + 13*x^2 + 4*x^3 + x^4],x]","\frac{1}{2} \log \left(9+4 x+2 x^2+2 \sqrt{16+18 x+13 x^2+4 x^3+x^4}\right)","\frac{1}{2} \log \left(9+4 x+2 x^2+2 \sqrt{16+18 x+13 x^2+4 x^3+x^4}\right)",1,"Log[9 + 4*x + 2*x^2 + 2*Sqrt[16 + 18*x + 13*x^2 + 4*x^3 + x^4]]/2","A",1
509,1,39,39,1.0247749,"\int \frac{\sqrt{-1+x^5} \left(2+3 x^5\right)}{x^2 \left(-1-a x^2+x^5\right)} \, dx","IntegrateAlgebraic[(Sqrt[-1 + x^5]*(2 + 3*x^5))/(x^2*(-1 - a*x^2 + x^5)),x]","\frac{2 \sqrt{-1+x^5}}{x}-2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{-1+x^5}}\right)","\frac{2 \sqrt{-1+x^5}}{x}-2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{-1+x^5}}\right)",1,"(2*Sqrt[-1 + x^5])/x - 2*Sqrt[a]*ArcTanh[(Sqrt[a]*x)/Sqrt[-1 + x^5]]","A",1
510,1,39,39,0.2619087,"\int x \sqrt{-x+x^6} \, dx","IntegrateAlgebraic[x*Sqrt[-x + x^6],x]","\frac{1}{5} x^2 \sqrt{-x+x^6}-\frac{1}{5} \tanh ^{-1}\left(\frac{x^3}{\sqrt{-x+x^6}}\right)","\frac{1}{5} x^2 \sqrt{-x+x^6}-\frac{1}{5} \tanh ^{-1}\left(\frac{x^3}{\sqrt{-x+x^6}}\right)",1,"(x^2*Sqrt[-x + x^6])/5 - ArcTanh[x^3/Sqrt[-x + x^6]]/5","A",1
511,1,39,39,0.5880942,"\int \frac{\left(1+x^5\right) \left(-1+4 x^5\right)}{x \left(1-a x+x^5\right) \sqrt{x+x^6}} \, dx","IntegrateAlgebraic[((1 + x^5)*(-1 + 4*x^5))/(x*(1 - a*x + x^5)*Sqrt[x + x^6]),x]","\frac{2 \sqrt{x+x^6}}{x}-2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{x+x^6}}\right)","\frac{2 \sqrt{x+x^6}}{x}-2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{x+x^6}}\right)",1,"(2*Sqrt[x + x^6])/x - 2*Sqrt[a]*ArcTanh[(Sqrt[a]*x)/Sqrt[x + x^6]]","A",1
512,1,39,39,0.0618179,"\int \frac{1}{\sqrt{x-\sqrt{-1+x^2}}} \, dx","IntegrateAlgebraic[1/Sqrt[x - Sqrt[-1 + x^2]],x]","\frac{1}{3 \left(x-\sqrt{-1+x^2}\right)^{3/2}}+\sqrt{x-\sqrt{-1+x^2}}","\frac{1}{3 \left(x-\sqrt{-1+x^2}\right)^{3/2}}+\sqrt{x-\sqrt{-1+x^2}}",1,"1/(3*(x - Sqrt[-1 + x^2])^(3/2)) + Sqrt[x - Sqrt[-1 + x^2]]","A",1
513,1,40,40,0.0903619,"\int \frac{1+x^2}{\left(-1+2 x+x^2\right) \sqrt{-x-x^2+x^3}} \, dx","IntegrateAlgebraic[(1 + x^2)/((-1 + 2*x + x^2)*Sqrt[-x - x^2 + x^3]),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} \sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)}{\sqrt{3}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} \sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)}{\sqrt{3}}",1,"(-2*ArcTan[(Sqrt[3]*Sqrt[-x - x^2 + x^3])/(-1 - x + x^2)])/Sqrt[3]","A",1
514,1,40,40,0.2899089,"\int \frac{\left(2+x^3\right) \sqrt{-1+x^2+x^3}}{\left(-1+x^3\right)^2} \, dx","IntegrateAlgebraic[((2 + x^3)*Sqrt[-1 + x^2 + x^3])/(-1 + x^3)^2,x]","-\frac{x \sqrt{-1+x^2+x^3}}{-1+x^3}-\tanh ^{-1}\left(\frac{x}{\sqrt{-1+x^2+x^3}}\right)","-\frac{x \sqrt{-1+x^2+x^3}}{-1+x^3}-\tanh ^{-1}\left(\frac{x}{\sqrt{-1+x^2+x^3}}\right)",1,"-((x*Sqrt[-1 + x^2 + x^3])/(-1 + x^3)) - ArcTanh[x/Sqrt[-1 + x^2 + x^3]]","A",1
515,1,40,40,0.0937999,"\int \frac{2+x^2}{\left(-2+x^2\right) \sqrt{-2 x+2 x^2+x^3}} \, dx","IntegrateAlgebraic[(2 + x^2)/((-2 + x^2)*Sqrt[-2*x + 2*x^2 + x^3]),x]","-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{-2 x+2 x^2+x^3}}{-2+2 x+x^2}\right)","-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{-2 x+2 x^2+x^3}}{-2+2 x+x^2}\right)",1,"-(Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[-2*x + 2*x^2 + x^3])/(-2 + 2*x + x^2)])","A",1
516,1,40,40,0.1029908,"\int \frac{2+x}{(-1+x) \sqrt{-1+3 x+a x^2+x^3}} \, dx","IntegrateAlgebraic[(2 + x)/((-1 + x)*Sqrt[-1 + 3*x + a*x^2 + x^3]),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{-3-a} x}{\sqrt{-1+3 x+a x^2+x^3}}\right)}{\sqrt{-3-a}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{-3-a} x}{\sqrt{-1+3 x+a x^2+x^3}}\right)}{\sqrt{-3-a}}",1,"(-2*ArcTan[(Sqrt[-3 - a]*x)/Sqrt[-1 + 3*x + a*x^2 + x^3]])/Sqrt[-3 - a]","A",1
517,1,40,40,0.1078876,"\int \frac{-2+x}{(1+x) \sqrt{1+3 x+a x^2+x^3}} \, dx","IntegrateAlgebraic[(-2 + x)/((1 + x)*Sqrt[1 + 3*x + a*x^2 + x^3]),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{3-a} x}{\sqrt{1+3 x+a x^2+x^3}}\right)}{\sqrt{3-a}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{3-a} x}{\sqrt{1+3 x+a x^2+x^3}}\right)}{\sqrt{3-a}}",1,"(-2*ArcTan[(Sqrt[3 - a]*x)/Sqrt[1 + 3*x + a*x^2 + x^3]])/Sqrt[3 - a]","A",1
518,1,40,40,0.5235695,"\int \frac{x \left(3 a b-2 (a+b) x+x^2\right)}{\sqrt{x (-a+x) (-b+x)} \left(-a b d+(a+b) d x-d x^2+x^3\right)} \, dx","IntegrateAlgebraic[(x*(3*a*b - 2*(a + b)*x + x^2))/(Sqrt[x*(-a + x)*(-b + x)]*(-(a*b*d) + (a + b)*d*x - d*x^2 + x^3)),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a b x+(-a-b) x^2+x^3}}{x^2}\right)}{\sqrt{d}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a b x+(-a-b) x^2+x^3}}{x^2}\right)}{\sqrt{d}}",1,"(-2*ArcTanh[(Sqrt[d]*Sqrt[a*b*x + (-a - b)*x^2 + x^3])/x^2])/Sqrt[d]","A",1
519,1,40,40,0.4753278,"\int \frac{3 a b x-2 (a+b) x^2+x^3}{\sqrt{x (-a+x) (-b+x)} \left(-a b+(a+b) x-x^2+d x^3\right)} \, dx","IntegrateAlgebraic[(3*a*b*x - 2*(a + b)*x^2 + x^3)/(Sqrt[x*(-a + x)*(-b + x)]*(-(a*b) + (a + b)*x - x^2 + d*x^3)),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a b x+(-a-b) x^2+x^3}}{\sqrt{d} x^2}\right)}{\sqrt{d}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a b x+(-a-b) x^2+x^3}}{\sqrt{d} x^2}\right)}{\sqrt{d}}",1,"(-2*ArcTanh[Sqrt[a*b*x + (-a - b)*x^2 + x^3]/(Sqrt[d]*x^2)])/Sqrt[d]","A",1
520,1,29,40,0.2033695,"\int \frac{\left(-1+x^2\right) \sqrt{1+2 x^2+x^4}}{\left(1+x^2\right) \left(1+x^4\right)} \, dx","IntegrateAlgebraic[((-1 + x^2)*Sqrt[1 + 2*x^2 + x^4])/((1 + x^2)*(1 + x^4)),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+2 x^2+x^4}}\right)}{\sqrt{2}}","-\frac{\sqrt{\left(1+x^2\right)^2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{1+x^2}\right)}{\sqrt{2} \left(1+x^2\right)}",1,"-(ArcTanh[(Sqrt[2]*x)/Sqrt[1 + 2*x^2 + x^4]]/Sqrt[2])","A",1
521,1,40,40,0.1294835,"\int \frac{-1+x}{\sqrt{-7+4 x+14 x^2-12 x^3+x^4}} \, dx","IntegrateAlgebraic[(-1 + x)/Sqrt[-7 + 4*x + 14*x^2 - 12*x^3 + x^4],x]","\log (-1+x)-\log \left(-5+6 x-x^2+\sqrt{-7+4 x+14 x^2-12 x^3+x^4}\right)","\log (-1+x)-\log \left(-5+6 x-x^2+\sqrt{-7+4 x+14 x^2-12 x^3+x^4}\right)",1,"Log[-1 + x] - Log[-5 + 6*x - x^2 + Sqrt[-7 + 4*x + 14*x^2 - 12*x^3 + x^4]]","A",1
522,1,40,40,5.6333288,"\int \frac{1-2 k^2 x^2+k^2 x^4}{x^2 \sqrt{\left(1-x^2\right) \left(1-k^2 x^2\right)} \left(-1+k^2 x^2\right)} \, dx","IntegrateAlgebraic[(1 - 2*k^2*x^2 + k^2*x^4)/(x^2*Sqrt[(1 - x^2)*(1 - k^2*x^2)]*(-1 + k^2*x^2)),x]","-\frac{\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}{x \left(-1+k^2 x^2\right)}","-\frac{\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}{x \left(-1+k^2 x^2\right)}",1,"-(Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4]/(x*(-1 + k^2*x^2)))","A",1
523,1,40,40,0.0364225,"\int \frac{-2+x^6}{x \sqrt{-1+x^6} \left(2+x^6\right)} \, dx","IntegrateAlgebraic[(-2 + x^6)/(x*Sqrt[-1 + x^6]*(2 + x^6)),x]","-\frac{1}{3} \tan ^{-1}\left(\sqrt{-1+x^6}\right)+\frac{2 \tan ^{-1}\left(\frac{\sqrt{-1+x^6}}{\sqrt{3}}\right)}{3 \sqrt{3}}","-\frac{1}{3} \tan ^{-1}\left(\sqrt{-1+x^6}\right)+\frac{2 \tan ^{-1}\left(\frac{\sqrt{-1+x^6}}{\sqrt{3}}\right)}{3 \sqrt{3}}",1,"-1/3*ArcTan[Sqrt[-1 + x^6]] + (2*ArcTan[Sqrt[-1 + x^6]/Sqrt[3]])/(3*Sqrt[3])","A",1
524,1,40,40,0.0363972,"\int \frac{\left(2+x^3\right) \left(1+x^3+x^6\right)}{x \sqrt{1+x^3}} \, dx","IntegrateAlgebraic[((2 + x^3)*(1 + x^3 + x^6))/(x*Sqrt[1 + x^3]),x]","\frac{2}{45} \sqrt{1+x^3} \left(23+11 x^3+3 x^6\right)-\frac{4}{3} \tanh ^{-1}\left(\sqrt{1+x^3}\right)","\frac{2}{45} \sqrt{1+x^3} \left(23+11 x^3+3 x^6\right)-\frac{4}{3} \tanh ^{-1}\left(\sqrt{1+x^3}\right)",1,"(2*Sqrt[1 + x^3]*(23 + 11*x^3 + 3*x^6))/45 - (4*ArcTanh[Sqrt[1 + x^3]])/3","A",1
525,1,40,40,0.1731625,"\int \frac{4 x+3 x^2}{\sqrt{-5+4 x^2+2 x^3+4 x^4+4 x^5+x^6}} \, dx","IntegrateAlgebraic[(4*x + 3*x^2)/Sqrt[-5 + 4*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + x^6],x]","\log \left(1+2 x^2+x^3+\sqrt{-5+4 x^2+2 x^3+4 x^4+4 x^5+x^6}\right)","\log \left(1+2 x^2+x^3+\sqrt{-5+4 x^2+2 x^3+4 x^4+4 x^5+x^6}\right)",1,"Log[1 + 2*x^2 + x^3 + Sqrt[-5 + 4*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + x^6]]","A",1
526,1,40,40,3.15689,"\int \frac{x^2 \sqrt{q+p x^5} \left(-2 q+3 p x^5\right)}{b x^6+a \left(q+p x^5\right)^3} \, dx","IntegrateAlgebraic[(x^2*Sqrt[q + p*x^5]*(-2*q + 3*p*x^5))/(b*x^6 + a*(q + p*x^5)^3),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{b} x^3}{\sqrt{a} \left(q+p x^5\right)^{3/2}}\right)}{3 \sqrt{a} \sqrt{b}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{b} x^3}{\sqrt{a} \left(q+p x^5\right)^{3/2}}\right)}{3 \sqrt{a} \sqrt{b}}",1,"(-2*ArcTan[(Sqrt[b]*x^3)/(Sqrt[a]*(q + p*x^5)^(3/2))])/(3*Sqrt[a]*Sqrt[b])","A",1
527,1,41,41,0.0340893,"\int \frac{x^8}{\sqrt{-b+a x^3}} \, dx","IntegrateAlgebraic[x^8/Sqrt[-b + a*x^3],x]","\frac{2 \sqrt{-b+a x^3} \left(8 b^2+4 a b x^3+3 a^2 x^6\right)}{45 a^3}","\frac{2 \sqrt{-b+a x^3} \left(8 b^2+4 a b x^3+3 a^2 x^6\right)}{45 a^3}",1,"(2*Sqrt[-b + a*x^3]*(8*b^2 + 4*a*b*x^3 + 3*a^2*x^6))/(45*a^3)","A",1
528,1,41,41,0.0264704,"\int x^5 \sqrt{-b+a x^3} \, dx","IntegrateAlgebraic[x^5*Sqrt[-b + a*x^3],x]","\frac{2 \sqrt{-b+a x^3} \left(-2 b^2-a b x^3+3 a^2 x^6\right)}{45 a^2}","\frac{2 \sqrt{-b+a x^3} \left(-2 b^2-a b x^3+3 a^2 x^6\right)}{45 a^2}",1,"(2*Sqrt[-b + a*x^3]*(-2*b^2 - a*b*x^3 + 3*a^2*x^6))/(45*a^2)","A",1
529,1,41,41,0.5898882,"\int \frac{2 b+a x^3}{\sqrt{-b+a x^3} \left(-2 b-3 x^2+2 a x^3\right)} \, dx","IntegrateAlgebraic[(2*b + a*x^3)/(Sqrt[-b + a*x^3]*(-2*b - 3*x^2 + 2*a*x^3)),x]","\sqrt{\frac{2}{3}} \tanh ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x \sqrt{-b+a x^3}}{b-a x^3}\right)","\sqrt{\frac{2}{3}} \tanh ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x \sqrt{-b+a x^3}}{b-a x^3}\right)",1,"Sqrt[2/3]*ArcTanh[(Sqrt[3/2]*x*Sqrt[-b + a*x^3])/(b - a*x^3)]","A",1
530,1,41,41,0.7057959,"\int \frac{\left(-1+x^4\right) \sqrt{1+x^4} \left(1+x^2+x^4\right)}{x^4 \left(1-x^2+x^4\right)} \, dx","IntegrateAlgebraic[((-1 + x^4)*Sqrt[1 + x^4]*(1 + x^2 + x^4))/(x^4*(1 - x^2 + x^4)),x]","\frac{\sqrt{1+x^4} \left(1+6 x^2+x^4\right)}{3 x^3}-2 \tanh ^{-1}\left(\frac{x}{\sqrt{1+x^4}}\right)","\frac{\sqrt{1+x^4} \left(1+6 x^2+x^4\right)}{3 x^3}-2 \tanh ^{-1}\left(\frac{x}{\sqrt{1+x^4}}\right)",1,"(Sqrt[1 + x^4]*(1 + 6*x^2 + x^4))/(3*x^3) - 2*ArcTanh[x/Sqrt[1 + x^4]]","A",1
531,1,41,41,0.4122041,"\int \frac{4 c+3 b x+2 a x^2}{\sqrt[4]{c+b x+a x^2} \left(-c-b x-a x^2+x^4\right)} \, dx","IntegrateAlgebraic[(4*c + 3*b*x + 2*a*x^2)/((c + b*x + a*x^2)^(1/4)*(-c - b*x - a*x^2 + x^4)),x]","2 \tan ^{-1}\left(\frac{\sqrt[4]{c+b x+a x^2}}{x}\right)-2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{c+b x+a x^2}}\right)","2 \tan ^{-1}\left(\frac{\sqrt[4]{c+b x+a x^2}}{x}\right)-2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{c+b x+a x^2}}\right)",1,"2*ArcTan[(c + b*x + a*x^2)^(1/4)/x] - 2*ArcTanh[x/(c + b*x + a*x^2)^(1/4)]","A",1
532,1,41,41,0.0490959,"\int \frac{\left(2+x^3\right) \left(1+x^3+x^6\right)}{x^4 \sqrt{1+x^3}} \, dx","IntegrateAlgebraic[((2 + x^3)*(1 + x^3 + x^6))/(x^4*Sqrt[1 + x^3]),x]","\frac{2 \sqrt{1+x^3} \left(-3+7 x^3+x^6\right)}{9 x^3}-\frac{4}{3} \tanh ^{-1}\left(\sqrt{1+x^3}\right)","\frac{2 \sqrt{1+x^3} \left(-3+7 x^3+x^6\right)}{9 x^3}-\frac{4}{3} \tanh ^{-1}\left(\sqrt{1+x^3}\right)",1,"(2*Sqrt[1 + x^3]*(-3 + 7*x^3 + x^6))/(9*x^3) - (4*ArcTanh[Sqrt[1 + x^3]])/3","A",1
533,1,41,41,0.0573928,"\int \frac{\sqrt{-1+x^3} \left(-2+x^3+2 x^6\right)}{x^{10}} \, dx","IntegrateAlgebraic[(Sqrt[-1 + x^3]*(-2 + x^3 + 2*x^6))/x^10,x]","-\frac{2 \sqrt{-1+x^3} \left(-1+x^3+3 x^6\right)}{9 x^9}+\frac{2}{3} \tan ^{-1}\left(\sqrt{-1+x^3}\right)","-\frac{2 \sqrt{-1+x^3} \left(-1+x^3+3 x^6\right)}{9 x^9}+\frac{2}{3} \tan ^{-1}\left(\sqrt{-1+x^3}\right)",1,"(-2*Sqrt[-1 + x^3]*(-1 + x^3 + 3*x^6))/(9*x^9) + (2*ArcTan[Sqrt[-1 + x^3]])/3","A",1
534,1,41,41,0.0387823,"\int \frac{-4+13 x^6}{x \sqrt{-1+x^6} \left(-1+4 x^6\right)} \, dx","IntegrateAlgebraic[(-4 + 13*x^6)/(x*Sqrt[-1 + x^6]*(-1 + 4*x^6)),x]","\frac{4}{3} \tan ^{-1}\left(\sqrt{-1+x^6}\right)-\frac{\tan ^{-1}\left(\frac{2 \sqrt{-1+x^6}}{\sqrt{3}}\right)}{2 \sqrt{3}}","\frac{4}{3} \tan ^{-1}\left(\sqrt{-1+x^6}\right)-\frac{\tan ^{-1}\left(\frac{2 \sqrt{-1+x^6}}{\sqrt{3}}\right)}{2 \sqrt{3}}",1,"(4*ArcTan[Sqrt[-1 + x^6]])/3 - ArcTan[(2*Sqrt[-1 + x^6])/Sqrt[3]]/(2*Sqrt[3])","A",1
535,1,41,41,2.7223518,"\int \frac{-b+2 a x^3}{\left(b-x+a x^3\right) \sqrt[4]{b x^3+a x^6}} \, dx","IntegrateAlgebraic[(-b + 2*a*x^3)/((b - x + a*x^3)*(b*x^3 + a*x^6)^(1/4)),x]","-2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{b x^3+a x^6}}\right)-2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{b x^3+a x^6}}\right)","-2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{b x^3+a x^6}}\right)-2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{b x^3+a x^6}}\right)",1,"-2*ArcTan[x/(b*x^3 + a*x^6)^(1/4)] - 2*ArcTanh[x/(b*x^3 + a*x^6)^(1/4)]","A",1
536,1,41,41,19.8938205,"\int \frac{x^3 \left(b+2 a x^5\right)}{\sqrt[4]{b x+a x^6} \left(-1+b x^5+a x^{10}\right)} \, dx","IntegrateAlgebraic[(x^3*(b + 2*a*x^5))/((b*x + a*x^6)^(1/4)*(-1 + b*x^5 + a*x^10)),x]","\frac{2}{5} \tan ^{-1}\left(x \sqrt[4]{b x+a x^6}\right)-\frac{2}{5} \tanh ^{-1}\left(x \sqrt[4]{b x+a x^6}\right)","\frac{2}{5} \tan ^{-1}\left(x \sqrt[4]{b x+a x^6}\right)-\frac{2}{5} \tanh ^{-1}\left(x \sqrt[4]{b x+a x^6}\right)",1,"(2*ArcTan[x*(b*x + a*x^6)^(1/4)])/5 - (2*ArcTanh[x*(b*x + a*x^6)^(1/4)])/5","A",1
537,1,42,42,0.0254692,"\int \frac{\sqrt[4]{1+x^3}}{x} \, dx","IntegrateAlgebraic[(1 + x^3)^(1/4)/x,x]","\frac{4}{3} \sqrt[4]{1+x^3}-\frac{2}{3} \tan ^{-1}\left(\sqrt[4]{1+x^3}\right)-\frac{2}{3} \tanh ^{-1}\left(\sqrt[4]{1+x^3}\right)","\frac{4}{3} \sqrt[4]{1+x^3}-\frac{2}{3} \tan ^{-1}\left(\sqrt[4]{1+x^3}\right)-\frac{2}{3} \tanh ^{-1}\left(\sqrt[4]{1+x^3}\right)",1,"(4*(1 + x^3)^(1/4))/3 - (2*ArcTan[(1 + x^3)^(1/4)])/3 - (2*ArcTanh[(1 + x^3)^(1/4)])/3","A",1
538,1,42,42,6.6952135,"\int \frac{-2 a q+3 b p x^2+a p x^3}{\sqrt{q+p x^3} \left(b^2 c+d q+2 a b c x+a^2 c x^2+d p x^3\right)} \, dx","IntegrateAlgebraic[(-2*a*q + 3*b*p*x^2 + a*p*x^3)/(Sqrt[q + p*x^3]*(b^2*c + d*q + 2*a*b*c*x + a^2*c*x^2 + d*p*x^3)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{q+p x^3}}{\sqrt{c} (b+a x)}\right)}{\sqrt{c} \sqrt{d}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{q+p x^3}}{\sqrt{c} (b+a x)}\right)}{\sqrt{c} \sqrt{d}}",1,"(2*ArcTan[(Sqrt[d]*Sqrt[q + p*x^3])/(Sqrt[c]*(b + a*x))])/(Sqrt[c]*Sqrt[d])","A",1
539,1,44,42,0.1810502,"\int \frac{\left(-1+x^3\right) \sqrt{-1+x^6}}{x^7 \left(1+x^3\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)*Sqrt[-1 + x^6])/(x^7*(1 + x^3)),x]","\frac{\left(1-4 x^3\right) \sqrt{-1+x^6}}{6 x^6}-\tan ^{-1}\left(\frac{\sqrt{-1+x^6}}{-1+x^3}\right)","\frac{\left(1-4 x^3\right) \sqrt{-1+x^6}}{6 x^6}-\tan ^{-1}\left(\frac{1+x^3}{\sqrt{-1+x^6}}\right)",1,"((1 - 4*x^3)*Sqrt[-1 + x^6])/(6*x^6) - ArcTan[Sqrt[-1 + x^6]/(-1 + x^3)]","A",1
540,1,44,42,0.1626878,"\int \frac{\left(1+x^3\right) \sqrt{-1+x^6}}{x^7 \left(-1+x^3\right)} \, dx","IntegrateAlgebraic[((1 + x^3)*Sqrt[-1 + x^6])/(x^7*(-1 + x^3)),x]","\frac{\left(1+4 x^3\right) \sqrt{-1+x^6}}{6 x^6}-\tan ^{-1}\left(\frac{\sqrt{-1+x^6}}{-1+x^3}\right)","\frac{\left(1+4 x^3\right) \sqrt{-1+x^6}}{6 x^6}-\tan ^{-1}\left(\frac{1+x^3}{\sqrt{-1+x^6}}\right)",1,"((1 + 4*x^3)*Sqrt[-1 + x^6])/(6*x^6) - ArcTan[Sqrt[-1 + x^6]/(-1 + x^3)]","A",1
541,1,42,42,0.0251881,"\int \frac{\sqrt[4]{1+x^6}}{x} \, dx","IntegrateAlgebraic[(1 + x^6)^(1/4)/x,x]","\frac{2}{3} \sqrt[4]{1+x^6}-\frac{1}{3} \tan ^{-1}\left(\sqrt[4]{1+x^6}\right)-\frac{1}{3} \tanh ^{-1}\left(\sqrt[4]{1+x^6}\right)","\frac{2}{3} \sqrt[4]{1+x^6}-\frac{1}{3} \tan ^{-1}\left(\sqrt[4]{1+x^6}\right)-\frac{1}{3} \tanh ^{-1}\left(\sqrt[4]{1+x^6}\right)",1,"(2*(1 + x^6)^(1/4))/3 - ArcTan[(1 + x^6)^(1/4)]/3 - ArcTanh[(1 + x^6)^(1/4)]/3","A",1
542,1,42,42,0.2538049,"\int \frac{1+x^{12}}{x^4 \sqrt{-1+x^6}} \, dx","IntegrateAlgebraic[(1 + x^12)/(x^4*Sqrt[-1 + x^6]),x]","\frac{\sqrt{-1+x^6} \left(2+x^6\right)}{6 x^3}+\frac{1}{3} \tanh ^{-1}\left(\frac{1+x^3}{\sqrt{-1+x^6}}\right)","\frac{\sqrt{-1+x^6} \left(2+x^6\right)}{6 x^3}+\frac{1}{3} \tanh ^{-1}\left(\frac{1+x^3}{\sqrt{-1+x^6}}\right)",1,"(Sqrt[-1 + x^6]*(2 + x^6))/(6*x^3) + ArcTanh[(1 + x^3)/Sqrt[-1 + x^6]]/3","A",1
543,1,43,43,0.0498207,"\int \frac{1}{x^{10} \sqrt{1+x^3}} \, dx","IntegrateAlgebraic[1/(x^10*Sqrt[1 + x^3]),x]","\frac{\sqrt{1+x^3} \left(-8+10 x^3-15 x^6\right)}{72 x^9}+\frac{5}{24} \tanh ^{-1}\left(\sqrt{1+x^3}\right)","\frac{\sqrt{1+x^3} \left(-8+10 x^3-15 x^6\right)}{72 x^9}+\frac{5}{24} \tanh ^{-1}\left(\sqrt{1+x^3}\right)",1,"(Sqrt[1 + x^3]*(-8 + 10*x^3 - 15*x^6))/(72*x^9) + (5*ArcTanh[Sqrt[1 + x^3]])/24","A",1
544,1,43,43,0.2754404,"\int \frac{-1+x^3}{x^6 \sqrt[3]{x^2+x^3}} \, dx","IntegrateAlgebraic[(-1 + x^3)/(x^6*(x^2 + x^3)^(1/3)),x]","-\frac{3 \left(x^2+x^3\right)^{2/3} \left(-3080+3300 x-3600 x^2+10595 x^3-12714 x^4+19071 x^5\right)}{52360 x^7}","-\frac{3 \left(x^2+x^3\right)^{2/3} \left(-3080+3300 x-3600 x^2+10595 x^3-12714 x^4+19071 x^5\right)}{52360 x^7}",1,"(-3*(x^2 + x^3)^(2/3)*(-3080 + 3300*x - 3600*x^2 + 10595*x^3 - 12714*x^4 + 19071*x^5))/(52360*x^7)","A",1
545,1,43,43,0.5957781,"\int \frac{\left(-2+x^3\right) \sqrt{1+x^3} \left(2+x^2+2 x^3\right)}{x^4 \left(1+x^2+x^3\right)} \, dx","IntegrateAlgebraic[((-2 + x^3)*Sqrt[1 + x^3]*(2 + x^2 + 2*x^3))/(x^4*(1 + x^2 + x^3)),x]","\frac{2 \sqrt{1+x^3} \left(2-3 x^2+2 x^3\right)}{3 x^3}-2 \tan ^{-1}\left(\frac{x}{\sqrt{1+x^3}}\right)","\frac{2 \sqrt{1+x^3} \left(2-3 x^2+2 x^3\right)}{3 x^3}-2 \tan ^{-1}\left(\frac{x}{\sqrt{1+x^3}}\right)",1,"(2*Sqrt[1 + x^3]*(2 - 3*x^2 + 2*x^3))/(3*x^3) - 2*ArcTan[x/Sqrt[1 + x^3]]","A",1
546,1,43,43,0.3570753,"\int \frac{\left(-1+x^4\right) \sqrt{1+x^2+x^4}}{\left(1+x^4\right) \left(1-x^2+x^4\right)} \, dx","IntegrateAlgebraic[((-1 + x^4)*Sqrt[1 + x^2 + x^4])/((1 + x^4)*(1 - x^2 + x^4)),x]","\tanh ^{-1}\left(\frac{x}{\sqrt{1+x^2+x^4}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^2+x^4}}\right)","\tanh ^{-1}\left(\frac{x}{\sqrt{1+x^2+x^4}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^2+x^4}}\right)",1,"ArcTanh[x/Sqrt[1 + x^2 + x^4]] - Sqrt[2]*ArcTanh[(Sqrt[2]*x)/Sqrt[1 + x^2 + x^4]]","A",1
547,1,43,43,0.1385438,"\int \frac{\sqrt{1+x+x^2+x^4} \left(-2-x+2 x^4\right)}{\left(1+x+x^4\right)^2} \, dx","IntegrateAlgebraic[(Sqrt[1 + x + x^2 + x^4]*(-2 - x + 2*x^4))/(1 + x + x^4)^2,x]","-\frac{x \sqrt{1+x+x^2+x^4}}{1+x+x^4}-\tanh ^{-1}\left(\frac{x}{\sqrt{1+x+x^2+x^4}}\right)","-\frac{x \sqrt{1+x+x^2+x^4}}{1+x+x^4}-\tanh ^{-1}\left(\frac{x}{\sqrt{1+x+x^2+x^4}}\right)",1,"-((x*Sqrt[1 + x + x^2 + x^4])/(1 + x + x^4)) - ArcTanh[x/Sqrt[1 + x + x^2 + x^4]]","A",1
548,1,43,43,0.3423745,"\int \frac{\left(-1+x^4\right) \left(1+3 x^4\right)}{x \left(-1-a x+x^4\right) \sqrt{-x+x^5}} \, dx","IntegrateAlgebraic[((-1 + x^4)*(1 + 3*x^4))/(x*(-1 - a*x + x^4)*Sqrt[-x + x^5]),x]","\frac{2 \sqrt{-x+x^5}}{x}-2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{-x+x^5}}\right)","\frac{2 \sqrt{-x+x^5}}{x}-2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{-x+x^5}}\right)",1,"(2*Sqrt[-x + x^5])/x - 2*Sqrt[a]*ArcTanh[(Sqrt[a]*x)/Sqrt[-x + x^5]]","A",1
549,1,43,43,0.0329016,"\int \frac{1}{x^{19} \sqrt{-1+x^6}} \, dx","IntegrateAlgebraic[1/(x^19*Sqrt[-1 + x^6]),x]","\frac{\sqrt{-1+x^6} \left(8+10 x^6+15 x^{12}\right)}{144 x^{18}}+\frac{5}{48} \tan ^{-1}\left(\sqrt{-1+x^6}\right)","\frac{\sqrt{-1+x^6} \left(8+10 x^6+15 x^{12}\right)}{144 x^{18}}+\frac{5}{48} \tan ^{-1}\left(\sqrt{-1+x^6}\right)",1,"(Sqrt[-1 + x^6]*(8 + 10*x^6 + 15*x^12))/(144*x^18) + (5*ArcTan[Sqrt[-1 + x^6]])/48","A",1
550,1,43,43,0.171046,"\int \frac{x^{14}}{\sqrt{-1+x^6}} \, dx","IntegrateAlgebraic[x^14/Sqrt[-1 + x^6],x]","\frac{1}{24} \sqrt{-1+x^6} \left(3 x^3+2 x^9\right)+\frac{1}{8} \log \left(x^3+\sqrt{-1+x^6}\right)","\frac{1}{24} \sqrt{-1+x^6} \left(3 x^3+2 x^9\right)+\frac{1}{8} \log \left(x^3+\sqrt{-1+x^6}\right)",1,"(Sqrt[-1 + x^6]*(3*x^3 + 2*x^9))/24 + Log[x^3 + Sqrt[-1 + x^6]]/8","A",1
551,1,43,43,0.0352143,"\int \frac{\sqrt{-1+x^6}}{x^{19}} \, dx","IntegrateAlgebraic[Sqrt[-1 + x^6]/x^19,x]","\frac{\sqrt{-1+x^6} \left(-8+2 x^6+3 x^{12}\right)}{144 x^{18}}+\frac{1}{48} \tan ^{-1}\left(\sqrt{-1+x^6}\right)","\frac{\sqrt{-1+x^6} \left(-8+2 x^6+3 x^{12}\right)}{144 x^{18}}+\frac{1}{48} \tan ^{-1}\left(\sqrt{-1+x^6}\right)",1,"(Sqrt[-1 + x^6]*(-8 + 2*x^6 + 3*x^12))/(144*x^18) + ArcTan[Sqrt[-1 + x^6]]/48","A",1
552,1,43,43,0.1127792,"\int x^8 \sqrt{-1+x^6} \, dx","IntegrateAlgebraic[x^8*Sqrt[-1 + x^6],x]","\frac{1}{24} \sqrt{-1+x^6} \left(-x^3+2 x^9\right)-\frac{1}{24} \log \left(x^3+\sqrt{-1+x^6}\right)","\frac{1}{24} \sqrt{-1+x^6} \left(-x^3+2 x^9\right)-\frac{1}{24} \log \left(x^3+\sqrt{-1+x^6}\right)",1,"(Sqrt[-1 + x^6]*(-x^3 + 2*x^9))/24 - Log[x^3 + Sqrt[-1 + x^6]]/24","A",1
553,1,43,43,0.6982714,"\int \frac{-1+x^6}{\sqrt{1+x^4} \left(1+x^6\right)} \, dx","IntegrateAlgebraic[(-1 + x^6)/(Sqrt[1 + x^4]*(1 + x^6)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{3 \sqrt{2}}-\frac{2}{3} \tanh ^{-1}\left(\frac{x}{\sqrt{1+x^4}}\right)","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{3 \sqrt{2}}-\frac{2}{3} \tanh ^{-1}\left(\frac{x}{\sqrt{1+x^4}}\right)",1,"-1/3*ArcTan[(Sqrt[2]*x)/Sqrt[1 + x^4]]/Sqrt[2] - (2*ArcTanh[x/Sqrt[1 + x^4]])/3","A",1
554,1,43,43,0.1602216,"\int \frac{x^{14}}{\sqrt{1+x^6}} \, dx","IntegrateAlgebraic[x^14/Sqrt[1 + x^6],x]","\frac{1}{24} \sqrt{1+x^6} \left(-3 x^3+2 x^9\right)+\frac{1}{8} \log \left(x^3+\sqrt{1+x^6}\right)","\frac{1}{24} \sqrt{1+x^6} \left(-3 x^3+2 x^9\right)+\frac{1}{8} \log \left(x^3+\sqrt{1+x^6}\right)",1,"(Sqrt[1 + x^6]*(-3*x^3 + 2*x^9))/24 + Log[x^3 + Sqrt[1 + x^6]]/8","A",1
555,1,43,43,0.0595837,"\int \frac{-1+x^6}{x^{19} \sqrt{1+x^6}} \, dx","IntegrateAlgebraic[(-1 + x^6)/(x^19*Sqrt[1 + x^6]),x]","\frac{\sqrt{1+x^6} \left(8-22 x^6+33 x^{12}\right)}{144 x^{18}}-\frac{11}{48} \tanh ^{-1}\left(\sqrt{1+x^6}\right)","\frac{\sqrt{1+x^6} \left(8-22 x^6+33 x^{12}\right)}{144 x^{18}}-\frac{11}{48} \tanh ^{-1}\left(\sqrt{1+x^6}\right)",1,"(Sqrt[1 + x^6]*(8 - 22*x^6 + 33*x^12))/(144*x^18) - (11*ArcTanh[Sqrt[1 + x^6]])/48","A",1
556,1,43,43,0.6938679,"\int \frac{1+x^6}{\sqrt{1+x^4} \left(1-x^6\right)} \, dx","IntegrateAlgebraic[(1 + x^6)/(Sqrt[1 + x^4]*(1 - x^6)),x]","\frac{2}{3} \tan ^{-1}\left(\frac{x}{\sqrt{1+x^4}}\right)+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{3 \sqrt{2}}","\frac{2}{3} \tan ^{-1}\left(\frac{x}{\sqrt{1+x^4}}\right)+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{3 \sqrt{2}}",1,"(2*ArcTan[x/Sqrt[1 + x^4]])/3 + ArcTanh[(Sqrt[2]*x)/Sqrt[1 + x^4]]/(3*Sqrt[2])","A",1
557,1,43,43,0.7038591,"\int \frac{1+x^6}{\sqrt{1+x^4} \left(-1+x^6\right)} \, dx","IntegrateAlgebraic[(1 + x^6)/(Sqrt[1 + x^4]*(-1 + x^6)),x]","-\frac{2}{3} \tan ^{-1}\left(\frac{x}{\sqrt{1+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{3 \sqrt{2}}","-\frac{2}{3} \tan ^{-1}\left(\frac{x}{\sqrt{1+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{3 \sqrt{2}}",1,"(-2*ArcTan[x/Sqrt[1 + x^4]])/3 - ArcTanh[(Sqrt[2]*x)/Sqrt[1 + x^4]]/(3*Sqrt[2])","A",1
558,1,43,43,0.5660066,"\int \frac{\left(-1+x^5\right) \left(1+4 x^5\right)}{x \left(-1-a x+x^5\right) \sqrt{-x+x^6}} \, dx","IntegrateAlgebraic[((-1 + x^5)*(1 + 4*x^5))/(x*(-1 - a*x + x^5)*Sqrt[-x + x^6]),x]","\frac{2 \sqrt{-x+x^6}}{x}-2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{-x+x^6}}\right)","\frac{2 \sqrt{-x+x^6}}{x}-2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{-x+x^6}}\right)",1,"(2*Sqrt[-x + x^6])/x - 2*Sqrt[a]*ArcTanh[(Sqrt[a]*x)/Sqrt[-x + x^6]]","A",1
559,1,43,43,0.0468079,"\int \frac{\left(2+x^3\right) \left(1+x^3+x^6\right)}{x^7 \sqrt{1+x^3}} \, dx","IntegrateAlgebraic[((2 + x^3)*(1 + x^3 + x^6))/(x^7*Sqrt[1 + x^3]),x]","\frac{\sqrt{1+x^3} \left(-2-3 x^3+4 x^6\right)}{6 x^6}-\frac{3}{2} \tanh ^{-1}\left(\sqrt{1+x^3}\right)","\frac{\sqrt{1+x^3} \left(-2-3 x^3+4 x^6\right)}{6 x^6}-\frac{3}{2} \tanh ^{-1}\left(\sqrt{1+x^3}\right)",1,"(Sqrt[1 + x^3]*(-2 - 3*x^3 + 4*x^6))/(6*x^6) - (3*ArcTanh[Sqrt[1 + x^3]])/2","A",1
560,1,43,43,2.6983739,"\int \frac{-x+4 x^6}{\left(1+x^5\right) \left(a-x+a x^5\right) \sqrt{x+x^6}} \, dx","IntegrateAlgebraic[(-x + 4*x^6)/((1 + x^5)*(a - x + a*x^5)*Sqrt[x + x^6]),x]","\frac{2 \sqrt{x+x^6}}{1+x^5}-2 \sqrt{a} \tanh ^{-1}\left(\frac{x}{\sqrt{a} \sqrt{x+x^6}}\right)","\frac{2 \sqrt{x+x^6}}{1+x^5}-2 \sqrt{a} \tanh ^{-1}\left(\frac{x}{\sqrt{a} \sqrt{x+x^6}}\right)",1,"(2*Sqrt[x + x^6])/(1 + x^5) - 2*Sqrt[a]*ArcTanh[x/(Sqrt[a]*Sqrt[x + x^6])]","A",1
561,1,43,43,15.6099672,"\int \frac{2+x^3+x^6}{x \sqrt[4]{1+x^6} \left(-4+5 x^3-4 x^6+x^9\right)} \, dx","IntegrateAlgebraic[(2 + x^3 + x^6)/(x*(1 + x^6)^(1/4)*(-4 + 5*x^3 - 4*x^6 + x^9)),x]","\frac{1}{3} \tan ^{-1}\left(\frac{1-x^3}{\sqrt[4]{1+x^6}}\right)-\frac{1}{3} \tanh ^{-1}\left(\frac{-1+x^3}{\sqrt[4]{1+x^6}}\right)","\frac{1}{3} \tan ^{-1}\left(\frac{1-x^3}{\sqrt[4]{1+x^6}}\right)-\frac{1}{3} \tanh ^{-1}\left(\frac{-1+x^3}{\sqrt[4]{1+x^6}}\right)",1,"ArcTan[(1 - x^3)/(1 + x^6)^(1/4)]/3 - ArcTanh[(-1 + x^3)/(1 + x^6)^(1/4)]/3","A",1
562,1,43,43,19.0049963,"\int \sqrt[5]{243-5265 x+47250 x^2-225810 x^3+615255 x^4-954733 x^5+820340 x^6-401440 x^7+112000 x^8-16640 x^9+1024 x^{10}} \, dx","IntegrateAlgebraic[(243 - 5265*x + 47250*x^2 - 225810*x^3 + 615255*x^4 - 954733*x^5 + 820340*x^6 - 401440*x^7 + 112000*x^8 - 16640*x^9 + 1024*x^10)^(1/5),x]","\frac{x \sqrt[5]{\left(3-13 x+4 x^2\right)^5} \left(18-39 x+8 x^2\right)}{6 \left(3-13 x+4 x^2\right)}","\frac{x \sqrt[5]{\left(3-13 x+4 x^2\right)^5} \left(18-39 x+8 x^2\right)}{6 \left(3-13 x+4 x^2\right)}",1,"(x*((3 - 13*x + 4*x^2)^5)^(1/5)*(18 - 39*x + 8*x^2))/(6*(3 - 13*x + 4*x^2))","A",1
563,1,43,43,19.9693806,"\int \frac{-b x^3+2 a x^8}{\sqrt[4]{-b x+a x^6} \left(-1-b x^5+a x^{10}\right)} \, dx","IntegrateAlgebraic[(-(b*x^3) + 2*a*x^8)/((-(b*x) + a*x^6)^(1/4)*(-1 - b*x^5 + a*x^10)),x]","\frac{2}{5} \tan ^{-1}\left(x \sqrt[4]{-b x+a x^6}\right)-\frac{2}{5} \tanh ^{-1}\left(x \sqrt[4]{-b x+a x^6}\right)","\frac{2}{5} \tan ^{-1}\left(x \sqrt[4]{-b x+a x^6}\right)-\frac{2}{5} \tanh ^{-1}\left(x \sqrt[4]{-b x+a x^6}\right)",1,"(2*ArcTan[x*(-(b*x) + a*x^6)^(1/4)])/5 - (2*ArcTanh[x*(-(b*x) + a*x^6)^(1/4)])/5","A",1
564,1,43,43,18.4689139,"\int \frac{x^5 \left(-7 b+10 a x^3\right)}{\sqrt[4]{-b x^3+a x^6} \left(-1-b x^7+a x^{10}\right)} \, dx","IntegrateAlgebraic[(x^5*(-7*b + 10*a*x^3))/((-(b*x^3) + a*x^6)^(1/4)*(-1 - b*x^7 + a*x^10)),x]","2 \tan ^{-1}\left(x \sqrt[4]{-b x^3+a x^6}\right)-2 \tanh ^{-1}\left(x \sqrt[4]{-b x^3+a x^6}\right)","2 \tan ^{-1}\left(x \sqrt[4]{-b x^3+a x^6}\right)-2 \tanh ^{-1}\left(x \sqrt[4]{-b x^3+a x^6}\right)",1,"2*ArcTan[x*(-(b*x^3) + a*x^6)^(1/4)] - 2*ArcTanh[x*(-(b*x^3) + a*x^6)^(1/4)]","A",1
565,1,43,43,2.8867592,"\int \frac{1}{x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}} \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}} \, dx","IntegrateAlgebraic[1/(x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]]),x]","\frac{2 b \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{a x}","\frac{2 b \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{a x}",1,"(2*b*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]])/(a*x)","A",1
566,1,44,44,0.3492875,"\int \frac{a b-2 a x+x^2}{\sqrt{x (-a+x) (-b+x)} \left(a d-(b+d) x+x^2\right)} \, dx","IntegrateAlgebraic[(a*b - 2*a*x + x^2)/(Sqrt[x*(-a + x)*(-b + x)]*(a*d - (b + d)*x + x^2)),x]","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a b x+(-a-b) x^2+x^3}}{\sqrt{d} (a-x)}\right)}{\sqrt{d}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a b x+(-a-b) x^2+x^3}}{\sqrt{d} (a-x)}\right)}{\sqrt{d}}",1,"(2*ArcTanh[Sqrt[a*b*x + (-a - b)*x^2 + x^3]/(Sqrt[d]*(a - x))])/Sqrt[d]","A",1
567,1,44,44,0.3223643,"\int \frac{a b-2 a x+x^2}{\sqrt{x (-a+x) (-b+x)} \left(a-(1+b d) x+d x^2\right)} \, dx","IntegrateAlgebraic[(a*b - 2*a*x + x^2)/(Sqrt[x*(-a + x)*(-b + x)]*(a - (1 + b*d)*x + d*x^2)),x]","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a b x+(-a-b) x^2+x^3}}{a-x}\right)}{\sqrt{d}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a b x+(-a-b) x^2+x^3}}{a-x}\right)}{\sqrt{d}}",1,"(2*ArcTanh[(Sqrt[d]*Sqrt[a*b*x + (-a - b)*x^2 + x^3])/(a - x)])/Sqrt[d]","A",1
568,1,44,44,0.285398,"\int \frac{\left(-2+x^3\right) \sqrt{1-x^2+x^3}}{\left(1+x^3\right)^2} \, dx","IntegrateAlgebraic[((-2 + x^3)*Sqrt[1 - x^2 + x^3])/(1 + x^3)^2,x]","-\frac{x \sqrt{1-x^2+x^3}}{1+x^3}-\tan ^{-1}\left(\frac{x}{\sqrt{1-x^2+x^3}}\right)","-\frac{x \sqrt{1-x^2+x^3}}{1+x^3}-\tan ^{-1}\left(\frac{x}{\sqrt{1-x^2+x^3}}\right)",1,"-((x*Sqrt[1 - x^2 + x^3])/(1 + x^3)) - ArcTan[x/Sqrt[1 - x^2 + x^3]]","A",1
569,1,44,44,0.6200446,"\int \frac{a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt{x (-a+x) (-b+x)} \left(-a^3+\left(3 a^2+b d\right) x-(3 a+d) x^2+x^3\right)} \, dx","IntegrateAlgebraic[(a^2*b - a*(2*a - b)*x - (-a + 2*b)*x^2 + x^3)/(Sqrt[x*(-a + x)*(-b + x)]*(-a^3 + (3*a^2 + b*d)*x - (3*a + d)*x^2 + x^3)),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a b x+(-a-b) x^2+x^3}}{(a-x)^2}\right)}{\sqrt{d}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a b x+(-a-b) x^2+x^3}}{(a-x)^2}\right)}{\sqrt{d}}",1,"(-2*ArcTanh[(Sqrt[d]*Sqrt[a*b*x + (-a - b)*x^2 + x^3])/(a - x)^2])/Sqrt[d]","A",1
570,1,44,44,0.6547863,"\int \frac{a^2 b-a (2 a-b) x-(-a+2 b) x^2+x^3}{\sqrt{x (-a+x) (-b+x)} \left(-a^3 d+\left(b+3 a^2 d\right) x-(1+3 a d) x^2+d x^3\right)} \, dx","IntegrateAlgebraic[(a^2*b - a*(2*a - b)*x - (-a + 2*b)*x^2 + x^3)/(Sqrt[x*(-a + x)*(-b + x)]*(-(a^3*d) + (b + 3*a^2*d)*x - (1 + 3*a*d)*x^2 + d*x^3)),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a b x+(-a-b) x^2+x^3}}{\sqrt{d} (a-x)^2}\right)}{\sqrt{d}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a b x+(-a-b) x^2+x^3}}{\sqrt{d} (a-x)^2}\right)}{\sqrt{d}}",1,"(-2*ArcTanh[Sqrt[a*b*x + (-a - b)*x^2 + x^3]/(Sqrt[d]*(a - x)^2)])/Sqrt[d]","A",1
571,1,44,44,0.3988482,"\int \frac{-3 x+2 x^2}{\left(-2+2 x+x^3\right) \sqrt{-2 x+2 x^2+3 x^4}} \, dx","IntegrateAlgebraic[(-3*x + 2*x^2)/((-2 + 2*x + x^3)*Sqrt[-2*x + 2*x^2 + 3*x^4]),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{-2 x+2 x^2+3 x^4}}{-2+2 x+3 x^3}\right)}{\sqrt{2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{-2 x+2 x^2+3 x^4}}{-2+2 x+3 x^3}\right)}{\sqrt{2}}",1,"ArcTanh[(Sqrt[2]*x*Sqrt[-2*x + 2*x^2 + 3*x^4])/(-2 + 2*x + 3*x^3)]/Sqrt[2]","A",1
572,1,44,44,18.3914439,"\int \frac{4 a q x-3 b p x^2+a p x^4}{\sqrt{q+p x^3} \left(b^2 c+d q+2 a b c x^2+d p x^3+a^2 c x^4\right)} \, dx","IntegrateAlgebraic[(4*a*q*x - 3*b*p*x^2 + a*p*x^4)/(Sqrt[q + p*x^3]*(b^2*c + d*q + 2*a*b*c*x^2 + d*p*x^3 + a^2*c*x^4)),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{q+p x^3}}{\sqrt{c} \left(b+a x^2\right)}\right)}{\sqrt{c} \sqrt{d}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{q+p x^3}}{\sqrt{c} \left(b+a x^2\right)}\right)}{\sqrt{c} \sqrt{d}}",1,"(-2*ArcTan[(Sqrt[d]*Sqrt[q + p*x^3])/(Sqrt[c]*(b + a*x^2))])/(Sqrt[c]*Sqrt[d])","A",1
573,1,44,44,0.3050491,"\int \frac{1+x^{12}}{x^{10} \sqrt{-1+x^6}} \, dx","IntegrateAlgebraic[(1 + x^12)/(x^10*Sqrt[-1 + x^6]),x]","\frac{\sqrt{-1+x^6} \left(1+2 x^6\right)}{9 x^9}+\frac{2}{3} \tanh ^{-1}\left(\frac{1+x^3}{\sqrt{-1+x^6}}\right)","\frac{\sqrt{-1+x^6} \left(1+2 x^6\right)}{9 x^9}+\frac{2}{3} \tanh ^{-1}\left(\frac{1+x^3}{\sqrt{-1+x^6}}\right)",1,"(Sqrt[-1 + x^6]*(1 + 2*x^6))/(9*x^9) + (2*ArcTanh[(1 + x^3)/Sqrt[-1 + x^6]])/3","A",1
574,1,44,44,20.5612838,"\int \frac{-1+x^2}{\left(1+x^2\right) \sqrt[10]{1+5 x^4+10 x^8+10 x^{12}+5 x^{16}+x^{20}}} \, dx","IntegrateAlgebraic[(-1 + x^2)/((1 + x^2)*(1 + 5*x^4 + 10*x^8 + 10*x^12 + 5*x^16 + x^20)^(1/10)),x]","-\frac{\left(\left(1+x^4\right)^5\right)^{9/10} \tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{\sqrt{2} \left(1+x^4\right)^{9/2}}","-\frac{\left(\left(1+x^4\right)^5\right)^{9/10} \tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{\sqrt{2} \left(1+x^4\right)^{9/2}}",1,"-((((1 + x^4)^5)^(9/10)*ArcTan[(Sqrt[2]*x)/Sqrt[1 + x^4]])/(Sqrt[2]*(1 + x^4)^(9/2)))","A",1
575,1,44,44,0.1237252,"\int \frac{\sqrt{b+\sqrt{b^2+a x^2}}}{b^2+a x^2} \, dx","IntegrateAlgebraic[Sqrt[b + Sqrt[b^2 + a*x^2]]/(b^2 + a*x^2),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{\sqrt{a} \sqrt{b}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{\sqrt{a} \sqrt{b}}",1,"(2*ArcTan[(Sqrt[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/(Sqrt[a]*Sqrt[b])","A",1
576,1,44,44,0.1825449,"\int \frac{\sqrt{x^2+\sqrt{1+x^4}}}{\sqrt{1+x^4}} \, dx","IntegrateAlgebraic[Sqrt[x^2 + Sqrt[1 + x^4]]/Sqrt[1 + x^4],x]","\frac{\log \left(x^2+\sqrt{1+x^4}+\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}\right)}{\sqrt{2}}","\frac{\log \left(x^2+\sqrt{1+x^4}+\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}\right)}{\sqrt{2}}",1,"Log[x^2 + Sqrt[1 + x^4] + Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]]]/Sqrt[2]","A",1
577,1,45,45,0.0464368,"\int \frac{1}{x^3 \left(1+x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x^3*(1 + x^2)^(3/4)),x]","-\frac{\sqrt[4]{1+x^2}}{2 x^2}+\frac{3}{4} \tan ^{-1}\left(\sqrt[4]{1+x^2}\right)+\frac{3}{4} \tanh ^{-1}\left(\sqrt[4]{1+x^2}\right)","-\frac{\sqrt[4]{1+x^2}}{2 x^2}+\frac{3}{4} \tan ^{-1}\left(\sqrt[4]{1+x^2}\right)+\frac{3}{4} \tanh ^{-1}\left(\sqrt[4]{1+x^2}\right)",1,"-1/2*(1 + x^2)^(1/4)/x^2 + (3*ArcTan[(1 + x^2)^(1/4)])/4 + (3*ArcTanh[(1 + x^2)^(1/4)])/4","A",1
578,1,45,45,0.0483965,"\int \frac{1}{x^4 \sqrt[4]{1+x^3}} \, dx","IntegrateAlgebraic[1/(x^4*(1 + x^3)^(1/4)),x]","-\frac{\left(1+x^3\right)^{3/4}}{3 x^3}-\frac{1}{6} \tan ^{-1}\left(\sqrt[4]{1+x^3}\right)+\frac{1}{6} \tanh ^{-1}\left(\sqrt[4]{1+x^3}\right)","-\frac{\left(1+x^3\right)^{3/4}}{3 x^3}-\frac{1}{6} \tan ^{-1}\left(\sqrt[4]{1+x^3}\right)+\frac{1}{6} \tanh ^{-1}\left(\sqrt[4]{1+x^3}\right)",1,"-1/3*(1 + x^3)^(3/4)/x^3 - ArcTan[(1 + x^3)^(1/4)]/6 + ArcTanh[(1 + x^3)^(1/4)]/6","A",1
579,1,45,45,0.0346927,"\int \frac{\sqrt[4]{1+x^3}}{x^4} \, dx","IntegrateAlgebraic[(1 + x^3)^(1/4)/x^4,x]","-\frac{\sqrt[4]{1+x^3}}{3 x^3}-\frac{1}{6} \tan ^{-1}\left(\sqrt[4]{1+x^3}\right)-\frac{1}{6} \tanh ^{-1}\left(\sqrt[4]{1+x^3}\right)","-\frac{\sqrt[4]{1+x^3}}{3 x^3}-\frac{1}{6} \tan ^{-1}\left(\sqrt[4]{1+x^3}\right)-\frac{1}{6} \tanh ^{-1}\left(\sqrt[4]{1+x^3}\right)",1,"-1/3*(1 + x^3)^(1/4)/x^3 - ArcTan[(1 + x^3)^(1/4)]/6 - ArcTanh[(1 + x^3)^(1/4)]/6","A",1
580,1,45,45,0.3834745,"\int x^3 \sqrt{-x+x^4} \, dx","IntegrateAlgebraic[x^3*Sqrt[-x + x^4],x]","\frac{1}{12} \sqrt{-x+x^4} \left(-x+2 x^4\right)-\frac{1}{12} \tanh ^{-1}\left(\frac{x^2}{\sqrt{-x+x^4}}\right)","\frac{1}{12} \sqrt{-x+x^4} \left(-x+2 x^4\right)-\frac{1}{12} \tanh ^{-1}\left(\frac{x^2}{\sqrt{-x+x^4}}\right)",1,"(Sqrt[-x + x^4]*(-x + 2*x^4))/12 - ArcTanh[x^2/Sqrt[-x + x^4]]/12","A",1
581,1,45,45,0.3350201,"\int \frac{\left(1+x^4\right) \left(-1+3 x^4\right)}{x \left(1-a x+x^4\right) \sqrt{x+x^5}} \, dx","IntegrateAlgebraic[((1 + x^4)*(-1 + 3*x^4))/(x*(1 - a*x + x^4)*Sqrt[x + x^5]),x]","\frac{2 \sqrt{x+x^5}}{x}-2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{x+x^5}}{1+x^4}\right)","\frac{2 \sqrt{x+x^5}}{x}-2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{x+x^5}}{1+x^4}\right)",1,"(2*Sqrt[x + x^5])/x - 2*Sqrt[a]*ArcTanh[(Sqrt[a]*Sqrt[x + x^5])/(1 + x^4)]","A",1
582,1,45,45,2.6494663,"\int \frac{\sqrt{1+x^2+x^5} \left(-2+3 x^5\right)}{\left(1+x^5\right) \left(1-x^2+x^5\right)} \, dx","IntegrateAlgebraic[(Sqrt[1 + x^2 + x^5]*(-2 + 3*x^5))/((1 + x^5)*(1 - x^2 + x^5)),x]","2 \tanh ^{-1}\left(\frac{x}{\sqrt{1+x^2+x^5}}\right)-2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^2+x^5}}\right)","2 \tanh ^{-1}\left(\frac{x}{\sqrt{1+x^2+x^5}}\right)-2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^2+x^5}}\right)",1,"2*ArcTanh[x/Sqrt[1 + x^2 + x^5]] - 2*Sqrt[2]*ArcTanh[(Sqrt[2]*x)/Sqrt[1 + x^2 + x^5]]","A",1
583,1,45,45,0.0323107,"\int \frac{\sqrt[4]{1+x^6}}{x^7} \, dx","IntegrateAlgebraic[(1 + x^6)^(1/4)/x^7,x]","-\frac{\sqrt[4]{1+x^6}}{6 x^6}-\frac{1}{12} \tan ^{-1}\left(\sqrt[4]{1+x^6}\right)-\frac{1}{12} \tanh ^{-1}\left(\sqrt[4]{1+x^6}\right)","-\frac{\sqrt[4]{1+x^6}}{6 x^6}-\frac{1}{12} \tan ^{-1}\left(\sqrt[4]{1+x^6}\right)-\frac{1}{12} \tanh ^{-1}\left(\sqrt[4]{1+x^6}\right)",1,"-1/6*(1 + x^6)^(1/4)/x^6 - ArcTan[(1 + x^6)^(1/4)]/12 - ArcTanh[(1 + x^6)^(1/4)]/12","A",1
584,1,45,45,4.4488073,"\int \frac{(-1+x)^2 \left(-10-8 x+5 x^2+5 x^3\right)}{\left(\frac{1+x}{-2+x^2}\right)^{3/4} \left(-2+x^2\right) \left(-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x)^2*(-10 - 8*x + 5*x^2 + 5*x^3))/(((1 + x)/(-2 + x^2))^(3/4)*(-2 + x^2)*(-3 + 7*x - 11*x^2 + 4*x^3 + 4*x^4 - 4*x^5 + x^6)),x]","2 \tan ^{-1}\left(\frac{-1+x}{\sqrt[4]{\frac{1+x}{-2+x^2}}}\right)-2 \tanh ^{-1}\left(\frac{-1+x}{\sqrt[4]{\frac{1+x}{-2+x^2}}}\right)","2 \tan ^{-1}\left(\frac{-1+x}{\sqrt[4]{\frac{1+x}{-2+x^2}}}\right)-2 \tanh ^{-1}\left(\frac{-1+x}{\sqrt[4]{\frac{1+x}{-2+x^2}}}\right)",1,"2*ArcTan[(-1 + x)/((1 + x)/(-2 + x^2))^(1/4)] - 2*ArcTanh[(-1 + x)/((1 + x)/(-2 + x^2))^(1/4)]","A",1
585,1,45,45,1.7173819,"\int \frac{x^2 (10 b+9 a x)}{\sqrt[4]{b x^2+a x^3} \left(-b-a x+x^{10}\right)} \, dx","IntegrateAlgebraic[(x^2*(10*b + 9*a*x))/((b*x^2 + a*x^3)^(1/4)*(-b - a*x + x^10)),x]","2 \tan ^{-1}\left(\frac{\sqrt[4]{b x^2+a x^3}}{x^3}\right)-2 \tanh ^{-1}\left(\frac{x^3}{\sqrt[4]{b x^2+a x^3}}\right)","2 \tan ^{-1}\left(\frac{\sqrt[4]{b x^2+a x^3}}{x^3}\right)-2 \tanh ^{-1}\left(\frac{x^3}{\sqrt[4]{b x^2+a x^3}}\right)",1,"2*ArcTan[(b*x^2 + a*x^3)^(1/4)/x^3] - 2*ArcTanh[x^3/(b*x^2 + a*x^3)^(1/4)]","A",1
586,1,45,45,0.300898,"\int \frac{-1+x^{16}}{\sqrt{1+x^4} \left(1+x^{16}\right)} \, dx","IntegrateAlgebraic[(-1 + x^16)/(Sqrt[1 + x^4]*(1 + x^16)),x]","\frac{1}{8} \text{RootSum}\left[2-4 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt{1+x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{1}{8} \text{RootSum}\left[2-4 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt{1+x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"RootSum[2 - 4*#1^4 + #1^8 & , (-Log[x] + Log[Sqrt[1 + x^4] - x*#1])/#1 & ]/8","A",1
587,1,45,45,0.3494383,"\int \frac{-1+x^{16}}{\sqrt{-1+x^4} \left(1-x^8+x^{16}\right)} \, dx","IntegrateAlgebraic[(-1 + x^16)/(Sqrt[-1 + x^4]*(1 - x^8 + x^16)),x]","\frac{1}{8} \text{RootSum}\left[1+4 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt{-1+x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{1}{8} \text{RootSum}\left[1+4 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt{-1+x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"RootSum[1 + 4*#1^4 + #1^8 & , (-Log[x] + Log[Sqrt[-1 + x^4] - x*#1])/#1 & ]/8","A",1
588,1,45,45,1.3120376,"\int \frac{\sqrt{x^2+x \sqrt{-1+x^2}}}{x \sqrt{-1+x^2}} \, dx","IntegrateAlgebraic[Sqrt[x^2 + x*Sqrt[-1 + x^2]]/(x*Sqrt[-1 + x^2]),x]","-\sqrt{2} \log \left(x+\sqrt{-1+x^2}-\sqrt{2} \sqrt{x^2+x \sqrt{-1+x^2}}\right)","-\sqrt{2} \log \left(x+\sqrt{-1+x^2}-\sqrt{2} \sqrt{x^2+x \sqrt{-1+x^2}}\right)",1,"-(Sqrt[2]*Log[x + Sqrt[-1 + x^2] - Sqrt[2]*Sqrt[x^2 + x*Sqrt[-1 + x^2]]])","A",1
589,1,46,46,0.4244635,"\int x^6 \sqrt{x+x^4} \, dx","IntegrateAlgebraic[x^6*Sqrt[x + x^4],x]","\frac{1}{72} \sqrt{x+x^4} \left(-3 x+2 x^4+8 x^7\right)+\frac{1}{24} \tanh ^{-1}\left(\frac{x^2}{\sqrt{x+x^4}}\right)","\frac{1}{72} \sqrt{x+x^4} \left(-3 x+2 x^4+8 x^7\right)+\frac{1}{24} \tanh ^{-1}\left(\frac{x^2}{\sqrt{x+x^4}}\right)",1,"(Sqrt[x + x^4]*(-3*x + 2*x^4 + 8*x^7))/72 + ArcTanh[x^2/Sqrt[x + x^4]]/24","A",1
590,1,46,46,0.5760512,"\int \frac{(-1+x) (1+x)^3}{\left(1+x^2\right)^2 \sqrt{1+x^2+x^4}} \, dx","IntegrateAlgebraic[((-1 + x)*(1 + x)^3)/((1 + x^2)^2*Sqrt[1 + x^2 + x^4]),x]","\frac{2 \sqrt{1+x^2+x^4}}{1+x^2}-2 \tan ^{-1}\left(\frac{x}{1+x^2+\sqrt{1+x^2+x^4}}\right)","\frac{2 \sqrt{1+x^2+x^4}}{1+x^2}-2 \tan ^{-1}\left(\frac{x}{1+x^2+\sqrt{1+x^2+x^4}}\right)",1,"(2*Sqrt[1 + x^2 + x^4])/(1 + x^2) - 2*ArcTan[x/(1 + x^2 + Sqrt[1 + x^2 + x^4])]","A",1
591,1,46,46,0.1713972,"\int \frac{-1+x^2}{\left(1+x^2\right) \sqrt{1-x-x^2-x^3+x^4}} \, dx","IntegrateAlgebraic[(-1 + x^2)/((1 + x^2)*Sqrt[1 - x - x^2 - x^3 + x^4]),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} x}{1+x^2-\sqrt{1-x-x^2-x^3+x^4}}\right)}{\sqrt{3}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} x}{1+x^2-\sqrt{1-x-x^2-x^3+x^4}}\right)}{\sqrt{3}}",1,"(2*ArcTan[(Sqrt[3]*x)/(1 + x^2 - Sqrt[1 - x - x^2 - x^3 + x^4])])/Sqrt[3]","A",1
592,1,46,46,3.736457,"\int \frac{-1-2 x+3 x^2}{\sqrt{-3-2 x-x^2+4 x^3-x^4-2 x^5+x^6}} \, dx","IntegrateAlgebraic[(-1 - 2*x + 3*x^2)/Sqrt[-3 - 2*x - x^2 + 4*x^3 - x^4 - 2*x^5 + x^6],x]","-\log \left(-1+x+x^2-x^3+\sqrt{-3-2 x-x^2+4 x^3-x^4-2 x^5+x^6}\right)","-\log \left(-1+x+x^2-x^3+\sqrt{-3-2 x-x^2+4 x^3-x^4-2 x^5+x^6}\right)",1,"-Log[-1 + x + x^2 - x^3 + Sqrt[-3 - 2*x - x^2 + 4*x^3 - x^4 - 2*x^5 + x^6]]","A",1
593,1,46,46,0.385378,"\int \frac{-x+x^2}{\sqrt{-2 x+4 x^2-2 x^3+x^4-2 x^5+x^6}} \, dx","IntegrateAlgebraic[(-x + x^2)/Sqrt[-2*x + 4*x^2 - 2*x^3 + x^4 - 2*x^5 + x^6],x]","\frac{2}{3} \tanh ^{-1}\left(\frac{(-1+x) \left(-2+x^3\right)}{x \sqrt{-2 x+4 x^2-2 x^3+x^4-2 x^5+x^6}}\right)","\frac{2}{3} \tanh ^{-1}\left(\frac{(-1+x) \left(-2+x^3\right)}{x \sqrt{-2 x+4 x^2-2 x^3+x^4-2 x^5+x^6}}\right)",1,"(2*ArcTanh[((-1 + x)*(-2 + x^3))/(x*Sqrt[-2*x + 4*x^2 - 2*x^3 + x^4 - 2*x^5 + x^6])])/3","A",1
594,1,47,47,0.1398321,"\int \frac{\sqrt[4]{-1+x^4}}{x^2} \, dx","IntegrateAlgebraic[(-1 + x^4)^(1/4)/x^2,x]","-\frac{\sqrt[4]{-1+x^4}}{x}-\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)+\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)","-\frac{\sqrt[4]{-1+x^4}}{x}-\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)+\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)",1,"-((-1 + x^4)^(1/4)/x) - ArcTan[x/(-1 + x^4)^(1/4)]/2 + ArcTanh[x/(-1 + x^4)^(1/4)]/2","A",1
595,1,47,47,0.0436872,"\int \frac{\left(-1+x^4\right) \sqrt[4]{1+x^4}}{x} \, dx","IntegrateAlgebraic[((-1 + x^4)*(1 + x^4)^(1/4))/x,x]","\frac{1}{5} \left(-4+x^4\right) \sqrt[4]{1+x^4}+\frac{1}{2} \tan ^{-1}\left(\sqrt[4]{1+x^4}\right)+\frac{1}{2} \tanh ^{-1}\left(\sqrt[4]{1+x^4}\right)","\frac{1}{5} \left(-4+x^4\right) \sqrt[4]{1+x^4}+\frac{1}{2} \tan ^{-1}\left(\sqrt[4]{1+x^4}\right)+\frac{1}{2} \tanh ^{-1}\left(\sqrt[4]{1+x^4}\right)",1,"((-4 + x^4)*(1 + x^4)^(1/4))/5 + ArcTan[(1 + x^4)^(1/4)]/2 + ArcTanh[(1 + x^4)^(1/4)]/2","A",1
596,1,47,47,0.2268168,"\int \frac{1}{\sqrt{3+4 x+x^4}} \, dx","IntegrateAlgebraic[1/Sqrt[3 + 4*x + x^4],x]","\sqrt{\frac{2}{3}} \tanh ^{-1}\left(\frac{\sqrt{6}+\sqrt{6} x}{1+2 x+x^2-\sqrt{3+4 x+x^4}}\right)","\sqrt{\frac{2}{3}} \tanh ^{-1}\left(\frac{\sqrt{6}+\sqrt{6} x}{1+2 x+x^2-\sqrt{3+4 x+x^4}}\right)",1,"Sqrt[2/3]*ArcTanh[(Sqrt[6] + Sqrt[6]*x)/(1 + 2*x + x^2 - Sqrt[3 + 4*x + x^4])]","A",1
597,1,47,47,0.3939341,"\int \frac{\left(1+x^4\right) \sqrt{-1+2 x^2+x^4}}{\left(-1+x^4\right) \left(-1+x^2+x^4\right)} \, dx","IntegrateAlgebraic[((1 + x^4)*Sqrt[-1 + 2*x^2 + x^4])/((-1 + x^4)*(-1 + x^2 + x^4)),x]","\tanh ^{-1}\left(\frac{x}{\sqrt{-1+2 x^2+x^4}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{-1+2 x^2+x^4}}\right)","\tanh ^{-1}\left(\frac{x}{\sqrt{-1+2 x^2+x^4}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{-1+2 x^2+x^4}}\right)",1,"ArcTanh[x/Sqrt[-1 + 2*x^2 + x^4]] - Sqrt[2]*ArcTanh[(Sqrt[2]*x)/Sqrt[-1 + 2*x^2 + x^4]]","A",1
598,1,47,47,0.1360721,"\int \frac{\sqrt{-1+x-x^2+x^4} \left(2-x+2 x^4\right)}{\left(-1+x+x^4\right)^2} \, dx","IntegrateAlgebraic[(Sqrt[-1 + x - x^2 + x^4]*(2 - x + 2*x^4))/(-1 + x + x^4)^2,x]","-\frac{x \sqrt{-1+x-x^2+x^4}}{-1+x+x^4}-\tan ^{-1}\left(\frac{x}{\sqrt{-1+x-x^2+x^4}}\right)","-\frac{x \sqrt{-1+x-x^2+x^4}}{-1+x+x^4}-\tan ^{-1}\left(\frac{x}{\sqrt{-1+x-x^2+x^4}}\right)",1,"-((x*Sqrt[-1 + x - x^2 + x^4])/(-1 + x + x^4)) - ArcTan[x/Sqrt[-1 + x - x^2 + x^4]]","A",1
599,1,47,47,2.017696,"\int \frac{x+3 x^5}{\left(-1+x^4\right) \left(-a-x+a x^4\right) \sqrt{-x+x^5}} \, dx","IntegrateAlgebraic[(x + 3*x^5)/((-1 + x^4)*(-a - x + a*x^4)*Sqrt[-x + x^5]),x]","\frac{2 \sqrt{-x+x^5}}{-1+x^4}-2 \sqrt{a} \tanh ^{-1}\left(\frac{x}{\sqrt{a} \sqrt{-x+x^5}}\right)","\frac{2 \sqrt{-x+x^5}}{-1+x^4}-2 \sqrt{a} \tanh ^{-1}\left(\frac{x}{\sqrt{a} \sqrt{-x+x^5}}\right)",1,"(2*Sqrt[-x + x^5])/(-1 + x^4) - 2*Sqrt[a]*ArcTanh[x/(Sqrt[a]*Sqrt[-x + x^5])]","A",1
600,1,47,47,2.7323499,"\int \frac{\sqrt{1+x^2+x^6} \left(-1+2 x^6\right)}{\left(1+x^6\right) \left(2-x^2+2 x^6\right)} \, dx","IntegrateAlgebraic[(Sqrt[1 + x^2 + x^6]*(-1 + 2*x^6))/((1 + x^6)*(2 - x^2 + 2*x^6)),x]","\tanh ^{-1}\left(\frac{x}{\sqrt{1+x^2+x^6}}\right)-\sqrt{\frac{3}{2}} \tanh ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt{1+x^2+x^6}}\right)","\tanh ^{-1}\left(\frac{x}{\sqrt{1+x^2+x^6}}\right)-\sqrt{\frac{3}{2}} \tanh ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt{1+x^2+x^6}}\right)",1,"ArcTanh[x/Sqrt[1 + x^2 + x^6]] - Sqrt[3/2]*ArcTanh[(Sqrt[3/2]*x)/Sqrt[1 + x^2 + x^6]]","A",1
601,1,47,47,2.6999611,"\int \frac{x+4 x^6}{\left(-1+x^5\right) \left(-a-x+a x^5\right) \sqrt{-x+x^6}} \, dx","IntegrateAlgebraic[(x + 4*x^6)/((-1 + x^5)*(-a - x + a*x^5)*Sqrt[-x + x^6]),x]","\frac{2 \sqrt{-x+x^6}}{-1+x^5}-2 \sqrt{a} \tanh ^{-1}\left(\frac{x}{\sqrt{a} \sqrt{-x+x^6}}\right)","\frac{2 \sqrt{-x+x^6}}{-1+x^5}-2 \sqrt{a} \tanh ^{-1}\left(\frac{x}{\sqrt{a} \sqrt{-x+x^6}}\right)",1,"(2*Sqrt[-x + x^6])/(-1 + x^5) - 2*Sqrt[a]*ArcTanh[x/(Sqrt[a]*Sqrt[-x + x^6])]","A",1
602,1,47,47,0.4317292,"\int \frac{\left(-1+6 x^4\right) \sqrt{x+2 x^5}}{\left(1+2 x^4\right) \left(1-x^2+4 x^4+4 x^8\right)} \, dx","IntegrateAlgebraic[((-1 + 6*x^4)*Sqrt[x + 2*x^5])/((1 + 2*x^4)*(1 - x^2 + 4*x^4 + 4*x^8)),x]","\tan ^{-1}\left(\frac{\sqrt{x+2 x^5}}{1+2 x^4}\right)-\tanh ^{-1}\left(\frac{\sqrt{x+2 x^5}}{1+2 x^4}\right)","\tan ^{-1}\left(\frac{\sqrt{x+2 x^5}}{1+2 x^4}\right)-\tanh ^{-1}\left(\frac{\sqrt{x+2 x^5}}{1+2 x^4}\right)",1,"ArcTan[Sqrt[x + 2*x^5]/(1 + 2*x^4)] - ArcTanh[Sqrt[x + 2*x^5]/(1 + 2*x^4)]","A",1
603,1,28,47,0.0139804,"\int \sqrt{1+\sqrt{1+x}} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[1 + x]],x]","\frac{4}{15} \left(1+\sqrt{1+x}\right)^{3/2} \left(-2+3 \sqrt{1+x}\right)","\frac{4}{15} \sqrt{1+x} \sqrt{1+\sqrt{1+x}}+\frac{4}{15} (1+3 x) \sqrt{1+\sqrt{1+x}}",1,"(4*(1 + Sqrt[1 + x])^(3/2)*(-2 + 3*Sqrt[1 + x]))/15","A",1
604,1,47,47,0.077934,"\int \frac{\sqrt{1+\sqrt{1+x^2}}}{x} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[1 + x^2]]/x,x]","2 \sqrt{1+\sqrt{1+x^2}}-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{1+\sqrt{1+x^2}}}{\sqrt{2}}\right)","2 \sqrt{1+\sqrt{1+x^2}}-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{1+\sqrt{1+x^2}}}{\sqrt{2}}\right)",1,"2*Sqrt[1 + Sqrt[1 + x^2]] - Sqrt[2]*ArcTanh[Sqrt[1 + Sqrt[1 + x^2]]/Sqrt[2]]","A",1
605,1,47,47,0.0881557,"\int \frac{1}{1+\sqrt[4]{9-6 x+x^2}} \, dx","IntegrateAlgebraic[(1 + (9 - 6*x + x^2)^(1/4))^(-1),x]","\frac{\sqrt{(-3+x)^2} \left(2 \sqrt[4]{9-6 x+x^2}-2 \log \left(1+\sqrt[4]{9-6 x+x^2}\right)\right)}{-3+x}","\frac{\sqrt{(-3+x)^2} \left(2 \sqrt[4]{9-6 x+x^2}-2 \log \left(1+\sqrt[4]{9-6 x+x^2}\right)\right)}{-3+x}",1,"(Sqrt[(-3 + x)^2]*(2*(9 - 6*x + x^2)^(1/4) - 2*Log[1 + (9 - 6*x + x^2)^(1/4)]))/(-3 + x)","A",1
606,1,48,48,2.0247787,"\int \frac{x^2}{\left(-2+x^2\right) \left(-1+x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[x^2/((-2 + x^2)*(-1 + x^2)^(3/4)),x]","\frac{\tan ^{-1}\left(\frac{x}{\sqrt{2} \sqrt[4]{-1+x^2}}\right)}{\sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{2} \sqrt[4]{-1+x^2}}\right)}{\sqrt{2}}","\frac{\tan ^{-1}\left(\frac{x}{\sqrt{2} \sqrt[4]{-1+x^2}}\right)}{\sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{2} \sqrt[4]{-1+x^2}}\right)}{\sqrt{2}}",1,"ArcTan[x/(Sqrt[2]*(-1 + x^2)^(1/4))]/Sqrt[2] - ArcTanh[x/(Sqrt[2]*(-1 + x^2)^(1/4))]/Sqrt[2]","A",1
607,1,48,48,0.3719859,"\int \frac{\left(-1+x^2\right) \sqrt{1+x^4}}{\left(1+x^2\right)^3} \, dx","IntegrateAlgebraic[((-1 + x^2)*Sqrt[1 + x^4])/(1 + x^2)^3,x]","-\frac{x \sqrt{1+x^4}}{2 \left(1+x^2\right)^2}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{2 \sqrt{2}}","-\frac{x \sqrt{1+x^4}}{2 \left(1+x^2\right)^2}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{2 \sqrt{2}}",1,"-1/2*(x*Sqrt[1 + x^4])/(1 + x^2)^2 - ArcTan[(Sqrt[2]*x)/Sqrt[1 + x^4]]/(2*Sqrt[2])","A",1
608,1,48,48,0.1764832,"\int \frac{77-46 x+5 x^2}{\left(-23+82 x-23 x^2\right) \sqrt{-60+83 x-21 x^2-3 x^3+x^4}} \, dx","IntegrateAlgebraic[(77 - 46*x + 5*x^2)/((-23 + 82*x - 23*x^2)*Sqrt[-60 + 83*x - 21*x^2 - 3*x^3 + x^4]),x]","\frac{\tan ^{-1}\left(\frac{2 \sqrt{42} \sqrt{-60+83 x-21 x^2-3 x^3+x^4}}{103-86 x+19 x^2}\right)}{\sqrt{42}}","\frac{\tan ^{-1}\left(\frac{2 \sqrt{42} \sqrt{-60+83 x-21 x^2-3 x^3+x^4}}{103-86 x+19 x^2}\right)}{\sqrt{42}}",1,"ArcTan[(2*Sqrt[42]*Sqrt[-60 + 83*x - 21*x^2 - 3*x^3 + x^4])/(103 - 86*x + 19*x^2)]/Sqrt[42]","A",1
609,1,48,48,0.2449887,"\int \frac{(-1+x) \sqrt[4]{x^3+x^4}}{x (1+x)} \, dx","IntegrateAlgebraic[((-1 + x)*(x^3 + x^4)^(1/4))/(x*(1 + x)),x]","\sqrt[4]{x^3+x^4}+\frac{7}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)-\frac{7}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)","\sqrt[4]{x^3+x^4}+\frac{7}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)-\frac{7}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)",1,"(x^3 + x^4)^(1/4) + (7*ArcTan[x/(x^3 + x^4)^(1/4)])/2 - (7*ArcTanh[x/(x^3 + x^4)^(1/4)])/2","A",1
610,1,48,48,0.3298833,"\int \frac{\left(-1+x^4\right) \sqrt[4]{x^3+x^4}}{x^8} \, dx","IntegrateAlgebraic[((-1 + x^4)*(x^3 + x^4)^(1/4))/x^8,x]","\frac{4 \sqrt[4]{x^3+x^4} \left(13923+663 x-780 x^2+960 x^3-39955 x^4-5687 x^5+22748 x^6\right)}{348075 x^7}","\frac{4 \sqrt[4]{x^3+x^4} \left(13923+663 x-780 x^2+960 x^3-39955 x^4-5687 x^5+22748 x^6\right)}{348075 x^7}",1,"(4*(x^3 + x^4)^(1/4)*(13923 + 663*x - 780*x^2 + 960*x^3 - 39955*x^4 - 5687*x^5 + 22748*x^6))/(348075*x^7)","A",1
611,1,48,48,0.3083407,"\int \frac{-1+4 x-4 x^2+4 x^4}{\sqrt{\frac{1-2 x^2}{1+2 x^2}} \left(1+2 x^2\right) \left(-1-4 x+12 x^2-8 x^3+4 x^4\right)} \, dx","IntegrateAlgebraic[(-1 + 4*x - 4*x^2 + 4*x^4)/(Sqrt[(1 - 2*x^2)/(1 + 2*x^2)]*(1 + 2*x^2)*(-1 - 4*x + 12*x^2 - 8*x^3 + 4*x^4)),x]","\frac{\tanh ^{-1}\left(\frac{-\sqrt{\frac{2}{3}}+\sqrt{\frac{2}{3}} x}{\sqrt{\frac{1-2 x^2}{1+2 x^2}}}\right)}{\sqrt{6}}","\frac{\tanh ^{-1}\left(\frac{-\sqrt{\frac{2}{3}}+\sqrt{\frac{2}{3}} x}{\sqrt{\frac{1-2 x^2}{1+2 x^2}}}\right)}{\sqrt{6}}",1,"ArcTanh[(-Sqrt[2/3] + Sqrt[2/3]*x)/Sqrt[(1 - 2*x^2)/(1 + 2*x^2)]]/Sqrt[6]","A",1
612,1,48,48,2.6342775,"\int \frac{\left(-3+x^4\right) \left(1-2 x^3+x^4\right) \left(1-x^3+x^4\right)}{x^6 \left(1+x^4\right) \sqrt[4]{x+x^5}} \, dx","IntegrateAlgebraic[((-3 + x^4)*(1 - 2*x^3 + x^4)*(1 - x^3 + x^4))/(x^6*(1 + x^4)*(x + x^5)^(1/4)),x]","\frac{4 \left(x+x^5\right)^{3/4} \left(1-7 x^3+2 x^4-14 x^6-7 x^7+x^8\right)}{7 x^6 \left(1+x^4\right)}","\frac{4 \left(x+x^5\right)^{3/4} \left(1-7 x^3+2 x^4-14 x^6-7 x^7+x^8\right)}{7 x^6 \left(1+x^4\right)}",1,"(4*(x + x^5)^(3/4)*(1 - 7*x^3 + 2*x^4 - 14*x^6 - 7*x^7 + x^8))/(7*x^6*(1 + x^4))","A",1
613,1,48,48,0.1793945,"\int \frac{x^{20}}{\sqrt{-1+x^6}} \, dx","IntegrateAlgebraic[x^20/Sqrt[-1 + x^6],x]","\frac{1}{144} \sqrt{-1+x^6} \left(15 x^3+10 x^9+8 x^{15}\right)+\frac{5}{48} \log \left(x^3+\sqrt{-1+x^6}\right)","\frac{1}{144} \sqrt{-1+x^6} \left(15 x^3+10 x^9+8 x^{15}\right)+\frac{5}{48} \log \left(x^3+\sqrt{-1+x^6}\right)",1,"(Sqrt[-1 + x^6]*(15*x^3 + 10*x^9 + 8*x^15))/144 + (5*Log[x^3 + Sqrt[-1 + x^6]])/48","A",1
614,1,48,48,0.1309175,"\int x^{14} \sqrt{-1+x^6} \, dx","IntegrateAlgebraic[x^14*Sqrt[-1 + x^6],x]","\frac{1}{144} \sqrt{-1+x^6} \left(-3 x^3-2 x^9+8 x^{15}\right)-\frac{1}{48} \log \left(x^3+\sqrt{-1+x^6}\right)","\frac{1}{144} \sqrt{-1+x^6} \left(-3 x^3-2 x^9+8 x^{15}\right)-\frac{1}{48} \log \left(x^3+\sqrt{-1+x^6}\right)",1,"(Sqrt[-1 + x^6]*(-3*x^3 - 2*x^9 + 8*x^15))/144 - Log[x^3 + Sqrt[-1 + x^6]]/48","A",1
615,1,48,48,0.1654107,"\int \frac{x^{20}}{\sqrt{1+x^6}} \, dx","IntegrateAlgebraic[x^20/Sqrt[1 + x^6],x]","\frac{1}{144} \sqrt{1+x^6} \left(15 x^3-10 x^9+8 x^{15}\right)-\frac{5}{48} \log \left(x^3+\sqrt{1+x^6}\right)","\frac{1}{144} \sqrt{1+x^6} \left(15 x^3-10 x^9+8 x^{15}\right)-\frac{5}{48} \log \left(x^3+\sqrt{1+x^6}\right)",1,"(Sqrt[1 + x^6]*(15*x^3 - 10*x^9 + 8*x^15))/144 - (5*Log[x^3 + Sqrt[1 + x^6]])/48","A",1
616,1,48,48,2.7071271,"\int \frac{x-4 x^6}{\sqrt{x+x^6} \left(1-a x^2+2 x^5+x^{10}\right)} \, dx","IntegrateAlgebraic[(x - 4*x^6)/(Sqrt[x + x^6]*(1 - a*x^2 + 2*x^5 + x^10)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt{x+x^6}}\right)}{a^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt{x+x^6}}\right)}{a^{3/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt{x+x^6}}\right)}{a^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt{x+x^6}}\right)}{a^{3/4}}",1,"-(ArcTan[(a^(1/4)*x)/Sqrt[x + x^6]]/a^(3/4)) + ArcTanh[(a^(1/4)*x)/Sqrt[x + x^6]]/a^(3/4)","A",1
617,1,48,48,2.7097964,"\int \frac{-x+4 x^6}{\sqrt{x+x^6} \left(a-x^2+2 a x^5+a x^{10}\right)} \, dx","IntegrateAlgebraic[(-x + 4*x^6)/(Sqrt[x + x^6]*(a - x^2 + 2*a*x^5 + a*x^10)),x]","\frac{\tan ^{-1}\left(\frac{x}{\sqrt[4]{a} \sqrt{x+x^6}}\right)}{\sqrt[4]{a}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[4]{a} \sqrt{x+x^6}}\right)}{\sqrt[4]{a}}","\frac{\tan ^{-1}\left(\frac{x}{\sqrt[4]{a} \sqrt{x+x^6}}\right)}{\sqrt[4]{a}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[4]{a} \sqrt{x+x^6}}\right)}{\sqrt[4]{a}}",1,"ArcTan[x/(a^(1/4)*Sqrt[x + x^6])]/a^(1/4) - ArcTanh[x/(a^(1/4)*Sqrt[x + x^6])]/a^(1/4)","A",1
618,1,48,48,0.0960034,"\int \frac{\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\sqrt{1+x^2}} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/Sqrt[1 + x^2],x]","4 \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-4 \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)","4 \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-4 \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)",1,"4*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - 4*ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]]","A",1
619,1,49,49,0.183605,"\int \frac{-3 b+2 a x}{\sqrt[4]{-b x+a x^2} \left(b-a x+x^3\right)} \, dx","IntegrateAlgebraic[(-3*b + 2*a*x)/((-(b*x) + a*x^2)^(1/4)*(b - a*x + x^3)),x]","2 \tan ^{-1}\left(\frac{\sqrt[4]{-b x+a x^2}}{x}\right)-2 \tanh ^{-1}\left(\frac{\left(-b x+a x^2\right)^{3/4}}{-b+a x}\right)","2 \tan ^{-1}\left(\frac{\sqrt[4]{-b x+a x^2}}{x}\right)-2 \tanh ^{-1}\left(\frac{\left(-b x+a x^2\right)^{3/4}}{-b+a x}\right)",1,"2*ArcTan[(-(b*x) + a*x^2)^(1/4)/x] - 2*ArcTanh[(-(b*x) + a*x^2)^(3/4)/(-b + a*x)]","A",1
620,1,49,49,0.3513116,"\int \frac{\left(-4+x^3\right) \sqrt{2-x^2+x^3}}{\left(2+x^3\right) \left(2+x^2+x^3\right)} \, dx","IntegrateAlgebraic[((-4 + x^3)*Sqrt[2 - x^2 + x^3])/((2 + x^3)*(2 + x^2 + x^3)),x]","2 \tan ^{-1}\left(\frac{x}{\sqrt{2-x^2+x^3}}\right)-2 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{2-x^2+x^3}}\right)","2 \tan ^{-1}\left(\frac{x}{\sqrt{2-x^2+x^3}}\right)-2 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{2-x^2+x^3}}\right)",1,"2*ArcTan[x/Sqrt[2 - x^2 + x^3]] - 2*Sqrt[2]*ArcTan[(Sqrt[2]*x)/Sqrt[2 - x^2 + x^3]]","A",1
621,1,49,49,0.1491644,"\int x^2 \sqrt[4]{-1+x^4} \, dx","IntegrateAlgebraic[x^2*(-1 + x^4)^(1/4),x]","\frac{1}{4} x^3 \sqrt[4]{-1+x^4}+\frac{1}{8} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)-\frac{1}{8} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)","\frac{1}{4} x^3 \sqrt[4]{-1+x^4}+\frac{1}{8} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)-\frac{1}{8} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)",1,"(x^3*(-1 + x^4)^(1/4))/4 + ArcTan[x/(-1 + x^4)^(1/4)]/8 - ArcTanh[x/(-1 + x^4)^(1/4)]/8","A",1
622,1,49,49,0.1519274,"\int x^2 \sqrt[4]{1+x^4} \, dx","IntegrateAlgebraic[x^2*(1 + x^4)^(1/4),x]","\frac{1}{4} x^3 \sqrt[4]{1+x^4}-\frac{1}{8} \tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)+\frac{1}{8} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)","\frac{1}{4} x^3 \sqrt[4]{1+x^4}-\frac{1}{8} \tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)+\frac{1}{8} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)",1,"(x^3*(1 + x^4)^(1/4))/4 - ArcTan[x/(1 + x^4)^(1/4)]/8 + ArcTanh[x/(1 + x^4)^(1/4)]/8","A",1
623,1,49,49,0.4499693,"\int \frac{\left(-b+a x^3\right) \sqrt{x+x^4}}{x^3} \, dx","IntegrateAlgebraic[((-b + a*x^3)*Sqrt[x + x^4])/x^3,x]","\frac{\left(2 b+a x^3\right) \sqrt{x+x^4}}{3 x^2}+\frac{1}{3} (a-2 b) \tanh ^{-1}\left(\frac{x^2}{\sqrt{x+x^4}}\right)","\frac{\left(2 b+a x^3\right) \sqrt{x+x^4}}{3 x^2}+\frac{1}{3} (a-2 b) \tanh ^{-1}\left(\frac{x^2}{\sqrt{x+x^4}}\right)",1,"((2*b + a*x^3)*Sqrt[x + x^4])/(3*x^2) + ((a - 2*b)*ArcTanh[x^2/Sqrt[x + x^4]])/3","A",1
624,1,49,49,0.1509041,"\int \frac{1}{\left(-2+x^4\right) \sqrt[4]{x^2+x^4}} \, dx","IntegrateAlgebraic[1/((-2 + x^4)*(x^2 + x^4)^(1/4)),x]","\frac{1}{8} \text{RootSum}\left[1-4 \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{x^2+x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{1}{8} \text{RootSum}\left[1-4 \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{x^2+x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"RootSum[1 - 4*#1^4 + 2*#1^8 & , (-Log[x] + Log[(x^2 + x^4)^(1/4) - x*#1])/#1 & ]/8","A",1
625,1,49,49,0.6088315,"\int \frac{-4 b+a x^3}{\sqrt[4]{-b+a x^3} \left(b-a x^3+x^4\right)} \, dx","IntegrateAlgebraic[(-4*b + a*x^3)/((-b + a*x^3)^(1/4)*(b - a*x^3 + x^4)),x]","2 \tan ^{-1}\left(\frac{\sqrt[4]{-b+a x^3}}{x}\right)+2 \tanh ^{-1}\left(\frac{x \left(-b+a x^3\right)^{3/4}}{b-a x^3}\right)","2 \tan ^{-1}\left(\frac{\sqrt[4]{-b+a x^3}}{x}\right)+2 \tanh ^{-1}\left(\frac{x \left(-b+a x^3\right)^{3/4}}{b-a x^3}\right)",1,"2*ArcTan[(-b + a*x^3)^(1/4)/x] + 2*ArcTanh[(x*(-b + a*x^3)^(3/4))/(b - a*x^3)]","A",1
626,1,49,49,1.8699938,"\int \frac{-x+3 x^5}{\left(1+x^4\right) \left(a-x+a x^4\right) \sqrt{x+x^5}} \, dx","IntegrateAlgebraic[(-x + 3*x^5)/((1 + x^4)*(a - x + a*x^4)*Sqrt[x + x^5]),x]","\frac{2 \sqrt{x+x^5}}{1+x^4}-2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{x+x^5}}{\sqrt{a} \left(1+x^4\right)}\right)","\frac{2 \sqrt{x+x^5}}{1+x^4}-2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{x+x^5}}{\sqrt{a} \left(1+x^4\right)}\right)",1,"(2*Sqrt[x + x^5])/(1 + x^4) - 2*Sqrt[a]*ArcTanh[Sqrt[x + x^5]/(Sqrt[a]*(1 + x^4))]","A",1
627,1,51,49,0.1539674,"\int \frac{\left(-1+x^3\right) \sqrt{-1+x^6}}{x^{10}} \, dx","IntegrateAlgebraic[((-1 + x^3)*Sqrt[-1 + x^6])/x^10,x]","\frac{\left(2-3 x^3-2 x^6\right) \sqrt{-1+x^6}}{18 x^9}-\frac{1}{3} \tan ^{-1}\left(\frac{\sqrt{-1+x^6}}{-1+x^3}\right)","\frac{\left(2-3 x^3-2 x^6\right) \sqrt{-1+x^6}}{18 x^9}-\frac{1}{3} \tan ^{-1}\left(\frac{1+x^3}{\sqrt{-1+x^6}}\right)",1,"((2 - 3*x^3 - 2*x^6)*Sqrt[-1 + x^6])/(18*x^9) - ArcTan[Sqrt[-1 + x^6]/(-1 + x^3)]/3","A",1
628,1,49,49,2.6725337,"\int \frac{1+x^6}{\sqrt[4]{-x^3+x^5} \left(1+x^3-x^6\right)} \, dx","IntegrateAlgebraic[(1 + x^6)/((-x^3 + x^5)^(1/4)*(1 + x^3 - x^6)),x]","-\frac{1}{3} \text{RootSum}\left[-1+3 \text{$\#$1}^4+\text{$\#$1}^{12}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-x^3+x^5}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","-\frac{1}{3} \text{RootSum}\left[-1+3 \text{$\#$1}^4+\text{$\#$1}^{12}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-x^3+x^5}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"-1/3*RootSum[-1 + 3*#1^4 + #1^12 & , (-Log[x] + Log[(-x^3 + x^5)^(1/4) - x*#1])/#1 & ]","A",1
629,1,49,49,0.0014786,"\int \frac{1+x^6}{\sqrt[4]{-x^3+x^5} \left(1+x^3-x^6\right)} \, dx","IntegrateAlgebraic[(1 + x^6)/((-x^3 + x^5)^(1/4)*(1 + x^3 - x^6)),x]","-\frac{1}{3} \text{RootSum}\left[-1+3 \text{$\#$1}^4+\text{$\#$1}^{12}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-x^3+x^5}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","-\frac{1}{3} \text{RootSum}\left[-1+3 \text{$\#$1}^4+\text{$\#$1}^{12}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-x^3+x^5}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"-1/3*RootSum[-1 + 3*#1^4 + #1^12 & , (-Log[x] + Log[(-x^3 + x^5)^(1/4) - x*#1])/#1 & ]","A",1
630,1,49,49,0.6125153,"\int \frac{x (6 b+5 a x)}{\sqrt[4]{b x^2+a x^3} \left(-b-a x+x^6\right)} \, dx","IntegrateAlgebraic[(x*(6*b + 5*a*x))/((b*x^2 + a*x^3)^(1/4)*(-b - a*x + x^6)),x]","2 \tan ^{-1}\left(\frac{\sqrt[4]{b x^2+a x^3}}{x^2}\right)-2 \tanh ^{-1}\left(\frac{\left(b x^2+a x^3\right)^{3/4}}{b+a x}\right)","2 \tan ^{-1}\left(\frac{\sqrt[4]{b x^2+a x^3}}{x^2}\right)-2 \tanh ^{-1}\left(\frac{\left(b x^2+a x^3\right)^{3/4}}{b+a x}\right)",1,"2*ArcTan[(b*x^2 + a*x^3)^(1/4)/x^2] - 2*ArcTanh[(b*x^2 + a*x^3)^(3/4)/(b + a*x)]","A",1
631,1,49,49,3.5890954,"\int \frac{\sqrt{-1+x^5} \left(2+3 x^5\right)}{1-a x^4-2 x^5+x^{10}} \, dx","IntegrateAlgebraic[(Sqrt[-1 + x^5]*(2 + 3*x^5))/(1 - a*x^4 - 2*x^5 + x^10),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt{-1+x^5}}\right)}{\sqrt[4]{a}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt{-1+x^5}}\right)}{\sqrt[4]{a}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt{-1+x^5}}\right)}{\sqrt[4]{a}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt{-1+x^5}}\right)}{\sqrt[4]{a}}",1,"-(ArcTan[(a^(1/4)*x)/Sqrt[-1 + x^5]]/a^(1/4)) - ArcTanh[(a^(1/4)*x)/Sqrt[-1 + x^5]]/a^(1/4)","A",1
632,1,49,49,6.9251726,"\int \frac{\sqrt{-1+x^5} \left(2+3 x^5\right)}{a-x^4-2 a x^5+a x^{10}} \, dx","IntegrateAlgebraic[(Sqrt[-1 + x^5]*(2 + 3*x^5))/(a - x^4 - 2*a*x^5 + a*x^10),x]","-\frac{\tan ^{-1}\left(\frac{x}{\sqrt[4]{a} \sqrt{-1+x^5}}\right)}{a^{3/4}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[4]{a} \sqrt{-1+x^5}}\right)}{a^{3/4}}","-\frac{\tan ^{-1}\left(\frac{x}{\sqrt[4]{a} \sqrt{-1+x^5}}\right)}{a^{3/4}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[4]{a} \sqrt{-1+x^5}}\right)}{a^{3/4}}",1,"-(ArcTan[x/(a^(1/4)*Sqrt[-1 + x^5])]/a^(3/4)) - ArcTanh[x/(a^(1/4)*Sqrt[-1 + x^5])]/a^(3/4)","A",1
633,1,49,49,7.2754741,"\int \frac{\sqrt{1+x^5} \left(-2+3 x^5\right)}{a-x^4+2 a x^5+a x^{10}} \, dx","IntegrateAlgebraic[(Sqrt[1 + x^5]*(-2 + 3*x^5))/(a - x^4 + 2*a*x^5 + a*x^10),x]","-\frac{\tan ^{-1}\left(\frac{x}{\sqrt[4]{a} \sqrt{1+x^5}}\right)}{a^{3/4}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[4]{a} \sqrt{1+x^5}}\right)}{a^{3/4}}","-\frac{\tan ^{-1}\left(\frac{x}{\sqrt[4]{a} \sqrt{1+x^5}}\right)}{a^{3/4}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[4]{a} \sqrt{1+x^5}}\right)}{a^{3/4}}",1,"-(ArcTan[x/(a^(1/4)*Sqrt[1 + x^5])]/a^(3/4)) - ArcTanh[x/(a^(1/4)*Sqrt[1 + x^5])]/a^(3/4)","A",1
634,1,49,49,0.1023029,"\int \frac{1}{\sqrt{1+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[1/Sqrt[1 + Sqrt[1 + x^2]],x]","\frac{2 x}{\sqrt{1+\sqrt{1+x^2}}}-\sqrt{2} \tan ^{-1}\left(\frac{x}{\sqrt{2} \sqrt{1+\sqrt{1+x^2}}}\right)","\frac{2 x}{\sqrt{1+\sqrt{1+x^2}}}-\sqrt{2} \tan ^{-1}\left(\frac{x}{\sqrt{2} \sqrt{1+\sqrt{1+x^2}}}\right)",1,"(2*x)/Sqrt[1 + Sqrt[1 + x^2]] - Sqrt[2]*ArcTan[x/(Sqrt[2]*Sqrt[1 + Sqrt[1 + x^2]])]","A",1
635,1,50,50,0.4279388,"\int x^6 \sqrt{-x+x^4} \, dx","IntegrateAlgebraic[x^6*Sqrt[-x + x^4],x]","\frac{1}{72} \sqrt{-x+x^4} \left(-3 x-2 x^4+8 x^7\right)-\frac{1}{24} \tanh ^{-1}\left(\frac{x^2}{\sqrt{-x+x^4}}\right)","\frac{1}{72} \sqrt{-x+x^4} \left(-3 x-2 x^4+8 x^7\right)-\frac{1}{24} \tanh ^{-1}\left(\frac{x^2}{\sqrt{-x+x^4}}\right)",1,"(Sqrt[-x + x^4]*(-3*x - 2*x^4 + 8*x^7))/72 - ArcTanh[x^2/Sqrt[-x + x^4]]/24","A",1
636,1,50,50,0.2723612,"\int \frac{\left(-1+x^2\right) \sqrt[4]{-x^3+x^4}}{x^8} \, dx","IntegrateAlgebraic[((-1 + x^2)*(-x^3 + x^4)^(1/4))/x^8,x]","\frac{4 \sqrt[4]{-x^3+x^4} \left(13923-663 x-21255 x^2+615 x^3+820 x^4+1312 x^5+5248 x^6\right)}{348075 x^7}","\frac{4 \sqrt[4]{-x^3+x^4} \left(13923-663 x-21255 x^2+615 x^3+820 x^4+1312 x^5+5248 x^6\right)}{348075 x^7}",1,"(4*(-x^3 + x^4)^(1/4)*(13923 - 663*x - 21255*x^2 + 615*x^3 + 820*x^4 + 1312*x^5 + 5248*x^6))/(348075*x^7)","A",1
637,1,54,50,0.1463376,"\int \frac{\left(-1+x^3\right) \sqrt{-1+x^6}}{x \left(1+x^3\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)*Sqrt[-1 + x^6])/(x*(1 + x^3)),x]","\frac{1}{3} \sqrt{-1+x^6}-\frac{2}{3} \tan ^{-1}\left(x^3-\sqrt{-1+x^6}\right)+\frac{2}{3} \log \left(-x^3+\sqrt{-1+x^6}\right)","\frac{1}{3} \sqrt{-1+x^6}+\frac{2}{3} \tan ^{-1}\left(x^3+\sqrt{-1+x^6}\right)-\frac{2}{3} \log \left(x^3+\sqrt{-1+x^6}\right)",1,"Sqrt[-1 + x^6]/3 - (2*ArcTan[x^3 - Sqrt[-1 + x^6]])/3 + (2*Log[-x^3 + Sqrt[-1 + x^6]])/3","A",1
638,1,54,50,0.1450233,"\int \frac{\left(1+x^3\right) \sqrt{-1+x^6}}{x \left(-1+x^3\right)} \, dx","IntegrateAlgebraic[((1 + x^3)*Sqrt[-1 + x^6])/(x*(-1 + x^3)),x]","\frac{1}{3} \sqrt{-1+x^6}-\frac{2}{3} \tan ^{-1}\left(x^3-\sqrt{-1+x^6}\right)-\frac{2}{3} \log \left(-x^3+\sqrt{-1+x^6}\right)","\frac{1}{3} \sqrt{-1+x^6}+\frac{2}{3} \tan ^{-1}\left(x^3+\sqrt{-1+x^6}\right)+\frac{2}{3} \log \left(x^3+\sqrt{-1+x^6}\right)",1,"Sqrt[-1 + x^6]/3 - (2*ArcTan[x^3 - Sqrt[-1 + x^6]])/3 - (2*Log[-x^3 + Sqrt[-1 + x^6]])/3","A",1
639,1,50,50,0.6760345,"\int \frac{2 b+a x^4}{\sqrt[4]{b+a x^4} \left(-2 b-2 a x^4+x^8\right)} \, dx","IntegrateAlgebraic[(2*b + a*x^4)/((b + a*x^4)^(1/4)*(-2*b - 2*a*x^4 + x^8)),x]","\frac{1}{8} \text{RootSum}\left[b+2 a \text{$\#$1}^4-2 \text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{1}{8} \text{RootSum}\left[b+2 a \text{$\#$1}^4-2 \text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"RootSum[b + 2*a*#1^4 - 2*#1^8 & , (-Log[x] + Log[(b + a*x^4)^(1/4) - x*#1])/#1 & ]/8","A",1
640,1,50,50,0.0019149,"\int \frac{2 b+a x^4}{\sqrt[4]{b+a x^4} \left(-2 b-2 a x^4+x^8\right)} \, dx","IntegrateAlgebraic[(2*b + a*x^4)/((b + a*x^4)^(1/4)*(-2*b - 2*a*x^4 + x^8)),x]","\frac{1}{8} \text{RootSum}\left[b+2 a \text{$\#$1}^4-2 \text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{1}{8} \text{RootSum}\left[b+2 a \text{$\#$1}^4-2 \text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"RootSum[b + 2*a*#1^4 - 2*#1^8 & , (-Log[x] + Log[(b + a*x^4)^(1/4) - x*#1])/#1 & ]/8","A",1
641,1,50,50,0.209003,"\int \frac{\left(-1+x^6\right) \left(1+x^6\right)}{\sqrt[4]{x-x^4+x^7} \left(1+3 x^6+x^{12}\right)} \, dx","IntegrateAlgebraic[((-1 + x^6)*(1 + x^6))/((x - x^4 + x^7)^(1/4)*(1 + 3*x^6 + x^12)),x]","\frac{1}{6} \text{RootSum}\left[2+2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{x-x^4+x^7}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{1}{6} \text{RootSum}\left[2+2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{x-x^4+x^7}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"RootSum[2 + 2*#1^4 + #1^8 & , (-Log[x] + Log[(x - x^4 + x^7)^(1/4) - x*#1])/#1 & ]/6","A",1
642,1,50,50,0.0802941,"\int \sqrt{1+\sqrt{1+x^2}} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[1 + x^2]],x]","\frac{4 x}{3 \sqrt{1+\sqrt{1+x^2}}}+\frac{2 x \sqrt{1+x^2}}{3 \sqrt{1+\sqrt{1+x^2}}}","\frac{4 x}{3 \sqrt{1+\sqrt{1+x^2}}}+\frac{2 x \sqrt{1+x^2}}{3 \sqrt{1+\sqrt{1+x^2}}}",1,"(4*x)/(3*Sqrt[1 + Sqrt[1 + x^2]]) + (2*x*Sqrt[1 + x^2])/(3*Sqrt[1 + Sqrt[1 + x^2]])","A",1
643,1,50,50,0.1095057,"\int \frac{\sqrt{1+\sqrt{1+x^2}}}{x^2} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[1 + x^2]]/x^2,x]","-\frac{\sqrt{1+\sqrt{1+x^2}}}{x}+\frac{\tan ^{-1}\left(\frac{x}{\sqrt{2} \sqrt{1+\sqrt{1+x^2}}}\right)}{\sqrt{2}}","-\frac{\sqrt{1+\sqrt{1+x^2}}}{x}+\frac{\tan ^{-1}\left(\frac{x}{\sqrt{2} \sqrt{1+\sqrt{1+x^2}}}\right)}{\sqrt{2}}",1,"-(Sqrt[1 + Sqrt[1 + x^2]]/x) + ArcTan[x/(Sqrt[2]*Sqrt[1 + Sqrt[1 + x^2]])]/Sqrt[2]","A",1
644,1,51,51,0.0622447,"\int \frac{1}{x \left(b+a x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x*(b + a*x^2)^(3/4)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b+a x^2}}{\sqrt[4]{b}}\right)}{b^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^2}}{\sqrt[4]{b}}\right)}{b^{3/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b+a x^2}}{\sqrt[4]{b}}\right)}{b^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^2}}{\sqrt[4]{b}}\right)}{b^{3/4}}",1,"-(ArcTan[(b + a*x^2)^(1/4)/b^(1/4)]/b^(3/4)) - ArcTanh[(b + a*x^2)^(1/4)/b^(1/4)]/b^(3/4)","A",1
645,1,51,51,0.1288281,"\int \frac{-7+x}{(-11+5 x) \sqrt{-60+83 x-21 x^2-3 x^3+x^4}} \, dx","IntegrateAlgebraic[(-7 + x)/((-11 + 5*x)*Sqrt[-60 + 83*x - 21*x^2 - 3*x^3 + x^4]),x]","-\frac{\tanh ^{-1}\left(\frac{6 \sqrt{6} \sqrt{-60+83 x-21 x^2-3 x^3+x^4}}{41-106 x+29 x^2}\right)}{3 \sqrt{6}}","-\frac{\tanh ^{-1}\left(\frac{6 \sqrt{6} \sqrt{-60+83 x-21 x^2-3 x^3+x^4}}{41-106 x+29 x^2}\right)}{3 \sqrt{6}}",1,"-1/3*ArcTanh[(6*Sqrt[6]*Sqrt[-60 + 83*x - 21*x^2 - 3*x^3 + x^4])/(41 - 106*x + 29*x^2)]/Sqrt[6]","A",1
646,1,51,51,0.0387527,"\int \frac{\left(-2+x^6\right) \sqrt{-1+x^6}}{x \left(2+x^6\right)} \, dx","IntegrateAlgebraic[((-2 + x^6)*Sqrt[-1 + x^6])/(x*(2 + x^6)),x]","\frac{1}{3} \sqrt{-1+x^6}+\frac{1}{3} \tan ^{-1}\left(\sqrt{-1+x^6}\right)-\frac{2 \tan ^{-1}\left(\frac{\sqrt{-1+x^6}}{\sqrt{3}}\right)}{\sqrt{3}}","\frac{1}{3} \sqrt{-1+x^6}+\frac{1}{3} \tan ^{-1}\left(\sqrt{-1+x^6}\right)-\frac{2 \tan ^{-1}\left(\frac{\sqrt{-1+x^6}}{\sqrt{3}}\right)}{\sqrt{3}}",1,"Sqrt[-1 + x^6]/3 + ArcTan[Sqrt[-1 + x^6]]/3 - (2*ArcTan[Sqrt[-1 + x^6]/Sqrt[3]])/Sqrt[3]","A",1
647,1,51,51,2.4880719,"\int \frac{\sqrt{-1-x^2+x^6} \left(1+2 x^6\right)}{\left(-1+x^6\right) \left(-2+x^2+2 x^6\right)} \, dx","IntegrateAlgebraic[(Sqrt[-1 - x^2 + x^6]*(1 + 2*x^6))/((-1 + x^6)*(-2 + x^2 + 2*x^6)),x]","\tan ^{-1}\left(\frac{x}{\sqrt{-1-x^2+x^6}}\right)-\sqrt{\frac{3}{2}} \tan ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt{-1-x^2+x^6}}\right)","\tan ^{-1}\left(\frac{x}{\sqrt{-1-x^2+x^6}}\right)-\sqrt{\frac{3}{2}} \tan ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt{-1-x^2+x^6}}\right)",1,"ArcTan[x/Sqrt[-1 - x^2 + x^6]] - Sqrt[3/2]*ArcTan[(Sqrt[3/2]*x)/Sqrt[-1 - x^2 + x^6]]","A",1
648,1,51,51,0.8817252,"\int \frac{x \left(-8 b+5 a x^3\right)}{\sqrt[4]{-b+a x^3} \left(b-a x^3+x^8\right)} \, dx","IntegrateAlgebraic[(x*(-8*b + 5*a*x^3))/((-b + a*x^3)^(1/4)*(b - a*x^3 + x^8)),x]","2 \tan ^{-1}\left(\frac{\sqrt[4]{-b+a x^3}}{x^2}\right)+2 \tanh ^{-1}\left(\frac{x^2 \left(-b+a x^3\right)^{3/4}}{b-a x^3}\right)","2 \tan ^{-1}\left(\frac{\sqrt[4]{-b+a x^3}}{x^2}\right)+2 \tanh ^{-1}\left(\frac{x^2 \left(-b+a x^3\right)^{3/4}}{b-a x^3}\right)",1,"2*ArcTan[(-b + a*x^3)^(1/4)/x^2] + 2*ArcTanh[(x^2*(-b + a*x^3)^(3/4))/(b - a*x^3)]","A",1
649,1,51,51,0.1973793,"\int \frac{2+16 x-x^2-9 x^3}{\sqrt[4]{\frac{1+x}{-2+x^2}} \left(-2+x^2\right) \left(-3+2 x+7 x^2-7 x^3-9 x^4+9 x^5+5 x^6-5 x^7-x^8+x^9\right)} \, dx","IntegrateAlgebraic[(2 + 16*x - x^2 - 9*x^3)/(((1 + x)/(-2 + x^2))^(1/4)*(-2 + x^2)*(-3 + 2*x + 7*x^2 - 7*x^3 - 9*x^4 + 9*x^5 + 5*x^6 - 5*x^7 - x^8 + x^9)),x]","-2 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1+x}{-2+x^2}}}{-1+x^2}\right)+2 \tanh ^{-1}\left(\frac{-1+x^2}{\sqrt[4]{\frac{1+x}{-2+x^2}}}\right)","-2 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1+x}{-2+x^2}}}{-1+x^2}\right)+2 \tanh ^{-1}\left(\frac{-1+x^2}{\sqrt[4]{\frac{1+x}{-2+x^2}}}\right)",1,"-2*ArcTan[((1 + x)/(-2 + x^2))^(1/4)/(-1 + x^2)] + 2*ArcTanh[(-1 + x^2)/((1 + x)/(-2 + x^2))^(1/4)]","A",1
650,1,51,51,0.5425104,"\int \frac{-1+7 x^8}{\left(1+x^8\right) \sqrt{3-x+x^2+6 x^8-x^9+3 x^{16}}} \, dx","IntegrateAlgebraic[(-1 + 7*x^8)/((1 + x^8)*Sqrt[3 - x + x^2 + 6*x^8 - x^9 + 3*x^16]),x]","2 \tanh ^{-1}\left(\frac{x}{\sqrt{3}+\sqrt{3} x^8-\sqrt{3-x+x^2+6 x^8-x^9+3 x^{16}}}\right)","2 \tanh ^{-1}\left(\frac{x}{\sqrt{3}+\sqrt{3} x^8-\sqrt{3-x+x^2+6 x^8-x^9+3 x^{16}}}\right)",1,"2*ArcTanh[x/(Sqrt[3] + Sqrt[3]*x^8 - Sqrt[3 - x + x^2 + 6*x^8 - x^9 + 3*x^16])]","A",1
651,1,51,51,0.0828313,"\int \frac{\sqrt{1+\sqrt{1+x^2}}}{x^2 \sqrt{1+x^2}} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[1 + x^2]]/(x^2*Sqrt[1 + x^2]),x]","-\frac{\sqrt{1+\sqrt{1+x^2}}}{x}-\frac{\tan ^{-1}\left(\frac{x}{\sqrt{2} \sqrt{1+\sqrt{1+x^2}}}\right)}{\sqrt{2}}","-\frac{\sqrt{1+\sqrt{1+x^2}}}{x}-\frac{\tan ^{-1}\left(\frac{x}{\sqrt{2} \sqrt{1+\sqrt{1+x^2}}}\right)}{\sqrt{2}}",1,"-(Sqrt[1 + Sqrt[1 + x^2]]/x) - ArcTan[x/(Sqrt[2]*Sqrt[1 + Sqrt[1 + x^2]])]/Sqrt[2]","A",1
652,1,52,51,0.0850045,"\int \frac{x}{x+\sqrt{1+\sqrt{1+x}}} \, dx","IntegrateAlgebraic[x/(x + Sqrt[1 + Sqrt[1 + x]]),x]","1+x-4 \sqrt{1+\sqrt{1+x}}+\frac{8 \tanh ^{-1}\left(\frac{1}{\sqrt{5}}+\frac{2 \sqrt{1+\sqrt{1+x}}}{\sqrt{5}}\right)}{\sqrt{5}}","x-4 \sqrt{1+\sqrt{1+x}}+\frac{8 \tanh ^{-1}\left(\frac{1}{\sqrt{5}}+\frac{2 \sqrt{1+\sqrt{1+x}}}{\sqrt{5}}\right)}{\sqrt{5}}",1,"1 + x - 4*Sqrt[1 + Sqrt[1 + x]] + (8*ArcTanh[1/Sqrt[5] + (2*Sqrt[1 + Sqrt[1 + x]])/Sqrt[5]])/Sqrt[5]","A",1
653,1,52,52,0.1258159,"\int \frac{a+2 x}{\sqrt[4]{b+a x+x^2} \left(-1+2 b+2 a x+2 x^2\right)} \, dx","IntegrateAlgebraic[(a + 2*x)/((b + a*x + x^2)^(1/4)*(-1 + 2*b + 2*a*x + 2*x^2)),x]","\sqrt[4]{2} \tan ^{-1}\left(\sqrt[4]{2} \sqrt[4]{b+a x+x^2}\right)-\sqrt[4]{2} \tanh ^{-1}\left(\sqrt[4]{2} \sqrt[4]{b+a x+x^2}\right)","\sqrt[4]{2} \tan ^{-1}\left(\sqrt[4]{2} \sqrt[4]{b+a x+x^2}\right)-\sqrt[4]{2} \tanh ^{-1}\left(\sqrt[4]{2} \sqrt[4]{b+a x+x^2}\right)",1,"2^(1/4)*ArcTan[2^(1/4)*(b + a*x + x^2)^(1/4)] - 2^(1/4)*ArcTanh[2^(1/4)*(b + a*x + x^2)^(1/4)]","A",1
654,1,52,52,0.0573028,"\int \frac{1}{x^7 \sqrt[4]{1+x^3}} \, dx","IntegrateAlgebraic[1/(x^7*(1 + x^3)^(1/4)),x]","\frac{\left(1+x^3\right)^{3/4} \left(-4+5 x^3\right)}{24 x^6}+\frac{5}{48} \tan ^{-1}\left(\sqrt[4]{1+x^3}\right)-\frac{5}{48} \tanh ^{-1}\left(\sqrt[4]{1+x^3}\right)","\frac{\left(1+x^3\right)^{3/4} \left(-4+5 x^3\right)}{24 x^6}+\frac{5}{48} \tan ^{-1}\left(\sqrt[4]{1+x^3}\right)-\frac{5}{48} \tanh ^{-1}\left(\sqrt[4]{1+x^3}\right)",1,"((1 + x^3)^(3/4)*(-4 + 5*x^3))/(24*x^6) + (5*ArcTan[(1 + x^3)^(1/4)])/48 - (5*ArcTanh[(1 + x^3)^(1/4)])/48","A",1
655,1,52,52,0.0627114,"\int \frac{\sqrt[4]{1+x^3}}{x^7} \, dx","IntegrateAlgebraic[(1 + x^3)^(1/4)/x^7,x]","\frac{\left(-4-x^3\right) \sqrt[4]{1+x^3}}{24 x^6}+\frac{1}{16} \tan ^{-1}\left(\sqrt[4]{1+x^3}\right)+\frac{1}{16} \tanh ^{-1}\left(\sqrt[4]{1+x^3}\right)","\frac{\left(-4-x^3\right) \sqrt[4]{1+x^3}}{24 x^6}+\frac{1}{16} \tan ^{-1}\left(\sqrt[4]{1+x^3}\right)+\frac{1}{16} \tanh ^{-1}\left(\sqrt[4]{1+x^3}\right)",1,"((-4 - x^3)*(1 + x^3)^(1/4))/(24*x^6) + ArcTan[(1 + x^3)^(1/4)]/16 + ArcTanh[(1 + x^3)^(1/4)]/16","A",1
656,1,52,52,0.0451535,"\int \frac{\sqrt{b+a x^3} \left(2 b+a x^3\right)}{x} \, dx","IntegrateAlgebraic[(Sqrt[b + a*x^3]*(2*b + a*x^3))/x,x]","\frac{2}{9} \sqrt{b+a x^3} \left(7 b+a x^3\right)-\frac{4}{3} b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b+a x^3}}{\sqrt{b}}\right)","\frac{2}{9} \sqrt{b+a x^3} \left(7 b+a x^3\right)-\frac{4}{3} b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b+a x^3}}{\sqrt{b}}\right)",1,"(2*Sqrt[b + a*x^3]*(7*b + a*x^3))/9 - (4*b^(3/2)*ArcTanh[Sqrt[b + a*x^3]/Sqrt[b]])/3","A",1
657,1,54,52,0.2115058,"\int \frac{\left(1+x^3\right) \sqrt{-1+x^6}}{x^{13} \left(-1+x^3\right)} \, dx","IntegrateAlgebraic[((1 + x^3)*Sqrt[-1 + x^6])/(x^13*(-1 + x^3)),x]","\frac{\sqrt{-1+x^6} \left(6+16 x^3+21 x^6+32 x^9\right)}{72 x^{12}}-\frac{7}{12} \tan ^{-1}\left(x^3-\sqrt{-1+x^6}\right)","\frac{\sqrt{-1+x^6} \left(6+16 x^3+21 x^6+32 x^9\right)}{72 x^{12}}+\frac{7}{12} \tan ^{-1}\left(x^3+\sqrt{-1+x^6}\right)",1,"(Sqrt[-1 + x^6]*(6 + 16*x^3 + 21*x^6 + 32*x^9))/(72*x^12) - (7*ArcTan[x^3 - Sqrt[-1 + x^6]])/12","A",1
658,1,52,52,0.4910325,"\int \frac{\sqrt{x+x^4} \left(-b+a x^6\right)}{x^6} \, dx","IntegrateAlgebraic[(Sqrt[x + x^4]*(-b + a*x^6))/x^6,x]","\frac{\sqrt{x+x^4} \left(2 b+2 b x^3+3 a x^6\right)}{9 x^5}+\frac{1}{3} a \tanh ^{-1}\left(\frac{x^2}{\sqrt{x+x^4}}\right)","\frac{\sqrt{x+x^4} \left(2 b+2 b x^3+3 a x^6\right)}{9 x^5}+\frac{1}{3} a \tanh ^{-1}\left(\frac{x^2}{\sqrt{x+x^4}}\right)",1,"(Sqrt[x + x^4]*(2*b + 2*b*x^3 + 3*a*x^6))/(9*x^5) + (a*ArcTanh[x^2/Sqrt[x + x^4]])/3","A",1
659,1,52,52,2.2351405,"\int \frac{-7 b x+5 a x^3}{\sqrt[4]{-b x+a x^3} \left(b-a x^2+x^7\right)} \, dx","IntegrateAlgebraic[(-7*b*x + 5*a*x^3)/((-(b*x) + a*x^3)^(1/4)*(b - a*x^2 + x^7)),x]","2 \tan ^{-1}\left(\frac{\sqrt[4]{-b x+a x^3}}{x^2}\right)-2 \tanh ^{-1}\left(\frac{x \left(-b x+a x^3\right)^{3/4}}{-b+a x^2}\right)","2 \tan ^{-1}\left(\frac{\sqrt[4]{-b x+a x^3}}{x^2}\right)-2 \tanh ^{-1}\left(\frac{x \left(-b x+a x^3\right)^{3/4}}{-b+a x^2}\right)",1,"2*ArcTan[(-(b*x) + a*x^3)^(1/4)/x^2] - 2*ArcTanh[(x*(-(b*x) + a*x^3)^(3/4))/(-b + a*x^2)]","A",1
660,1,52,52,2.1712405,"\int \frac{x+3 x^5}{\sqrt{-x+x^5} \left(1-a x^2-2 x^4+x^8\right)} \, dx","IntegrateAlgebraic[(x + 3*x^5)/(Sqrt[-x + x^5]*(1 - a*x^2 - 2*x^4 + x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt{-x+x^5}}\right)}{a^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt{-x+x^5}}\right)}{a^{3/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt{-x+x^5}}\right)}{a^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt{-x+x^5}}\right)}{a^{3/4}}",1,"ArcTan[(a^(1/4)*x)/Sqrt[-x + x^5]]/a^(3/4) - ArcTanh[(a^(1/4)*x)/Sqrt[-x + x^5]]/a^(3/4)","A",1
661,1,52,52,2.6791417,"\int \frac{x+3 x^5}{\sqrt{-x+x^5} \left(a-x^2-2 a x^4+a x^8\right)} \, dx","IntegrateAlgebraic[(x + 3*x^5)/(Sqrt[-x + x^5]*(a - x^2 - 2*a*x^4 + a*x^8)),x]","\frac{\tan ^{-1}\left(\frac{x}{\sqrt[4]{a} \sqrt{-x+x^5}}\right)}{\sqrt[4]{a}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[4]{a} \sqrt{-x+x^5}}\right)}{\sqrt[4]{a}}","\frac{\tan ^{-1}\left(\frac{x}{\sqrt[4]{a} \sqrt{-x+x^5}}\right)}{\sqrt[4]{a}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[4]{a} \sqrt{-x+x^5}}\right)}{\sqrt[4]{a}}",1,"ArcTan[x/(a^(1/4)*Sqrt[-x + x^5])]/a^(1/4) - ArcTanh[x/(a^(1/4)*Sqrt[-x + x^5])]/a^(1/4)","A",1
662,1,52,52,2.7282606,"\int \frac{x+4 x^6}{\sqrt{-x+x^6} \left(1-a x^2-2 x^5+x^{10}\right)} \, dx","IntegrateAlgebraic[(x + 4*x^6)/(Sqrt[-x + x^6]*(1 - a*x^2 - 2*x^5 + x^10)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt{-x+x^6}}\right)}{a^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt{-x+x^6}}\right)}{a^{3/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt{-x+x^6}}\right)}{a^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt{-x+x^6}}\right)}{a^{3/4}}",1,"ArcTan[(a^(1/4)*x)/Sqrt[-x + x^6]]/a^(3/4) - ArcTanh[(a^(1/4)*x)/Sqrt[-x + x^6]]/a^(3/4)","A",1
663,1,52,52,2.7227595,"\int \frac{x+4 x^6}{\sqrt{-x+x^6} \left(a-x^2-2 a x^5+a x^{10}\right)} \, dx","IntegrateAlgebraic[(x + 4*x^6)/(Sqrt[-x + x^6]*(a - x^2 - 2*a*x^5 + a*x^10)),x]","\frac{\tan ^{-1}\left(\frac{x}{\sqrt[4]{a} \sqrt{-x+x^6}}\right)}{\sqrt[4]{a}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[4]{a} \sqrt{-x+x^6}}\right)}{\sqrt[4]{a}}","\frac{\tan ^{-1}\left(\frac{x}{\sqrt[4]{a} \sqrt{-x+x^6}}\right)}{\sqrt[4]{a}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[4]{a} \sqrt{-x+x^6}}\right)}{\sqrt[4]{a}}",1,"ArcTan[x/(a^(1/4)*Sqrt[-x + x^6])]/a^(1/4) - ArcTanh[x/(a^(1/4)*Sqrt[-x + x^6])]/a^(1/4)","A",1
664,1,56,53,4.7073827,"\int \frac{1}{(-5+2 x)^2 \sqrt[4]{4-4 x+x^2}} \, dx","IntegrateAlgebraic[1/((-5 + 2*x)^2*(4 - 4*x + x^2)^(1/4)),x]","\frac{\left((-2+x)^2\right)^{3/4} \left(-\frac{\sqrt{-2+x}}{-1+2 (-2+x)}+\frac{\tanh ^{-1}\left(\sqrt{2} \sqrt{-2+x}\right)}{\sqrt{2}}\right)}{(-2+x)^{3/2}}","\frac{\left((-2+x)^2\right)^{3/4} \left(\frac{\sqrt{-2+x}}{5-2 x}+\frac{\tanh ^{-1}\left(\sqrt{2} \sqrt{-2+x}\right)}{\sqrt{2}}\right)}{(-2+x)^{3/2}}",1,"(((-2 + x)^2)^(3/4)*(-(Sqrt[-2 + x]/(-1 + 2*(-2 + x))) + ArcTanh[Sqrt[2]*Sqrt[-2 + x]]/Sqrt[2]))/(-2 + x)^(3/2)","A",1
665,1,53,53,0.6726153,"\int \frac{\left(-2+x^3\right) \sqrt{1+x^3} \left(2-x^2+2 x^3\right)}{x^4 \left(1-3 x^2+x^3\right)} \, dx","IntegrateAlgebraic[((-2 + x^3)*Sqrt[1 + x^3]*(2 - x^2 + 2*x^3))/(x^4*(1 - 3*x^2 + x^3)),x]","\frac{2 \sqrt{1+x^3} \left(2+15 x^2+2 x^3\right)}{3 x^3}-10 \sqrt{3} \tanh ^{-1}\left(\frac{\sqrt{3} x}{\sqrt{1+x^3}}\right)","\frac{2 \sqrt{1+x^3} \left(2+15 x^2+2 x^3\right)}{3 x^3}-10 \sqrt{3} \tanh ^{-1}\left(\frac{\sqrt{3} x}{\sqrt{1+x^3}}\right)",1,"(2*Sqrt[1 + x^3]*(2 + 15*x^2 + 2*x^3))/(3*x^3) - 10*Sqrt[3]*ArcTanh[(Sqrt[3]*x)/Sqrt[1 + x^3]]","A",1
666,1,53,53,0.0440561,"\int \frac{\left(-b+a x^3\right) \sqrt{b+a x^3}}{x} \, dx","IntegrateAlgebraic[((-b + a*x^3)*Sqrt[b + a*x^3])/x,x]","-\frac{2}{9} \left(2 b-a x^3\right) \sqrt{b+a x^3}+\frac{2}{3} b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b+a x^3}}{\sqrt{b}}\right)","-\frac{2}{9} \left(2 b-a x^3\right) \sqrt{b+a x^3}+\frac{2}{3} b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b+a x^3}}{\sqrt{b}}\right)",1,"(-2*(2*b - a*x^3)*Sqrt[b + a*x^3])/9 + (2*b^(3/2)*ArcTanh[Sqrt[b + a*x^3]/Sqrt[b]])/3","A",1
667,1,53,53,0.2663949,"\int \frac{x^2}{\left(-1+x^4\right) \sqrt{1+x^4}} \, dx","IntegrateAlgebraic[x^2/((-1 + x^4)*Sqrt[1 + x^4]),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{4 \sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{4 \sqrt{2}}","\frac{\tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{4 \sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{4 \sqrt{2}}",1,"ArcTan[(Sqrt[2]*x)/Sqrt[1 + x^4]]/(4*Sqrt[2]) - ArcTanh[(Sqrt[2]*x)/Sqrt[1 + x^4]]/(4*Sqrt[2])","A",1
668,1,53,53,0.232277,"\int \frac{\sqrt{1+x^4}}{-1+x^4} \, dx","IntegrateAlgebraic[Sqrt[1 + x^4]/(-1 + x^4),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{2 \sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{2 \sqrt{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{2 \sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{2 \sqrt{2}}",1,"-1/2*ArcTan[(Sqrt[2]*x)/Sqrt[1 + x^4]]/Sqrt[2] - ArcTanh[(Sqrt[2]*x)/Sqrt[1 + x^4]]/(2*Sqrt[2])","A",1
669,1,53,53,0.4670294,"\int \left(b+a x^3\right) \sqrt{x+x^4} \, dx","IntegrateAlgebraic[(b + a*x^3)*Sqrt[x + x^4],x]","\frac{1}{12} \sqrt{x+x^4} \left(a x+4 b x+2 a x^4\right)+\frac{1}{12} (-a+4 b) \tanh ^{-1}\left(\frac{x^2}{\sqrt{x+x^4}}\right)","\frac{1}{12} \sqrt{x+x^4} \left(a x+4 b x+2 a x^4\right)+\frac{1}{12} (-a+4 b) \tanh ^{-1}\left(\frac{x^2}{\sqrt{x+x^4}}\right)",1,"(Sqrt[x + x^4]*(a*x + 4*b*x + 2*a*x^4))/12 + ((-a + 4*b)*ArcTanh[x^2/Sqrt[x + x^4]])/12","A",1
670,1,53,53,0.1292626,"\int \sqrt[4]{x^2+x^4} \, dx","IntegrateAlgebraic[(x^2 + x^4)^(1/4),x]","\frac{1}{2} x \sqrt[4]{x^2+x^4}-\frac{1}{4} \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^2+x^4}}\right)+\frac{1}{4} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^2+x^4}}\right)","\frac{1}{2} x \sqrt[4]{x^2+x^4}-\frac{1}{4} \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^2+x^4}}\right)+\frac{1}{4} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^2+x^4}}\right)",1,"(x*(x^2 + x^4)^(1/4))/2 - ArcTan[x/(x^2 + x^4)^(1/4)]/4 + ArcTanh[x/(x^2 + x^4)^(1/4)]/4","A",1
671,1,53,53,0.6320598,"\int \frac{\left(-1+x^4\right) \sqrt{1+x^4}}{\left(1+3 x^2+x^4\right)^2} \, dx","IntegrateAlgebraic[((-1 + x^4)*Sqrt[1 + x^4])/(1 + 3*x^2 + x^4)^2,x]","-\frac{x \sqrt{1+x^4}}{2 \left(1+3 x^2+x^4\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{\sqrt{1+x^4}}\right)}{2 \sqrt{3}}","-\frac{x \sqrt{1+x^4}}{2 \left(1+3 x^2+x^4\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{\sqrt{1+x^4}}\right)}{2 \sqrt{3}}",1,"-1/2*(x*Sqrt[1 + x^4])/(1 + 3*x^2 + x^4) - ArcTan[(Sqrt[3]*x)/Sqrt[1 + x^4]]/(2*Sqrt[3])","A",1
672,1,53,53,0.2749103,"\int \frac{1}{x^8 \sqrt[4]{x^3+x^4}} \, dx","IntegrateAlgebraic[1/(x^8*(x^3 + x^4)^(1/4)),x]","\frac{4 \left(x^3+x^4\right)^{3/4} \left(-129789+134596 x-140448 x^2+147840 x^3-157696 x^4+172032 x^5-196608 x^6+262144 x^7\right)}{4023459 x^{10}}","\frac{4 \left(x^3+x^4\right)^{3/4} \left(-129789+134596 x-140448 x^2+147840 x^3-157696 x^4+172032 x^5-196608 x^6+262144 x^7\right)}{4023459 x^{10}}",1,"(4*(x^3 + x^4)^(3/4)*(-129789 + 134596*x - 140448*x^2 + 147840*x^3 - 157696*x^4 + 172032*x^5 - 196608*x^6 + 262144*x^7))/(4023459*x^10)","A",1
673,1,53,53,4.8154159,"\int \frac{1+4 x}{\sqrt{1-2 x+3 x^2+2 x^3+x^4}} \, dx","IntegrateAlgebraic[(1 + 4*x)/Sqrt[1 - 2*x + 3*x^2 + 2*x^3 + x^4],x]","-\log \left(-2 x-5 x^2-3 x^3-x^4+\left(2+2 x+x^2\right) \sqrt{1-2 x+3 x^2+2 x^3+x^4}\right)","-\log \left(-2 x-5 x^2-3 x^3-x^4+\left(2+2 x+x^2\right) \sqrt{1-2 x+3 x^2+2 x^3+x^4}\right)",1,"-Log[-2*x - 5*x^2 - 3*x^3 - x^4 + (2 + 2*x + x^2)*Sqrt[1 - 2*x + 3*x^2 + 2*x^3 + x^4]]","A",1
674,1,53,53,0.2210201,"\int \frac{1}{\sqrt[4]{-1+x^4} \left(1+3 x^4\right)} \, dx","IntegrateAlgebraic[1/((-1 + x^4)^(1/4)*(1 + 3*x^4)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt[4]{-1+x^4}}\right)}{2 \sqrt{2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt[4]{-1+x^4}}\right)}{2 \sqrt{2}}","\frac{\tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt[4]{-1+x^4}}\right)}{2 \sqrt{2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt[4]{-1+x^4}}\right)}{2 \sqrt{2}}",1,"ArcTan[(Sqrt[2]*x)/(-1 + x^4)^(1/4)]/(2*Sqrt[2]) + ArcTanh[(Sqrt[2]*x)/(-1 + x^4)^(1/4)]/(2*Sqrt[2])","A",1
675,1,53,53,1.1579123,"\int \frac{-3 b+a x^4}{\left(b-x^3+a x^4\right) \sqrt[4]{b x+a x^5}} \, dx","IntegrateAlgebraic[(-3*b + a*x^4)/((b - x^3 + a*x^4)*(b*x + a*x^5)^(1/4)),x]","-2 \tan ^{-1}\left(\frac{\left(b x+a x^5\right)^{3/4}}{b+a x^4}\right)-2 \tanh ^{-1}\left(\frac{\left(b x+a x^5\right)^{3/4}}{b+a x^4}\right)","-2 \tan ^{-1}\left(\frac{\left(b x+a x^5\right)^{3/4}}{b+a x^4}\right)-2 \tanh ^{-1}\left(\frac{\left(b x+a x^5\right)^{3/4}}{b+a x^4}\right)",1,"-2*ArcTan[(b*x + a*x^5)^(3/4)/(b + a*x^4)] - 2*ArcTanh[(b*x + a*x^5)^(3/4)/(b + a*x^4)]","A",1
676,1,53,53,0.1472184,"\int x^{20} \sqrt{-1+x^6} \, dx","IntegrateAlgebraic[x^20*Sqrt[-1 + x^6],x]","\frac{\sqrt{-1+x^6} \left(-15 x^3-10 x^9-8 x^{15}+48 x^{21}\right)}{1152}-\frac{5}{384} \log \left(x^3+\sqrt{-1+x^6}\right)","\frac{\sqrt{-1+x^6} \left(-15 x^3-10 x^9-8 x^{15}+48 x^{21}\right)}{1152}-\frac{5}{384} \log \left(x^3+\sqrt{-1+x^6}\right)",1,"(Sqrt[-1 + x^6]*(-15*x^3 - 10*x^9 - 8*x^15 + 48*x^21))/1152 - (5*Log[x^3 + Sqrt[-1 + x^6]])/384","A",1
677,1,57,53,0.1714795,"\int \frac{\left(-1+x^3\right) \sqrt{-1+x^6}}{x^4 \left(1+x^3\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)*Sqrt[-1 + x^6])/(x^4*(1 + x^3)),x]","\frac{\sqrt{-1+x^6}}{3 x^3}+\frac{4}{3} \tan ^{-1}\left(x^3-\sqrt{-1+x^6}\right)-\frac{1}{3} \log \left(-x^3+\sqrt{-1+x^6}\right)","\frac{\sqrt{-1+x^6}}{3 x^3}-\frac{4}{3} \tan ^{-1}\left(x^3+\sqrt{-1+x^6}\right)+\frac{1}{3} \log \left(x^3+\sqrt{-1+x^6}\right)",1,"Sqrt[-1 + x^6]/(3*x^3) + (4*ArcTan[x^3 - Sqrt[-1 + x^6]])/3 - Log[-x^3 + Sqrt[-1 + x^6]]/3","A",1
678,1,53,53,0.2771748,"\int \frac{1+x^6}{\sqrt{1-x^2+x^4} \left(1-x^6\right)} \, dx","IntegrateAlgebraic[(1 + x^6)/(Sqrt[1 - x^2 + x^4]*(1 - x^6)),x]","\frac{1}{3} \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1-x^2+x^4}}\right)+\frac{1}{3} \tanh ^{-1}\left(\frac{x}{\sqrt{1-x^2+x^4}}\right)","\frac{1}{3} \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1-x^2+x^4}}\right)+\frac{1}{3} \tanh ^{-1}\left(\frac{x}{\sqrt{1-x^2+x^4}}\right)",1,"(Sqrt[2]*ArcTan[(Sqrt[2]*x)/Sqrt[1 - x^2 + x^4]])/3 + ArcTanh[x/Sqrt[1 - x^2 + x^4]]/3","A",1
679,1,53,53,0.0587326,"\int \frac{\left(-2+x^6\right) \sqrt{-1+x^6}}{x^7 \left(2+x^6\right)} \, dx","IntegrateAlgebraic[((-2 + x^6)*Sqrt[-1 + x^6])/(x^7*(2 + x^6)),x]","\frac{\sqrt{-1+x^6}}{6 x^6}-\frac{1}{2} \tan ^{-1}\left(\sqrt{-1+x^6}\right)+\frac{\tan ^{-1}\left(\frac{\sqrt{-1+x^6}}{\sqrt{3}}\right)}{\sqrt{3}}","\frac{\sqrt{-1+x^6}}{6 x^6}-\frac{1}{2} \tan ^{-1}\left(\sqrt{-1+x^6}\right)+\frac{\tan ^{-1}\left(\frac{\sqrt{-1+x^6}}{\sqrt{3}}\right)}{\sqrt{3}}",1,"Sqrt[-1 + x^6]/(6*x^6) - ArcTan[Sqrt[-1 + x^6]]/2 + ArcTan[Sqrt[-1 + x^6]/Sqrt[3]]/Sqrt[3]","A",1
680,1,53,53,0.660852,"\int \frac{x (-6 b+5 a x)}{\sqrt[4]{-b x^2+a x^3} \left(b-a x+x^6\right)} \, dx","IntegrateAlgebraic[(x*(-6*b + 5*a*x))/((-(b*x^2) + a*x^3)^(1/4)*(b - a*x + x^6)),x]","2 \tan ^{-1}\left(\frac{\sqrt[4]{-b x^2+a x^3}}{x^2}\right)-2 \tanh ^{-1}\left(\frac{\left(-b x^2+a x^3\right)^{3/4}}{-b+a x}\right)","2 \tan ^{-1}\left(\frac{\sqrt[4]{-b x^2+a x^3}}{x^2}\right)-2 \tanh ^{-1}\left(\frac{\left(-b x^2+a x^3\right)^{3/4}}{-b+a x}\right)",1,"2*ArcTan[(-(b*x^2) + a*x^3)^(1/4)/x^2] - 2*ArcTanh[(-(b*x^2) + a*x^3)^(3/4)/(-b + a*x)]","A",1
681,1,53,53,0.0668245,"\int \frac{\sqrt{1+x^3} \left(2+2 x^3+x^6\right)}{x \left(-1+x^6\right)} \, dx","IntegrateAlgebraic[(Sqrt[1 + x^3]*(2 + 2*x^3 + x^6))/(x*(-1 + x^6)),x]","\frac{2 \sqrt{1+x^3}}{3}+\frac{4}{3} \tanh ^{-1}\left(\sqrt{1+x^3}\right)-\frac{5}{3} \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{1+x^3}}{\sqrt{2}}\right)","\frac{2 \sqrt{1+x^3}}{3}+\frac{4}{3} \tanh ^{-1}\left(\sqrt{1+x^3}\right)-\frac{5}{3} \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{1+x^3}}{\sqrt{2}}\right)",1,"(2*Sqrt[1 + x^3])/3 + (4*ArcTanh[Sqrt[1 + x^3]])/3 - (5*Sqrt[2]*ArcTanh[Sqrt[1 + x^3]/Sqrt[2]])/3","A",1
682,1,53,53,0.2868996,"\int \frac{\left(-1+x^4\right) \sqrt{1+x^4}}{1+x^8} \, dx","IntegrateAlgebraic[((-1 + x^4)*Sqrt[1 + x^4])/(1 + x^8),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{1+x^4}}\right)}{2 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{1+x^4}}\right)}{2 \sqrt[4]{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{1+x^4}}\right)}{2 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{1+x^4}}\right)}{2 \sqrt[4]{2}}",1,"-1/2*ArcTan[(2^(1/4)*x)/Sqrt[1 + x^4]]/2^(1/4) - ArcTanh[(2^(1/4)*x)/Sqrt[1 + x^4]]/(2*2^(1/4))","A",1
683,1,53,53,0.2739642,"\int \frac{-1+x^8}{\sqrt{1+x^4} \left(1+x^8\right)} \, dx","IntegrateAlgebraic[(-1 + x^8)/(Sqrt[1 + x^4]*(1 + x^8)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{1+x^4}}\right)}{2 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{1+x^4}}\right)}{2 \sqrt[4]{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{1+x^4}}\right)}{2 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{1+x^4}}\right)}{2 \sqrt[4]{2}}",1,"-1/2*ArcTan[(2^(1/4)*x)/Sqrt[1 + x^4]]/2^(1/4) - ArcTanh[(2^(1/4)*x)/Sqrt[1 + x^4]]/(2*2^(1/4))","A",1
684,1,53,53,0.3183725,"\int \frac{\sqrt{-2+x^4} \left(2+x^4\right)}{4-6 x^4+x^8} \, dx","IntegrateAlgebraic[(Sqrt[-2 + x^4]*(2 + x^4))/(4 - 6*x^4 + x^8),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{-2+x^4}}\right)}{2 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{-2+x^4}}\right)}{2 \sqrt[4]{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{-2+x^4}}\right)}{2 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{-2+x^4}}\right)}{2 \sqrt[4]{2}}",1,"-1/2*ArcTan[(2^(1/4)*x)/Sqrt[-2 + x^4]]/2^(1/4) - ArcTanh[(2^(1/4)*x)/Sqrt[-2 + x^4]]/(2*2^(1/4))","A",1
685,1,53,53,0.6977476,"\int \frac{-2 b+a x^4}{\sqrt[4]{-b+a x^4} \left(-b+a x^4+2 x^8\right)} \, dx","IntegrateAlgebraic[(-2*b + a*x^4)/((-b + a*x^4)^(1/4)*(-b + a*x^4 + 2*x^8)),x]","-\frac{1}{4} \text{RootSum}\left[2 b+a \text{$\#$1}^4-\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","-\frac{1}{4} \text{RootSum}\left[2 b+a \text{$\#$1}^4-\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"-1/4*RootSum[2*b + a*#1^4 - #1^8 & , (-Log[x] + Log[(-b + a*x^4)^(1/4) - x*#1])/#1 & ]","A",1
686,1,53,53,0.1229305,"\int \frac{1}{x^2 \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[1/(x^2*Sqrt[x + Sqrt[1 + x^2]]),x]","-\frac{1}{x \sqrt{x+\sqrt{1+x^2}}}+\tan ^{-1}\left(\sqrt{x+\sqrt{1+x^2}}\right)+\tanh ^{-1}\left(\sqrt{x+\sqrt{1+x^2}}\right)","-\frac{1}{x \sqrt{x+\sqrt{1+x^2}}}+\tan ^{-1}\left(\sqrt{x+\sqrt{1+x^2}}\right)+\tanh ^{-1}\left(\sqrt{x+\sqrt{1+x^2}}\right)",1,"-(1/(x*Sqrt[x + Sqrt[1 + x^2]])) + ArcTan[Sqrt[x + Sqrt[1 + x^2]]] + ArcTanh[Sqrt[x + Sqrt[1 + x^2]]]","A",1
687,1,54,54,3.3581423,"\int \frac{-1+\sqrt{k} x}{\left(1+\sqrt{k} x\right) \sqrt{\left(1-x^2\right) \left(1-k^2 x^2\right)}} \, dx","IntegrateAlgebraic[(-1 + Sqrt[k]*x)/((1 + Sqrt[k]*x)*Sqrt[(1 - x^2)*(1 - k^2*x^2)]),x]","-\frac{2 \tan ^{-1}\left(\frac{(-1+k) x}{1+2 \sqrt{k} x+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{-1+k}","-\frac{2 \tan ^{-1}\left(\frac{(-1+k) x}{1+2 \sqrt{k} x+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{-1+k}",1,"(-2*ArcTan[((-1 + k)*x)/(1 + 2*Sqrt[k]*x + k*x^2 + Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4])])/(-1 + k)","A",1
688,1,54,54,3.3443484,"\int \frac{1+\sqrt{k} x}{\left(-1+\sqrt{k} x\right) \sqrt{\left(1-x^2\right) \left(1-k^2 x^2\right)}} \, dx","IntegrateAlgebraic[(1 + Sqrt[k]*x)/((-1 + Sqrt[k]*x)*Sqrt[(1 - x^2)*(1 - k^2*x^2)]),x]","-\frac{2 \tan ^{-1}\left(\frac{(-1+k) x}{1-2 \sqrt{k} x+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{-1+k}","-\frac{2 \tan ^{-1}\left(\frac{(-1+k) x}{1-2 \sqrt{k} x+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{-1+k}",1,"(-2*ArcTan[((-1 + k)*x)/(1 - 2*Sqrt[k]*x + k*x^2 + Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4])])/(-1 + k)","A",1
689,1,54,54,0.1077673,"\int \frac{2+x}{(-1+x) \sqrt{-1+3 x-a x^2+x^3}} \, dx","IntegrateAlgebraic[(2 + x)/((-1 + x)*Sqrt[-1 + 3*x - a*x^2 + x^3]),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt{-3+a} x \sqrt{-1+3 x-a x^2+x^3}}{1-3 x+a x^2-x^3}\right)}{\sqrt{-3+a}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{-3+a} x \sqrt{-1+3 x-a x^2+x^3}}{1-3 x+a x^2-x^3}\right)}{\sqrt{-3+a}}",1,"(2*ArcTan[(Sqrt[-3 + a]*x*Sqrt[-1 + 3*x - a*x^2 + x^3])/(1 - 3*x + a*x^2 - x^3)])/Sqrt[-3 + a]","A",1
690,1,54,54,0.1095712,"\int \frac{-2+x}{(1+x) \sqrt{1+3 x-a x^2+x^3}} \, dx","IntegrateAlgebraic[(-2 + x)/((1 + x)*Sqrt[1 + 3*x - a*x^2 + x^3]),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt{3+a} x \sqrt{1+3 x-a x^2+x^3}}{-1-3 x+a x^2-x^3}\right)}{\sqrt{3+a}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{3+a} x \sqrt{1+3 x-a x^2+x^3}}{-1-3 x+a x^2-x^3}\right)}{\sqrt{3+a}}",1,"(2*ArcTan[(Sqrt[3 + a]*x*Sqrt[1 + 3*x - a*x^2 + x^3])/(-1 - 3*x + a*x^2 - x^3)])/Sqrt[3 + a]","A",1
691,1,54,54,0.1725323,"\int \frac{\left(-1+x^4\right) \sqrt[4]{1+x^4}}{x^2} \, dx","IntegrateAlgebraic[((-1 + x^4)*(1 + x^4)^(1/4))/x^2,x]","\frac{\sqrt[4]{1+x^4} \left(4+x^4\right)}{4 x}+\frac{3}{8} \tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\frac{3}{8} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)","\frac{\sqrt[4]{1+x^4} \left(4+x^4\right)}{4 x}+\frac{3}{8} \tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\frac{3}{8} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)",1,"((1 + x^4)^(1/4)*(4 + x^4))/(4*x) + (3*ArcTan[x/(1 + x^4)^(1/4)])/8 - (3*ArcTanh[x/(1 + x^4)^(1/4)])/8","A",1
692,1,54,54,0.2048743,"\int \frac{2 b+a x^2}{\sqrt[4]{b+a x^2} \left(-2 b-2 a x^2+x^4\right)} \, dx","IntegrateAlgebraic[(2*b + a*x^2)/((b + a*x^2)^(1/4)*(-2*b - 2*a*x^2 + x^4)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{b+a x^2}}{x}\right)}{2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[4]{2} \sqrt[4]{b+a x^2}}\right)}{2^{3/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{b+a x^2}}{x}\right)}{2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[4]{2} \sqrt[4]{b+a x^2}}\right)}{2^{3/4}}",1,"ArcTan[(2^(1/4)*(b + a*x^2)^(1/4))/x]/2^(3/4) - ArcTanh[x/(2^(1/4)*(b + a*x^2)^(1/4))]/2^(3/4)","A",1
693,1,54,54,0.224846,"\int \frac{\sqrt[4]{-x^3+x^4}}{x} \, dx","IntegrateAlgebraic[(-x^3 + x^4)^(1/4)/x,x]","\sqrt[4]{-x^3+x^4}+\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)-\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)","\sqrt[4]{-x^3+x^4}+\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)-\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)",1,"(-x^3 + x^4)^(1/4) + ArcTan[x/(-x^3 + x^4)^(1/4)]/2 - ArcTanh[x/(-x^3 + x^4)^(1/4)]/2","A",1
694,1,56,54,0.1699815,"\int \frac{\left(-1+x^3\right) \sqrt{-1+x^6}}{x^{13}} \, dx","IntegrateAlgebraic[((-1 + x^3)*Sqrt[-1 + x^6])/x^13,x]","\frac{\sqrt{-1+x^6} \left(6-8 x^3-3 x^6+8 x^9\right)}{72 x^{12}}+\frac{1}{12} \tan ^{-1}\left(\frac{\sqrt{-1+x^6}}{-1+x^3}\right)","\frac{\sqrt{-1+x^6} \left(6-8 x^3-3 x^6+8 x^9\right)}{72 x^{12}}+\frac{1}{12} \tan ^{-1}\left(\frac{1+x^3}{\sqrt{-1+x^6}}\right)",1,"(Sqrt[-1 + x^6]*(6 - 8*x^3 - 3*x^6 + 8*x^9))/(72*x^12) + ArcTan[Sqrt[-1 + x^6]/(-1 + x^3)]/12","A",1
695,1,54,54,0.4643099,"\int \frac{\left(2+x^3\right) \sqrt[3]{x-x^4}}{1+x^2-2 x^3-x^4-x^5+x^6} \, dx","IntegrateAlgebraic[((2 + x^3)*(x - x^4)^(1/3))/(1 + x^2 - 2*x^3 - x^4 - x^5 + x^6),x]","-\text{RootSum}\left[-1+\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{x-x^4}-x \text{$\#$1}\right) \text{$\#$1}}{1+2 \text{$\#$1}^3}\&\right]","-\text{RootSum}\left[-1+\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{x-x^4}-x \text{$\#$1}\right) \text{$\#$1}}{1+2 \text{$\#$1}^3}\&\right]",1,"-RootSum[-1 + #1^3 + #1^6 & , (-(Log[x]*#1) + Log[(x - x^4)^(1/3) - x*#1]*#1)/(1 + 2*#1^3) & ]","A",1
696,1,54,54,0.001304,"\int \frac{\left(2+x^3\right) \sqrt[3]{x-x^4}}{1+x^2-2 x^3-x^4-x^5+x^6} \, dx","IntegrateAlgebraic[((2 + x^3)*(x - x^4)^(1/3))/(1 + x^2 - 2*x^3 - x^4 - x^5 + x^6),x]","-\text{RootSum}\left[-1+\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{x-x^4}-x \text{$\#$1}\right) \text{$\#$1}}{1+2 \text{$\#$1}^3}\&\right]","-\text{RootSum}\left[-1+\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{x-x^4}-x \text{$\#$1}\right) \text{$\#$1}}{1+2 \text{$\#$1}^3}\&\right]",1,"-RootSum[-1 + #1^3 + #1^6 & , (-(Log[x]*#1) + Log[(x - x^4)^(1/3) - x*#1]*#1)/(1 + 2*#1^3) & ]","A",1
697,1,54,54,16.2040437,"\int \frac{1}{\sqrt[8]{1+2 x^4+x^8}} \, dx","IntegrateAlgebraic[(1 + 2*x^4 + x^8)^(-1/8),x]","\frac{\left(\left(1+x^4\right)^2\right)^{7/8} \left(\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)+\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)\right)}{\left(1+x^4\right)^{7/4}}","\frac{\left(\left(1+x^4\right)^2\right)^{7/8} \left(\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)+\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)\right)}{\left(1+x^4\right)^{7/4}}",1,"(((1 + x^4)^2)^(7/8)*(ArcTan[x/(1 + x^4)^(1/4)]/2 + ArcTanh[x/(1 + x^4)^(1/4)]/2))/(1 + x^4)^(7/4)","A",1
698,1,54,54,0.0757613,"\int \frac{1}{x \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[1/(x*Sqrt[x + Sqrt[1 + x^2]]),x]","\frac{2}{\sqrt{x+\sqrt{1+x^2}}}+2 \tan ^{-1}\left(\sqrt{x+\sqrt{1+x^2}}\right)-2 \tanh ^{-1}\left(\sqrt{x+\sqrt{1+x^2}}\right)","\frac{2}{\sqrt{x+\sqrt{1+x^2}}}+2 \tan ^{-1}\left(\sqrt{x+\sqrt{1+x^2}}\right)-2 \tanh ^{-1}\left(\sqrt{x+\sqrt{1+x^2}}\right)",1,"2/Sqrt[x + Sqrt[1 + x^2]] + 2*ArcTan[Sqrt[x + Sqrt[1 + x^2]]] - 2*ArcTanh[Sqrt[x + Sqrt[1 + x^2]]]","A",1
699,1,55,55,0.228047,"\int \frac{1-2 x+k^2 x^2}{\sqrt{(1-x) x \left(1-k^2 x\right)} \left(-1+2 x+\left(-2+k^2\right) x^2\right)} \, dx","IntegrateAlgebraic[(1 - 2*x + k^2*x^2)/(Sqrt[(1 - x)*x*(1 - k^2*x)]*(-1 + 2*x + (-2 + k^2)*x^2)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt{-2+k^2} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{-1+k^2 x}\right)}{\sqrt{-2+k^2}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{-2+k^2} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{-1+k^2 x}\right)}{\sqrt{-2+k^2}}",1,"(2*ArcTan[(Sqrt[-2 + k^2]*Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3])/(-1 + k^2*x)])/Sqrt[-2 + k^2]","A",1
700,1,55,55,0.0532851,"\int \frac{1}{x \left(b+a x^3\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x*(b + a*x^3)^(3/4)),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^3}}{\sqrt[4]{b}}\right)}{3 b^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^3}}{\sqrt[4]{b}}\right)}{3 b^{3/4}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^3}}{\sqrt[4]{b}}\right)}{3 b^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^3}}{\sqrt[4]{b}}\right)}{3 b^{3/4}}",1,"(-2*ArcTan[(b + a*x^3)^(1/4)/b^(1/4)])/(3*b^(3/4)) - (2*ArcTanh[(b + a*x^3)^(1/4)/b^(1/4)])/(3*b^(3/4))","A",1
701,1,55,55,0.0681676,"\int \frac{\left(-b+a x^3\right) \sqrt{b+a x^3}}{x^4} \, dx","IntegrateAlgebraic[((-b + a*x^3)*Sqrt[b + a*x^3])/x^4,x]","\frac{\sqrt{b+a x^3} \left(b+2 a x^3\right)}{3 x^3}-\frac{1}{3} a \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b+a x^3}}{\sqrt{b}}\right)","\frac{\sqrt{b+a x^3} \left(b+2 a x^3\right)}{3 x^3}-\frac{1}{3} a \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b+a x^3}}{\sqrt{b}}\right)",1,"(Sqrt[b + a*x^3]*(b + 2*a*x^3))/(3*x^3) - (a*Sqrt[b]*ArcTanh[Sqrt[b + a*x^3]/Sqrt[b]])/3","A",1
702,1,55,55,0.1974836,"\int \frac{1-2 x+k^2 x^2}{\left(-1+2 x-2 x^2+k^2 x^2\right) \sqrt{x-x^2-k^2 x^2+k^2 x^3}} \, dx","IntegrateAlgebraic[(1 - 2*x + k^2*x^2)/((-1 + 2*x - 2*x^2 + k^2*x^2)*Sqrt[x - x^2 - k^2*x^2 + k^2*x^3]),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt{-2+k^2} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{-1+k^2 x}\right)}{\sqrt{-2+k^2}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{-2+k^2} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{-1+k^2 x}\right)}{\sqrt{-2+k^2}}",1,"(2*ArcTan[(Sqrt[-2 + k^2]*Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3])/(-1 + k^2*x)])/Sqrt[-2 + k^2]","A",1
703,1,55,55,4.2787982,"\int \frac{2+x}{x \sqrt[4]{1+x^4}} \, dx","IntegrateAlgebraic[(2 + x)/(x*(1 + x^4)^(1/4)),x]","\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)+\tan ^{-1}\left(\sqrt[4]{1+x^4}\right)+\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\tanh ^{-1}\left(\sqrt[4]{1+x^4}\right)","\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)+\tan ^{-1}\left(\sqrt[4]{1+x^4}\right)+\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\tanh ^{-1}\left(\sqrt[4]{1+x^4}\right)",1,"ArcTan[x/(1 + x^4)^(1/4)]/2 + ArcTan[(1 + x^4)^(1/4)] + ArcTanh[x/(1 + x^4)^(1/4)]/2 - ArcTanh[(1 + x^4)^(1/4)]","A",1
704,1,55,55,0.1755929,"\int x^6 \sqrt[4]{1+x^4} \, dx","IntegrateAlgebraic[x^6*(1 + x^4)^(1/4),x]","\frac{1}{32} \sqrt[4]{1+x^4} \left(x^3+4 x^7\right)+\frac{3}{64} \tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\frac{3}{64} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)","\frac{1}{32} \sqrt[4]{1+x^4} \left(x^3+4 x^7\right)+\frac{3}{64} \tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\frac{3}{64} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)",1,"((1 + x^4)^(1/4)*(x^3 + 4*x^7))/32 + (3*ArcTan[x/(1 + x^4)^(1/4)])/64 - (3*ArcTanh[x/(1 + x^4)^(1/4)])/64","A",1
705,1,55,55,0.214585,"\int \frac{\sqrt[4]{-x+x^4}}{x^2} \, dx","IntegrateAlgebraic[(-x + x^4)^(1/4)/x^2,x]","-\frac{4 \sqrt[4]{-x+x^4}}{3 x}-\frac{2}{3} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x+x^4}}\right)+\frac{2}{3} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x+x^4}}\right)","-\frac{4 \sqrt[4]{-x+x^4}}{3 x}-\frac{2}{3} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x+x^4}}\right)+\frac{2}{3} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x+x^4}}\right)",1,"(-4*(-x + x^4)^(1/4))/(3*x) - (2*ArcTan[x/(-x + x^4)^(1/4)])/3 + (2*ArcTanh[x/(-x + x^4)^(1/4)])/3","A",1
706,1,55,55,0.2200785,"\int x \sqrt[4]{-x+x^4} \, dx","IntegrateAlgebraic[x*(-x + x^4)^(1/4),x]","\frac{1}{3} x^2 \sqrt[4]{-x+x^4}+\frac{1}{6} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x+x^4}}\right)-\frac{1}{6} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x+x^4}}\right)","\frac{1}{3} x^2 \sqrt[4]{-x+x^4}+\frac{1}{6} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x+x^4}}\right)-\frac{1}{6} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x+x^4}}\right)",1,"(x^2*(-x + x^4)^(1/4))/3 + ArcTan[x/(-x + x^4)^(1/4)]/6 - ArcTanh[x/(-x + x^4)^(1/4)]/6","A",1
707,1,55,55,0.4588303,"\int \frac{\left(-b+a x^3\right) \sqrt{-x+x^4}}{x^3} \, dx","IntegrateAlgebraic[((-b + a*x^3)*Sqrt[-x + x^4])/x^3,x]","\frac{\left(2 b+a x^3\right) \sqrt{-x+x^4}}{3 x^2}+\frac{1}{3} (-a-2 b) \tanh ^{-1}\left(\frac{x^2}{\sqrt{-x+x^4}}\right)","\frac{\left(2 b+a x^3\right) \sqrt{-x+x^4}}{3 x^2}+\frac{1}{3} (-a-2 b) \tanh ^{-1}\left(\frac{x^2}{\sqrt{-x+x^4}}\right)",1,"((2*b + a*x^3)*Sqrt[-x + x^4])/(3*x^2) + ((-a - 2*b)*ArcTanh[x^2/Sqrt[-x + x^4]])/3","A",1
708,1,55,55,0.2158566,"\int \frac{\sqrt[4]{-x^3+x^4}}{x^2} \, dx","IntegrateAlgebraic[(-x^3 + x^4)^(1/4)/x^2,x]","-\frac{4 \sqrt[4]{-x^3+x^4}}{x}-2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)+2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)","-\frac{4 \sqrt[4]{-x^3+x^4}}{x}-2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)+2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)",1,"(-4*(-x^3 + x^4)^(1/4))/x - 2*ArcTan[x/(-x^3 + x^4)^(1/4)] + 2*ArcTanh[x/(-x^3 + x^4)^(1/4)]","A",1
709,1,55,55,4.7226186,"\int \frac{x}{\sqrt{-17+18 x-11 x^2+6 x^3+x^4}} \, dx","IntegrateAlgebraic[x/Sqrt[-17 + 18*x - 11*x^2 + 6*x^3 + x^4],x]","-\frac{1}{4} \log \left(-11-68 x^2-24 x^3-2 x^4+\left(34+18 x+2 x^2\right) \sqrt{-17+18 x-11 x^2+6 x^3+x^4}\right)","-\frac{1}{4} \log \left(-11-68 x^2-24 x^3-2 x^4+\left(34+18 x+2 x^2\right) \sqrt{-17+18 x-11 x^2+6 x^3+x^4}\right)",1,"-1/4*Log[-11 - 68*x^2 - 24*x^3 - 2*x^4 + (34 + 18*x + 2*x^2)*Sqrt[-17 + 18*x - 11*x^2 + 6*x^3 + x^4]]","A",1
710,1,55,55,0.2236808,"\int \frac{\sqrt{-1-x-x^2+x^4} \left(2+x+2 x^4\right)}{\left(-1-x+x^4\right) \left(-1-x+x^2+x^4\right)} \, dx","IntegrateAlgebraic[(Sqrt[-1 - x - x^2 + x^4]*(2 + x + 2*x^4))/((-1 - x + x^4)*(-1 - x + x^2 + x^4)),x]","2 \tan ^{-1}\left(\frac{x}{\sqrt{-1-x-x^2+x^4}}\right)-2 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{-1-x-x^2+x^4}}\right)","2 \tan ^{-1}\left(\frac{x}{\sqrt{-1-x-x^2+x^4}}\right)-2 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{-1-x-x^2+x^4}}\right)",1,"2*ArcTan[x/Sqrt[-1 - x - x^2 + x^4]] - 2*Sqrt[2]*ArcTan[(Sqrt[2]*x)/Sqrt[-1 - x - x^2 + x^4]]","A",1
711,1,55,55,0.0505283,"\int \frac{1}{x \left(b+a x^4\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x*(b + a*x^4)^(3/4)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b+a x^4}}{\sqrt[4]{b}}\right)}{2 b^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^4}}{\sqrt[4]{b}}\right)}{2 b^{3/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b+a x^4}}{\sqrt[4]{b}}\right)}{2 b^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^4}}{\sqrt[4]{b}}\right)}{2 b^{3/4}}",1,"-1/2*ArcTan[(b + a*x^4)^(1/4)/b^(1/4)]/b^(3/4) - ArcTanh[(b + a*x^4)^(1/4)/b^(1/4)]/(2*b^(3/4))","A",1
712,1,55,55,0.049485,"\int \frac{1}{x \sqrt[4]{b+a x^4}} \, dx","IntegrateAlgebraic[1/(x*(b + a*x^4)^(1/4)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b+a x^4}}{\sqrt[4]{b}}\right)}{2 \sqrt[4]{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^4}}{\sqrt[4]{b}}\right)}{2 \sqrt[4]{b}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b+a x^4}}{\sqrt[4]{b}}\right)}{2 \sqrt[4]{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^4}}{\sqrt[4]{b}}\right)}{2 \sqrt[4]{b}}",1,"ArcTan[(b + a*x^4)^(1/4)/b^(1/4)]/(2*b^(1/4)) - ArcTanh[(b + a*x^4)^(1/4)/b^(1/4)]/(2*b^(1/4))","A",1
713,1,55,55,0.0550632,"\int \frac{1}{x \left(b+a x^5\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x*(b + a*x^5)^(3/4)),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^5}}{\sqrt[4]{b}}\right)}{5 b^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^5}}{\sqrt[4]{b}}\right)}{5 b^{3/4}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^5}}{\sqrt[4]{b}}\right)}{5 b^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^5}}{\sqrt[4]{b}}\right)}{5 b^{3/4}}",1,"(-2*ArcTan[(b + a*x^5)^(1/4)/b^(1/4)])/(5*b^(3/4)) - (2*ArcTanh[(b + a*x^5)^(1/4)/b^(1/4)])/(5*b^(3/4))","A",1
714,1,63,55,0.1540515,"\int \frac{\left(1+2 x^3\right) \sqrt{-1+x^6}}{x} \, dx","IntegrateAlgebraic[((1 + 2*x^3)*Sqrt[-1 + x^6])/x,x]","\frac{1}{3} \left(1+x^3\right) \sqrt{-1+x^6}+\frac{2}{3} \tan ^{-1}\left(\frac{\sqrt{-1+x^6}}{-1+x^3}\right)-\frac{2}{3} \tanh ^{-1}\left(\frac{\sqrt{-1+x^6}}{-1+x^3}\right)","\frac{1}{3} \left(1+x^3\right) \sqrt{-1+x^6}-\frac{2}{3} \tan ^{-1}\left(x^3+\sqrt{-1+x^6}\right)-\frac{1}{3} \log \left(x^3+\sqrt{-1+x^6}\right)",1,"((1 + x^3)*Sqrt[-1 + x^6])/3 + (2*ArcTan[Sqrt[-1 + x^6]/(-1 + x^3)])/3 - (2*ArcTanh[Sqrt[-1 + x^6]/(-1 + x^3)])/3","A",1
715,1,55,55,0.1292576,"\int \frac{\left(-2+x^3\right) \sqrt[3]{x+x^3+x^4}}{1+x^2+2 x^3+x^4+x^5+x^6} \, dx","IntegrateAlgebraic[((-2 + x^3)*(x + x^3 + x^4)^(1/3))/(1 + x^2 + 2*x^3 + x^4 + x^5 + x^6),x]","\text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{x+x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-1+2 \text{$\#$1}^3}\&\right]","\text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{x+x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-1+2 \text{$\#$1}^3}\&\right]",1,"RootSum[1 - #1^3 + #1^6 & , (-(Log[x]*#1) + Log[(x + x^3 + x^4)^(1/3) - x*#1]*#1)/(-1 + 2*#1^3) & ]","A",1
716,1,55,55,0.0012739,"\int \frac{\left(-2+x^3\right) \sqrt[3]{x+x^3+x^4}}{1+x^2+2 x^3+x^4+x^5+x^6} \, dx","IntegrateAlgebraic[((-2 + x^3)*(x + x^3 + x^4)^(1/3))/(1 + x^2 + 2*x^3 + x^4 + x^5 + x^6),x]","\text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{x+x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-1+2 \text{$\#$1}^3}\&\right]","\text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{x+x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-1+2 \text{$\#$1}^3}\&\right]",1,"RootSum[1 - #1^3 + #1^6 & , (-(Log[x]*#1) + Log[(x + x^3 + x^4)^(1/3) - x*#1]*#1)/(-1 + 2*#1^3) & ]","A",1
717,1,55,55,0.0466612,"\int \frac{1}{x \left(b+a x^6\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x*(b + a*x^6)^(3/4)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b+a x^6}}{\sqrt[4]{b}}\right)}{3 b^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^6}}{\sqrt[4]{b}}\right)}{3 b^{3/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b+a x^6}}{\sqrt[4]{b}}\right)}{3 b^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^6}}{\sqrt[4]{b}}\right)}{3 b^{3/4}}",1,"-1/3*ArcTan[(b + a*x^6)^(1/4)/b^(1/4)]/b^(3/4) - ArcTanh[(b + a*x^6)^(1/4)/b^(1/4)]/(3*b^(3/4))","A",1
718,1,55,55,0.805389,"\int \frac{-2 b+a x^4}{\sqrt[4]{b+a x^4} \left(-b+a x^4+x^8\right)} \, dx","IntegrateAlgebraic[(-2*b + a*x^4)/((b + a*x^4)^(1/4)*(-b + a*x^4 + x^8)),x]","-\frac{1}{4} \text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","-\frac{1}{4} \text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"-1/4*RootSum[2*a^2 - b - 3*a*#1^4 + #1^8 & , (-Log[x] + Log[(b + a*x^4)^(1/4) - x*#1])/#1 & ]","A",1
719,1,55,55,0.0015555,"\int \frac{-2 b+a x^4}{\sqrt[4]{b+a x^4} \left(-b+a x^4+x^8\right)} \, dx","IntegrateAlgebraic[(-2*b + a*x^4)/((b + a*x^4)^(1/4)*(-b + a*x^4 + x^8)),x]","-\frac{1}{4} \text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","-\frac{1}{4} \text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"-1/4*RootSum[2*a^2 - b - 3*a*#1^4 + #1^8 & , (-Log[x] + Log[(b + a*x^4)^(1/4) - x*#1])/#1 & ]","A",1
720,1,55,55,0.4330432,"\int \frac{x^2}{\left(b+a x^4\right)^{3/4} \left(b^2+a^2 x^8\right)} \, dx","IntegrateAlgebraic[x^2/((b + a*x^4)^(3/4)*(b^2 + a^2*x^8)),x]","-\frac{\text{RootSum}\left[2 a^2-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}^3}\&\right]}{8 b^2}","-\frac{\text{RootSum}\left[2 a^2-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}^3}\&\right]}{8 b^2}",1,"-1/8*RootSum[2*a^2 - 2*a*#1^4 + #1^8 & , (-Log[x] + Log[(b + a*x^4)^(1/4) - x*#1])/#1^3 & ]/b^2","A",1
721,1,60,56,0.2088317,"\int \frac{1}{(-1+x) \sqrt{-\sqrt{x}+x}} \, dx","IntegrateAlgebraic[1/((-1 + x)*Sqrt[-Sqrt[x] + x]),x]","-\frac{2 \sqrt{-\sqrt{x}+x}}{-1+\sqrt{x}}+\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{-\sqrt{x}+x}}{-1+\sqrt{x}}\right)","-\frac{2 \sqrt{-\sqrt{x}+x}}{-1+\sqrt{x}}+\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{-\sqrt{x}+x}}{\sqrt{2} \sqrt{x}}\right)",1,"(-2*Sqrt[-Sqrt[x] + x])/(-1 + Sqrt[x]) + Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[-Sqrt[x] + x])/(-1 + Sqrt[x])]","A",1
722,1,56,56,0.1060401,"\int \frac{1}{(1+2 x) \sqrt[4]{1+2 x+2 x^2}} \, dx","IntegrateAlgebraic[1/((1 + 2*x)*(1 + 2*x + 2*x^2)^(1/4)),x]","\frac{\tan ^{-1}\left(\sqrt[4]{2} \sqrt[4]{1+2 x+2 x^2}\right)}{2^{3/4}}-\frac{\tanh ^{-1}\left(\sqrt[4]{2} \sqrt[4]{1+2 x+2 x^2}\right)}{2^{3/4}}","\frac{\tan ^{-1}\left(\sqrt[4]{2} \sqrt[4]{1+2 x+2 x^2}\right)}{2^{3/4}}-\frac{\tanh ^{-1}\left(\sqrt[4]{2} \sqrt[4]{1+2 x+2 x^2}\right)}{2^{3/4}}",1,"ArcTan[2^(1/4)*(1 + 2*x + 2*x^2)^(1/4)]/2^(3/4) - ArcTanh[2^(1/4)*(1 + 2*x + 2*x^2)^(1/4)]/2^(3/4)","A",1
723,1,56,56,0.0618999,"\int \frac{\sqrt{b+a x^3} \left(2 b+a x^3\right)}{x^4} \, dx","IntegrateAlgebraic[(Sqrt[b + a*x^3]*(2*b + a*x^3))/x^4,x]","\frac{2 \left(-b+a x^3\right) \sqrt{b+a x^3}}{3 x^3}-\frac{4}{3} a \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b+a x^3}}{\sqrt{b}}\right)","\frac{2 \left(-b+a x^3\right) \sqrt{b+a x^3}}{3 x^3}-\frac{4}{3} a \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b+a x^3}}{\sqrt{b}}\right)",1,"(2*(-b + a*x^3)*Sqrt[b + a*x^3])/(3*x^3) - (4*a*Sqrt[b]*ArcTanh[Sqrt[b + a*x^3]/Sqrt[b]])/3","A",1
724,1,56,56,0.553662,"\int \frac{\left(-2 q+p x^3\right) \sqrt{q+p x^3}}{x^2 \left(a q+b x^2+a p x^3\right)} \, dx","IntegrateAlgebraic[((-2*q + p*x^3)*Sqrt[q + p*x^3])/(x^2*(a*q + b*x^2 + a*p*x^3)),x]","\frac{2 \sqrt{q+p x^3}}{a x}+\frac{2 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a} \sqrt{q+p x^3}}\right)}{a^{3/2}}","\frac{2 \sqrt{q+p x^3}}{a x}+\frac{2 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a} \sqrt{q+p x^3}}\right)}{a^{3/2}}",1,"(2*Sqrt[q + p*x^3])/(a*x) + (2*Sqrt[b]*ArcTan[(Sqrt[b]*x)/(Sqrt[a]*Sqrt[q + p*x^3])])/a^(3/2)","A",1
725,1,56,56,4.8444887,"\int \frac{-1+2 x}{\sqrt{-8+4 x-3 x^2-10 x^3+x^4}} \, dx","IntegrateAlgebraic[(-1 + 2*x)/Sqrt[-8 + 4*x - 3*x^2 - 10*x^3 + x^4],x]","-\frac{1}{2} \log \left(-90-76 x+89 x^2-18 x^3+x^4+\left(-38+13 x-x^2\right) \sqrt{-8+4 x-3 x^2-10 x^3+x^4}\right)","-\frac{1}{2} \log \left(-90-76 x+89 x^2-18 x^3+x^4+\left(-38+13 x-x^2\right) \sqrt{-8+4 x-3 x^2-10 x^3+x^4}\right)",1,"-1/2*Log[-90 - 76*x + 89*x^2 - 18*x^3 + x^4 + (-38 + 13*x - x^2)*Sqrt[-8 + 4*x - 3*x^2 - 10*x^3 + x^4]]","A",1
726,1,56,56,4.8227092,"\int \frac{-1+x}{\sqrt{4-16 x+12 x^2-8 x^3+x^4}} \, dx","IntegrateAlgebraic[(-1 + x)/Sqrt[4 - 16*x + 12*x^2 - 8*x^3 + x^4],x]","-\frac{1}{4} \log \left(36-56 x+44 x^2-12 x^3+x^4+\left(-14+8 x-x^2\right) \sqrt{4-16 x+12 x^2-8 x^3+x^4}\right)","-\frac{1}{4} \log \left(36-56 x+44 x^2-12 x^3+x^4+\left(-14+8 x-x^2\right) \sqrt{4-16 x+12 x^2-8 x^3+x^4}\right)",1,"-1/4*Log[36 - 56*x + 44*x^2 - 12*x^3 + x^4 + (-14 + 8*x - x^2)*Sqrt[4 - 16*x + 12*x^2 - 8*x^3 + x^4]]","A",1
727,1,56,56,0.7791667,"\int \frac{\sqrt{q+p x^5} \left(-2 q+3 p x^5\right)}{x^2 \left(a q+b x^2+a p x^5\right)} \, dx","IntegrateAlgebraic[(Sqrt[q + p*x^5]*(-2*q + 3*p*x^5))/(x^2*(a*q + b*x^2 + a*p*x^5)),x]","\frac{2 \sqrt{q+p x^5}}{a x}+\frac{2 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a} \sqrt{q+p x^5}}\right)}{a^{3/2}}","\frac{2 \sqrt{q+p x^5}}{a x}+\frac{2 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a} \sqrt{q+p x^5}}\right)}{a^{3/2}}",1,"(2*Sqrt[q + p*x^5])/(a*x) + (2*Sqrt[b]*ArcTan[(Sqrt[b]*x)/(Sqrt[a]*Sqrt[q + p*x^5])])/a^(3/2)","A",1
728,1,56,56,2.7333708,"\int \frac{x^2 \left(-1+4 x^5\right)}{\left(1+x^5\right)^2 \left(a-x+a x^5\right) \sqrt{x+x^6}} \, dx","IntegrateAlgebraic[(x^2*(-1 + 4*x^5))/((1 + x^5)^2*(a - x + a*x^5)*Sqrt[x + x^6]),x]","\frac{2 \left(3 a+x+3 a x^5\right) \sqrt{x+x^6}}{3 \left(1+x^5\right)^2}-2 a^{3/2} \tanh ^{-1}\left(\frac{x}{\sqrt{a} \sqrt{x+x^6}}\right)","\frac{2 \left(3 a+x+3 a x^5\right) \sqrt{x+x^6}}{3 \left(1+x^5\right)^2}-2 a^{3/2} \tanh ^{-1}\left(\frac{x}{\sqrt{a} \sqrt{x+x^6}}\right)",1,"(2*(3*a + x + 3*a*x^5)*Sqrt[x + x^6])/(3*(1 + x^5)^2) - 2*a^(3/2)*ArcTanh[x/(Sqrt[a]*Sqrt[x + x^6])]","A",1
729,1,56,56,0.4757196,"\int \frac{\sqrt{-x+x^4} \left(-b+a x^6\right)}{x^6} \, dx","IntegrateAlgebraic[(Sqrt[-x + x^4]*(-b + a*x^6))/x^6,x]","\frac{\sqrt{-x+x^4} \left(2 b-2 b x^3+3 a x^6\right)}{9 x^5}-\frac{1}{3} a \tanh ^{-1}\left(\frac{x^2}{\sqrt{-x+x^4}}\right)","\frac{\sqrt{-x+x^4} \left(2 b-2 b x^3+3 a x^6\right)}{9 x^5}-\frac{1}{3} a \tanh ^{-1}\left(\frac{x^2}{\sqrt{-x+x^4}}\right)",1,"(Sqrt[-x + x^4]*(2*b - 2*b*x^3 + 3*a*x^6))/(9*x^5) - (a*ArcTanh[x^2/Sqrt[-x + x^4]])/3","A",1
730,1,56,56,11.5527098,"\int \frac{\left(-2 q+p x^6\right) \sqrt{q+p x^6}}{x^3 \left(a q+b x^4+a p x^6\right)} \, dx","IntegrateAlgebraic[((-2*q + p*x^6)*Sqrt[q + p*x^6])/(x^3*(a*q + b*x^4 + a*p*x^6)),x]","\frac{\sqrt{q+p x^6}}{a x^2}+\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a} \sqrt{q+p x^6}}\right)}{a^{3/2}}","\frac{\sqrt{q+p x^6}}{a x^2}+\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x^2}{\sqrt{a} \sqrt{q+p x^6}}\right)}{a^{3/2}}",1,"Sqrt[q + p*x^6]/(a*x^2) + (Sqrt[b]*ArcTan[(Sqrt[b]*x^2)/(Sqrt[a]*Sqrt[q + p*x^6])])/a^(3/2)","A",1
731,1,56,56,18.7877689,"\int \frac{2 \left(-2 q+p x^6\right) \sqrt{q+p x^6} \left(a q+b x^4+a p x^6\right)}{x^{11}} \, dx","IntegrateAlgebraic[(2*(-2*q + p*x^6)*Sqrt[q + p*x^6]*(a*q + b*x^4 + a*p*x^6))/x^11,x]","\frac{2 \sqrt{q+p x^6} \left(3 a q^2+5 b q x^4+6 a p q x^6+5 b p x^{10}+3 a p^2 x^{12}\right)}{15 x^{10}}","\frac{2 \sqrt{q+p x^6} \left(3 a q^2+5 b q x^4+6 a p q x^6+5 b p x^{10}+3 a p^2 x^{12}\right)}{15 x^{10}}",1,"(2*Sqrt[q + p*x^6]*(3*a*q^2 + 5*b*q*x^4 + 6*a*p*q*x^6 + 5*b*p*x^10 + 3*a*p^2*x^12))/(15*x^10)","A",1
732,1,56,56,11.4991044,"\int \frac{1}{x^3 \sqrt[8]{256-256 x^2+96 x^4-16 x^6+x^8}} \, dx","IntegrateAlgebraic[1/(x^3*(256 - 256*x^2 + 96*x^4 - 16*x^6 + x^8)^(1/8)),x]","\frac{\left(\left(-4+x^2\right)^4\right)^{7/8} \left(\frac{\sqrt{-4+x^2}}{8 x^2}+\frac{1}{16} \tan ^{-1}\left(\frac{1}{2} \sqrt{-4+x^2}\right)\right)}{\left(-4+x^2\right)^{7/2}}","\frac{\left(\left(-4+x^2\right)^4\right)^{7/8} \left(\frac{\sqrt{-4+x^2}}{8 x^2}+\frac{1}{16} \tan ^{-1}\left(\frac{1}{2} \sqrt{-4+x^2}\right)\right)}{\left(-4+x^2\right)^{7/2}}",1,"(((-4 + x^2)^4)^(7/8)*(Sqrt[-4 + x^2]/(8*x^2) + ArcTan[Sqrt[-4 + x^2]/2]/16))/(-4 + x^2)^(7/2)","A",1
733,1,56,56,13.172129,"\int \frac{x^3 \left(5 b+8 a x^3\right)}{\sqrt[4]{b x+a x^4} \left(-2+b x^5+a x^8\right)} \, dx","IntegrateAlgebraic[(x^3*(5*b + 8*a*x^3))/((b*x + a*x^4)^(1/4)*(-2 + b*x^5 + a*x^8)),x]","2^{3/4} \tan ^{-1}\left(\frac{x \sqrt[4]{b x+a x^4}}{\sqrt[4]{2}}\right)-2^{3/4} \tanh ^{-1}\left(\frac{x \sqrt[4]{b x+a x^4}}{\sqrt[4]{2}}\right)","2^{3/4} \tan ^{-1}\left(\frac{x \sqrt[4]{b x+a x^4}}{\sqrt[4]{2}}\right)-2^{3/4} \tanh ^{-1}\left(\frac{x \sqrt[4]{b x+a x^4}}{\sqrt[4]{2}}\right)",1,"2^(3/4)*ArcTan[(x*(b*x + a*x^4)^(1/4))/2^(1/4)] - 2^(3/4)*ArcTanh[(x*(b*x + a*x^4)^(1/4))/2^(1/4)]","A",1
734,1,56,56,0.0615216,"\int \sqrt{1-x^2-y^4} \, dx","IntegrateAlgebraic[Sqrt[1 - x^2 - y^4],x]","\frac{1}{2} x \sqrt{1-x^2-y^4}-\frac{1}{2} i \left(-1+y^4\right) \log \left(-i x+\sqrt{1-x^2-y^4}\right)","\frac{1}{2} x \sqrt{1-x^2-y^4}-\frac{1}{2} i \left(-1+y^4\right) \log \left(-i x+\sqrt{1-x^2-y^4}\right)",1,"(x*Sqrt[1 - x^2 - y^4])/2 - (I/2)*(-1 + y^4)*Log[(-I)*x + Sqrt[1 - x^2 - y^4]]","A",1
735,1,57,57,0.3316776,"\int \frac{1-2 k^2 x+k^2 x^2}{\sqrt{(1-x) x \left(1-k^2 x\right)} \left(-a-b x+\left(a k^2+b k^2\right) x^2\right)} \, dx","IntegrateAlgebraic[(1 - 2*k^2*x + k^2*x^2)/(Sqrt[(1 - x)*x*(1 - k^2*x)]*(-a - b*x + (a*k^2 + b*k^2)*x^2)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a+b} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{\sqrt{a} (-1+x)}\right)}{\sqrt{a} \sqrt{a+b}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a+b} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{\sqrt{a} (-1+x)}\right)}{\sqrt{a} \sqrt{a+b}}",1,"(2*ArcTan[(Sqrt[a + b]*Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3])/(Sqrt[a]*(-1 + x))])/(Sqrt[a]*Sqrt[a + b])","A",1
736,1,57,57,0.8721065,"\int \frac{a b c-(a+b+c) x^2+2 x^3}{\sqrt{x (-a+x) (-b+x) (-c+x)} \left(-a b c d+(-1+a b d+a c d+b c d) x-(a+b+c) d x^2+d x^3\right)} \, dx","IntegrateAlgebraic[(a*b*c - (a + b + c)*x^2 + 2*x^3)/(Sqrt[x*(-a + x)*(-b + x)*(-c + x)]*(-(a*b*c*d) + (-1 + a*b*d + a*c*d + b*c*d)*x - (a + b + c)*d*x^2 + d*x^3)),x]","-\frac{2 \tanh ^{-1}\left(\frac{x}{\sqrt{d} \sqrt{-a b c x+(a b+a c+b c) x^2+(-a-b-c) x^3+x^4}}\right)}{\sqrt{d}}","-\frac{2 \tanh ^{-1}\left(\frac{x}{\sqrt{d} \sqrt{-a b c x+(a b+a c+b c) x^2+(-a-b-c) x^3+x^4}}\right)}{\sqrt{d}}",1,"(-2*ArcTanh[x/(Sqrt[d]*Sqrt[-(a*b*c*x) + (a*b + a*c + b*c)*x^2 + (-a - b - c)*x^3 + x^4])])/Sqrt[d]","A",1
737,1,57,57,0.1954486,"\int x^6 \sqrt[4]{-1+x^4} \, dx","IntegrateAlgebraic[x^6*(-1 + x^4)^(1/4),x]","\frac{1}{32} \sqrt[4]{-1+x^4} \left(-x^3+4 x^7\right)+\frac{3}{64} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)-\frac{3}{64} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)","\frac{1}{32} \sqrt[4]{-1+x^4} \left(-x^3+4 x^7\right)+\frac{3}{64} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)-\frac{3}{64} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)",1,"((-1 + x^4)^(1/4)*(-x^3 + 4*x^7))/32 + (3*ArcTan[x/(-1 + x^4)^(1/4)])/64 - (3*ArcTanh[x/(-1 + x^4)^(1/4)])/64","A",1
738,1,57,57,0.2854094,"\int \frac{(-1+2 x) \sqrt[4]{x^3+x^4}}{x} \, dx","IntegrateAlgebraic[((-1 + 2*x)*(x^3 + x^4)^(1/4))/x,x]","\frac{1}{4} (-3+4 x) \sqrt[4]{x^3+x^4}+\frac{7}{8} \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)-\frac{7}{8} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)","\frac{1}{4} (-3+4 x) \sqrt[4]{x^3+x^4}+\frac{7}{8} \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)-\frac{7}{8} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)",1,"((-3 + 4*x)*(x^3 + x^4)^(1/4))/4 + (7*ArcTan[x/(x^3 + x^4)^(1/4)])/8 - (7*ArcTanh[x/(x^3 + x^4)^(1/4)])/8","A",1
739,1,57,57,0.4387998,"\int \frac{\sqrt{-1+x^2-2 x^3+x^4} \left(1-x^3+x^4\right)}{\left(-1-2 x^3+x^4\right) \left(-2-x^2-4 x^3+2 x^4\right)} \, dx","IntegrateAlgebraic[(Sqrt[-1 + x^2 - 2*x^3 + x^4]*(1 - x^3 + x^4))/((-1 - 2*x^3 + x^4)*(-2 - x^2 - 4*x^3 + 2*x^4)),x]","\tanh ^{-1}\left(\frac{x}{\sqrt{-1+x^2-2 x^3+x^4}}\right)-\sqrt{\frac{3}{2}} \tanh ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt{-1+x^2-2 x^3+x^4}}\right)","\tanh ^{-1}\left(\frac{x}{\sqrt{-1+x^2-2 x^3+x^4}}\right)-\sqrt{\frac{3}{2}} \tanh ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt{-1+x^2-2 x^3+x^4}}\right)",1,"ArcTanh[x/Sqrt[-1 + x^2 - 2*x^3 + x^4]] - Sqrt[3/2]*ArcTanh[(Sqrt[3/2]*x)/Sqrt[-1 + x^2 - 2*x^3 + x^4]]","A",1
740,1,57,57,0.2897889,"\int \frac{x^2}{\left(b+a x^4\right)^{3/4}} \, dx","IntegrateAlgebraic[x^2/(b + a*x^4)^(3/4),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2 a^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2 a^{3/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2 a^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2 a^{3/4}}",1,"-1/2*ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(3/4) + ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/(2*a^(3/4))","A",1
741,1,57,57,0.4036063,"\int \frac{1}{\sqrt[4]{b+a x^4} \left(-b+a x^8\right)} \, dx","IntegrateAlgebraic[1/((b + a*x^4)^(1/4)*(-b + a*x^8)),x]","\frac{\text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{8 b}","\frac{\text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{8 b}",1,"RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (-Log[x] + Log[(b + a*x^4)^(1/4) - x*#1])/#1 & ]/(8*b)","A",1
742,1,57,57,12.9061013,"\int \frac{-b+a x^8}{\sqrt[4]{b+a x^8} \left(b-c x^4+a x^8\right)} \, dx","IntegrateAlgebraic[(-b + a*x^8)/((b + a*x^8)^(1/4)*(b - c*x^4 + a*x^8)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{b+a x^8}}\right)}{2 \sqrt[4]{c}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{b+a x^8}}\right)}{2 \sqrt[4]{c}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{b+a x^8}}\right)}{2 \sqrt[4]{c}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{b+a x^8}}\right)}{2 \sqrt[4]{c}}",1,"-1/2*ArcTan[(c^(1/4)*x)/(b + a*x^8)^(1/4)]/c^(1/4) - ArcTanh[(c^(1/4)*x)/(b + a*x^8)^(1/4)]/(2*c^(1/4))","A",1
743,1,57,57,2.7870565,"\int \frac{\left(-3+2 x^5\right) \left(1+2 x^5+x^6+x^{10}\right)}{x^6 \left(1-x^3+x^5\right) \sqrt[4]{x+x^6}} \, dx","IntegrateAlgebraic[((-3 + 2*x^5)*(1 + 2*x^5 + x^6 + x^10))/(x^6*(1 - x^3 + x^5)*(x + x^6)^(1/4)),x]","\frac{4 \left(3+7 x^3+3 x^5\right) \left(x+x^6\right)^{3/4}}{21 x^6}-4 \tan ^{-1}\left(\frac{x}{\sqrt[4]{x+x^6}}\right)-4 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x+x^6}}\right)","\frac{4 \left(3+7 x^3+3 x^5\right) \left(x+x^6\right)^{3/4}}{21 x^6}-4 \tan ^{-1}\left(\frac{x}{\sqrt[4]{x+x^6}}\right)-4 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x+x^6}}\right)",1,"(4*(3 + 7*x^3 + 3*x^5)*(x + x^6)^(3/4))/(21*x^6) - 4*ArcTan[x/(x + x^6)^(1/4)] - 4*ArcTanh[x/(x + x^6)^(1/4)]","A",1
744,1,57,57,0.1083495,"\int \sqrt{x^2+\sqrt{1+x^4}} \, dx","IntegrateAlgebraic[Sqrt[x^2 + Sqrt[1 + x^4]],x]","\frac{1}{2} x \sqrt{x^2+\sqrt{1+x^4}}+\frac{\tan ^{-1}\left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}\right)}{2 \sqrt{2}}","\frac{1}{2} x \sqrt{x^2+\sqrt{1+x^4}}+\frac{\tan ^{-1}\left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}\right)}{2 \sqrt{2}}",1,"(x*Sqrt[x^2 + Sqrt[1 + x^4]])/2 + ArcTan[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]]]/(2*Sqrt[2])","A",1
745,1,58,58,0.2718112,"\int \frac{-1-x+x^2}{\left(1+x^2\right) \sqrt{-x+x^3}} \, dx","IntegrateAlgebraic[(-1 - x + x^2)/((1 + x^2)*Sqrt[-x + x^3]),x]","-\frac{1}{4} \tan ^{-1}\left(\frac{2 \sqrt{-x+x^3}}{-1-2 x+x^2}\right)-\frac{3}{4} \tanh ^{-1}\left(\frac{-\frac{1}{2}+x+\frac{x^2}{2}}{\sqrt{-x+x^3}}\right)","-\frac{1}{4} \tan ^{-1}\left(\frac{2 \sqrt{-x+x^3}}{-1-2 x+x^2}\right)-\frac{3}{4} \tanh ^{-1}\left(\frac{-\frac{1}{2}+x+\frac{x^2}{2}}{\sqrt{-x+x^3}}\right)",1,"-1/4*ArcTan[(2*Sqrt[-x + x^3])/(-1 - 2*x + x^2)] - (3*ArcTanh[(-1/2 + x + x^2/2)/Sqrt[-x + x^3]])/4","A",1
746,1,58,58,0.2707499,"\int \frac{-1+x+x^2}{\left(1+x^2\right) \sqrt{-x+x^3}} \, dx","IntegrateAlgebraic[(-1 + x + x^2)/((1 + x^2)*Sqrt[-x + x^3]),x]","-\frac{3}{4} \tan ^{-1}\left(\frac{2 \sqrt{-x+x^3}}{-1-2 x+x^2}\right)-\frac{1}{4} \tanh ^{-1}\left(\frac{-\frac{1}{2}+x+\frac{x^2}{2}}{\sqrt{-x+x^3}}\right)","-\frac{3}{4} \tan ^{-1}\left(\frac{2 \sqrt{-x+x^3}}{-1-2 x+x^2}\right)-\frac{1}{4} \tanh ^{-1}\left(\frac{-\frac{1}{2}+x+\frac{x^2}{2}}{\sqrt{-x+x^3}}\right)",1,"(-3*ArcTan[(2*Sqrt[-x + x^3])/(-1 - 2*x + x^2)])/4 - ArcTanh[(-1/2 + x + x^2/2)/Sqrt[-x + x^3]]/4","A",1
747,1,58,58,0.2790256,"\int \frac{-7+x+7 x^2}{\left(1+x^2\right) \sqrt{-x+x^3}} \, dx","IntegrateAlgebraic[(-7 + x + 7*x^2)/((1 + x^2)*Sqrt[-x + x^3]),x]","-\frac{15}{4} \tan ^{-1}\left(\frac{2 \sqrt{-x+x^3}}{-1-2 x+x^2}\right)-\frac{13}{4} \tanh ^{-1}\left(\frac{-\frac{1}{2}+x+\frac{x^2}{2}}{\sqrt{-x+x^3}}\right)","-\frac{15}{4} \tan ^{-1}\left(\frac{2 \sqrt{-x+x^3}}{-1-2 x+x^2}\right)-\frac{13}{4} \tanh ^{-1}\left(\frac{-\frac{1}{2}+x+\frac{x^2}{2}}{\sqrt{-x+x^3}}\right)",1,"(-15*ArcTan[(2*Sqrt[-x + x^3])/(-1 - 2*x + x^2)])/4 - (13*ArcTanh[(-1/2 + x + x^2/2)/Sqrt[-x + x^3]])/4","A",1
748,1,58,58,0.2095641,"\int \frac{1}{(-2+x) \sqrt[4]{-x^2+x^3}} \, dx","IntegrateAlgebraic[1/((-2 + x)*(-x^2 + x^3)^(1/4)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-x^2+x^3}}{x}\right)}{\sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{2} \sqrt[4]{-x^2+x^3}}\right)}{\sqrt{2}}","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-x^2+x^3}}{x}\right)}{\sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{2} \sqrt[4]{-x^2+x^3}}\right)}{\sqrt{2}}",1,"ArcTan[(Sqrt[2]*(-x^2 + x^3)^(1/4))/x]/Sqrt[2] - ArcTanh[x/(Sqrt[2]*(-x^2 + x^3)^(1/4))]/Sqrt[2]","A",1
749,1,58,58,0.4563439,"\int \left(b+a x^3\right) \sqrt{-x+x^4} \, dx","IntegrateAlgebraic[(b + a*x^3)*Sqrt[-x + x^4],x]","\frac{1}{12} \sqrt{-x+x^4} \left(-a x+4 b x+2 a x^4\right)+\frac{1}{12} (-a-4 b) \tanh ^{-1}\left(\frac{x^2}{\sqrt{-x+x^4}}\right)","\frac{1}{12} \sqrt{-x+x^4} \left(-a x+4 b x+2 a x^4\right)+\frac{1}{12} (-a-4 b) \tanh ^{-1}\left(\frac{x^2}{\sqrt{-x+x^4}}\right)",1,"(Sqrt[-x + x^4]*(-(a*x) + 4*b*x + 2*a*x^4))/12 + ((-a - 4*b)*ArcTanh[x^2/Sqrt[-x + x^4]])/12","A",1
750,1,58,58,0.5584219,"\int \frac{-2-x+2 x^2}{\left(1+x^2\right) \sqrt{1+x^2+x^4}} \, dx","IntegrateAlgebraic[(-2 - x + 2*x^2)/((1 + x^2)*Sqrt[1 + x^2 + x^4]),x]","-2 \tan ^{-1}\left(\frac{x}{\sqrt{1+x^2+x^4}}\right)+\frac{1}{2} \tanh ^{-1}\left(\frac{2 x^2}{1+2 x^2+x^4+\left(-1+x^2\right) \sqrt{1+x^2+x^4}}\right)","-2 \tan ^{-1}\left(\frac{x}{\sqrt{1+x^2+x^4}}\right)+\frac{1}{2} \tanh ^{-1}\left(\frac{2 x^2}{1+2 x^2+x^4+\left(-1+x^2\right) \sqrt{1+x^2+x^4}}\right)",1,"-2*ArcTan[x/Sqrt[1 + x^2 + x^4]] + ArcTanh[(2*x^2)/(1 + 2*x^2 + x^4 + (-1 + x^2)*Sqrt[1 + x^2 + x^4])]/2","A",1
751,1,58,58,15.2988984,"\int \frac{x^3 \left(-5 b+9 a x^4\right)}{\sqrt[4]{-b x+a x^5} \left(-2-b x^5+a x^9\right)} \, dx","IntegrateAlgebraic[(x^3*(-5*b + 9*a*x^4))/((-(b*x) + a*x^5)^(1/4)*(-2 - b*x^5 + a*x^9)),x]","2^{3/4} \tan ^{-1}\left(\frac{x \sqrt[4]{-b x+a x^5}}{\sqrt[4]{2}}\right)-2^{3/4} \tanh ^{-1}\left(\frac{x \sqrt[4]{-b x+a x^5}}{\sqrt[4]{2}}\right)","2^{3/4} \tan ^{-1}\left(\frac{x \sqrt[4]{-b x+a x^5}}{\sqrt[4]{2}}\right)-2^{3/4} \tanh ^{-1}\left(\frac{x \sqrt[4]{-b x+a x^5}}{\sqrt[4]{2}}\right)",1,"2^(3/4)*ArcTan[(x*(-(b*x) + a*x^5)^(1/4))/2^(1/4)] - 2^(3/4)*ArcTanh[(x*(-(b*x) + a*x^5)^(1/4))/2^(1/4)]","A",1
752,1,59,59,0.1099037,"\int \frac{1+x^2}{\left(-1+x^2\right) \sqrt{x+x^2+x^3}} \, dx","IntegrateAlgebraic[(1 + x^2)/((-1 + x^2)*Sqrt[x + x^2 + x^3]),x]","-\tan ^{-1}\left(\frac{\sqrt{x+x^2+x^3}}{1+x+x^2}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt{3} \sqrt{x+x^2+x^3}}{1+x+x^2}\right)}{\sqrt{3}}","-\tan ^{-1}\left(\frac{\sqrt{x+x^2+x^3}}{1+x+x^2}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt{3} \sqrt{x+x^2+x^3}}{1+x+x^2}\right)}{\sqrt{3}}",1,"-ArcTan[Sqrt[x + x^2 + x^3]/(1 + x + x^2)] - ArcTanh[(Sqrt[3]*Sqrt[x + x^2 + x^3])/(1 + x + x^2)]/Sqrt[3]","A",1
753,1,59,59,0.6163468,"\int \frac{a b c-(a+b+c) x^2+2 x^3}{\sqrt{x (-a+x) (-b+x) (-c+x)} \left(-a b c+(a b+a c+b c-d) x-(a+b+c) x^2+x^3\right)} \, dx","IntegrateAlgebraic[(a*b*c - (a + b + c)*x^2 + 2*x^3)/(Sqrt[x*(-a + x)*(-b + x)*(-c + x)]*(-(a*b*c) + (a*b + a*c + b*c - d)*x - (a + b + c)*x^2 + x^3)),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{-a b c x+(a b+a c+b c) x^2+(-a-b-c) x^3+x^4}}{\sqrt{d} x}\right)}{\sqrt{d}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{-a b c x+(a b+a c+b c) x^2+(-a-b-c) x^3+x^4}}{\sqrt{d} x}\right)}{\sqrt{d}}",1,"(-2*ArcTanh[Sqrt[-(a*b*c*x) + (a*b + a*c + b*c)*x^2 + (-a - b - c)*x^3 + x^4]/(Sqrt[d]*x)])/Sqrt[d]","A",1
754,1,59,59,0.1099199,"\int \frac{\left(-b+a x^3\right) \sqrt{b+a x^3}}{x^7} \, dx","IntegrateAlgebraic[((-b + a*x^3)*Sqrt[b + a*x^3])/x^7,x]","\frac{\left(2 b-3 a x^3\right) \sqrt{b+a x^3}}{12 x^6}-\frac{5 a^2 \tanh ^{-1}\left(\frac{\sqrt{b+a x^3}}{\sqrt{b}}\right)}{12 \sqrt{b}}","\frac{\left(2 b-3 a x^3\right) \sqrt{b+a x^3}}{12 x^6}-\frac{5 a^2 \tanh ^{-1}\left(\frac{\sqrt{b+a x^3}}{\sqrt{b}}\right)}{12 \sqrt{b}}",1,"((2*b - 3*a*x^3)*Sqrt[b + a*x^3])/(12*x^6) - (5*a^2*ArcTanh[Sqrt[b + a*x^3]/Sqrt[b]])/(12*Sqrt[b])","A",1
755,1,59,59,0.0954117,"\int \frac{\sqrt{b+a x^3} \left(2 b+a x^3\right)}{x^7} \, dx","IntegrateAlgebraic[(Sqrt[b + a*x^3]*(2*b + a*x^3))/x^7,x]","\frac{\left(-2 b-3 a x^3\right) \sqrt{b+a x^3}}{6 x^6}-\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{b+a x^3}}{\sqrt{b}}\right)}{6 \sqrt{b}}","\frac{\left(-2 b-3 a x^3\right) \sqrt{b+a x^3}}{6 x^6}-\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{b+a x^3}}{\sqrt{b}}\right)}{6 \sqrt{b}}",1,"((-2*b - 3*a*x^3)*Sqrt[b + a*x^3])/(6*x^6) - (a^2*ArcTanh[Sqrt[b + a*x^3]/Sqrt[b]])/(6*Sqrt[b])","A",1
756,1,59,59,0.7311399,"\int \frac{3 a b c x-2 (a b+a c+b c) x^2+(a+b+c) x^3}{\sqrt{x (-a+x) (-b+x) (-c+x)} \left(a b c-(a b+a c+b c) x+(a+b+c) x^2+(-1+d) x^3\right)} \, dx","IntegrateAlgebraic[(3*a*b*c*x - 2*(a*b + a*c + b*c)*x^2 + (a + b + c)*x^3)/(Sqrt[x*(-a + x)*(-b + x)*(-c + x)]*(a*b*c - (a*b + a*c + b*c)*x + (a + b + c)*x^2 + (-1 + d)*x^3)),x]","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{-a b c x+(a b+a c+b c) x^2+(-a-b-c) x^3+x^4}}{\sqrt{d} x^2}\right)}{\sqrt{d}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{-a b c x+(a b+a c+b c) x^2+(-a-b-c) x^3+x^4}}{\sqrt{d} x^2}\right)}{\sqrt{d}}",1,"(2*ArcTanh[Sqrt[-(a*b*c*x) + (a*b + a*c + b*c)*x^2 + (-a - b - c)*x^3 + x^4]/(Sqrt[d]*x^2)])/Sqrt[d]","A",1
757,1,59,59,0.9662658,"\int \frac{x \left(3 a b c-2 (a b+a c+b c) x+(a+b+c) x^2\right)}{\sqrt{x (-a+x) (-b+x) (-c+x)} \left(-a b c d+(a b+a c+b c) d x-(a+b+c) d x^2+(-1+d) x^3\right)} \, dx","IntegrateAlgebraic[(x*(3*a*b*c - 2*(a*b + a*c + b*c)*x + (a + b + c)*x^2))/(Sqrt[x*(-a + x)*(-b + x)*(-c + x)]*(-(a*b*c*d) + (a*b + a*c + b*c)*d*x - (a + b + c)*d*x^2 + (-1 + d)*x^3)),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{-a b c x+(a b+a c+b c) x^2+(-a-b-c) x^3+x^4}}{x^2}\right)}{\sqrt{d}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{-a b c x+(a b+a c+b c) x^2+(-a-b-c) x^3+x^4}}{x^2}\right)}{\sqrt{d}}",1,"(-2*ArcTanh[(Sqrt[d]*Sqrt[-(a*b*c*x) + (a*b + a*c + b*c)*x^2 + (-a - b - c)*x^3 + x^4])/x^2])/Sqrt[d]","A",1
758,1,59,59,0.1145216,"\int \sqrt[4]{-x^2+x^4} \, dx","IntegrateAlgebraic[(-x^2 + x^4)^(1/4),x]","\frac{1}{2} x \sqrt[4]{-x^2+x^4}+\frac{1}{4} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)-\frac{1}{4} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)","\frac{1}{2} x \sqrt[4]{-x^2+x^4}+\frac{1}{4} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)-\frac{1}{4} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)",1,"(x*(-x^2 + x^4)^(1/4))/2 + ArcTan[x/(-x^2 + x^4)^(1/4)]/4 - ArcTanh[x/(-x^2 + x^4)^(1/4)]/4","A",1
759,1,59,59,0.1360735,"\int x^2 \sqrt[4]{x^2+x^4} \, dx","IntegrateAlgebraic[x^2*(x^2 + x^4)^(1/4),x]","\frac{1}{16} \left(x+4 x^3\right) \sqrt[4]{x^2+x^4}+\frac{3}{32} \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^2+x^4}}\right)-\frac{3}{32} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^2+x^4}}\right)","\frac{1}{16} \left(x+4 x^3\right) \sqrt[4]{x^2+x^4}+\frac{3}{32} \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^2+x^4}}\right)-\frac{3}{32} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^2+x^4}}\right)",1,"((x + 4*x^3)*(x^2 + x^4)^(1/4))/16 + (3*ArcTan[x/(x^2 + x^4)^(1/4)])/32 - (3*ArcTanh[x/(x^2 + x^4)^(1/4)])/32","A",1
760,1,59,59,0.6388496,"\int \frac{\left(-1+2 x^4\right) \sqrt{1+3 x^2+2 x^4}}{\left(1+2 x^2+2 x^4\right)^2} \, dx","IntegrateAlgebraic[((-1 + 2*x^4)*Sqrt[1 + 3*x^2 + 2*x^4])/(1 + 2*x^2 + 2*x^4)^2,x]","-\frac{x \sqrt{1+3 x^2+2 x^4}}{2 \left(1+2 x^2+2 x^4\right)}-\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt{1+3 x^2+2 x^4}}\right)","-\frac{x \sqrt{1+3 x^2+2 x^4}}{2 \left(1+2 x^2+2 x^4\right)}-\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt{1+3 x^2+2 x^4}}\right)",1,"-1/2*(x*Sqrt[1 + 3*x^2 + 2*x^4])/(1 + 2*x^2 + 2*x^4) - ArcTanh[x/Sqrt[1 + 3*x^2 + 2*x^4]]/2","A",1
761,1,59,59,0.2271726,"\int \frac{\sqrt{-2+x^2-2 x^3+2 x^4} \left(2-x^3+2 x^4\right)}{\left(-1-x^3+x^4\right) \left(-2-x^2-2 x^3+2 x^4\right)} \, dx","IntegrateAlgebraic[(Sqrt[-2 + x^2 - 2*x^3 + 2*x^4]*(2 - x^3 + 2*x^4))/((-1 - x^3 + x^4)*(-2 - x^2 - 2*x^3 + 2*x^4)),x]","2 \tanh ^{-1}\left(\frac{x}{\sqrt{-2+x^2-2 x^3+2 x^4}}\right)-2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{-2+x^2-2 x^3+2 x^4}}\right)","2 \tanh ^{-1}\left(\frac{x}{\sqrt{-2+x^2-2 x^3+2 x^4}}\right)-2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{-2+x^2-2 x^3+2 x^4}}\right)",1,"2*ArcTanh[x/Sqrt[-2 + x^2 - 2*x^3 + 2*x^4]] - 2*Sqrt[2]*ArcTanh[(Sqrt[2]*x)/Sqrt[-2 + x^2 - 2*x^3 + 2*x^4]]","A",1
762,1,50,59,0.4771335,"\int \frac{-b+a x^2}{\left(b+a x^2\right) \sqrt{b^2+a^2 x^4}} \, dx","IntegrateAlgebraic[(-b + a*x^2)/((b + a*x^2)*Sqrt[b^2 + a^2*x^4]),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{b} x}{\sqrt{b^2+a^2 x^4}}\right)}{\sqrt{2} \sqrt{a} \sqrt{b}}","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{b} x}{b+a x^2+\sqrt{b^2+a^2 x^4}}\right)}{\sqrt{a} \sqrt{b}}",1,"-(ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[b]*x)/Sqrt[b^2 + a^2*x^4]]/(Sqrt[2]*Sqrt[a]*Sqrt[b]))","A",1
763,1,59,59,1.2041999,"\int \frac{3 b+a x^4}{\left(-b-x^3+a x^4\right) \sqrt[4]{-b x+a x^5}} \, dx","IntegrateAlgebraic[(3*b + a*x^4)/((-b - x^3 + a*x^4)*(-(b*x) + a*x^5)^(1/4)),x]","-2 \tan ^{-1}\left(\frac{\left(-b x+a x^5\right)^{3/4}}{-b+a x^4}\right)-2 \tanh ^{-1}\left(\frac{\left(-b x+a x^5\right)^{3/4}}{-b+a x^4}\right)","-2 \tan ^{-1}\left(\frac{\left(-b x+a x^5\right)^{3/4}}{-b+a x^4}\right)-2 \tanh ^{-1}\left(\frac{\left(-b x+a x^5\right)^{3/4}}{-b+a x^4}\right)",1,"-2*ArcTan[(-(b*x) + a*x^5)^(3/4)/(-b + a*x^4)] - 2*ArcTanh[(-(b*x) + a*x^5)^(3/4)/(-b + a*x^4)]","A",1
764,1,63,59,0.1600318,"\int \frac{\left(-1+x^3\right) \sqrt{-1+x^6}}{x} \, dx","IntegrateAlgebraic[((-1 + x^3)*Sqrt[-1 + x^6])/x,x]","\frac{1}{6} \left(-2+x^3\right) \sqrt{-1+x^6}-\frac{2}{3} \tan ^{-1}\left(\frac{\sqrt{-1+x^6}}{-1+x^3}\right)-\frac{1}{3} \tanh ^{-1}\left(\frac{\sqrt{-1+x^6}}{-1+x^3}\right)","\frac{1}{6} \left(-2+x^3\right) \sqrt{-1+x^6}-\frac{2}{3} \tan ^{-1}\left(\frac{1+x^3}{\sqrt{-1+x^6}}\right)-\frac{1}{3} \tanh ^{-1}\left(\frac{1+x^3}{\sqrt{-1+x^6}}\right)",1,"((-2 + x^3)*Sqrt[-1 + x^6])/6 - (2*ArcTan[Sqrt[-1 + x^6]/(-1 + x^3)])/3 - ArcTanh[Sqrt[-1 + x^6]/(-1 + x^3)]/3","A",1
765,1,59,59,0.2919678,"\int \frac{x (3+4 x) \sqrt[3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6} \, dx","IntegrateAlgebraic[(x*(3 + 4*x)*(-1 - 2*x + x^3)^(1/3))/(-2 - 8*x - 8*x^2 + x^6),x]","-\frac{1}{4} \text{RootSum}\left[1-4 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{-1-2 x+x^3}-x \text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^3}\&\right]","-\frac{1}{4} \text{RootSum}\left[1-4 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{-1-2 x+x^3}-x \text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^3}\&\right]",1,"-1/4*RootSum[1 - 4*#1^3 + 2*#1^6 & , (-(Log[x]*#1) + Log[(-1 - 2*x + x^3)^(1/3) - x*#1]*#1)/(-1 + #1^3) & ]","A",1
766,1,59,59,0.001319,"\int \frac{x (3+4 x) \sqrt[3]{-1-2 x+x^3}}{-2-8 x-8 x^2+x^6} \, dx","IntegrateAlgebraic[(x*(3 + 4*x)*(-1 - 2*x + x^3)^(1/3))/(-2 - 8*x - 8*x^2 + x^6),x]","-\frac{1}{4} \text{RootSum}\left[1-4 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{-1-2 x+x^3}-x \text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^3}\&\right]","-\frac{1}{4} \text{RootSum}\left[1-4 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{-1-2 x+x^3}-x \text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^3}\&\right]",1,"-1/4*RootSum[1 - 4*#1^3 + 2*#1^6 & , (-(Log[x]*#1) + Log[(-1 - 2*x + x^3)^(1/3) - x*#1]*#1)/(-1 + #1^3) & ]","A",1
767,1,59,59,0.3028931,"\int \frac{(3+2 x) \left(1+x+x^3\right)^{2/3}}{1+2 x+x^2+x^3+x^4+x^6} \, dx","IntegrateAlgebraic[((3 + 2*x)*(1 + x + x^3)^(2/3))/(1 + 2*x + x^2 + x^3 + x^4 + x^6),x]","-\text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{1+x+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-1+2 \text{$\#$1}^3}\&\right]","-\text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{1+x+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-1+2 \text{$\#$1}^3}\&\right]",1,"-RootSum[1 - #1^3 + #1^6 & , (-(Log[x]*#1^2) + Log[(1 + x + x^3)^(1/3) - x*#1]*#1^2)/(-1 + 2*#1^3) & ]","A",1
768,1,59,59,0.0014377,"\int \frac{(3+2 x) \left(1+x+x^3\right)^{2/3}}{1+2 x+x^2+x^3+x^4+x^6} \, dx","IntegrateAlgebraic[((3 + 2*x)*(1 + x + x^3)^(2/3))/(1 + 2*x + x^2 + x^3 + x^4 + x^6),x]","-\text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{1+x+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-1+2 \text{$\#$1}^3}\&\right]","-\text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{1+x+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-1+2 \text{$\#$1}^3}\&\right]",1,"-RootSum[1 - #1^3 + #1^6 & , (-(Log[x]*#1^2) + Log[(1 + x + x^3)^(1/3) - x*#1]*#1^2)/(-1 + 2*#1^3) & ]","A",1
769,1,59,59,0.1339505,"\int \frac{\left(2+x^3\right) \sqrt[3]{x+x^3-x^4}}{1+x^2-2 x^3+x^4-x^5+x^6} \, dx","IntegrateAlgebraic[((2 + x^3)*(x + x^3 - x^4)^(1/3))/(1 + x^2 - 2*x^3 + x^4 - x^5 + x^6),x]","-\text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{x+x^3-x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-1+2 \text{$\#$1}^3}\&\right]","-\text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{x+x^3-x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-1+2 \text{$\#$1}^3}\&\right]",1,"-RootSum[1 - #1^3 + #1^6 & , (-(Log[x]*#1) + Log[(x + x^3 - x^4)^(1/3) - x*#1]*#1)/(-1 + 2*#1^3) & ]","A",1
770,1,59,59,0.0015111,"\int \frac{\left(2+x^3\right) \sqrt[3]{x+x^3-x^4}}{1+x^2-2 x^3+x^4-x^5+x^6} \, dx","IntegrateAlgebraic[((2 + x^3)*(x + x^3 - x^4)^(1/3))/(1 + x^2 - 2*x^3 + x^4 - x^5 + x^6),x]","-\text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{x+x^3-x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-1+2 \text{$\#$1}^3}\&\right]","-\text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{x+x^3-x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-1+2 \text{$\#$1}^3}\&\right]",1,"-RootSum[1 - #1^3 + #1^6 & , (-(Log[x]*#1) + Log[(x + x^3 - x^4)^(1/3) - x*#1]*#1)/(-1 + 2*#1^3) & ]","A",1
771,1,59,59,0.3083941,"\int \frac{\left(3+x^2\right) \left(1+x^2+x^3\right)^{2/3}}{-1-2 x^2+x^3-x^4+x^5+x^6} \, dx","IntegrateAlgebraic[((3 + x^2)*(1 + x^2 + x^3)^(2/3))/(-1 - 2*x^2 + x^3 - x^4 + x^5 + x^6),x]","\text{RootSum}\left[1-3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{1+x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-3+2 \text{$\#$1}^3}\&\right]","\text{RootSum}\left[1-3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{1+x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-3+2 \text{$\#$1}^3}\&\right]",1,"RootSum[1 - 3*#1^3 + #1^6 & , (-(Log[x]*#1^2) + Log[(1 + x^2 + x^3)^(1/3) - x*#1]*#1^2)/(-3 + 2*#1^3) & ]","A",1
772,1,59,59,0.0014334,"\int \frac{\left(3+x^2\right) \left(1+x^2+x^3\right)^{2/3}}{-1-2 x^2+x^3-x^4+x^5+x^6} \, dx","IntegrateAlgebraic[((3 + x^2)*(1 + x^2 + x^3)^(2/3))/(-1 - 2*x^2 + x^3 - x^4 + x^5 + x^6),x]","\text{RootSum}\left[1-3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{1+x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-3+2 \text{$\#$1}^3}\&\right]","\text{RootSum}\left[1-3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{1+x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-3+2 \text{$\#$1}^3}\&\right]",1,"RootSum[1 - 3*#1^3 + #1^6 & , (-(Log[x]*#1^2) + Log[(1 + x^2 + x^3)^(1/3) - x*#1]*#1^2)/(-3 + 2*#1^3) & ]","A",1
773,1,59,59,12.3078723,"\int \frac{b+a x^8}{\sqrt[4]{b-a x^8} \left(-b+c x^4+a x^8\right)} \, dx","IntegrateAlgebraic[(b + a*x^8)/((b - a*x^8)^(1/4)*(-b + c*x^4 + a*x^8)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{b-a x^8}}\right)}{2 \sqrt[4]{c}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{b-a x^8}}\right)}{2 \sqrt[4]{c}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{b-a x^8}}\right)}{2 \sqrt[4]{c}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{b-a x^8}}\right)}{2 \sqrt[4]{c}}",1,"-1/2*ArcTan[(c^(1/4)*x)/(b - a*x^8)^(1/4)]/c^(1/4) - ArcTanh[(c^(1/4)*x)/(b - a*x^8)^(1/4)]/(2*c^(1/4))","A",1
774,1,59,59,2.6875187,"\int \frac{x \sqrt[3]{2-x^3+x^8} \left(-6+5 x^8\right)}{4+x^6+4 x^8+x^{16}} \, dx","IntegrateAlgebraic[(x*(2 - x^3 + x^8)^(1/3)*(-6 + 5*x^8))/(4 + x^6 + 4*x^8 + x^16),x]","\frac{1}{2} \text{RootSum}\left[2+2 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{2-x^3+x^8}-x \text{$\#$1}\right) \text{$\#$1}}{1+\text{$\#$1}^3}\&\right]","\frac{1}{2} \text{RootSum}\left[2+2 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{2-x^3+x^8}-x \text{$\#$1}\right) \text{$\#$1}}{1+\text{$\#$1}^3}\&\right]",1,"RootSum[2 + 2*#1^3 + #1^6 & , (-(Log[x]*#1) + Log[(2 - x^3 + x^8)^(1/3) - x*#1]*#1)/(1 + #1^3) & ]/2","A",1
775,1,60,60,2.8088821,"\int \frac{a b+a c-b c-2 a x+x^2}{\sqrt{(-a+x) (-b+x) (-c+x)} \left(b c+a d-(b+c+d) x+x^2\right)} \, dx","IntegrateAlgebraic[(a*b + a*c - b*c - 2*a*x + x^2)/(Sqrt[(-a + x)*(-b + x)*(-c + x)]*(b*c + a*d - (b + c + d)*x + x^2)),x]","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{\sqrt{d} (a-x)}\right)}{\sqrt{d}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{\sqrt{d} (a-x)}\right)}{\sqrt{d}}",1,"(2*ArcTanh[Sqrt[-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3]/(Sqrt[d]*(a - x))])/Sqrt[d]","A",1
776,1,60,60,2.8108287,"\int \frac{a b+a c-b c-2 a x+x^2}{\sqrt{(-a+x) (-b+x) (-c+x)} \left(a+b c d-(1+b d+c d) x+d x^2\right)} \, dx","IntegrateAlgebraic[(a*b + a*c - b*c - 2*a*x + x^2)/(Sqrt[(-a + x)*(-b + x)*(-c + x)]*(a + b*c*d - (1 + b*d + c*d)*x + d*x^2)),x]","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{a-x}\right)}{\sqrt{d}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{a-x}\right)}{\sqrt{d}}",1,"(2*ArcTanh[(Sqrt[d]*Sqrt[-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3])/(a - x)])/Sqrt[d]","A",1
777,1,60,60,0.1322669,"\int \frac{-1+k^2 x^2}{\sqrt{(1-x) x \left(1-k^2 x\right)} \left(1+k^2 x^2\right)} \, dx","IntegrateAlgebraic[(-1 + k^2*x^2)/(Sqrt[(1 - x)*x*(1 - k^2*x)]*(1 + k^2*x^2)),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{1+k^2} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{(-1+x) \left(-1+k^2 x\right)}\right)}{\sqrt{1+k^2}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{1+k^2} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{(-1+x) \left(-1+k^2 x\right)}\right)}{\sqrt{1+k^2}}",1,"(-2*ArcTan[(Sqrt[1 + k^2]*Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3])/((-1 + x)*(-1 + k^2*x))])/Sqrt[1 + k^2]","A",1
778,1,60,60,3.1462582,"\int \frac{a (a b+a c-3 b c)+\left(-2 a^2+a b+a c+3 b c\right) x+(a-2 b-2 c) x^2+x^3}{\sqrt{(-a+x) (-b+x) (-c+x)} \left(-a^3-b c d+\left(3 a^2+b d+c d\right) x-(3 a+d) x^2+x^3\right)} \, dx","IntegrateAlgebraic[(a*(a*b + a*c - 3*b*c) + (-2*a^2 + a*b + a*c + 3*b*c)*x + (a - 2*b - 2*c)*x^2 + x^3)/(Sqrt[(-a + x)*(-b + x)*(-c + x)]*(-a^3 - b*c*d + (3*a^2 + b*d + c*d)*x - (3*a + d)*x^2 + x^3)),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{(a-x)^2}\right)}{\sqrt{d}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{(a-x)^2}\right)}{\sqrt{d}}",1,"(-2*ArcTanh[(Sqrt[d]*Sqrt[-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3])/(a - x)^2])/Sqrt[d]","A",1
779,1,60,60,0.4945795,"\int \frac{\left(-2 c+a x^3\right) \sqrt{c+b x^2+a x^3}}{\left(c+a x^3\right)^2} \, dx","IntegrateAlgebraic[((-2*c + a*x^3)*Sqrt[c + b*x^2 + a*x^3])/(c + a*x^3)^2,x]","-\frac{x \sqrt{c+b x^2+a x^3}}{c+a x^3}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{c+b x^2+a x^3}}\right)}{\sqrt{b}}","-\frac{x \sqrt{c+b x^2+a x^3}}{c+a x^3}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{c+b x^2+a x^3}}\right)}{\sqrt{b}}",1,"-((x*Sqrt[c + b*x^2 + a*x^3])/(c + a*x^3)) - ArcTanh[(Sqrt[b]*x)/Sqrt[c + b*x^2 + a*x^3]]/Sqrt[b]","A",1
780,1,60,60,0.3393467,"\int \frac{-b+a^2 x^2}{\left(b+2 a b x+a^2 x^2\right) \sqrt{b x+a^2 x^3}} \, dx","IntegrateAlgebraic[(-b + a^2*x^2)/((b + 2*a*b*x + a^2*x^2)*Sqrt[b*x + a^2*x^3]),x]","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{b} \sqrt{b x+a^2 x^3}}{b+a^2 x^2}\right)}{\sqrt{a} \sqrt{b}}","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{b} \sqrt{b x+a^2 x^3}}{b+a^2 x^2}\right)}{\sqrt{a} \sqrt{b}}",1,"-((Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[b]*Sqrt[b*x + a^2*x^3])/(b + a^2*x^2)])/(Sqrt[a]*Sqrt[b]))","A",1
781,1,60,60,3.131207,"\int \frac{a (a b+a c-3 b c)+\left(-2 a^2+a b+a c+3 b c\right) x+(a-2 b-2 c) x^2+x^3}{\sqrt{(-a+x) (-b+x) (-c+x)} \left(-b c-a^3 d+\left(b+c+3 a^2 d\right) x-(1+3 a d) x^2+d x^3\right)} \, dx","IntegrateAlgebraic[(a*(a*b + a*c - 3*b*c) + (-2*a^2 + a*b + a*c + 3*b*c)*x + (a - 2*b - 2*c)*x^2 + x^3)/(Sqrt[(-a + x)*(-b + x)*(-c + x)]*(-(b*c) - a^3*d + (b + c + 3*a^2*d)*x - (1 + 3*a*d)*x^2 + d*x^3)),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{\sqrt{d} (a-x)^2}\right)}{\sqrt{d}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{\sqrt{d} (a-x)^2}\right)}{\sqrt{d}}",1,"(-2*ArcTanh[Sqrt[-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3]/(Sqrt[d]*(a - x)^2)])/Sqrt[d]","A",1
782,1,64,60,0.169765,"\int \frac{\left(-1+x^2\right) \sqrt{1+x^4}}{x^5} \, dx","IntegrateAlgebraic[((-1 + x^2)*Sqrt[1 + x^4])/x^5,x]","\frac{\left(1-2 x^2\right) \sqrt{1+x^4}}{4 x^4}-\frac{1}{2} \tanh ^{-1}\left(x^2-\sqrt{1+x^4}\right)-\frac{1}{2} \log \left(-x^2+\sqrt{1+x^4}\right)","\frac{\left(1-2 x^2\right) \sqrt{1+x^4}}{4 x^4}+\frac{1}{2} \tanh ^{-1}\left(x^2+\sqrt{1+x^4}\right)+\frac{1}{2} \log \left(x^2+\sqrt{1+x^4}\right)",1,"((1 - 2*x^2)*Sqrt[1 + x^4])/(4*x^4) - ArcTanh[x^2 - Sqrt[1 + x^4]]/2 - Log[-x^2 + Sqrt[1 + x^4]]/2","A",1
783,1,58,60,0.1355246,"\int \frac{\left(-1+x^2\right) \sqrt{1+x^4}}{x^3} \, dx","IntegrateAlgebraic[((-1 + x^2)*Sqrt[1 + x^4])/x^3,x]","\frac{\left(1+x^2\right) \sqrt{1+x^4}}{2 x^2}+\tanh ^{-1}\left(x^2-\sqrt{1+x^4}\right)+\frac{1}{2} \log \left(-x^2+\sqrt{1+x^4}\right)","\frac{\left(1+x^2\right) \sqrt{1+x^4}}{2 x^2}+\tanh ^{-1}\left(1-2 x^2-2 \sqrt{1+x^4}\right)-\frac{1}{2} \log \left(1+x^2+\sqrt{1+x^4}\right)",1,"((1 + x^2)*Sqrt[1 + x^4])/(2*x^2) + ArcTanh[x^2 - Sqrt[1 + x^4]] + Log[-x^2 + Sqrt[1 + x^4]]/2","A",1
784,1,68,60,0.1752309,"\int \frac{\left(1+2 x^3\right) \sqrt{-1+x^6}}{x^7} \, dx","IntegrateAlgebraic[((1 + 2*x^3)*Sqrt[-1 + x^6])/x^7,x]","\frac{\left(-1-4 x^3\right) \sqrt{-1+x^6}}{6 x^6}-\frac{1}{3} \tan ^{-1}\left(\frac{\sqrt{-1+x^6}}{-1+x^3}\right)+\frac{4}{3} \tanh ^{-1}\left(\frac{\sqrt{-1+x^6}}{-1+x^3}\right)","\frac{\left(-1-4 x^3\right) \sqrt{-1+x^6}}{6 x^6}+\frac{1}{3} \tan ^{-1}\left(x^3+\sqrt{-1+x^6}\right)+\frac{2}{3} \log \left(x^3+\sqrt{-1+x^6}\right)",1,"((-1 - 4*x^3)*Sqrt[-1 + x^6])/(6*x^6) - ArcTan[Sqrt[-1 + x^6]/(-1 + x^3)]/3 + (4*ArcTanh[Sqrt[-1 + x^6]/(-1 + x^3)])/3","A",1
785,1,68,60,0.1818502,"\int \frac{\left(1+2 x^3\right) \sqrt{-1+x^6}}{x^4} \, dx","IntegrateAlgebraic[((1 + 2*x^3)*Sqrt[-1 + x^6])/x^4,x]","\frac{\left(-1+2 x^3\right) \sqrt{-1+x^6}}{3 x^3}+\frac{4}{3} \tan ^{-1}\left(\frac{\sqrt{-1+x^6}}{-1+x^3}\right)+\frac{2}{3} \tanh ^{-1}\left(\frac{\sqrt{-1+x^6}}{-1+x^3}\right)","\frac{\left(-1+2 x^3\right) \sqrt{-1+x^6}}{3 x^3}-\frac{4}{3} \tan ^{-1}\left(x^3+\sqrt{-1+x^6}\right)+\frac{1}{3} \log \left(x^3+\sqrt{-1+x^6}\right)",1,"((-1 + 2*x^3)*Sqrt[-1 + x^6])/(3*x^3) + (4*ArcTan[Sqrt[-1 + x^6]/(-1 + x^3)])/3 + (2*ArcTanh[Sqrt[-1 + x^6]/(-1 + x^3)])/3","A",1
786,1,60,60,1.0103628,"\int \frac{b^6+a^6 x^6}{\sqrt{-b^2 x+a^2 x^3} \left(-b^6+c x^3+a^6 x^6\right)} \, dx","IntegrateAlgebraic[(b^6 + a^6*x^6)/(Sqrt[-(b^2*x) + a^2*x^3]*(-b^6 + c*x^3 + a^6*x^6)),x]","\frac{1}{3} \text{RootSum}\left[c+3 a^2 b^2 \text{$\#$1}^2+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt{-b^2 x+a^2 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{1}{3} \text{RootSum}\left[c+3 a^2 b^2 \text{$\#$1}^2+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt{-b^2 x+a^2 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"RootSum[c + 3*a^2*b^2*#1^2 + #1^6 & , (-Log[x] + Log[Sqrt[-(b^2*x) + a^2*x^3] - x*#1])/#1 & ]/3","A",1
787,1,60,60,0.0020732,"\int \frac{b^6+a^6 x^6}{\sqrt{-b^2 x+a^2 x^3} \left(-b^6+c x^3+a^6 x^6\right)} \, dx","IntegrateAlgebraic[(b^6 + a^6*x^6)/(Sqrt[-(b^2*x) + a^2*x^3]*(-b^6 + c*x^3 + a^6*x^6)),x]","\frac{1}{3} \text{RootSum}\left[c+3 a^2 b^2 \text{$\#$1}^2+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt{-b^2 x+a^2 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{1}{3} \text{RootSum}\left[c+3 a^2 b^2 \text{$\#$1}^2+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt{-b^2 x+a^2 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"RootSum[c + 3*a^2*b^2*#1^2 + #1^6 & , (-Log[x] + Log[Sqrt[-(b^2*x) + a^2*x^3] - x*#1])/#1 & ]/3","A",1
788,1,60,60,1.9043347,"\int \frac{x-3 x^5}{\sqrt{x+x^5} \left(1-a x^2+2 x^4+x^8\right)} \, dx","IntegrateAlgebraic[(x - 3*x^5)/(Sqrt[x + x^5]*(1 - a*x^2 + 2*x^4 + x^8)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} \sqrt{x+x^5}}{1+x^4}\right)}{a^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} \sqrt{x+x^5}}{1+x^4}\right)}{a^{3/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} \sqrt{x+x^5}}{1+x^4}\right)}{a^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} \sqrt{x+x^5}}{1+x^4}\right)}{a^{3/4}}",1,"-(ArcTan[(a^(1/4)*Sqrt[x + x^5])/(1 + x^4)]/a^(3/4)) + ArcTanh[(a^(1/4)*Sqrt[x + x^5])/(1 + x^4)]/a^(3/4)","A",1
789,1,60,60,22.2169454,"\int \frac{\sqrt{1-2 x^8} \left(-1+2 x^8\right) \left(1+2 x^8\right)}{x^7 \left(-1+x^4+2 x^8\right)} \, dx","IntegrateAlgebraic[(Sqrt[1 - 2*x^8]*(-1 + 2*x^8)*(1 + 2*x^8))/(x^7*(-1 + x^4 + 2*x^8)),x]","\frac{\sqrt{1-2 x^8} \left(-1-3 x^4+2 x^8\right)}{6 x^6}-\frac{1}{2} \tanh ^{-1}\left(\frac{x^2 \sqrt{1-2 x^8}}{-1+2 x^8}\right)","\frac{\sqrt{1-2 x^8} \left(-1-3 x^4+2 x^8\right)}{6 x^6}-\frac{1}{2} \tanh ^{-1}\left(\frac{x^2 \sqrt{1-2 x^8}}{-1+2 x^8}\right)",1,"(Sqrt[1 - 2*x^8]*(-1 - 3*x^4 + 2*x^8))/(6*x^6) - ArcTanh[(x^2*Sqrt[1 - 2*x^8])/(-1 + 2*x^8)]/2","A",1
790,1,60,60,2.6606978,"\int \frac{-x+3 x^5}{\sqrt{x+x^5} \left(a-x^2+2 a x^4+a x^8\right)} \, dx","IntegrateAlgebraic[(-x + 3*x^5)/(Sqrt[x + x^5]*(a - x^2 + 2*a*x^4 + a*x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{x+x^5}}{\sqrt[4]{a} \left(1+x^4\right)}\right)}{\sqrt[4]{a}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{x+x^5}}{\sqrt[4]{a} \left(1+x^4\right)}\right)}{\sqrt[4]{a}}","\frac{\tan ^{-1}\left(\frac{\sqrt{x+x^5}}{\sqrt[4]{a} \left(1+x^4\right)}\right)}{\sqrt[4]{a}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{x+x^5}}{\sqrt[4]{a} \left(1+x^4\right)}\right)}{\sqrt[4]{a}}",1,"ArcTan[Sqrt[x + x^5]/(a^(1/4)*(1 + x^4))]/a^(1/4) - ArcTanh[Sqrt[x + x^5]/(a^(1/4)*(1 + x^4))]/a^(1/4)","A",1
791,1,60,60,15.8748891,"\int \frac{x^5 \left(7 b+9 a x^2\right)}{\sqrt[4]{b x^3+a x^5} \left(-2+b x^7+a x^9\right)} \, dx","IntegrateAlgebraic[(x^5*(7*b + 9*a*x^2))/((b*x^3 + a*x^5)^(1/4)*(-2 + b*x^7 + a*x^9)),x]","2^{3/4} \tan ^{-1}\left(\frac{x \sqrt[4]{b x^3+a x^5}}{\sqrt[4]{2}}\right)-2^{3/4} \tanh ^{-1}\left(\frac{x \sqrt[4]{b x^3+a x^5}}{\sqrt[4]{2}}\right)","2^{3/4} \tan ^{-1}\left(\frac{x \sqrt[4]{b x^3+a x^5}}{\sqrt[4]{2}}\right)-2^{3/4} \tanh ^{-1}\left(\frac{x \sqrt[4]{b x^3+a x^5}}{\sqrt[4]{2}}\right)",1,"2^(3/4)*ArcTan[(x*(b*x^3 + a*x^5)^(1/4))/2^(1/4)] - 2^(3/4)*ArcTanh[(x*(b*x^3 + a*x^5)^(1/4))/2^(1/4)]","A",1
792,1,60,60,19.2735588,"\int \frac{x^2}{\left(1+x^2\right) \sqrt[5]{243-5265 x+47250 x^2-225810 x^3+615255 x^4-954733 x^5+820340 x^6-401440 x^7+112000 x^8-16640 x^9+1024 x^{10}}} \, dx","IntegrateAlgebraic[x^2/((1 + x^2)*(243 - 5265*x + 47250*x^2 - 225810*x^3 + 615255*x^4 - 954733*x^5 + 820340*x^6 - 401440*x^7 + 112000*x^8 - 16640*x^9 + 1024*x^10)^(1/5)),x]","\frac{(-3+x) (-1+4 x) \left(\frac{1}{170} \tan ^{-1}(x)+\frac{9}{110} \log (-3+x)-\frac{1}{187} \log (-1+4 x)-\frac{13}{340} \log \left(1+x^2\right)\right)}{\sqrt[5]{\left(3-13 x+4 x^2\right)^5}}","\frac{(-3+x) (-1+4 x) \left(\frac{1}{170} \tan ^{-1}(x)+\frac{9}{110} \log (-3+x)-\frac{1}{187} \log (-1+4 x)-\frac{13}{340} \log \left(1+x^2\right)\right)}{\sqrt[5]{\left(3-13 x+4 x^2\right)^5}}",1,"((-3 + x)*(-1 + 4*x)*(ArcTan[x]/170 + (9*Log[-3 + x])/110 - Log[-1 + 4*x]/187 - (13*Log[1 + x^2])/340))/((3 - 13*x + 4*x^2)^5)^(1/5)","A",1
793,1,61,61,4.1917582,"\int \frac{\sqrt[3]{1-2 x+x^2}}{2+x^2} \, dx","IntegrateAlgebraic[(1 - 2*x + x^2)^(1/3)/(2 + x^2),x]","\frac{\sqrt[3]{(-1+x)^2} \text{RootSum}\left[3+2 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{\log \left(\sqrt[3]{-1+x}-\text{$\#$1}\right) \text{$\#$1}^2}{1+\text{$\#$1}^3}\&\right]}{2 (-1+x)^{2/3}}","\frac{\sqrt[3]{(-1+x)^2} \text{RootSum}\left[3+2 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{\log \left(\sqrt[3]{-1+x}-\text{$\#$1}\right) \text{$\#$1}^2}{1+\text{$\#$1}^3}\&\right]}{2 (-1+x)^{2/3}}",1,"(((-1 + x)^2)^(1/3)*RootSum[3 + 2*#1^3 + #1^6 & , (Log[(-1 + x)^(1/3) - #1]*#1^2)/(1 + #1^3) & ])/(2*(-1 + x)^(2/3))","A",1
794,1,61,61,4.2574937,"\int \frac{\sqrt[3]{1+2 x+x^2}}{3+x^2} \, dx","IntegrateAlgebraic[(1 + 2*x + x^2)^(1/3)/(3 + x^2),x]","\frac{\sqrt[3]{(1+x)^2} \text{RootSum}\left[4-2 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{\log \left(\sqrt[3]{1+x}-\text{$\#$1}\right) \text{$\#$1}^2}{-1+\text{$\#$1}^3}\&\right]}{2 (1+x)^{2/3}}","\frac{\sqrt[3]{(1+x)^2} \text{RootSum}\left[4-2 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{\log \left(\sqrt[3]{1+x}-\text{$\#$1}\right) \text{$\#$1}^2}{-1+\text{$\#$1}^3}\&\right]}{2 (1+x)^{2/3}}",1,"(((1 + x)^2)^(1/3)*RootSum[4 - 2*#1^3 + #1^6 & , (Log[(1 + x)^(1/3) - #1]*#1^2)/(-1 + #1^3) & ])/(2*(1 + x)^(2/3))","A",1
795,1,61,61,0.2560983,"\int \frac{1}{(-1+x) \sqrt[4]{x+x^3}} \, dx","IntegrateAlgebraic[1/((-1 + x)*(x + x^3)^(1/4)),x]","\frac{\tan ^{-1}\left(\frac{2^{3/4} \sqrt[4]{x+x^3}}{1+x}\right)}{2 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{2^{3/4} \sqrt[4]{x+x^3}}{1+x}\right)}{2 \sqrt[4]{2}}","\frac{\tan ^{-1}\left(\frac{2^{3/4} \sqrt[4]{x+x^3}}{1+x}\right)}{2 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{2^{3/4} \sqrt[4]{x+x^3}}{1+x}\right)}{2 \sqrt[4]{2}}",1,"ArcTan[(2^(3/4)*(x + x^3)^(1/4))/(1 + x)]/(2*2^(1/4)) - ArcTanh[(2^(3/4)*(x + x^3)^(1/4))/(1 + x)]/(2*2^(1/4))","A",1
796,1,61,61,0.2844681,"\int \frac{-1+x^4}{\sqrt{x+x^3} \left(1+x^4\right)} \, dx","IntegrateAlgebraic[(-1 + x^4)/(Sqrt[x + x^3]*(1 + x^4)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt{x+x^3}}{1+x^2}\right)}{\sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt{x+x^3}}{1+x^2}\right)}{\sqrt[4]{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt{x+x^3}}{1+x^2}\right)}{\sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt{x+x^3}}{1+x^2}\right)}{\sqrt[4]{2}}",1,"-(ArcTan[(2^(1/4)*Sqrt[x + x^3])/(1 + x^2)]/2^(1/4)) - ArcTanh[(2^(1/4)*Sqrt[x + x^3])/(1 + x^2)]/2^(1/4)","A",1
797,1,61,61,1.0730663,"\int \frac{1+x^3}{\left(-1+x^3\right) \sqrt{1+x^4}} \, dx","IntegrateAlgebraic[(1 + x^3)/((-1 + x^3)*Sqrt[1 + x^4]),x]","-\frac{4}{3} \tan ^{-1}\left(\frac{x}{1+x+x^2+\sqrt{1+x^4}}\right)-\frac{1}{3} \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{1-2 x+x^2+\sqrt{1+x^4}}\right)","-\frac{4}{3} \tan ^{-1}\left(\frac{x}{1+x+x^2+\sqrt{1+x^4}}\right)-\frac{1}{3} \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{1-2 x+x^2+\sqrt{1+x^4}}\right)",1,"(-4*ArcTan[x/(1 + x + x^2 + Sqrt[1 + x^4])])/3 - (Sqrt[2]*ArcTanh[(Sqrt[2]*x)/(1 - 2*x + x^2 + Sqrt[1 + x^4])])/3","A",1
798,1,61,61,4.2469844,"\int \frac{-1+x}{x \sqrt[4]{1+x^4}} \, dx","IntegrateAlgebraic[(-1 + x)/(x*(1 + x^4)^(1/4)),x]","\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\frac{1}{2} \tan ^{-1}\left(\sqrt[4]{1+x^4}\right)+\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)+\frac{1}{2} \tanh ^{-1}\left(\sqrt[4]{1+x^4}\right)","\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\frac{1}{2} \tan ^{-1}\left(\sqrt[4]{1+x^4}\right)+\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)+\frac{1}{2} \tanh ^{-1}\left(\sqrt[4]{1+x^4}\right)",1,"ArcTan[x/(1 + x^4)^(1/4)]/2 - ArcTan[(1 + x^4)^(1/4)]/2 + ArcTanh[x/(1 + x^4)^(1/4)]/2 + ArcTanh[(1 + x^4)^(1/4)]/2","A",1
799,1,61,61,0.282937,"\int \frac{x^2}{\left(-b+a x^4\right)^{3/4}} \, dx","IntegrateAlgebraic[x^2/(-b + a*x^4)^(3/4),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2 a^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2 a^{3/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2 a^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2 a^{3/4}}",1,"-1/2*ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/a^(3/4) + ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(2*a^(3/4))","A",1
800,1,61,61,0.168687,"\int \frac{\left(6+x^4\right) \sqrt{-2 x+x^4+x^5}}{\left(-2+x^4\right) \left(-2-x^3+x^4\right)} \, dx","IntegrateAlgebraic[((6 + x^4)*Sqrt[-2*x + x^4 + x^5])/((-2 + x^4)*(-2 - x^3 + x^4)),x]","2 \tanh ^{-1}\left(\frac{x^2}{\sqrt{-2 x+x^4+x^5}}\right)-2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{-2 x+x^4+x^5}}{-2+x^3+x^4}\right)","2 \tanh ^{-1}\left(\frac{x^2}{\sqrt{-2 x+x^4+x^5}}\right)-2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{-2 x+x^4+x^5}}{-2+x^3+x^4}\right)",1,"2*ArcTanh[x^2/Sqrt[-2*x + x^4 + x^5]] - 2*Sqrt[2]*ArcTanh[(Sqrt[2]*x*Sqrt[-2*x + x^4 + x^5])/(-2 + x^3 + x^4)]","A",1
801,1,61,61,0.3219984,"\int \frac{\left(-3+x^2\right) \left(1-x^2+x^3\right)^{2/3}}{1-2 x^2-x^3+x^4+x^5+x^6} \, dx","IntegrateAlgebraic[((-3 + x^2)*(1 - x^2 + x^3)^(2/3))/(1 - 2*x^2 - x^3 + x^4 + x^5 + x^6),x]","\text{RootSum}\left[3-3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{1-x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-3+2 \text{$\#$1}^3}\&\right]","\text{RootSum}\left[3-3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{1-x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-3+2 \text{$\#$1}^3}\&\right]",1,"RootSum[3 - 3*#1^3 + #1^6 & , (-(Log[x]*#1^2) + Log[(1 - x^2 + x^3)^(1/3) - x*#1]*#1^2)/(-3 + 2*#1^3) & ]","A",1
802,1,61,61,0.0013302,"\int \frac{\left(-3+x^2\right) \left(1-x^2+x^3\right)^{2/3}}{1-2 x^2-x^3+x^4+x^5+x^6} \, dx","IntegrateAlgebraic[((-3 + x^2)*(1 - x^2 + x^3)^(2/3))/(1 - 2*x^2 - x^3 + x^4 + x^5 + x^6),x]","\text{RootSum}\left[3-3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{1-x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-3+2 \text{$\#$1}^3}\&\right]","\text{RootSum}\left[3-3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{1-x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-3+2 \text{$\#$1}^3}\&\right]",1,"RootSum[3 - 3*#1^3 + #1^6 & , (-(Log[x]*#1^2) + Log[(1 - x^2 + x^3)^(1/3) - x*#1]*#1^2)/(-3 + 2*#1^3) & ]","A",1
803,1,61,61,0.0526766,"\int \frac{\sqrt{-1+x^6} \left(-1+2 x^6\right)^2}{x \left(-1+4 x^6\right)} \, dx","IntegrateAlgebraic[(Sqrt[-1 + x^6]*(-1 + 2*x^6)^2)/(x*(-1 + 4*x^6)),x]","\frac{1}{36} \sqrt{-1+x^6} \left(-13+4 x^6\right)+\frac{1}{3} \tan ^{-1}\left(\sqrt{-1+x^6}\right)-\frac{\tan ^{-1}\left(\frac{2 \sqrt{-1+x^6}}{\sqrt{3}}\right)}{8 \sqrt{3}}","\frac{1}{36} \sqrt{-1+x^6} \left(-13+4 x^6\right)+\frac{1}{3} \tan ^{-1}\left(\sqrt{-1+x^6}\right)-\frac{\tan ^{-1}\left(\frac{2 \sqrt{-1+x^6}}{\sqrt{3}}\right)}{8 \sqrt{3}}",1,"(Sqrt[-1 + x^6]*(-13 + 4*x^6))/36 + ArcTan[Sqrt[-1 + x^6]]/3 - ArcTan[(2*Sqrt[-1 + x^6])/Sqrt[3]]/(8*Sqrt[3])","A",1
804,1,61,61,0.7752438,"\int \frac{1}{\sqrt[3]{x+x^3} \left(b+a x^6\right)} \, dx","IntegrateAlgebraic[1/((x + x^3)^(1/3)*(b + a*x^6)),x]","-\frac{\text{RootSum}\left[a-b+3 b \text{$\#$1}^3-3 b \text{$\#$1}^6+b \text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{x+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 b}","-\frac{\text{RootSum}\left[a-b+3 b \text{$\#$1}^3-3 b \text{$\#$1}^6+b \text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{x+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 b}",1,"-1/6*RootSum[a - b + 3*b*#1^3 - 3*b*#1^6 + b*#1^9 & , (-Log[x] + Log[(x + x^3)^(1/3) - x*#1])/#1 & ]/b","A",1
805,1,61,61,0.0010563,"\int \frac{1}{\sqrt[3]{x+x^3} \left(b+a x^6\right)} \, dx","IntegrateAlgebraic[1/((x + x^3)^(1/3)*(b + a*x^6)),x]","-\frac{\text{RootSum}\left[a-b+3 b \text{$\#$1}^3-3 b \text{$\#$1}^6+b \text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{x+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 b}","-\frac{\text{RootSum}\left[a-b+3 b \text{$\#$1}^3-3 b \text{$\#$1}^6+b \text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{x+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 b}",1,"-1/6*RootSum[a - b + 3*b*#1^3 - 3*b*#1^6 + b*#1^9 & , (-Log[x] + Log[(x + x^3)^(1/3) - x*#1])/#1 & ]/b","A",1
806,1,61,61,1.3903735,"\int \frac{-b+a x^4}{\left(b-2 x^2+a x^4\right) \sqrt[4]{b x^2+a x^6}} \, dx","IntegrateAlgebraic[(-b + a*x^4)/((b - 2*x^2 + a*x^4)*(b*x^2 + a*x^6)^(1/4)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{b x^2+a x^6}}\right)}{\sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{b x^2+a x^6}}\right)}{\sqrt[4]{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{b x^2+a x^6}}\right)}{\sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{b x^2+a x^6}}\right)}{\sqrt[4]{2}}",1,"-(ArcTan[(2^(1/4)*x)/(b*x^2 + a*x^6)^(1/4)]/2^(1/4)) - ArcTanh[(2^(1/4)*x)/(b*x^2 + a*x^6)^(1/4)]/2^(1/4)","A",1
807,1,61,61,0.3253802,"\int \frac{\left(4+x^5\right) \sqrt{1-2 x^5+x^8+x^{10}}}{x^9} \, dx","IntegrateAlgebraic[((4 + x^5)*Sqrt[1 - 2*x^5 + x^8 + x^10])/x^9,x]","\frac{\left(-1+x^5\right) \sqrt{1-2 x^5+x^8+x^{10}}}{2 x^8}-2 \log (x)+\frac{1}{2} \log \left(-1+x^5+\sqrt{1-2 x^5+x^8+x^{10}}\right)","\frac{\left(-1+x^5\right) \sqrt{1-2 x^5+x^8+x^{10}}}{2 x^8}-2 \log (x)+\frac{1}{2} \log \left(-1+x^5+\sqrt{1-2 x^5+x^8+x^{10}}\right)",1,"((-1 + x^5)*Sqrt[1 - 2*x^5 + x^8 + x^10])/(2*x^8) - 2*Log[x] + Log[-1 + x^5 + Sqrt[1 - 2*x^5 + x^8 + x^10]]/2","A",1
808,1,61,61,0.5034192,"\int \frac{\left(1+x^6\right) \left(-1+x^3+x^6\right)^{2/3}}{1-x^6+x^{12}} \, dx","IntegrateAlgebraic[((1 + x^6)*(-1 + x^3 + x^6)^(2/3))/(1 - x^6 + x^12),x]","\frac{1}{6} \text{RootSum}\left[2-2 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{-1+x^3+x^6}-x \text{$\#$1}\right) \text{$\#$1}^2}{-1+\text{$\#$1}^3}\&\right]","\frac{1}{6} \text{RootSum}\left[2-2 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{-1+x^3+x^6}-x \text{$\#$1}\right) \text{$\#$1}^2}{-1+\text{$\#$1}^3}\&\right]",1,"RootSum[2 - 2*#1^3 + #1^6 & , (-(Log[x]*#1^2) + Log[(-1 + x^3 + x^6)^(1/3) - x*#1]*#1^2)/(-1 + #1^3) & ]/6","A",1
809,1,61,61,0.0012923,"\int \frac{\left(1+x^6\right) \left(-1+x^3+x^6\right)^{2/3}}{1-x^6+x^{12}} \, dx","IntegrateAlgebraic[((1 + x^6)*(-1 + x^3 + x^6)^(2/3))/(1 - x^6 + x^12),x]","\frac{1}{6} \text{RootSum}\left[2-2 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{-1+x^3+x^6}-x \text{$\#$1}\right) \text{$\#$1}^2}{-1+\text{$\#$1}^3}\&\right]","\frac{1}{6} \text{RootSum}\left[2-2 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{-1+x^3+x^6}-x \text{$\#$1}\right) \text{$\#$1}^2}{-1+\text{$\#$1}^3}\&\right]",1,"RootSum[2 - 2*#1^3 + #1^6 & , (-(Log[x]*#1^2) + Log[(-1 + x^3 + x^6)^(1/3) - x*#1]*#1^2)/(-1 + #1^3) & ]/6","A",1
810,1,61,61,0.1906648,"\int \frac{\left(-2+5 x^7\right) \sqrt[3]{2 x+x^3+2 x^8}}{4+x^4+8 x^7+4 x^{14}} \, dx","IntegrateAlgebraic[((-2 + 5*x^7)*(2*x + x^3 + 2*x^8)^(1/3))/(4 + x^4 + 8*x^7 + 4*x^14),x]","\frac{1}{4} \text{RootSum}\left[2-2 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{2 x+x^3+2 x^8}-x \text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^3}\&\right]","\frac{1}{4} \text{RootSum}\left[2-2 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{2 x+x^3+2 x^8}-x \text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^3}\&\right]",1,"RootSum[2 - 2*#1^3 + #1^6 & , (-(Log[x]*#1) + Log[(2*x + x^3 + 2*x^8)^(1/3) - x*#1]*#1)/(-1 + #1^3) & ]/4","A",1
811,1,61,61,0.0012557,"\int \frac{\left(-2+5 x^7\right) \sqrt[3]{2 x+x^3+2 x^8}}{4+x^4+8 x^7+4 x^{14}} \, dx","IntegrateAlgebraic[((-2 + 5*x^7)*(2*x + x^3 + 2*x^8)^(1/3))/(4 + x^4 + 8*x^7 + 4*x^14),x]","\frac{1}{4} \text{RootSum}\left[2-2 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{2 x+x^3+2 x^8}-x \text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^3}\&\right]","\frac{1}{4} \text{RootSum}\left[2-2 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{2 x+x^3+2 x^8}-x \text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^3}\&\right]",1,"RootSum[2 - 2*#1^3 + #1^6 & , (-(Log[x]*#1) + Log[(2*x + x^3 + 2*x^8)^(1/3) - x*#1]*#1)/(-1 + #1^3) & ]/4","A",1
812,1,61,61,0.0997728,"\int \frac{1}{\sqrt{a x+\sqrt{b^2+a^2 x^2}}} \, dx","IntegrateAlgebraic[1/Sqrt[a*x + Sqrt[b^2 + a^2*x^2]],x]","-\frac{b^2}{3 a \left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}+\frac{\sqrt{a x+\sqrt{b^2+a^2 x^2}}}{a}","-\frac{b^2}{3 a \left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}+\frac{\sqrt{a x+\sqrt{b^2+a^2 x^2}}}{a}",1,"-1/3*b^2/(a*(a*x + Sqrt[b^2 + a^2*x^2])^(3/2)) + Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]/a","A",1
813,1,62,62,0.1875857,"\int x^{10} \sqrt[4]{-1+x^4} \, dx","IntegrateAlgebraic[x^10*(-1 + x^4)^(1/4),x]","\frac{1}{384} \sqrt[4]{-1+x^4} \left(-7 x^3-4 x^7+32 x^{11}\right)+\frac{7}{256} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)-\frac{7}{256} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)","\frac{1}{384} \sqrt[4]{-1+x^4} \left(-7 x^3-4 x^7+32 x^{11}\right)+\frac{7}{256} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)-\frac{7}{256} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)",1,"((-1 + x^4)^(1/4)*(-7*x^3 - 4*x^7 + 32*x^11))/384 + (7*ArcTan[x/(-1 + x^4)^(1/4)])/256 - (7*ArcTanh[x/(-1 + x^4)^(1/4)])/256","A",1
814,1,62,62,0.1884402,"\int x^{10} \sqrt[4]{1+x^4} \, dx","IntegrateAlgebraic[x^10*(1 + x^4)^(1/4),x]","\frac{1}{384} \sqrt[4]{1+x^4} \left(-7 x^3+4 x^7+32 x^{11}\right)-\frac{7}{256} \tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)+\frac{7}{256} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)","\frac{1}{384} \sqrt[4]{1+x^4} \left(-7 x^3+4 x^7+32 x^{11}\right)-\frac{7}{256} \tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)+\frac{7}{256} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)",1,"((1 + x^4)^(1/4)*(-7*x^3 + 4*x^7 + 32*x^11))/384 - (7*ArcTan[x/(1 + x^4)^(1/4)])/256 + (7*ArcTanh[x/(1 + x^4)^(1/4)])/256","A",1
815,1,62,62,0.1993131,"\int \frac{1+x^2}{\left(1-x^2+x^4\right) \sqrt[4]{x^2+x^4}} \, dx","IntegrateAlgebraic[(1 + x^2)/((1 - x^2 + x^4)*(x^2 + x^4)^(1/4)),x]","-\frac{1}{2} \text{RootSum}\left[3-3 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}^3+\log \left(\sqrt[4]{x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}^3}{-3+2 \text{$\#$1}^4}\&\right]","-\frac{1}{2} \text{RootSum}\left[3-3 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}^3+\log \left(\sqrt[4]{x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}^3}{-3+2 \text{$\#$1}^4}\&\right]",1,"-1/2*RootSum[3 - 3*#1^4 + #1^8 & , (-(Log[x]*#1^3) + Log[(x^2 + x^4)^(1/4) - x*#1]*#1^3)/(-3 + 2*#1^4) & ]","A",1
816,1,62,62,0.3829097,"\int \frac{1}{\left(-b+a x^4\right) \sqrt[4]{-b x^2+a x^4}} \, dx","IntegrateAlgebraic[1/((-b + a*x^4)*(-(b*x^2) + a*x^4)^(1/4)),x]","\frac{\text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{4 b}","\frac{\text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{4 b}",1,"RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (-Log[x] + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1])/#1 & ]/(4*b)","A",1
817,1,62,62,0.0019287,"\int \frac{1}{\left(-b+a x^4\right) \sqrt[4]{-b x^2+a x^4}} \, dx","IntegrateAlgebraic[1/((-b + a*x^4)*(-(b*x^2) + a*x^4)^(1/4)),x]","\frac{\text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{4 b}","\frac{\text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{4 b}",1,"RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (-Log[x] + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1])/#1 & ]/(4*b)","A",1
818,1,62,62,0.3085432,"\int \frac{1-x^2}{x \sqrt{a+b x+c x^2+b x^3+a x^4}} \, dx","IntegrateAlgebraic[(1 - x^2)/(x*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]),x]","-\frac{\log (x)}{\sqrt{a}}+\frac{\log \left(-2 a-b x-2 a x^2+2 \sqrt{a} \sqrt{a+b x+c x^2+b x^3+a x^4}\right)}{\sqrt{a}}","-\frac{\log (x)}{\sqrt{a}}+\frac{\log \left(-2 a-b x-2 a x^2+2 \sqrt{a} \sqrt{a+b x+c x^2+b x^3+a x^4}\right)}{\sqrt{a}}",1,"-(Log[x]/Sqrt[a]) + Log[-2*a - b*x - 2*a*x^2 + 2*Sqrt[a]*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]]/Sqrt[a]","A",1
819,1,62,62,2.6647018,"\int \frac{x^2 \left(-1+3 x^4\right)}{\left(1+x^4\right)^2 \left(a-x+a x^4\right) \sqrt{x+x^5}} \, dx","IntegrateAlgebraic[(x^2*(-1 + 3*x^4))/((1 + x^4)^2*(a - x + a*x^4)*Sqrt[x + x^5]),x]","\frac{2 \left(3 a+x+3 a x^4\right) \sqrt{x+x^5}}{3 \left(1+x^4\right)^2}-2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{x+x^5}}{\sqrt{a} \left(1+x^4\right)}\right)","\frac{2 \left(3 a+x+3 a x^4\right) \sqrt{x+x^5}}{3 \left(1+x^4\right)^2}-2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{x+x^5}}{\sqrt{a} \left(1+x^4\right)}\right)",1,"(2*(3*a + x + 3*a*x^4)*Sqrt[x + x^5])/(3*(1 + x^4)^2) - 2*a^(3/2)*ArcTanh[Sqrt[x + x^5]/(Sqrt[a]*(1 + x^4))]","A",1
820,1,62,62,2.6750738,"\int \frac{\left(-3+x^4\right) \left(1-x^3+2 x^4-x^6-x^7+x^8\right)}{x^6 \left(1-x^3+x^4\right) \sqrt[4]{x+x^5}} \, dx","IntegrateAlgebraic[((-3 + x^4)*(1 - x^3 + 2*x^4 - x^6 - x^7 + x^8))/(x^6*(1 - x^3 + x^4)*(x + x^5)^(1/4)),x]","\frac{4 \left(1+x^4\right) \left(x+x^5\right)^{3/4}}{7 x^6}+2 \tan ^{-1}\left(\frac{\left(x+x^5\right)^{3/4}}{1+x^4}\right)+2 \tanh ^{-1}\left(\frac{\left(x+x^5\right)^{3/4}}{1+x^4}\right)","\frac{4 \left(1+x^4\right) \left(x+x^5\right)^{3/4}}{7 x^6}+2 \tan ^{-1}\left(\frac{\left(x+x^5\right)^{3/4}}{1+x^4}\right)+2 \tanh ^{-1}\left(\frac{\left(x+x^5\right)^{3/4}}{1+x^4}\right)",1,"(4*(1 + x^4)*(x + x^5)^(3/4))/(7*x^6) + 2*ArcTan[(x + x^5)^(3/4)/(1 + x^4)] + 2*ArcTanh[(x + x^5)^(3/4)/(1 + x^4)]","A",1
821,1,58,62,0.0354941,"\int \frac{1+x^2}{\sqrt{1+\sqrt{1+x}}} \, dx","IntegrateAlgebraic[(1 + x^2)/Sqrt[1 + Sqrt[1 + x]],x]","\frac{4 \sqrt{1+\sqrt{1+x}} \left(-2732+1366 \sqrt{1+x}+708 (1+x)-590 (1+x)^{3/2}-350 (1+x)^2+315 (1+x)^{5/2}\right)}{3465}","-\frac{8 \left(1187-4 x+175 x^2\right) \sqrt{1+\sqrt{1+x}}}{3465}+\frac{4 \sqrt{1+x} \left(1091+40 x+315 x^2\right) \sqrt{1+\sqrt{1+x}}}{3465}",1,"(4*Sqrt[1 + Sqrt[1 + x]]*(-2732 + 1366*Sqrt[1 + x] + 708*(1 + x) - 590*(1 + x)^(3/2) - 350*(1 + x)^2 + 315*(1 + x)^(5/2)))/3465","A",1
822,1,62,62,0.4288558,"\int \frac{-1+2 x}{(1+x) \sqrt{-a^2 x^2+(1+x)^6}} \, dx","IntegrateAlgebraic[(-1 + 2*x)/((1 + x)*Sqrt[-(a^2*x^2) + (1 + x)^6]),x]","-\frac{2 \tan ^{-1}\left(\frac{a x}{1+3 x+3 x^2+x^3+\sqrt{1+6 x+\left(15-a^2\right) x^2+20 x^3+15 x^4+6 x^5+x^6}}\right)}{a}","-\frac{2 \tan ^{-1}\left(\frac{a x}{1+3 x+3 x^2+x^3+\sqrt{1+6 x+\left(15-a^2\right) x^2+20 x^3+15 x^4+6 x^5+x^6}}\right)}{a}",1,"(-2*ArcTan[(a*x)/(1 + 3*x + 3*x^2 + x^3 + Sqrt[1 + 6*x + (15 - a^2)*x^2 + 20*x^3 + 15*x^4 + 6*x^5 + x^6])])/a","A",1
823,1,62,62,0.1185371,"\int \frac{x}{\sqrt{-b+a^2 x^2} \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}} \, dx","IntegrateAlgebraic[x/(Sqrt[-b + a^2*x^2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4)),x]","-\frac{2 b}{5 a^2 \left(a x+\sqrt{-b+a^2 x^2}\right)^{5/4}}+\frac{2 \left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4}}{3 a^2}","-\frac{2 b}{5 a^2 \left(a x+\sqrt{-b+a^2 x^2}\right)^{5/4}}+\frac{2 \left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4}}{3 a^2}",1,"(-2*b)/(5*a^2*(a*x + Sqrt[-b + a^2*x^2])^(5/4)) + (2*(a*x + Sqrt[-b + a^2*x^2])^(3/4))/(3*a^2)","A",1
824,1,62,62,0.0819217,"\int \sqrt{a x+\sqrt{b^2+a^2 x^2}} \, dx","IntegrateAlgebraic[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]],x]","-\frac{b^2}{a \sqrt{a x+\sqrt{b^2+a^2 x^2}}}+\frac{\left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}{3 a}","-\frac{b^2}{a \sqrt{a x+\sqrt{b^2+a^2 x^2}}}+\frac{\left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}{3 a}",1,"-(b^2/(a*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])) + (a*x + Sqrt[b^2 + a^2*x^2])^(3/2)/(3*a)","A",1
825,1,63,63,0.204416,"\int \frac{1}{\left(1+\sqrt{x}\right) \sqrt{-\sqrt{x}+x}} \, dx","IntegrateAlgebraic[1/((1 + Sqrt[x])*Sqrt[-Sqrt[x] + x]),x]","4 \tanh ^{-1}\left(\frac{\sqrt{-\sqrt{x}+x}}{-1+\sqrt{x}}\right)-2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{-\sqrt{x}+x}}{-1+\sqrt{x}}\right)","4 \tanh ^{-1}\left(\frac{\sqrt{-\sqrt{x}+x}}{-1+\sqrt{x}}\right)-2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{-\sqrt{x}+x}}{-1+\sqrt{x}}\right)",1,"4*ArcTanh[Sqrt[-Sqrt[x] + x]/(-1 + Sqrt[x])] - 2*Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[-Sqrt[x] + x])/(-1 + Sqrt[x])]","A",1
826,1,63,63,0.1085929,"\int \frac{x}{\left(-1+x^2\right) \sqrt{x+x^2+x^3}} \, dx","IntegrateAlgebraic[x/((-1 + x^2)*Sqrt[x + x^2 + x^3]),x]","\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{x+x^2+x^3}}{1+x+x^2}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt{3} \sqrt{x+x^2+x^3}}{1+x+x^2}\right)}{2 \sqrt{3}}","\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{x+x^2+x^3}}{1+x+x^2}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt{3} \sqrt{x+x^2+x^3}}{1+x+x^2}\right)}{2 \sqrt{3}}",1,"ArcTan[Sqrt[x + x^2 + x^3]/(1 + x + x^2)]/2 - ArcTanh[(Sqrt[3]*Sqrt[x + x^2 + x^3])/(1 + x + x^2)]/(2*Sqrt[3])","A",1
827,1,63,63,0.7024019,"\int \frac{-a b c+2 a (b+c) x-(3 a+b+c) x^2+2 x^3}{\sqrt{x (-a+x) (-b+x) (-c+x)} \left(a d+(b c-d) x-(b+c) x^2+x^3\right)} \, dx","IntegrateAlgebraic[(-(a*b*c) + 2*a*(b + c)*x - (3*a + b + c)*x^2 + 2*x^3)/(Sqrt[x*(-a + x)*(-b + x)*(-c + x)]*(a*d + (b*c - d)*x - (b + c)*x^2 + x^3)),x]","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{-a b c x+(a b+a c+b c) x^2+(-a-b-c) x^3+x^4}}{\sqrt{d} (a-x)}\right)}{\sqrt{d}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{-a b c x+(a b+a c+b c) x^2+(-a-b-c) x^3+x^4}}{\sqrt{d} (a-x)}\right)}{\sqrt{d}}",1,"(2*ArcTanh[Sqrt[-(a*b*c*x) + (a*b + a*c + b*c)*x^2 + (-a - b - c)*x^3 + x^4]/(Sqrt[d]*(a - x))])/Sqrt[d]","A",1
828,1,63,63,0.06745,"\int \frac{3 b+a x^3}{x \left(-b+a x^3\right) \sqrt{b+a x^3}} \, dx","IntegrateAlgebraic[(3*b + a*x^3)/(x*(-b + a*x^3)*Sqrt[b + a*x^3]),x]","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{b+a x^3}}{\sqrt{b}}\right)}{\sqrt{b}}-\frac{4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{b+a x^3}}{\sqrt{2} \sqrt{b}}\right)}{3 \sqrt{b}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{b+a x^3}}{\sqrt{b}}\right)}{\sqrt{b}}-\frac{4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{b+a x^3}}{\sqrt{2} \sqrt{b}}\right)}{3 \sqrt{b}}",1,"(2*ArcTanh[Sqrt[b + a*x^3]/Sqrt[b]])/Sqrt[b] - (4*Sqrt[2]*ArcTanh[Sqrt[b + a*x^3]/(Sqrt[2]*Sqrt[b])])/(3*Sqrt[b])","A",1
829,1,63,63,2.9022412,"\int \frac{\left(2 c-a x^3\right) \sqrt{c+b x^2+a x^3}}{\left(c+(-3+b) x^2+a x^3\right) \left(c+(-2+b) x^2+a x^3\right)} \, dx","IntegrateAlgebraic[((2*c - a*x^3)*Sqrt[c + b*x^2 + a*x^3])/((c + (-3 + b)*x^2 + a*x^3)*(c + (-2 + b)*x^2 + a*x^3)),x]","-2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{c+b x^2+a x^3}}\right)+2 \sqrt{3} \tanh ^{-1}\left(\frac{\sqrt{3} x}{\sqrt{c+b x^2+a x^3}}\right)","-2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{c+b x^2+a x^3}}\right)+2 \sqrt{3} \tanh ^{-1}\left(\frac{\sqrt{3} x}{\sqrt{c+b x^2+a x^3}}\right)",1,"-2*Sqrt[2]*ArcTanh[(Sqrt[2]*x)/Sqrt[c + b*x^2 + a*x^3]] + 2*Sqrt[3]*ArcTanh[(Sqrt[3]*x)/Sqrt[c + b*x^2 + a*x^3]]","A",1
830,1,63,63,0.5364202,"\int \frac{-a b c+2 a (b+c) x-(3 a+b+c) x^2+2 x^3}{\sqrt{x (-a+x) (-b+x) (-c+x)} \left(a+(-1+b c d) x-(b+c) d x^2+d x^3\right)} \, dx","IntegrateAlgebraic[(-(a*b*c) + 2*a*(b + c)*x - (3*a + b + c)*x^2 + 2*x^3)/(Sqrt[x*(-a + x)*(-b + x)*(-c + x)]*(a + (-1 + b*c*d)*x - (b + c)*d*x^2 + d*x^3)),x]","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{-a b c x+(a b+a c+b c) x^2+(-a-b-c) x^3+x^4}}{a-x}\right)}{\sqrt{d}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{-a b c x+(a b+a c+b c) x^2+(-a-b-c) x^3+x^4}}{a-x}\right)}{\sqrt{d}}",1,"(2*ArcTanh[(Sqrt[d]*Sqrt[-(a*b*c*x) + (a*b + a*c + b*c)*x^2 + (-a - b - c)*x^3 + x^4])/(a - x)])/Sqrt[d]","A",1
831,1,63,63,1.0464327,"\int \frac{-1+x^3}{\left(1+x^3\right) \sqrt{1+x^4}} \, dx","IntegrateAlgebraic[(-1 + x^3)/((1 + x^3)*Sqrt[1 + x^4]),x]","-\frac{4}{3} \tan ^{-1}\left(\frac{x}{1-x+x^2+\sqrt{1+x^4}}\right)-\frac{1}{3} \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{1+2 x+x^2+\sqrt{1+x^4}}\right)","-\frac{4}{3} \tan ^{-1}\left(\frac{x}{1-x+x^2+\sqrt{1+x^4}}\right)-\frac{1}{3} \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{1+2 x+x^2+\sqrt{1+x^4}}\right)",1,"(-4*ArcTan[x/(1 - x + x^2 + Sqrt[1 + x^4])])/3 - (Sqrt[2]*ArcTanh[(Sqrt[2]*x)/(1 + 2*x + x^2 + Sqrt[1 + x^4])])/3","A",1
832,1,63,63,0.257085,"\int x^4 \sqrt[4]{-x+x^4} \, dx","IntegrateAlgebraic[x^4*(-x + x^4)^(1/4),x]","\frac{1}{24} \sqrt[4]{-x+x^4} \left(-x^2+4 x^5\right)+\frac{1}{16} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x+x^4}}\right)-\frac{1}{16} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x+x^4}}\right)","\frac{1}{24} \sqrt[4]{-x+x^4} \left(-x^2+4 x^5\right)+\frac{1}{16} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x+x^4}}\right)-\frac{1}{16} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x+x^4}}\right)",1,"((-x + x^4)^(1/4)*(-x^2 + 4*x^5))/24 + ArcTan[x/(-x + x^4)^(1/4)]/16 - ArcTanh[x/(-x + x^4)^(1/4)]/16","A",1
833,1,63,63,0.2465745,"\int \sqrt[4]{-x^3+x^4} \, dx","IntegrateAlgebraic[(-x^3 + x^4)^(1/4),x]","\frac{1}{8} (-1+4 x) \sqrt[4]{-x^3+x^4}+\frac{3}{16} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)-\frac{3}{16} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)","\frac{1}{8} (-1+4 x) \sqrt[4]{-x^3+x^4}+\frac{3}{16} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)-\frac{3}{16} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)",1,"((-1 + 4*x)*(-x^3 + x^4)^(1/4))/8 + (3*ArcTan[x/(-x^3 + x^4)^(1/4)])/16 - (3*ArcTanh[x/(-x^3 + x^4)^(1/4)])/16","A",1
834,1,63,63,1.3455306,"\int \frac{\sqrt{2-x^3-x^4} \left(4+x^3+2 x^4\right)}{\left(-2-3 x^2+x^3+x^4\right) \left(-2-x^2+x^3+x^4\right)} \, dx","IntegrateAlgebraic[(Sqrt[2 - x^3 - x^4]*(4 + x^3 + 2*x^4))/((-2 - 3*x^2 + x^3 + x^4)*(-2 - x^2 + x^3 + x^4)),x]","-\tan ^{-1}\left(\frac{x}{\sqrt{2-x^3-x^4}}\right)-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x \sqrt{2-x^3-x^4}}{-2+x^3+x^4}\right)","-\tan ^{-1}\left(\frac{x}{\sqrt{2-x^3-x^4}}\right)-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x \sqrt{2-x^3-x^4}}{-2+x^3+x^4}\right)",1,"-ArcTan[x/Sqrt[2 - x^3 - x^4]] - Sqrt[3]*ArcTan[(Sqrt[3]*x*Sqrt[2 - x^3 - x^4])/(-2 + x^3 + x^4)]","A",1
835,1,63,63,0.4367982,"\int \frac{\sqrt[4]{b x^3+a x^4}}{x \left(b+a x+x^2\right)} \, dx","IntegrateAlgebraic[(b*x^3 + a*x^4)^(1/4)/(x*(b + a*x + x^2)),x]","-\text{RootSum}\left[b-a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+2 \text{$\#$1}^4}\&\right]","-\text{RootSum}\left[b-a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+2 \text{$\#$1}^4}\&\right]",1,"-RootSum[b - a*#1^4 + #1^8 & , (-(Log[x]*#1) + Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1)/(-a + 2*#1^4) & ]","A",1
836,1,63,63,0.0015743,"\int \frac{\sqrt[4]{b x^3+a x^4}}{x \left(b+a x+x^2\right)} \, dx","IntegrateAlgebraic[(b*x^3 + a*x^4)^(1/4)/(x*(b + a*x + x^2)),x]","-\text{RootSum}\left[b-a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+2 \text{$\#$1}^4}\&\right]","-\text{RootSum}\left[b-a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+2 \text{$\#$1}^4}\&\right]",1,"-RootSum[b - a*#1^4 + #1^8 & , (-(Log[x]*#1) + Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1)/(-a + 2*#1^4) & ]","A",1
837,1,67,63,0.1476165,"\int \frac{x^2 \left(-4+x^6\right)}{\sqrt{-1+x^6} \left(2+x^6\right)} \, dx","IntegrateAlgebraic[(x^2*(-4 + x^6))/(Sqrt[-1 + x^6]*(2 + x^6)),x]","-\sqrt{\frac{2}{3}} \tanh ^{-1}\left(\sqrt{\frac{2}{3}}+\frac{x^6}{\sqrt{6}}-\frac{x^3 \sqrt{-1+x^6}}{\sqrt{6}}\right)-\frac{1}{3} \log \left(-x^3+\sqrt{-1+x^6}\right)","\sqrt{\frac{2}{3}} \tanh ^{-1}\left(\sqrt{\frac{2}{3}}+\frac{x^6}{\sqrt{6}}+\frac{x^3 \sqrt{-1+x^6}}{\sqrt{6}}\right)+\frac{1}{3} \log \left(x^3+\sqrt{-1+x^6}\right)",1,"-(Sqrt[2/3]*ArcTanh[Sqrt[2/3] + x^6/Sqrt[6] - (x^3*Sqrt[-1 + x^6])/Sqrt[6]]) - Log[-x^3 + Sqrt[-1 + x^6]]/3","A",1
838,1,63,63,2.7383143,"\int \frac{\left(-1+x^5\right) \left(-1+x^3+x^5\right) \left(3+2 x^5\right)}{x^6 \left(-1-x^3+x^5\right) \sqrt[4]{-x+x^6}} \, dx","IntegrateAlgebraic[((-1 + x^5)*(-1 + x^3 + x^5)*(3 + 2*x^5))/(x^6*(-1 - x^3 + x^5)*(-x + x^6)^(1/4)),x]","\frac{4 \left(-3+14 x^3+3 x^5\right) \left(-x+x^6\right)^{3/4}}{21 x^6}-4 \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x+x^6}}\right)-4 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x+x^6}}\right)","\frac{4 \left(-3+14 x^3+3 x^5\right) \left(-x+x^6\right)^{3/4}}{21 x^6}-4 \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x+x^6}}\right)-4 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x+x^6}}\right)",1,"(4*(-3 + 14*x^3 + 3*x^5)*(-x + x^6)^(3/4))/(21*x^6) - 4*ArcTan[x/(-x + x^6)^(1/4)] - 4*ArcTanh[x/(-x + x^6)^(1/4)]","A",1
839,1,63,63,0.091776,"\int \frac{\sqrt{1+x^3} \left(2+2 x^3+x^6\right)}{x^7 \left(-1+x^6\right)} \, dx","IntegrateAlgebraic[(Sqrt[1 + x^3]*(2 + 2*x^3 + x^6))/(x^7*(-1 + x^6)),x]","\frac{\sqrt{1+x^3} \left(2+5 x^3\right)}{6 x^6}+\frac{5}{2} \tanh ^{-1}\left(\sqrt{1+x^3}\right)-\frac{5}{3} \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{1+x^3}}{\sqrt{2}}\right)","\frac{\sqrt{1+x^3} \left(2+5 x^3\right)}{6 x^6}+\frac{5}{2} \tanh ^{-1}\left(\sqrt{1+x^3}\right)-\frac{5}{3} \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{1+x^3}}{\sqrt{2}}\right)",1,"(Sqrt[1 + x^3]*(2 + 5*x^3))/(6*x^6) + (5*ArcTanh[Sqrt[1 + x^3]])/2 - (5*Sqrt[2]*ArcTanh[Sqrt[1 + x^3]/Sqrt[2]])/3","A",1
840,1,63,63,2.8623358,"\int \frac{\sqrt{-1+x^2+x^5} \left(2+3 x^5\right)}{1+x^4-2 x^5+x^{10}} \, dx","IntegrateAlgebraic[(Sqrt[-1 + x^2 + x^5]*(2 + 3*x^5))/(1 + x^4 - 2*x^5 + x^10),x]","-\sqrt{1+i} \tan ^{-1}\left(\frac{\sqrt{-1-i} x}{\sqrt{-1+x^2+x^5}}\right)-\sqrt{1-i} \tan ^{-1}\left(\frac{\sqrt{-1+i} x}{\sqrt{-1+x^2+x^5}}\right)","-\sqrt{1+i} \tan ^{-1}\left(\frac{\sqrt{-1-i} x}{\sqrt{-1+x^2+x^5}}\right)-\sqrt{1-i} \tan ^{-1}\left(\frac{\sqrt{-1+i} x}{\sqrt{-1+x^2+x^5}}\right)",1,"-(Sqrt[1 + I]*ArcTan[(Sqrt[-1 - I]*x)/Sqrt[-1 + x^2 + x^5]]) - Sqrt[1 - I]*ArcTan[(Sqrt[-1 + I]*x)/Sqrt[-1 + x^2 + x^5]]","A",1
841,1,63,63,3.8656511,"\int \frac{1}{x \sqrt{x+x^2} \sqrt{x^2+x \sqrt{x+x^2}}} \, dx","IntegrateAlgebraic[1/(x*Sqrt[x + x^2]*Sqrt[x^2 + x*Sqrt[x + x^2]]),x]","\frac{4 (-3+4 x) \sqrt{x \left(x+\sqrt{x+x^2}\right)}}{15 x^2}+\frac{16 \sqrt{x+x^2} \sqrt{x \left(x+\sqrt{x+x^2}\right)}}{15 x^2}","\frac{4 (-3+4 x) \sqrt{x \left(x+\sqrt{x+x^2}\right)}}{15 x^2}+\frac{16 \sqrt{x+x^2} \sqrt{x \left(x+\sqrt{x+x^2}\right)}}{15 x^2}",1,"(4*(-3 + 4*x)*Sqrt[x*(x + Sqrt[x + x^2])])/(15*x^2) + (16*Sqrt[x + x^2]*Sqrt[x*(x + Sqrt[x + x^2])])/(15*x^2)","A",1
842,1,63,63,0.1205471,"\int \sqrt{b+\sqrt{b^2+a x^2}} \, dx","IntegrateAlgebraic[Sqrt[b + Sqrt[b^2 + a*x^2]],x]","\frac{4 b x}{3 \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{2 x \sqrt{b^2+a x^2}}{3 \sqrt{b+\sqrt{b^2+a x^2}}}","\frac{4 b x}{3 \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{2 x \sqrt{b^2+a x^2}}{3 \sqrt{b+\sqrt{b^2+a x^2}}}",1,"(4*b*x)/(3*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (2*x*Sqrt[b^2 + a*x^2])/(3*Sqrt[b + Sqrt[b^2 + a*x^2]])","A",1
843,1,63,63,0.1247212,"\int \frac{\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\sqrt{1+x^2} \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/(Sqrt[1 + x^2]*Sqrt[x + Sqrt[1 + x^2]]),x]","-\frac{2 \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\sqrt{x+\sqrt{1+x^2}}}-2 \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)","-\frac{2 \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\sqrt{x+\sqrt{1+x^2}}}-2 \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)",1,"(-2*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/Sqrt[x + Sqrt[1 + x^2]] - 2*ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]]","A",1
844,1,64,64,0.279453,"\int \frac{-b+a x}{(b+a x) \sqrt{b^2 x+a^2 x^3}} \, dx","IntegrateAlgebraic[(-b + a*x)/((b + a*x)*Sqrt[b^2*x + a^2*x^3]),x]","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{\sqrt{a} \sqrt{b}}","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{\sqrt{a} \sqrt{b}}",1,"-((Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[b]*Sqrt[b^2*x + a^2*x^3])/(b^2 + a^2*x^2)])/(Sqrt[a]*Sqrt[b]))","A",1
845,1,64,64,0.2780067,"\int \frac{b+a x}{(-b+a x) \sqrt{b^2 x+a^2 x^3}} \, dx","IntegrateAlgebraic[(b + a*x)/((-b + a*x)*Sqrt[b^2*x + a^2*x^3]),x]","-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{\sqrt{a} \sqrt{b}}","-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{\sqrt{a} \sqrt{b}}",1,"-((Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[b]*Sqrt[b^2*x + a^2*x^3])/(b^2 + a^2*x^2)])/(Sqrt[a]*Sqrt[b]))","A",1
846,1,64,64,0.2905314,"\int \frac{-b x^2+a x^3}{\left(b x^2+a x^3\right) \sqrt{b^2 x+a^2 x^3}} \, dx","IntegrateAlgebraic[(-(b*x^2) + a*x^3)/((b*x^2 + a*x^3)*Sqrt[b^2*x + a^2*x^3]),x]","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{\sqrt{a} \sqrt{b}}","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{\sqrt{a} \sqrt{b}}",1,"-((Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[b]*Sqrt[b^2*x + a^2*x^3])/(b^2 + a^2*x^2)])/(Sqrt[a]*Sqrt[b]))","A",1
847,1,64,64,5.15627,"\int \frac{2+5 x}{\sqrt{-12+20 x+5 x^2+2 x^3+x^4}} \, dx","IntegrateAlgebraic[(2 + 5*x)/Sqrt[-12 + 20*x + 5*x^2 + 2*x^3 + x^4],x]","-\log \left(-16-16 x-24 x^2-8 x^3-3 x^4-x^5+\left(8+4 x+2 x^2+x^3\right) \sqrt{-12+20 x+5 x^2+2 x^3+x^4}\right)","-\log \left(-16-16 x-24 x^2-8 x^3-3 x^4-x^5+\left(8+4 x+2 x^2+x^3\right) \sqrt{-12+20 x+5 x^2+2 x^3+x^4}\right)",1,"-Log[-16 - 16*x - 24*x^2 - 8*x^3 - 3*x^4 - x^5 + (8 + 4*x + 2*x^2 + x^3)*Sqrt[-12 + 20*x + 5*x^2 + 2*x^3 + x^4]]","A",1
848,1,68,64,0.1959235,"\int \frac{\left(-1+x^3\right) \sqrt{-1+x^6}}{x^7} \, dx","IntegrateAlgebraic[((-1 + x^3)*Sqrt[-1 + x^6])/x^7,x]","\frac{\left(1-2 x^3\right) \sqrt{-1+x^6}}{6 x^6}+\frac{1}{3} \tan ^{-1}\left(\frac{\sqrt{-1+x^6}}{-1+x^3}\right)+\frac{2}{3} \tanh ^{-1}\left(\frac{\sqrt{-1+x^6}}{-1+x^3}\right)","\frac{\left(1-2 x^3\right) \sqrt{-1+x^6}}{6 x^6}+\frac{1}{3} \tan ^{-1}\left(\frac{1+x^3}{\sqrt{-1+x^6}}\right)+\frac{2}{3} \tanh ^{-1}\left(\frac{1+x^3}{\sqrt{-1+x^6}}\right)",1,"((1 - 2*x^3)*Sqrt[-1 + x^6])/(6*x^6) + ArcTan[Sqrt[-1 + x^6]/(-1 + x^3)]/3 + (2*ArcTanh[Sqrt[-1 + x^6]/(-1 + x^3)])/3","A",1
849,1,64,64,0.0782037,"\int \frac{\sqrt{-1+x^6} \left(-1+2 x^6\right)^2}{x^7 \left(-1+4 x^6\right)} \, dx","IntegrateAlgebraic[(Sqrt[-1 + x^6]*(-1 + 2*x^6)^2)/(x^7*(-1 + 4*x^6)),x]","\frac{\sqrt{-1+x^6} \left(1+2 x^6\right)}{6 x^6}-\frac{1}{6} \tan ^{-1}\left(\sqrt{-1+x^6}\right)-\frac{\tan ^{-1}\left(\frac{2 \sqrt{-1+x^6}}{\sqrt{3}}\right)}{2 \sqrt{3}}","\frac{\sqrt{-1+x^6} \left(1+2 x^6\right)}{6 x^6}-\frac{1}{6} \tan ^{-1}\left(\sqrt{-1+x^6}\right)-\frac{\tan ^{-1}\left(\frac{2 \sqrt{-1+x^6}}{\sqrt{3}}\right)}{2 \sqrt{3}}",1,"(Sqrt[-1 + x^6]*(1 + 2*x^6))/(6*x^6) - ArcTan[Sqrt[-1 + x^6]]/6 - ArcTan[(2*Sqrt[-1 + x^6])/Sqrt[3]]/(2*Sqrt[3])","A",1
850,1,64,64,0.3277363,"\int \frac{-2+x^4}{\sqrt[4]{1+x^4} \left(-1+x^4+2 x^8\right)} \, dx","IntegrateAlgebraic[(-2 + x^4)/((1 + x^4)^(1/4)*(-1 + x^4 + 2*x^8)),x]","\frac{x}{\sqrt[4]{1+x^4}}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{3} x}{\sqrt[4]{1+x^4}}\right)}{2 \sqrt[4]{3}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{3} x}{\sqrt[4]{1+x^4}}\right)}{2 \sqrt[4]{3}}","\frac{x}{\sqrt[4]{1+x^4}}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{3} x}{\sqrt[4]{1+x^4}}\right)}{2 \sqrt[4]{3}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{3} x}{\sqrt[4]{1+x^4}}\right)}{2 \sqrt[4]{3}}",1,"x/(1 + x^4)^(1/4) + ArcTan[(3^(1/4)*x)/(1 + x^4)^(1/4)]/(2*3^(1/4)) + ArcTanh[(3^(1/4)*x)/(1 + x^4)^(1/4)]/(2*3^(1/4))","A",1
851,1,66,64,0.1008394,"\int \frac{-x^2+10 x^8}{\sqrt{-1+x^6} \left(-1+4 x^6\right)} \, dx","IntegrateAlgebraic[(-x^2 + 10*x^8)/(Sqrt[-1 + x^6]*(-1 + 4*x^6)),x]","\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{4 x^6}{\sqrt{3}}+\frac{4 x^3 \sqrt{-1+x^6}}{\sqrt{3}}\right)}{2 \sqrt{3}}-\frac{5}{6} \log \left(-x^3+\sqrt{-1+x^6}\right)","-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{4 x^6}{\sqrt{3}}-\frac{4 x^3 \sqrt{-1+x^6}}{\sqrt{3}}\right)}{2 \sqrt{3}}+\frac{5}{6} \log \left(x^3+\sqrt{-1+x^6}\right)",1,"ArcTan[1/Sqrt[3] - (4*x^6)/Sqrt[3] + (4*x^3*Sqrt[-1 + x^6])/Sqrt[3]]/(2*Sqrt[3]) - (5*Log[-x^3 + Sqrt[-1 + x^6]])/6","A",1
852,1,65,65,0.1680551,"\int \frac{1+x^2}{\left(-1+x^2\right) \left(1+2 x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(1 + x^2)/((-1 + x^2)*(1 + 2*x^2)^(3/2)),x]","-\frac{x}{3 \sqrt{1+2 x^2}}-\frac{2 \tanh ^{-1}\left(\sqrt{\frac{2}{3}}-\sqrt{\frac{2}{3}} x^2+\frac{x \sqrt{1+2 x^2}}{\sqrt{3}}\right)}{3 \sqrt{3}}","-\frac{x}{3 \sqrt{1+2 x^2}}-\frac{2 \tanh ^{-1}\left(\sqrt{\frac{2}{3}}-\sqrt{\frac{2}{3}} x^2+\frac{x \sqrt{1+2 x^2}}{\sqrt{3}}\right)}{3 \sqrt{3}}",1,"-1/3*x/Sqrt[1 + 2*x^2] - (2*ArcTanh[Sqrt[2/3] - Sqrt[2/3]*x^2 + (x*Sqrt[1 + 2*x^2])/Sqrt[3]])/(3*Sqrt[3])","A",1
853,1,65,65,0.0637469,"\int \frac{\left(b+a x^2\right)^{3/4}}{x} \, dx","IntegrateAlgebraic[(b + a*x^2)^(3/4)/x,x]","\frac{2}{3} \left(b+a x^2\right)^{3/4}+b^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^2}}{\sqrt[4]{b}}\right)-b^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^2}}{\sqrt[4]{b}}\right)","\frac{2}{3} \left(b+a x^2\right)^{3/4}+b^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^2}}{\sqrt[4]{b}}\right)-b^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^2}}{\sqrt[4]{b}}\right)",1,"(2*(b + a*x^2)^(3/4))/3 + b^(3/4)*ArcTan[(b + a*x^2)^(1/4)/b^(1/4)] - b^(3/4)*ArcTanh[(b + a*x^2)^(1/4)/b^(1/4)]","A",1
854,1,65,65,0.2668617,"\int \frac{1-x+x^2}{\left(-1+x^2\right) \sqrt{x+x^3}} \, dx","IntegrateAlgebraic[(1 - x + x^2)/((-1 + x^2)*Sqrt[x + x^3]),x]","-\frac{3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{x+x^3}}{1+x^2}\right)}{2 \sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{x+x^3}}{1+x^2}\right)}{2 \sqrt{2}}","-\frac{3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{x+x^3}}{1+x^2}\right)}{2 \sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{x+x^3}}{1+x^2}\right)}{2 \sqrt{2}}",1,"(-3*ArcTan[(Sqrt[2]*Sqrt[x + x^3])/(1 + x^2)])/(2*Sqrt[2]) - ArcTanh[(Sqrt[2]*Sqrt[x + x^3])/(1 + x^2)]/(2*Sqrt[2])","A",1
855,1,65,65,3.8253321,"\int \frac{-1+2 x^2}{\left(1+x^2\right) \sqrt{-1-x^2+x^4}} \, dx","IntegrateAlgebraic[(-1 + 2*x^2)/((1 + x^2)*Sqrt[-1 - x^2 + x^4]),x]","-\frac{1}{2} i \log \left(\frac{-i-2 i x^2+x \sqrt{-1-x^2+x^4}}{-i-2 i x^2-x \sqrt{-1-x^2+x^4}}\right)","-\frac{1}{2} i \log \left(\frac{-i-2 i x^2+x \sqrt{-1-x^2+x^4}}{-i-2 i x^2-x \sqrt{-1-x^2+x^4}}\right)",1,"(-1/2*I)*Log[(-I - (2*I)*x^2 + x*Sqrt[-1 - x^2 + x^4])/(-I - (2*I)*x^2 - x*Sqrt[-1 - x^2 + x^4])]","A",1
856,1,65,65,0.4076382,"\int \frac{1}{\left(-2 b+a x^4\right) \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[1/((-2*b + a*x^4)*(b*x^2 + a*x^4)^(1/4)),x]","\frac{\text{RootSum}\left[2 a^2-a b-4 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{8 b}","\frac{\text{RootSum}\left[2 a^2-a b-4 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{8 b}",1,"RootSum[2*a^2 - a*b - 4*a*#1^4 + 2*#1^8 & , (-Log[x] + Log[(b*x^2 + a*x^4)^(1/4) - x*#1])/#1 & ]/(8*b)","A",1
857,1,65,65,0.0017883,"\int \frac{1}{\left(-2 b+a x^4\right) \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[1/((-2*b + a*x^4)*(b*x^2 + a*x^4)^(1/4)),x]","\frac{\text{RootSum}\left[2 a^2-a b-4 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{8 b}","\frac{\text{RootSum}\left[2 a^2-a b-4 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{8 b}",1,"RootSum[2*a^2 - a*b - 4*a*#1^4 + 2*#1^8 & , (-Log[x] + Log[(b*x^2 + a*x^4)^(1/4) - x*#1])/#1 & ]/(8*b)","A",1
858,1,65,65,0.4553042,"\int \frac{-2 b+c x^2}{\left(-b+c x^2\right) \sqrt[4]{-b+c x^2+a x^4}} \, dx","IntegrateAlgebraic[(-2*b + c*x^2)/((-b + c*x^2)*(-b + c*x^2 + a*x^4)^(1/4)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+c x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+c x^2+a x^4}}\right)}{\sqrt[4]{a}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+c x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+c x^2+a x^4}}\right)}{\sqrt[4]{a}}",1,"ArcTan[(a^(1/4)*x)/(-b + c*x^2 + a*x^4)^(1/4)]/a^(1/4) + ArcTanh[(a^(1/4)*x)/(-b + c*x^2 + a*x^4)^(1/4)]/a^(1/4)","A",1
859,1,65,65,0.8104804,"\int \frac{-4 b+a x^3}{\left(-b+a x^3\right) \sqrt[4]{b-a x^3+c x^4}} \, dx","IntegrateAlgebraic[(-4*b + a*x^3)/((-b + a*x^3)*(b - a*x^3 + c*x^4)^(1/4)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{b-a x^3+c x^4}}\right)}{\sqrt[4]{c}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{b-a x^3+c x^4}}\right)}{\sqrt[4]{c}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{b-a x^3+c x^4}}\right)}{\sqrt[4]{c}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{b-a x^3+c x^4}}\right)}{\sqrt[4]{c}}",1,"(2*ArcTan[(c^(1/4)*x)/(b - a*x^3 + c*x^4)^(1/4)])/c^(1/4) + (2*ArcTanh[(c^(1/4)*x)/(b - a*x^3 + c*x^4)^(1/4)])/c^(1/4)","A",1
860,1,65,65,2.7522012,"\int \frac{-b+a x^6}{\left(b+a x^6\right) \sqrt[3]{-b+a^3 x^3+a x^6}} \, dx","IntegrateAlgebraic[(-b + a*x^6)/((b + a*x^6)*(-b + a^3*x^3 + a*x^6)^(1/3)),x]","\frac{1}{6} \text{RootSum}\left[a^6+4 a b-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b+a^3 x^3+a x^6}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{1}{6} \text{RootSum}\left[a^6+4 a b-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b+a^3 x^3+a x^6}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"RootSum[a^6 + 4*a*b - 2*a^3*#1^3 + #1^6 & , (-Log[x] + Log[(-b + a^3*x^3 + a*x^6)^(1/3) - x*#1])/#1 & ]/6","A",1
861,1,65,65,0.0020988,"\int \frac{-b+a x^6}{\left(b+a x^6\right) \sqrt[3]{-b+a^3 x^3+a x^6}} \, dx","IntegrateAlgebraic[(-b + a*x^6)/((b + a*x^6)*(-b + a^3*x^3 + a*x^6)^(1/3)),x]","\frac{1}{6} \text{RootSum}\left[a^6+4 a b-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b+a^3 x^3+a x^6}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{1}{6} \text{RootSum}\left[a^6+4 a b-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b+a^3 x^3+a x^6}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"RootSum[a^6 + 4*a*b - 2*a^3*#1^3 + #1^6 & , (-Log[x] + Log[(-b + a^3*x^3 + a*x^6)^(1/3) - x*#1])/#1 & ]/6","A",1
862,1,57,65,0.2687042,"\int \frac{\sqrt{1-x^4} \left(1+x^4\right)}{4-7 x^4+4 x^8} \, dx","IntegrateAlgebraic[(Sqrt[1 - x^4]*(1 + x^4))/(4 - 7*x^4 + 4*x^8),x]","\left(\frac{1}{8}-\frac{i}{8}\right) \tan ^{-1}\left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) x}{\sqrt{1-x^4}}\right)-\left(\frac{1}{8}+\frac{i}{8}\right) \tan ^{-1}\left(\frac{(1+i) \sqrt{1-x^4}}{x}\right)","\frac{1}{8} \tan ^{-1}\left(\frac{-1+\frac{x^2}{2}+x^4}{x \sqrt{1-x^4}}\right)-\frac{1}{8} \tanh ^{-1}\left(\frac{-1-\frac{x^2}{2}+x^4}{x \sqrt{1-x^4}}\right)",1,"(1/8 - I/8)*ArcTan[((1/2 + I/2)*x)/Sqrt[1 - x^4]] - (1/8 + I/8)*ArcTan[((1 + I)*Sqrt[1 - x^4])/x]","C",1
863,1,64,65,0.4987284,"\int \frac{x^4}{\sqrt[4]{-b+a x^4} \left(-b+a x^8\right)} \, dx","IntegrateAlgebraic[x^4/((-b + a*x^4)^(1/4)*(-b + a*x^8)),x]","\frac{1}{8} \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{\log (x)-\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)}{-a \text{$\#$1}+\text{$\#$1}^5}\&\right]","\frac{1}{8} \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)}{a \text{$\#$1}-\text{$\#$1}^5}\&\right]",1,"RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (Log[x] - Log[(-b + a*x^4)^(1/4) - x*#1])/(-(a*#1) + #1^5) & ]/8","A",1
864,1,65,65,0.1475434,"\int \frac{\sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{1+x^2} \, dx","IntegrateAlgebraic[(Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 + x^2),x]","\text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^2}\&\right]","\text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^2}\&\right]",1,"RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(-1 + #1^2) & ]","A",1
865,1,65,65,0.0018466,"\int \frac{\sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{1+x^2} \, dx","IntegrateAlgebraic[(Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 + x^2),x]","\text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^2}\&\right]","\text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^2}\&\right]",1,"RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(-1 + #1^2) & ]","A",1
866,1,66,66,0.0589945,"\int \frac{\sqrt{-b+a x^3}}{x \left(2 b+a x^3\right)} \, dx","IntegrateAlgebraic[Sqrt[-b + a*x^3]/(x*(2*b + a*x^3)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{-b+a x^3}}{\sqrt{b}}\right)}{3 \sqrt{b}}+\frac{\tan ^{-1}\left(\frac{\sqrt{-b+a x^3}}{\sqrt{3} \sqrt{b}}\right)}{\sqrt{3} \sqrt{b}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{-b+a x^3}}{\sqrt{b}}\right)}{3 \sqrt{b}}+\frac{\tan ^{-1}\left(\frac{\sqrt{-b+a x^3}}{\sqrt{3} \sqrt{b}}\right)}{\sqrt{3} \sqrt{b}}",1,"-1/3*ArcTan[Sqrt[-b + a*x^3]/Sqrt[b]]/Sqrt[b] + ArcTan[Sqrt[-b + a*x^3]/(Sqrt[3]*Sqrt[b])]/(Sqrt[3]*Sqrt[b])","A",1
867,1,66,66,0.0586688,"\int \frac{-b+4 a x^3}{x \sqrt{-b+a x^3} \left(2 b+a x^3\right)} \, dx","IntegrateAlgebraic[(-b + 4*a*x^3)/(x*Sqrt[-b + a*x^3]*(2*b + a*x^3)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{-b+a x^3}}{\sqrt{b}}\right)}{3 \sqrt{b}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{-b+a x^3}}{\sqrt{3} \sqrt{b}}\right)}{\sqrt{b}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{-b+a x^3}}{\sqrt{b}}\right)}{3 \sqrt{b}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{-b+a x^3}}{\sqrt{3} \sqrt{b}}\right)}{\sqrt{b}}",1,"-1/3*ArcTan[Sqrt[-b + a*x^3]/Sqrt[b]]/Sqrt[b] + (Sqrt[3]*ArcTan[Sqrt[-b + a*x^3]/(Sqrt[3]*Sqrt[b])])/Sqrt[b]","A",1
868,1,66,66,0.6347699,"\int \frac{\sqrt{1+6 x^2+x^4}}{(-1+x) (1+x)^3} \, dx","IntegrateAlgebraic[Sqrt[1 + 6*x^2 + x^4]/((-1 + x)*(1 + x)^3),x]","\frac{\sqrt{1+6 x^2+x^4}}{4 (1+x)^2}-\frac{\tanh ^{-1}\left(\frac{2 \sqrt{2} x}{1-2 x+x^2+\sqrt{1+6 x^2+x^4}}\right)}{2 \sqrt{2}}","\frac{\sqrt{1+6 x^2+x^4}}{4 (1+x)^2}-\frac{\tanh ^{-1}\left(\frac{2 \sqrt{2} x}{1-2 x+x^2+\sqrt{1+6 x^2+x^4}}\right)}{2 \sqrt{2}}",1,"Sqrt[1 + 6*x^2 + x^4]/(4*(1 + x)^2) - ArcTanh[(2*Sqrt[2]*x)/(1 - 2*x + x^2 + Sqrt[1 + 6*x^2 + x^4])]/(2*Sqrt[2])","A",1
869,1,66,66,0.058292,"\int \frac{\sqrt[4]{b+a x^4}}{x} \, dx","IntegrateAlgebraic[(b + a*x^4)^(1/4)/x,x]","\sqrt[4]{b+a x^4}-\frac{1}{2} \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^4}}{\sqrt[4]{b}}\right)-\frac{1}{2} \sqrt[4]{b} \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^4}}{\sqrt[4]{b}}\right)","\sqrt[4]{b+a x^4}-\frac{1}{2} \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^4}}{\sqrt[4]{b}}\right)-\frac{1}{2} \sqrt[4]{b} \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^4}}{\sqrt[4]{b}}\right)",1,"(b + a*x^4)^(1/4) - (b^(1/4)*ArcTan[(b + a*x^4)^(1/4)/b^(1/4)])/2 - (b^(1/4)*ArcTanh[(b + a*x^4)^(1/4)/b^(1/4)])/2","A",1
870,1,66,66,0.3775774,"\int \frac{1}{\left(-2 b+a x^4\right) \sqrt[4]{-b x^2+a x^4}} \, dx","IntegrateAlgebraic[1/((-2*b + a*x^4)*(-(b*x^2) + a*x^4)^(1/4)),x]","\frac{\text{RootSum}\left[2 a^2-a b-4 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{8 b}","\frac{\text{RootSum}\left[2 a^2-a b-4 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{8 b}",1,"RootSum[2*a^2 - a*b - 4*a*#1^4 + 2*#1^8 & , (-Log[x] + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1])/#1 & ]/(8*b)","A",1
871,1,66,66,0.0018073,"\int \frac{1}{\left(-2 b+a x^4\right) \sqrt[4]{-b x^2+a x^4}} \, dx","IntegrateAlgebraic[1/((-2*b + a*x^4)*(-(b*x^2) + a*x^4)^(1/4)),x]","\frac{\text{RootSum}\left[2 a^2-a b-4 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{8 b}","\frac{\text{RootSum}\left[2 a^2-a b-4 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{8 b}",1,"RootSum[2*a^2 - a*b - 4*a*#1^4 + 2*#1^8 & , (-Log[x] + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1])/#1 & ]/(8*b)","A",1
872,1,65,66,0.6418735,"\int \frac{x^6}{\left(b+a x^4\right)^{3/4} \left(b^2+a^2 x^8\right)} \, dx","IntegrateAlgebraic[x^6/((b + a*x^4)^(3/4)*(b^2 + a^2*x^8)),x]","\frac{\text{RootSum}\left[2 a^2-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{\log (x)-\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)}{-a \text{$\#$1}^3+\text{$\#$1}^7}\&\right]}{8 b}","\frac{\text{RootSum}\left[2 a^2-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)}{a \text{$\#$1}^3-\text{$\#$1}^7}\&\right]}{8 b}",1,"RootSum[2*a^2 - 2*a*#1^4 + #1^8 & , (Log[x] - Log[(b + a*x^4)^(1/4) - x*#1])/(-(a*#1^3) + #1^7) & ]/(8*b)","A",1
873,1,66,66,3.2020032,"\int \sqrt{x^3+x^2 \sqrt{-1+x^2}} \, dx","IntegrateAlgebraic[Sqrt[x^3 + x^2*Sqrt[-1 + x^2]],x]","-\frac{2}{15} \sqrt{-1+x^2} \sqrt{x^2 \left(x+\sqrt{-1+x^2}\right)}+\frac{4 \left(-1+2 x^2\right) \sqrt{x^2 \left(x+\sqrt{-1+x^2}\right)}}{15 x}","-\frac{2}{15} \sqrt{-1+x^2} \sqrt{x^2 \left(x+\sqrt{-1+x^2}\right)}+\frac{4 \left(-1+2 x^2\right) \sqrt{x^2 \left(x+\sqrt{-1+x^2}\right)}}{15 x}",1,"(-2*Sqrt[-1 + x^2]*Sqrt[x^2*(x + Sqrt[-1 + x^2])])/15 + (4*(-1 + 2*x^2)*Sqrt[x^2*(x + Sqrt[-1 + x^2])])/(15*x)","A",1
874,1,67,67,0.1783198,"\int \frac{(-a+x) (-b+x) \left(-a b+x^2\right)}{x \sqrt{x (-a+x) (-b+x)} \left(a b-(a+b+d) x+x^2\right)} \, dx","IntegrateAlgebraic[((-a + x)*(-b + x)*(-(a*b) + x^2))/(x*Sqrt[x*(-a + x)*(-b + x)]*(a*b - (a + b + d)*x + x^2)),x]","\frac{2 \sqrt{a b x+(-a-b) x^2+x^3}}{x}-2 \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{a b x+(-a-b) x^2+x^3}}\right)","\frac{2 \sqrt{a b x+(-a-b) x^2+x^3}}{x}-2 \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{a b x+(-a-b) x^2+x^3}}\right)",1,"(2*Sqrt[a*b*x + (-a - b)*x^2 + x^3])/x - 2*Sqrt[d]*ArcTanh[(Sqrt[d]*x)/Sqrt[a*b*x + (-a - b)*x^2 + x^3]]","A",1
875,1,67,67,0.3333397,"\int \frac{1}{\left(b+a^3 x^2\right) \sqrt[3]{-b x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[1/((b + a^3*x^2)*(-(b*x^2) + a^3*x^3)^(1/3)),x]","-\frac{\text{RootSum}\left[a^6+a^3 b-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{2 b}","-\frac{\text{RootSum}\left[a^6+a^3 b-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{2 b}",1,"-1/2*RootSum[a^6 + a^3*b - 2*a^3*#1^3 + #1^6 & , (-Log[x] + Log[(-(b*x^2) + a^3*x^3)^(1/3) - x*#1])/#1 & ]/b","A",1
876,1,67,67,0.0019211,"\int \frac{1}{\left(b+a^3 x^2\right) \sqrt[3]{-b x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[1/((b + a^3*x^2)*(-(b*x^2) + a^3*x^3)^(1/3)),x]","-\frac{\text{RootSum}\left[a^6+a^3 b-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{2 b}","-\frac{\text{RootSum}\left[a^6+a^3 b-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{2 b}",1,"-1/2*RootSum[a^6 + a^3*b - 2*a^3*#1^3 + #1^6 & , (-Log[x] + Log[(-(b*x^2) + a^3*x^3)^(1/3) - x*#1])/#1 & ]/b","A",1
877,1,53,67,0.1895656,"\int \frac{\sqrt{-1+x^4}}{1+x^4} \, dx","IntegrateAlgebraic[Sqrt[-1 + x^4]/(1 + x^4),x]","\left(-\frac{1}{4}+\frac{i}{4}\right) \tan ^{-1}\left(\frac{(1+i) x}{\sqrt{-1+x^4}}\right)+\left(\frac{1}{4}+\frac{i}{4}\right) \tan ^{-1}\left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{-1+x^4}}{x}\right)","\frac{1}{4} \tan ^{-1}\left(\frac{-\frac{1}{2}-x^2+\frac{x^4}{2}}{x \sqrt{-1+x^4}}\right)-\frac{1}{4} \tanh ^{-1}\left(\frac{-\frac{1}{2}+x^2+\frac{x^4}{2}}{x \sqrt{-1+x^4}}\right)",1,"(-1/4 + I/4)*ArcTan[((1 + I)*x)/Sqrt[-1 + x^4]] + (1/4 + I/4)*ArcTan[((1/2 + I/2)*Sqrt[-1 + x^4])/x]","C",1
878,1,67,67,0.1535394,"\int x^2 \sqrt[4]{-x^2+x^4} \, dx","IntegrateAlgebraic[x^2*(-x^2 + x^4)^(1/4),x]","\frac{1}{16} \left(-x+4 x^3\right) \sqrt[4]{-x^2+x^4}+\frac{3}{32} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)-\frac{3}{32} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)","\frac{1}{16} \left(-x+4 x^3\right) \sqrt[4]{-x^2+x^4}+\frac{3}{32} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)-\frac{3}{32} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)",1,"((-x + 4*x^3)*(-x^2 + x^4)^(1/4))/16 + (3*ArcTan[x/(-x^2 + x^4)^(1/4)])/32 - (3*ArcTanh[x/(-x^2 + x^4)^(1/4)])/32","A",1
879,1,67,67,13.5608733,"\int \frac{x}{\sqrt{1+4 x+3 x^2-2 x^3+x^4}} \, dx","IntegrateAlgebraic[x/Sqrt[1 + 4*x + 3*x^2 - 2*x^3 + x^4],x]","\frac{1}{3} \tanh ^{-1}\left(\frac{(1+x) \sqrt{1+4 x+3 x^2-2 x^3+x^4}}{x^3}\right)+\tanh ^{-1}\left(\frac{-1-2 x+\sqrt{1+4 x+3 x^2-2 x^3+x^4}}{x^2}\right)","\frac{1}{3} \tanh ^{-1}\left(\frac{(1+x) \sqrt{1+4 x+3 x^2-2 x^3+x^4}}{x^3}\right)+\tanh ^{-1}\left(\frac{-1-2 x+\sqrt{1+4 x+3 x^2-2 x^3+x^4}}{x^2}\right)",1,"ArcTanh[((1 + x)*Sqrt[1 + 4*x + 3*x^2 - 2*x^3 + x^4])/x^3]/3 + ArcTanh[(-1 - 2*x + Sqrt[1 + 4*x + 3*x^2 - 2*x^3 + x^4])/x^2]","A",1
880,1,67,67,14.1738067,"\int \frac{x}{\sqrt{1-4 x+3 x^2+2 x^3+x^4}} \, dx","IntegrateAlgebraic[x/Sqrt[1 - 4*x + 3*x^2 + 2*x^3 + x^4],x]","\frac{1}{3} \tanh ^{-1}\left(\frac{(-1+x) \sqrt{1-4 x+3 x^2+2 x^3+x^4}}{x^3}\right)+\tanh ^{-1}\left(\frac{-1+2 x+\sqrt{1-4 x+3 x^2+2 x^3+x^4}}{x^2}\right)","\frac{1}{3} \tanh ^{-1}\left(\frac{(-1+x) \sqrt{1-4 x+3 x^2+2 x^3+x^4}}{x^3}\right)+\tanh ^{-1}\left(\frac{-1+2 x+\sqrt{1-4 x+3 x^2+2 x^3+x^4}}{x^2}\right)",1,"ArcTanh[((-1 + x)*Sqrt[1 - 4*x + 3*x^2 + 2*x^3 + x^4])/x^3]/3 + ArcTanh[(-1 + 2*x + Sqrt[1 - 4*x + 3*x^2 + 2*x^3 + x^4])/x^2]","A",1
881,1,67,67,0.4436817,"\int \frac{3-3 x^2+2 x^4}{\sqrt[4]{-1+x^2} \left(2-3 x^2+x^4\right)} \, dx","IntegrateAlgebraic[(3 - 3*x^2 + 2*x^4)/((-1 + x^2)^(1/4)*(2 - 3*x^2 + x^4)),x]","\frac{4 x}{\sqrt[4]{-1+x^2}}+\frac{5 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^2}}{x}\right)}{2 \sqrt{2}}-\frac{5 \tanh ^{-1}\left(\frac{x}{\sqrt{2} \sqrt[4]{-1+x^2}}\right)}{2 \sqrt{2}}","\frac{4 x}{\sqrt[4]{-1+x^2}}+\frac{5 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^2}}{x}\right)}{2 \sqrt{2}}-\frac{5 \tanh ^{-1}\left(\frac{x}{\sqrt{2} \sqrt[4]{-1+x^2}}\right)}{2 \sqrt{2}}",1,"(4*x)/(-1 + x^2)^(1/4) + (5*ArcTan[(Sqrt[2]*(-1 + x^2)^(1/4))/x])/(2*Sqrt[2]) - (5*ArcTanh[x/(Sqrt[2]*(-1 + x^2)^(1/4))])/(2*Sqrt[2])","A",1
882,1,66,67,0.3981568,"\int \frac{x^2}{\left(-b+a x^4\right) \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[x^2/((-b + a*x^4)*(b*x^2 + a*x^4)^(1/4)),x]","-\frac{1}{4} \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{\log (x)-\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{-a \text{$\#$1}+\text{$\#$1}^5}\&\right]","-\frac{1}{4} \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{a \text{$\#$1}-\text{$\#$1}^5}\&\right]",1,"-1/4*RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (Log[x] - Log[(b*x^2 + a*x^4)^(1/4) - x*#1])/(-(a*#1) + #1^5) & ]","A",1
883,1,66,67,0.0017864,"\int \frac{x^2}{\left(-b+a x^4\right) \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[x^2/((-b + a*x^4)*(b*x^2 + a*x^4)^(1/4)),x]","-\frac{1}{4} \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{\log (x)-\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{-a \text{$\#$1}+\text{$\#$1}^5}\&\right]","-\frac{1}{4} \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{a \text{$\#$1}-\text{$\#$1}^5}\&\right]",1,"-1/4*RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (Log[x] - Log[(b*x^2 + a*x^4)^(1/4) - x*#1])/(-(a*#1) + #1^5) & ]","A",1
884,1,67,67,0.5278309,"\int \frac{4 b+x^3}{\left(b+x^3\right) \sqrt[4]{-b-x^3+a x^4}} \, dx","IntegrateAlgebraic[(4*b + x^3)/((b + x^3)*(-b - x^3 + a*x^4)^(1/4)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b-x^3+a x^4}}\right)}{\sqrt[4]{a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b-x^3+a x^4}}\right)}{\sqrt[4]{a}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b-x^3+a x^4}}\right)}{\sqrt[4]{a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b-x^3+a x^4}}\right)}{\sqrt[4]{a}}",1,"(2*ArcTan[(a^(1/4)*x)/(-b - x^3 + a*x^4)^(1/4)])/a^(1/4) + (2*ArcTanh[(a^(1/4)*x)/(-b - x^3 + a*x^4)^(1/4)])/a^(1/4)","A",1
885,1,67,67,3.1150015,"\int \frac{4 b+a x^5}{\left(-b+a x^5\right) \sqrt[4]{-b+c x^4+a x^5}} \, dx","IntegrateAlgebraic[(4*b + a*x^5)/((-b + a*x^5)*(-b + c*x^4 + a*x^5)^(1/4)),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{-b+c x^4+a x^5}}\right)}{\sqrt[4]{c}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{-b+c x^4+a x^5}}\right)}{\sqrt[4]{c}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{-b+c x^4+a x^5}}\right)}{\sqrt[4]{c}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{-b+c x^4+a x^5}}\right)}{\sqrt[4]{c}}",1,"(-2*ArcTan[(c^(1/4)*x)/(-b + c*x^4 + a*x^5)^(1/4)])/c^(1/4) - (2*ArcTanh[(c^(1/4)*x)/(-b + c*x^4 + a*x^5)^(1/4)])/c^(1/4)","A",1
886,1,67,67,2.8003975,"\int \frac{\sqrt{1+x^2-2 x^6} \left(1+4 x^6\right)}{\left(-1-4 x^2+2 x^6\right) \left(-1-2 x^2+2 x^6\right)} \, dx","IntegrateAlgebraic[(Sqrt[1 + x^2 - 2*x^6]*(1 + 4*x^6))/((-1 - 4*x^2 + 2*x^6)*(-1 - 2*x^2 + 2*x^6)),x]","-\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt{1+x^2-2 x^6}}\right)-\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x \sqrt{1+x^2-2 x^6}}{-1-x^2+2 x^6}\right)","-\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt{1+x^2-2 x^6}}\right)-\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x \sqrt{1+x^2-2 x^6}}{-1-x^2+2 x^6}\right)",1,"-1/2*ArcTan[x/Sqrt[1 + x^2 - 2*x^6]] - (Sqrt[3]*ArcTan[(Sqrt[3]*x*Sqrt[1 + x^2 - 2*x^6])/(-1 - x^2 + 2*x^6)])/2","A",1
887,1,67,67,0.3245703,"\int \frac{x^4}{\sqrt[4]{-1+x^4} \left(-1+x^8\right)} \, dx","IntegrateAlgebraic[x^4/((-1 + x^4)^(1/4)*(-1 + x^8)),x]","-\frac{x}{2 \sqrt[4]{-1+x^4}}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-1+x^4}}\right)}{4 \sqrt[4]{2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-1+x^4}}\right)}{4 \sqrt[4]{2}}","-\frac{x}{2 \sqrt[4]{-1+x^4}}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-1+x^4}}\right)}{4 \sqrt[4]{2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-1+x^4}}\right)}{4 \sqrt[4]{2}}",1,"-1/2*x/(-1 + x^4)^(1/4) + ArcTan[(2^(1/4)*x)/(-1 + x^4)^(1/4)]/(4*2^(1/4)) + ArcTanh[(2^(1/4)*x)/(-1 + x^4)^(1/4)]/(4*2^(1/4))","A",1
888,1,67,67,0.3183347,"\int \frac{1}{\sqrt[4]{1+x^4} \left(-1+x^8\right)} \, dx","IntegrateAlgebraic[1/((1 + x^4)^(1/4)*(-1 + x^8)),x]","-\frac{x}{2 \sqrt[4]{1+x^4}}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{1+x^4}}\right)}{4 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{1+x^4}}\right)}{4 \sqrt[4]{2}}","-\frac{x}{2 \sqrt[4]{1+x^4}}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{1+x^4}}\right)}{4 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{1+x^4}}\right)}{4 \sqrt[4]{2}}",1,"-1/2*x/(1 + x^4)^(1/4) - ArcTan[(2^(1/4)*x)/(1 + x^4)^(1/4)]/(4*2^(1/4)) - ArcTanh[(2^(1/4)*x)/(1 + x^4)^(1/4)]/(4*2^(1/4))","A",1
889,1,67,67,0.3422474,"\int \frac{-1+2 x^4}{\sqrt[4]{-1+x^4} \left(-1+x^8\right)} \, dx","IntegrateAlgebraic[(-1 + 2*x^4)/((-1 + x^4)^(1/4)*(-1 + x^8)),x]","-\frac{x}{2 \sqrt[4]{-1+x^4}}+\frac{3 \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-1+x^4}}\right)}{4 \sqrt[4]{2}}+\frac{3 \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-1+x^4}}\right)}{4 \sqrt[4]{2}}","-\frac{x}{2 \sqrt[4]{-1+x^4}}+\frac{3 \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-1+x^4}}\right)}{4 \sqrt[4]{2}}+\frac{3 \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-1+x^4}}\right)}{4 \sqrt[4]{2}}",1,"-1/2*x/(-1 + x^4)^(1/4) + (3*ArcTan[(2^(1/4)*x)/(-1 + x^4)^(1/4)])/(4*2^(1/4)) + (3*ArcTanh[(2^(1/4)*x)/(-1 + x^4)^(1/4)])/(4*2^(1/4))","A",1
890,1,67,67,0.392299,"\int \frac{1+x^8}{\sqrt{1+x^4} \left(-1+x^8\right)} \, dx","IntegrateAlgebraic[(1 + x^8)/(Sqrt[1 + x^4]*(-1 + x^8)),x]","-\frac{x}{2 \sqrt{1+x^4}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{4 \sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{4 \sqrt{2}}","-\frac{x}{2 \sqrt{1+x^4}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{4 \sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{4 \sqrt{2}}",1,"-1/2*x/Sqrt[1 + x^4] - ArcTan[(Sqrt[2]*x)/Sqrt[1 + x^4]]/(4*Sqrt[2]) - ArcTanh[(Sqrt[2]*x)/Sqrt[1 + x^4]]/(4*Sqrt[2])","A",1
891,1,67,67,0.5199782,"\int \frac{-1+x^{12}}{\sqrt{1+x^4} \left(1+x^{12}\right)} \, dx","IntegrateAlgebraic[(-1 + x^12)/(Sqrt[1 + x^4]*(1 + x^12)),x]","-\frac{x}{3 \sqrt{1+x^4}}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{3} x}{\sqrt{1+x^4}}\right)}{3 \sqrt[4]{3}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{3} x}{\sqrt{1+x^4}}\right)}{3 \sqrt[4]{3}}","-\frac{x}{3 \sqrt{1+x^4}}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{3} x}{\sqrt{1+x^4}}\right)}{3 \sqrt[4]{3}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{3} x}{\sqrt{1+x^4}}\right)}{3 \sqrt[4]{3}}",1,"-1/3*x/Sqrt[1 + x^4] - ArcTan[(3^(1/4)*x)/Sqrt[1 + x^4]]/(3*3^(1/4)) - ArcTanh[(3^(1/4)*x)/Sqrt[1 + x^4]]/(3*3^(1/4))","A",1
892,1,67,67,0.0848724,"\int \frac{-1+x^2}{\sqrt{1+x^2} \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[(-1 + x^2)/(Sqrt[1 + x^2]*Sqrt[x + Sqrt[1 + x^2]]),x]","\frac{4 \sqrt{1+x^2} \left(5 x+x^3\right)}{3 \left(x+\sqrt{1+x^2}\right)^{5/2}}+\frac{2 \left(23+55 x^2+10 x^4\right)}{15 \left(x+\sqrt{1+x^2}\right)^{5/2}}","\frac{4 \sqrt{1+x^2} \left(5 x+x^3\right)}{3 \left(x+\sqrt{1+x^2}\right)^{5/2}}+\frac{2 \left(23+55 x^2+10 x^4\right)}{15 \left(x+\sqrt{1+x^2}\right)^{5/2}}",1,"(4*Sqrt[1 + x^2]*(5*x + x^3))/(3*(x + Sqrt[1 + x^2])^(5/2)) + (2*(23 + 55*x^2 + 10*x^4))/(15*(x + Sqrt[1 + x^2])^(5/2))","A",1
893,1,67,67,0.0787503,"\int \frac{\sqrt{1+x^2}}{\sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[Sqrt[1 + x^2]/Sqrt[x + Sqrt[1 + x^2]],x]","\frac{4 \sqrt{1+x^2} \left(-x+x^3\right)}{3 \left(x+\sqrt{1+x^2}\right)^{5/2}}+\frac{2 \left(-7-5 x^2+10 x^4\right)}{15 \left(x+\sqrt{1+x^2}\right)^{5/2}}","\frac{4 \sqrt{1+x^2} \left(-x+x^3\right)}{3 \left(x+\sqrt{1+x^2}\right)^{5/2}}+\frac{2 \left(-7-5 x^2+10 x^4\right)}{15 \left(x+\sqrt{1+x^2}\right)^{5/2}}",1,"(4*Sqrt[1 + x^2]*(-x + x^3))/(3*(x + Sqrt[1 + x^2])^(5/2)) + (2*(-7 - 5*x^2 + 10*x^4))/(15*(x + Sqrt[1 + x^2])^(5/2))","A",1
894,1,68,68,0.3478619,"\int \frac{1}{\left(-b+a^3 x^2\right) \sqrt[3]{-b x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[1/((-b + a^3*x^2)*(-(b*x^2) + a^3*x^3)^(1/3)),x]","\frac{\text{RootSum}\left[a^6-a^3 b-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{2 b}","\frac{\text{RootSum}\left[a^6-a^3 b-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{2 b}",1,"RootSum[a^6 - a^3*b - 2*a^3*#1^3 + #1^6 & , (-Log[x] + Log[(-(b*x^2) + a^3*x^3)^(1/3) - x*#1])/#1 & ]/(2*b)","A",1
895,1,68,68,0.0020953,"\int \frac{1}{\left(-b+a^3 x^2\right) \sqrt[3]{-b x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[1/((-b + a^3*x^2)*(-(b*x^2) + a^3*x^3)^(1/3)),x]","\frac{\text{RootSum}\left[a^6-a^3 b-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{2 b}","\frac{\text{RootSum}\left[a^6-a^3 b-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{2 b}",1,"RootSum[a^6 - a^3*b - 2*a^3*#1^3 + #1^6 & , (-Log[x] + Log[(-(b*x^2) + a^3*x^3)^(1/3) - x*#1])/#1 & ]/(2*b)","A",1
896,1,68,68,0.3255318,"\int \frac{1}{\left(b+a x^2\right) \sqrt[3]{b^2 x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[1/((b + a*x^2)*(b^2*x^2 + a^3*x^3)^(1/3)),x]","-\frac{\text{RootSum}\left[a^6+a b^3-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{2 b}","-\frac{\text{RootSum}\left[a^6+a b^3-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{2 b}",1,"-1/2*RootSum[a^6 + a*b^3 - 2*a^3*#1^3 + #1^6 & , (-Log[x] + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1])/#1 & ]/b","A",1
897,1,68,68,0.0024909,"\int \frac{1}{\left(b+a x^2\right) \sqrt[3]{b^2 x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[1/((b + a*x^2)*(b^2*x^2 + a^3*x^3)^(1/3)),x]","-\frac{\text{RootSum}\left[a^6+a b^3-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{2 b}","-\frac{\text{RootSum}\left[a^6+a b^3-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{2 b}",1,"-1/2*RootSum[a^6 + a*b^3 - 2*a^3*#1^3 + #1^6 & , (-Log[x] + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1])/#1 & ]/b","A",1
898,1,68,68,0.2804455,"\int x^7 \sqrt[4]{-x+x^4} \, dx","IntegrateAlgebraic[x^7*(-x + x^4)^(1/4),x]","\frac{1}{288} \sqrt[4]{-x+x^4} \left(-7 x^2-4 x^5+32 x^8\right)+\frac{7}{192} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x+x^4}}\right)-\frac{7}{192} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x+x^4}}\right)","\frac{1}{288} \sqrt[4]{-x+x^4} \left(-7 x^2-4 x^5+32 x^8\right)+\frac{7}{192} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x+x^4}}\right)-\frac{7}{192} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x+x^4}}\right)",1,"((-x + x^4)^(1/4)*(-7*x^2 - 4*x^5 + 32*x^8))/288 + (7*ArcTan[x/(-x + x^4)^(1/4)])/192 - (7*ArcTanh[x/(-x + x^4)^(1/4)])/192","A",1
899,1,68,68,0.2796091,"\int \frac{\left(2+x^2\right) \sqrt[4]{-1-x^2+x^4} \left(1+x^2+x^4\right)}{x^6 \left(1+x^2\right)} \, dx","IntegrateAlgebraic[((2 + x^2)*(-1 - x^2 + x^4)^(1/4)*(1 + x^2 + x^4))/(x^6*(1 + x^2)),x]","-\frac{2 \sqrt[4]{-1-x^2+x^4} \left(1+x^2+4 x^4\right)}{5 x^5}-\tan ^{-1}\left(\frac{x}{\sqrt[4]{-1-x^2+x^4}}\right)+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{-1-x^2+x^4}}\right)","-\frac{2 \sqrt[4]{-1-x^2+x^4} \left(1+x^2+4 x^4\right)}{5 x^5}-\tan ^{-1}\left(\frac{x}{\sqrt[4]{-1-x^2+x^4}}\right)+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{-1-x^2+x^4}}\right)",1,"(-2*(-1 - x^2 + x^4)^(1/4)*(1 + x^2 + 4*x^4))/(5*x^5) - ArcTan[x/(-1 - x^2 + x^4)^(1/4)] + ArcTanh[x/(-1 - x^2 + x^4)^(1/4)]","A",1
900,1,68,68,0.3615894,"\int \frac{\sqrt[4]{x^3+x^4}}{x^2 \left(-1+x^2\right)} \, dx","IntegrateAlgebraic[(x^3 + x^4)^(1/4)/(x^2*(-1 + x^2)),x]","\frac{4 \sqrt[4]{x^3+x^4}}{x}+\sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^4}}\right)-\sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^4}}\right)","\frac{4 \sqrt[4]{x^3+x^4}}{x}+\sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^4}}\right)-\sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^4}}\right)",1,"(4*(x^3 + x^4)^(1/4))/x + 2^(1/4)*ArcTan[(2^(1/4)*x)/(x^3 + x^4)^(1/4)] - 2^(1/4)*ArcTanh[(2^(1/4)*x)/(x^3 + x^4)^(1/4)]","A",1
901,1,68,68,0.3573866,"\int \frac{b+2 a x^4}{x^2 \left(b+a x^4\right)^{3/4}} \, dx","IntegrateAlgebraic[(b + 2*a*x^4)/(x^2*(b + a*x^4)^(3/4)),x]","-\frac{\sqrt[4]{b+a x^4}}{x}-\sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)+\sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)","-\frac{\sqrt[4]{b+a x^4}}{x}-\sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)+\sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)",1,"-((b + a*x^4)^(1/4)/x) - a^(1/4)*ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)] + a^(1/4)*ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)]","A",1
902,1,68,68,0.4153463,"\int \frac{\left(1+x^2\right) \sqrt[4]{x^2+x^6}}{x^2 \left(-1+x^2\right)} \, dx","IntegrateAlgebraic[((1 + x^2)*(x^2 + x^6)^(1/4))/(x^2*(-1 + x^2)),x]","\frac{2 \sqrt[4]{x^2+x^6}}{x}+\sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)-\sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)","\frac{2 \sqrt[4]{x^2+x^6}}{x}+\sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)-\sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)",1,"(2*(x^2 + x^6)^(1/4))/x + 2^(1/4)*ArcTan[(2^(1/4)*x)/(x^2 + x^6)^(1/4)] - 2^(1/4)*ArcTanh[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]","A",1
903,1,68,68,0.094139,"\int \frac{\sqrt{-1+x^3} \left(1-x^3+x^6\right)}{x^{10} \left(2+x^3\right)} \, dx","IntegrateAlgebraic[(Sqrt[-1 + x^3]*(1 - x^3 + x^6))/(x^10*(2 + x^3)),x]","\frac{\sqrt{-1+x^3} \left(-2+5 x^3-12 x^6\right)}{36 x^9}+\frac{13}{24} \tan ^{-1}\left(\sqrt{-1+x^3}\right)-\frac{7 \tan ^{-1}\left(\frac{\sqrt{-1+x^3}}{\sqrt{3}}\right)}{8 \sqrt{3}}","\frac{\sqrt{-1+x^3} \left(-2+5 x^3-12 x^6\right)}{36 x^9}+\frac{13}{24} \tan ^{-1}\left(\sqrt{-1+x^3}\right)-\frac{7 \tan ^{-1}\left(\frac{\sqrt{-1+x^3}}{\sqrt{3}}\right)}{8 \sqrt{3}}",1,"(Sqrt[-1 + x^3]*(-2 + 5*x^3 - 12*x^6))/(36*x^9) + (13*ArcTan[Sqrt[-1 + x^3]])/24 - (7*ArcTan[Sqrt[-1 + x^3]/Sqrt[3]])/(8*Sqrt[3])","A",1
904,1,68,68,4.0351191,"\int \frac{\sqrt{x^2+x \sqrt{x+x^2}}}{x \sqrt{x+x^2}} \, dx","IntegrateAlgebraic[Sqrt[x^2 + x*Sqrt[x + x^2]]/(x*Sqrt[x + x^2]),x]","\frac{2 \sqrt{2} \sqrt{-x+\sqrt{x+x^2}} \sqrt{x \left(x+\sqrt{x+x^2}\right)} \tanh ^{-1}\left(\sqrt{2} \sqrt{-x+\sqrt{x+x^2}}\right)}{x}","\frac{2 \sqrt{2} \sqrt{-x+\sqrt{x+x^2}} \sqrt{x \left(x+\sqrt{x+x^2}\right)} \tanh ^{-1}\left(\sqrt{2} \sqrt{-x+\sqrt{x+x^2}}\right)}{x}",1,"(2*Sqrt[2]*Sqrt[-x + Sqrt[x + x^2]]*Sqrt[x*(x + Sqrt[x + x^2])]*ArcTanh[Sqrt[2]*Sqrt[-x + Sqrt[x + x^2]]])/x","A",1
905,1,69,69,0.3228816,"\int \frac{1}{\left(-b+a x^2\right) \sqrt[3]{b^2 x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[1/((-b + a*x^2)*(b^2*x^2 + a^3*x^3)^(1/3)),x]","\frac{\text{RootSum}\left[a^6-a b^3-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{2 b}","\frac{\text{RootSum}\left[a^6-a b^3-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{2 b}",1,"RootSum[a^6 - a*b^3 - 2*a^3*#1^3 + #1^6 & , (-Log[x] + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1])/#1 & ]/(2*b)","A",1
906,1,69,69,0.0038175,"\int \frac{1}{\left(-b+a x^2\right) \sqrt[3]{b^2 x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[1/((-b + a*x^2)*(b^2*x^2 + a^3*x^3)^(1/3)),x]","\frac{\text{RootSum}\left[a^6-a b^3-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{2 b}","\frac{\text{RootSum}\left[a^6-a b^3-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{2 b}",1,"RootSum[a^6 - a*b^3 - 2*a^3*#1^3 + #1^6 & , (-Log[x] + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1])/#1 & ]/(2*b)","A",1
907,-1,0,69,180.0021638,"\int \frac{(-a+x) (-b+x) \left(3 a b-2 (a+b) x+x^2\right)}{x^2 \sqrt{x (-a+x) (-b+x)} \left(-a b+(a+b) x-x^2+d x^3\right)} \, dx","IntegrateAlgebraic[((-a + x)*(-b + x)*(3*a*b - 2*(a + b)*x + x^2))/(x^2*Sqrt[x*(-a + x)*(-b + x)]*(-(a*b) + (a + b)*x - x^2 + d*x^3)),x]","\text{\$Aborted}","\frac{2 \sqrt{a b x+(-a-b) x^2+x^3}}{x^2}-2 \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{a b x+(-a-b) x^2+x^3}}{\sqrt{d} x^2}\right)",1,"$Aborted","F",-1
908,1,69,69,0.3720407,"\int \frac{1}{\left(1+x^3\right) \sqrt[4]{-x+x^4}} \, dx","IntegrateAlgebraic[1/((1 + x^3)*(-x + x^4)^(1/4)),x]","\frac{1}{3} 2^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \left(-x+x^4\right)^{3/4}}{-1+x^3}\right)+\frac{1}{3} 2^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \left(-x+x^4\right)^{3/4}}{-1+x^3}\right)","\frac{1}{3} 2^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \left(-x+x^4\right)^{3/4}}{-1+x^3}\right)+\frac{1}{3} 2^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \left(-x+x^4\right)^{3/4}}{-1+x^3}\right)",1,"(2^(3/4)*ArcTan[(2^(1/4)*(-x + x^4)^(3/4))/(-1 + x^3)])/3 + (2^(3/4)*ArcTanh[(2^(1/4)*(-x + x^4)^(3/4))/(-1 + x^3)])/3","A",1
909,1,80,69,0.3070731,"\int \frac{x \sqrt{-x^2+x^4}}{-3+2 x^2} \, dx","IntegrateAlgebraic[(x*Sqrt[-x^2 + x^4])/(-3 + 2*x^2),x]","\frac{1}{4} \sqrt{-x^2+x^4}+\frac{1}{2} \tanh ^{-1}\left(\frac{\sqrt{-x^2+x^4}}{-1+x^2}\right)-\frac{1}{4} \sqrt{3} \tanh ^{-1}\left(\frac{\sqrt{-x^2+x^4}}{\sqrt{3} \left(-1+x^2\right)}\right)","\frac{1}{4} \sqrt{-x^2+x^4}+\tanh ^{-1}\left(\frac{(-1+x) x}{\sqrt{-x^2+x^4}}\right)-\frac{1}{4} \sqrt{3} \tanh ^{-1}\left(\frac{\sqrt{3} \sqrt{-x^2+x^4}}{x^2}\right)",1,"Sqrt[-x^2 + x^4]/4 + ArcTanh[Sqrt[-x^2 + x^4]/(-1 + x^2)]/2 - (Sqrt[3]*ArcTanh[Sqrt[-x^2 + x^4]/(Sqrt[3]*(-1 + x^2))])/4","A",1
910,1,69,69,0.8867678,"\int \frac{\left(-1+x^4\right) \left(1+x^4\right) \sqrt{-1-x^2+x^4}}{\left(-2-x^2+2 x^4\right)^2 \left(-2+x^2+2 x^4\right)} \, dx","IntegrateAlgebraic[((-1 + x^4)*(1 + x^4)*Sqrt[-1 - x^2 + x^4])/((-2 - x^2 + 2*x^4)^2*(-2 + x^2 + 2*x^4)),x]","-\frac{x \sqrt{-1-x^2+x^4}}{16 \left(-2-x^2+2 x^4\right)}-\frac{1}{16} \sqrt{\frac{3}{2}} \tan ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt{-1-x^2+x^4}}\right)","-\frac{x \sqrt{-1-x^2+x^4}}{16 \left(-2-x^2+2 x^4\right)}-\frac{1}{16} \sqrt{\frac{3}{2}} \tan ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt{-1-x^2+x^4}}\right)",1,"-1/16*(x*Sqrt[-1 - x^2 + x^4])/(-2 - x^2 + 2*x^4) - (Sqrt[3/2]*ArcTan[(Sqrt[3/2]*x)/Sqrt[-1 - x^2 + x^4]])/16","A",1
911,1,69,69,0.5889502,"\int \frac{b+a x^2}{\left(b+a x^2+x^4\right) \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[(b + a*x^2)/((b + a*x^2 + x^4)*(b*x^2 + a*x^4)^(1/4)),x]","-\frac{1}{2} \text{RootSum}\left[b-a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}^3+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^3}{-a+2 \text{$\#$1}^4}\&\right]","-\frac{1}{2} \text{RootSum}\left[b-a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}^3+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^3}{-a+2 \text{$\#$1}^4}\&\right]",1,"-1/2*RootSum[b - a*#1^4 + #1^8 & , (-(Log[x]*#1^3) + Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1^3)/(-a + 2*#1^4) & ]","A",1
912,1,69,69,0.0021273,"\int \frac{b+a x^2}{\left(b+a x^2+x^4\right) \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[(b + a*x^2)/((b + a*x^2 + x^4)*(b*x^2 + a*x^4)^(1/4)),x]","-\frac{1}{2} \text{RootSum}\left[b-a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}^3+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^3}{-a+2 \text{$\#$1}^4}\&\right]","-\frac{1}{2} \text{RootSum}\left[b-a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}^3+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^3}{-a+2 \text{$\#$1}^4}\&\right]",1,"-1/2*RootSum[b - a*#1^4 + #1^8 & , (-(Log[x]*#1^3) + Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1^3)/(-a + 2*#1^4) & ]","A",1
913,1,69,69,0.450586,"\int \frac{\left(-b+a x^2\right) \sqrt{b^2+a^2 x^4}}{x^2 \left(b+a x^2\right)} \, dx","IntegrateAlgebraic[((-b + a*x^2)*Sqrt[b^2 + a^2*x^4])/(x^2*(b + a*x^2)),x]","\frac{\sqrt{b^2+a^2 x^4}}{x}+\sqrt{2} \sqrt{a} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{b} x}{\sqrt{b^2+a^2 x^4}}\right)","\frac{\sqrt{b^2+a^2 x^4}}{x}+\sqrt{2} \sqrt{a} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{b} x}{\sqrt{b^2+a^2 x^4}}\right)",1,"Sqrt[b^2 + a^2*x^4]/x + Sqrt[2]*Sqrt[a]*Sqrt[b]*ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[b]*x)/Sqrt[b^2 + a^2*x^4]]","A",1
914,1,69,69,0.9241816,"\int \frac{4 b+a x^3}{\left(b+a x^3\right) \sqrt[4]{-b-a x^3+c x^4}} \, dx","IntegrateAlgebraic[(4*b + a*x^3)/((b + a*x^3)*(-b - a*x^3 + c*x^4)^(1/4)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{-b-a x^3+c x^4}}\right)}{\sqrt[4]{c}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{-b-a x^3+c x^4}}\right)}{\sqrt[4]{c}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{-b-a x^3+c x^4}}\right)}{\sqrt[4]{c}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{-b-a x^3+c x^4}}\right)}{\sqrt[4]{c}}",1,"(2*ArcTan[(c^(1/4)*x)/(-b - a*x^3 + c*x^4)^(1/4)])/c^(1/4) + (2*ArcTanh[(c^(1/4)*x)/(-b - a*x^3 + c*x^4)^(1/4)])/c^(1/4)","A",1
915,1,69,69,2.6877047,"\int \frac{\left(-3+x^4\right) \left(1+x^4\right) \left(1+x^3+x^4\right)}{x^6 \left(1-x^3+x^4\right) \sqrt[4]{x+x^5}} \, dx","IntegrateAlgebraic[((-3 + x^4)*(1 + x^4)*(1 + x^3 + x^4))/(x^6*(1 - x^3 + x^4)*(x + x^5)^(1/4)),x]","\frac{4 \left(3+14 x^3+3 x^4\right) \left(x+x^5\right)^{3/4}}{21 x^6}-4 \tan ^{-1}\left(\frac{\left(x+x^5\right)^{3/4}}{1+x^4}\right)-4 \tanh ^{-1}\left(\frac{\left(x+x^5\right)^{3/4}}{1+x^4}\right)","\frac{4 \left(3+14 x^3+3 x^4\right) \left(x+x^5\right)^{3/4}}{21 x^6}-4 \tan ^{-1}\left(\frac{\left(x+x^5\right)^{3/4}}{1+x^4}\right)-4 \tanh ^{-1}\left(\frac{\left(x+x^5\right)^{3/4}}{1+x^4}\right)",1,"(4*(3 + 14*x^3 + 3*x^4)*(x + x^5)^(3/4))/(21*x^6) - 4*ArcTan[(x + x^5)^(3/4)/(1 + x^4)] - 4*ArcTanh[(x + x^5)^(3/4)/(1 + x^4)]","A",1
916,1,69,69,2.7070256,"\int \frac{\left(-3+x^4\right) \left(1+2 x^4+x^6+x^8\right)}{x^6 \left(1-x^3+x^4\right) \sqrt[4]{x+x^5}} \, dx","IntegrateAlgebraic[((-3 + x^4)*(1 + 2*x^4 + x^6 + x^8))/(x^6*(1 - x^3 + x^4)*(x + x^5)^(1/4)),x]","\frac{4 \left(3+7 x^3+3 x^4\right) \left(x+x^5\right)^{3/4}}{21 x^6}-4 \tan ^{-1}\left(\frac{\left(x+x^5\right)^{3/4}}{1+x^4}\right)-4 \tanh ^{-1}\left(\frac{\left(x+x^5\right)^{3/4}}{1+x^4}\right)","\frac{4 \left(3+7 x^3+3 x^4\right) \left(x+x^5\right)^{3/4}}{21 x^6}-4 \tan ^{-1}\left(\frac{\left(x+x^5\right)^{3/4}}{1+x^4}\right)-4 \tanh ^{-1}\left(\frac{\left(x+x^5\right)^{3/4}}{1+x^4}\right)",1,"(4*(3 + 7*x^3 + 3*x^4)*(x + x^5)^(3/4))/(21*x^6) - 4*ArcTan[(x + x^5)^(3/4)/(1 + x^4)] - 4*ArcTanh[(x + x^5)^(3/4)/(1 + x^4)]","A",1
917,1,70,70,0.0840281,"\int \frac{\left(b+a x^3\right)^{3/4}}{x} \, dx","IntegrateAlgebraic[(b + a*x^3)^(3/4)/x,x]","\frac{4}{9} \left(b+a x^3\right)^{3/4}+\frac{2}{3} b^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^3}}{\sqrt[4]{b}}\right)-\frac{2}{3} b^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^3}}{\sqrt[4]{b}}\right)","\frac{4}{9} \left(b+a x^3\right)^{3/4}+\frac{2}{3} b^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^3}}{\sqrt[4]{b}}\right)-\frac{2}{3} b^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^3}}{\sqrt[4]{b}}\right)",1,"(4*(b + a*x^3)^(3/4))/9 + (2*b^(3/4)*ArcTan[(b + a*x^3)^(1/4)/b^(1/4)])/3 - (2*b^(3/4)*ArcTanh[(b + a*x^3)^(1/4)/b^(1/4)])/3","A",1
918,1,70,70,5.1490712,"\int \frac{1+3 x+3 x^4}{x \sqrt[4]{1+x^4}} \, dx","IntegrateAlgebraic[(1 + 3*x + 3*x^4)/(x*(1 + x^4)^(1/4)),x]","\left(1+x^4\right)^{3/4}+\frac{3}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)+\frac{1}{2} \tan ^{-1}\left(\sqrt[4]{1+x^4}\right)+\frac{3}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\frac{1}{2} \tanh ^{-1}\left(\sqrt[4]{1+x^4}\right)","\left(1+x^4\right)^{3/4}+\frac{3}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)+\frac{1}{2} \tan ^{-1}\left(\sqrt[4]{1+x^4}\right)+\frac{3}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\frac{1}{2} \tanh ^{-1}\left(\sqrt[4]{1+x^4}\right)",1,"(1 + x^4)^(3/4) + (3*ArcTan[x/(1 + x^4)^(1/4)])/2 + ArcTan[(1 + x^4)^(1/4)]/2 + (3*ArcTanh[x/(1 + x^4)^(1/4)])/2 - ArcTanh[(1 + x^4)^(1/4)]/2","A",1
919,1,70,70,0.0699254,"\int \frac{\left(b+a x^4\right)^{3/4}}{x} \, dx","IntegrateAlgebraic[(b + a*x^4)^(3/4)/x,x]","\frac{1}{3} \left(b+a x^4\right)^{3/4}+\frac{1}{2} b^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^4}}{\sqrt[4]{b}}\right)-\frac{1}{2} b^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^4}}{\sqrt[4]{b}}\right)","\frac{1}{3} \left(b+a x^4\right)^{3/4}+\frac{1}{2} b^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^4}}{\sqrt[4]{b}}\right)-\frac{1}{2} b^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^4}}{\sqrt[4]{b}}\right)",1,"(b + a*x^4)^(3/4)/3 + (b^(3/4)*ArcTan[(b + a*x^4)^(1/4)/b^(1/4)])/2 - (b^(3/4)*ArcTanh[(b + a*x^4)^(1/4)/b^(1/4)])/2","A",1
920,1,70,70,0.5903803,"\int \frac{-1+x^2}{\left(1+x^2\right) \sqrt{a+b x+c x^2+b x^3+a x^4}} \, dx","IntegrateAlgebraic[(-1 + x^2)/((1 + x^2)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt{2 a-c} x}{\sqrt{a}+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{\sqrt{2 a-c}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{2 a-c} x}{\sqrt{a}+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{\sqrt{2 a-c}}",1,"(2*ArcTan[(Sqrt[2*a - c]*x)/(Sqrt[a] + Sqrt[a]*x^2 - Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])])/Sqrt[2*a - c]","A",1
921,1,70,70,0.457117,"\int \frac{\left(b+a x^2\right) \sqrt{b^2+a^2 x^4}}{x^2 \left(-b+a x^2\right)} \, dx","IntegrateAlgebraic[((b + a*x^2)*Sqrt[b^2 + a^2*x^4])/(x^2*(-b + a*x^2)),x]","\frac{\sqrt{b^2+a^2 x^4}}{x}-\sqrt{2} \sqrt{a} \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{b} x}{\sqrt{b^2+a^2 x^4}}\right)","\frac{\sqrt{b^2+a^2 x^4}}{x}-\sqrt{2} \sqrt{a} \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{b} x}{\sqrt{b^2+a^2 x^4}}\right)",1,"Sqrt[b^2 + a^2*x^4]/x - Sqrt[2]*Sqrt[a]*Sqrt[b]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[b]*x)/Sqrt[b^2 + a^2*x^4]]","A",1
922,1,70,70,0.0862299,"\int \frac{\left(b+a x^5\right)^{3/4}}{x} \, dx","IntegrateAlgebraic[(b + a*x^5)^(3/4)/x,x]","\frac{4}{15} \left(b+a x^5\right)^{3/4}+\frac{2}{5} b^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^5}}{\sqrt[4]{b}}\right)-\frac{2}{5} b^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^5}}{\sqrt[4]{b}}\right)","\frac{4}{15} \left(b+a x^5\right)^{3/4}+\frac{2}{5} b^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^5}}{\sqrt[4]{b}}\right)-\frac{2}{5} b^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^5}}{\sqrt[4]{b}}\right)",1,"(4*(b + a*x^5)^(3/4))/15 + (2*b^(3/4)*ArcTan[(b + a*x^5)^(1/4)/b^(1/4)])/5 - (2*b^(3/4)*ArcTanh[(b + a*x^5)^(1/4)/b^(1/4)])/5","A",1
923,1,70,70,0.288753,"\int \frac{\left(3+2 x^5\right) \sqrt{x-2 x^4-x^6}}{\left(-1+x^5\right)^2} \, dx","IntegrateAlgebraic[((3 + 2*x^5)*Sqrt[x - 2*x^4 - x^6])/(-1 + x^5)^2,x]","-\frac{x \sqrt{x-2 x^4-x^6}}{-1+x^5}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x \sqrt{x-2 x^4-x^6}}{-1+2 x^3+x^5}\right)}{\sqrt{2}}","-\frac{x \sqrt{x-2 x^4-x^6}}{-1+x^5}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x \sqrt{x-2 x^4-x^6}}{-1+2 x^3+x^5}\right)}{\sqrt{2}}",1,"-((x*Sqrt[x - 2*x^4 - x^6])/(-1 + x^5)) - ArcTan[(Sqrt[2]*x*Sqrt[x - 2*x^4 - x^6])/(-1 + 2*x^3 + x^5)]/Sqrt[2]","A",1
924,1,70,70,2.7003991,"\int \frac{\left(-2+x^6\right) \left(1+x^6\right) \sqrt[4]{1-x^4+x^6}}{x^6 \left(1-2 x^4+x^6\right)} \, dx","IntegrateAlgebraic[((-2 + x^6)*(1 + x^6)*(1 - x^4 + x^6)^(1/4))/(x^6*(1 - 2*x^4 + x^6)),x]","\frac{2 \sqrt[4]{1-x^4+x^6} \left(1+9 x^4+x^6\right)}{5 x^5}+2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{1-x^4+x^6}}\right)-2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{1-x^4+x^6}}\right)","\frac{2 \sqrt[4]{1-x^4+x^6} \left(1+9 x^4+x^6\right)}{5 x^5}+2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{1-x^4+x^6}}\right)-2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{1-x^4+x^6}}\right)",1,"(2*(1 - x^4 + x^6)^(1/4)*(1 + 9*x^4 + x^6))/(5*x^5) + 2*ArcTan[x/(1 - x^4 + x^6)^(1/4)] - 2*ArcTanh[x/(1 - x^4 + x^6)^(1/4)]","A",1
925,1,70,70,0.0777,"\int \frac{\left(b+a x^6\right)^{3/4}}{x} \, dx","IntegrateAlgebraic[(b + a*x^6)^(3/4)/x,x]","\frac{2}{9} \left(b+a x^6\right)^{3/4}+\frac{1}{3} b^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^6}}{\sqrt[4]{b}}\right)-\frac{1}{3} b^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^6}}{\sqrt[4]{b}}\right)","\frac{2}{9} \left(b+a x^6\right)^{3/4}+\frac{1}{3} b^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^6}}{\sqrt[4]{b}}\right)-\frac{1}{3} b^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^6}}{\sqrt[4]{b}}\right)",1,"(2*(b + a*x^6)^(3/4))/9 + (b^(3/4)*ArcTan[(b + a*x^6)^(1/4)/b^(1/4)])/3 - (b^(3/4)*ArcTanh[(b + a*x^6)^(1/4)/b^(1/4)])/3","A",1
926,1,60,70,0.1544425,"\int \frac{\sqrt{1+x}}{\sqrt{x+\sqrt{1+x}}} \, dx","IntegrateAlgebraic[Sqrt[1 + x]/Sqrt[x + Sqrt[1 + x]],x]","\frac{1}{2} \sqrt{x+\sqrt{1+x}} \left(-3+2 \sqrt{1+x}\right)-\frac{7}{4} \log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}\right)","-\frac{3}{2} \sqrt{x+\sqrt{1+x}}+\sqrt{1+x} \sqrt{x+\sqrt{1+x}}-\frac{7}{4} \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}\right)",1,"(Sqrt[x + Sqrt[1 + x]]*(-3 + 2*Sqrt[1 + x]))/2 - (7*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]])/4","A",1
927,1,60,70,0.1295169,"\int \frac{\sqrt{x+\sqrt{1+x}}}{\sqrt{1+x}} \, dx","IntegrateAlgebraic[Sqrt[x + Sqrt[1 + x]]/Sqrt[1 + x],x]","\frac{1}{2} \sqrt{x+\sqrt{1+x}} \left(1+2 \sqrt{1+x}\right)+\frac{5}{4} \log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}\right)","\frac{1}{2} \sqrt{x+\sqrt{1+x}}+\sqrt{1+x} \sqrt{x+\sqrt{1+x}}+\frac{5}{4} \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}\right)",1,"(Sqrt[x + Sqrt[1 + x]]*(1 + 2*Sqrt[1 + x]))/2 + (5*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]])/4","A",1
928,1,70,70,0.1318451,"\int \frac{\left(-1+x^2\right) \sqrt{1+\sqrt{1+x^2}}}{1+x^2} \, dx","IntegrateAlgebraic[((-1 + x^2)*Sqrt[1 + Sqrt[1 + x^2]])/(1 + x^2),x]","\frac{4 x}{3 \sqrt{1+\sqrt{1+x^2}}}+\frac{2 x \sqrt{1+x^2}}{3 \sqrt{1+\sqrt{1+x^2}}}-4 \tan ^{-1}\left(\frac{x}{\sqrt{1+\sqrt{1+x^2}}}\right)","\frac{4 x}{3 \sqrt{1+\sqrt{1+x^2}}}+\frac{2 x \sqrt{1+x^2}}{3 \sqrt{1+\sqrt{1+x^2}}}-4 \tan ^{-1}\left(\frac{x}{\sqrt{1+\sqrt{1+x^2}}}\right)",1,"(4*x)/(3*Sqrt[1 + Sqrt[1 + x^2]]) + (2*x*Sqrt[1 + x^2])/(3*Sqrt[1 + Sqrt[1 + x^2]]) - 4*ArcTan[x/Sqrt[1 + Sqrt[1 + x^2]]]","A",1
929,1,70,70,0.3783426,"\int \frac{1}{\sqrt{x^2+\sqrt{1+x^4}}} \, dx","IntegrateAlgebraic[1/Sqrt[x^2 + Sqrt[1 + x^4]],x]","\frac{x}{2 \sqrt{x^2+\sqrt{1+x^4}}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)}{\sqrt{2}}","\frac{x}{2 \sqrt{x^2+\sqrt{1+x^4}}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)}{\sqrt{2}}",1,"x/(2*Sqrt[x^2 + Sqrt[1 + x^4]]) + ArcTanh[(Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2 + Sqrt[1 + x^4])]/Sqrt[2]","A",1
930,1,70,70,0.3725277,"\int \frac{\sqrt{x^2+\sqrt{1+x^4}}}{x^2} \, dx","IntegrateAlgebraic[Sqrt[x^2 + Sqrt[1 + x^4]]/x^2,x]","-\frac{\sqrt{x^2+\sqrt{1+x^4}}}{x}+\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)","-\frac{\sqrt{x^2+\sqrt{1+x^4}}}{x}+\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)",1,"-(Sqrt[x^2 + Sqrt[1 + x^4]]/x) + Sqrt[2]*ArcTanh[(Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2 + Sqrt[1 + x^4])]","A",1
931,1,71,71,0.3816764,"\int \frac{1}{\left(\sqrt{-1+x}+2 \sqrt{x}\right)^2 \sqrt{-1+x}} \, dx","IntegrateAlgebraic[1/((Sqrt[-1 + x] + 2*Sqrt[x])^2*Sqrt[-1 + x]),x]","-\frac{2 \sqrt{-1+x}}{3 (1+3 x)}+\frac{4 \sqrt{x}}{3 (1+3 x)}+\frac{8 \tan ^{-1}\left(\frac{\sqrt{-1+x}}{\sqrt{3}}-\frac{\sqrt{x}}{\sqrt{3}}\right)}{3 \sqrt{3}}","-\frac{2 \sqrt{-1+x}}{3 (1+3 x)}+\frac{4 \sqrt{x}}{3 (1+3 x)}+\frac{8 \tan ^{-1}\left(\frac{\sqrt{-1+x}}{\sqrt{3}}-\frac{\sqrt{x}}{\sqrt{3}}\right)}{3 \sqrt{3}}",1,"(-2*Sqrt[-1 + x])/(3*(1 + 3*x)) + (4*Sqrt[x])/(3*(1 + 3*x)) + (8*ArcTan[Sqrt[-1 + x]/Sqrt[3] - Sqrt[x]/Sqrt[3]])/(3*Sqrt[3])","A",1
932,1,71,71,0.1365489,"\int \frac{1}{\sqrt{-1+x^2} \left(-4+3 x^2\right)^2} \, dx","IntegrateAlgebraic[1/(Sqrt[-1 + x^2]*(-4 + 3*x^2)^2),x]","-\frac{3 x \sqrt{-1+x^2}}{8 \left(-4+3 x^2\right)}+\frac{5}{32} \log \left(2-x^2+x \sqrt{-1+x^2}\right)-\frac{5}{32} \log \left(2-3 x^2+3 x \sqrt{-1+x^2}\right)","-\frac{3 x \sqrt{-1+x^2}}{8 \left(-4+3 x^2\right)}+\frac{5}{32} \log \left(2-x^2+x \sqrt{-1+x^2}\right)-\frac{5}{32} \log \left(2-3 x^2+3 x \sqrt{-1+x^2}\right)",1,"(-3*x*Sqrt[-1 + x^2])/(8*(-4 + 3*x^2)) + (5*Log[2 - x^2 + x*Sqrt[-1 + x^2]])/32 - (5*Log[2 - 3*x^2 + 3*x*Sqrt[-1 + x^2]])/32","A",1
933,1,71,71,0.0679208,"\int \frac{1}{x \sqrt[3]{1+x^3}} \, dx","IntegrateAlgebraic[1/(x*(1 + x^3)^(1/3)),x]","\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^3}}{\sqrt{3}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(-1+\sqrt[3]{1+x^3}\right)-\frac{1}{6} \log \left(1+\sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)","\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^3}}{\sqrt{3}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(-1+\sqrt[3]{1+x^3}\right)-\frac{1}{6} \log \left(1+\sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)",1,"ArcTan[1/Sqrt[3] + (2*(1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3] + Log[-1 + (1 + x^3)^(1/3)]/3 - Log[1 + (1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/6","A",1
934,1,71,71,3.3253648,"\int \frac{x^2 \left(3-2 (1+k) x+k x^2\right)}{((1-x) x (1-k x))^{3/4} \left(-1+(1+k) x-k x^2+d x^3\right)} \, dx","IntegrateAlgebraic[(x^2*(3 - 2*(1 + k)*x + k*x^2))/(((1 - x)*x*(1 - k*x))^(3/4)*(-1 + (1 + k)*x - k*x^2 + d*x^3)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{x+(-1-k) x^2+k x^3}}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{x+(-1-k) x^2+k x^3}}\right)}{d^{3/4}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{x+(-1-k) x^2+k x^3}}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{x+(-1-k) x^2+k x^3}}\right)}{d^{3/4}}",1,"(2*ArcTan[(d^(1/4)*x)/(x + (-1 - k)*x^2 + k*x^3)^(1/4)])/d^(3/4) - (2*ArcTanh[(d^(1/4)*x)/(x + (-1 - k)*x^2 + k*x^3)^(1/4)])/d^(3/4)","A",1
935,1,71,71,0.638561,"\int \frac{-b+a x^2}{\left(-b+a x^2+x^4\right) \sqrt[4]{-b x^2+a x^4}} \, dx","IntegrateAlgebraic[(-b + a*x^2)/((-b + a*x^2 + x^4)*(-(b*x^2) + a*x^4)^(1/4)),x]","-\frac{1}{2} \text{RootSum}\left[b+a \text{$\#$1}^4-\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}^3+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^3}{-a+2 \text{$\#$1}^4}\&\right]","-\frac{1}{2} \text{RootSum}\left[b+a \text{$\#$1}^4-\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}^3+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^3}{-a+2 \text{$\#$1}^4}\&\right]",1,"-1/2*RootSum[b + a*#1^4 - #1^8 & , (-(Log[x]*#1^3) + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^3)/(-a + 2*#1^4) & ]","A",1
936,1,71,71,0.003261,"\int \frac{-b+a x^2}{\left(-b+a x^2+x^4\right) \sqrt[4]{-b x^2+a x^4}} \, dx","IntegrateAlgebraic[(-b + a*x^2)/((-b + a*x^2 + x^4)*(-(b*x^2) + a*x^4)^(1/4)),x]","-\frac{1}{2} \text{RootSum}\left[b+a \text{$\#$1}^4-\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}^3+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^3}{-a+2 \text{$\#$1}^4}\&\right]","-\frac{1}{2} \text{RootSum}\left[b+a \text{$\#$1}^4-\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}^3+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^3}{-a+2 \text{$\#$1}^4}\&\right]",1,"-1/2*RootSum[b + a*#1^4 - #1^8 & , (-(Log[x]*#1^3) + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^3)/(-a + 2*#1^4) & ]","A",1
937,1,71,71,0.3605844,"\int \frac{\sqrt[3]{-x+x^3}}{b+a x^6} \, dx","IntegrateAlgebraic[(-x + x^3)^(1/3)/(b + a*x^6),x]","\frac{\text{RootSum}\left[a+b-3 b \text{$\#$1}^3+3 b \text{$\#$1}^6-b \text{$\#$1}^9\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{-x+x^3}-x \text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^3}\&\right]}{6 b}","\frac{\text{RootSum}\left[a+b-3 b \text{$\#$1}^3+3 b \text{$\#$1}^6-b \text{$\#$1}^9\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{-x+x^3}-x \text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^3}\&\right]}{6 b}",1,"RootSum[a + b - 3*b*#1^3 + 3*b*#1^6 - b*#1^9 & , (-(Log[x]*#1) + Log[(-x + x^3)^(1/3) - x*#1]*#1)/(-1 + #1^3) & ]/(6*b)","A",1
938,1,71,71,0.0015964,"\int \frac{\sqrt[3]{-x+x^3}}{b+a x^6} \, dx","IntegrateAlgebraic[(-x + x^3)^(1/3)/(b + a*x^6),x]","\frac{\text{RootSum}\left[a+b-3 b \text{$\#$1}^3+3 b \text{$\#$1}^6-b \text{$\#$1}^9\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{-x+x^3}-x \text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^3}\&\right]}{6 b}","\frac{\text{RootSum}\left[a+b-3 b \text{$\#$1}^3+3 b \text{$\#$1}^6-b \text{$\#$1}^9\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{-x+x^3}-x \text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^3}\&\right]}{6 b}",1,"RootSum[a + b - 3*b*#1^3 + 3*b*#1^6 - b*#1^9 & , (-(Log[x]*#1) + Log[(-x + x^3)^(1/3) - x*#1]*#1)/(-1 + #1^3) & ]/(6*b)","A",1
939,1,71,71,3.4748668,"\int \frac{\left(-2 b+a x^6\right) \left(b+a x^6\right)^{3/4}}{x^4 \left(b-c x^4+a x^6\right)} \, dx","IntegrateAlgebraic[((-2*b + a*x^6)*(b + a*x^6)^(3/4))/(x^4*(b - c*x^4 + a*x^6)),x]","\frac{2 \left(b+a x^6\right)^{3/4}}{3 x^3}-c^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{b+a x^6}}\right)-c^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{b+a x^6}}\right)","\frac{2 \left(b+a x^6\right)^{3/4}}{3 x^3}-c^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{b+a x^6}}\right)-c^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{b+a x^6}}\right)",1,"(2*(b + a*x^6)^(3/4))/(3*x^3) - c^(3/4)*ArcTan[(c^(1/4)*x)/(b + a*x^6)^(1/4)] - c^(3/4)*ArcTanh[(c^(1/4)*x)/(b + a*x^6)^(1/4)]","A",1
940,1,71,71,2.864461,"\int \frac{b+a x^8}{\left(-b+a x^8\right) \sqrt[4]{-b+c x^4+a x^8}} \, dx","IntegrateAlgebraic[(b + a*x^8)/((-b + a*x^8)*(-b + c*x^4 + a*x^8)^(1/4)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{-b+c x^4+a x^8}}\right)}{2 \sqrt[4]{c}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{-b+c x^4+a x^8}}\right)}{2 \sqrt[4]{c}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{-b+c x^4+a x^8}}\right)}{2 \sqrt[4]{c}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{-b+c x^4+a x^8}}\right)}{2 \sqrt[4]{c}}",1,"-1/2*ArcTan[(c^(1/4)*x)/(-b + c*x^4 + a*x^8)^(1/4)]/c^(1/4) - ArcTanh[(c^(1/4)*x)/(-b + c*x^4 + a*x^8)^(1/4)]/(2*c^(1/4))","A",1
941,1,82,71,5.6446966,"\int \frac{\sqrt{1+x^5} \left(-2+3 x^5\right)}{1+x^4+2 x^5+x^{10}} \, dx","IntegrateAlgebraic[(Sqrt[1 + x^5]*(-2 + 3*x^5))/(1 + x^4 + 2*x^5 + x^10),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x \sqrt{1+x^5}}{1-x^2+x^5}\right)}{\sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{x^5}{\sqrt{2}}}{x \sqrt{1+x^5}}\right)}{\sqrt{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x \sqrt{1+x^5}}{1-x^2+x^5}\right)}{\sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{1+x^5}}{1+x^2+x^5}\right)}{\sqrt{2}}",1,"-(ArcTan[(Sqrt[2]*x*Sqrt[1 + x^5])/(1 - x^2 + x^5)]/Sqrt[2]) - ArcTanh[(1/Sqrt[2] + x^2/Sqrt[2] + x^5/Sqrt[2])/(x*Sqrt[1 + x^5])]/Sqrt[2]","A",1
942,1,71,71,0.4154511,"\int \frac{1}{\left(2 x+\sqrt{1+x^2}\right)^2} \, dx","IntegrateAlgebraic[(2*x + Sqrt[1 + x^2])^(-2),x]","-\frac{4 x}{3 \left(-1+3 x^2\right)}+\frac{2 \sqrt{1+x^2}}{3 \left(-1+3 x^2\right)}-\frac{2 \tanh ^{-1}\left(\frac{x}{\sqrt{3}}-\frac{\sqrt{1+x^2}}{\sqrt{3}}\right)}{3 \sqrt{3}}","-\frac{4 x}{3 \left(-1+3 x^2\right)}+\frac{2 \sqrt{1+x^2}}{3 \left(-1+3 x^2\right)}-\frac{2 \tanh ^{-1}\left(\frac{x}{\sqrt{3}}-\frac{\sqrt{1+x^2}}{\sqrt{3}}\right)}{3 \sqrt{3}}",1,"(-4*x)/(3*(-1 + 3*x^2)) + (2*Sqrt[1 + x^2])/(3*(-1 + 3*x^2)) - (2*ArcTanh[x/Sqrt[3] - Sqrt[1 + x^2]/Sqrt[3]])/(3*Sqrt[3])","A",1
943,1,71,71,2.5610506,"\int \frac{\sqrt{a x^2+x \sqrt{-b+a^2 x^2}}}{x \sqrt{-b+a^2 x^2}} \, dx","IntegrateAlgebraic[Sqrt[a*x^2 + x*Sqrt[-b + a^2*x^2]]/(x*Sqrt[-b + a^2*x^2]),x]","-\frac{\sqrt{2} \log \left(a x+\sqrt{-b+a^2 x^2}-\sqrt{2} \sqrt{a} \sqrt{a x^2+x \sqrt{-b+a^2 x^2}}\right)}{\sqrt{a}}","-\frac{\sqrt{2} \log \left(a x+\sqrt{-b+a^2 x^2}-\sqrt{2} \sqrt{a} \sqrt{a x^2+x \sqrt{-b+a^2 x^2}}\right)}{\sqrt{a}}",1,"-((Sqrt[2]*Log[a*x + Sqrt[-b + a^2*x^2] - Sqrt[2]*Sqrt[a]*Sqrt[a*x^2 + x*Sqrt[-b + a^2*x^2]]])/Sqrt[a])","A",1
944,1,71,71,0.204242,"\int \frac{\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{1+x^2} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/(1 + x^2),x]","\text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{1-2 \text{$\#$1}^2+\text{$\#$1}^4}\&\right]","\text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{1-2 \text{$\#$1}^2+\text{$\#$1}^4}\&\right]",1,"RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(1 - 2*#1^2 + #1^4) & ]","A",1
945,1,71,71,0.0024536,"\int \frac{\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{1+x^2} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/(1 + x^2),x]","\text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{1-2 \text{$\#$1}^2+\text{$\#$1}^4}\&\right]","\text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{1-2 \text{$\#$1}^2+\text{$\#$1}^4}\&\right]",1,"RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(1 - 2*#1^2 + #1^4) & ]","A",1
946,1,72,72,7.1165029,"\int \frac{\sqrt[3]{1+2 x+x^2}}{4+x+x^2+x^3} \, dx","IntegrateAlgebraic[(1 + 2*x + x^2)^(1/3)/(4 + x + x^2 + x^3),x]","\frac{\sqrt[3]{(1+x)^2} \text{RootSum}\left[3+2 \text{$\#$1}^3-2 \text{$\#$1}^6+\text{$\#$1}^9\&,\frac{\log \left(\sqrt[3]{1+x}-\text{$\#$1}\right) \text{$\#$1}^2}{2-4 \text{$\#$1}^3+3 \text{$\#$1}^6}\&\right]}{(1+x)^{2/3}}","\frac{\sqrt[3]{(1+x)^2} \text{RootSum}\left[3+2 \text{$\#$1}^3-2 \text{$\#$1}^6+\text{$\#$1}^9\&,\frac{\log \left(\sqrt[3]{1+x}-\text{$\#$1}\right) \text{$\#$1}^2}{2-4 \text{$\#$1}^3+3 \text{$\#$1}^6}\&\right]}{(1+x)^{2/3}}",1,"(((1 + x)^2)^(1/3)*RootSum[3 + 2*#1^3 - 2*#1^6 + #1^9 & , (Log[(1 + x)^(1/3) - #1]*#1^2)/(2 - 4*#1^3 + 3*#1^6) & ])/(1 + x)^(2/3)","A",1
947,1,72,72,1.8138486,"\int \frac{\left(-1+x^2\right) \sqrt{1+x^4}}{\left(1-x+x^2\right) \left(1+x+x^2\right)^2} \, dx","IntegrateAlgebraic[((-1 + x^2)*Sqrt[1 + x^4])/((1 - x + x^2)*(1 + x + x^2)^2),x]","\frac{\sqrt{1+x^4}}{2 \left(1+x+x^2\right)}+\frac{1}{2} \tan ^{-1}\left(\frac{x}{1-x+x^2+\sqrt{1+x^4}}\right)-\frac{3}{2} \tan ^{-1}\left(\frac{x}{1+x+x^2+\sqrt{1+x^4}}\right)","\frac{\sqrt{1+x^4}}{2 \left(1+x+x^2\right)}+\frac{1}{2} \tan ^{-1}\left(\frac{x}{1-x+x^2+\sqrt{1+x^4}}\right)-\frac{3}{2} \tan ^{-1}\left(\frac{x}{1+x+x^2+\sqrt{1+x^4}}\right)",1,"Sqrt[1 + x^4]/(2*(1 + x + x^2)) + ArcTan[x/(1 - x + x^2 + Sqrt[1 + x^4])]/2 - (3*ArcTan[x/(1 + x + x^2 + Sqrt[1 + x^4])])/2","A",1
948,1,72,72,0.2117508,"\int x^4 \sqrt[4]{-x^2+x^4} \, dx","IntegrateAlgebraic[x^4*(-x^2 + x^4)^(1/4),x]","\frac{1}{192} \sqrt[4]{-x^2+x^4} \left(-7 x-4 x^3+32 x^5\right)+\frac{7}{128} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)-\frac{7}{128} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)","\frac{1}{192} \sqrt[4]{-x^2+x^4} \left(-7 x-4 x^3+32 x^5\right)+\frac{7}{128} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)-\frac{7}{128} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)",1,"((-x^2 + x^4)^(1/4)*(-7*x - 4*x^3 + 32*x^5))/192 + (7*ArcTan[x/(-x^2 + x^4)^(1/4)])/128 - (7*ArcTanh[x/(-x^2 + x^4)^(1/4)])/128","A",1
949,1,72,72,0.4105631,"\int \frac{x^4 \sqrt[4]{x^3+x^4}}{1+x} \, dx","IntegrateAlgebraic[(x^4*(x^3 + x^4)^(1/4))/(1 + x),x]","\frac{\sqrt[4]{x^3+x^4} \left(7315-4180 x+3040 x^2-2432 x^3+2048 x^4\right)}{10240}+\frac{4389 \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)}{4096}-\frac{4389 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)}{4096}","\frac{\sqrt[4]{x^3+x^4} \left(7315-4180 x+3040 x^2-2432 x^3+2048 x^4\right)}{10240}+\frac{4389 \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)}{4096}-\frac{4389 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)}{4096}",1,"((x^3 + x^4)^(1/4)*(7315 - 4180*x + 3040*x^2 - 2432*x^3 + 2048*x^4))/10240 + (4389*ArcTan[x/(x^3 + x^4)^(1/4)])/4096 - (4389*ArcTanh[x/(x^3 + x^4)^(1/4)])/4096","A",1
950,1,72,72,1.9145495,"\int \frac{\left(4 b+a x^3\right) \left(-b-a x^3+x^4\right)}{x^4 \sqrt[4]{b+a x^3} \left(-b-a x^3+2 x^4\right)} \, dx","IntegrateAlgebraic[((4*b + a*x^3)*(-b - a*x^3 + x^4))/(x^4*(b + a*x^3)^(1/4)*(-b - a*x^3 + 2*x^4)),x]","-\frac{4 \left(b+a x^3\right)^{3/4}}{3 x^3}-2^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^3}}{\sqrt[4]{2} x}\right)+2^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{b+a x^3}}\right)","-\frac{4 \left(b+a x^3\right)^{3/4}}{3 x^3}-2^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^3}}{\sqrt[4]{2} x}\right)+2^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{b+a x^3}}\right)",1,"(-4*(b + a*x^3)^(3/4))/(3*x^3) - 2^(3/4)*ArcTan[(b + a*x^3)^(1/4)/(2^(1/4)*x)] + 2^(3/4)*ArcTanh[(2^(1/4)*x)/(b + a*x^3)^(1/4)]","A",1
951,1,72,72,0.5874806,"\int \frac{b+a x^2}{\left(b+a x^4\right) \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[(b + a*x^2)/((b + a*x^4)*(b*x^2 + a*x^4)^(1/4)),x]","-\frac{1}{4} \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}^3+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^3}{-a+\text{$\#$1}^4}\&\right]","-\frac{1}{4} \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}^3+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^3}{-a+\text{$\#$1}^4}\&\right]",1,"-1/4*RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (-(Log[x]*#1^3) + Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1^3)/(-a + #1^4) & ]","A",1
952,1,72,72,0.0033177,"\int \frac{b+a x^2}{\left(b+a x^4\right) \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[(b + a*x^2)/((b + a*x^4)*(b*x^2 + a*x^4)^(1/4)),x]","-\frac{1}{4} \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}^3+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^3}{-a+\text{$\#$1}^4}\&\right]","-\frac{1}{4} \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}^3+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^3}{-a+\text{$\#$1}^4}\&\right]",1,"-1/4*RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (-(Log[x]*#1^3) + Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1^3)/(-a + #1^4) & ]","A",1
953,1,72,72,0.4048313,"\int \frac{1-x}{\sqrt{3+2 x-5 x^2-4 x^3+x^4+2 x^5+x^6}} \, dx","IntegrateAlgebraic[(1 - x)/Sqrt[3 + 2*x - 5*x^2 - 4*x^3 + x^4 + 2*x^5 + x^6],x]","-\sqrt{2} \tanh ^{-1}\left(\frac{(1+x) \left(-\sqrt{2}+\sqrt{2} x\right)}{-1-x+x^2+x^3-\sqrt{3+2 x-5 x^2-4 x^3+x^4+2 x^5+x^6}}\right)","-\sqrt{2} \tanh ^{-1}\left(\frac{(1+x) \left(-\sqrt{2}+\sqrt{2} x\right)}{-1-x+x^2+x^3-\sqrt{3+2 x-5 x^2-4 x^3+x^4+2 x^5+x^6}}\right)",1,"-(Sqrt[2]*ArcTanh[((1 + x)*(-Sqrt[2] + Sqrt[2]*x))/(-1 - x + x^2 + x^3 - Sqrt[3 + 2*x - 5*x^2 - 4*x^3 + x^4 + 2*x^5 + x^6])])","A",1
954,1,72,72,0.516469,"\int \frac{\left(-1+3 x^4\right) \sqrt{1+x^2+2 x^4+x^8}}{\left(1-x+x^4\right)^2 \left(1+x+x^4\right)} \, dx","IntegrateAlgebraic[((-1 + 3*x^4)*Sqrt[1 + x^2 + 2*x^4 + x^8])/((1 - x + x^4)^2*(1 + x + x^4)),x]","-\frac{\sqrt{1+x^2+2 x^4+x^8}}{2 \left(1-x+x^4\right)}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x}{1+x+x^4+\sqrt{1+x^2+2 x^4+x^8}}\right)}{\sqrt{2}}","-\frac{\sqrt{1+x^2+2 x^4+x^8}}{2 \left(1-x+x^4\right)}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x}{1+x+x^4+\sqrt{1+x^2+2 x^4+x^8}}\right)}{\sqrt{2}}",1,"-1/2*Sqrt[1 + x^2 + 2*x^4 + x^8]/(1 - x + x^4) - ArcTanh[(Sqrt[2]*x)/(1 + x + x^4 + Sqrt[1 + x^2 + 2*x^4 + x^8])]/Sqrt[2]","A",1
955,1,77,73,0.2512768,"\int \sqrt{c+b x+a x^2} \, dx","IntegrateAlgebraic[Sqrt[c + b*x + a*x^2],x]","\frac{(b+2 a x) \sqrt{c+b x+a x^2}}{4 a}+\frac{\left(b^2-4 a c\right) \log \left(a b+2 a^2 x-2 a^{3/2} \sqrt{c+b x+a x^2}\right)}{8 a^{3/2}}","\frac{(b+2 a x) \sqrt{c+b x+a x^2}}{4 a}+\frac{\left(b^2-4 a c\right) \log \left(b+2 a x-2 \sqrt{a} \sqrt{c+b x+a x^2}\right)}{8 a^{3/2}}",1,"((b + 2*a*x)*Sqrt[c + b*x + a*x^2])/(4*a) + ((b^2 - 4*a*c)*Log[a*b + 2*a^2*x - 2*a^(3/2)*Sqrt[c + b*x + a*x^2]])/(8*a^(3/2))","A",1
956,1,85,73,0.7390012,"\int \frac{-1+x}{\left(-1-2 x+x^2\right) \sqrt{-x+x^3}} \, dx","IntegrateAlgebraic[(-1 + x)/((-1 - 2*x + x^2)*Sqrt[-x + x^3]),x]","\frac{1}{4} \left(2+\sqrt{2}\right) \tanh ^{-1}\left(\frac{1-\sqrt{2}+\left(-1+\sqrt{2}\right) x}{\sqrt{-x+x^3}}\right)+\frac{1}{4} \left(-2+\sqrt{2}\right) \tanh ^{-1}\left(\frac{-1-\sqrt{2}+\left(1+\sqrt{2}\right) x}{\sqrt{-x+x^3}}\right)","\frac{1}{4} \left(-2+\sqrt{2}\right) \tanh ^{-1}\left(\frac{-1+x}{\left(-1+\sqrt{2}\right) \sqrt{-x+x^3}}\right)+\frac{1}{4} \left(2+\sqrt{2}\right) \tanh ^{-1}\left(\frac{-1+x}{\left(1+\sqrt{2}\right) \sqrt{-x+x^3}}\right)",1,"((2 + Sqrt[2])*ArcTanh[(1 - Sqrt[2] + (-1 + Sqrt[2])*x)/Sqrt[-x + x^3]])/4 + ((-2 + Sqrt[2])*ArcTanh[(-1 - Sqrt[2] + (1 + Sqrt[2])*x)/Sqrt[-x + x^3]])/4","A",1
957,1,85,73,0.7377852,"\int \frac{-2+3 x+x^2}{\left(-1-2 x+x^2\right) \sqrt{-x+x^3}} \, dx","IntegrateAlgebraic[(-2 + 3*x + x^2)/((-1 - 2*x + x^2)*Sqrt[-x + x^3]),x]","\frac{1}{4} \left(6+\sqrt{2}\right) \tanh ^{-1}\left(\frac{1-\sqrt{2}+\left(-1+\sqrt{2}\right) x}{\sqrt{-x+x^3}}\right)+\frac{1}{4} \left(-6+\sqrt{2}\right) \tanh ^{-1}\left(\frac{-1-\sqrt{2}+\left(1+\sqrt{2}\right) x}{\sqrt{-x+x^3}}\right)","\frac{1}{4} \left(-6+\sqrt{2}\right) \tanh ^{-1}\left(\frac{-1+x}{\left(-1+\sqrt{2}\right) \sqrt{-x+x^3}}\right)+\frac{1}{4} \left(6+\sqrt{2}\right) \tanh ^{-1}\left(\frac{-1+x}{\left(1+\sqrt{2}\right) \sqrt{-x+x^3}}\right)",1,"((6 + Sqrt[2])*ArcTanh[(1 - Sqrt[2] + (-1 + Sqrt[2])*x)/Sqrt[-x + x^3]])/4 + ((-6 + Sqrt[2])*ArcTanh[(-1 - Sqrt[2] + (1 + Sqrt[2])*x)/Sqrt[-x + x^3]])/4","A",1
958,1,73,73,2.0839179,"\int \frac{x \left(-3+x^2\right)}{\left(-1+x^2\right)^{2/3} \left(1-x^2+x^3\right)} \, dx","IntegrateAlgebraic[(x*(-3 + x^2))/((-1 + x^2)^(2/3)*(1 - x^2 + x^3)),x]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^2}}\right)+\log \left(-x+\sqrt[3]{-1+x^2}\right)-\frac{1}{2} \log \left(x^2+x \sqrt[3]{-1+x^2}+\left(-1+x^2\right)^{2/3}\right)","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^2}}\right)+\log \left(-x+\sqrt[3]{-1+x^2}\right)-\frac{1}{2} \log \left(x^2+x \sqrt[3]{-1+x^2}+\left(-1+x^2\right)^{2/3}\right)",1,"Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^2)^(1/3))] + Log[-x + (-1 + x^2)^(1/3)] - Log[x^2 + x*(-1 + x^2)^(1/3) + (-1 + x^2)^(2/3)]/2","A",1
959,1,73,73,0.3588711,"\int x^2 \sqrt[4]{-x^3+x^4} \, dx","IntegrateAlgebraic[x^2*(-x^3 + x^4)^(1/4),x]","\frac{\left(-77-44 x-32 x^2+384 x^3\right) \sqrt[4]{-x^3+x^4}}{1536}+\frac{77 \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)}{1024}-\frac{77 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)}{1024}","\frac{\left(-77-44 x-32 x^2+384 x^3\right) \sqrt[4]{-x^3+x^4}}{1536}+\frac{77 \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)}{1024}-\frac{77 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)}{1024}",1,"((-77 - 44*x - 32*x^2 + 384*x^3)*(-x^3 + x^4)^(1/4))/1536 + (77*ArcTan[x/(-x^3 + x^4)^(1/4)])/1024 - (77*ArcTanh[x/(-x^3 + x^4)^(1/4)])/1024","A",1
960,1,73,73,0.2213082,"\int \frac{\left(3+x^4\right) \sqrt{x+x^4-x^5}}{\left(-1+x^4\right) \left(-1+x^3+x^4\right)} \, dx","IntegrateAlgebraic[((3 + x^4)*Sqrt[x + x^4 - x^5])/((-1 + x^4)*(-1 + x^3 + x^4)),x]","2 \tanh ^{-1}\left(\frac{x \sqrt{x+x^4-x^5}}{-1-x^3+x^4}\right)-2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x+x^4-x^5}}{-1-x^3+x^4}\right)","2 \tanh ^{-1}\left(\frac{x \sqrt{x+x^4-x^5}}{-1-x^3+x^4}\right)-2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x+x^4-x^5}}{-1-x^3+x^4}\right)",1,"2*ArcTanh[(x*Sqrt[x + x^4 - x^5])/(-1 - x^3 + x^4)] - 2*Sqrt[2]*ArcTanh[(Sqrt[2]*x*Sqrt[x + x^4 - x^5])/(-1 - x^3 + x^4)]","A",1
961,1,73,73,0.47712,"\int \frac{\left(-b+a x^2\right) \sqrt{b x+a x^3}}{b^2 x+2 (-1+a b) x^3+a^2 x^5} \, dx","IntegrateAlgebraic[((-b + a*x^2)*Sqrt[b*x + a*x^3])/(b^2*x + 2*(-1 + a*b)*x^3 + a^2*x^5),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt{b x+a x^3}}{b+a x^2}\right)}{\sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt{b x+a x^3}}{b+a x^2}\right)}{\sqrt[4]{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt{b x+a x^3}}{b+a x^2}\right)}{\sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt{b x+a x^3}}{b+a x^2}\right)}{\sqrt[4]{2}}",1,"-(ArcTan[(2^(1/4)*Sqrt[b*x + a*x^3])/(b + a*x^2)]/2^(1/4)) - ArcTanh[(2^(1/4)*Sqrt[b*x + a*x^3])/(b + a*x^2)]/2^(1/4)","A",1
962,1,73,73,0.2317372,"\int \frac{\left(-2+x^3\right) \left(1+x^3\right)^{2/3}}{x^3 \left(2+x^3+x^6\right)} \, dx","IntegrateAlgebraic[((-2 + x^3)*(1 + x^3)^(2/3))/(x^3*(2 + x^3 + x^6)),x]","\frac{\left(1+x^3\right)^{2/3}}{2 x^2}-\frac{2}{3} \text{RootSum}\left[2-3 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)}{-3 \text{$\#$1}+4 \text{$\#$1}^4}\&\right]","\frac{\left(1+x^3\right)^{2/3}}{2 x^2}-\frac{2}{3} \text{RootSum}\left[2-3 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)}{-3 \text{$\#$1}+4 \text{$\#$1}^4}\&\right]",1,"(1 + x^3)^(2/3)/(2*x^2) - (2*RootSum[2 - 3*#1^3 + 2*#1^6 & , (-Log[x] + Log[(1 + x^3)^(1/3) - x*#1])/(-3*#1 + 4*#1^4) & ])/3","A",1
963,1,73,73,0.0022017,"\int \frac{\left(-2+x^3\right) \left(1+x^3\right)^{2/3}}{x^3 \left(2+x^3+x^6\right)} \, dx","IntegrateAlgebraic[((-2 + x^3)*(1 + x^3)^(2/3))/(x^3*(2 + x^3 + x^6)),x]","\frac{\left(1+x^3\right)^{2/3}}{2 x^2}-\frac{2}{3} \text{RootSum}\left[2-3 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)}{-3 \text{$\#$1}+4 \text{$\#$1}^4}\&\right]","\frac{\left(1+x^3\right)^{2/3}}{2 x^2}-\frac{2}{3} \text{RootSum}\left[2-3 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)}{-3 \text{$\#$1}+4 \text{$\#$1}^4}\&\right]",1,"(1 + x^3)^(2/3)/(2*x^2) - (2*RootSum[2 - 3*#1^3 + 2*#1^6 & , (-Log[x] + Log[(1 + x^3)^(1/3) - x*#1])/(-3*#1 + 4*#1^4) & ])/3","A",1
964,1,73,73,0.3840247,"\int \frac{\left(-3+2 x^5\right) \sqrt{x+2 x^4+x^6}}{\left(1+x^5\right) \left(1+x^3+x^5\right)} \, dx","IntegrateAlgebraic[((-3 + 2*x^5)*Sqrt[x + 2*x^4 + x^6])/((1 + x^5)*(1 + x^3 + x^5)),x]","2 \tanh ^{-1}\left(\frac{x \sqrt{x+2 x^4+x^6}}{1+2 x^3+x^5}\right)-2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x+2 x^4+x^6}}{1+2 x^3+x^5}\right)","2 \tanh ^{-1}\left(\frac{x \sqrt{x+2 x^4+x^6}}{1+2 x^3+x^5}\right)-2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x+2 x^4+x^6}}{1+2 x^3+x^5}\right)",1,"2*ArcTanh[(x*Sqrt[x + 2*x^4 + x^6])/(1 + 2*x^3 + x^5)] - 2*Sqrt[2]*ArcTanh[(Sqrt[2]*x*Sqrt[x + 2*x^4 + x^6])/(1 + 2*x^3 + x^5)]","A",1
965,1,73,73,0.5990901,"\int \frac{\sqrt{-1+x^2+x^4+x^6} \left(1+x^4+2 x^6\right)}{1-x^4-2 x^6+x^8+2 x^{10}+x^{12}} \, dx","IntegrateAlgebraic[(Sqrt[-1 + x^2 + x^4 + x^6]*(1 + x^4 + 2*x^6))/(1 - x^4 - 2*x^6 + x^8 + 2*x^10 + x^12),x]","-\frac{1}{2} \sqrt{1+i} \tan ^{-1}\left(\frac{\sqrt{-1-i} x}{\sqrt{-1+x^2+x^4+x^6}}\right)-\frac{1}{2} \sqrt{1-i} \tan ^{-1}\left(\frac{\sqrt{-1+i} x}{\sqrt{-1+x^2+x^4+x^6}}\right)","-\frac{1}{2} \sqrt{1+i} \tan ^{-1}\left(\frac{\sqrt{-1-i} x}{\sqrt{-1+x^2+x^4+x^6}}\right)-\frac{1}{2} \sqrt{1-i} \tan ^{-1}\left(\frac{\sqrt{-1+i} x}{\sqrt{-1+x^2+x^4+x^6}}\right)",1,"-1/2*(Sqrt[1 + I]*ArcTan[(Sqrt[-1 - I]*x)/Sqrt[-1 + x^2 + x^4 + x^6]]) - (Sqrt[1 - I]*ArcTan[(Sqrt[-1 + I]*x)/Sqrt[-1 + x^2 + x^4 + x^6]])/2","A",1
966,1,73,73,3.8783916,"\int \frac{\sqrt{x^2-x \sqrt{-x+x^2}}}{x^3} \, dx","IntegrateAlgebraic[Sqrt[x^2 - x*Sqrt[-x + x^2]]/x^3,x]","\frac{4 (-3+x) \sqrt{-x \left(-x+\sqrt{-x+x^2}\right)}}{15 x^2}-\frac{4 \sqrt{-x+x^2} \sqrt{-x \left(-x+\sqrt{-x+x^2}\right)}}{15 x^2}","\frac{4 (-3+x) \sqrt{-x \left(-x+\sqrt{-x+x^2}\right)}}{15 x^2}-\frac{4 \sqrt{-x+x^2} \sqrt{-x \left(-x+\sqrt{-x+x^2}\right)}}{15 x^2}",1,"(4*(-3 + x)*Sqrt[-(x*(-x + Sqrt[-x + x^2]))])/(15*x^2) - (4*Sqrt[-x + x^2]*Sqrt[-(x*(-x + Sqrt[-x + x^2]))])/(15*x^2)","A",1
967,1,74,74,0.0781686,"\int \frac{1}{x \left(1+x^2\right)^{2/3}} \, dx","IntegrateAlgebraic[1/(x*(1 + x^2)^(2/3)),x]","-\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^2}}{\sqrt{3}}\right)+\frac{1}{2} \log \left(-1+\sqrt[3]{1+x^2}\right)-\frac{1}{4} \log \left(1+\sqrt[3]{1+x^2}+\left(1+x^2\right)^{2/3}\right)","-\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^2}}{\sqrt{3}}\right)+\frac{1}{2} \log \left(-1+\sqrt[3]{1+x^2}\right)-\frac{1}{4} \log \left(1+\sqrt[3]{1+x^2}+\left(1+x^2\right)^{2/3}\right)",1,"-1/2*(Sqrt[3]*ArcTan[1/Sqrt[3] + (2*(1 + x^2)^(1/3))/Sqrt[3]]) + Log[-1 + (1 + x^2)^(1/3)]/2 - Log[1 + (1 + x^2)^(1/3) + (1 + x^2)^(2/3)]/4","A",1
968,1,74,74,0.0675607,"\int \frac{1}{x \sqrt[3]{1+x^2}} \, dx","IntegrateAlgebraic[1/(x*(1 + x^2)^(1/3)),x]","\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^2}}{\sqrt{3}}\right)+\frac{1}{2} \log \left(-1+\sqrt[3]{1+x^2}\right)-\frac{1}{4} \log \left(1+\sqrt[3]{1+x^2}+\left(1+x^2\right)^{2/3}\right)","\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^2}}{\sqrt{3}}\right)+\frac{1}{2} \log \left(-1+\sqrt[3]{1+x^2}\right)-\frac{1}{4} \log \left(1+\sqrt[3]{1+x^2}+\left(1+x^2\right)^{2/3}\right)",1,"(Sqrt[3]*ArcTan[1/Sqrt[3] + (2*(1 + x^2)^(1/3))/Sqrt[3]])/2 + Log[-1 + (1 + x^2)^(1/3)]/2 - Log[1 + (1 + x^2)^(1/3) + (1 + x^2)^(2/3)]/4","A",1
969,1,74,74,0.3104339,"\int \frac{1}{\left(-1+x^3\right) \sqrt[3]{-x^2+x^3}} \, dx","IntegrateAlgebraic[1/((-1 + x^3)*(-x^2 + x^3)^(1/3)),x]","\frac{\left(-x^2+x^3\right)^{2/3}}{(1-x) x}+\frac{1}{3} \text{RootSum}\left[3-3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{\left(-x^2+x^3\right)^{2/3}}{(1-x) x}+\frac{1}{3} \text{RootSum}\left[3-3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"(-x^2 + x^3)^(2/3)/((1 - x)*x) + RootSum[3 - 3*#1^3 + #1^6 & , (-Log[x] + Log[(-x^2 + x^3)^(1/3) - x*#1])/#1 & ]/3","A",1
970,1,74,74,0.0016284,"\int \frac{1}{\left(-1+x^3\right) \sqrt[3]{-x^2+x^3}} \, dx","IntegrateAlgebraic[1/((-1 + x^3)*(-x^2 + x^3)^(1/3)),x]","\frac{\left(-x^2+x^3\right)^{2/3}}{(1-x) x}+\frac{1}{3} \text{RootSum}\left[3-3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{\left(-x^2+x^3\right)^{2/3}}{(1-x) x}+\frac{1}{3} \text{RootSum}\left[3-3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"(-x^2 + x^3)^(2/3)/((1 - x)*x) + RootSum[3 - 3*#1^3 + #1^6 & , (-Log[x] + Log[(-x^2 + x^3)^(1/3) - x*#1])/#1 & ]/3","A",1
971,1,74,74,1.3623473,"\int \frac{\sqrt{-1+x^3} \left(2+x^3\right) \left(-1-x^2+x^3\right)^2}{x^6 \left(-2-3 x^2+2 x^3\right)} \, dx","IntegrateAlgebraic[(Sqrt[-1 + x^3]*(2 + x^3)*(-1 - x^2 + x^3)^2)/(x^6*(-2 - 3*x^2 + 2*x^3)),x]","\frac{\sqrt{-1+x^3} \left(12+10 x^2-24 x^3+15 x^4-10 x^5+12 x^6\right)}{60 x^5}-\frac{1}{4} \sqrt{\frac{3}{2}} \tanh ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt{-1+x^3}}\right)","\frac{\sqrt{-1+x^3} \left(12+10 x^2-24 x^3+15 x^4-10 x^5+12 x^6\right)}{60 x^5}-\frac{1}{4} \sqrt{\frac{3}{2}} \tanh ^{-1}\left(\frac{\sqrt{\frac{3}{2}} x}{\sqrt{-1+x^3}}\right)",1,"(Sqrt[-1 + x^3]*(12 + 10*x^2 - 24*x^3 + 15*x^4 - 10*x^5 + 12*x^6))/(60*x^5) - (Sqrt[3/2]*ArcTanh[(Sqrt[3/2]*x)/Sqrt[-1 + x^3]])/4","A",1
972,1,74,74,10.6414854,"\int \frac{\sqrt[4]{-1+3 x-3 x^2+x^3}}{-1-2 x+x^2+3 x^3} \, dx","IntegrateAlgebraic[(-1 + 3*x - 3*x^2 + x^3)^(1/4)/(-1 - 2*x + x^2 + 3*x^3),x]","\frac{\sqrt[4]{(-1+x)^3} \text{RootSum}\left[1+9 \text{$\#$1}^4+10 \text{$\#$1}^8+3 \text{$\#$1}^{12}\&,\frac{\log \left(\sqrt[4]{-1+x}-\text{$\#$1}\right) \text{$\#$1}^3}{9+20 \text{$\#$1}^4+9 \text{$\#$1}^8}\&\right]}{(-1+x)^{3/4}}","\frac{\sqrt[4]{(-1+x)^3} \text{RootSum}\left[1+9 \text{$\#$1}^4+10 \text{$\#$1}^8+3 \text{$\#$1}^{12}\&,\frac{\log \left(\sqrt[4]{-1+x}-\text{$\#$1}\right) \text{$\#$1}^3}{9+20 \text{$\#$1}^4+9 \text{$\#$1}^8}\&\right]}{(-1+x)^{3/4}}",1,"(((-1 + x)^3)^(1/4)*RootSum[1 + 9*#1^4 + 10*#1^8 + 3*#1^12 & , (Log[(-1 + x)^(1/4) - #1]*#1^3)/(9 + 20*#1^4 + 9*#1^8) & ])/(-1 + x)^(3/4)","A",1
973,1,74,74,1.4455142,"\int \frac{\left(1+x+x^2\right) \left(2 x+x^2\right) \sqrt{1+2 x+x^2-x^4}}{(1+x)^4} \, dx","IntegrateAlgebraic[((1 + x + x^2)*(2*x + x^2)*Sqrt[1 + 2*x + x^2 - x^4])/(1 + x)^4,x]","\frac{\sqrt{1+2 x+x^2-x^4} \left(-2-4 x+x^2+3 x^3+2 x^4\right)}{6 (1+x)^3}-\tan ^{-1}\left(\frac{\sqrt{1+2 x+x^2-x^4}}{1+x+x^2}\right)","\frac{\sqrt{1+2 x+x^2-x^4} \left(-2-4 x+x^2+3 x^3+2 x^4\right)}{6 (1+x)^3}-\tan ^{-1}\left(\frac{\sqrt{1+2 x+x^2-x^4}}{1+x+x^2}\right)",1,"(Sqrt[1 + 2*x + x^2 - x^4]*(-2 - 4*x + x^2 + 3*x^3 + 2*x^4))/(6*(1 + x)^3) - ArcTan[Sqrt[1 + 2*x + x^2 - x^4]/(1 + x + x^2)]","A",1
974,1,74,74,5.246684,"\int \frac{-1-x+x^4}{x \sqrt[4]{1+x^4}} \, dx","IntegrateAlgebraic[(-1 - x + x^4)/(x*(1 + x^4)^(1/4)),x]","\frac{1}{3} \left(1+x^4\right)^{3/4}-\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\frac{1}{2} \tan ^{-1}\left(\sqrt[4]{1+x^4}\right)-\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)+\frac{1}{2} \tanh ^{-1}\left(\sqrt[4]{1+x^4}\right)","\frac{1}{3} \left(1+x^4\right)^{3/4}-\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\frac{1}{2} \tan ^{-1}\left(\sqrt[4]{1+x^4}\right)-\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)+\frac{1}{2} \tanh ^{-1}\left(\sqrt[4]{1+x^4}\right)",1,"(1 + x^4)^(3/4)/3 - ArcTan[x/(1 + x^4)^(1/4)]/2 - ArcTan[(1 + x^4)^(1/4)]/2 - ArcTanh[x/(1 + x^4)^(1/4)]/2 + ArcTanh[(1 + x^4)^(1/4)]/2","A",1
975,1,74,74,0.4088233,"\int \frac{\sqrt[4]{-x^3+x^4}}{x^2 \left(-1+x^2\right)} \, dx","IntegrateAlgebraic[(-x^3 + x^4)^(1/4)/(x^2*(-1 + x^2)),x]","\frac{4 \sqrt[4]{-x^3+x^4}}{x}+\sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)-\sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)","\frac{4 \sqrt[4]{-x^3+x^4}}{x}+\sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)-\sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)",1,"(4*(-x^3 + x^4)^(1/4))/x + 2^(1/4)*ArcTan[(2^(1/4)*x)/(-x^3 + x^4)^(1/4)] - 2^(1/4)*ArcTanh[(2^(1/4)*x)/(-x^3 + x^4)^(1/4)]","A",1
976,1,74,74,1.2517514,"\int \frac{-3+x^4}{\sqrt[3]{1+x^4} \left(1-x^3+x^4\right)} \, dx","IntegrateAlgebraic[(-3 + x^4)/((1 + x^4)^(1/3)*(1 - x^3 + x^4)),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^4}}\right)+\log \left(-x+\sqrt[3]{1+x^4}\right)-\frac{1}{2} \log \left(x^2+x \sqrt[3]{1+x^4}+\left(1+x^4\right)^{2/3}\right)","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^4}}\right)+\log \left(-x+\sqrt[3]{1+x^4}\right)-\frac{1}{2} \log \left(x^2+x \sqrt[3]{1+x^4}+\left(1+x^4\right)^{2/3}\right)",1,"-(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^4)^(1/3))]) + Log[-x + (1 + x^4)^(1/3)] - Log[x^2 + x*(1 + x^4)^(1/3) + (1 + x^4)^(2/3)]/2","A",1
977,1,74,74,0.8375882,"\int \frac{b+a x^2}{\left(b-a x^2+x^4\right) \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[(b + a*x^2)/((b - a*x^2 + x^4)*(b*x^2 + a*x^4)^(1/4)),x]","-\frac{1}{2} \text{RootSum}\left[2 a^2+b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}^3+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^3}{-3 a+2 \text{$\#$1}^4}\&\right]","-\frac{1}{2} \text{RootSum}\left[2 a^2+b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}^3+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^3}{-3 a+2 \text{$\#$1}^4}\&\right]",1,"-1/2*RootSum[2*a^2 + b - 3*a*#1^4 + #1^8 & , (-(Log[x]*#1^3) + Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1^3)/(-3*a + 2*#1^4) & ]","A",1
978,1,74,74,0.0043877,"\int \frac{b+a x^2}{\left(b-a x^2+x^4\right) \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[(b + a*x^2)/((b - a*x^2 + x^4)*(b*x^2 + a*x^4)^(1/4)),x]","-\frac{1}{2} \text{RootSum}\left[2 a^2+b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}^3+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^3}{-3 a+2 \text{$\#$1}^4}\&\right]","-\frac{1}{2} \text{RootSum}\left[2 a^2+b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}^3+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^3}{-3 a+2 \text{$\#$1}^4}\&\right]",1,"-1/2*RootSum[2*a^2 + b - 3*a*#1^4 + #1^8 & , (-(Log[x]*#1^3) + Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1^3)/(-3*a + 2*#1^4) & ]","A",1
979,1,74,74,0.4388294,"\int \frac{-b+2 a x^4}{x^2 \left(-b+a x^4\right)^{3/4}} \, dx","IntegrateAlgebraic[(-b + 2*a*x^4)/(x^2*(-b + a*x^4)^(3/4)),x]","-\frac{\sqrt[4]{-b+a x^4}}{x}-\sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)+\sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)","-\frac{\sqrt[4]{-b+a x^4}}{x}-\sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)+\sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)",1,"-((-b + a*x^4)^(1/4)/x) - a^(1/4)*ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)] + a^(1/4)*ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]","A",1
980,1,74,74,0.060214,"\int \frac{1}{x \sqrt[3]{1+x^6}} \, dx","IntegrateAlgebraic[1/(x*(1 + x^6)^(1/3)),x]","\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^6}}{\sqrt{3}}\right)}{2 \sqrt{3}}+\frac{1}{6} \log \left(-1+\sqrt[3]{1+x^6}\right)-\frac{1}{12} \log \left(1+\sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)","\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^6}}{\sqrt{3}}\right)}{2 \sqrt{3}}+\frac{1}{6} \log \left(-1+\sqrt[3]{1+x^6}\right)-\frac{1}{12} \log \left(1+\sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)",1,"ArcTan[1/Sqrt[3] + (2*(1 + x^6)^(1/3))/Sqrt[3]]/(2*Sqrt[3]) + Log[-1 + (1 + x^6)^(1/3)]/6 - Log[1 + (1 + x^6)^(1/3) + (1 + x^6)^(2/3)]/12","A",1
981,1,74,74,0.153905,"\int \frac{3-9 x^4+2 x^6}{x \left(1+x^2\right)^2 \left(-1+2 x^2\right) \sqrt{\frac{1-2 x^2}{1+2 x^2}} \left(1+2 x^2\right)} \, dx","IntegrateAlgebraic[(3 - 9*x^4 + 2*x^6)/(x*(1 + x^2)^2*(-1 + 2*x^2)*Sqrt[(1 - 2*x^2)/(1 + 2*x^2)]*(1 + 2*x^2)),x]","-\frac{2 \sqrt{\frac{1-2 x^2}{1+2 x^2}} \left(-1-x^2+2 x^4\right)}{3 \left(-1+x^2+2 x^4\right)}+3 \tanh ^{-1}\left(\sqrt{\frac{1-2 x^2}{1+2 x^2}}\right)","-\frac{2 \sqrt{\frac{1-2 x^2}{1+2 x^2}} \left(-1-x^2+2 x^4\right)}{3 \left(-1+x^2+2 x^4\right)}+3 \tanh ^{-1}\left(\sqrt{\frac{1-2 x^2}{1+2 x^2}}\right)",1,"(-2*Sqrt[(1 - 2*x^2)/(1 + 2*x^2)]*(-1 - x^2 + 2*x^4))/(3*(-1 + x^2 + 2*x^4)) + 3*ArcTanh[Sqrt[(1 - 2*x^2)/(1 + 2*x^2)]]","A",1
982,1,74,74,0.3434891,"\int \frac{\sqrt[4]{-1+2 x^4} \left(-2+x^8\right)}{x^6 \left(-1+x^4\right)^2} \, dx","IntegrateAlgebraic[((-1 + 2*x^4)^(1/4)*(-2 + x^8))/(x^6*(-1 + x^4)^2),x]","\frac{\sqrt[4]{-1+2 x^4} \left(-8-56 x^4+69 x^8\right)}{20 x^5 \left(-1+x^4\right)}+\frac{15}{8} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-1+2 x^4}}\right)-\frac{15}{8} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-1+2 x^4}}\right)","\frac{\sqrt[4]{-1+2 x^4} \left(-8-56 x^4+69 x^8\right)}{20 x^5 \left(-1+x^4\right)}+\frac{15}{8} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-1+2 x^4}}\right)-\frac{15}{8} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-1+2 x^4}}\right)",1,"((-1 + 2*x^4)^(1/4)*(-8 - 56*x^4 + 69*x^8))/(20*x^5*(-1 + x^4)) + (15*ArcTan[x/(-1 + 2*x^4)^(1/4)])/8 - (15*ArcTanh[x/(-1 + 2*x^4)^(1/4)])/8","A",1
983,1,75,75,0.1984176,"\int \frac{\left(b+a x^2\right)^{3/4}}{x^3} \, dx","IntegrateAlgebraic[(b + a*x^2)^(3/4)/x^3,x]","-\frac{\left(b+a x^2\right)^{3/4}}{2 x^2}+\frac{3 a \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^2}}{\sqrt[4]{b}}\right)}{4 \sqrt[4]{b}}-\frac{3 a \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^2}}{\sqrt[4]{b}}\right)}{4 \sqrt[4]{b}}","-\frac{\left(b+a x^2\right)^{3/4}}{2 x^2}+\frac{3 a \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^2}}{\sqrt[4]{b}}\right)}{4 \sqrt[4]{b}}-\frac{3 a \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^2}}{\sqrt[4]{b}}\right)}{4 \sqrt[4]{b}}",1,"-1/2*(b + a*x^2)^(3/4)/x^2 + (3*a*ArcTan[(b + a*x^2)^(1/4)/b^(1/4)])/(4*b^(1/4)) - (3*a*ArcTanh[(b + a*x^2)^(1/4)/b^(1/4)])/(4*b^(1/4))","A",1
984,1,75,75,0.3230195,"\int \frac{3-2 (1+k) x+k x^2}{\sqrt[4]{(1-x) x (1-k x)} \left(-d+d (1+k) x-d k x^2+x^3\right)} \, dx","IntegrateAlgebraic[(3 - 2*(1 + k)*x + k*x^2)/(((1 - x)*x*(1 - k*x))^(1/4)*(-d + d*(1 + k)*x - d*k*x^2 + x^3)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{x+(-1-k) x^2+k x^3}}{x}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{x+(-1-k) x^2+k x^3}}{x}\right)}{d^{3/4}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{x+(-1-k) x^2+k x^3}}{x}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{x+(-1-k) x^2+k x^3}}{x}\right)}{d^{3/4}}",1,"(2*ArcTan[(d^(1/4)*(x + (-1 - k)*x^2 + k*x^3)^(1/4))/x])/d^(3/4) - (2*ArcTanh[(d^(1/4)*(x + (-1 - k)*x^2 + k*x^3)^(1/4))/x])/d^(3/4)","A",1
985,1,75,75,10.4253165,"\int \frac{1}{\sqrt[4]{-1+3 x-3 x^2+x^3} \left(-1-2 x+x^2+3 x^3\right)} \, dx","IntegrateAlgebraic[1/((-1 + 3*x - 3*x^2 + x^3)^(1/4)*(-1 - 2*x + x^2 + 3*x^3)),x]","\frac{\left((-1+x)^3\right)^{3/4} \text{RootSum}\left[1+9 \text{$\#$1}^4+10 \text{$\#$1}^8+3 \text{$\#$1}^{12}\&,\frac{\log \left(\sqrt[4]{-1+x}-\text{$\#$1}\right)}{9 \text{$\#$1}^3+20 \text{$\#$1}^7+9 \text{$\#$1}^{11}}\&\right]}{(-1+x)^{9/4}}","\frac{\left((-1+x)^3\right)^{3/4} \text{RootSum}\left[1+9 \text{$\#$1}^4+10 \text{$\#$1}^8+3 \text{$\#$1}^{12}\&,\frac{\log \left(\sqrt[4]{-1+x}-\text{$\#$1}\right)}{9 \text{$\#$1}^3+20 \text{$\#$1}^7+9 \text{$\#$1}^{11}}\&\right]}{(-1+x)^{9/4}}",1,"(((-1 + x)^3)^(3/4)*RootSum[1 + 9*#1^4 + 10*#1^8 + 3*#1^12 & , Log[(-1 + x)^(1/4) - #1]/(9*#1^3 + 20*#1^7 + 9*#1^11) & ])/(-1 + x)^(9/4)","A",1
986,1,75,75,0.1029184,"\int \frac{\left(b+a x^3\right)^{3/4}}{x^4} \, dx","IntegrateAlgebraic[(b + a*x^3)^(3/4)/x^4,x]","-\frac{\left(b+a x^3\right)^{3/4}}{3 x^3}+\frac{a \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^3}}{\sqrt[4]{b}}\right)}{2 \sqrt[4]{b}}-\frac{a \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^3}}{\sqrt[4]{b}}\right)}{2 \sqrt[4]{b}}","-\frac{\left(b+a x^3\right)^{3/4}}{3 x^3}+\frac{a \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^3}}{\sqrt[4]{b}}\right)}{2 \sqrt[4]{b}}-\frac{a \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^3}}{\sqrt[4]{b}}\right)}{2 \sqrt[4]{b}}",1,"-1/3*(b + a*x^3)^(3/4)/x^3 + (a*ArcTan[(b + a*x^3)^(1/4)/b^(1/4)])/(2*b^(1/4)) - (a*ArcTanh[(b + a*x^3)^(1/4)/b^(1/4)])/(2*b^(1/4))","A",1
987,1,75,75,0.3291737,"\int \frac{\sqrt[4]{x^2+x^4}}{x^4 \left(-1+x^4\right)} \, dx","IntegrateAlgebraic[(x^2 + x^4)^(1/4)/(x^4*(-1 + x^4)),x]","\frac{2 \left(1+x^2\right) \sqrt[4]{x^2+x^4}}{5 x^3}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2^{3/4}}","\frac{2 \left(1+x^2\right) \sqrt[4]{x^2+x^4}}{5 x^3}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2^{3/4}}",1,"(2*(1 + x^2)*(x^2 + x^4)^(1/4))/(5*x^3) + ArcTan[(2^(1/4)*x)/(x^2 + x^4)^(1/4)]/2^(3/4) - ArcTanh[(2^(1/4)*x)/(x^2 + x^4)^(1/4)]/2^(3/4)","A",1
988,1,75,75,1.6985316,"\int \frac{x \left(-3+x^4\right)}{\left(1+x^4\right)^{2/3} \left(1+x^3+x^4\right)} \, dx","IntegrateAlgebraic[(x*(-3 + x^4))/((1 + x^4)^(2/3)*(1 + x^3 + x^4)),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{1+x^4}}\right)+\log \left(x+\sqrt[3]{1+x^4}\right)-\frac{1}{2} \log \left(x^2-x \sqrt[3]{1+x^4}+\left(1+x^4\right)^{2/3}\right)","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{1+x^4}}\right)+\log \left(x+\sqrt[3]{1+x^4}\right)-\frac{1}{2} \log \left(x^2-x \sqrt[3]{1+x^4}+\left(1+x^4\right)^{2/3}\right)",1,"-(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(1 + x^4)^(1/3))]) + Log[x + (1 + x^4)^(1/3)] - Log[x^2 - x*(1 + x^4)^(1/3) + (1 + x^4)^(2/3)]/2","A",1
989,1,75,75,0.5047105,"\int \frac{\sqrt{2-x^2-4 x^4} \left(1+2 x^4\right)}{\left(-1+2 x^4\right) \left(-1-x^2+2 x^4\right)} \, dx","IntegrateAlgebraic[(Sqrt[2 - x^2 - 4*x^4]*(1 + 2*x^4))/((-1 + 2*x^4)*(-1 - x^2 + 2*x^4)),x]","\tan ^{-1}\left(\frac{x \sqrt{2-x^2-4 x^4}}{-2+x^2+4 x^4}\right)-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x \sqrt{2-x^2-4 x^4}}{-2+x^2+4 x^4}\right)","\tan ^{-1}\left(\frac{x \sqrt{2-x^2-4 x^4}}{-2+x^2+4 x^4}\right)-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x \sqrt{2-x^2-4 x^4}}{-2+x^2+4 x^4}\right)",1,"ArcTan[(x*Sqrt[2 - x^2 - 4*x^4])/(-2 + x^2 + 4*x^4)] - Sqrt[3]*ArcTan[(Sqrt[3]*x*Sqrt[2 - x^2 - 4*x^4])/(-2 + x^2 + 4*x^4)]","A",1
990,1,75,75,0.3549033,"\int \left(b+a x^4\right)^{3/4} \, dx","IntegrateAlgebraic[(b + a*x^4)^(3/4),x]","\frac{1}{4} x \left(b+a x^4\right)^{3/4}+\frac{3 b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{8 \sqrt[4]{a}}+\frac{3 b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{8 \sqrt[4]{a}}","\frac{1}{4} x \left(b+a x^4\right)^{3/4}+\frac{3 b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{8 \sqrt[4]{a}}+\frac{3 b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{8 \sqrt[4]{a}}",1,"(x*(b + a*x^4)^(3/4))/4 + (3*b*ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)])/(8*a^(1/4)) + (3*b*ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)])/(8*a^(1/4))","A",1
991,1,75,75,0.2843494,"\int \frac{\left(b+a x^4\right)^{3/4}}{x^4} \, dx","IntegrateAlgebraic[(b + a*x^4)^(3/4)/x^4,x]","-\frac{\left(b+a x^4\right)^{3/4}}{3 x^3}+\frac{1}{2} a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)+\frac{1}{2} a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)","-\frac{\left(b+a x^4\right)^{3/4}}{3 x^3}+\frac{1}{2} a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)+\frac{1}{2} a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)",1,"-1/3*(b + a*x^4)^(3/4)/x^3 + (a^(3/4)*ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)])/2 + (a^(3/4)*ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)])/2","A",1
992,1,75,75,0.1250097,"\int \frac{\left(b+a x^5\right)^{3/4}}{x^6} \, dx","IntegrateAlgebraic[(b + a*x^5)^(3/4)/x^6,x]","-\frac{\left(b+a x^5\right)^{3/4}}{5 x^5}+\frac{3 a \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^5}}{\sqrt[4]{b}}\right)}{10 \sqrt[4]{b}}-\frac{3 a \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^5}}{\sqrt[4]{b}}\right)}{10 \sqrt[4]{b}}","-\frac{\left(b+a x^5\right)^{3/4}}{5 x^5}+\frac{3 a \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^5}}{\sqrt[4]{b}}\right)}{10 \sqrt[4]{b}}-\frac{3 a \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^5}}{\sqrt[4]{b}}\right)}{10 \sqrt[4]{b}}",1,"-1/5*(b + a*x^5)^(3/4)/x^5 + (3*a*ArcTan[(b + a*x^5)^(1/4)/b^(1/4)])/(10*b^(1/4)) - (3*a*ArcTanh[(b + a*x^5)^(1/4)/b^(1/4)])/(10*b^(1/4))","A",1
993,1,75,75,2.749259,"\int \frac{2-3 x^5}{\left(1-x^2+x^5\right) \sqrt[3]{x+x^6}} \, dx","IntegrateAlgebraic[(2 - 3*x^5)/((1 - x^2 + x^5)*(x + x^6)^(1/3)),x]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^6}}\right)-\log \left(-x+\sqrt[3]{x+x^6}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{x+x^6}+\left(x+x^6\right)^{2/3}\right)","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^6}}\right)-\log \left(-x+\sqrt[3]{x+x^6}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{x+x^6}+\left(x+x^6\right)^{2/3}\right)",1,"Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^6)^(1/3))] - Log[-x + (x + x^6)^(1/3)] + Log[x^2 + x*(x + x^6)^(1/3) + (x + x^6)^(2/3)]/2","A",1
994,1,75,75,0.1220498,"\int \frac{\left(b+a x^6\right)^{3/4}}{x^7} \, dx","IntegrateAlgebraic[(b + a*x^6)^(3/4)/x^7,x]","-\frac{\left(b+a x^6\right)^{3/4}}{6 x^6}+\frac{a \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^6}}{\sqrt[4]{b}}\right)}{4 \sqrt[4]{b}}-\frac{a \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^6}}{\sqrt[4]{b}}\right)}{4 \sqrt[4]{b}}","-\frac{\left(b+a x^6\right)^{3/4}}{6 x^6}+\frac{a \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^6}}{\sqrt[4]{b}}\right)}{4 \sqrt[4]{b}}-\frac{a \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^6}}{\sqrt[4]{b}}\right)}{4 \sqrt[4]{b}}",1,"-1/6*(b + a*x^6)^(3/4)/x^6 + (a*ArcTan[(b + a*x^6)^(1/4)/b^(1/4)])/(4*b^(1/4)) - (a*ArcTanh[(b + a*x^6)^(1/4)/b^(1/4)])/(4*b^(1/4))","A",1
995,1,75,75,5.2024299,"\int \frac{2 b+a x^6}{\sqrt[4]{-b+a x^6} \left(-b-2 x^4+a x^6\right)} \, dx","IntegrateAlgebraic[(2*b + a*x^6)/((-b + a*x^6)^(1/4)*(-b - 2*x^4 + a*x^6)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x \left(-b+a x^6\right)^{3/4}}{b-a x^6}\right)}{\sqrt[4]{2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x \left(-b+a x^6\right)^{3/4}}{b-a x^6}\right)}{\sqrt[4]{2}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x \left(-b+a x^6\right)^{3/4}}{b-a x^6}\right)}{\sqrt[4]{2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x \left(-b+a x^6\right)^{3/4}}{b-a x^6}\right)}{\sqrt[4]{2}}",1,"ArcTan[(2^(1/4)*x*(-b + a*x^6)^(3/4))/(b - a*x^6)]/2^(1/4) + ArcTanh[(2^(1/4)*x*(-b + a*x^6)^(3/4))/(b - a*x^6)]/2^(1/4)","A",1
996,1,75,75,0.3638661,"\int \frac{\left(-1+x^4\right) \sqrt{1+x^4}}{1+3 x^4+x^8} \, dx","IntegrateAlgebraic[((-1 + x^4)*Sqrt[1 + x^4])/(1 + 3*x^4 + x^8),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x \sqrt{1+x^4}}{1-x^2+x^4}\right)}{2 \sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{1+x^4}}{1+x^2+x^4}\right)}{2 \sqrt{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x \sqrt{1+x^4}}{1-x^2+x^4}\right)}{2 \sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{1+x^4}}{1+x^2+x^4}\right)}{2 \sqrt{2}}",1,"-1/2*ArcTan[(Sqrt[2]*x*Sqrt[1 + x^4])/(1 - x^2 + x^4)]/Sqrt[2] - ArcTanh[(Sqrt[2]*x*Sqrt[1 + x^4])/(1 + x^2 + x^4)]/(2*Sqrt[2])","A",1
997,1,76,76,0.2577687,"\int \frac{-1+k^2 x^2}{\sqrt{(1-x) x \left(1-k^2 x\right)} \left(a+b x+a k^2 x^2\right)} \, dx","IntegrateAlgebraic[(-1 + k^2*x^2)/(Sqrt[(1 - x)*x*(1 - k^2*x)]*(a + b*x + a*k^2*x^2)),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a+b+a k^2} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{\sqrt{a} (-1+x) \left(-1+k^2 x\right)}\right)}{\sqrt{a} \sqrt{a+b+a k^2}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a+b+a k^2} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{\sqrt{a} (-1+x) \left(-1+k^2 x\right)}\right)}{\sqrt{a} \sqrt{a+b+a k^2}}",1,"(-2*ArcTan[(Sqrt[a + b + a*k^2]*Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3])/(Sqrt[a]*(-1 + x)*(-1 + k^2*x))])/(Sqrt[a]*Sqrt[a + b + a*k^2])","A",1
998,1,76,76,0.0762043,"\int \frac{1}{x \sqrt[3]{-1+x^4}} \, dx","IntegrateAlgebraic[1/(x*(-1 + x^4)^(1/3)),x]","-\frac{1}{4} \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^4}}{\sqrt{3}}\right)-\frac{1}{4} \log \left(1+\sqrt[3]{-1+x^4}\right)+\frac{1}{8} \log \left(1-\sqrt[3]{-1+x^4}+\left(-1+x^4\right)^{2/3}\right)","-\frac{1}{4} \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^4}}{\sqrt{3}}\right)-\frac{1}{4} \log \left(1+\sqrt[3]{-1+x^4}\right)+\frac{1}{8} \log \left(1-\sqrt[3]{-1+x^4}+\left(-1+x^4\right)^{2/3}\right)",1,"-1/4*(Sqrt[3]*ArcTan[1/Sqrt[3] - (2*(-1 + x^4)^(1/3))/Sqrt[3]]) - Log[1 + (-1 + x^4)^(1/3)]/4 + Log[1 - (-1 + x^4)^(1/3) + (-1 + x^4)^(2/3)]/8","A",1
999,1,76,76,3.130669,"\int \frac{x^2 \left(-2 b+a x^2\right)}{\left(-b+a x^2\right)^{3/4} \left(4 b-4 a x^2+x^4\right)} \, dx","IntegrateAlgebraic[(x^2*(-2*b + a*x^2))/((-b + a*x^2)^(3/4)*(4*b - 4*a*x^2 + x^4)),x]","-\frac{\tan ^{-1}\left(\frac{x \left(-b+a x^2\right)^{3/4}}{\sqrt{2} \left(b-a x^2\right)}\right)}{\sqrt{2}}+\frac{\tanh ^{-1}\left(\frac{x \left(-b+a x^2\right)^{3/4}}{\sqrt{2} \left(b-a x^2\right)}\right)}{\sqrt{2}}","-\frac{\tan ^{-1}\left(\frac{x \left(-b+a x^2\right)^{3/4}}{\sqrt{2} \left(b-a x^2\right)}\right)}{\sqrt{2}}+\frac{\tanh ^{-1}\left(\frac{x \left(-b+a x^2\right)^{3/4}}{\sqrt{2} \left(b-a x^2\right)}\right)}{\sqrt{2}}",1,"-(ArcTan[(x*(-b + a*x^2)^(3/4))/(Sqrt[2]*(b - a*x^2))]/Sqrt[2]) + ArcTanh[(x*(-b + a*x^2)^(3/4))/(Sqrt[2]*(b - a*x^2))]/Sqrt[2]","A",1
1000,1,76,76,0.3242105,"\int \frac{-a b x+x^3}{\sqrt{x (-a+x) (-b+x)} \left(a^2 b^2-2 a b (a+b) x+\left(a^2+4 a b+b^2-d\right) x^2-2 (a+b) x^3+x^4\right)} \, dx","IntegrateAlgebraic[(-(a*b*x) + x^3)/(Sqrt[x*(-a + x)*(-b + x)]*(a^2*b^2 - 2*a*b*(a + b)*x + (a^2 + 4*a*b + b^2 - d)*x^2 - 2*(a + b)*x^3 + x^4)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt{a b x+(-a-b) x^2+x^3}}\right)}{d^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt{a b x+(-a-b) x^2+x^3}}\right)}{d^{3/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt{a b x+(-a-b) x^2+x^3}}\right)}{d^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt{a b x+(-a-b) x^2+x^3}}\right)}{d^{3/4}}",1,"ArcTan[(d^(1/4)*x)/Sqrt[a*b*x + (-a - b)*x^2 + x^3]]/d^(3/4) - ArcTanh[(d^(1/4)*x)/Sqrt[a*b*x + (-a - b)*x^2 + x^3]]/d^(3/4)","A",1
1001,1,76,76,0.2573845,"\int \frac{\sqrt[4]{-1-x^4} \left(-1+x^4\right)}{x^6 \left(1+2 x^4\right)} \, dx","IntegrateAlgebraic[((-1 - x^4)^(1/4)*(-1 + x^4))/(x^6*(1 + 2*x^4)),x]","\frac{\left(1-14 x^4\right) \sqrt[4]{-1-x^4}}{5 x^5}+\frac{3}{2} \tan ^{-1}\left(\frac{x \left(-1-x^4\right)^{3/4}}{1+x^4}\right)-\frac{3}{2} \tanh ^{-1}\left(\frac{x \left(-1-x^4\right)^{3/4}}{1+x^4}\right)","\frac{\left(1-14 x^4\right) \sqrt[4]{-1-x^4}}{5 x^5}+\frac{3}{2} \tan ^{-1}\left(\frac{x \left(-1-x^4\right)^{3/4}}{1+x^4}\right)-\frac{3}{2} \tanh ^{-1}\left(\frac{x \left(-1-x^4\right)^{3/4}}{1+x^4}\right)",1,"((1 - 14*x^4)*(-1 - x^4)^(1/4))/(5*x^5) + (3*ArcTan[(x*(-1 - x^4)^(3/4))/(1 + x^4)])/2 - (3*ArcTanh[(x*(-1 - x^4)^(3/4))/(1 + x^4)])/2","A",1
1002,1,76,76,0.4017614,"\int \frac{a b x-x^3}{\sqrt{x (-a+x) (-b+x)} \left(a^2 b^2 d-2 a b (a+b) d x+\left(-1+a^2 d+4 a b d+b^2 d\right) x^2-2 (a+b) d x^3+d x^4\right)} \, dx","IntegrateAlgebraic[(a*b*x - x^3)/(Sqrt[x*(-a + x)*(-b + x)]*(a^2*b^2*d - 2*a*b*(a + b)*d*x + (-1 + a^2*d + 4*a*b*d + b^2*d)*x^2 - 2*(a + b)*d*x^3 + d*x^4)),x]","-\frac{\tan ^{-1}\left(\frac{x}{\sqrt[4]{d} \sqrt{a b x+(-a-b) x^2+x^3}}\right)}{\sqrt[4]{d}}+\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[4]{d} \sqrt{a b x+(-a-b) x^2+x^3}}\right)}{\sqrt[4]{d}}","-\frac{\tan ^{-1}\left(\frac{x}{\sqrt[4]{d} \sqrt{a b x+(-a-b) x^2+x^3}}\right)}{\sqrt[4]{d}}+\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[4]{d} \sqrt{a b x+(-a-b) x^2+x^3}}\right)}{\sqrt[4]{d}}",1,"-(ArcTan[x/(d^(1/4)*Sqrt[a*b*x + (-a - b)*x^2 + x^3])]/d^(1/4)) + ArcTanh[x/(d^(1/4)*Sqrt[a*b*x + (-a - b)*x^2 + x^3])]/d^(1/4)","A",1
1003,1,76,76,0.3623921,"\int \frac{-a b x+x^3}{\sqrt{x (-a+x) (-b+x)} \left(a^2 b^2 d-2 a b (a+b) d x+\left(-1+a^2 d+4 a b d+b^2 d\right) x^2-2 (a+b) d x^3+d x^4\right)} \, dx","IntegrateAlgebraic[(-(a*b*x) + x^3)/(Sqrt[x*(-a + x)*(-b + x)]*(a^2*b^2*d - 2*a*b*(a + b)*d*x + (-1 + a^2*d + 4*a*b*d + b^2*d)*x^2 - 2*(a + b)*d*x^3 + d*x^4)),x]","\frac{\tan ^{-1}\left(\frac{x}{\sqrt[4]{d} \sqrt{a b x+(-a-b) x^2+x^3}}\right)}{\sqrt[4]{d}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[4]{d} \sqrt{a b x+(-a-b) x^2+x^3}}\right)}{\sqrt[4]{d}}","\frac{\tan ^{-1}\left(\frac{x}{\sqrt[4]{d} \sqrt{a b x+(-a-b) x^2+x^3}}\right)}{\sqrt[4]{d}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[4]{d} \sqrt{a b x+(-a-b) x^2+x^3}}\right)}{\sqrt[4]{d}}",1,"ArcTan[x/(d^(1/4)*Sqrt[a*b*x + (-a - b)*x^2 + x^3])]/d^(1/4) - ArcTanh[x/(d^(1/4)*Sqrt[a*b*x + (-a - b)*x^2 + x^3])]/d^(1/4)","A",1
1004,1,76,76,0.0627082,"\int \frac{1}{x \sqrt[3]{-1+x^6}} \, dx","IntegrateAlgebraic[1/(x*(-1 + x^6)^(1/3)),x]","-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^6}}{\sqrt{3}}\right)}{2 \sqrt{3}}-\frac{1}{6} \log \left(1+\sqrt[3]{-1+x^6}\right)+\frac{1}{12} \log \left(1-\sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)","-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^6}}{\sqrt{3}}\right)}{2 \sqrt{3}}-\frac{1}{6} \log \left(1+\sqrt[3]{-1+x^6}\right)+\frac{1}{12} \log \left(1-\sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)",1,"-1/2*ArcTan[1/Sqrt[3] - (2*(-1 + x^6)^(1/3))/Sqrt[3]]/Sqrt[3] - Log[1 + (-1 + x^6)^(1/3)]/6 + Log[1 - (-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)]/12","A",1
1005,1,76,76,0.2420216,"\int \frac{\left(1+x^3\right)^{2/3} \left(2+x^3+2 x^6\right)}{x^6 \left(1+x^3+x^6\right)} \, dx","IntegrateAlgebraic[((1 + x^3)^(2/3)*(2 + x^3 + 2*x^6))/(x^6*(1 + x^3 + x^6)),x]","\frac{\left(-4+x^3\right) \left(1+x^3\right)^{2/3}}{10 x^5}-\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)}{-\text{$\#$1}+2 \text{$\#$1}^4}\&\right]","\frac{\left(-4+x^3\right) \left(1+x^3\right)^{2/3}}{10 x^5}-\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)}{-\text{$\#$1}+2 \text{$\#$1}^4}\&\right]",1,"((-4 + x^3)*(1 + x^3)^(2/3))/(10*x^5) - RootSum[1 - #1^3 + #1^6 & , (-Log[x] + Log[(1 + x^3)^(1/3) - x*#1])/(-#1 + 2*#1^4) & ]/3","A",1
1006,1,76,76,0.0017251,"\int \frac{\left(1+x^3\right)^{2/3} \left(2+x^3+2 x^6\right)}{x^6 \left(1+x^3+x^6\right)} \, dx","IntegrateAlgebraic[((1 + x^3)^(2/3)*(2 + x^3 + 2*x^6))/(x^6*(1 + x^3 + x^6)),x]","\frac{\left(-4+x^3\right) \left(1+x^3\right)^{2/3}}{10 x^5}-\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)}{-\text{$\#$1}+2 \text{$\#$1}^4}\&\right]","\frac{\left(-4+x^3\right) \left(1+x^3\right)^{2/3}}{10 x^5}-\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)}{-\text{$\#$1}+2 \text{$\#$1}^4}\&\right]",1,"((-4 + x^3)*(1 + x^3)^(2/3))/(10*x^5) - RootSum[1 - #1^3 + #1^6 & , (-Log[x] + Log[(1 + x^3)^(1/3) - x*#1])/(-#1 + 2*#1^4) & ]/3","A",1
1007,1,76,76,0.9103831,"\int \frac{\left(-1+x^4\right) \left(1-x^2+x^4\right) \sqrt{4-x^2+4 x^4}}{\left(1+x^4\right) \left(4+7 x^4+4 x^8\right)} \, dx","IntegrateAlgebraic[((-1 + x^4)*(1 - x^2 + x^4)*Sqrt[4 - x^2 + 4*x^4])/((1 + x^4)*(4 + 7*x^4 + 4*x^8)),x]","\tan ^{-1}\left(\frac{x}{\sqrt{4-x^2+4 x^4}}\right)-\frac{3}{4} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{\sqrt{4-x^2+4 x^4}}\right)+\frac{1}{4} \tanh ^{-1}\left(\frac{x}{\sqrt{4-x^2+4 x^4}}\right)","\tan ^{-1}\left(\frac{x}{\sqrt{4-x^2+4 x^4}}\right)-\frac{3}{4} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{\sqrt{4-x^2+4 x^4}}\right)+\frac{1}{4} \tanh ^{-1}\left(\frac{x}{\sqrt{4-x^2+4 x^4}}\right)",1,"ArcTan[x/Sqrt[4 - x^2 + 4*x^4]] - (3*Sqrt[3]*ArcTan[(Sqrt[3]*x)/Sqrt[4 - x^2 + 4*x^4]])/4 + ArcTanh[x/Sqrt[4 - x^2 + 4*x^4]]/4","A",1
1008,1,71,76,0.0983397,"\int \sqrt[3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}} \, dx","IntegrateAlgebraic[(1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(1/3),x]","\frac{\left(-229-6 x-21 x^2-70 x^3\right) \left(1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}\right)^{4/3}}{910 (1+x)^4 \left(1-x+x^2\right)^4}","\frac{\left(-229+681 x-3 x^2-7 x^3+210 x^4\right) \sqrt[3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}}{910 (1+x) \left(1-x+x^2\right)}",1,"((-229 - 6*x - 21*x^2 - 70*x^3)*(1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(4/3))/(910*(1 + x)^4*(1 - x + x^2)^4)","A",1
1009,1,77,77,0.8828599,"\int \frac{x (-b+x) \left(a b-2 a x+x^2\right)}{(-a+x) \sqrt{x (-a+x) (-b+x)} \left(a d+(-b-d) x+x^2\right)} \, dx","IntegrateAlgebraic[(x*(-b + x)*(a*b - 2*a*x + x^2))/((-a + x)*Sqrt[x*(-a + x)*(-b + x)]*(a*d + (-b - d)*x + x^2)),x]","-\frac{2 \sqrt{a b x+(-a-b) x^2+x^3}}{a-x}+2 \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{a b x+(-a-b) x^2+x^3}}{\sqrt{d} (a-x)}\right)","-\frac{2 \sqrt{a b x+(-a-b) x^2+x^3}}{a-x}+2 \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{a b x+(-a-b) x^2+x^3}}{\sqrt{d} (a-x)}\right)",1,"(-2*Sqrt[a*b*x + (-a - b)*x^2 + x^3])/(a - x) + 2*Sqrt[d]*ArcTanh[Sqrt[a*b*x + (-a - b)*x^2 + x^3]/(Sqrt[d]*(a - x))]","A",1
1010,1,77,77,0.26093,"\int \frac{1+k^2 x^2}{\sqrt{(1-x) x \left(1-k^2 x\right)} \left(-1+k^2 x^2\right)} \, dx","IntegrateAlgebraic[(1 + k^2*x^2)/(Sqrt[(1 - x)*x*(1 - k^2*x)]*(-1 + k^2*x^2)),x]","\frac{\tan ^{-1}\left(\frac{(-1+k) x}{\sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}\right)}{1-k}+\frac{\tan ^{-1}\left(\frac{(1+k) x}{\sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}\right)}{-1-k}","\frac{\tan ^{-1}\left(\frac{(-1+k) x}{\sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}\right)}{1-k}+\frac{\tan ^{-1}\left(\frac{(1+k) x}{\sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}\right)}{-1-k}",1,"ArcTan[((-1 + k)*x)/Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3]]/(1 - k) + ArcTan[((1 + k)*x)/Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3]]/(-1 - k)","A",1
1011,1,77,77,0.1762151,"\int \frac{1}{\sqrt[3]{1+x^3}} \, dx","IntegrateAlgebraic[(1 + x^3)^(-1/3),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(-x+\sqrt[3]{1+x^3}\right)+\frac{1}{6} \log \left(x^2+x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)","\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(-x+\sqrt[3]{1+x^3}\right)+\frac{1}{6} \log \left(x^2+x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)",1,"ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^3)^(1/3))]/Sqrt[3] - Log[-x + (1 + x^3)^(1/3)]/3 + Log[x^2 + x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/6","A",1
1012,1,89,77,0.4992887,"\int \frac{x+x^2}{\left(-1-2 x+x^2\right) \sqrt{-x+x^3}} \, dx","IntegrateAlgebraic[(x + x^2)/((-1 - 2*x + x^2)*Sqrt[-x + x^3]),x]","\frac{1}{4} \left(2-\sqrt{2}\right) \tanh ^{-1}\left(\frac{1-\sqrt{2}+\left(-1+\sqrt{2}\right) x}{\sqrt{-x+x^3}}\right)+\frac{1}{4} \left(-2-\sqrt{2}\right) \tanh ^{-1}\left(\frac{-1-\sqrt{2}+\left(1+\sqrt{2}\right) x}{\sqrt{-x+x^3}}\right)","\frac{1}{4} \left(-2-\sqrt{2}\right) \tanh ^{-1}\left(\frac{-1+x}{\left(-1+\sqrt{2}\right) \sqrt{-x+x^3}}\right)+\frac{1}{4} \left(2-\sqrt{2}\right) \tanh ^{-1}\left(\frac{-1+x}{\left(1+\sqrt{2}\right) \sqrt{-x+x^3}}\right)",1,"((2 - Sqrt[2])*ArcTanh[(1 - Sqrt[2] + (-1 + Sqrt[2])*x)/Sqrt[-x + x^3]])/4 + ((-2 - Sqrt[2])*ArcTanh[(-1 - Sqrt[2] + (1 + Sqrt[2])*x)/Sqrt[-x + x^3]])/4","A",1
1013,1,77,77,3.4558359,"\int \frac{x^2 \left(3 a b-2 (a+b) x+x^2\right)}{(x (-a+x) (-b+x))^{3/4} \left(-a b+(a+b) x-x^2+d x^3\right)} \, dx","IntegrateAlgebraic[(x^2*(3*a*b - 2*(a + b)*x + x^2))/((x*(-a + x)*(-b + x))^(3/4)*(-(a*b) + (a + b)*x - x^2 + d*x^3)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{a b x+(-a-b) x^2+x^3}}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{a b x+(-a-b) x^2+x^3}}\right)}{d^{3/4}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{a b x+(-a-b) x^2+x^3}}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{a b x+(-a-b) x^2+x^3}}\right)}{d^{3/4}}",1,"(2*ArcTan[(d^(1/4)*x)/(a*b*x + (-a - b)*x^2 + x^3)^(1/4)])/d^(3/4) - (2*ArcTanh[(d^(1/4)*x)/(a*b*x + (-a - b)*x^2 + x^3)^(1/4)])/d^(3/4)","A",1
1014,1,82,77,0.1106202,"\int \frac{1}{x \sqrt[4]{-1+x^4}} \, dx","IntegrateAlgebraic[1/(x*(-1 + x^4)^(1/4)),x]","\frac{\tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{\sqrt{-1+x^4}}{\sqrt{2}}}{\sqrt[4]{-1+x^4}}\right)}{2 \sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^4}}{1+\sqrt{-1+x^4}}\right)}{2 \sqrt{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^4}}{-1+\sqrt{-1+x^4}}\right)}{2 \sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^4}}{1+\sqrt{-1+x^4}}\right)}{2 \sqrt{2}}",1,"ArcTan[(-(1/Sqrt[2]) + Sqrt[-1 + x^4]/Sqrt[2])/(-1 + x^4)^(1/4)]/(2*Sqrt[2]) - ArcTanh[(Sqrt[2]*(-1 + x^4)^(1/4))/(1 + Sqrt[-1 + x^4])]/(2*Sqrt[2])","A",1
1015,1,77,77,0.3594156,"\int \frac{(-a+x) (-b+x) \left(-a b+x^2\right)}{\sqrt{x (-a+x) (-b+x)} \left(a^2 b^2-2 a b (a+b) x+\left(a^2+4 a b+b^2-d\right) x^2-2 (a+b) x^3+x^4\right)} \, dx","IntegrateAlgebraic[((-a + x)*(-b + x)*(-(a*b) + x^2))/(Sqrt[x*(-a + x)*(-b + x)]*(a^2*b^2 - 2*a*b*(a + b)*x + (a^2 + 4*a*b + b^2 - d)*x^2 - 2*(a + b)*x^3 + x^4)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt{a b x+(-a-b) x^2+x^3}}\right)}{\sqrt[4]{d}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt{a b x+(-a-b) x^2+x^3}}\right)}{\sqrt[4]{d}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt{a b x+(-a-b) x^2+x^3}}\right)}{\sqrt[4]{d}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt{a b x+(-a-b) x^2+x^3}}\right)}{\sqrt[4]{d}}",1,"-(ArcTan[(d^(1/4)*x)/Sqrt[a*b*x + (-a - b)*x^2 + x^3]]/d^(1/4)) - ArcTanh[(d^(1/4)*x)/Sqrt[a*b*x + (-a - b)*x^2 + x^3]]/d^(1/4)","A",1
1016,1,77,77,0.5060544,"\int \frac{x^2 \left(-b+a x^4\right)}{\left(b+a x^4\right)^{3/4}} \, dx","IntegrateAlgebraic[(x^2*(-b + a*x^4))/(b + a*x^4)^(3/4),x]","\frac{1}{4} x^3 \sqrt[4]{b+a x^4}+\frac{7 b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{8 a^{3/4}}-\frac{7 b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{8 a^{3/4}}","\frac{1}{4} x^3 \sqrt[4]{b+a x^4}+\frac{7 b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{8 a^{3/4}}-\frac{7 b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{8 a^{3/4}}",1,"(x^3*(b + a*x^4)^(1/4))/4 + (7*b*ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)])/(8*a^(3/4)) - (7*b*ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)])/(8*a^(3/4))","A",1
1017,1,77,77,0.3007222,"\int \frac{1}{\sqrt[4]{b x+a x^4}} \, dx","IntegrateAlgebraic[(b*x + a*x^4)^(-1/4),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 \sqrt[4]{a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 \sqrt[4]{a}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 \sqrt[4]{a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 \sqrt[4]{a}}",1,"(2*ArcTan[(a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/(3*a^(1/4)) + (2*ArcTanh[(a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/(3*a^(1/4))","A",1
1018,1,77,77,0.3565837,"\int \frac{(-a+x) (-b+x) \left(-a b+x^2\right)}{\sqrt{x (-a+x) (-b+x)} \left(a^2 b^2 d-2 a b (a+b) d x+\left(-1+a^2 d+4 a b d+b^2 d\right) x^2-2 (a+b) d x^3+d x^4\right)} \, dx","IntegrateAlgebraic[((-a + x)*(-b + x)*(-(a*b) + x^2))/(Sqrt[x*(-a + x)*(-b + x)]*(a^2*b^2*d - 2*a*b*(a + b)*d*x + (-1 + a^2*d + 4*a*b*d + b^2*d)*x^2 - 2*(a + b)*d*x^3 + d*x^4)),x]","-\frac{\tan ^{-1}\left(\frac{x}{\sqrt[4]{d} \sqrt{a b x+(-a-b) x^2+x^3}}\right)}{d^{3/4}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[4]{d} \sqrt{a b x+(-a-b) x^2+x^3}}\right)}{d^{3/4}}","-\frac{\tan ^{-1}\left(\frac{x}{\sqrt[4]{d} \sqrt{a b x+(-a-b) x^2+x^3}}\right)}{d^{3/4}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[4]{d} \sqrt{a b x+(-a-b) x^2+x^3}}\right)}{d^{3/4}}",1,"-(ArcTan[x/(d^(1/4)*Sqrt[a*b*x + (-a - b)*x^2 + x^3])]/d^(3/4)) - ArcTanh[x/(d^(1/4)*Sqrt[a*b*x + (-a - b)*x^2 + x^3])]/d^(3/4)","A",1
1019,1,82,77,0.086584,"\int \frac{1}{x \sqrt[4]{-1+x^6}} \, dx","IntegrateAlgebraic[1/(x*(-1 + x^6)^(1/4)),x]","\frac{\tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{\sqrt{-1+x^6}}{\sqrt{2}}}{\sqrt[4]{-1+x^6}}\right)}{3 \sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^6}}{1+\sqrt{-1+x^6}}\right)}{3 \sqrt{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^6}}{-1+\sqrt{-1+x^6}}\right)}{3 \sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^6}}{1+\sqrt{-1+x^6}}\right)}{3 \sqrt{2}}",1,"ArcTan[(-(1/Sqrt[2]) + Sqrt[-1 + x^6]/Sqrt[2])/(-1 + x^6)^(1/4)]/(3*Sqrt[2]) - ArcTanh[(Sqrt[2]*(-1 + x^6)^(1/4))/(1 + Sqrt[-1 + x^6])]/(3*Sqrt[2])","A",1
1020,1,77,77,0.5814031,"\int \frac{1}{\sqrt[3]{-b x+a x^3} \left(d+c x^6\right)} \, dx","IntegrateAlgebraic[1/((-(b*x) + a*x^3)^(1/3)*(d + c*x^6)),x]","-\frac{\text{RootSum}\left[b^3 c+a^3 d-3 a^2 d \text{$\#$1}^3+3 a d \text{$\#$1}^6-d \text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x+a x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 d}","-\frac{\text{RootSum}\left[b^3 c+a^3 d-3 a^2 d \text{$\#$1}^3+3 a d \text{$\#$1}^6-d \text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x+a x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 d}",1,"-1/6*RootSum[b^3*c + a^3*d - 3*a^2*d*#1^3 + 3*a*d*#1^6 - d*#1^9 & , (-Log[x] + Log[(-(b*x) + a*x^3)^(1/3) - x*#1])/#1 & ]/d","A",1
1021,1,77,77,0.0029895,"\int \frac{1}{\sqrt[3]{-b x+a x^3} \left(d+c x^6\right)} \, dx","IntegrateAlgebraic[1/((-(b*x) + a*x^3)^(1/3)*(d + c*x^6)),x]","-\frac{\text{RootSum}\left[b^3 c+a^3 d-3 a^2 d \text{$\#$1}^3+3 a d \text{$\#$1}^6-d \text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x+a x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 d}","-\frac{\text{RootSum}\left[b^3 c+a^3 d-3 a^2 d \text{$\#$1}^3+3 a d \text{$\#$1}^6-d \text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x+a x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 d}",1,"-1/6*RootSum[b^3*c + a^3*d - 3*a^2*d*#1^3 + 3*a*d*#1^6 - d*#1^9 & , (-Log[x] + Log[(-(b*x) + a*x^3)^(1/3) - x*#1])/#1 & ]/d","A",1
1022,1,77,77,0.4131581,"\int \left(-1+x^2\right) \left(-1+x \sqrt{-1+3 x^2-x^4}\right) \, dx","IntegrateAlgebraic[(-1 + x^2)*(-1 + x*Sqrt[-1 + 3*x^2 - x^4]),x]","x-\frac{x^3}{3}+\frac{1}{48} \sqrt{-1+3 x^2-x^4} \left(-1-18 x^2+8 x^4\right)+\frac{5}{32} i \log \left(3 i-2 i x^2+2 \sqrt{-1+3 x^2-x^4}\right)","x-\frac{x^3}{3}+\frac{1}{48} \sqrt{-1+3 x^2-x^4} \left(-1-18 x^2+8 x^4\right)+\frac{5}{32} i \log \left(3 i-2 i x^2+2 \sqrt{-1+3 x^2-x^4}\right)",1,"x - x^3/3 + (Sqrt[-1 + 3*x^2 - x^4]*(-1 - 18*x^2 + 8*x^4))/48 + ((5*I)/32)*Log[3*I - (2*I)*x^2 + 2*Sqrt[-1 + 3*x^2 - x^4]]","A",1
1023,1,77,77,0.1896814,"\int \frac{\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1+x^2\right) \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((1 + x^2)*Sqrt[x + Sqrt[1 + x^2]]),x]","\text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+3 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]","\text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+3 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]",1,"RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(-1 + 3*#1^2 - 3*#1^4 + #1^6) & ]","A",1
1024,1,77,77,0.0029582,"\int \frac{\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1+x^2\right) \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((1 + x^2)*Sqrt[x + Sqrt[1 + x^2]]),x]","\text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+3 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]","\text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+3 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]",1,"RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(-1 + 3*#1^2 - 3*#1^4 + #1^6) & ]","A",1
1025,1,77,77,0.2821514,"\int \frac{\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{\sqrt{b+a^2 x^2}} \, dx","IntegrateAlgebraic[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/Sqrt[b + a^2*x^2],x]","\frac{4 \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{a}-\frac{4 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{\sqrt{c}}\right)}{a}","\frac{4 \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{a}-\frac{4 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{\sqrt{c}}\right)}{a}",1,"(4*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]])/a - (4*Sqrt[c]*ArcTanh[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/Sqrt[c]])/a","A",1
1026,1,78,78,0.3704712,"\int \frac{\sqrt{-1+x^2}}{(-i+x)^2} \, dx","IntegrateAlgebraic[Sqrt[-1 + x^2]/(-I + x)^2,x]","\frac{\sqrt{-1+x^2}}{i-x}-\sqrt{2} \tanh ^{-1}\left(\frac{1}{\sqrt{2}}+\frac{i x}{\sqrt{2}}-\frac{i \sqrt{-1+x^2}}{\sqrt{2}}\right)-\log \left(-x+\sqrt{-1+x^2}\right)","\frac{\sqrt{-1+x^2}}{i-x}-\sqrt{2} \tanh ^{-1}\left(\frac{1}{\sqrt{2}}+\frac{i x}{\sqrt{2}}-\frac{i \sqrt{-1+x^2}}{\sqrt{2}}\right)-\log \left(-x+\sqrt{-1+x^2}\right)",1,"Sqrt[-1 + x^2]/(I - x) - Sqrt[2]*ArcTanh[1/Sqrt[2] + (I*x)/Sqrt[2] - (I*Sqrt[-1 + x^2])/Sqrt[2]] - Log[-x + Sqrt[-1 + x^2]]","A",1
1027,1,78,78,0.1570453,"\int \frac{1}{x^3 \left(b+a x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x^3*(b + a*x^2)^(3/4)),x]","-\frac{\sqrt[4]{b+a x^2}}{2 b x^2}+\frac{3 a \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^2}}{\sqrt[4]{b}}\right)}{4 b^{7/4}}+\frac{3 a \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^2}}{\sqrt[4]{b}}\right)}{4 b^{7/4}}","-\frac{\sqrt[4]{b+a x^2}}{2 b x^2}+\frac{3 a \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^2}}{\sqrt[4]{b}}\right)}{4 b^{7/4}}+\frac{3 a \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^2}}{\sqrt[4]{b}}\right)}{4 b^{7/4}}",1,"-1/2*(b + a*x^2)^(1/4)/(b*x^2) + (3*a*ArcTan[(b + a*x^2)^(1/4)/b^(1/4)])/(4*b^(7/4)) + (3*a*ArcTanh[(b + a*x^2)^(1/4)/b^(1/4)])/(4*b^(7/4))","A",1
1028,1,138,78,5.0834807,"\int \frac{\left(-1-2 x+x^2+3 x^3\right)^4}{\sqrt[4]{-1+3 x-3 x^2+x^3}} \, dx","IntegrateAlgebraic[(-1 - 2*x + x^2 + 3*x^3)^4/(-1 + 3*x - 3*x^2 + x^3)^(1/4),x]","\frac{4 \left(1179090487575 \sqrt[4]{-1+x}+8489451510540 (-1+x)^{5/4}+68911288496050 (-1+x)^{9/4}+363522667246200 (-1+x)^{13/4}+1193308931689875 (-1+x)^{17/4}+2427578872418700 (-1+x)^{21/4}+3201560819102646 (-1+x)^{25/4}+2835184064813100 (-1+x)^{29/4}+1709824127042850 (-1+x)^{33/4}+695599653048300 (-1+x)^{37/4}+183247916751900 (-1+x)^{41/4}+28298171701800 (-1+x)^{45/4}+1949108765175 (-1+x)^{49/4}\right) \left((-1+x)^3\right)^{3/4}}{1179090487575 (-1+x)^{9/4}}","\frac{4 \left((-1+x)^3\right)^{3/4} \left(1308401597431+32327777464 x-1158885626660 x^2-2834315032620 x^3-2008108342110 x^4+4070651298324 x^5+9260757242646 x^6+3131067556500 x^7-8805988591725 x^8-9283999210200 x^9+609206533650 x^{10}+4908866519700 x^{11}+1949108765175 x^{12}\right)}{1179090487575 (-1+x)^2}",1,"(4*(1179090487575*(-1 + x)^(1/4) + 8489451510540*(-1 + x)^(5/4) + 68911288496050*(-1 + x)^(9/4) + 363522667246200*(-1 + x)^(13/4) + 1193308931689875*(-1 + x)^(17/4) + 2427578872418700*(-1 + x)^(21/4) + 3201560819102646*(-1 + x)^(25/4) + 2835184064813100*(-1 + x)^(29/4) + 1709824127042850*(-1 + x)^(33/4) + 695599653048300*(-1 + x)^(37/4) + 183247916751900*(-1 + x)^(41/4) + 28298171701800*(-1 + x)^(45/4) + 1949108765175*(-1 + x)^(49/4))*((-1 + x)^3)^(3/4))/(1179090487575*(-1 + x)^(9/4))","A",1
1029,1,78,78,0.0632397,"\int \frac{1}{x^4 \left(b+a x^3\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x^4*(b + a*x^3)^(3/4)),x]","-\frac{\sqrt[4]{b+a x^3}}{3 b x^3}+\frac{a \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^3}}{\sqrt[4]{b}}\right)}{2 b^{7/4}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^3}}{\sqrt[4]{b}}\right)}{2 b^{7/4}}","-\frac{\sqrt[4]{b+a x^3}}{3 b x^3}+\frac{a \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^3}}{\sqrt[4]{b}}\right)}{2 b^{7/4}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^3}}{\sqrt[4]{b}}\right)}{2 b^{7/4}}",1,"-1/3*(b + a*x^3)^(1/4)/(b*x^3) + (a*ArcTan[(b + a*x^3)^(1/4)/b^(1/4)])/(2*b^(7/4)) + (a*ArcTanh[(b + a*x^3)^(1/4)/b^(1/4)])/(2*b^(7/4))","A",1
1030,1,78,78,5.1723677,"\int \frac{x}{\sqrt{-71-96 x+10 x^2+x^4}} \, dx","IntegrateAlgebraic[x/Sqrt[-71 - 96*x + 10*x^2 + x^4],x]","-\frac{1}{8} \log \left(-10001-3124 x^2+1408 x^3-54 x^4+128 x^5-20 x^6-x^8+\sqrt{-71-96 x+10 x^2+x^4} \left(781-528 x+27 x^2-80 x^3+15 x^4+x^6\right)\right)","-\frac{1}{8} \log \left(-10001-3124 x^2+1408 x^3-54 x^4+128 x^5-20 x^6-x^8+\sqrt{-71-96 x+10 x^2+x^4} \left(781-528 x+27 x^2-80 x^3+15 x^4+x^6\right)\right)",1,"-1/8*Log[-10001 - 3124*x^2 + 1408*x^3 - 54*x^4 + 128*x^5 - 20*x^6 - x^8 + Sqrt[-71 - 96*x + 10*x^2 + x^4]*(781 - 528*x + 27*x^2 - 80*x^3 + 15*x^4 + x^6)]","A",1
1031,1,78,78,0.5245457,"\int \frac{(8+3 x) \sqrt[4]{-2-x+2 x^4}}{x^2 \left(2+x+x^4\right)} \, dx","IntegrateAlgebraic[((8 + 3*x)*(-2 - x + 2*x^4)^(1/4))/(x^2*(2 + x + x^4)),x]","-\frac{4 \sqrt[4]{-2-x+2 x^4}}{x}-2 \sqrt[4]{3} \tan ^{-1}\left(\frac{\sqrt[4]{3} x}{\sqrt[4]{-2-x+2 x^4}}\right)+2 \sqrt[4]{3} \tanh ^{-1}\left(\frac{\sqrt[4]{3} x}{\sqrt[4]{-2-x+2 x^4}}\right)","-\frac{4 \sqrt[4]{-2-x+2 x^4}}{x}-2 \sqrt[4]{3} \tan ^{-1}\left(\frac{\sqrt[4]{3} x}{\sqrt[4]{-2-x+2 x^4}}\right)+2 \sqrt[4]{3} \tanh ^{-1}\left(\frac{\sqrt[4]{3} x}{\sqrt[4]{-2-x+2 x^4}}\right)",1,"(-4*(-2 - x + 2*x^4)^(1/4))/x - 2*3^(1/4)*ArcTan[(3^(1/4)*x)/(-2 - x + 2*x^4)^(1/4)] + 2*3^(1/4)*ArcTanh[(3^(1/4)*x)/(-2 - x + 2*x^4)^(1/4)]","A",1
1032,1,78,78,0.0665788,"\int \frac{1}{x^5 \left(b+a x^4\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x^5*(b + a*x^4)^(3/4)),x]","-\frac{\sqrt[4]{b+a x^4}}{4 b x^4}+\frac{3 a \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^4}}{\sqrt[4]{b}}\right)}{8 b^{7/4}}+\frac{3 a \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^4}}{\sqrt[4]{b}}\right)}{8 b^{7/4}}","-\frac{\sqrt[4]{b+a x^4}}{4 b x^4}+\frac{3 a \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^4}}{\sqrt[4]{b}}\right)}{8 b^{7/4}}+\frac{3 a \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^4}}{\sqrt[4]{b}}\right)}{8 b^{7/4}}",1,"-1/4*(b + a*x^4)^(1/4)/(b*x^4) + (3*a*ArcTan[(b + a*x^4)^(1/4)/b^(1/4)])/(8*b^(7/4)) + (3*a*ArcTanh[(b + a*x^4)^(1/4)/b^(1/4)])/(8*b^(7/4))","A",1
1033,1,78,78,0.0662001,"\int \frac{1}{x^6 \left(b+a x^5\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x^6*(b + a*x^5)^(3/4)),x]","-\frac{\sqrt[4]{b+a x^5}}{5 b x^5}+\frac{3 a \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^5}}{\sqrt[4]{b}}\right)}{10 b^{7/4}}+\frac{3 a \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^5}}{\sqrt[4]{b}}\right)}{10 b^{7/4}}","-\frac{\sqrt[4]{b+a x^5}}{5 b x^5}+\frac{3 a \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^5}}{\sqrt[4]{b}}\right)}{10 b^{7/4}}+\frac{3 a \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^5}}{\sqrt[4]{b}}\right)}{10 b^{7/4}}",1,"-1/5*(b + a*x^5)^(1/4)/(b*x^5) + (3*a*ArcTan[(b + a*x^5)^(1/4)/b^(1/4)])/(10*b^(7/4)) + (3*a*ArcTanh[(b + a*x^5)^(1/4)/b^(1/4)])/(10*b^(7/4))","A",1
1034,1,78,78,0.2103319,"\int \frac{\left(-1+x^3\right)^{2/3} \left(1-2 x^3+x^6\right)}{x^6 \left(1-x^3+x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(1 - 2*x^3 + x^6))/(x^6*(1 - x^3 + x^6)),x]","\frac{\left(-1+x^3\right)^{2/3} \left(-2+7 x^3\right)}{10 x^5}-\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right)}{-\text{$\#$1}+2 \text{$\#$1}^4}\&\right]","\frac{\left(-1+x^3\right)^{2/3} \left(-2+7 x^3\right)}{10 x^5}-\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right)}{-\text{$\#$1}+2 \text{$\#$1}^4}\&\right]",1,"((-1 + x^3)^(2/3)*(-2 + 7*x^3))/(10*x^5) - RootSum[1 - #1^3 + #1^6 & , (-Log[x] + Log[(-1 + x^3)^(1/3) - x*#1])/(-#1 + 2*#1^4) & ]/3","A",1
1035,1,78,78,0.0012426,"\int \frac{\left(-1+x^3\right)^{2/3} \left(1-2 x^3+x^6\right)}{x^6 \left(1-x^3+x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(1 - 2*x^3 + x^6))/(x^6*(1 - x^3 + x^6)),x]","\frac{\left(-1+x^3\right)^{2/3} \left(-2+7 x^3\right)}{10 x^5}-\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right)}{-\text{$\#$1}+2 \text{$\#$1}^4}\&\right]","\frac{\left(-1+x^3\right)^{2/3} \left(-2+7 x^3\right)}{10 x^5}-\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right)}{-\text{$\#$1}+2 \text{$\#$1}^4}\&\right]",1,"((-1 + x^3)^(2/3)*(-2 + 7*x^3))/(10*x^5) - RootSum[1 - #1^3 + #1^6 & , (-Log[x] + Log[(-1 + x^3)^(1/3) - x*#1])/(-#1 + 2*#1^4) & ]/3","A",1
1036,1,78,78,0.0649694,"\int \frac{1}{x^7 \left(b+a x^6\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x^7*(b + a*x^6)^(3/4)),x]","-\frac{\sqrt[4]{b+a x^6}}{6 b x^6}+\frac{a \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^6}}{\sqrt[4]{b}}\right)}{4 b^{7/4}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^6}}{\sqrt[4]{b}}\right)}{4 b^{7/4}}","-\frac{\sqrt[4]{b+a x^6}}{6 b x^6}+\frac{a \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^6}}{\sqrt[4]{b}}\right)}{4 b^{7/4}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^6}}{\sqrt[4]{b}}\right)}{4 b^{7/4}}",1,"-1/6*(b + a*x^6)^(1/4)/(b*x^6) + (a*ArcTan[(b + a*x^6)^(1/4)/b^(1/4)])/(4*b^(7/4)) + (a*ArcTanh[(b + a*x^6)^(1/4)/b^(1/4)])/(4*b^(7/4))","A",1
1037,1,78,78,0.2390204,"\int \frac{\sqrt[4]{-1+x^4} \left(-1+x^8\right)}{x^6 \left(1+x^8\right)} \, dx","IntegrateAlgebraic[((-1 + x^4)^(1/4)*(-1 + x^8))/(x^6*(1 + x^8)),x]","\frac{\left(1-x^4\right) \sqrt[4]{-1+x^4}}{5 x^5}+\frac{1}{4} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^4}\&\right]","\frac{\left(1-x^4\right) \sqrt[4]{-1+x^4}}{5 x^5}+\frac{1}{4} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^4}\&\right]",1,"((1 - x^4)*(-1 + x^4)^(1/4))/(5*x^5) + RootSum[2 - 2*#1^4 + #1^8 & , (-(Log[x]*#1) + Log[(-1 + x^4)^(1/4) - x*#1]*#1)/(-1 + #1^4) & ]/4","A",1
1038,1,78,78,0.2426435,"\int \frac{-1+x^8}{\sqrt[4]{1+x^4} \left(1+x^8\right)} \, dx","IntegrateAlgebraic[(-1 + x^8)/((1 + x^4)^(1/4)*(1 + x^8)),x]","\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)+\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)+\frac{1}{4} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{1+x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)+\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)+\frac{1}{4} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{1+x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"ArcTan[x/(1 + x^4)^(1/4)]/2 + ArcTanh[x/(1 + x^4)^(1/4)]/2 + RootSum[2 - 2*#1^4 + #1^8 & , (-Log[x] + Log[(1 + x^4)^(1/4) - x*#1])/#1 & ]/4","A",1
1039,1,78,78,0.0008843,"\int \frac{-1+x^8}{\sqrt[4]{1+x^4} \left(1+x^8\right)} \, dx","IntegrateAlgebraic[(-1 + x^8)/((1 + x^4)^(1/4)*(1 + x^8)),x]","\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)+\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)+\frac{1}{4} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{1+x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)+\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)+\frac{1}{4} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{1+x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"ArcTan[x/(1 + x^4)^(1/4)]/2 + ArcTanh[x/(1 + x^4)^(1/4)]/2 + RootSum[2 - 2*#1^4 + #1^8 & , (-Log[x] + Log[(1 + x^4)^(1/4) - x*#1])/#1 & ]/4","A",1
1040,1,78,78,0.253105,"\int \frac{\sqrt[4]{-1+x^4} \left(1-x^4+x^8\right)}{x^6 \left(1+2 x^8\right)} \, dx","IntegrateAlgebraic[((-1 + x^4)^(1/4)*(1 - x^4 + x^8))/(x^6*(1 + 2*x^8)),x]","\frac{\sqrt[4]{-1+x^4} \left(-1+6 x^4\right)}{5 x^5}-\frac{3}{8} \text{RootSum}\left[3-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right)}{-\text{$\#$1}^3+\text{$\#$1}^7}\&\right]","\frac{\sqrt[4]{-1+x^4} \left(-1+6 x^4\right)}{5 x^5}-\frac{3}{8} \text{RootSum}\left[3-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right)}{-\text{$\#$1}^3+\text{$\#$1}^7}\&\right]",1,"((-1 + x^4)^(1/4)*(-1 + 6*x^4))/(5*x^5) - (3*RootSum[3 - 2*#1^4 + #1^8 & , (-Log[x] + Log[(-1 + x^4)^(1/4) - x*#1])/(-#1^3 + #1^7) & ])/8","A",1
1041,1,78,78,0.0009649,"\int \frac{\sqrt[4]{-1+x^4} \left(1-x^4+x^8\right)}{x^6 \left(1+2 x^8\right)} \, dx","IntegrateAlgebraic[((-1 + x^4)^(1/4)*(1 - x^4 + x^8))/(x^6*(1 + 2*x^8)),x]","\frac{\sqrt[4]{-1+x^4} \left(-1+6 x^4\right)}{5 x^5}-\frac{3}{8} \text{RootSum}\left[3-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right)}{-\text{$\#$1}^3+\text{$\#$1}^7}\&\right]","\frac{\sqrt[4]{-1+x^4} \left(-1+6 x^4\right)}{5 x^5}-\frac{3}{8} \text{RootSum}\left[3-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right)}{-\text{$\#$1}^3+\text{$\#$1}^7}\&\right]",1,"((-1 + x^4)^(1/4)*(-1 + 6*x^4))/(5*x^5) - (3*RootSum[3 - 2*#1^4 + #1^8 & , (-Log[x] + Log[(-1 + x^4)^(1/4) - x*#1])/(-#1^3 + #1^7) & ])/8","A",1
1042,1,81,78,0.1594571,"\int \frac{\left(1+2 x^4\right) \sqrt{1+2 x^8}}{x} \, dx","IntegrateAlgebraic[((1 + 2*x^4)*Sqrt[1 + 2*x^8])/x,x]","\frac{1}{4} \left(1+x^4\right) \sqrt{1+2 x^8}+\frac{1}{2} \tanh ^{-1}\left(\sqrt{2} x^4-\sqrt{1+2 x^8}\right)-\frac{\log \left(-\sqrt{2} x^4+\sqrt{1+2 x^8}\right)}{4 \sqrt{2}}","\frac{1}{4} \left(1+x^4\right) \sqrt{1+2 x^8}-\frac{1}{2} \tanh ^{-1}\left(\sqrt{2} x^4+\sqrt{1+2 x^8}\right)+\frac{\log \left(\sqrt{2} x^4+\sqrt{1+2 x^8}\right)}{4 \sqrt{2}}",1,"((1 + x^4)*Sqrt[1 + 2*x^8])/4 + ArcTanh[Sqrt[2]*x^4 - Sqrt[1 + 2*x^8]]/2 - Log[-(Sqrt[2]*x^4) + Sqrt[1 + 2*x^8]]/(4*Sqrt[2])","A",1
1043,1,65,78,0.1143741,"\int \sqrt{x+\sqrt{1+x}} \, dx","IntegrateAlgebraic[Sqrt[x + Sqrt[1 + x]],x]","\frac{1}{12} \sqrt{x+\sqrt{1+x}} \left(-11+2 \sqrt{1+x}+8 (1+x)\right)-\frac{5}{8} \log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}\right)","\frac{1}{6} \sqrt{1+x} \sqrt{x+\sqrt{1+x}}+\frac{1}{12} (-3+8 x) \sqrt{x+\sqrt{1+x}}-\frac{5}{8} \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}\right)",1,"(Sqrt[x + Sqrt[1 + x]]*(-11 + 2*Sqrt[1 + x] + 8*(1 + x)))/12 - (5*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]])/8","A",1
1044,1,79,79,0.0702849,"\int \frac{2 b+a x^2}{x \left(b^2+a^2 x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[(2*b + a*x^2)/(x*(b^2 + a^2*x^2)^(3/4)),x]","\frac{2 \sqrt[4]{b^2+a^2 x^2}}{a}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{b^2+a^2 x^2}}{\sqrt{b}}\right)}{\sqrt{b}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{b^2+a^2 x^2}}{\sqrt{b}}\right)}{\sqrt{b}}","\frac{2 \sqrt[4]{b^2+a^2 x^2}}{a}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{b^2+a^2 x^2}}{\sqrt{b}}\right)}{\sqrt{b}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{b^2+a^2 x^2}}{\sqrt{b}}\right)}{\sqrt{b}}",1,"(2*(b^2 + a^2*x^2)^(1/4))/a - (2*ArcTan[(b^2 + a^2*x^2)^(1/4)/Sqrt[b]])/Sqrt[b] - (2*ArcTanh[(b^2 + a^2*x^2)^(1/4)/Sqrt[b]])/Sqrt[b]","A",1
1045,1,79,79,0.2896815,"\int \frac{-a b x+x^3}{(-a+x) (-b+x) \sqrt{x (-a+x) (-b+x)} \left(a b d-(1+a d+b d) x+d x^2\right)} \, dx","IntegrateAlgebraic[(-(a*b*x) + x^3)/((-a + x)*(-b + x)*Sqrt[x*(-a + x)*(-b + x)]*(a*b*d - (1 + a*d + b*d)*x + d*x^2)),x]","\frac{2 \sqrt{a b x-a x^2-b x^2+x^3}}{(-a+x) (-b+x)}-2 \sqrt{d} \tanh ^{-1}\left(\frac{x}{\sqrt{d} \sqrt{a b x+(-a-b) x^2+x^3}}\right)","\frac{2 \sqrt{a b x-a x^2-b x^2+x^3}}{(-a+x) (-b+x)}-2 \sqrt{d} \tanh ^{-1}\left(\frac{x}{\sqrt{d} \sqrt{a b x+(-a-b) x^2+x^3}}\right)",1,"(2*Sqrt[a*b*x - a*x^2 - b*x^2 + x^3])/((-a + x)*(-b + x)) - 2*Sqrt[d]*ArcTanh[x/(Sqrt[d]*Sqrt[a*b*x + (-a - b)*x^2 + x^3])]","A",1
1046,1,79,79,0.2629556,"\int \frac{-1+2 (-1+k) x+k x^2}{\sqrt[4]{(1-x) x (1-k x)} \left(-1+(3+d) x-(3+d k) x^2+x^3\right)} \, dx","IntegrateAlgebraic[(-1 + 2*(-1 + k)*x + k*x^2)/(((1 - x)*x*(1 - k*x))^(1/4)*(-1 + (3 + d)*x - (3 + d*k)*x^2 + x^3)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{x+(-1-k) x^2+k x^3}}{-1+x}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{x+(-1-k) x^2+k x^3}}{-1+x}\right)}{d^{3/4}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{x+(-1-k) x^2+k x^3}}{-1+x}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{x+(-1-k) x^2+k x^3}}{-1+x}\right)}{d^{3/4}}",1,"(2*ArcTan[(d^(1/4)*(x + (-1 - k)*x^2 + k*x^3)^(1/4))/(-1 + x)])/d^(3/4) - (2*ArcTanh[(d^(1/4)*(x + (-1 - k)*x^2 + k*x^3)^(1/4))/(-1 + x)])/d^(3/4)","A",1
1047,1,79,79,0.2904428,"\int \frac{1+x^4}{\sqrt{-x+x^3} \left(-1+x^4\right)} \, dx","IntegrateAlgebraic[(1 + x^4)/(Sqrt[-x + x^3]*(-1 + x^4)),x]","\frac{\sqrt{-x+x^3}}{1-x^2}-\frac{1}{4} \tan ^{-1}\left(\frac{2 \sqrt{-x+x^3}}{-1-2 x+x^2}\right)-\frac{1}{4} \tanh ^{-1}\left(\frac{-\frac{1}{2}+x+\frac{x^2}{2}}{\sqrt{-x+x^3}}\right)","\frac{\sqrt{-x+x^3}}{1-x^2}-\frac{1}{4} \tan ^{-1}\left(\frac{2 \sqrt{-x+x^3}}{-1-2 x+x^2}\right)-\frac{1}{4} \tanh ^{-1}\left(\frac{-\frac{1}{2}+x+\frac{x^2}{2}}{\sqrt{-x+x^3}}\right)",1,"Sqrt[-x + x^3]/(1 - x^2) - ArcTan[(2*Sqrt[-x + x^3])/(-1 - 2*x + x^2)]/4 - ArcTanh[(-1/2 + x + x^2/2)/Sqrt[-x + x^3]]/4","A",1
1048,1,79,79,0.5993421,"\int \frac{\left(2+x^2\right) \sqrt{4-5 x^2+x^4}}{x^2 \left(-2+2 x+x^2\right)} \, dx","IntegrateAlgebraic[((2 + x^2)*Sqrt[4 - 5*x^2 + x^4])/(x^2*(-2 + 2*x + x^2)),x]","\frac{\sqrt{4-5 x^2+x^4}}{x}-4 \tanh ^{-1}\left(\frac{-2+x+x^2}{\sqrt{4-5 x^2+x^4}}\right)+2 \sqrt{3} \tanh ^{-1}\left(\frac{\sqrt{4-5 x^2+x^4}}{\sqrt{3} \left(-2+x+x^2\right)}\right)","\frac{\sqrt{4-5 x^2+x^4}}{x}-4 \tanh ^{-1}\left(\frac{-2+x+x^2}{\sqrt{4-5 x^2+x^4}}\right)+2 \sqrt{3} \tanh ^{-1}\left(\frac{\sqrt{4-5 x^2+x^4}}{\sqrt{3} \left(-2+x+x^2\right)}\right)",1,"Sqrt[4 - 5*x^2 + x^4]/x - 4*ArcTanh[(-2 + x + x^2)/Sqrt[4 - 5*x^2 + x^4]] + 2*Sqrt[3]*ArcTanh[Sqrt[4 - 5*x^2 + x^4]/(Sqrt[3]*(-2 + x + x^2))]","A",1
1049,1,79,79,0.2530354,"\int \frac{x^2}{\left(-1+x^4\right) \sqrt[4]{x^2+x^4}} \, dx","IntegrateAlgebraic[x^2/((-1 + x^4)*(x^2 + x^4)^(1/4)),x]","\frac{\left(x^2+x^4\right)^{3/4}}{x \left(1+x^2\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2 \sqrt[4]{2}}","\frac{\left(x^2+x^4\right)^{3/4}}{x \left(1+x^2\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2 \sqrt[4]{2}}",1,"(x^2 + x^4)^(3/4)/(x*(1 + x^2)) - ArcTan[(2^(1/4)*x)/(x^2 + x^4)^(1/4)]/(2*2^(1/4)) - ArcTanh[(2^(1/4)*x)/(x^2 + x^4)^(1/4)]/(2*2^(1/4))","A",1
1050,1,79,79,0.3572845,"\int \frac{\left(-2+x^6\right) \sqrt{-1+x^6}}{x^4 \left(2+x^6\right)} \, dx","IntegrateAlgebraic[((-2 + x^6)*Sqrt[-1 + x^6])/(x^4*(2 + x^6)),x]","\frac{\sqrt{-1+x^6}}{3 x^3}+\sqrt{\frac{2}{3}} \tanh ^{-1}\left(\sqrt{\frac{2}{3}}+\frac{x^6}{\sqrt{6}}+\frac{x^3 \sqrt{-1+x^6}}{\sqrt{6}}\right)+\frac{1}{3} \log \left(x^3+\sqrt{-1+x^6}\right)","\frac{\sqrt{-1+x^6}}{3 x^3}+\sqrt{\frac{2}{3}} \tanh ^{-1}\left(\sqrt{\frac{2}{3}}+\frac{x^6}{\sqrt{6}}+\frac{x^3 \sqrt{-1+x^6}}{\sqrt{6}}\right)+\frac{1}{3} \log \left(x^3+\sqrt{-1+x^6}\right)",1,"Sqrt[-1 + x^6]/(3*x^3) + Sqrt[2/3]*ArcTanh[Sqrt[2/3] + x^6/Sqrt[6] + (x^3*Sqrt[-1 + x^6])/Sqrt[6]] + Log[x^3 + Sqrt[-1 + x^6]]/3","A",1
1051,1,79,79,3.1261979,"\int \frac{\left(-4+x^6\right) \left(2-x^4+x^6\right)^{5/2}}{x^7 \left(2+x^6\right)^2} \, dx","IntegrateAlgebraic[((-4 + x^6)*(2 - x^4 + x^6)^(5/2))/(x^7*(2 + x^6)^2),x]","\frac{\sqrt{2-x^4+x^6} \left(8-28 x^4+8 x^6-3 x^8-14 x^{10}+2 x^{12}\right)}{6 x^6 \left(2+x^6\right)}-\frac{5}{2} \tan ^{-1}\left(\frac{x^2}{\sqrt{2-x^4+x^6}}\right)","\frac{\sqrt{2-x^4+x^6} \left(8-28 x^4+8 x^6-3 x^8-14 x^{10}+2 x^{12}\right)}{6 x^6 \left(2+x^6\right)}-\frac{5}{2} \tan ^{-1}\left(\frac{x^2}{\sqrt{2-x^4+x^6}}\right)",1,"(Sqrt[2 - x^4 + x^6]*(8 - 28*x^4 + 8*x^6 - 3*x^8 - 14*x^10 + 2*x^12))/(6*x^6*(2 + x^6)) - (5*ArcTan[x^2/Sqrt[2 - x^4 + x^6]])/2","A",1
1052,1,79,79,0.1960642,"\int \frac{\left(-1+x^3\right)^{2/3} \left(2+x^3\right)}{x^6 \left(2+x^3+2 x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(2 + x^3))/(x^6*(2 + x^3 + 2*x^6)),x]","\frac{\left(-1+x^3\right)^{5/3}}{5 x^5}-\frac{2}{3} \text{RootSum}\left[5-5 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-5+4 \text{$\#$1}^3}\&\right]","\frac{\left(-1+x^3\right)^{5/3}}{5 x^5}-\frac{2}{3} \text{RootSum}\left[5-5 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-5+4 \text{$\#$1}^3}\&\right]",1,"(-1 + x^3)^(5/3)/(5*x^5) - (2*RootSum[5 - 5*#1^3 + 2*#1^6 & , (-(Log[x]*#1^2) + Log[(-1 + x^3)^(1/3) - x*#1]*#1^2)/(-5 + 4*#1^3) & ])/3","A",1
1053,1,79,79,0.0019278,"\int \frac{\left(-1+x^3\right)^{2/3} \left(2+x^3\right)}{x^6 \left(2+x^3+2 x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(2 + x^3))/(x^6*(2 + x^3 + 2*x^6)),x]","\frac{\left(-1+x^3\right)^{5/3}}{5 x^5}-\frac{2}{3} \text{RootSum}\left[5-5 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-5+4 \text{$\#$1}^3}\&\right]","\frac{\left(-1+x^3\right)^{5/3}}{5 x^5}-\frac{2}{3} \text{RootSum}\left[5-5 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-5+4 \text{$\#$1}^3}\&\right]",1,"(-1 + x^3)^(5/3)/(5*x^5) - (2*RootSum[5 - 5*#1^3 + 2*#1^6 & , (-(Log[x]*#1^2) + Log[(-1 + x^3)^(1/3) - x*#1]*#1^2)/(-5 + 4*#1^3) & ])/3","A",1
1054,1,79,79,1.7416639,"\int \frac{x+x^7}{\left(-1+x^6\right)^{2/3} \left(-1+x^3+x^6\right)} \, dx","IntegrateAlgebraic[(x + x^7)/((-1 + x^6)^(2/3)*(-1 + x^3 + x^6)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-1+x^6}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(x+\sqrt[3]{-1+x^6}\right)-\frac{1}{6} \log \left(x^2-x \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-1+x^6}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(x+\sqrt[3]{-1+x^6}\right)-\frac{1}{6} \log \left(x^2-x \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)",1,"-(ArcTan[(Sqrt[3]*x)/(-x + 2*(-1 + x^6)^(1/3))]/Sqrt[3]) + Log[x + (-1 + x^6)^(1/3)]/3 - Log[x^2 - x*(-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)]/6","A",1
1055,1,91,79,5.1164736,"\int \frac{\sqrt{1-x^6} \left(1+2 x^6\right)}{1+x^4-2 x^6+x^{12}} \, dx","IntegrateAlgebraic[(Sqrt[1 - x^6]*(1 + 2*x^6))/(1 + x^4 - 2*x^6 + x^12),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x \sqrt{1-x^6}}{-1+x^2+x^6}\right)}{2 \sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}-\frac{x^2}{\sqrt{2}}+\frac{x^6}{\sqrt{2}}}{x \sqrt{1-x^6}}\right)}{2 \sqrt{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x \sqrt{1-x^6}}{-1+x^2+x^6}\right)}{2 \sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{1-x^6}}{-1-x^2+x^6}\right)}{2 \sqrt{2}}",1,"-1/2*ArcTan[(Sqrt[2]*x*Sqrt[1 - x^6])/(-1 + x^2 + x^6)]/Sqrt[2] - ArcTanh[(-(1/Sqrt[2]) - x^2/Sqrt[2] + x^6/Sqrt[2])/(x*Sqrt[1 - x^6])]/(2*Sqrt[2])","A",1
1056,1,79,79,0.1360059,"\int \frac{\sqrt{1+x^2}}{\sqrt{1+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[Sqrt[1 + x^2]/Sqrt[1 + Sqrt[1 + x^2]],x]","-\frac{2 x}{3 \sqrt{1+\sqrt{1+x^2}}}+\frac{2 x \sqrt{1+x^2}}{3 \sqrt{1+\sqrt{1+x^2}}}+\sqrt{2} \tan ^{-1}\left(\frac{x}{\sqrt{2} \sqrt{1+\sqrt{1+x^2}}}\right)","-\frac{2 x}{3 \sqrt{1+\sqrt{1+x^2}}}+\frac{2 x \sqrt{1+x^2}}{3 \sqrt{1+\sqrt{1+x^2}}}+\sqrt{2} \tan ^{-1}\left(\frac{x}{\sqrt{2} \sqrt{1+\sqrt{1+x^2}}}\right)",1,"(-2*x)/(3*Sqrt[1 + Sqrt[1 + x^2]]) + (2*x*Sqrt[1 + x^2])/(3*Sqrt[1 + Sqrt[1 + x^2]]) + Sqrt[2]*ArcTan[x/(Sqrt[2]*Sqrt[1 + Sqrt[1 + x^2]])]","A",1
1057,1,79,79,0.0984445,"\int \frac{1+x^2}{\sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[(1 + x^2)/Sqrt[x + Sqrt[1 + x^2]],x]","\frac{2 \sqrt{1+x^2} \left(7 x+19 x^3+4 x^5\right)}{5 \left(x+\sqrt{1+x^2}\right)^{7/2}}+\frac{2 \left(9+112 x^2+147 x^4+28 x^6\right)}{35 \left(x+\sqrt{1+x^2}\right)^{7/2}}","\frac{2 \sqrt{1+x^2} \left(7 x+19 x^3+4 x^5\right)}{5 \left(x+\sqrt{1+x^2}\right)^{7/2}}+\frac{2 \left(9+112 x^2+147 x^4+28 x^6\right)}{35 \left(x+\sqrt{1+x^2}\right)^{7/2}}",1,"(2*Sqrt[1 + x^2]*(7*x + 19*x^3 + 4*x^5))/(5*(x + Sqrt[1 + x^2])^(7/2)) + (2*(9 + 112*x^2 + 147*x^4 + 28*x^6))/(35*(x + Sqrt[1 + x^2])^(7/2))","A",1
1058,1,79,79,0.1488963,"\int \frac{x^2}{\sqrt{x^2+\sqrt{1+x^4}}} \, dx","IntegrateAlgebraic[x^2/Sqrt[x^2 + Sqrt[1 + x^4]],x]","-\frac{x}{8 \left(x^2+\sqrt{1+x^4}\right)^{3/2}}+\frac{1}{4} x \sqrt{x^2+\sqrt{1+x^4}}-\frac{\tan ^{-1}\left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}\right)}{8 \sqrt{2}}","-\frac{x}{8 \left(x^2+\sqrt{1+x^4}\right)^{3/2}}+\frac{1}{4} x \sqrt{x^2+\sqrt{1+x^4}}-\frac{\tan ^{-1}\left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}\right)}{8 \sqrt{2}}",1,"-1/8*x/(x^2 + Sqrt[1 + x^4])^(3/2) + (x*Sqrt[x^2 + Sqrt[1 + x^4]])/4 - ArcTan[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]]]/(8*Sqrt[2])","A",1
1059,1,79,79,0.1560304,"\int \frac{\sqrt{1+x^4}}{\sqrt{x^2+\sqrt{1+x^4}}} \, dx","IntegrateAlgebraic[Sqrt[1 + x^4]/Sqrt[x^2 + Sqrt[1 + x^4]],x]","\frac{x}{8 \left(x^2+\sqrt{1+x^4}\right)^{3/2}}+\frac{1}{4} x \sqrt{x^2+\sqrt{1+x^4}}+\frac{5 \tan ^{-1}\left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}\right)}{8 \sqrt{2}}","\frac{x}{8 \left(x^2+\sqrt{1+x^4}\right)^{3/2}}+\frac{1}{4} x \sqrt{x^2+\sqrt{1+x^4}}+\frac{5 \tan ^{-1}\left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}\right)}{8 \sqrt{2}}",1,"x/(8*(x^2 + Sqrt[1 + x^4])^(3/2)) + (x*Sqrt[x^2 + Sqrt[1 + x^4]])/4 + (5*ArcTan[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]]])/(8*Sqrt[2])","A",1
1060,1,80,80,0.1292025,"\int \frac{-3+x^2}{\sqrt[3]{-1+x^2} \left(-1+x^2+x^3\right)} \, dx","IntegrateAlgebraic[(-3 + x^2)/((-1 + x^2)^(1/3)*(-1 + x^2 + x^3)),x]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-1+x^2}}{-2 x+\sqrt[3]{-1+x^2}}\right)+\log \left(x+\sqrt[3]{-1+x^2}\right)-\frac{1}{2} \log \left(x^2-x \sqrt[3]{-1+x^2}+\left(-1+x^2\right)^{2/3}\right)","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-1+x^2}}{-2 x+\sqrt[3]{-1+x^2}}\right)+\log \left(x+\sqrt[3]{-1+x^2}\right)-\frac{1}{2} \log \left(x^2-x \sqrt[3]{-1+x^2}+\left(-1+x^2\right)^{2/3}\right)",1,"Sqrt[3]*ArcTan[(Sqrt[3]*(-1 + x^2)^(1/3))/(-2*x + (-1 + x^2)^(1/3))] + Log[x + (-1 + x^2)^(1/3)] - Log[x^2 - x*(-1 + x^2)^(1/3) + (-1 + x^2)^(2/3)]/2","A",1
1061,1,83,80,1.0268962,"\int \frac{-a^2 b+a (2 a+b) x-3 a x^2+x^3}{x (-b+x) \sqrt{x (-a+x) (-b+x)} \left(a-(1+b d) x+d x^2\right)} \, dx","IntegrateAlgebraic[(-(a^2*b) + a*(2*a + b)*x - 3*a*x^2 + x^3)/(x*(-b + x)*Sqrt[x*(-a + x)*(-b + x)]*(a - (1 + b*d)*x + d*x^2)),x]","\frac{2 \sqrt{a b x+(-a-b) x^2+x^3}}{x (-b+x)}-2 \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{a b x+(-a-b) x^2+x^3}}{\sqrt{d} x (-b+x)}\right)","\frac{2 \sqrt{a b x+(-a-b) x^2+x^3}}{x (-b+x)}+2 \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a b x+(-a-b) x^2+x^3}}{a-x}\right)",1,"(2*Sqrt[a*b*x + (-a - b)*x^2 + x^3])/(x*(-b + x)) - 2*Sqrt[d]*ArcTanh[Sqrt[a*b*x + (-a - b)*x^2 + x^3]/(Sqrt[d]*x*(-b + x))]","A",1
1062,1,80,80,0.3641378,"\int \frac{1}{\left(b+a^3 x^3\right) \sqrt[3]{-b x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[1/((b + a^3*x^3)*(-(b*x^2) + a^3*x^3)^(1/3)),x]","-\frac{\text{RootSum}\left[a^9+a^3 b^2-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{3 b}","-\frac{\text{RootSum}\left[a^9+a^3 b^2-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{3 b}",1,"-1/3*RootSum[a^9 + a^3*b^2 - 3*a^6*#1^3 + 3*a^3*#1^6 - #1^9 & , (-Log[x] + Log[(-(b*x^2) + a^3*x^3)^(1/3) - x*#1])/#1 & ]/b","A",1
1063,1,80,80,0.0020124,"\int \frac{1}{\left(b+a^3 x^3\right) \sqrt[3]{-b x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[1/((b + a^3*x^3)*(-(b*x^2) + a^3*x^3)^(1/3)),x]","-\frac{\text{RootSum}\left[a^9+a^3 b^2-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{3 b}","-\frac{\text{RootSum}\left[a^9+a^3 b^2-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{3 b}",1,"-1/3*RootSum[a^9 + a^3*b^2 - 3*a^6*#1^3 + 3*a^3*#1^6 - #1^9 & , (-Log[x] + Log[(-(b*x^2) + a^3*x^3)^(1/3) - x*#1])/#1 & ]/b","A",1
1064,1,80,80,2.7254561,"\int \frac{x^2 \left(-4+x^3\right)}{\left(-1+x^3\right)^{3/4} \left(-1+x^3+x^4\right)} \, dx","IntegrateAlgebraic[(x^2*(-4 + x^3))/((-1 + x^3)^(3/4)*(-1 + x^3 + x^4)),x]","\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-1+x^3}}{-x^2+\sqrt{-1+x^3}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-1+x^3}}{x^2+\sqrt{-1+x^3}}\right)","\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-1+x^3}}{-x^2+\sqrt{-1+x^3}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-1+x^3}}{x^2+\sqrt{-1+x^3}}\right)",1,"Sqrt[2]*ArcTan[(Sqrt[2]*x*(-1 + x^3)^(1/4))/(-x^2 + Sqrt[-1 + x^3])] - Sqrt[2]*ArcTanh[(Sqrt[2]*x*(-1 + x^3)^(1/4))/(x^2 + Sqrt[-1 + x^3])]","A",1
1065,1,80,80,0.4135287,"\int \frac{x^6}{\left(b+a x^4\right)^{3/4}} \, dx","IntegrateAlgebraic[x^6/(b + a*x^4)^(3/4),x]","\frac{x^3 \sqrt[4]{b+a x^4}}{4 a}+\frac{3 b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{8 a^{7/4}}-\frac{3 b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{8 a^{7/4}}","\frac{x^3 \sqrt[4]{b+a x^4}}{4 a}+\frac{3 b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{8 a^{7/4}}-\frac{3 b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{8 a^{7/4}}",1,"(x^3*(b + a*x^4)^(1/4))/(4*a) + (3*b*ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)])/(8*a^(7/4)) - (3*b*ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)])/(8*a^(7/4))","A",1
1066,1,80,80,0.2122899,"\int \frac{\sqrt[4]{b x^2+a x^4}}{x^2} \, dx","IntegrateAlgebraic[(b*x^2 + a*x^4)^(1/4)/x^2,x]","-\frac{2 \sqrt[4]{b x^2+a x^4}}{x}-\sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)+\sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)","-\frac{2 \sqrt[4]{b x^2+a x^4}}{x}-\sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)+\sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)",1,"(-2*(b*x^2 + a*x^4)^(1/4))/x - a^(1/4)*ArcTan[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)] + a^(1/4)*ArcTanh[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)]","A",1
1067,1,80,80,0.2696855,"\int \frac{\sqrt[4]{2+x^4} \left(-4+x^8\right)}{x^6 \left(-4-2 x^4+x^8\right)} \, dx","IntegrateAlgebraic[((2 + x^4)^(1/4)*(-4 + x^8))/(x^6*(-4 - 2*x^4 + x^8)),x]","\frac{\sqrt[4]{2+x^4} \left(-1+2 x^4\right)}{5 x^5}+\frac{1}{8} \text{RootSum}\left[-1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{2+x^4}-x \text{$\#$1}\right)}{-\text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]","\frac{\sqrt[4]{2+x^4} \left(-1+2 x^4\right)}{5 x^5}+\frac{1}{8} \text{RootSum}\left[-1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{2+x^4}-x \text{$\#$1}\right)}{-\text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]",1,"((2 + x^4)^(1/4)*(-1 + 2*x^4))/(5*x^5) + RootSum[-1 - #1^4 + #1^8 & , (-Log[x] + Log[(2 + x^4)^(1/4) - x*#1])/(-#1^3 + 2*#1^7) & ]/8","A",1
1068,1,80,80,0.0014543,"\int \frac{\sqrt[4]{2+x^4} \left(-4+x^8\right)}{x^6 \left(-4-2 x^4+x^8\right)} \, dx","IntegrateAlgebraic[((2 + x^4)^(1/4)*(-4 + x^8))/(x^6*(-4 - 2*x^4 + x^8)),x]","\frac{\sqrt[4]{2+x^4} \left(-1+2 x^4\right)}{5 x^5}+\frac{1}{8} \text{RootSum}\left[-1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{2+x^4}-x \text{$\#$1}\right)}{-\text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]","\frac{\sqrt[4]{2+x^4} \left(-1+2 x^4\right)}{5 x^5}+\frac{1}{8} \text{RootSum}\left[-1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{2+x^4}-x \text{$\#$1}\right)}{-\text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]",1,"((2 + x^4)^(1/4)*(-1 + 2*x^4))/(5*x^5) + RootSum[-1 - #1^4 + #1^8 & , (-Log[x] + Log[(2 + x^4)^(1/4) - x*#1])/(-#1^3 + 2*#1^7) & ]/8","A",1
1069,1,80,80,0.2891234,"\int \frac{\sqrt[4]{-1+x^4} \left(-1+x^4+2 x^8\right)}{x^6 \left(1-x^4+x^8\right)} \, dx","IntegrateAlgebraic[((-1 + x^4)^(1/4)*(-1 + x^4 + 2*x^8))/(x^6*(1 - x^4 + x^8)),x]","\frac{\left(1-x^4\right) \sqrt[4]{-1+x^4}}{5 x^5}+\frac{3}{4} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-1+2 \text{$\#$1}^4}\&\right]","\frac{\left(1-x^4\right) \sqrt[4]{-1+x^4}}{5 x^5}+\frac{3}{4} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-1+2 \text{$\#$1}^4}\&\right]",1,"((1 - x^4)*(-1 + x^4)^(1/4))/(5*x^5) + (3*RootSum[1 - #1^4 + #1^8 & , (-(Log[x]*#1) + Log[(-1 + x^4)^(1/4) - x*#1]*#1)/(-1 + 2*#1^4) & ])/4","A",1
1070,1,80,80,0.0011642,"\int \frac{\sqrt[4]{-1+x^4} \left(-1+x^4+2 x^8\right)}{x^6 \left(1-x^4+x^8\right)} \, dx","IntegrateAlgebraic[((-1 + x^4)^(1/4)*(-1 + x^4 + 2*x^8))/(x^6*(1 - x^4 + x^8)),x]","\frac{\left(1-x^4\right) \sqrt[4]{-1+x^4}}{5 x^5}+\frac{3}{4} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-1+2 \text{$\#$1}^4}\&\right]","\frac{\left(1-x^4\right) \sqrt[4]{-1+x^4}}{5 x^5}+\frac{3}{4} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-1+2 \text{$\#$1}^4}\&\right]",1,"((1 - x^4)*(-1 + x^4)^(1/4))/(5*x^5) + (3*RootSum[1 - #1^4 + #1^8 & , (-(Log[x]*#1) + Log[(-1 + x^4)^(1/4) - x*#1]*#1)/(-1 + 2*#1^4) & ])/4","A",1
1071,1,81,81,0.3716104,"\int \frac{-2 a b+(a+b) x}{\sqrt[4]{x^2 (-a+x) (-b+x)} \left(a b d-(a+b) d x+(-1+d) x^2\right)} \, dx","IntegrateAlgebraic[(-2*a*b + (a + b)*x)/((x^2*(-a + x)*(-b + x))^(1/4)*(a*b*d - (a + b)*d*x + (-1 + d)*x^2)),x]","-\frac{2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{d} \sqrt[4]{a b x^2+(-a-b) x^3+x^4}}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{d} \sqrt[4]{a b x^2+(-a-b) x^3+x^4}}\right)}{d^{3/4}}","-\frac{2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{d} \sqrt[4]{a b x^2+(-a-b) x^3+x^4}}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{d} \sqrt[4]{a b x^2+(-a-b) x^3+x^4}}\right)}{d^{3/4}}",1,"(-2*ArcTan[x/(d^(1/4)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/4))])/d^(3/4) - (2*ArcTanh[x/(d^(1/4)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/4))])/d^(3/4)","A",1
1072,1,81,81,0.0551232,"\int \frac{\sqrt[3]{1+x^3}}{x} \, dx","IntegrateAlgebraic[(1 + x^3)^(1/3)/x,x]","\sqrt[3]{1+x^3}-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^3}}{\sqrt{3}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(-1+\sqrt[3]{1+x^3}\right)-\frac{1}{6} \log \left(1+\sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)","\sqrt[3]{1+x^3}-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^3}}{\sqrt{3}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(-1+\sqrt[3]{1+x^3}\right)-\frac{1}{6} \log \left(1+\sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)",1,"(1 + x^3)^(1/3) - ArcTan[1/Sqrt[3] + (2*(1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3] + Log[-1 + (1 + x^3)^(1/3)]/3 - Log[1 + (1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/6","A",1
1073,1,81,81,0.284841,"\int \frac{3 a b-2 (a+b) x+x^2}{\sqrt[4]{x (-a+x) (-b+x)} \left(-a b d+(a+b) d x-d x^2+x^3\right)} \, dx","IntegrateAlgebraic[(3*a*b - 2*(a + b)*x + x^2)/((x*(-a + x)*(-b + x))^(1/4)*(-(a*b*d) + (a + b)*d*x - d*x^2 + x^3)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b x+(-a-b) x^2+x^3}}{x}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b x+(-a-b) x^2+x^3}}{x}\right)}{d^{3/4}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b x+(-a-b) x^2+x^3}}{x}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b x+(-a-b) x^2+x^3}}{x}\right)}{d^{3/4}}",1,"(2*ArcTan[(d^(1/4)*(a*b*x + (-a - b)*x^2 + x^3)^(1/4))/x])/d^(3/4) - (2*ArcTanh[(d^(1/4)*(a*b*x + (-a - b)*x^2 + x^3)^(1/4))/x])/d^(3/4)","A",1
1074,1,81,81,0.360444,"\int \frac{1}{\left(-b+a^3 x^3\right) \sqrt[3]{-b x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[1/((-b + a^3*x^3)*(-(b*x^2) + a^3*x^3)^(1/3)),x]","\frac{\text{RootSum}\left[a^9-a^3 b^2-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{3 b}","\frac{\text{RootSum}\left[a^9-a^3 b^2-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{3 b}",1,"RootSum[a^9 - a^3*b^2 - 3*a^6*#1^3 + 3*a^3*#1^6 - #1^9 & , (-Log[x] + Log[(-(b*x^2) + a^3*x^3)^(1/3) - x*#1])/#1 & ]/(3*b)","A",1
1075,1,81,81,0.0024275,"\int \frac{1}{\left(-b+a^3 x^3\right) \sqrt[3]{-b x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[1/((-b + a^3*x^3)*(-(b*x^2) + a^3*x^3)^(1/3)),x]","\frac{\text{RootSum}\left[a^9-a^3 b^2-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{3 b}","\frac{\text{RootSum}\left[a^9-a^3 b^2-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{3 b}",1,"RootSum[a^9 - a^3*b^2 - 3*a^6*#1^3 + 3*a^3*#1^6 - #1^9 & , (-Log[x] + Log[(-(b*x^2) + a^3*x^3)^(1/3) - x*#1])/#1 & ]/(3*b)","A",1
1076,1,81,81,2.7271475,"\int \frac{-2 a b x^2+(a+b) x^3}{\left(x^2 (-a+x) (-b+x)\right)^{3/4} \left(-a b+(a+b) x+(-1+d) x^2\right)} \, dx","IntegrateAlgebraic[(-2*a*b*x^2 + (a + b)*x^3)/((x^2*(-a + x)*(-b + x))^(3/4)*(-(a*b) + (a + b)*x + (-1 + d)*x^2)),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{a b x^2+(-a-b) x^3+x^4}}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{a b x^2+(-a-b) x^3+x^4}}\right)}{d^{3/4}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{a b x^2+(-a-b) x^3+x^4}}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{a b x^2+(-a-b) x^3+x^4}}\right)}{d^{3/4}}",1,"(-2*ArcTan[(d^(1/4)*x)/(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/4)])/d^(3/4) + (2*ArcTanh[(d^(1/4)*x)/(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/4)])/d^(3/4)","A",1
1077,1,81,81,0.3010575,"\int \frac{\left(-1+x^4\right)^{3/4}}{1+x^4} \, dx","IntegrateAlgebraic[(-1 + x^4)^(3/4)/(1 + x^4),x]","\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-1+x^4}}\right)}{\sqrt[4]{2}}+\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-1+x^4}}\right)}{\sqrt[4]{2}}","\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-1+x^4}}\right)}{\sqrt[4]{2}}+\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-1+x^4}}\right)}{\sqrt[4]{2}}",1,"ArcTan[x/(-1 + x^4)^(1/4)]/2 - ArcTan[(2^(1/4)*x)/(-1 + x^4)^(1/4)]/2^(1/4) + ArcTanh[x/(-1 + x^4)^(1/4)]/2 - ArcTanh[(2^(1/4)*x)/(-1 + x^4)^(1/4)]/2^(1/4)","A",1
1078,1,81,81,0.2536913,"\int \frac{\sqrt[4]{-x^2+x^4}}{x^4 \left(-1+x^4\right)} \, dx","IntegrateAlgebraic[(-x^2 + x^4)^(1/4)/(x^4*(-1 + x^4)),x]","-\frac{2 \left(-1+x^2\right) \sqrt[4]{-x^2+x^4}}{5 x^3}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{2^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{2^{3/4}}","-\frac{2 \left(-1+x^2\right) \sqrt[4]{-x^2+x^4}}{5 x^3}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{2^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{2^{3/4}}",1,"(-2*(-1 + x^2)*(-x^2 + x^4)^(1/4))/(5*x^3) - ArcTan[(2^(1/4)*x)/(-x^2 + x^4)^(1/4)]/2^(3/4) + ArcTanh[(2^(1/4)*x)/(-x^2 + x^4)^(1/4)]/2^(3/4)","A",1
1079,1,81,81,5.2948796,"\int \frac{\left(1-x^2\right)^2}{\left(1+x^2\right) \left(1+6 x^2+x^4\right)^{3/4}} \, dx","IntegrateAlgebraic[(1 - x^2)^2/((1 + x^2)*(1 + 6*x^2 + x^4)^(3/4)),x]","-\tan ^{-1}\left(\frac{-1+x}{\sqrt[4]{1+6 x^2+x^4}}\right)-\tan ^{-1}\left(\frac{1+x}{\sqrt[4]{1+6 x^2+x^4}}\right)+\tanh ^{-1}\left(\frac{-1+x}{\sqrt[4]{1+6 x^2+x^4}}\right)+\tanh ^{-1}\left(\frac{1+x}{\sqrt[4]{1+6 x^2+x^4}}\right)","-\tan ^{-1}\left(\frac{-1+x}{\sqrt[4]{1+6 x^2+x^4}}\right)-\tan ^{-1}\left(\frac{1+x}{\sqrt[4]{1+6 x^2+x^4}}\right)+\tanh ^{-1}\left(\frac{-1+x}{\sqrt[4]{1+6 x^2+x^4}}\right)+\tanh ^{-1}\left(\frac{1+x}{\sqrt[4]{1+6 x^2+x^4}}\right)",1,"-ArcTan[(-1 + x)/(1 + 6*x^2 + x^4)^(1/4)] - ArcTan[(1 + x)/(1 + 6*x^2 + x^4)^(1/4)] + ArcTanh[(-1 + x)/(1 + 6*x^2 + x^4)^(1/4)] + ArcTanh[(1 + x)/(1 + 6*x^2 + x^4)^(1/4)]","A",1
1080,1,81,81,0.9932193,"\int \frac{1}{(1+x) \sqrt[4]{1+6 x^2+x^4}} \, dx","IntegrateAlgebraic[1/((1 + x)*(1 + 6*x^2 + x^4)^(1/4)),x]","\frac{\tan ^{-1}\left(\frac{-\frac{1}{\sqrt[4]{2}}+\frac{x}{\sqrt[4]{2}}}{\sqrt[4]{1+6 x^2+x^4}}\right)}{2\ 2^{3/4}}+\frac{\tanh ^{-1}\left(\frac{-\frac{1}{\sqrt[4]{2}}+\frac{x}{\sqrt[4]{2}}}{\sqrt[4]{1+6 x^2+x^4}}\right)}{2\ 2^{3/4}}","\frac{\tan ^{-1}\left(\frac{-\frac{1}{\sqrt[4]{2}}+\frac{x}{\sqrt[4]{2}}}{\sqrt[4]{1+6 x^2+x^4}}\right)}{2\ 2^{3/4}}+\frac{\tanh ^{-1}\left(\frac{-\frac{1}{\sqrt[4]{2}}+\frac{x}{\sqrt[4]{2}}}{\sqrt[4]{1+6 x^2+x^4}}\right)}{2\ 2^{3/4}}",1,"ArcTan[(-2^(-1/4) + x/2^(1/4))/(1 + 6*x^2 + x^4)^(1/4)]/(2*2^(3/4)) + ArcTanh[(-2^(-1/4) + x/2^(1/4))/(1 + 6*x^2 + x^4)^(1/4)]/(2*2^(3/4))","A",1
1081,1,81,81,0.2825628,"\int \frac{1+x}{\sqrt{-7+4 x+14 x^2-12 x^3+x^4}} \, dx","IntegrateAlgebraic[(1 + x)/Sqrt[-7 + 4*x + 14*x^2 - 12*x^3 + x^4],x]","\tan ^{-1}\left(\frac{-4+4 x}{1-2 x+x^2-\sqrt{-7+4 x+14 x^2-12 x^3+x^4}}\right)+\log (-1+x)-\log \left(-5+6 x-x^2+\sqrt{-7+4 x+14 x^2-12 x^3+x^4}\right)","\tan ^{-1}\left(\frac{-4+4 x}{1-2 x+x^2-\sqrt{-7+4 x+14 x^2-12 x^3+x^4}}\right)+\log (-1+x)-\log \left(-5+6 x-x^2+\sqrt{-7+4 x+14 x^2-12 x^3+x^4}\right)",1,"ArcTan[(-4 + 4*x)/(1 - 2*x + x^2 - Sqrt[-7 + 4*x + 14*x^2 - 12*x^3 + x^4])] + Log[-1 + x] - Log[-5 + 6*x - x^2 + Sqrt[-7 + 4*x + 14*x^2 - 12*x^3 + x^4]]","A",1
1082,1,81,81,1.1303911,"\int \frac{3+x^4}{\sqrt[3]{-1+x^4} \left(-1-8 x^3+x^4\right)} \, dx","IntegrateAlgebraic[(3 + x^4)/((-1 + x^4)^(1/3)*(-1 - 8*x^3 + x^4)),x]","-\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+\sqrt[3]{-1+x^4}}\right)+\frac{1}{2} \log \left(-2 x+\sqrt[3]{-1+x^4}\right)-\frac{1}{4} \log \left(4 x^2+2 x \sqrt[3]{-1+x^4}+\left(-1+x^4\right)^{2/3}\right)","-\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+\sqrt[3]{-1+x^4}}\right)+\frac{1}{2} \log \left(-2 x+\sqrt[3]{-1+x^4}\right)-\frac{1}{4} \log \left(4 x^2+2 x \sqrt[3]{-1+x^4}+\left(-1+x^4\right)^{2/3}\right)",1,"-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + (-1 + x^4)^(1/3))]) + Log[-2*x + (-1 + x^4)^(1/3)]/2 - Log[4*x^2 + 2*x*(-1 + x^4)^(1/3) + (-1 + x^4)^(2/3)]/4","A",1
1083,1,81,81,0.4422637,"\int \frac{(-3+2 x) \sqrt{-2 x+2 x^2+3 x^4}}{\left(-2+2 x+x^3\right)^2} \, dx","IntegrateAlgebraic[((-3 + 2*x)*Sqrt[-2*x + 2*x^2 + 3*x^4])/(-2 + 2*x + x^3)^2,x]","\frac{x \sqrt{-2 x+2 x^2+3 x^4}}{2 \left(-2+2 x+x^3\right)}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{-2 x+2 x^2+3 x^4}}{-2+2 x+3 x^3}\right)}{2 \sqrt{2}}","\frac{x \sqrt{-2 x+2 x^2+3 x^4}}{2 \left(-2+2 x+x^3\right)}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{-2 x+2 x^2+3 x^4}}{-2+2 x+3 x^3}\right)}{2 \sqrt{2}}",1,"(x*Sqrt[-2*x + 2*x^2 + 3*x^4])/(2*(-2 + 2*x + x^3)) + ArcTanh[(Sqrt[2]*x*Sqrt[-2*x + 2*x^2 + 3*x^4])/(-2 + 2*x + 3*x^3)]/(2*Sqrt[2])","A",1
1084,1,81,81,0.2869562,"\int \frac{-2 b+a x^4}{x^4 \sqrt[4]{-b+a x^4}} \, dx","IntegrateAlgebraic[(-2*b + a*x^4)/(x^4*(-b + a*x^4)^(1/4)),x]","-\frac{2 \left(-b+a x^4\right)^{3/4}}{3 x^3}+\frac{1}{2} a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)+\frac{1}{2} a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)","-\frac{2 \left(-b+a x^4\right)^{3/4}}{3 x^3}+\frac{1}{2} a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)+\frac{1}{2} a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)",1,"(-2*(-b + a*x^4)^(3/4))/(3*x^3) + (a^(3/4)*ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)])/2 + (a^(3/4)*ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)])/2","A",1
1085,1,81,81,0.2853671,"\int \left(-b+a x^4\right)^{3/4} \, dx","IntegrateAlgebraic[(-b + a*x^4)^(3/4),x]","\frac{1}{4} x \left(-b+a x^4\right)^{3/4}-\frac{3 b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{8 \sqrt[4]{a}}-\frac{3 b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{8 \sqrt[4]{a}}","\frac{1}{4} x \left(-b+a x^4\right)^{3/4}-\frac{3 b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{8 \sqrt[4]{a}}-\frac{3 b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{8 \sqrt[4]{a}}",1,"(x*(-b + a*x^4)^(3/4))/4 - (3*b*ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)])/(8*a^(1/4)) - (3*b*ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)])/(8*a^(1/4))","A",1
1086,1,81,81,0.2312413,"\int \frac{\left(-b+a x^4\right)^{3/4}}{x^4} \, dx","IntegrateAlgebraic[(-b + a*x^4)^(3/4)/x^4,x]","-\frac{\left(-b+a x^4\right)^{3/4}}{3 x^3}+\frac{1}{2} a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)+\frac{1}{2} a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)","-\frac{\left(-b+a x^4\right)^{3/4}}{3 x^3}+\frac{1}{2} a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)+\frac{1}{2} a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)",1,"-1/3*(-b + a*x^4)^(3/4)/x^3 + (a^(3/4)*ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)])/2 + (a^(3/4)*ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)])/2","A",1
1087,1,81,81,0.9292381,"\int \frac{1}{\sqrt[3]{-b x^2+a x^3} \left(b+a x^4\right)} \, dx","IntegrateAlgebraic[1/((-(b*x^2) + a*x^3)^(1/3)*(b + a*x^4)),x]","-\frac{\text{RootSum}\left[a^4+a b^3-4 a^3 \text{$\#$1}^3+6 a^2 \text{$\#$1}^6-4 a \text{$\#$1}^9+\text{$\#$1}^{12}\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{4 b}","-\frac{\text{RootSum}\left[a^4+a b^3-4 a^3 \text{$\#$1}^3+6 a^2 \text{$\#$1}^6-4 a \text{$\#$1}^9+\text{$\#$1}^{12}\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{4 b}",1,"-1/4*RootSum[a^4 + a*b^3 - 4*a^3*#1^3 + 6*a^2*#1^6 - 4*a*#1^9 + #1^12 & , (-Log[x] + Log[(-(b*x^2) + a*x^3)^(1/3) - x*#1])/#1 & ]/b","A",1
1088,1,81,81,0.0023777,"\int \frac{1}{\sqrt[3]{-b x^2+a x^3} \left(b+a x^4\right)} \, dx","IntegrateAlgebraic[1/((-(b*x^2) + a*x^3)^(1/3)*(b + a*x^4)),x]","-\frac{\text{RootSum}\left[a^4+a b^3-4 a^3 \text{$\#$1}^3+6 a^2 \text{$\#$1}^6-4 a \text{$\#$1}^9+\text{$\#$1}^{12}\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{4 b}","-\frac{\text{RootSum}\left[a^4+a b^3-4 a^3 \text{$\#$1}^3+6 a^2 \text{$\#$1}^6-4 a \text{$\#$1}^9+\text{$\#$1}^{12}\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{4 b}",1,"-1/4*RootSum[a^4 + a*b^3 - 4*a^3*#1^3 + 6*a^2*#1^6 - 4*a*#1^9 + #1^12 & , (-Log[x] + Log[(-(b*x^2) + a*x^3)^(1/3) - x*#1])/#1 & ]/b","A",1
1089,1,81,81,0.3065256,"\int \frac{\sqrt[4]{b x^3+a x^4}}{x^2} \, dx","IntegrateAlgebraic[(b*x^3 + a*x^4)^(1/4)/x^2,x]","-\frac{4 \sqrt[4]{b x^3+a x^4}}{x}-2 \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)+2 \sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)","-\frac{4 \sqrt[4]{b x^3+a x^4}}{x}-2 \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)+2 \sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)",1,"(-4*(b*x^3 + a*x^4)^(1/4))/x - 2*a^(1/4)*ArcTan[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)] + 2*a^(1/4)*ArcTanh[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)]","A",1
1090,1,81,81,0.1928906,"\int \frac{\left(1-x^3+x^4+x^6\right)^{3/4} \left(-4+x^3+2 x^6\right)}{\left(1-x^3+x^6\right)^2} \, dx","IntegrateAlgebraic[((1 - x^3 + x^4 + x^6)^(3/4)*(-4 + x^3 + 2*x^6))/(1 - x^3 + x^6)^2,x]","-\frac{x \left(1-x^3+x^4+x^6\right)^{3/4}}{1-x^3+x^6}-\frac{3}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{1-x^3+x^4+x^6}}\right)-\frac{3}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{1-x^3+x^4+x^6}}\right)","-\frac{x \left(1-x^3+x^4+x^6\right)^{3/4}}{1-x^3+x^6}-\frac{3}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{1-x^3+x^4+x^6}}\right)-\frac{3}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{1-x^3+x^4+x^6}}\right)",1,"-((x*(1 - x^3 + x^4 + x^6)^(3/4))/(1 - x^3 + x^6)) - (3*ArcTan[x/(1 - x^3 + x^4 + x^6)^(1/4)])/2 - (3*ArcTanh[x/(1 - x^3 + x^4 + x^6)^(1/4)])/2","A",1
1091,1,81,81,0.3889492,"\int \frac{1+2 x^4}{\sqrt[4]{1+x^4} \left(-2-x^4+x^8\right)} \, dx","IntegrateAlgebraic[(1 + 2*x^4)/((1 + x^4)^(1/4)*(-2 - x^4 + x^8)),x]","\frac{x}{3 \sqrt[4]{1+x^4}}-\frac{5 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{3}{2}} x}{\sqrt[4]{1+x^4}}\right)}{6\ 2^{3/4} \sqrt[4]{3}}-\frac{5 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{3}{2}} x}{\sqrt[4]{1+x^4}}\right)}{6\ 2^{3/4} \sqrt[4]{3}}","\frac{x}{3 \sqrt[4]{1+x^4}}-\frac{5 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{3}{2}} x}{\sqrt[4]{1+x^4}}\right)}{6\ 2^{3/4} \sqrt[4]{3}}-\frac{5 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{3}{2}} x}{\sqrt[4]{1+x^4}}\right)}{6\ 2^{3/4} \sqrt[4]{3}}",1,"x/(3*(1 + x^4)^(1/4)) - (5*ArcTan[((3/2)^(1/4)*x)/(1 + x^4)^(1/4)])/(6*2^(3/4)*3^(1/4)) - (5*ArcTanh[((3/2)^(1/4)*x)/(1 + x^4)^(1/4)])/(6*2^(3/4)*3^(1/4))","A",1
1092,1,81,81,2.3360499,"\int \frac{1}{\sqrt[4]{-b x^2+a x^4} \left(b+a x^8\right)} \, dx","IntegrateAlgebraic[1/((-(b*x^2) + a*x^4)^(1/4)*(b + a*x^8)),x]","-\frac{\text{RootSum}\left[a^4+a b^3-4 a^3 \text{$\#$1}^4+6 a^2 \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{8 b}","-\frac{\text{RootSum}\left[a^4+a b^3-4 a^3 \text{$\#$1}^4+6 a^2 \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{8 b}",1,"-1/8*RootSum[a^4 + a*b^3 - 4*a^3*#1^4 + 6*a^2*#1^8 - 4*a*#1^12 + #1^16 & , (-Log[x] + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1])/#1 & ]/b","A",1
1093,1,81,81,0.0017024,"\int \frac{1}{\sqrt[4]{-b x^2+a x^4} \left(b+a x^8\right)} \, dx","IntegrateAlgebraic[1/((-(b*x^2) + a*x^4)^(1/4)*(b + a*x^8)),x]","-\frac{\text{RootSum}\left[a^4+a b^3-4 a^3 \text{$\#$1}^4+6 a^2 \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{8 b}","-\frac{\text{RootSum}\left[a^4+a b^3-4 a^3 \text{$\#$1}^4+6 a^2 \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{8 b}",1,"-1/8*RootSum[a^4 + a*b^3 - 4*a^3*#1^4 + 6*a^2*#1^8 - 4*a*#1^12 + #1^16 & , (-Log[x] + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1])/#1 & ]/b","A",1
1094,1,81,81,0.4339106,"\int \frac{\left(-b^4+a^4 x^4\right) \sqrt{b^4+a^4 x^4}}{b^8+a^8 x^8} \, dx","IntegrateAlgebraic[((-b^4 + a^4*x^4)*Sqrt[b^4 + a^4*x^4])/(b^8 + a^8*x^8),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} a b x}{\sqrt{b^4+a^4 x^4}}\right)}{2 \sqrt[4]{2} a b}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} a b x}{\sqrt{b^4+a^4 x^4}}\right)}{2 \sqrt[4]{2} a b}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} a b x}{\sqrt{b^4+a^4 x^4}}\right)}{2 \sqrt[4]{2} a b}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} a b x}{\sqrt{b^4+a^4 x^4}}\right)}{2 \sqrt[4]{2} a b}",1,"-1/2*ArcTan[(2^(1/4)*a*b*x)/Sqrt[b^4 + a^4*x^4]]/(2^(1/4)*a*b) - ArcTanh[(2^(1/4)*a*b*x)/Sqrt[b^4 + a^4*x^4]]/(2*2^(1/4)*a*b)","A",1
1095,1,81,81,0.5290955,"\int \frac{-b^8+a^8 x^8}{\sqrt{b^4+a^4 x^4} \left(b^8+a^8 x^8\right)} \, dx","IntegrateAlgebraic[(-b^8 + a^8*x^8)/(Sqrt[b^4 + a^4*x^4]*(b^8 + a^8*x^8)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} a b x}{\sqrt{b^4+a^4 x^4}}\right)}{2 \sqrt[4]{2} a b}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} a b x}{\sqrt{b^4+a^4 x^4}}\right)}{2 \sqrt[4]{2} a b}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} a b x}{\sqrt{b^4+a^4 x^4}}\right)}{2 \sqrt[4]{2} a b}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} a b x}{\sqrt{b^4+a^4 x^4}}\right)}{2 \sqrt[4]{2} a b}",1,"-1/2*ArcTan[(2^(1/4)*a*b*x)/Sqrt[b^4 + a^4*x^4]]/(2^(1/4)*a*b) - ArcTanh[(2^(1/4)*a*b*x)/Sqrt[b^4 + a^4*x^4]]/(2*2^(1/4)*a*b)","A",1
1096,1,81,81,0.2159606,"\int \frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\left(b^2+a x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[Sqrt[b + Sqrt[b^2 + a*x^2]]/(b^2 + a*x^2)^(3/2),x]","\frac{x}{b \sqrt{b^2+a x^2} \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{\sqrt{a} b^{3/2}}","\frac{x}{b \sqrt{b^2+a x^2} \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{\sqrt{a} b^{3/2}}",1,"x/(b*Sqrt[b^2 + a*x^2]*Sqrt[b + Sqrt[b^2 + a*x^2]]) + ArcTan[(Sqrt[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])]/(Sqrt[a]*b^(3/2))","A",1
1097,1,81,81,0.3880398,"\int \frac{\sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{b+a^2 x^4}} \, dx","IntegrateAlgebraic[Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/Sqrt[b + a^2*x^4],x]","\frac{\log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}+i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{\sqrt{2} \sqrt{a}}","\frac{\log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}+i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{\sqrt{2} \sqrt{a}}",1,"Log[I*a^(3/2)*x^2 + I*Sqrt[a]*Sqrt[b + a^2*x^4] + I*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]]/(Sqrt[2]*Sqrt[a])","A",1
1098,1,82,82,0.9209654,"\int \frac{1}{\sqrt[3]{-b x^2+a x^3} \left(-b+a x^4\right)} \, dx","IntegrateAlgebraic[1/((-(b*x^2) + a*x^3)^(1/3)*(-b + a*x^4)),x]","\frac{\text{RootSum}\left[a^4-a b^3-4 a^3 \text{$\#$1}^3+6 a^2 \text{$\#$1}^6-4 a \text{$\#$1}^9+\text{$\#$1}^{12}\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{4 b}","\frac{\text{RootSum}\left[a^4-a b^3-4 a^3 \text{$\#$1}^3+6 a^2 \text{$\#$1}^6-4 a \text{$\#$1}^9+\text{$\#$1}^{12}\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{4 b}",1,"RootSum[a^4 - a*b^3 - 4*a^3*#1^3 + 6*a^2*#1^6 - 4*a*#1^9 + #1^12 & , (-Log[x] + Log[(-(b*x^2) + a*x^3)^(1/3) - x*#1])/#1 & ]/(4*b)","A",1
1099,1,82,82,0.0024424,"\int \frac{1}{\sqrt[3]{-b x^2+a x^3} \left(-b+a x^4\right)} \, dx","IntegrateAlgebraic[1/((-(b*x^2) + a*x^3)^(1/3)*(-b + a*x^4)),x]","\frac{\text{RootSum}\left[a^4-a b^3-4 a^3 \text{$\#$1}^3+6 a^2 \text{$\#$1}^6-4 a \text{$\#$1}^9+\text{$\#$1}^{12}\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{4 b}","\frac{\text{RootSum}\left[a^4-a b^3-4 a^3 \text{$\#$1}^3+6 a^2 \text{$\#$1}^6-4 a \text{$\#$1}^9+\text{$\#$1}^{12}\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{4 b}",1,"RootSum[a^4 - a*b^3 - 4*a^3*#1^3 + 6*a^2*#1^6 - 4*a*#1^9 + #1^12 & , (-Log[x] + Log[(-(b*x^2) + a*x^3)^(1/3) - x*#1])/#1 & ]/(4*b)","A",1
1100,1,82,82,0.7293582,"\int \frac{\sqrt[4]{-b x^3+a x^4}}{x \left(b+a x^3\right)} \, dx","IntegrateAlgebraic[(-(b*x^3) + a*x^4)^(1/4)/(x*(b + a*x^3)),x]","\frac{1}{3} \text{RootSum}\left[a^3+a b^2-3 a^2 \text{$\#$1}^4+3 a \text{$\#$1}^8-\text{$\#$1}^{12}\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]","\frac{1}{3} \text{RootSum}\left[a^3+a b^2-3 a^2 \text{$\#$1}^4+3 a \text{$\#$1}^8-\text{$\#$1}^{12}\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]",1,"RootSum[a^3 + a*b^2 - 3*a^2*#1^4 + 3*a*#1^8 - #1^12 & , (-(Log[x]*#1) + Log[(-(b*x^3) + a*x^4)^(1/4) - x*#1]*#1)/(-a + #1^4) & ]/3","A",1
1101,1,82,82,0.0015534,"\int \frac{\sqrt[4]{-b x^3+a x^4}}{x \left(b+a x^3\right)} \, dx","IntegrateAlgebraic[(-(b*x^3) + a*x^4)^(1/4)/(x*(b + a*x^3)),x]","\frac{1}{3} \text{RootSum}\left[a^3+a b^2-3 a^2 \text{$\#$1}^4+3 a \text{$\#$1}^8-\text{$\#$1}^{12}\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]","\frac{1}{3} \text{RootSum}\left[a^3+a b^2-3 a^2 \text{$\#$1}^4+3 a \text{$\#$1}^8-\text{$\#$1}^{12}\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]",1,"RootSum[a^3 + a*b^2 - 3*a^2*#1^4 + 3*a*#1^8 - #1^12 & , (-(Log[x]*#1) + Log[(-(b*x^3) + a*x^4)^(1/4) - x*#1]*#1)/(-a + #1^4) & ]/3","A",1
1102,1,82,82,0.3785832,"\int \frac{\sqrt[4]{b x^3+a x^4}}{x} \, dx","IntegrateAlgebraic[(b*x^3 + a*x^4)^(1/4)/x,x]","\sqrt[4]{b x^3+a x^4}-\frac{b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{2 a^{3/4}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{2 a^{3/4}}","\sqrt[4]{b x^3+a x^4}-\frac{b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{2 a^{3/4}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{2 a^{3/4}}",1,"(b*x^3 + a*x^4)^(1/4) - (b*ArcTan[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(2*a^(3/4)) + (b*ArcTanh[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(2*a^(3/4))","A",1
1103,1,82,82,0.4059844,"\int \frac{(-b+a x) \sqrt[4]{b x^3+a x^4}}{x (b+a x)} \, dx","IntegrateAlgebraic[((-b + a*x)*(b*x^3 + a*x^4)^(1/4))/(x*(b + a*x)),x]","\sqrt[4]{b x^3+a x^4}+\frac{7 b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{2 a^{3/4}}-\frac{7 b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{2 a^{3/4}}","\sqrt[4]{b x^3+a x^4}+\frac{7 b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{2 a^{3/4}}-\frac{7 b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{2 a^{3/4}}",1,"(b*x^3 + a*x^4)^(1/4) + (7*b*ArcTan[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(2*a^(3/4)) - (7*b*ArcTanh[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(2*a^(3/4))","A",1
1104,1,82,82,2.6614337,"\int \frac{-1+3 x^4}{\left(1-x+x^4\right) \sqrt[3]{x^2+x^6}} \, dx","IntegrateAlgebraic[(-1 + 3*x^4)/((1 - x + x^4)*(x^2 + x^6)^(1/3)),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^6}}\right)+\log \left(-x+\sqrt[3]{x^2+x^6}\right)-\frac{1}{2} \log \left(x^2+x \sqrt[3]{x^2+x^6}+\left(x^2+x^6\right)^{2/3}\right)","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^6}}\right)+\log \left(-x+\sqrt[3]{x^2+x^6}\right)-\frac{1}{2} \log \left(x^2+x \sqrt[3]{x^2+x^6}+\left(x^2+x^6\right)^{2/3}\right)",1,"-(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(x^2 + x^6)^(1/3))]) + Log[-x + (x^2 + x^6)^(1/3)] - Log[x^2 + x*(x^2 + x^6)^(1/3) + (x^2 + x^6)^(2/3)]/2","A",1
1105,1,82,82,2.3192542,"\int \frac{1}{\sqrt[4]{-b x^2+a x^4} \left(-b+a x^8\right)} \, dx","IntegrateAlgebraic[1/((-(b*x^2) + a*x^4)^(1/4)*(-b + a*x^8)),x]","\frac{\text{RootSum}\left[a^4-a b^3-4 a^3 \text{$\#$1}^4+6 a^2 \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{8 b}","\frac{\text{RootSum}\left[a^4-a b^3-4 a^3 \text{$\#$1}^4+6 a^2 \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{8 b}",1,"RootSum[a^4 - a*b^3 - 4*a^3*#1^4 + 6*a^2*#1^8 - 4*a*#1^12 + #1^16 & , (-Log[x] + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1])/#1 & ]/(8*b)","A",1
1106,1,82,82,0.0017378,"\int \frac{1}{\sqrt[4]{-b x^2+a x^4} \left(-b+a x^8\right)} \, dx","IntegrateAlgebraic[1/((-(b*x^2) + a*x^4)^(1/4)*(-b + a*x^8)),x]","\frac{\text{RootSum}\left[a^4-a b^3-4 a^3 \text{$\#$1}^4+6 a^2 \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{8 b}","\frac{\text{RootSum}\left[a^4-a b^3-4 a^3 \text{$\#$1}^4+6 a^2 \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{8 b}",1,"RootSum[a^4 - a*b^3 - 4*a^3*#1^4 + 6*a^2*#1^8 - 4*a*#1^12 + #1^16 & , (-Log[x] + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1])/#1 & ]/(8*b)","A",1
1107,1,82,82,0.5581306,"\int \frac{-b+a x^8}{x^2 \left(b+a x^4\right)^{3/4}} \, dx","IntegrateAlgebraic[(-b + a*x^8)/(x^2*(b + a*x^4)^(3/4)),x]","\frac{\left(4+x^4\right) \sqrt[4]{b+a x^4}}{4 x}+\frac{3 b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{8 a^{3/4}}-\frac{3 b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{8 a^{3/4}}","\frac{\left(4+x^4\right) \sqrt[4]{b+a x^4}}{4 x}+\frac{3 b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{8 a^{3/4}}-\frac{3 b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{8 a^{3/4}}",1,"((4 + x^4)*(b + a*x^4)^(1/4))/(4*x) + (3*b*ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)])/(8*a^(3/4)) - (3*b*ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)])/(8*a^(3/4))","A",1
1108,1,82,82,18.4736823,"\int \frac{x^5 \left(-4 b+5 a x^2\right)}{\sqrt[4]{-b+a x^2} \left(b-b x^8+a x^{10}\right)} \, dx","IntegrateAlgebraic[(x^5*(-4*b + 5*a*x^2))/((-b + a*x^2)^(1/4)*(b - b*x^8 + a*x^10)),x]","\frac{1}{2} a \text{RootSum}\left[a^4 b+b^4 \text{$\#$1}^4+4 b^3 \text{$\#$1}^8+6 b^2 \text{$\#$1}^{12}+4 b \text{$\#$1}^{16}+\text{$\#$1}^{20}\&,\frac{\log \left(\sqrt[4]{-b+a x^2}-\text{$\#$1}\right)}{b \text{$\#$1}+\text{$\#$1}^5}\&\right]","\frac{1}{2} a \text{RootSum}\left[a^4 b+b^4 \text{$\#$1}^4+4 b^3 \text{$\#$1}^8+6 b^2 \text{$\#$1}^{12}+4 b \text{$\#$1}^{16}+\text{$\#$1}^{20}\&,\frac{\log \left(\sqrt[4]{-b+a x^2}-\text{$\#$1}\right)}{b \text{$\#$1}+\text{$\#$1}^5}\&\right]",1,"(a*RootSum[a^4*b + b^4*#1^4 + 4*b^3*#1^8 + 6*b^2*#1^12 + 4*b*#1^16 + #1^20 & , Log[(-b + a*x^2)^(1/4) - #1]/(b*#1 + #1^5) & ])/2","A",1
1109,1,83,83,0.1479083,"\int \frac{1}{\sqrt[3]{x^2+x^3}} \, dx","IntegrateAlgebraic[(x^2 + x^3)^(-1/3),x]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{x^2+x^3}+\left(x^2+x^3\right)^{2/3}\right)","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{x^2+x^3}+\left(x^2+x^3\right)^{2/3}\right)",1,"Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(x^2 + x^3)^(1/3))] - Log[-x + (x^2 + x^3)^(1/3)] + Log[x^2 + x*(x^2 + x^3)^(1/3) + (x^2 + x^3)^(2/3)]/2","A",1
1110,1,83,83,0.363978,"\int \frac{a-3 b+2 x}{\sqrt[4]{(-a+x) (-b+x)} \left(-a^3+b d-\left(-3 a^2+d\right) x-3 a x^2+x^3\right)} \, dx","IntegrateAlgebraic[(a - 3*b + 2*x)/(((-a + x)*(-b + x))^(1/4)*(-a^3 + b*d - (-3*a^2 + d)*x - 3*a*x^2 + x^3)),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b+(-a-b) x+x^2}}{a-x}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b+(-a-b) x+x^2}}{a-x}\right)}{d^{3/4}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b+(-a-b) x+x^2}}{a-x}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b+(-a-b) x+x^2}}{a-x}\right)}{d^{3/4}}",1,"(-2*ArcTan[(d^(1/4)*(a*b + (-a - b)*x + x^2)^(1/4))/(a - x)])/d^(3/4) + (2*ArcTanh[(d^(1/4)*(a*b + (-a - b)*x + x^2)^(1/4))/(a - x)])/d^(3/4)","A",1
1111,1,83,83,0.6038687,"\int \frac{(a-3 b+2 x) \left(a^2-2 a x+x^2\right)}{((-a+x) (-b+x))^{3/4} \left(-b+a^3 d+\left(1-3 a^2 d\right) x+3 a d x^2-d x^3\right)} \, dx","IntegrateAlgebraic[((a - 3*b + 2*x)*(a^2 - 2*a*x + x^2))/(((-a + x)*(-b + x))^(3/4)*(-b + a^3*d + (1 - 3*a^2*d)*x + 3*a*d*x^2 - d*x^3)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(a b+(-a-b) x+x^2\right)^{3/4}}{b-x}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \left(a b+(-a-b) x+x^2\right)^{3/4}}{b-x}\right)}{d^{3/4}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(a b+(-a-b) x+x^2\right)^{3/4}}{b-x}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \left(a b+(-a-b) x+x^2\right)^{3/4}}{b-x}\right)}{d^{3/4}}",1,"(2*ArcTan[(d^(1/4)*(a*b + (-a - b)*x + x^2)^(3/4))/(b - x)])/d^(3/4) - (2*ArcTanh[(d^(1/4)*(a*b + (-a - b)*x + x^2)^(3/4))/(b - x)])/d^(3/4)","A",1
1112,1,83,83,0.2497053,"\int \frac{-1-2 (-1+k) x+k x^2}{\sqrt[4]{(1-x) x (1-k x)} \left(-1+(d+3 k) x-\left(d+3 k^2\right) x^2+k^3 x^3\right)} \, dx","IntegrateAlgebraic[(-1 - 2*(-1 + k)*x + k*x^2)/(((1 - x)*x*(1 - k*x))^(1/4)*(-1 + (d + 3*k)*x - (d + 3*k^2)*x^2 + k^3*x^3)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{x+(-1-k) x^2+k x^3}}{-1+k x}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{x+(-1-k) x^2+k x^3}}{-1+k x}\right)}{d^{3/4}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{x+(-1-k) x^2+k x^3}}{-1+k x}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{x+(-1-k) x^2+k x^3}}{-1+k x}\right)}{d^{3/4}}",1,"(2*ArcTan[(d^(1/4)*(x + (-1 - k)*x^2 + k*x^3)^(1/4))/(-1 + k*x)])/d^(3/4) - (2*ArcTanh[(d^(1/4)*(x + (-1 - k)*x^2 + k*x^3)^(1/4))/(-1 + k*x)])/d^(3/4)","A",1
1113,1,83,83,0.177521,"\int \frac{\sqrt{1+6 x^2+x^4}}{x \left(1+x^2\right)} \, dx","IntegrateAlgebraic[Sqrt[1 + 6*x^2 + x^4]/(x*(1 + x^2)),x]","-2 \tan ^{-1}\left(\frac{1}{2}+\frac{x^2}{2}-\frac{1}{2} \sqrt{1+6 x^2+x^4}\right)-\tanh ^{-1}\left(2+x^2-\sqrt{1+6 x^2+x^4}\right)-\frac{1}{2} \log \left(1-x^2+\sqrt{1+6 x^2+x^4}\right)","-2 \tan ^{-1}\left(\frac{1}{2}+\frac{x^2}{2}-\frac{1}{2} \sqrt{1+6 x^2+x^4}\right)-\tanh ^{-1}\left(2+x^2-\sqrt{1+6 x^2+x^4}\right)-\frac{1}{2} \log \left(1-x^2+\sqrt{1+6 x^2+x^4}\right)",1,"-2*ArcTan[1/2 + x^2/2 - Sqrt[1 + 6*x^2 + x^4]/2] - ArcTanh[2 + x^2 - Sqrt[1 + 6*x^2 + x^4]] - Log[1 - x^2 + Sqrt[1 + 6*x^2 + x^4]]/2","A",1
1114,1,83,83,0.217287,"\int \frac{\left(-1+x^2\right) \sqrt[4]{-1+2 x^2+2 x^4}}{x^2 \left(-1+2 x^2\right)} \, dx","IntegrateAlgebraic[((-1 + x^2)*(-1 + 2*x^2 + 2*x^4)^(1/4))/(x^2*(-1 + 2*x^2)),x]","-\frac{\sqrt[4]{-1+2 x^2+2 x^4}}{x}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-1+2 x^2+2 x^4}}\right)}{2^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-1+2 x^2+2 x^4}}\right)}{2^{3/4}}","-\frac{\sqrt[4]{-1+2 x^2+2 x^4}}{x}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-1+2 x^2+2 x^4}}\right)}{2^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-1+2 x^2+2 x^4}}\right)}{2^{3/4}}",1,"-((-1 + 2*x^2 + 2*x^4)^(1/4)/x) - ArcTan[(2^(1/4)*x)/(-1 + 2*x^2 + 2*x^4)^(1/4)]/2^(3/4) + ArcTanh[(2^(1/4)*x)/(-1 + 2*x^2 + 2*x^4)^(1/4)]/2^(3/4)","A",1
1115,1,83,83,0.3251064,"\int \frac{1}{\sqrt[4]{b+a x^4} \left(2 b+a x^4\right)} \, dx","IntegrateAlgebraic[1/((b + a*x^4)^(1/4)*(2*b + a*x^4)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{2} \sqrt[4]{b+a x^4}}\right)}{2\ 2^{3/4} \sqrt[4]{a} b}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{2} \sqrt[4]{b+a x^4}}\right)}{2\ 2^{3/4} \sqrt[4]{a} b}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{2} \sqrt[4]{b+a x^4}}\right)}{2\ 2^{3/4} \sqrt[4]{a} b}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{2} \sqrt[4]{b+a x^4}}\right)}{2\ 2^{3/4} \sqrt[4]{a} b}",1,"ArcTan[(a^(1/4)*x)/(2^(1/4)*(b + a*x^4)^(1/4))]/(2*2^(3/4)*a^(1/4)*b) + ArcTanh[(a^(1/4)*x)/(2^(1/4)*(b + a*x^4)^(1/4))]/(2*2^(3/4)*a^(1/4)*b)","A",1
1116,1,83,83,0.1998504,"\int \frac{\sqrt[4]{-b x^2+a x^4}}{x^2} \, dx","IntegrateAlgebraic[(-(b*x^2) + a*x^4)^(1/4)/x^2,x]","-\frac{2 \sqrt[4]{-b x^2+a x^4}}{x}-\sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)+\sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)","-\frac{2 \sqrt[4]{-b x^2+a x^4}}{x}-\sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)+\sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)",1,"(-2*(-(b*x^2) + a*x^4)^(1/4))/x - a^(1/4)*ArcTan[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)] + a^(1/4)*ArcTanh[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)]","A",1
1117,1,83,83,4.5779986,"\int \frac{1}{\sqrt[4]{-b x^2+a x^4} \left(a+b x^8\right)} \, dx","IntegrateAlgebraic[1/((-(b*x^2) + a*x^4)^(1/4)*(a + b*x^8)),x]","-\frac{\text{RootSum}\left[a^5+b^5-4 a^4 \text{$\#$1}^4+6 a^3 \text{$\#$1}^8-4 a^2 \text{$\#$1}^{12}+a \text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{8 a}","-\frac{\text{RootSum}\left[a^5+b^5-4 a^4 \text{$\#$1}^4+6 a^3 \text{$\#$1}^8-4 a^2 \text{$\#$1}^{12}+a \text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{8 a}",1,"-1/8*RootSum[a^5 + b^5 - 4*a^4*#1^4 + 6*a^3*#1^8 - 4*a^2*#1^12 + a*#1^16 & , (-Log[x] + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1])/#1 & ]/a","A",1
1118,1,83,83,0.0017521,"\int \frac{1}{\sqrt[4]{-b x^2+a x^4} \left(a+b x^8\right)} \, dx","IntegrateAlgebraic[1/((-(b*x^2) + a*x^4)^(1/4)*(a + b*x^8)),x]","-\frac{\text{RootSum}\left[a^5+b^5-4 a^4 \text{$\#$1}^4+6 a^3 \text{$\#$1}^8-4 a^2 \text{$\#$1}^{12}+a \text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{8 a}","-\frac{\text{RootSum}\left[a^5+b^5-4 a^4 \text{$\#$1}^4+6 a^3 \text{$\#$1}^8-4 a^2 \text{$\#$1}^{12}+a \text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{8 a}",1,"-1/8*RootSum[a^5 + b^5 - 4*a^4*#1^4 + 6*a^3*#1^8 - 4*a^2*#1^12 + a*#1^16 & , (-Log[x] + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1])/#1 & ]/a","A",1
1119,1,74,83,0.1621784,"\int \frac{(-1+x) \sqrt{x+\sqrt{1+x}}}{\sqrt{1+x}} \, dx","IntegrateAlgebraic[((-1 + x)*Sqrt[x + Sqrt[1 + x]])/Sqrt[1 + x],x]","\frac{1}{96} \sqrt{x+\sqrt{1+x}} \left(-29-226 \sqrt{1+x}+8 (1+x)+48 (1+x)^{3/2}\right)-\frac{115}{64} \log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}\right)","\frac{1}{96} (-21+8 x) \sqrt{x+\sqrt{1+x}}+\frac{1}{48} \sqrt{1+x} (-89+24 x) \sqrt{x+\sqrt{1+x}}-\frac{115}{64} \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}\right)",1,"(Sqrt[x + Sqrt[1 + x]]*(-29 - 226*Sqrt[1 + x] + 8*(1 + x) + 48*(1 + x)^(3/2)))/96 - (115*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]])/64","A",1
1120,1,74,83,0.1765174,"\int \frac{x \sqrt{x+\sqrt{1+x}}}{\sqrt{1+x}} \, dx","IntegrateAlgebraic[(x*Sqrt[x + Sqrt[1 + x]])/Sqrt[1 + x],x]","\frac{1}{96} \sqrt{x+\sqrt{1+x}} \left(19-130 \sqrt{1+x}+8 (1+x)+48 (1+x)^{3/2}\right)-\frac{35}{64} \log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}\right)","\frac{1}{96} (27+8 x) \sqrt{x+\sqrt{1+x}}+\frac{1}{48} \sqrt{1+x} (-41+24 x) \sqrt{x+\sqrt{1+x}}-\frac{35}{64} \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}\right)",1,"(Sqrt[x + Sqrt[1 + x]]*(19 - 130*Sqrt[1 + x] + 8*(1 + x) + 48*(1 + x)^(3/2)))/96 - (35*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]])/64","A",1
1121,1,74,83,0.1217508,"\int \sqrt{1+x} \sqrt{x+\sqrt{1+x}} \, dx","IntegrateAlgebraic[Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]],x]","\frac{1}{96} \sqrt{x+\sqrt{1+x}} \left(67-34 \sqrt{1+x}+8 (1+x)+48 (1+x)^{3/2}\right)+\frac{45}{64} \log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}\right)","\frac{1}{96} (75+8 x) \sqrt{x+\sqrt{1+x}}+\frac{1}{48} \sqrt{1+x} (7+24 x) \sqrt{x+\sqrt{1+x}}+\frac{45}{64} \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}\right)",1,"(Sqrt[x + Sqrt[1 + x]]*(67 - 34*Sqrt[1 + x] + 8*(1 + x) + 48*(1 + x)^(3/2)))/96 + (45*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]])/64","A",1
1122,1,83,83,0.0926838,"\int \frac{1+\sqrt{1+x^2}}{\sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[(1 + Sqrt[1 + x^2])/Sqrt[x + Sqrt[1 + x^2]],x]","\frac{2 \sqrt{1+x^2} \left(1-2 x+6 x^2+2 x^3\right)}{3 \left(x+\sqrt{1+x^2}\right)^{5/2}}+\frac{2 \left(-7+20 x-5 x^2+30 x^3+10 x^4\right)}{15 \left(x+\sqrt{1+x^2}\right)^{5/2}}","\frac{2 \sqrt{1+x^2} \left(1-2 x+6 x^2+2 x^3\right)}{3 \left(x+\sqrt{1+x^2}\right)^{5/2}}+\frac{2 \left(-7+20 x-5 x^2+30 x^3+10 x^4\right)}{15 \left(x+\sqrt{1+x^2}\right)^{5/2}}",1,"(2*Sqrt[1 + x^2]*(1 - 2*x + 6*x^2 + 2*x^3))/(3*(x + Sqrt[1 + x^2])^(5/2)) + (2*(-7 + 20*x - 5*x^2 + 30*x^3 + 10*x^4))/(15*(x + Sqrt[1 + x^2])^(5/2))","A",1
1123,0,0,83,0.0018944,"\int \frac{\sqrt{1-4 x+x^2}+\left(1-4 x+x^2\right)^{3/2}}{\sqrt{1-4 x+x^2}+\left(1-4 x+x^2\right)^{3/2}-\left(1-4 x+x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[(Sqrt[1 - 4*x + x^2] + (1 - 4*x + x^2)^(3/2))/(Sqrt[1 - 4*x + x^2] + (1 - 4*x + x^2)^(3/2) - (1 - 4*x + x^2)^(5/2)),x]","\int \frac{\sqrt{1-4 x+x^2}+\left(1-4 x+x^2\right)^{3/2}}{\sqrt{1-4 x+x^2}+\left(1-4 x+x^2\right)^{3/2}-\left(1-4 x+x^2\right)^{5/2}} \, dx","-\frac{1}{2} \text{RootSum}\left[-1-4 \text{$\#$1}+17 \text{$\#$1}^2-8 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{2 \log (x-\text{$\#$1})-4 \log (x-\text{$\#$1}) \text{$\#$1}+\log (x-\text{$\#$1}) \text{$\#$1}^2}{-2+17 \text{$\#$1}-12 \text{$\#$1}^2+2 \text{$\#$1}^3}\&\right]",1,"Could not integrate","F",-1
1124,1,83,83,4.0552002,"\int \frac{x^2}{\sqrt{-b x+a^2 x^2} \left(a x^2+x \sqrt{-b x+a^2 x^2}\right)^{3/2}} \, dx","IntegrateAlgebraic[x^2/(Sqrt[-(b*x) + a^2*x^2]*(a*x^2 + x*Sqrt[-(b*x) + a^2*x^2])^(3/2)),x]","-\frac{4 a \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{b^2}+\frac{4 \sqrt{-b x+a^2 x^2} \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{b^2 x}","-\frac{4 a \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{b^2}+\frac{4 \sqrt{-b x+a^2 x^2} \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{b^2 x}",1,"(-4*a*Sqrt[x*(a*x + Sqrt[-(b*x) + a^2*x^2])])/b^2 + (4*Sqrt[-(b*x) + a^2*x^2]*Sqrt[x*(a*x + Sqrt[-(b*x) + a^2*x^2])])/(b^2*x)","A",1
1125,1,84,84,1.0085411,"\int \frac{a^2 b-a (2 a+b) x+3 a x^2-x^3}{x (-b+x) \sqrt{x (-a+x) (-b+x)} \left(a+(-1-b d) x+d x^2\right)} \, dx","IntegrateAlgebraic[(a^2*b - a*(2*a + b)*x + 3*a*x^2 - x^3)/(x*(-b + x)*Sqrt[x*(-a + x)*(-b + x)]*(a + (-1 - b*d)*x + d*x^2)),x]","\frac{2 \sqrt{a b x-a x^2-b x^2+x^3}}{(b-x) x}+2 \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{a b x+(-a-b) x^2+x^3}}{\sqrt{d} x (-b+x)}\right)","\frac{2 \sqrt{a b x-a x^2-b x^2+x^3}}{(b-x) x}+2 \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{a b x+(-a-b) x^2+x^3}}{\sqrt{d} x (-b+x)}\right)",1,"(2*Sqrt[a*b*x - a*x^2 - b*x^2 + x^3])/((b - x)*x) + 2*Sqrt[d]*ArcTanh[Sqrt[a*b*x + (-a - b)*x^2 + x^3]/(Sqrt[d]*x*(-b + x))]","A",1
1126,1,84,84,0.0486488,"\int \frac{\left(1+x^3\right)^{2/3}}{x} \, dx","IntegrateAlgebraic[(1 + x^3)^(2/3)/x,x]","\frac{1}{2} \left(1+x^3\right)^{2/3}+\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^3}}{\sqrt{3}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(-1+\sqrt[3]{1+x^3}\right)-\frac{1}{6} \log \left(1+\sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)","\frac{1}{2} \left(1+x^3\right)^{2/3}+\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^3}}{\sqrt{3}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(-1+\sqrt[3]{1+x^3}\right)-\frac{1}{6} \log \left(1+\sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)",1,"(1 + x^3)^(2/3)/2 + ArcTan[1/Sqrt[3] + (2*(1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3] + Log[-1 + (1 + x^3)^(1/3)]/3 - Log[1 + (1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/6","A",1
1127,1,84,84,0.3063226,"\int \frac{\left(-b+2 a x^2\right) \sqrt[4]{b x^2+a x^4}}{b+a x^2} \, dx","IntegrateAlgebraic[((-b + 2*a*x^2)*(b*x^2 + a*x^4)^(1/4))/(b + a*x^2),x]","x \sqrt[4]{b x^2+a x^4}+\frac{5 b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{2 a^{3/4}}-\frac{5 b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{2 a^{3/4}}","x \sqrt[4]{b x^2+a x^4}+\frac{5 b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{2 a^{3/4}}-\frac{5 b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{2 a^{3/4}}",1,"x*(b*x^2 + a*x^4)^(1/4) + (5*b*ArcTan[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/(2*a^(3/4)) - (5*b*ArcTanh[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/(2*a^(3/4))","A",1
1128,1,84,84,1.4943116,"\int \frac{\left(-1+x^4\right) \sqrt{1+x^2+x^4} \left(1+x^2+3 x^4+x^6+x^8\right)}{\left(1+x^4\right)^3 \left(1-x^2+x^4\right)} \, dx","IntegrateAlgebraic[((-1 + x^4)*Sqrt[1 + x^2 + x^4]*(1 + x^2 + 3*x^4 + x^6 + x^8))/((1 + x^4)^3*(1 - x^2 + x^4)),x]","\frac{\sqrt{1+x^2+x^4} \left(9 x+2 x^3+9 x^5\right)}{8 \left(1+x^4\right)^2}+\frac{31}{8} \tanh ^{-1}\left(\frac{x}{\sqrt{1+x^2+x^4}}\right)-3 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^2+x^4}}\right)","\frac{\sqrt{1+x^2+x^4} \left(9 x+2 x^3+9 x^5\right)}{8 \left(1+x^4\right)^2}+\frac{31}{8} \tanh ^{-1}\left(\frac{x}{\sqrt{1+x^2+x^4}}\right)-3 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^2+x^4}}\right)",1,"(Sqrt[1 + x^2 + x^4]*(9*x + 2*x^3 + 9*x^5))/(8*(1 + x^4)^2) + (31*ArcTanh[x/Sqrt[1 + x^2 + x^4]])/8 - 3*Sqrt[2]*ArcTanh[(Sqrt[2]*x)/Sqrt[1 + x^2 + x^4]]","A",1
1129,1,84,84,37.0484622,"\int \frac{\left(1+x^6\right) \left(-1+x^3+x^6\right) \sqrt{1+x^{12}}}{x^7 \left(-1-x^3+x^6\right)} \, dx","IntegrateAlgebraic[((1 + x^6)*(-1 + x^3 + x^6)*Sqrt[1 + x^12])/(x^7*(-1 - x^3 + x^6)),x]","\frac{\left(-1+4 x^3+x^6\right) \sqrt{1+x^{12}}}{6 x^6}-\frac{4 \tanh ^{-1}\left(\frac{\sqrt{3} x^3}{-1-x^3+x^6+\sqrt{1+x^{12}}}\right)}{\sqrt{3}}-3 \log (x)+\log \left(-1+x^6+\sqrt{1+x^{12}}\right)","\frac{\left(-1+4 x^3+x^6\right) \sqrt{1+x^{12}}}{6 x^6}-\frac{4 \tanh ^{-1}\left(\frac{\sqrt{3} x^3}{-1-x^3+x^6+\sqrt{1+x^{12}}}\right)}{\sqrt{3}}-3 \log (x)+\log \left(-1+x^6+\sqrt{1+x^{12}}\right)",1,"((-1 + 4*x^3 + x^6)*Sqrt[1 + x^12])/(6*x^6) - (4*ArcTanh[(Sqrt[3]*x^3)/(-1 - x^3 + x^6 + Sqrt[1 + x^12])])/Sqrt[3] - 3*Log[x] + Log[-1 + x^6 + Sqrt[1 + x^12]]","A",1
1130,1,55,84,0.031392,"\int \sqrt{d+c \sqrt{b+a x}} \, dx","IntegrateAlgebraic[Sqrt[d + c*Sqrt[b + a*x]],x]","\frac{4 \sqrt{d+c \sqrt{b+a x}} \left(-2 d^2+c d \sqrt{b+a x}+3 c^2 (b+a x)\right)}{15 a c^2}","\frac{4 d \sqrt{b+a x} \sqrt{d+c \sqrt{b+a x}}}{15 a c}+\frac{4 \left(3 b c^2-2 d^2+3 a c^2 x\right) \sqrt{d+c \sqrt{b+a x}}}{15 a c^2}",1,"(4*Sqrt[d + c*Sqrt[b + a*x]]*(-2*d^2 + c*d*Sqrt[b + a*x] + 3*c^2*(b + a*x)))/(15*a*c^2)","A",1
1131,1,85,85,0.0499969,"\int \frac{1+2 x^2}{x \left(1+x^2\right)^{2/3}} \, dx","IntegrateAlgebraic[(1 + 2*x^2)/(x*(1 + x^2)^(2/3)),x]","3 \sqrt[3]{1+x^2}-\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^2}}{\sqrt{3}}\right)+\frac{1}{2} \log \left(-1+\sqrt[3]{1+x^2}\right)-\frac{1}{4} \log \left(1+\sqrt[3]{1+x^2}+\left(1+x^2\right)^{2/3}\right)","3 \sqrt[3]{1+x^2}-\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^2}}{\sqrt{3}}\right)+\frac{1}{2} \log \left(-1+\sqrt[3]{1+x^2}\right)-\frac{1}{4} \log \left(1+\sqrt[3]{1+x^2}+\left(1+x^2\right)^{2/3}\right)",1,"3*(1 + x^2)^(1/3) - (Sqrt[3]*ArcTan[1/Sqrt[3] + (2*(1 + x^2)^(1/3))/Sqrt[3]])/2 + Log[-1 + (1 + x^2)^(1/3)]/2 - Log[1 + (1 + x^2)^(1/3) + (1 + x^2)^(2/3)]/4","A",1
1132,1,85,85,0.186061,"\int \frac{x}{\sqrt{(1-x) x \left(1-k^2 x\right)} \left(-1+k^2 x^2\right)} \, dx","IntegrateAlgebraic[x/(Sqrt[(1 - x)*x*(1 - k^2*x)]*(-1 + k^2*x^2)),x]","-\frac{\tan ^{-1}\left(\frac{(-1+k) x}{\sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}\right)}{2 (-1+k) k}+\frac{\tan ^{-1}\left(\frac{(1+k) x}{\sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}\right)}{2 k (1+k)}","-\frac{\tan ^{-1}\left(\frac{(-1+k) x}{\sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}\right)}{2 (-1+k) k}+\frac{\tan ^{-1}\left(\frac{(1+k) x}{\sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}\right)}{2 k (1+k)}",1,"-1/2*ArcTan[((-1 + k)*x)/Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3]]/((-1 + k)*k) + ArcTan[((1 + k)*x)/Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3]]/(2*k*(1 + k))","A",1
1133,1,73,85,0.6051149,"\int \frac{1+x}{\left(-1+2 x+x^2\right) \sqrt{-x+x^3}} \, dx","IntegrateAlgebraic[(1 + x)/((-1 + 2*x + x^2)*Sqrt[-x + x^3]),x]","\frac{1}{4} \left(-2+\sqrt{2}\right) \tan ^{-1}\left(\frac{\left(-1+\sqrt{2}\right) \sqrt{-x+x^3}}{1+x}\right)+\frac{1}{4} \left(2+\sqrt{2}\right) \tan ^{-1}\left(\frac{\left(1+\sqrt{2}\right) \sqrt{-x+x^3}}{1+x}\right)","\frac{1}{4} \left(-2+\sqrt{2}\right) \tan ^{-1}\left(\frac{\sqrt{3-2 \sqrt{2}} \sqrt{-x+x^3}}{1+x}\right)+\frac{1}{4} \left(2+\sqrt{2}\right) \tan ^{-1}\left(\frac{\sqrt{3+2 \sqrt{2}} \sqrt{-x+x^3}}{1+x}\right)",1,"((-2 + Sqrt[2])*ArcTan[((-1 + Sqrt[2])*Sqrt[-x + x^3])/(1 + x)])/4 + ((2 + Sqrt[2])*ArcTan[((1 + Sqrt[2])*Sqrt[-x + x^3])/(1 + x)])/4","A",1
1134,1,73,85,0.3876714,"\int \frac{-x+x^2}{\left(-1+2 x+x^2\right) \sqrt{-x+x^3}} \, dx","IntegrateAlgebraic[(-x + x^2)/((-1 + 2*x + x^2)*Sqrt[-x + x^3]),x]","\frac{1}{4} \left(2+\sqrt{2}\right) \tan ^{-1}\left(\frac{\left(-1+\sqrt{2}\right) \sqrt{-x+x^3}}{1+x}\right)+\frac{1}{4} \left(-2+\sqrt{2}\right) \tan ^{-1}\left(\frac{\left(1+\sqrt{2}\right) \sqrt{-x+x^3}}{1+x}\right)","\frac{1}{4} \left(2+\sqrt{2}\right) \tan ^{-1}\left(\frac{\sqrt{3-2 \sqrt{2}} \sqrt{-x+x^3}}{1+x}\right)+\frac{1}{4} \left(-2+\sqrt{2}\right) \tan ^{-1}\left(\frac{\sqrt{3+2 \sqrt{2}} \sqrt{-x+x^3}}{1+x}\right)",1,"((2 + Sqrt[2])*ArcTan[((-1 + Sqrt[2])*Sqrt[-x + x^3])/(1 + x)])/4 + ((-2 + Sqrt[2])*ArcTan[((1 + Sqrt[2])*Sqrt[-x + x^3])/(1 + x)])/4","A",1
1135,1,85,85,0.1232159,"\int \frac{3+x^2}{\sqrt[3]{1+x^2} \left(1+x^2+x^3\right)} \, dx","IntegrateAlgebraic[(3 + x^2)/((1 + x^2)^(1/3)*(1 + x^2 + x^3)),x]","-\sqrt{3} \tan ^{-1}\left(\frac{-\frac{2 x}{\sqrt{3}}+\frac{\sqrt[3]{1+x^2}}{\sqrt{3}}}{\sqrt[3]{1+x^2}}\right)+\log \left(x+\sqrt[3]{1+x^2}\right)-\frac{1}{2} \log \left(x^2-x \sqrt[3]{1+x^2}+\left(1+x^2\right)^{2/3}\right)","-\sqrt{3} \tan ^{-1}\left(\frac{-\frac{2 x}{\sqrt{3}}+\frac{\sqrt[3]{1+x^2}}{\sqrt{3}}}{\sqrt[3]{1+x^2}}\right)+\log \left(x+\sqrt[3]{1+x^2}\right)-\frac{1}{2} \log \left(x^2-x \sqrt[3]{1+x^2}+\left(1+x^2\right)^{2/3}\right)",1,"-(Sqrt[3]*ArcTan[((-2*x)/Sqrt[3] + (1 + x^2)^(1/3)/Sqrt[3])/(1 + x^2)^(1/3)]) + Log[x + (1 + x^2)^(1/3)] - Log[x^2 - x*(1 + x^2)^(1/3) + (1 + x^2)^(2/3)]/2","A",1
1136,1,85,85,0.1927628,"\int \frac{x}{\left(-1+x^3\right) \left(-1+2 x^3\right)^{2/3}} \, dx","IntegrateAlgebraic[x/((-1 + x^3)*(-1 + 2*x^3)^(2/3)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+2 x^3}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(-x+\sqrt[3]{-1+2 x^3}\right)-\frac{1}{6} \log \left(x^2+x \sqrt[3]{-1+2 x^3}+\left(-1+2 x^3\right)^{2/3}\right)","\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+2 x^3}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(-x+\sqrt[3]{-1+2 x^3}\right)-\frac{1}{6} \log \left(x^2+x \sqrt[3]{-1+2 x^3}+\left(-1+2 x^3\right)^{2/3}\right)",1,"ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + 2*x^3)^(1/3))]/Sqrt[3] + Log[-x + (-1 + 2*x^3)^(1/3)]/3 - Log[x^2 + x*(-1 + 2*x^3)^(1/3) + (-1 + 2*x^3)^(2/3)]/6","A",1
1137,1,85,85,3.1623768,"\int \frac{\left(-1-2 (-1+k) x+k x^2\right) \left(1-2 k x+k^2 x^2\right)}{((1-x) x (1-k x))^{3/4} \left(-d+(1+3 d k) x-\left(1+3 d k^2\right) x^2+d k^3 x^3\right)} \, dx","IntegrateAlgebraic[((-1 - 2*(-1 + k)*x + k*x^2)*(1 - 2*k*x + k^2*x^2))/(((1 - x)*x*(1 - k*x))^(3/4)*(-d + (1 + 3*d*k)*x - (1 + 3*d*k^2)*x^2 + d*k^3*x^3)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(x+(-1-k) x^2+k x^3\right)^{3/4}}{(-1+x) x}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \left(x+(-1-k) x^2+k x^3\right)^{3/4}}{(-1+x) x}\right)}{d^{3/4}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(x+(-1-k) x^2+k x^3\right)^{3/4}}{(-1+x) x}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \left(x+(-1-k) x^2+k x^3\right)^{3/4}}{(-1+x) x}\right)}{d^{3/4}}",1,"(2*ArcTan[(d^(1/4)*(x + (-1 - k)*x^2 + k*x^3)^(3/4))/((-1 + x)*x)])/d^(3/4) - (2*ArcTanh[(d^(1/4)*(x + (-1 - k)*x^2 + k*x^3)^(3/4))/((-1 + x)*x)])/d^(3/4)","A",1
1138,1,85,85,0.3533239,"\int \frac{-1+x^4}{\sqrt{-x+x^3} \left(1+x^4\right)} \, dx","IntegrateAlgebraic[(-1 + x^4)/(Sqrt[-x + x^3]*(1 + x^4)),x]","\frac{\tan ^{-1}\left(\frac{2^{3/4} \sqrt{-x+x^3}}{1+\sqrt{2} x-x^2}\right)}{2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{-\frac{1}{2^{3/4}}+\frac{x}{\sqrt[4]{2}}+\frac{x^2}{2^{3/4}}}{\sqrt{-x+x^3}}\right)}{2^{3/4}}","\frac{\tan ^{-1}\left(\frac{2^{3/4} \sqrt{-x+x^3}}{1+\sqrt{2} x-x^2}\right)}{2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{-\frac{1}{2^{3/4}}+\frac{x}{\sqrt[4]{2}}+\frac{x^2}{2^{3/4}}}{\sqrt{-x+x^3}}\right)}{2^{3/4}}",1,"ArcTan[(2^(3/4)*Sqrt[-x + x^3])/(1 + Sqrt[2]*x - x^2)]/2^(3/4) - ArcTanh[(-2^(-3/4) + x/2^(1/4) + x^2/2^(3/4))/Sqrt[-x + x^3]]/2^(3/4)","A",1
1139,1,85,85,0.3197427,"\int \frac{-1+x^2}{\left(1+x+x^2\right) \sqrt[3]{x^2+x^4}} \, dx","IntegrateAlgebraic[(-1 + x^2)/((1 + x + x^2)*(x^2 + x^4)^(1/3)),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{x^2+x^4}}\right)-\log \left(x+\sqrt[3]{x^2+x^4}\right)+\frac{1}{2} \log \left(x^2-x \sqrt[3]{x^2+x^4}+\left(x^2+x^4\right)^{2/3}\right)","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{x^2+x^4}}\right)-\log \left(x+\sqrt[3]{x^2+x^4}\right)+\frac{1}{2} \log \left(x^2-x \sqrt[3]{x^2+x^4}+\left(x^2+x^4\right)^{2/3}\right)",1,"-(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(x^2 + x^4)^(1/3))]) - Log[x + (x^2 + x^4)^(1/3)] + Log[x^2 - x*(x^2 + x^4)^(1/3) + (x^2 + x^4)^(2/3)]/2","A",1
1140,1,85,85,0.2746431,"\int \frac{\left(-2 b+a x^4\right) \left(b+a x^4\right)^{3/4}}{x^8} \, dx","IntegrateAlgebraic[((-2*b + a*x^4)*(b + a*x^4)^(3/4))/x^8,x]","\frac{\left(6 b-a x^4\right) \left(b+a x^4\right)^{3/4}}{21 x^7}+\frac{1}{2} a^{7/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)+\frac{1}{2} a^{7/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)","\frac{\left(6 b-a x^4\right) \left(b+a x^4\right)^{3/4}}{21 x^7}+\frac{1}{2} a^{7/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)+\frac{1}{2} a^{7/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)",1,"((6*b - a*x^4)*(b + a*x^4)^(3/4))/(21*x^7) + (a^(7/4)*ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)])/2 + (a^(7/4)*ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)])/2","A",1
1141,1,85,85,0.2658601,"\int \frac{\left(-b+a x^4\right)^{3/4} \left(-b+2 a x^4\right)}{x^8} \, dx","IntegrateAlgebraic[((-b + a*x^4)^(3/4)*(-b + 2*a*x^4))/x^8,x]","\frac{\left(3 b-17 a x^4\right) \left(-b+a x^4\right)^{3/4}}{21 x^7}+a^{7/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)+a^{7/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)","\frac{\left(3 b-17 a x^4\right) \left(-b+a x^4\right)^{3/4}}{21 x^7}+a^{7/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)+a^{7/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)",1,"((3*b - 17*a*x^4)*(-b + a*x^4)^(3/4))/(21*x^7) + a^(7/4)*ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)] + a^(7/4)*ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]","A",1
1142,1,85,85,1.162242,"\int \frac{\left(-q+p x^2\right) \left(a q+b x+a p x^2\right) \sqrt{q^2+p^2 x^4}}{x^4} \, dx","IntegrateAlgebraic[((-q + p*x^2)*(a*q + b*x + a*p*x^2)*Sqrt[q^2 + p^2*x^4])/x^4,x]","\frac{\sqrt{q^2+p^2 x^4} \left(2 a q^2+3 b q x+3 b p x^3+2 a p^2 x^4\right)}{6 x^3}+b p q \log (x)-b p q \log \left(q+p x^2+\sqrt{q^2+p^2 x^4}\right)","\frac{\sqrt{q^2+p^2 x^4} \left(2 a q^2+3 b q x+3 b p x^3+2 a p^2 x^4\right)}{6 x^3}+b p q \log (x)-b p q \log \left(q+p x^2+\sqrt{q^2+p^2 x^4}\right)",1,"(Sqrt[q^2 + p^2*x^4]*(2*a*q^2 + 3*b*q*x + 3*b*p*x^3 + 2*a*p^2*x^4))/(6*x^3) + b*p*q*Log[x] - b*p*q*Log[q + p*x^2 + Sqrt[q^2 + p^2*x^4]]","A",1
1143,1,85,85,0.9306808,"\int \frac{-1+2 x^3}{\left(1+x+x^3\right) \sqrt[3]{x^2+x^5}} \, dx","IntegrateAlgebraic[(-1 + 2*x^3)/((1 + x + x^3)*(x^2 + x^5)^(1/3)),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{x^2+x^5}}\right)-\log \left(x+\sqrt[3]{x^2+x^5}\right)+\frac{1}{2} \log \left(x^2-x \sqrt[3]{x^2+x^5}+\left(x^2+x^5\right)^{2/3}\right)","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{x^2+x^5}}\right)-\log \left(x+\sqrt[3]{x^2+x^5}\right)+\frac{1}{2} \log \left(x^2-x \sqrt[3]{x^2+x^5}+\left(x^2+x^5\right)^{2/3}\right)",1,"-(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(x^2 + x^5)^(1/3))]) - Log[x + (x^2 + x^5)^(1/3)] + Log[x^2 - x*(x^2 + x^5)^(1/3) + (x^2 + x^5)^(2/3)]/2","A",1
1144,1,85,85,2.684365,"\int \frac{2+3 x^5}{\left(-1+x^2+x^5\right) \sqrt[3]{-x+x^6}} \, dx","IntegrateAlgebraic[(2 + 3*x^5)/((-1 + x^2 + x^5)*(-x + x^6)^(1/3)),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-x+x^6}}\right)-\log \left(x+\sqrt[3]{-x+x^6}\right)+\frac{1}{2} \log \left(x^2-x \sqrt[3]{-x+x^6}+\left(-x+x^6\right)^{2/3}\right)","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-x+x^6}}\right)-\log \left(x+\sqrt[3]{-x+x^6}\right)+\frac{1}{2} \log \left(x^2-x \sqrt[3]{-x+x^6}+\left(-x+x^6\right)^{2/3}\right)",1,"-(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(-x + x^6)^(1/3))]) - Log[x + (-x + x^6)^(1/3)] + Log[x^2 - x*(-x + x^6)^(1/3) + (-x + x^6)^(2/3)]/2","A",1
1145,1,87,85,0.1913772,"\int \frac{\sqrt{-1+x^6} \left(-1+2 x^6\right)^2}{x^4 \left(-1+4 x^6\right)} \, dx","IntegrateAlgebraic[(Sqrt[-1 + x^6]*(-1 + 2*x^6)^2)/(x^4*(-1 + 4*x^6)),x]","\frac{\sqrt{-1+x^6} \left(2+x^6\right)}{6 x^3}-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{4 x^6}{\sqrt{3}}+\frac{4 x^3 \sqrt{-1+x^6}}{\sqrt{3}}\right)}{4 \sqrt{3}}+\frac{5}{12} \log \left(-x^3+\sqrt{-1+x^6}\right)","\frac{\sqrt{-1+x^6} \left(2+x^6\right)}{6 x^3}+\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{4 x^6}{\sqrt{3}}-\frac{4 x^3 \sqrt{-1+x^6}}{\sqrt{3}}\right)}{4 \sqrt{3}}-\frac{5}{12} \log \left(x^3+\sqrt{-1+x^6}\right)",1,"(Sqrt[-1 + x^6]*(2 + x^6))/(6*x^3) - ArcTan[1/Sqrt[3] - (4*x^6)/Sqrt[3] + (4*x^3*Sqrt[-1 + x^6])/Sqrt[3]]/(4*Sqrt[3]) + (5*Log[-x^3 + Sqrt[-1 + x^6]])/12","A",1
1146,1,85,85,0.2687973,"\int \frac{1+x^4}{\sqrt[4]{-1+x^4} \left(1+x^4+x^8\right)} \, dx","IntegrateAlgebraic[(1 + x^4)/((-1 + x^4)^(1/4)*(1 + x^4 + x^8)),x]","\frac{1}{4} \text{RootSum}\left[3-3 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 \log (x)+2 \log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 \text{$\#$1}+2 \text{$\#$1}^5}\&\right]","\frac{1}{4} \text{RootSum}\left[3-3 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 \log (x)+2 \log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 \text{$\#$1}+2 \text{$\#$1}^5}\&\right]",1,"RootSum[3 - 3*#1^4 + #1^8 & , (-2*Log[x] + 2*Log[(-1 + x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-1 + x^4)^(1/4) - x*#1]*#1^4)/(-3*#1 + 2*#1^5) & ]/4","A",1
1147,1,85,85,0.2749679,"\int \frac{-1+2 x^4}{\sqrt[4]{-1+x^4} \left(-2-x^4+2 x^8\right)} \, dx","IntegrateAlgebraic[(-1 + 2*x^4)/((-1 + x^4)^(1/4)*(-2 - x^4 + 2*x^8)),x]","\frac{1}{4} \text{RootSum}\left[1-5 \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{\log (x)-\log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-5 \text{$\#$1}+4 \text{$\#$1}^5}\&\right]","\frac{1}{4} \text{RootSum}\left[1-5 \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{\log (x)-\log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-5 \text{$\#$1}+4 \text{$\#$1}^5}\&\right]",1,"RootSum[1 - 5*#1^4 + 2*#1^8 & , (Log[x] - Log[(-1 + x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-1 + x^4)^(1/4) - x*#1]*#1^4)/(-5*#1 + 4*#1^5) & ]/4","A",1
1148,1,85,85,0.3102398,"\int \frac{-1+x^4+2 x^8}{\sqrt[4]{-1+x^4} \left(1-x^4+x^8\right)} \, dx","IntegrateAlgebraic[(-1 + x^4 + 2*x^8)/((-1 + x^4)^(1/4)*(1 - x^4 + x^8)),x]","\tan ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)+\frac{3}{4} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}^3+\log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right) \text{$\#$1}^3}{-1+2 \text{$\#$1}^4}\&\right]","\tan ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)+\frac{3}{4} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}^3+\log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right) \text{$\#$1}^3}{-1+2 \text{$\#$1}^4}\&\right]",1,"ArcTan[x/(-1 + x^4)^(1/4)] + ArcTanh[x/(-1 + x^4)^(1/4)] + (3*RootSum[1 - #1^4 + #1^8 & , (-(Log[x]*#1^3) + Log[(-1 + x^4)^(1/4) - x*#1]*#1^3)/(-1 + 2*#1^4) & ])/4","A",1
1149,1,85,85,2.5743249,"\int \frac{1}{\sqrt[4]{b x^2+a x^4} \left(-2 b+a x^8\right)} \, dx","IntegrateAlgebraic[1/((b*x^2 + a*x^4)^(1/4)*(-2*b + a*x^8)),x]","\frac{\text{RootSum}\left[2 a^4-a b^3-8 a^3 \text{$\#$1}^4+12 a^2 \text{$\#$1}^8-8 a \text{$\#$1}^{12}+2 \text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{16 b}","\frac{\text{RootSum}\left[2 a^4-a b^3-8 a^3 \text{$\#$1}^4+12 a^2 \text{$\#$1}^8-8 a \text{$\#$1}^{12}+2 \text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{16 b}",1,"RootSum[2*a^4 - a*b^3 - 8*a^3*#1^4 + 12*a^2*#1^8 - 8*a*#1^12 + 2*#1^16 & , (-Log[x] + Log[(b*x^2 + a*x^4)^(1/4) - x*#1])/#1 & ]/(16*b)","A",1
1150,1,85,85,0.0015833,"\int \frac{1}{\sqrt[4]{b x^2+a x^4} \left(-2 b+a x^8\right)} \, dx","IntegrateAlgebraic[1/((b*x^2 + a*x^4)^(1/4)*(-2*b + a*x^8)),x]","\frac{\text{RootSum}\left[2 a^4-a b^3-8 a^3 \text{$\#$1}^4+12 a^2 \text{$\#$1}^8-8 a \text{$\#$1}^{12}+2 \text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{16 b}","\frac{\text{RootSum}\left[2 a^4-a b^3-8 a^3 \text{$\#$1}^4+12 a^2 \text{$\#$1}^8-8 a \text{$\#$1}^{12}+2 \text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{16 b}",1,"RootSum[2*a^4 - a*b^3 - 8*a^3*#1^4 + 12*a^2*#1^8 - 8*a*#1^12 + 2*#1^16 & , (-Log[x] + Log[(b*x^2 + a*x^4)^(1/4) - x*#1])/#1 & ]/(16*b)","A",1
1151,1,85,85,2.6634643,"\int \frac{\left(4+x^5\right) \sqrt[4]{-2+x^4+2 x^5} \left(2-4 x^5+x^8+2 x^{10}\right)}{x^{10} \left(-1+x^5\right)} \, dx","IntegrateAlgebraic[((4 + x^5)*(-2 + x^4 + 2*x^5)^(1/4)*(2 - 4*x^5 + x^8 + 2*x^10))/(x^10*(-1 + x^5)),x]","\frac{4 \sqrt[4]{-2+x^4+2 x^5} \left(10-x^4-20 x^5+43 x^8+x^9+10 x^{10}\right)}{45 x^9}+2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{-2+x^4+2 x^5}}\right)-2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-2+x^4+2 x^5}}\right)","\frac{4 \sqrt[4]{-2+x^4+2 x^5} \left(10-x^4-20 x^5+43 x^8+x^9+10 x^{10}\right)}{45 x^9}+2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{-2+x^4+2 x^5}}\right)-2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-2+x^4+2 x^5}}\right)",1,"(4*(-2 + x^4 + 2*x^5)^(1/4)*(10 - x^4 - 20*x^5 + 43*x^8 + x^9 + 10*x^10))/(45*x^9) + 2*ArcTan[x/(-2 + x^4 + 2*x^5)^(1/4)] - 2*ArcTanh[x/(-2 + x^4 + 2*x^5)^(1/4)]","A",1
1152,1,85,85,0.4244145,"\int \frac{1+x^{12}}{\sqrt{1+x^4} \left(-1+x^{12}\right)} \, dx","IntegrateAlgebraic[(1 + x^12)/(Sqrt[1 + x^4]*(-1 + x^12)),x]","-\frac{1}{3} \tan ^{-1}\left(\frac{x}{\sqrt{1+x^4}}\right)-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{6 \sqrt{2}}-\frac{1}{3} \tanh ^{-1}\left(\frac{x}{\sqrt{1+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{6 \sqrt{2}}","-\frac{1}{3} \tan ^{-1}\left(\frac{x}{\sqrt{1+x^4}}\right)-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{6 \sqrt{2}}-\frac{1}{3} \tanh ^{-1}\left(\frac{x}{\sqrt{1+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{6 \sqrt{2}}",1,"-1/3*ArcTan[x/Sqrt[1 + x^4]] - ArcTan[(Sqrt[2]*x)/Sqrt[1 + x^4]]/(6*Sqrt[2]) - ArcTanh[x/Sqrt[1 + x^4]]/3 - ArcTanh[(Sqrt[2]*x)/Sqrt[1 + x^4]]/(6*Sqrt[2])","A",1
1153,1,105,85,18.1827293,"\int \frac{\sqrt{1-x^6} \left(1+2 x^6\right) \left(1+x^2-x^4-2 x^6-x^8+x^{12}\right)}{\left(-1+x^6\right) \left(-1+2 x^6-3 x^{12}+x^{18}\right)} \, dx","IntegrateAlgebraic[(Sqrt[1 - x^6]*(1 + 2*x^6)*(1 + x^2 - x^4 - 2*x^6 - x^8 + x^12))/((-1 + x^6)*(-1 + 2*x^6 - 3*x^12 + x^18)),x]","-\frac{1}{3} \tan ^{-1}\left(\frac{x}{\sqrt{1-x^6}}\right)+\frac{1}{3} \left(1+i \sqrt{3}\right) \tan ^{-1}\left(\frac{\left(1-i \sqrt{3}\right) x}{2 \sqrt{1-x^6}}\right)+\frac{1}{3} \left(1-i \sqrt{3}\right) \tan ^{-1}\left(\frac{\left(1+i \sqrt{3}\right) x}{2 \sqrt{1-x^6}}\right)","-\frac{1}{3} \tan ^{-1}\left(\frac{x}{\sqrt{1-x^6}}\right)-\frac{1}{3} \tan ^{-1}\left(\frac{x \sqrt{1-x^6}}{-1+x^2+x^6}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt{3} x \sqrt{1-x^6}}{-1-x^2+x^6}\right)}{\sqrt{3}}",1,"-1/3*ArcTan[x/Sqrt[1 - x^6]] + ((1 + I*Sqrt[3])*ArcTan[((1 - I*Sqrt[3])*x)/(2*Sqrt[1 - x^6])])/3 + ((1 - I*Sqrt[3])*ArcTan[((1 + I*Sqrt[3])*x)/(2*Sqrt[1 - x^6])])/3","C",1
1154,1,85,85,0.4090198,"\int \frac{x^2-\sqrt{1+x^2}}{\sqrt{1+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[(x^2 - Sqrt[1 + x^2])/Sqrt[1 + Sqrt[1 + x^2]],x]","-\frac{4 x \sqrt{1+x^2}}{5 \sqrt{1+\sqrt{1+x^2}}}+\frac{2 x \left(2+x^2\right)}{5 \sqrt{1+\sqrt{1+x^2}}}-\sqrt{2} \tan ^{-1}\left(\frac{x}{\sqrt{2} \sqrt{1+\sqrt{1+x^2}}}\right)","-\frac{4 x \sqrt{1+x^2}}{5 \sqrt{1+\sqrt{1+x^2}}}+\frac{2 x \left(2+x^2\right)}{5 \sqrt{1+\sqrt{1+x^2}}}-\sqrt{2} \tan ^{-1}\left(\frac{x}{\sqrt{2} \sqrt{1+\sqrt{1+x^2}}}\right)",1,"(-4*x*Sqrt[1 + x^2])/(5*Sqrt[1 + Sqrt[1 + x^2]]) + (2*x*(2 + x^2))/(5*Sqrt[1 + Sqrt[1 + x^2]]) - Sqrt[2]*ArcTan[x/(Sqrt[2]*Sqrt[1 + Sqrt[1 + x^2]])]","A",1
1155,1,86,86,0.0933773,"\int \frac{1}{(-1+x) \sqrt[3]{2-2 x+x^2}} \, dx","IntegrateAlgebraic[1/((-1 + x)*(2 - 2*x + x^2)^(1/3)),x]","\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{2-2 x+x^2}}{\sqrt{3}}\right)+\frac{1}{2} \log \left(-1+\sqrt[3]{2-2 x+x^2}\right)-\frac{1}{4} \log \left(1+\sqrt[3]{2-2 x+x^2}+\left(2-2 x+x^2\right)^{2/3}\right)","\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{2-2 x+x^2}}{\sqrt{3}}\right)+\frac{1}{2} \log \left(-1+\sqrt[3]{2-2 x+x^2}\right)-\frac{1}{4} \log \left(1+\sqrt[3]{2-2 x+x^2}+\left(2-2 x+x^2\right)^{2/3}\right)",1,"(Sqrt[3]*ArcTan[1/Sqrt[3] + (2*(2 - 2*x + x^2)^(1/3))/Sqrt[3]])/2 + Log[-1 + (2 - 2*x + x^2)^(1/3)]/2 - Log[1 + (2 - 2*x + x^2)^(1/3) + (2 - 2*x + x^2)^(2/3)]/4","A",1
1156,1,86,86,0.0879211,"\int \frac{1}{(1+x) \sqrt[3]{2+2 x+x^2}} \, dx","IntegrateAlgebraic[1/((1 + x)*(2 + 2*x + x^2)^(1/3)),x]","\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{2+2 x+x^2}}{\sqrt{3}}\right)+\frac{1}{2} \log \left(-1+\sqrt[3]{2+2 x+x^2}\right)-\frac{1}{4} \log \left(1+\sqrt[3]{2+2 x+x^2}+\left(2+2 x+x^2\right)^{2/3}\right)","\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{2+2 x+x^2}}{\sqrt{3}}\right)+\frac{1}{2} \log \left(-1+\sqrt[3]{2+2 x+x^2}\right)-\frac{1}{4} \log \left(1+\sqrt[3]{2+2 x+x^2}+\left(2+2 x+x^2\right)^{2/3}\right)",1,"(Sqrt[3]*ArcTan[1/Sqrt[3] + (2*(2 + 2*x + x^2)^(1/3))/Sqrt[3]])/2 + Log[-1 + (2 + 2*x + x^2)^(1/3)]/2 - Log[1 + (2 + 2*x + x^2)^(1/3) + (2 + 2*x + x^2)^(2/3)]/4","A",1
1157,1,86,86,0.061366,"\int \frac{\left(-1+x^3\right)^{2/3}}{x} \, dx","IntegrateAlgebraic[(-1 + x^3)^(2/3)/x,x]","\frac{1}{2} \left(-1+x^3\right)^{2/3}+\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^3}}{\sqrt{3}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(1+\sqrt[3]{-1+x^3}\right)-\frac{1}{6} \log \left(1-\sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)","\frac{1}{2} \left(-1+x^3\right)^{2/3}+\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^3}}{\sqrt{3}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(1+\sqrt[3]{-1+x^3}\right)-\frac{1}{6} \log \left(1-\sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)",1,"(-1 + x^3)^(2/3)/2 + ArcTan[1/Sqrt[3] - (2*(-1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3] + Log[1 + (-1 + x^3)^(1/3)]/3 - Log[1 - (-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)]/6","A",1
1158,1,86,86,0.1255475,"\int \frac{3+x^2}{\sqrt[3]{1+x^2} \left(-1-x^2+x^3\right)} \, dx","IntegrateAlgebraic[(3 + x^2)/((1 + x^2)^(1/3)*(-1 - x^2 + x^3)),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\frac{2 x}{\sqrt{3}}+\frac{\sqrt[3]{1+x^2}}{\sqrt{3}}}{\sqrt[3]{1+x^2}}\right)+\log \left(-x+\sqrt[3]{1+x^2}\right)-\frac{1}{2} \log \left(x^2+x \sqrt[3]{1+x^2}+\left(1+x^2\right)^{2/3}\right)","-\sqrt{3} \tan ^{-1}\left(\frac{\frac{2 x}{\sqrt{3}}+\frac{\sqrt[3]{1+x^2}}{\sqrt{3}}}{\sqrt[3]{1+x^2}}\right)+\log \left(-x+\sqrt[3]{1+x^2}\right)-\frac{1}{2} \log \left(x^2+x \sqrt[3]{1+x^2}+\left(1+x^2\right)^{2/3}\right)",1,"-(Sqrt[3]*ArcTan[((2*x)/Sqrt[3] + (1 + x^2)^(1/3)/Sqrt[3])/(1 + x^2)^(1/3)]) + Log[-x + (1 + x^2)^(1/3)] - Log[x^2 + x*(1 + x^2)^(1/3) + (1 + x^2)^(2/3)]/2","A",1
1159,1,86,86,12.924008,"\int \frac{-2-(-1+k) (1+k) x+2 k^2 x^2}{\sqrt[4]{\left(1-x^2\right) \left(1-k^2 x^2\right)} \left(-1+d-(3+d) x-\left(3+d k^2\right) x^2+\left(-1+d k^2\right) x^3\right)} \, dx","IntegrateAlgebraic[(-2 - (-1 + k)*(1 + k)*x + 2*k^2*x^2)/(((1 - x^2)*(1 - k^2*x^2))^(1/4)*(-1 + d - (3 + d)*x - (3 + d*k^2)*x^2 + (-1 + d*k^2)*x^3)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{1+x}\right)}{d^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{1+x}\right)}{d^{3/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{1+x}\right)}{d^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{1+x}\right)}{d^{3/4}}",1,"ArcTan[(d^(1/4)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/4))/(1 + x)]/d^(3/4) - ArcTanh[(d^(1/4)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/4))/(1 + x)]/d^(3/4)","A",1
1160,1,86,86,12.9430752,"\int \frac{-2+(-1+k) (1+k) x+2 k^2 x^2}{\sqrt[4]{\left(1-x^2\right) \left(1-k^2 x^2\right)} \left(1-d-(3+d) x+\left(3+d k^2\right) x^2+\left(-1+d k^2\right) x^3\right)} \, dx","IntegrateAlgebraic[(-2 + (-1 + k)*(1 + k)*x + 2*k^2*x^2)/(((1 - x^2)*(1 - k^2*x^2))^(1/4)*(1 - d - (3 + d)*x + (3 + d*k^2)*x^2 + (-1 + d*k^2)*x^3)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{-1+x}\right)}{d^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{-1+x}\right)}{d^{3/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{-1+x}\right)}{d^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{-1+x}\right)}{d^{3/4}}",1,"-(ArcTan[(d^(1/4)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/4))/(-1 + x)]/d^(3/4)) + ArcTanh[(d^(1/4)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/4))/(-1 + x)]/d^(3/4)","A",1
1161,1,91,86,0.1033975,"\int \frac{\sqrt[4]{-1+x^4}}{x} \, dx","IntegrateAlgebraic[(-1 + x^4)^(1/4)/x,x]","\sqrt[4]{-1+x^4}-\frac{\tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{\sqrt{-1+x^4}}{\sqrt{2}}}{\sqrt[4]{-1+x^4}}\right)}{2 \sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^4}}{1+\sqrt{-1+x^4}}\right)}{2 \sqrt{2}}","\sqrt[4]{-1+x^4}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^4}}{-1+\sqrt{-1+x^4}}\right)}{2 \sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^4}}{1+\sqrt{-1+x^4}}\right)}{2 \sqrt{2}}",1,"(-1 + x^4)^(1/4) - ArcTan[(-(1/Sqrt[2]) + Sqrt[-1 + x^4]/Sqrt[2])/(-1 + x^4)^(1/4)]/(2*Sqrt[2]) - ArcTanh[(Sqrt[2]*(-1 + x^4)^(1/4))/(1 + Sqrt[-1 + x^4])]/(2*Sqrt[2])","A",1
1162,1,86,86,0.5251499,"\int \frac{\sqrt{-x+x^4}}{-b+a x^3} \, dx","IntegrateAlgebraic[Sqrt[-x + x^4]/(-b + a*x^3),x]","\frac{2 \sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} x \sqrt{-x+x^4}}{\sqrt{b} (-1+x) \left(1+x+x^2\right)}\right)}{3 a \sqrt{b}}+\frac{2 \tanh ^{-1}\left(\frac{x^2}{\sqrt{-x+x^4}}\right)}{3 a}","\frac{2 \sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} x \sqrt{-x+x^4}}{\sqrt{b} (-1+x) \left(1+x+x^2\right)}\right)}{3 a \sqrt{b}}+\frac{2 \tanh ^{-1}\left(\frac{x^2}{\sqrt{-x+x^4}}\right)}{3 a}",1,"(2*Sqrt[a - b]*ArcTan[(Sqrt[a - b]*x*Sqrt[-x + x^4])/(Sqrt[b]*(-1 + x)*(1 + x + x^2))])/(3*a*Sqrt[b]) + (2*ArcTanh[x^2/Sqrt[-x + x^4]])/(3*a)","A",1
1163,1,86,86,0.2429484,"\int \frac{1}{\left(-1+x^4\right) \sqrt[4]{-x^2+x^4}} \, dx","IntegrateAlgebraic[1/((-1 + x^4)*(-x^2 + x^4)^(1/4)),x]","-\frac{\left(-x^2+x^4\right)^{3/4}}{x \left(-1+x^2\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{2 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{2 \sqrt[4]{2}}","-\frac{\left(-x^2+x^4\right)^{3/4}}{x \left(-1+x^2\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{2 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{2 \sqrt[4]{2}}",1,"-((-x^2 + x^4)^(3/4)/(x*(-1 + x^2))) - ArcTan[(2^(1/4)*x)/(-x^2 + x^4)^(1/4)]/(2*2^(1/4)) - ArcTanh[(2^(1/4)*x)/(-x^2 + x^4)^(1/4)]/(2*2^(1/4))","A",1
1164,1,86,86,0.276966,"\int \frac{\sqrt[4]{-x^3+x^4}}{-1-2 x+x^2} \, dx","IntegrateAlgebraic[(-x^3 + x^4)^(1/4)/(-1 - 2*x + x^2),x]","-2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)+2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)+\frac{1}{2} \text{RootSum}\left[2-4 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right)}{\text{$\#$1}^3}\&\right]","-2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)+2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)+\frac{1}{2} \text{RootSum}\left[2-4 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right)}{\text{$\#$1}^3}\&\right]",1,"-2*ArcTan[x/(-x^3 + x^4)^(1/4)] + 2*ArcTanh[x/(-x^3 + x^4)^(1/4)] + RootSum[2 - 4*#1^4 + #1^8 & , (-Log[x] + Log[(-x^3 + x^4)^(1/4) - x*#1])/#1^3 & ]/2","A",1
1165,1,86,86,0.0007981,"\int \frac{\sqrt[4]{-x^3+x^4}}{-1-2 x+x^2} \, dx","IntegrateAlgebraic[(-x^3 + x^4)^(1/4)/(-1 - 2*x + x^2),x]","-2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)+2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)+\frac{1}{2} \text{RootSum}\left[2-4 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right)}{\text{$\#$1}^3}\&\right]","-2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)+2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)+\frac{1}{2} \text{RootSum}\left[2-4 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right)}{\text{$\#$1}^3}\&\right]",1,"-2*ArcTan[x/(-x^3 + x^4)^(1/4)] + 2*ArcTanh[x/(-x^3 + x^4)^(1/4)] + RootSum[2 - 4*#1^4 + #1^8 & , (-Log[x] + Log[(-x^3 + x^4)^(1/4) - x*#1])/#1^3 & ]/2","A",1
1166,1,86,86,0.3825448,"\int \frac{x^6}{\left(-b+a x^4\right)^{3/4}} \, dx","IntegrateAlgebraic[x^6/(-b + a*x^4)^(3/4),x]","\frac{x^3 \sqrt[4]{-b+a x^4}}{4 a}-\frac{3 b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{8 a^{7/4}}+\frac{3 b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{8 a^{7/4}}","\frac{x^3 \sqrt[4]{-b+a x^4}}{4 a}-\frac{3 b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{8 a^{7/4}}+\frac{3 b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{8 a^{7/4}}",1,"(x^3*(-b + a*x^4)^(1/4))/(4*a) - (3*b*ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)])/(8*a^(7/4)) + (3*b*ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)])/(8*a^(7/4))","A",1
1167,1,86,86,0.3529425,"\int \frac{\left(-b+a x^4\right) \sqrt[4]{b+a x^4}}{x^2} \, dx","IntegrateAlgebraic[((-b + a*x^4)*(b + a*x^4)^(1/4))/x^2,x]","\frac{\sqrt[4]{b+a x^4} \left(4 b+a x^4\right)}{4 x}+\frac{3}{8} \sqrt[4]{a} b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)-\frac{3}{8} \sqrt[4]{a} b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)","\frac{\sqrt[4]{b+a x^4} \left(4 b+a x^4\right)}{4 x}+\frac{3}{8} \sqrt[4]{a} b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)-\frac{3}{8} \sqrt[4]{a} b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)",1,"((b + a*x^4)^(1/4)*(4*b + a*x^4))/(4*x) + (3*a^(1/4)*b*ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)])/8 - (3*a^(1/4)*b*ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)])/8","A",1
1168,1,86,86,0.1966765,"\int \frac{\left(-1+x^3\right)^{2/3} \left(2+x^3\right)}{x^6 \left(-4-2 x^3+x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(2 + x^3))/(x^6*(-4 - 2*x^3 + x^6)),x]","\frac{\left(1-x^3\right) \left(-1+x^3\right)^{2/3}}{10 x^5}-\frac{1}{12} \text{RootSum}\left[5-10 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-5+4 \text{$\#$1}^3}\&\right]","\frac{\left(1-x^3\right) \left(-1+x^3\right)^{2/3}}{10 x^5}-\frac{1}{12} \text{RootSum}\left[5-10 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-5+4 \text{$\#$1}^3}\&\right]",1,"((1 - x^3)*(-1 + x^3)^(2/3))/(10*x^5) - RootSum[5 - 10*#1^3 + 4*#1^6 & , (-(Log[x]*#1^2) + Log[(-1 + x^3)^(1/3) - x*#1]*#1^2)/(-5 + 4*#1^3) & ]/12","A",1
1169,1,86,86,0.0011067,"\int \frac{\left(-1+x^3\right)^{2/3} \left(2+x^3\right)}{x^6 \left(-4-2 x^3+x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(2 + x^3))/(x^6*(-4 - 2*x^3 + x^6)),x]","\frac{\left(1-x^3\right) \left(-1+x^3\right)^{2/3}}{10 x^5}-\frac{1}{12} \text{RootSum}\left[5-10 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-5+4 \text{$\#$1}^3}\&\right]","\frac{\left(1-x^3\right) \left(-1+x^3\right)^{2/3}}{10 x^5}-\frac{1}{12} \text{RootSum}\left[5-10 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-5+4 \text{$\#$1}^3}\&\right]",1,"((1 - x^3)*(-1 + x^3)^(2/3))/(10*x^5) - RootSum[5 - 10*#1^3 + 4*#1^6 & , (-(Log[x]*#1^2) + Log[(-1 + x^3)^(1/3) - x*#1]*#1^2)/(-5 + 4*#1^3) & ]/12","A",1
1170,1,86,86,0.2046405,"\int \frac{\left(-1+x^3\right)^{2/3} \left(4-2 x^3+x^6\right)}{x^6 \left(-8+4 x^3+x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(4 - 2*x^3 + x^6))/(x^6*(-8 + 4*x^3 + x^6)),x]","\frac{\left(1-x^3\right) \left(-1+x^3\right)^{2/3}}{10 x^5}-\frac{1}{8} \text{RootSum}\left[3-12 \text{$\#$1}^3+8 \text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-3+4 \text{$\#$1}^3}\&\right]","\frac{\left(1-x^3\right) \left(-1+x^3\right)^{2/3}}{10 x^5}-\frac{1}{8} \text{RootSum}\left[3-12 \text{$\#$1}^3+8 \text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-3+4 \text{$\#$1}^3}\&\right]",1,"((1 - x^3)*(-1 + x^3)^(2/3))/(10*x^5) - RootSum[3 - 12*#1^3 + 8*#1^6 & , (-(Log[x]*#1^2) + Log[(-1 + x^3)^(1/3) - x*#1]*#1^2)/(-3 + 4*#1^3) & ]/8","A",1
1171,1,86,86,0.0012637,"\int \frac{\left(-1+x^3\right)^{2/3} \left(4-2 x^3+x^6\right)}{x^6 \left(-8+4 x^3+x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(4 - 2*x^3 + x^6))/(x^6*(-8 + 4*x^3 + x^6)),x]","\frac{\left(1-x^3\right) \left(-1+x^3\right)^{2/3}}{10 x^5}-\frac{1}{8} \text{RootSum}\left[3-12 \text{$\#$1}^3+8 \text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-3+4 \text{$\#$1}^3}\&\right]","\frac{\left(1-x^3\right) \left(-1+x^3\right)^{2/3}}{10 x^5}-\frac{1}{8} \text{RootSum}\left[3-12 \text{$\#$1}^3+8 \text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-3+4 \text{$\#$1}^3}\&\right]",1,"((1 - x^3)*(-1 + x^3)^(2/3))/(10*x^5) - RootSum[3 - 12*#1^3 + 8*#1^6 & , (-(Log[x]*#1^2) + Log[(-1 + x^3)^(1/3) - x*#1]*#1^2)/(-3 + 4*#1^3) & ]/8","A",1
1172,1,86,86,0.2056956,"\int \frac{\left(-1+x^3\right)^{2/3} \left(2-2 x^3+x^6\right)}{x^6 \left(-4+4 x^3+x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(2 - 2*x^3 + x^6))/(x^6*(-4 + 4*x^3 + x^6)),x]","\frac{\left(1-x^3\right) \left(-1+x^3\right)^{2/3}}{10 x^5}-\frac{1}{8} \text{RootSum}\left[-1-4 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-1+2 \text{$\#$1}^3}\&\right]","\frac{\left(1-x^3\right) \left(-1+x^3\right)^{2/3}}{10 x^5}-\frac{1}{8} \text{RootSum}\left[-1-4 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-1+2 \text{$\#$1}^3}\&\right]",1,"((1 - x^3)*(-1 + x^3)^(2/3))/(10*x^5) - RootSum[-1 - 4*#1^3 + 4*#1^6 & , (-(Log[x]*#1^2) + Log[(-1 + x^3)^(1/3) - x*#1]*#1^2)/(-1 + 2*#1^3) & ]/8","A",1
1173,1,86,86,0.0011923,"\int \frac{\left(-1+x^3\right)^{2/3} \left(2-2 x^3+x^6\right)}{x^6 \left(-4+4 x^3+x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(2 - 2*x^3 + x^6))/(x^6*(-4 + 4*x^3 + x^6)),x]","\frac{\left(1-x^3\right) \left(-1+x^3\right)^{2/3}}{10 x^5}-\frac{1}{8} \text{RootSum}\left[-1-4 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-1+2 \text{$\#$1}^3}\&\right]","\frac{\left(1-x^3\right) \left(-1+x^3\right)^{2/3}}{10 x^5}-\frac{1}{8} \text{RootSum}\left[-1-4 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-1+2 \text{$\#$1}^3}\&\right]",1,"((1 - x^3)*(-1 + x^3)^(2/3))/(10*x^5) - RootSum[-1 - 4*#1^3 + 4*#1^6 & , (-(Log[x]*#1^2) + Log[(-1 + x^3)^(1/3) - x*#1]*#1^2)/(-1 + 2*#1^3) & ]/8","A",1
1174,1,86,86,0.288336,"\int \frac{2+x^4}{\sqrt[4]{-1+x^4} \left(-2+x^8\right)} \, dx","IntegrateAlgebraic[(2 + x^4)/((-1 + x^4)^(1/4)*(-2 + x^8)),x]","-\frac{1}{16} \text{RootSum}\left[1-4 \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-3 \log (x)+3 \log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right)+2 \log (x) \text{$\#$1}^4-2 \log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}+\text{$\#$1}^5}\&\right]","-\frac{1}{16} \text{RootSum}\left[1-4 \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-3 \log (x)+3 \log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right)+2 \log (x) \text{$\#$1}^4-2 \log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}+\text{$\#$1}^5}\&\right]",1,"-1/16*RootSum[1 - 4*#1^4 + 2*#1^8 & , (-3*Log[x] + 3*Log[(-1 + x^4)^(1/4) - x*#1] + 2*Log[x]*#1^4 - 2*Log[(-1 + x^4)^(1/4) - x*#1]*#1^4)/(-#1 + #1^5) & ]","A",1
1175,1,86,86,0.5583946,"\int \frac{-b+a x^8}{x^6 \left(b+a x^4\right)^{3/4}} \, dx","IntegrateAlgebraic[(-b + a*x^8)/(x^6*(b + a*x^4)^(3/4)),x]","\frac{\left(b-4 a x^4\right) \sqrt[4]{b+a x^4}}{5 b x^5}-\frac{1}{2} \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)+\frac{1}{2} \sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)","\frac{\left(b-4 a x^4\right) \sqrt[4]{b+a x^4}}{5 b x^5}-\frac{1}{2} \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)+\frac{1}{2} \sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)",1,"((b - 4*a*x^4)*(b + a*x^4)^(1/4))/(5*b*x^5) - (a^(1/4)*ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)])/2 + (a^(1/4)*ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)])/2","A",1
1176,1,87,87,0.2317685,"\int \frac{\sqrt{x}}{(-1+x) \sqrt{-\sqrt{x}+x}} \, dx","IntegrateAlgebraic[Sqrt[x]/((-1 + x)*Sqrt[-Sqrt[x] + x]),x]","-\frac{2 \sqrt{-\sqrt{x}+x}}{-1+\sqrt{x}}+4 \tanh ^{-1}\left(\frac{\sqrt{-\sqrt{x}+x}}{-1+\sqrt{x}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{-\sqrt{x}+x}}{-1+\sqrt{x}}\right)","-\frac{2 \sqrt{-\sqrt{x}+x}}{-1+\sqrt{x}}+4 \tanh ^{-1}\left(\frac{\sqrt{-\sqrt{x}+x}}{-1+\sqrt{x}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{-\sqrt{x}+x}}{-1+\sqrt{x}}\right)",1,"(-2*Sqrt[-Sqrt[x] + x])/(-1 + Sqrt[x]) + 4*ArcTanh[Sqrt[-Sqrt[x] + x]/(-1 + Sqrt[x])] - Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[-Sqrt[x] + x])/(-1 + Sqrt[x])]","A",1
1177,1,87,87,0.0835898,"\int \frac{1}{x^3 \left(1+x^2\right)^{2/3}} \, dx","IntegrateAlgebraic[1/(x^3*(1 + x^2)^(2/3)),x]","-\frac{\sqrt[3]{1+x^2}}{2 x^2}+\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^2}}{\sqrt{3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(-1+\sqrt[3]{1+x^2}\right)+\frac{1}{6} \log \left(1+\sqrt[3]{1+x^2}+\left(1+x^2\right)^{2/3}\right)","-\frac{\sqrt[3]{1+x^2}}{2 x^2}+\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^2}}{\sqrt{3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(-1+\sqrt[3]{1+x^2}\right)+\frac{1}{6} \log \left(1+\sqrt[3]{1+x^2}+\left(1+x^2\right)^{2/3}\right)",1,"-1/2*(1 + x^2)^(1/3)/x^2 + ArcTan[1/Sqrt[3] + (2*(1 + x^2)^(1/3))/Sqrt[3]]/Sqrt[3] - Log[-1 + (1 + x^2)^(1/3)]/3 + Log[1 + (1 + x^2)^(1/3) + (1 + x^2)^(2/3)]/6","A",1
1178,1,87,87,0.0758615,"\int \frac{\left(1+x^2\right)^{2/3}}{x^3} \, dx","IntegrateAlgebraic[(1 + x^2)^(2/3)/x^3,x]","-\frac{\left(1+x^2\right)^{2/3}}{2 x^2}+\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^2}}{\sqrt{3}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(-1+\sqrt[3]{1+x^2}\right)-\frac{1}{6} \log \left(1+\sqrt[3]{1+x^2}+\left(1+x^2\right)^{2/3}\right)","-\frac{\left(1+x^2\right)^{2/3}}{2 x^2}+\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^2}}{\sqrt{3}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(-1+\sqrt[3]{1+x^2}\right)-\frac{1}{6} \log \left(1+\sqrt[3]{1+x^2}+\left(1+x^2\right)^{2/3}\right)",1,"-1/2*(1 + x^2)^(2/3)/x^2 + ArcTan[1/Sqrt[3] + (2*(1 + x^2)^(1/3))/Sqrt[3]]/Sqrt[3] + Log[-1 + (1 + x^2)^(1/3)]/3 - Log[1 + (1 + x^2)^(1/3) + (1 + x^2)^(2/3)]/6","A",1
1179,1,87,87,0.0458032,"\int \frac{\left(1+x^2\right)^{2/3}}{x} \, dx","IntegrateAlgebraic[(1 + x^2)^(2/3)/x,x]","\frac{3}{4} \left(1+x^2\right)^{2/3}+\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^2}}{\sqrt{3}}\right)+\frac{1}{2} \log \left(-1+\sqrt[3]{1+x^2}\right)-\frac{1}{4} \log \left(1+\sqrt[3]{1+x^2}+\left(1+x^2\right)^{2/3}\right)","\frac{3}{4} \left(1+x^2\right)^{2/3}+\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^2}}{\sqrt{3}}\right)+\frac{1}{2} \log \left(-1+\sqrt[3]{1+x^2}\right)-\frac{1}{4} \log \left(1+\sqrt[3]{1+x^2}+\left(1+x^2\right)^{2/3}\right)",1,"(3*(1 + x^2)^(2/3))/4 + (Sqrt[3]*ArcTan[1/Sqrt[3] + (2*(1 + x^2)^(1/3))/Sqrt[3]])/2 + Log[-1 + (1 + x^2)^(1/3)]/2 - Log[1 + (1 + x^2)^(1/3) + (1 + x^2)^(2/3)]/4","A",1
1180,1,87,87,0.3916999,"\int \frac{(3+4 x) \sqrt{x+2 x^2-2 x^4}}{(1+2 x) \left(1+2 x+x^3\right)} \, dx","IntegrateAlgebraic[((3 + 4*x)*Sqrt[x + 2*x^2 - 2*x^4])/((1 + 2*x)*(1 + 2*x + x^3)),x]","2 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt{x+2 x^2-2 x^4}}{-1-2 x+2 x^3}\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x \sqrt{x+2 x^2-2 x^4}}{-1-2 x+2 x^3}\right)","2 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt{x+2 x^2-2 x^4}}{-1-2 x+2 x^3}\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x \sqrt{x+2 x^2-2 x^4}}{-1-2 x+2 x^3}\right)",1,"2*Sqrt[2]*ArcTan[(Sqrt[2]*x*Sqrt[x + 2*x^2 - 2*x^4])/(-1 - 2*x + 2*x^3)] - 2*Sqrt[3]*ArcTan[(Sqrt[3]*x*Sqrt[x + 2*x^2 - 2*x^4])/(-1 - 2*x + 2*x^3)]","A",1
1181,1,87,87,0.0460046,"\int \frac{\left(1+x^4\right)^{2/3}}{x} \, dx","IntegrateAlgebraic[(1 + x^4)^(2/3)/x,x]","\frac{3}{8} \left(1+x^4\right)^{2/3}+\frac{1}{4} \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^4}}{\sqrt{3}}\right)+\frac{1}{4} \log \left(-1+\sqrt[3]{1+x^4}\right)-\frac{1}{8} \log \left(1+\sqrt[3]{1+x^4}+\left(1+x^4\right)^{2/3}\right)","\frac{3}{8} \left(1+x^4\right)^{2/3}+\frac{1}{4} \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^4}}{\sqrt{3}}\right)+\frac{1}{4} \log \left(-1+\sqrt[3]{1+x^4}\right)-\frac{1}{8} \log \left(1+\sqrt[3]{1+x^4}+\left(1+x^4\right)^{2/3}\right)",1,"(3*(1 + x^4)^(2/3))/8 + (Sqrt[3]*ArcTan[1/Sqrt[3] + (2*(1 + x^4)^(1/3))/Sqrt[3]])/4 + Log[-1 + (1 + x^4)^(1/3)]/4 - Log[1 + (1 + x^4)^(1/3) + (1 + x^4)^(2/3)]/8","A",1
1182,1,87,87,0.8249624,"\int \frac{\sqrt[3]{-1+x^4} \left(3+x^4\right)}{x^2 \left(-1-x^3+x^4\right)} \, dx","IntegrateAlgebraic[((-1 + x^4)^(1/3)*(3 + x^4))/(x^2*(-1 - x^3 + x^4)),x]","\frac{3 \sqrt[3]{-1+x^4}}{x}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^4}}\right)+\log \left(-x+\sqrt[3]{-1+x^4}\right)-\frac{1}{2} \log \left(x^2+x \sqrt[3]{-1+x^4}+\left(-1+x^4\right)^{2/3}\right)","\frac{3 \sqrt[3]{-1+x^4}}{x}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^4}}\right)+\log \left(-x+\sqrt[3]{-1+x^4}\right)-\frac{1}{2} \log \left(x^2+x \sqrt[3]{-1+x^4}+\left(-1+x^4\right)^{2/3}\right)",1,"(3*(-1 + x^4)^(1/3))/x + Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^4)^(1/3))] + Log[-x + (-1 + x^4)^(1/3)] - Log[x^2 + x*(-1 + x^4)^(1/3) + (-1 + x^4)^(2/3)]/2","A",1
1183,1,87,87,2.5464541,"\int \frac{x^2 \left(4+x^3\right)}{\left(1+x^3\right)^{3/4} \left(1+x^3+x^4\right)} \, dx","IntegrateAlgebraic[(x^2*(4 + x^3))/((1 + x^3)^(3/4)*(1 + x^3 + x^4)),x]","-\sqrt{2} \tan ^{-1}\left(\frac{-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^3}}{\sqrt{2}}}{x \sqrt[4]{1+x^3}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{1+x^3}}{x^2+\sqrt{1+x^3}}\right)","-\sqrt{2} \tan ^{-1}\left(\frac{-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^3}}{\sqrt{2}}}{x \sqrt[4]{1+x^3}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{1+x^3}}{x^2+\sqrt{1+x^3}}\right)",1,"-(Sqrt[2]*ArcTan[(-(x^2/Sqrt[2]) + Sqrt[1 + x^3]/Sqrt[2])/(x*(1 + x^3)^(1/4))]) - Sqrt[2]*ArcTanh[(Sqrt[2]*x*(1 + x^3)^(1/4))/(x^2 + Sqrt[1 + x^3])]","A",1
1184,1,87,87,0.5370084,"\int \frac{x^2}{\left(-b+a x^4\right)^{3/4} \left(b+a x^4\right)} \, dx","IntegrateAlgebraic[x^2/((-b + a*x^4)^(3/4)*(b + a*x^4)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2\ 2^{3/4} a^{3/4} b}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2\ 2^{3/4} a^{3/4} b}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2\ 2^{3/4} a^{3/4} b}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2\ 2^{3/4} a^{3/4} b}",1,"-1/2*ArcTan[(2^(1/4)*a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(2^(3/4)*a^(3/4)*b) + ArcTanh[(2^(1/4)*a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(2*2^(3/4)*a^(3/4)*b)","A",1
1185,1,87,87,0.3192475,"\int \frac{1}{\sqrt[4]{-b+a x^4} \left(b+a x^4\right)} \, dx","IntegrateAlgebraic[1/((-b + a*x^4)^(1/4)*(b + a*x^4)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2 \sqrt[4]{2} \sqrt[4]{a} b}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2 \sqrt[4]{2} \sqrt[4]{a} b}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2 \sqrt[4]{2} \sqrt[4]{a} b}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2 \sqrt[4]{2} \sqrt[4]{a} b}",1,"ArcTan[(2^(1/4)*a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(2*2^(1/4)*a^(1/4)*b) + ArcTanh[(2^(1/4)*a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(2*2^(1/4)*a^(1/4)*b)","A",1
1186,1,90,87,1.221288,"\int \frac{-a^2 b+a (2 a+b) x-3 a x^2+x^3}{\sqrt{x (-a+x) (-b+x)} \left(-a^2+2 a x+\left(-1+b^2 d\right) x^2-2 b d x^3+d x^4\right)} \, dx","IntegrateAlgebraic[(-(a^2*b) + a*(2*a + b)*x - 3*a*x^2 + x^3)/(Sqrt[x*(-a + x)*(-b + x)]*(-a^2 + 2*a*x + (-1 + b^2*d)*x^2 - 2*b*d*x^3 + d*x^4)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a b x+(-a-b) x^2+x^3}}{\sqrt[4]{d} x (-b+x)}\right)}{\sqrt[4]{d}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{a b x+(-a-b) x^2+x^3}}{a-x}\right)}{\sqrt[4]{d}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{a b x+(-a-b) x^2+x^3}}{a-x}\right)}{\sqrt[4]{d}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{a b x+(-a-b) x^2+x^3}}{a-x}\right)}{\sqrt[4]{d}}",1,"ArcTan[Sqrt[a*b*x + (-a - b)*x^2 + x^3]/(d^(1/4)*x*(-b + x))]/d^(1/4) + ArcTanh[(d^(1/4)*Sqrt[a*b*x + (-a - b)*x^2 + x^3])/(a - x)]/d^(1/4)","A",1
1187,1,87,87,1.8476866,"\int \frac{1+k^2 x^4}{\sqrt{\left(1-x^2\right) \left(1-k^2 x^2\right)} \left(-1+k^2 x^4\right)} \, dx","IntegrateAlgebraic[(1 + k^2*x^4)/(Sqrt[(1 - x^2)*(1 - k^2*x^2)]*(-1 + k^2*x^4)),x]","-\frac{\tan ^{-1}\left(\frac{(-1+k) x}{\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{2 (-1+k)}+\frac{\tan ^{-1}\left(\frac{(1+k) x}{1+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{-1-k}","-\frac{\tan ^{-1}\left(\frac{(-1+k) x}{\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{2 (-1+k)}+\frac{\tan ^{-1}\left(\frac{(1+k) x}{1+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{-1-k}",1,"-1/2*ArcTan[((-1 + k)*x)/Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4]]/(-1 + k) + ArcTan[((1 + k)*x)/(1 + k*x^2 + Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4])]/(-1 - k)","A",1
1188,1,87,87,3.0518937,"\int \frac{x (-1+k x) (-1+(-1+2 k) x)}{\sqrt[3]{(1-x) x (1-k x)} \left(-1+(4-c) x+(-6+b+2 c+c k) x^2+(4-c-2 b k-2 c k) x^3+\left(-1+c k+b k^2\right) x^4\right)} \, dx","IntegrateAlgebraic[(x*(-1 + k*x)*(-1 + (-1 + 2*k)*x))/(((1 - x)*x*(1 - k*x))^(1/3)*(-1 + (4 - c)*x + (-6 + b + 2*c + c*k)*x^2 + (4 - c - 2*b*k - 2*c*k)*x^3 + (-1 + c*k + b*k^2)*x^4)),x]","\frac{1}{3} \text{RootSum}\left[b-c \text{$\#$1}^3-\text{$\#$1}^6\&,\frac{-\log \left(x+(-1-k) x^2+k x^3\right)+3 \log \left(-1+x+\sqrt[3]{x+(-1-k) x^2+k x^3} \text{$\#$1}\right)}{c \text{$\#$1}^2+2 \text{$\#$1}^5}\&\right]","\frac{1}{3} \text{RootSum}\left[b-c \text{$\#$1}^3-\text{$\#$1}^6\&,\frac{-\log \left(x+(-1-k) x^2+k x^3\right)+3 \log \left(-1+x+\sqrt[3]{x+(-1-k) x^2+k x^3} \text{$\#$1}\right)}{c \text{$\#$1}^2+2 \text{$\#$1}^5}\&\right]",1,"RootSum[b - c*#1^3 - #1^6 & , (-Log[x + (-1 - k)*x^2 + k*x^3] + 3*Log[-1 + x + (x + (-1 - k)*x^2 + k*x^3)^(1/3)*#1])/(c*#1^2 + 2*#1^5) & ]/3","A",1
1189,1,87,87,0.0435767,"\int \frac{\left(1+x^5\right)^{2/3}}{x} \, dx","IntegrateAlgebraic[(1 + x^5)^(2/3)/x,x]","\frac{3}{10} \left(1+x^5\right)^{2/3}+\frac{1}{5} \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^5}}{\sqrt{3}}\right)+\frac{1}{5} \log \left(-1+\sqrt[3]{1+x^5}\right)-\frac{1}{10} \log \left(1+\sqrt[3]{1+x^5}+\left(1+x^5\right)^{2/3}\right)","\frac{3}{10} \left(1+x^5\right)^{2/3}+\frac{1}{5} \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^5}}{\sqrt{3}}\right)+\frac{1}{5} \log \left(-1+\sqrt[3]{1+x^5}\right)-\frac{1}{10} \log \left(1+\sqrt[3]{1+x^5}+\left(1+x^5\right)^{2/3}\right)",1,"(3*(1 + x^5)^(2/3))/10 + (Sqrt[3]*ArcTan[1/Sqrt[3] + (2*(1 + x^5)^(1/3))/Sqrt[3]])/5 + Log[-1 + (1 + x^5)^(1/3)]/5 - Log[1 + (1 + x^5)^(1/3) + (1 + x^5)^(2/3)]/10","A",1
1190,1,87,87,0.0434221,"\int \frac{\left(1+x^6\right)^{2/3}}{x} \, dx","IntegrateAlgebraic[(1 + x^6)^(2/3)/x,x]","\frac{1}{4} \left(1+x^6\right)^{2/3}+\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^6}}{\sqrt{3}}\right)}{2 \sqrt{3}}+\frac{1}{6} \log \left(-1+\sqrt[3]{1+x^6}\right)-\frac{1}{12} \log \left(1+\sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)","\frac{1}{4} \left(1+x^6\right)^{2/3}+\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^6}}{\sqrt{3}}\right)}{2 \sqrt{3}}+\frac{1}{6} \log \left(-1+\sqrt[3]{1+x^6}\right)-\frac{1}{12} \log \left(1+\sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)",1,"(1 + x^6)^(2/3)/4 + ArcTan[1/Sqrt[3] + (2*(1 + x^6)^(1/3))/Sqrt[3]]/(2*Sqrt[3]) + Log[-1 + (1 + x^6)^(1/3)]/6 - Log[1 + (1 + x^6)^(1/3) + (1 + x^6)^(2/3)]/12","A",1
1191,1,87,87,0.1995225,"\int \frac{\left(1-x^2+2 x^4\right) \sqrt{1-x^2-x^4-x^6}}{\left(-1+x^2\right) \left(1+x^2\right) \left(-1+x^4+x^6\right)} \, dx","IntegrateAlgebraic[((1 - x^2 + 2*x^4)*Sqrt[1 - x^2 - x^4 - x^6])/((-1 + x^2)*(1 + x^2)*(-1 + x^4 + x^6)),x]","\tan ^{-1}\left(\frac{x \sqrt{1-x^2-x^4-x^6}}{-1+x^2+x^4+x^6}\right)-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt{1-x^2-x^4-x^6}}{-1+x^2+x^4+x^6}\right)","\tan ^{-1}\left(\frac{x \sqrt{1-x^2-x^4-x^6}}{-1+x^2+x^4+x^6}\right)-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt{1-x^2-x^4-x^6}}{-1+x^2+x^4+x^6}\right)",1,"ArcTan[(x*Sqrt[1 - x^2 - x^4 - x^6])/(-1 + x^2 + x^4 + x^6)] - Sqrt[2]*ArcTan[(Sqrt[2]*x*Sqrt[1 - x^2 - x^4 - x^6])/(-1 + x^2 + x^4 + x^6)]","A",1
1192,1,87,87,20.3271032,"\int \frac{\sqrt[3]{-1+x^8} \left(3+5 x^8\right)}{x^2 \left(-1-x^3+x^8\right)} \, dx","IntegrateAlgebraic[((-1 + x^8)^(1/3)*(3 + 5*x^8))/(x^2*(-1 - x^3 + x^8)),x]","\frac{3 \sqrt[3]{-1+x^8}}{x}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^8}}\right)+\log \left(-x+\sqrt[3]{-1+x^8}\right)-\frac{1}{2} \log \left(x^2+x \sqrt[3]{-1+x^8}+\left(-1+x^8\right)^{2/3}\right)","\frac{3 \sqrt[3]{-1+x^8}}{x}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^8}}\right)+\log \left(-x+\sqrt[3]{-1+x^8}\right)-\frac{1}{2} \log \left(x^2+x \sqrt[3]{-1+x^8}+\left(-1+x^8\right)^{2/3}\right)",1,"(3*(-1 + x^8)^(1/3))/x + Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^8)^(1/3))] + Log[-x + (-1 + x^8)^(1/3)] - Log[x^2 + x*(-1 + x^8)^(1/3) + (-1 + x^8)^(2/3)]/2","A",1
1193,1,87,87,0.1591047,"\int \frac{\left(2+x^8\right) \sqrt{4-2 x^8+x^{16}}}{x^9} \, dx","IntegrateAlgebraic[((2 + x^8)*Sqrt[4 - 2*x^8 + x^16])/x^9,x]","\frac{\left(-2+x^8\right) \sqrt{4-2 x^8+x^{16}}}{8 x^8}+\frac{1}{4} \tanh ^{-1}\left(\frac{1}{3}+\frac{2 x^8}{3}-\frac{2}{3} \sqrt{4-2 x^8+x^{16}}\right)-\frac{1}{8} \log \left(2-x^8+\sqrt{4-2 x^8+x^{16}}\right)","\frac{\left(-2+x^8\right) \sqrt{4-2 x^8+x^{16}}}{8 x^8}+\frac{1}{4} \tanh ^{-1}\left(\frac{1}{3}+\frac{2 x^8}{3}-\frac{2}{3} \sqrt{4-2 x^8+x^{16}}\right)-\frac{1}{8} \log \left(2-x^8+\sqrt{4-2 x^8+x^{16}}\right)",1,"((-2 + x^8)*Sqrt[4 - 2*x^8 + x^16])/(8*x^8) + ArcTanh[1/3 + (2*x^8)/3 - (2*Sqrt[4 - 2*x^8 + x^16])/3]/4 - Log[2 - x^8 + Sqrt[4 - 2*x^8 + x^16]]/8","A",1
1194,1,87,87,0.2391648,"\int \frac{\sqrt{x+\sqrt{1+x}}}{1-\sqrt{1+x}} \, dx","IntegrateAlgebraic[Sqrt[x + Sqrt[1 + x]]/(1 - Sqrt[1 + x]),x]","\frac{1}{2} \left(-5-2 \sqrt{1+x}\right) \sqrt{x+\sqrt{1+x}}+4 \tanh ^{-1}\left(1-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}\right)+\frac{7}{4} \log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}\right)","\frac{1}{2} \left(-5-2 \sqrt{1+x}\right) \sqrt{x+\sqrt{1+x}}+4 \tanh ^{-1}\left(1-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}\right)+\frac{7}{4} \log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}\right)",1,"((-5 - 2*Sqrt[1 + x])*Sqrt[x + Sqrt[1 + x]])/2 + 4*ArcTanh[1 - Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]]] + (7*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]])/4","A",1
1195,1,87,87,0.1458279,"\int \frac{\sqrt{c+\sqrt{b+a x}}}{d-\sqrt{b+a x}} \, dx","IntegrateAlgebraic[Sqrt[c + Sqrt[b + a*x]]/(d - Sqrt[b + a*x]),x]","-\frac{4 \sqrt{c+\sqrt{b+a x}} \left(c+3 d+\sqrt{b+a x}\right)}{3 a}-\frac{4 \sqrt{-c-d} d \tan ^{-1}\left(\frac{\sqrt{-c-d} \sqrt{c+\sqrt{b+a x}}}{c+d}\right)}{a}","-\frac{4 \sqrt{c+\sqrt{b+a x}} \left(c+3 d+\sqrt{b+a x}\right)}{3 a}-\frac{4 \sqrt{-c-d} d \tan ^{-1}\left(\frac{\sqrt{-c-d} \sqrt{c+\sqrt{b+a x}}}{c+d}\right)}{a}",1,"(-4*Sqrt[c + Sqrt[b + a*x]]*(c + 3*d + Sqrt[b + a*x]))/(3*a) - (4*Sqrt[-c - d]*d*ArcTan[(Sqrt[-c - d]*Sqrt[c + Sqrt[b + a*x]])/(c + d)])/a","A",1
1196,1,87,87,0.1727304,"\int \left(b^2+a x^2\right) \sqrt{b+\sqrt{b^2+a x^2}} \, dx","IntegrateAlgebraic[(b^2 + a*x^2)*Sqrt[b + Sqrt[b^2 + a*x^2]],x]","\frac{2 x \sqrt{b^2+a x^2} \left(13 b^2+5 a x^2\right)}{35 \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{4 x \left(11 b^3+3 a b x^2\right)}{35 \sqrt{b+\sqrt{b^2+a x^2}}}","\frac{2 x \sqrt{b^2+a x^2} \left(13 b^2+5 a x^2\right)}{35 \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{4 x \left(11 b^3+3 a b x^2\right)}{35 \sqrt{b+\sqrt{b^2+a x^2}}}",1,"(2*x*Sqrt[b^2 + a*x^2]*(13*b^2 + 5*a*x^2))/(35*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (4*x*(11*b^3 + 3*a*b*x^2))/(35*Sqrt[b + Sqrt[b^2 + a*x^2]])","A",1
1197,1,87,87,0.612223,"\int \frac{1}{x+\sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[(x + Sqrt[x + Sqrt[1 + x^2]])^(-1),x]","-\log \left(x+\sqrt{1+x^2}\right)+2 \text{RootSum}\left[-1+2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right)+\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}}{3+2 \text{$\#$1}}\&\right]","-\log \left(x+\sqrt{1+x^2}\right)+2 \text{RootSum}\left[-1+2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right)+\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}}{3+2 \text{$\#$1}}\&\right]",1,"-Log[x + Sqrt[1 + x^2]] + 2*RootSum[-1 + 2*#1^3 + #1^4 & , (Log[Sqrt[x + Sqrt[1 + x^2]] - #1] + Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1)/(3 + 2*#1) & ]","A",1
1198,1,87,87,0.0008595,"\int \frac{1}{x+\sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[(x + Sqrt[x + Sqrt[1 + x^2]])^(-1),x]","-\log \left(x+\sqrt{1+x^2}\right)+2 \text{RootSum}\left[-1+2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right)+\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}}{3+2 \text{$\#$1}}\&\right]","-\log \left(x+\sqrt{1+x^2}\right)+2 \text{RootSum}\left[-1+2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right)+\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}}{3+2 \text{$\#$1}}\&\right]",1,"-Log[x + Sqrt[1 + x^2]] + 2*RootSum[-1 + 2*#1^3 + #1^4 & , (Log[Sqrt[x + Sqrt[1 + x^2]] - #1] + Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1)/(3 + 2*#1) & ]","A",1
1199,1,88,88,0.0966023,"\int \frac{1}{(-2+x) \sqrt[3]{-4-4 x+x^2}} \, dx","IntegrateAlgebraic[1/((-2 + x)*(-4 - 4*x + x^2)^(1/3)),x]","-\frac{1}{4} \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{\sqrt[3]{-4-4 x+x^2}}{\sqrt{3}}\right)-\frac{1}{4} \log \left(2+\sqrt[3]{-4-4 x+x^2}\right)+\frac{1}{8} \log \left(4-2 \sqrt[3]{-4-4 x+x^2}+\left(-4-4 x+x^2\right)^{2/3}\right)","-\frac{1}{4} \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{\sqrt[3]{-4-4 x+x^2}}{\sqrt{3}}\right)-\frac{1}{4} \log \left(2+\sqrt[3]{-4-4 x+x^2}\right)+\frac{1}{8} \log \left(4-2 \sqrt[3]{-4-4 x+x^2}+\left(-4-4 x+x^2\right)^{2/3}\right)",1,"-1/4*(Sqrt[3]*ArcTan[1/Sqrt[3] - (-4 - 4*x + x^2)^(1/3)/Sqrt[3]]) - Log[2 + (-4 - 4*x + x^2)^(1/3)]/4 + Log[4 - 2*(-4 - 4*x + x^2)^(1/3) + (-4 - 4*x + x^2)^(2/3)]/8","A",1
1200,1,88,88,0.2975688,"\int \frac{-1+k x^2}{(a+b x) \sqrt{(1-x) x (1-k x)} (b+a k x)} \, dx","IntegrateAlgebraic[(-1 + k*x^2)/((a + b*x)*Sqrt[(1 - x)*x*(1 - k*x)]*(b + a*k*x)),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a+b} \sqrt{b+a k} \sqrt{x+(-1-k) x^2+k x^3}}{\sqrt{a} \sqrt{b} (-1+x) (-1+k x)}\right)}{\sqrt{a} \sqrt{b} \sqrt{a+b} \sqrt{b+a k}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a+b} \sqrt{b+a k} \sqrt{x+(-1-k) x^2+k x^3}}{\sqrt{a} \sqrt{b} (-1+x) (-1+k x)}\right)}{\sqrt{a} \sqrt{b} \sqrt{a+b} \sqrt{b+a k}}",1,"(-2*ArcTan[(Sqrt[a + b]*Sqrt[b + a*k]*Sqrt[x + (-1 - k)*x^2 + k*x^3])/(Sqrt[a]*Sqrt[b]*(-1 + x)*(-1 + k*x))])/(Sqrt[a]*Sqrt[b]*Sqrt[a + b]*Sqrt[b + a*k])","A",1
1201,1,88,88,0.9918787,"\int \frac{x (-b+x) \left(a b-2 a x+x^2\right)}{\sqrt{x (-a+x) (-b+x)} \left(-a^2+2 a x+\left(-1+b^2 d\right) x^2-2 b d x^3+d x^4\right)} \, dx","IntegrateAlgebraic[(x*(-b + x)*(a*b - 2*a*x + x^2))/(Sqrt[x*(-a + x)*(-b + x)]*(-a^2 + 2*a*x + (-1 + b^2*d)*x^2 - 2*b*d*x^3 + d*x^4)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{a b x+(-a-b) x^2+x^3}}{a-x}\right)}{d^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{a b x+(-a-b) x^2+x^3}}{a-x}\right)}{d^{3/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{a b x+(-a-b) x^2+x^3}}{a-x}\right)}{d^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{a b x+(-a-b) x^2+x^3}}{a-x}\right)}{d^{3/4}}",1,"-(ArcTan[(d^(1/4)*Sqrt[a*b*x + (-a - b)*x^2 + x^3])/(a - x)]/d^(3/4)) + ArcTanh[(d^(1/4)*Sqrt[a*b*x + (-a - b)*x^2 + x^3])/(a - x)]/d^(3/4)","A",1
1202,1,88,88,0.5010568,"\int \frac{1+x^4}{\left(-1-x^2+x^4\right) \sqrt[3]{-x+x^5}} \, dx","IntegrateAlgebraic[(1 + x^4)/((-1 - x^2 + x^4)*(-x + x^5)^(1/3)),x]","-\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^5}}\right)+\frac{1}{2} \log \left(-x+\sqrt[3]{-x+x^5}\right)-\frac{1}{4} \log \left(x^2+x \sqrt[3]{-x+x^5}+\left(-x+x^5\right)^{2/3}\right)","-\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^5}}\right)+\frac{1}{2} \log \left(-x+\sqrt[3]{-x+x^5}\right)-\frac{1}{4} \log \left(x^2+x \sqrt[3]{-x+x^5}+\left(-x+x^5\right)^{2/3}\right)",1,"-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-x + x^5)^(1/3))]) + Log[-x + (-x + x^5)^(1/3)]/2 - Log[x^2 + x*(-x + x^5)^(1/3) + (-x + x^5)^(2/3)]/4","A",1
1203,1,88,88,0.7477828,"\int \frac{x^3}{\left(-1+x^6\right)^{2/3}} \, dx","IntegrateAlgebraic[x^3/(-1 + x^6)^(2/3),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{-1+x^6}}\right)}{2 \sqrt{3}}-\frac{1}{6} \log \left(-x^2+\sqrt[3]{-1+x^6}\right)+\frac{1}{12} \log \left(x^4+x^2 \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{-1+x^6}}\right)}{2 \sqrt{3}}-\frac{1}{6} \log \left(-x^2+\sqrt[3]{-1+x^6}\right)+\frac{1}{12} \log \left(x^4+x^2 \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)",1,"-1/2*ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(-1 + x^6)^(1/3))]/Sqrt[3] - Log[-x^2 + (-1 + x^6)^(1/3)]/6 + Log[x^4 + x^2*(-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)]/12","A",1
1204,1,88,88,0.6064021,"\int \frac{x}{\sqrt[3]{-1+x^6}} \, dx","IntegrateAlgebraic[x/(-1 + x^6)^(1/3),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{-1+x^6}}\right)}{2 \sqrt{3}}-\frac{1}{6} \log \left(-x^2+\sqrt[3]{-1+x^6}\right)+\frac{1}{12} \log \left(x^4+x^2 \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)","\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{-1+x^6}}\right)}{2 \sqrt{3}}-\frac{1}{6} \log \left(-x^2+\sqrt[3]{-1+x^6}\right)+\frac{1}{12} \log \left(x^4+x^2 \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)",1,"ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(-1 + x^6)^(1/3))]/(2*Sqrt[3]) - Log[-x^2 + (-1 + x^6)^(1/3)]/6 + Log[x^4 + x^2*(-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)]/12","A",1
1205,1,88,88,0.7436014,"\int \frac{x^3}{\left(1+x^6\right)^{2/3}} \, dx","IntegrateAlgebraic[x^3/(1 + x^6)^(2/3),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{1+x^6}}\right)}{2 \sqrt{3}}-\frac{1}{6} \log \left(-x^2+\sqrt[3]{1+x^6}\right)+\frac{1}{12} \log \left(x^4+x^2 \sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{1+x^6}}\right)}{2 \sqrt{3}}-\frac{1}{6} \log \left(-x^2+\sqrt[3]{1+x^6}\right)+\frac{1}{12} \log \left(x^4+x^2 \sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)",1,"-1/2*ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(1 + x^6)^(1/3))]/Sqrt[3] - Log[-x^2 + (1 + x^6)^(1/3)]/6 + Log[x^4 + x^2*(1 + x^6)^(1/3) + (1 + x^6)^(2/3)]/12","A",1
1206,1,88,88,0.6026019,"\int \frac{x}{\sqrt[3]{1+x^6}} \, dx","IntegrateAlgebraic[x/(1 + x^6)^(1/3),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{1+x^6}}\right)}{2 \sqrt{3}}-\frac{1}{6} \log \left(-x^2+\sqrt[3]{1+x^6}\right)+\frac{1}{12} \log \left(x^4+x^2 \sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)","\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{1+x^6}}\right)}{2 \sqrt{3}}-\frac{1}{6} \log \left(-x^2+\sqrt[3]{1+x^6}\right)+\frac{1}{12} \log \left(x^4+x^2 \sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)",1,"ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(1 + x^6)^(1/3))]/(2*Sqrt[3]) - Log[-x^2 + (1 + x^6)^(1/3)]/6 + Log[x^4 + x^2*(1 + x^6)^(1/3) + (1 + x^6)^(2/3)]/12","A",1
1207,1,88,88,0.599517,"\int \frac{-b+a x^8}{x^2 \left(-b+a x^4\right)^{3/4}} \, dx","IntegrateAlgebraic[(-b + a*x^8)/(x^2*(-b + a*x^4)^(3/4)),x]","\frac{\left(-4+x^4\right) \sqrt[4]{-b+a x^4}}{4 x}-\frac{3 b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{8 a^{3/4}}+\frac{3 b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{8 a^{3/4}}","\frac{\left(-4+x^4\right) \sqrt[4]{-b+a x^4}}{4 x}-\frac{3 b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{8 a^{3/4}}+\frac{3 b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{8 a^{3/4}}",1,"((-4 + x^4)*(-b + a*x^4)^(1/4))/(4*x) - (3*b*ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)])/(8*a^(3/4)) + (3*b*ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)])/(8*a^(3/4))","A",1
1208,1,88,88,0.3929555,"\int \frac{-3 b+2 a x^8}{x^8 \sqrt[4]{-b+a x^4}} \, dx","IntegrateAlgebraic[(-3*b + 2*a*x^8)/(x^8*(-b + a*x^4)^(1/4)),x]","\frac{\left(-3 b-4 a x^4\right) \left(-b+a x^4\right)^{3/4}}{7 b x^7}+a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)+a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)","\frac{\left(-3 b-4 a x^4\right) \left(-b+a x^4\right)^{3/4}}{7 b x^7}+a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)+a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)",1,"((-3*b - 4*a*x^4)*(-b + a*x^4)^(3/4))/(7*b*x^7) + a^(3/4)*ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)] + a^(3/4)*ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]","A",1
1209,1,87,88,0.6311665,"\int \frac{-b^2+a^2 x^8}{x^2 \left(b+a x^4\right)^{3/4} \left(b^2+a^2 x^8\right)} \, dx","IntegrateAlgebraic[(-b^2 + a^2*x^8)/(x^2*(b + a*x^4)^(3/4)*(b^2 + a^2*x^8)),x]","\frac{\sqrt[4]{b+a x^4}}{b x}+\frac{a^2 \text{RootSum}\left[2 a^2-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{\log (x)-\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)}{-a \text{$\#$1}^3+\text{$\#$1}^7}\&\right]}{4 b}","\frac{\sqrt[4]{b+a x^4}}{b x}+\frac{a^2 \text{RootSum}\left[2 a^2-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)}{a \text{$\#$1}^3-\text{$\#$1}^7}\&\right]}{4 b}",1,"(b + a*x^4)^(1/4)/(b*x) + (a^2*RootSum[2*a^2 - 2*a*#1^4 + #1^8 & , (Log[x] - Log[(b + a*x^4)^(1/4) - x*#1])/(-(a*#1^3) + #1^7) & ])/(4*b)","A",1
1210,1,88,88,3.4694543,"\int \frac{\sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}} \, dx","IntegrateAlgebraic[Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]]/(x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]),x]","-\frac{\sqrt{2} b \log \left(-a x-b \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}+\sqrt{2} \sqrt{a} \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}\right)}{\sqrt{a}}","-\frac{\sqrt{2} b \log \left(-a x-b \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}+\sqrt{2} \sqrt{a} \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}\right)}{\sqrt{a}}",1,"-((Sqrt[2]*b*Log[-(a*x) - b*Sqrt[-(a/b^2) + (a^2*x^2)/b^2] + Sqrt[2]*Sqrt[a]*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]]])/Sqrt[a])","A",1
1211,1,89,89,0.0623248,"\int \frac{\left(-1+x^2\right)^{2/3}}{x} \, dx","IntegrateAlgebraic[(-1 + x^2)^(2/3)/x,x]","\frac{3}{4} \left(-1+x^2\right)^{2/3}+\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^2}}{\sqrt{3}}\right)+\frac{1}{2} \log \left(1+\sqrt[3]{-1+x^2}\right)-\frac{1}{4} \log \left(1-\sqrt[3]{-1+x^2}+\left(-1+x^2\right)^{2/3}\right)","\frac{3}{4} \left(-1+x^2\right)^{2/3}+\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^2}}{\sqrt{3}}\right)+\frac{1}{2} \log \left(1+\sqrt[3]{-1+x^2}\right)-\frac{1}{4} \log \left(1-\sqrt[3]{-1+x^2}+\left(-1+x^2\right)^{2/3}\right)",1,"(3*(-1 + x^2)^(2/3))/4 + (Sqrt[3]*ArcTan[1/Sqrt[3] - (2*(-1 + x^2)^(1/3))/Sqrt[3]])/2 + Log[1 + (-1 + x^2)^(1/3)]/2 - Log[1 - (-1 + x^2)^(1/3) + (-1 + x^2)^(2/3)]/4","A",1
1212,1,89,89,0.1314299,"\int \frac{-3+x}{\sqrt[3]{-1+x^2} \left(2+x+x^2\right)} \, dx","IntegrateAlgebraic[(-3 + x)/((-1 + x^2)^(1/3)*(2 + x + x^2)),x]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-1+x^2}}{2+2 x+\sqrt[3]{-1+x^2}}\right)+\log \left(-1-x+\sqrt[3]{-1+x^2}\right)-\frac{1}{2} \log \left(1+2 x+x^2+(1+x) \sqrt[3]{-1+x^2}+\left(-1+x^2\right)^{2/3}\right)","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-1+x^2}}{2+2 x+\sqrt[3]{-1+x^2}}\right)+\log \left(-1-x+\sqrt[3]{-1+x^2}\right)-\frac{1}{2} \log \left(1+2 x+x^2+(1+x) \sqrt[3]{-1+x^2}+\left(-1+x^2\right)^{2/3}\right)",1,"Sqrt[3]*ArcTan[(Sqrt[3]*(-1 + x^2)^(1/3))/(2 + 2*x + (-1 + x^2)^(1/3))] + Log[-1 - x + (-1 + x^2)^(1/3)] - Log[1 + 2*x + x^2 + (1 + x)*(-1 + x^2)^(1/3) + (-1 + x^2)^(2/3)]/2","A",1
1213,1,89,89,1.5882454,"\int \frac{2+x^2}{\left(-2-2 x+x^2\right) \sqrt{-1+x^3}} \, dx","IntegrateAlgebraic[(2 + x^2)/((-2 - 2*x + x^2)*Sqrt[-1 + x^3]),x]","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{-3+2 \sqrt{3}} \sqrt{-1+x^3}}{1+x+x^2}\right)}{3^{3/4}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} \sqrt{-1+x^3}}{1+x+x^2}\right)}{3^{3/4}}","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{-3+2 \sqrt{3}} \sqrt{-1+x^3}}{1+x+x^2}\right)}{3^{3/4}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} \sqrt{-1+x^3}}{1+x+x^2}\right)}{3^{3/4}}",1,"-((Sqrt[2]*ArcTan[(Sqrt[-3 + 2*Sqrt[3]]*Sqrt[-1 + x^3])/(1 + x + x^2)])/3^(3/4)) - (Sqrt[2]*ArcTanh[(Sqrt[3 + 2*Sqrt[3]]*Sqrt[-1 + x^3])/(1 + x + x^2)])/3^(3/4)","A",1
1214,1,89,89,0.0584131,"\int \frac{\left(-1+x^3\right) \sqrt[3]{1+x^3}}{x} \, dx","IntegrateAlgebraic[((-1 + x^3)*(1 + x^3)^(1/3))/x,x]","\frac{1}{4} \left(-3+x^3\right) \sqrt[3]{1+x^3}+\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^3}}{\sqrt{3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(-1+\sqrt[3]{1+x^3}\right)+\frac{1}{6} \log \left(1+\sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)","\frac{1}{4} \left(-3+x^3\right) \sqrt[3]{1+x^3}+\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^3}}{\sqrt{3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(-1+\sqrt[3]{1+x^3}\right)+\frac{1}{6} \log \left(1+\sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)",1,"((-3 + x^3)*(1 + x^3)^(1/3))/4 + ArcTan[1/Sqrt[3] + (2*(1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3] - Log[-1 + (1 + x^3)^(1/3)]/3 + Log[1 + (1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/6","A",1
1215,1,89,89,0.3069982,"\int \frac{-a b+2 (a-b) x+x^2}{\sqrt[4]{x (-a+x) (-b+x)} \left(-a^3+\left(3 a^2+b d\right) x-(3 a+d) x^2+x^3\right)} \, dx","IntegrateAlgebraic[(-(a*b) + 2*(a - b)*x + x^2)/((x*(-a + x)*(-b + x))^(1/4)*(-a^3 + (3*a^2 + b*d)*x - (3*a + d)*x^2 + x^3)),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b x+(-a-b) x^2+x^3}}{a-x}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b x+(-a-b) x^2+x^3}}{a-x}\right)}{d^{3/4}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b x+(-a-b) x^2+x^3}}{a-x}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b x+(-a-b) x^2+x^3}}{a-x}\right)}{d^{3/4}}",1,"(-2*ArcTan[(d^(1/4)*(a*b*x + (-a - b)*x^2 + x^3)^(1/4))/(a - x)])/d^(3/4) + (2*ArcTanh[(d^(1/4)*(a*b*x + (-a - b)*x^2 + x^3)^(1/4))/(a - x)])/d^(3/4)","A",1
1216,1,89,89,3.269719,"\int \frac{\left(1-2 x+x^2\right) \left(-1+2 (-1+k) x+k x^2\right)}{((1-x) x (1-k x))^{3/4} \left(-d+(1+3 d) x-(3 d+k) x^2+d x^3\right)} \, dx","IntegrateAlgebraic[((1 - 2*x + x^2)*(-1 + 2*(-1 + k)*x + k*x^2))/(((1 - x)*x*(1 - k*x))^(3/4)*(-d + (1 + 3*d)*x - (3*d + k)*x^2 + d*x^3)),x]","\frac{2 \tan ^{-1}\left(\frac{-\sqrt[4]{d}+\sqrt[4]{d} x}{\sqrt[4]{x+(-1-k) x^2+k x^3}}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{-\sqrt[4]{d}+\sqrt[4]{d} x}{\sqrt[4]{x+(-1-k) x^2+k x^3}}\right)}{d^{3/4}}","\frac{2 \tan ^{-1}\left(\frac{-\sqrt[4]{d}+\sqrt[4]{d} x}{\sqrt[4]{x+(-1-k) x^2+k x^3}}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{-\sqrt[4]{d}+\sqrt[4]{d} x}{\sqrt[4]{x+(-1-k) x^2+k x^3}}\right)}{d^{3/4}}",1,"(2*ArcTan[(-d^(1/4) + d^(1/4)*x)/(x + (-1 - k)*x^2 + k*x^3)^(1/4)])/d^(3/4) - (2*ArcTanh[(-d^(1/4) + d^(1/4)*x)/(x + (-1 - k)*x^2 + k*x^3)^(1/4)])/d^(3/4)","A",1
1217,1,89,89,0.1535457,"\int \frac{x^2}{\sqrt{x+x^2+x^3} \left(-1+x^4\right)} \, dx","IntegrateAlgebraic[x^2/(Sqrt[x + x^2 + x^3]*(-1 + x^4)),x]","-\frac{1}{4} \tan ^{-1}\left(\frac{\sqrt{x+x^2+x^3}}{1+x+x^2}\right)+\frac{1}{2} \tanh ^{-1}\left(\frac{\sqrt{x+x^2+x^3}}{1+x+x^2}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt{3} \sqrt{x+x^2+x^3}}{1+x+x^2}\right)}{4 \sqrt{3}}","-\frac{1}{4} \tan ^{-1}\left(\frac{\sqrt{x+x^2+x^3}}{1+x+x^2}\right)+\frac{1}{2} \tanh ^{-1}\left(\frac{\sqrt{x+x^2+x^3}}{1+x+x^2}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt{3} \sqrt{x+x^2+x^3}}{1+x+x^2}\right)}{4 \sqrt{3}}",1,"-1/4*ArcTan[Sqrt[x + x^2 + x^3]/(1 + x + x^2)] + ArcTanh[Sqrt[x + x^2 + x^3]/(1 + x + x^2)]/2 - ArcTanh[(Sqrt[3]*Sqrt[x + x^2 + x^3])/(1 + x + x^2)]/(4*Sqrt[3])","A",1
1218,1,89,89,0.1375467,"\int \frac{\sqrt{x+x^2+x^3}}{-1+x^4} \, dx","IntegrateAlgebraic[Sqrt[x + x^2 + x^3]/(-1 + x^4),x]","\frac{1}{4} \tan ^{-1}\left(\frac{\sqrt{x+x^2+x^3}}{1+x+x^2}\right)+\frac{1}{2} \tanh ^{-1}\left(\frac{\sqrt{x+x^2+x^3}}{1+x+x^2}\right)-\frac{1}{4} \sqrt{3} \tanh ^{-1}\left(\frac{\sqrt{3} \sqrt{x+x^2+x^3}}{1+x+x^2}\right)","\frac{1}{4} \tan ^{-1}\left(\frac{\sqrt{x+x^2+x^3}}{1+x+x^2}\right)+\frac{1}{2} \tanh ^{-1}\left(\frac{\sqrt{x+x^2+x^3}}{1+x+x^2}\right)-\frac{1}{4} \sqrt{3} \tanh ^{-1}\left(\frac{\sqrt{3} \sqrt{x+x^2+x^3}}{1+x+x^2}\right)",1,"ArcTan[Sqrt[x + x^2 + x^3]/(1 + x + x^2)]/4 + ArcTanh[Sqrt[x + x^2 + x^3]/(1 + x + x^2)]/2 - (Sqrt[3]*ArcTanh[(Sqrt[3]*Sqrt[x + x^2 + x^3])/(1 + x + x^2)])/4","A",1
1219,1,89,89,0.0545349,"\int \frac{\left(-1+x^4\right)^{2/3}}{x} \, dx","IntegrateAlgebraic[(-1 + x^4)^(2/3)/x,x]","\frac{3}{8} \left(-1+x^4\right)^{2/3}+\frac{1}{4} \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^4}}{\sqrt{3}}\right)+\frac{1}{4} \log \left(1+\sqrt[3]{-1+x^4}\right)-\frac{1}{8} \log \left(1-\sqrt[3]{-1+x^4}+\left(-1+x^4\right)^{2/3}\right)","\frac{3}{8} \left(-1+x^4\right)^{2/3}+\frac{1}{4} \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^4}}{\sqrt{3}}\right)+\frac{1}{4} \log \left(1+\sqrt[3]{-1+x^4}\right)-\frac{1}{8} \log \left(1-\sqrt[3]{-1+x^4}+\left(-1+x^4\right)^{2/3}\right)",1,"(3*(-1 + x^4)^(2/3))/8 + (Sqrt[3]*ArcTan[1/Sqrt[3] - (2*(-1 + x^4)^(1/3))/Sqrt[3]])/4 + Log[1 + (-1 + x^4)^(1/3)]/4 - Log[1 - (-1 + x^4)^(1/3) + (-1 + x^4)^(2/3)]/8","A",1
1220,1,89,89,0.1551904,"\int \frac{1-x^2+x^4}{\sqrt{x+x^2+x^3} \left(-1+x^4\right)} \, dx","IntegrateAlgebraic[(1 - x^2 + x^4)/(Sqrt[x + x^2 + x^3]*(-1 + x^4)),x]","-\frac{1}{4} \tan ^{-1}\left(\frac{\sqrt{x+x^2+x^3}}{1+x+x^2}\right)-\frac{3}{2} \tanh ^{-1}\left(\frac{\sqrt{x+x^2+x^3}}{1+x+x^2}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt{3} \sqrt{x+x^2+x^3}}{1+x+x^2}\right)}{4 \sqrt{3}}","-\frac{1}{4} \tan ^{-1}\left(\frac{\sqrt{x+x^2+x^3}}{1+x+x^2}\right)-\frac{3}{2} \tanh ^{-1}\left(\frac{\sqrt{x+x^2+x^3}}{1+x+x^2}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt{3} \sqrt{x+x^2+x^3}}{1+x+x^2}\right)}{4 \sqrt{3}}",1,"-1/4*ArcTan[Sqrt[x + x^2 + x^3]/(1 + x + x^2)] - (3*ArcTanh[Sqrt[x + x^2 + x^3]/(1 + x + x^2)])/2 - ArcTanh[(Sqrt[3]*Sqrt[x + x^2 + x^3])/(1 + x + x^2)]/(4*Sqrt[3])","A",1
1221,1,89,89,0.4376272,"\int \frac{1-x+x^2}{\left(-1+x^2\right) \sqrt{1-x+x^2-x^3+x^4}} \, dx","IntegrateAlgebraic[(1 - x + x^2)/((-1 + x^2)*Sqrt[1 - x + x^2 - x^3 + x^4]),x]","\frac{1}{2} \tanh ^{-1}\left(\frac{x}{1-2 x+x^2-\sqrt{1-x+x^2-x^3+x^4}}\right)+\frac{3 \tanh ^{-1}\left(\frac{\sqrt{5} x}{1+2 x+x^2-\sqrt{1-x+x^2-x^3+x^4}}\right)}{2 \sqrt{5}}","\frac{1}{2} \tanh ^{-1}\left(\frac{x}{1-2 x+x^2-\sqrt{1-x+x^2-x^3+x^4}}\right)+\frac{3 \tanh ^{-1}\left(\frac{\sqrt{5} x}{1+2 x+x^2-\sqrt{1-x+x^2-x^3+x^4}}\right)}{2 \sqrt{5}}",1,"ArcTanh[x/(1 - 2*x + x^2 - Sqrt[1 - x + x^2 - x^3 + x^4])]/2 + (3*ArcTanh[(Sqrt[5]*x)/(1 + 2*x + x^2 - Sqrt[1 - x + x^2 - x^3 + x^4])])/(2*Sqrt[5])","A",1
1222,1,89,89,0.4506076,"\int \frac{(-1+x)^2 \left(x-2 x^2+2 x^3\right)}{(-1+2 x) \sqrt{\frac{1-2 x}{1+2 x^2}} \left(-2+4 x+3 x^2-4 x^3+2 x^4\right)} \, dx","IntegrateAlgebraic[((-1 + x)^2*(x - 2*x^2 + 2*x^3))/((-1 + 2*x)*Sqrt[(1 - 2*x)/(1 + 2*x^2)]*(-2 + 4*x + 3*x^2 - 4*x^3 + 2*x^4)),x]","\frac{\sqrt{\frac{1-2 x}{1+2 x^2}} \left(1-x+2 x^2-2 x^3\right)}{3 (-1+2 x)}-\frac{\tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{3}}+\frac{x}{\sqrt{3}}}{\sqrt{\frac{1-2 x}{1+2 x^2}}}\right)}{\sqrt{3}}","\frac{\sqrt{\frac{1-2 x}{1+2 x^2}} \left(1-x+2 x^2-2 x^3\right)}{3 (-1+2 x)}-\frac{\tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{3}}+\frac{x}{\sqrt{3}}}{\sqrt{\frac{1-2 x}{1+2 x^2}}}\right)}{\sqrt{3}}",1,"(Sqrt[(1 - 2*x)/(1 + 2*x^2)]*(1 - x + 2*x^2 - 2*x^3))/(3*(-1 + 2*x)) - ArcTanh[(-(1/Sqrt[3]) + x/Sqrt[3])/Sqrt[(1 - 2*x)/(1 + 2*x^2)]]/Sqrt[3]","A",1
1223,1,89,89,0.0595451,"\int \frac{\left(-1+x^5\right)^{2/3}}{x} \, dx","IntegrateAlgebraic[(-1 + x^5)^(2/3)/x,x]","\frac{3}{10} \left(-1+x^5\right)^{2/3}+\frac{1}{5} \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^5}}{\sqrt{3}}\right)+\frac{1}{5} \log \left(1+\sqrt[3]{-1+x^5}\right)-\frac{1}{10} \log \left(1-\sqrt[3]{-1+x^5}+\left(-1+x^5\right)^{2/3}\right)","\frac{3}{10} \left(-1+x^5\right)^{2/3}+\frac{1}{5} \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^5}}{\sqrt{3}}\right)+\frac{1}{5} \log \left(1+\sqrt[3]{-1+x^5}\right)-\frac{1}{10} \log \left(1-\sqrt[3]{-1+x^5}+\left(-1+x^5\right)^{2/3}\right)",1,"(3*(-1 + x^5)^(2/3))/10 + (Sqrt[3]*ArcTan[1/Sqrt[3] - (2*(-1 + x^5)^(1/3))/Sqrt[3]])/5 + Log[1 + (-1 + x^5)^(1/3)]/5 - Log[1 - (-1 + x^5)^(1/3) + (-1 + x^5)^(2/3)]/10","A",1
1224,1,89,89,0.0531014,"\int \frac{\left(-1+x^6\right)^{2/3}}{x} \, dx","IntegrateAlgebraic[(-1 + x^6)^(2/3)/x,x]","\frac{1}{4} \left(-1+x^6\right)^{2/3}+\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^6}}{\sqrt{3}}\right)}{2 \sqrt{3}}+\frac{1}{6} \log \left(1+\sqrt[3]{-1+x^6}\right)-\frac{1}{12} \log \left(1-\sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)","\frac{1}{4} \left(-1+x^6\right)^{2/3}+\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^6}}{\sqrt{3}}\right)}{2 \sqrt{3}}+\frac{1}{6} \log \left(1+\sqrt[3]{-1+x^6}\right)-\frac{1}{12} \log \left(1-\sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)",1,"(-1 + x^6)^(2/3)/4 + ArcTan[1/Sqrt[3] - (2*(-1 + x^6)^(1/3))/Sqrt[3]]/(2*Sqrt[3]) + Log[1 + (-1 + x^6)^(1/3)]/6 - Log[1 - (-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)]/12","A",1
1225,1,89,89,0.1911451,"\int \sqrt{a x^2+\sqrt{b+a^2 x^4}} \, dx","IntegrateAlgebraic[Sqrt[a*x^2 + Sqrt[b + a^2*x^4]],x]","\frac{1}{2} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{b}}\right)}{2 \sqrt{2} \sqrt{a}}","\frac{1}{2} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{b}}\right)}{2 \sqrt{2} \sqrt{a}}",1,"(x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/2 + (Sqrt[b]*ArcTan[(Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[b]])/(2*Sqrt[2]*Sqrt[a])","A",1
1226,1,90,90,0.0978329,"\int \frac{\left(-1+x^2\right)^{2/3}}{x^3} \, dx","IntegrateAlgebraic[(-1 + x^2)^(2/3)/x^3,x]","-\frac{\left(-1+x^2\right)^{2/3}}{2 x^2}-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^2}}{\sqrt{3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(1+\sqrt[3]{-1+x^2}\right)+\frac{1}{6} \log \left(1-\sqrt[3]{-1+x^2}+\left(-1+x^2\right)^{2/3}\right)","-\frac{\left(-1+x^2\right)^{2/3}}{2 x^2}-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^2}}{\sqrt{3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(1+\sqrt[3]{-1+x^2}\right)+\frac{1}{6} \log \left(1-\sqrt[3]{-1+x^2}+\left(-1+x^2\right)^{2/3}\right)",1,"-1/2*(-1 + x^2)^(2/3)/x^2 - ArcTan[1/Sqrt[3] - (2*(-1 + x^2)^(1/3))/Sqrt[3]]/Sqrt[3] - Log[1 + (-1 + x^2)^(1/3)]/3 + Log[1 - (-1 + x^2)^(1/3) + (-1 + x^2)^(2/3)]/6","A",1
1227,1,86,90,19.9504999,"\int \frac{-2+x+x^2}{x^2 \left(-1+x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[(-2 + x + x^2)/(x^2*(-1 + x^2)^(3/4)),x]","-\frac{2 \sqrt[4]{-1+x^2}}{x}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^2}}{-1+\sqrt{-1+x^2}}\right)}{\sqrt{2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^2}}{1+\sqrt{-1+x^2}}\right)}{\sqrt{2}}","-\frac{2 \sqrt[4]{-1+x^2}}{x}+\frac{\tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{\sqrt{-1+x^2}}{\sqrt{2}}}{\sqrt[4]{-1+x^2}}\right)}{\sqrt{2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^2}}{1+\sqrt{-1+x^2}}\right)}{\sqrt{2}}",1,"(-2*(-1 + x^2)^(1/4))/x - ArcTan[(Sqrt[2]*(-1 + x^2)^(1/4))/(-1 + Sqrt[-1 + x^2])]/Sqrt[2] + ArcTanh[(Sqrt[2]*(-1 + x^2)^(1/4))/(1 + Sqrt[-1 + x^2])]/Sqrt[2]","A",1
1228,1,90,90,0.0750036,"\int \frac{\sqrt[3]{1+x^3}}{x^4} \, dx","IntegrateAlgebraic[(1 + x^3)^(1/3)/x^4,x]","-\frac{\sqrt[3]{1+x^3}}{3 x^3}-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^3}}{\sqrt{3}}\right)}{3 \sqrt{3}}+\frac{1}{9} \log \left(-1+\sqrt[3]{1+x^3}\right)-\frac{1}{18} \log \left(1+\sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)","-\frac{\sqrt[3]{1+x^3}}{3 x^3}-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^3}}{\sqrt{3}}\right)}{3 \sqrt{3}}+\frac{1}{9} \log \left(-1+\sqrt[3]{1+x^3}\right)-\frac{1}{18} \log \left(1+\sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)",1,"-1/3*(1 + x^3)^(1/3)/x^3 - ArcTan[1/Sqrt[3] + (2*(1 + x^3)^(1/3))/Sqrt[3]]/(3*Sqrt[3]) + Log[-1 + (1 + x^3)^(1/3)]/9 - Log[1 + (1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/18","A",1
1229,1,90,90,0.0706929,"\int \frac{\left(1+x^3\right)^{2/3}}{x^4} \, dx","IntegrateAlgebraic[(1 + x^3)^(2/3)/x^4,x]","-\frac{\left(1+x^3\right)^{2/3}}{3 x^3}+\frac{2 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^3}}{\sqrt{3}}\right)}{3 \sqrt{3}}+\frac{2}{9} \log \left(-1+\sqrt[3]{1+x^3}\right)-\frac{1}{9} \log \left(1+\sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)","-\frac{\left(1+x^3\right)^{2/3}}{3 x^3}+\frac{2 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^3}}{\sqrt{3}}\right)}{3 \sqrt{3}}+\frac{2}{9} \log \left(-1+\sqrt[3]{1+x^3}\right)-\frac{1}{9} \log \left(1+\sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)",1,"-1/3*(1 + x^3)^(2/3)/x^3 + (2*ArcTan[1/Sqrt[3] + (2*(1 + x^3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]) + (2*Log[-1 + (1 + x^3)^(1/3)])/9 - Log[1 + (1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/9","A",1
1230,1,90,90,12.9928117,"\int \frac{-2 k-(-1+k) (1+k) x+2 k x^2}{\sqrt[4]{\left(1-x^2\right) \left(1-k^2 x^2\right)} \left(-1+d+(3+d) k x-\left(d+3 k^2\right) x^2+k \left(-d+k^2\right) x^3\right)} \, dx","IntegrateAlgebraic[(-2*k - (-1 + k)*(1 + k)*x + 2*k*x^2)/(((1 - x^2)*(1 - k^2*x^2))^(1/4)*(-1 + d + (3 + d)*k*x - (d + 3*k^2)*x^2 + k*(-d + k^2)*x^3)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{-1+k x}\right)}{d^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{-1+k x}\right)}{d^{3/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{-1+k x}\right)}{d^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{-1+k x}\right)}{d^{3/4}}",1,"ArcTan[(d^(1/4)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/4))/(-1 + k*x)]/d^(3/4) - ArcTanh[(d^(1/4)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/4))/(-1 + k*x)]/d^(3/4)","A",1
1231,1,90,90,12.9750664,"\int \frac{-2 k+(-1+k) (1+k) x+2 k x^2}{\sqrt[4]{\left(1-x^2\right) \left(1-k^2 x^2\right)} \left(1-d+(3+d) k x+\left(d+3 k^2\right) x^2+k \left(-d+k^2\right) x^3\right)} \, dx","IntegrateAlgebraic[(-2*k + (-1 + k)*(1 + k)*x + 2*k*x^2)/(((1 - x^2)*(1 - k^2*x^2))^(1/4)*(1 - d + (3 + d)*k*x + (d + 3*k^2)*x^2 + k*(-d + k^2)*x^3)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{1+k x}\right)}{d^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{1+k x}\right)}{d^{3/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{1+k x}\right)}{d^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{1+k x}\right)}{d^{3/4}}",1,"-(ArcTan[(d^(1/4)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/4))/(1 + k*x)]/d^(3/4)) + ArcTanh[(d^(1/4)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/4))/(1 + k*x)]/d^(3/4)","A",1
1232,1,84,90,0.177908,"\int \frac{\left(1+x^2\right) \sqrt{1-2 x^4}}{x^5} \, dx","IntegrateAlgebraic[((1 + x^2)*Sqrt[1 - 2*x^4])/x^5,x]","\frac{\left(-1-2 x^2\right) \sqrt{1-2 x^4}}{4 x^4}-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x^2}{-1+\sqrt{1-2 x^4}}\right)+\frac{\log \left(x^2\right)}{2}-\frac{1}{2} \log \left(-1+\sqrt{1-2 x^4}\right)","\frac{\left(-1-2 x^2\right) \sqrt{1-2 x^4}}{4 x^4}+i \tan ^{-1}\left(\sqrt{2} x^2-i \sqrt{1-2 x^4}\right)+\frac{i \log \left(i \sqrt{2} x^2+\sqrt{1-2 x^4}\right)}{\sqrt{2}}",1,"((-1 - 2*x^2)*Sqrt[1 - 2*x^4])/(4*x^4) - Sqrt[2]*ArcTan[(Sqrt[2]*x^2)/(-1 + Sqrt[1 - 2*x^4])] + Log[x^2]/2 - Log[-1 + Sqrt[1 - 2*x^4]]/2","A",1
1233,1,90,90,0.058531,"\int \frac{\left(1+x^4\right)^{2/3}}{x^5} \, dx","IntegrateAlgebraic[(1 + x^4)^(2/3)/x^5,x]","-\frac{\left(1+x^4\right)^{2/3}}{4 x^4}+\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^4}}{\sqrt{3}}\right)}{2 \sqrt{3}}+\frac{1}{6} \log \left(-1+\sqrt[3]{1+x^4}\right)-\frac{1}{12} \log \left(1+\sqrt[3]{1+x^4}+\left(1+x^4\right)^{2/3}\right)","-\frac{\left(1+x^4\right)^{2/3}}{4 x^4}+\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^4}}{\sqrt{3}}\right)}{2 \sqrt{3}}+\frac{1}{6} \log \left(-1+\sqrt[3]{1+x^4}\right)-\frac{1}{12} \log \left(1+\sqrt[3]{1+x^4}+\left(1+x^4\right)^{2/3}\right)",1,"-1/4*(1 + x^4)^(2/3)/x^4 + ArcTan[1/Sqrt[3] + (2*(1 + x^4)^(1/3))/Sqrt[3]]/(2*Sqrt[3]) + Log[-1 + (1 + x^4)^(1/3)]/6 - Log[1 + (1 + x^4)^(1/3) + (1 + x^4)^(2/3)]/12","A",1
1234,1,90,90,1.0273411,"\int \frac{\left(-3+x^4\right) \left(1+x^4\right)^{2/3}}{x^3 \left(1-x^3+x^4\right)} \, dx","IntegrateAlgebraic[((-3 + x^4)*(1 + x^4)^(2/3))/(x^3*(1 - x^3 + x^4)),x]","\frac{3 \left(1+x^4\right)^{2/3}}{2 x^2}-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^4}}\right)+\log \left(-x+\sqrt[3]{1+x^4}\right)-\frac{1}{2} \log \left(x^2+x \sqrt[3]{1+x^4}+\left(1+x^4\right)^{2/3}\right)","\frac{3 \left(1+x^4\right)^{2/3}}{2 x^2}-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^4}}\right)+\log \left(-x+\sqrt[3]{1+x^4}\right)-\frac{1}{2} \log \left(x^2+x \sqrt[3]{1+x^4}+\left(1+x^4\right)^{2/3}\right)",1,"(3*(1 + x^4)^(2/3))/(2*x^2) - Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^4)^(1/3))] + Log[-x + (1 + x^4)^(1/3)] - Log[x^2 + x*(1 + x^4)^(1/3) + (1 + x^4)^(2/3)]/2","A",1
1235,1,90,90,0.8537248,"\int \frac{\sqrt[3]{-1+x^4} \left(3+x^4\right)}{x^2 \left(-1+x^3+x^4\right)} \, dx","IntegrateAlgebraic[((-1 + x^4)^(1/3)*(3 + x^4))/(x^2*(-1 + x^3 + x^4)),x]","\frac{3 \sqrt[3]{-1+x^4}}{x}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-1+x^4}}\right)-\log \left(x+\sqrt[3]{-1+x^4}\right)+\frac{1}{2} \log \left(x^2-x \sqrt[3]{-1+x^4}+\left(-1+x^4\right)^{2/3}\right)","\frac{3 \sqrt[3]{-1+x^4}}{x}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-1+x^4}}\right)-\log \left(x+\sqrt[3]{-1+x^4}\right)+\frac{1}{2} \log \left(x^2-x \sqrt[3]{-1+x^4}+\left(-1+x^4\right)^{2/3}\right)",1,"(3*(-1 + x^4)^(1/3))/x + Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(-1 + x^4)^(1/3))] - Log[x + (-1 + x^4)^(1/3)] + Log[x^2 - x*(-1 + x^4)^(1/3) + (-1 + x^4)^(2/3)]/2","A",1
1236,1,90,90,1.0231143,"\int \frac{\left(-1+x^4\right)^{2/3} \left(3+x^4\right)}{x^3 \left(-1+x^3+x^4\right)} \, dx","IntegrateAlgebraic[((-1 + x^4)^(2/3)*(3 + x^4))/(x^3*(-1 + x^3 + x^4)),x]","\frac{3 \left(-1+x^4\right)^{2/3}}{2 x^2}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-1+x^4}}\right)+\log \left(x+\sqrt[3]{-1+x^4}\right)-\frac{1}{2} \log \left(x^2-x \sqrt[3]{-1+x^4}+\left(-1+x^4\right)^{2/3}\right)","\frac{3 \left(-1+x^4\right)^{2/3}}{2 x^2}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-1+x^4}}\right)+\log \left(x+\sqrt[3]{-1+x^4}\right)-\frac{1}{2} \log \left(x^2-x \sqrt[3]{-1+x^4}+\left(-1+x^4\right)^{2/3}\right)",1,"(3*(-1 + x^4)^(2/3))/(2*x^2) + Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(-1 + x^4)^(1/3))] + Log[x + (-1 + x^4)^(1/3)] - Log[x^2 - x*(-1 + x^4)^(1/3) + (-1 + x^4)^(2/3)]/2","A",1
1237,1,90,90,0.8655998,"\int \frac{\left(-3+x^4\right) \sqrt[3]{1+x^4}}{x^2 \left(1+x^3+x^4\right)} \, dx","IntegrateAlgebraic[((-3 + x^4)*(1 + x^4)^(1/3))/(x^2*(1 + x^3 + x^4)),x]","\frac{3 \sqrt[3]{1+x^4}}{x}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{1+x^4}}\right)-\log \left(x+\sqrt[3]{1+x^4}\right)+\frac{1}{2} \log \left(x^2-x \sqrt[3]{1+x^4}+\left(1+x^4\right)^{2/3}\right)","\frac{3 \sqrt[3]{1+x^4}}{x}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{1+x^4}}\right)-\log \left(x+\sqrt[3]{1+x^4}\right)+\frac{1}{2} \log \left(x^2-x \sqrt[3]{1+x^4}+\left(1+x^4\right)^{2/3}\right)",1,"(3*(1 + x^4)^(1/3))/x + Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(1 + x^4)^(1/3))] - Log[x + (1 + x^4)^(1/3)] + Log[x^2 - x*(1 + x^4)^(1/3) + (1 + x^4)^(2/3)]/2","A",1
1238,1,90,90,2.3849084,"\int \frac{(4+3 x) \left(-1-x+x^4\right) \sqrt[4]{-1-x+2 x^4}}{x^6 \left(1+x+x^4\right)} \, dx","IntegrateAlgebraic[((4 + 3*x)*(-1 - x + x^4)*(-1 - x + 2*x^4)^(1/4))/(x^6*(1 + x + x^4)),x]","-\frac{4 \sqrt[4]{-1-x+2 x^4} \left(-1-x+12 x^4\right)}{5 x^5}-4 \sqrt[4]{3} \tan ^{-1}\left(\frac{\sqrt[4]{3} x}{\sqrt[4]{-1-x+2 x^4}}\right)+4 \sqrt[4]{3} \tanh ^{-1}\left(\frac{\sqrt[4]{3} x}{\sqrt[4]{-1-x+2 x^4}}\right)","-\frac{4 \sqrt[4]{-1-x+2 x^4} \left(-1-x+12 x^4\right)}{5 x^5}-4 \sqrt[4]{3} \tan ^{-1}\left(\frac{\sqrt[4]{3} x}{\sqrt[4]{-1-x+2 x^4}}\right)+4 \sqrt[4]{3} \tanh ^{-1}\left(\frac{\sqrt[4]{3} x}{\sqrt[4]{-1-x+2 x^4}}\right)",1,"(-4*(-1 - x + 2*x^4)^(1/4)*(-1 - x + 12*x^4))/(5*x^5) - 4*3^(1/4)*ArcTan[(3^(1/4)*x)/(-1 - x + 2*x^4)^(1/4)] + 4*3^(1/4)*ArcTanh[(3^(1/4)*x)/(-1 - x + 2*x^4)^(1/4)]","A",1
1239,1,77,90,0.4343479,"\int \frac{\sqrt{1+3 x^4}}{-1+3 x^4} \, dx","IntegrateAlgebraic[Sqrt[1 + 3*x^4]/(-1 + 3*x^4),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{3} x}{\sqrt{1+3 x^4}}\right)}{2 \sqrt{2} \sqrt[4]{3}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{3} x}{\sqrt{1+3 x^4}}\right)}{2 \sqrt{2} \sqrt[4]{3}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{1+3 x^4}}{1-\sqrt{2} \sqrt[4]{3} x+\sqrt{3} x^2}\right)}{\sqrt{2} \sqrt[4]{3}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{3} x}{\sqrt{1+3 x^4}}\right)}{2 \sqrt{2} \sqrt[4]{3}}",1,"-1/2*ArcTan[(Sqrt[2]*3^(1/4)*x)/Sqrt[1 + 3*x^4]]/(Sqrt[2]*3^(1/4)) - ArcTanh[(Sqrt[2]*3^(1/4)*x)/Sqrt[1 + 3*x^4]]/(2*Sqrt[2]*3^(1/4))","A",1
1240,1,90,90,0.3498261,"\int \frac{\left(b+a x^2\right) \sqrt[4]{-b x^2+a x^4}}{-b+a x^2} \, dx","IntegrateAlgebraic[((b + a*x^2)*(-(b*x^2) + a*x^4)^(1/4))/(-b + a*x^2),x]","\frac{1}{2} x \sqrt[4]{-b x^2+a x^4}-\frac{7 b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{4 a^{3/4}}+\frac{7 b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{4 a^{3/4}}","\frac{1}{2} x \sqrt[4]{-b x^2+a x^4}-\frac{7 b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{4 a^{3/4}}+\frac{7 b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{4 a^{3/4}}",1,"(x*(-(b*x^2) + a*x^4)^(1/4))/2 - (7*b*ArcTan[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/(4*a^(3/4)) + (7*b*ArcTanh[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/(4*a^(3/4))","A",1
1241,1,90,90,0.2775845,"\int \frac{-b+2 a x^2}{\left(b+a x^2\right) \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[(-b + 2*a*x^2)/((b + a*x^2)*(b*x^2 + a*x^4)^(1/4)),x]","-\frac{6 \left(b x^2+a x^4\right)^{3/4}}{x \left(b+a x^2\right)}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{\sqrt[4]{a}}","-\frac{6 \left(b x^2+a x^4\right)^{3/4}}{x \left(b+a x^2\right)}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{\sqrt[4]{a}}",1,"(-6*(b*x^2 + a*x^4)^(3/4))/(x*(b + a*x^2)) + (2*ArcTan[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/a^(1/4) + (2*ArcTanh[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/a^(1/4)","A",1
1242,1,90,90,0.0624934,"\int \frac{\left(1+x^5\right)^{2/3}}{x^6} \, dx","IntegrateAlgebraic[(1 + x^5)^(2/3)/x^6,x]","-\frac{\left(1+x^5\right)^{2/3}}{5 x^5}+\frac{2 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^5}}{\sqrt{3}}\right)}{5 \sqrt{3}}+\frac{2}{15} \log \left(-1+\sqrt[3]{1+x^5}\right)-\frac{1}{15} \log \left(1+\sqrt[3]{1+x^5}+\left(1+x^5\right)^{2/3}\right)","-\frac{\left(1+x^5\right)^{2/3}}{5 x^5}+\frac{2 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^5}}{\sqrt{3}}\right)}{5 \sqrt{3}}+\frac{2}{15} \log \left(-1+\sqrt[3]{1+x^5}\right)-\frac{1}{15} \log \left(1+\sqrt[3]{1+x^5}+\left(1+x^5\right)^{2/3}\right)",1,"-1/5*(1 + x^5)^(2/3)/x^5 + (2*ArcTan[1/Sqrt[3] + (2*(1 + x^5)^(1/3))/Sqrt[3]])/(5*Sqrt[3]) + (2*Log[-1 + (1 + x^5)^(1/3)])/15 - Log[1 + (1 + x^5)^(1/3) + (1 + x^5)^(2/3)]/15","A",1
1243,1,90,90,1.4955762,"\int \frac{\left(1+x^5\right)^{2/3} \left(-3+2 x^5\right)}{x^3 \left(1-x^3+x^5\right)} \, dx","IntegrateAlgebraic[((1 + x^5)^(2/3)*(-3 + 2*x^5))/(x^3*(1 - x^3 + x^5)),x]","\frac{3 \left(1+x^5\right)^{2/3}}{2 x^2}-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^5}}\right)+\log \left(-x+\sqrt[3]{1+x^5}\right)-\frac{1}{2} \log \left(x^2+x \sqrt[3]{1+x^5}+\left(1+x^5\right)^{2/3}\right)","\frac{3 \left(1+x^5\right)^{2/3}}{2 x^2}-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^5}}\right)+\log \left(-x+\sqrt[3]{1+x^5}\right)-\frac{1}{2} \log \left(x^2+x \sqrt[3]{1+x^5}+\left(1+x^5\right)^{2/3}\right)",1,"(3*(1 + x^5)^(2/3))/(2*x^2) - Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^5)^(1/3))] + Log[-x + (1 + x^5)^(1/3)] - Log[x^2 + x*(1 + x^5)^(1/3) + (1 + x^5)^(2/3)]/2","A",1
1244,1,90,90,1.5557669,"\int \frac{\left(-1+x^5\right)^{2/3} \left(3+2 x^5\right)}{x^3 \left(-1-x^3+x^5\right)} \, dx","IntegrateAlgebraic[((-1 + x^5)^(2/3)*(3 + 2*x^5))/(x^3*(-1 - x^3 + x^5)),x]","\frac{3 \left(-1+x^5\right)^{2/3}}{2 x^2}-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^5}}\right)+\log \left(-x+\sqrt[3]{-1+x^5}\right)-\frac{1}{2} \log \left(x^2+x \sqrt[3]{-1+x^5}+\left(-1+x^5\right)^{2/3}\right)","\frac{3 \left(-1+x^5\right)^{2/3}}{2 x^2}-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^5}}\right)+\log \left(-x+\sqrt[3]{-1+x^5}\right)-\frac{1}{2} \log \left(x^2+x \sqrt[3]{-1+x^5}+\left(-1+x^5\right)^{2/3}\right)",1,"(3*(-1 + x^5)^(2/3))/(2*x^2) - Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^5)^(1/3))] + Log[-x + (-1 + x^5)^(1/3)] - Log[x^2 + x*(-1 + x^5)^(1/3) + (-1 + x^5)^(2/3)]/2","A",1
1245,1,95,90,0.0973995,"\int \frac{\sqrt[4]{-1+x^6}}{x} \, dx","IntegrateAlgebraic[(-1 + x^6)^(1/4)/x,x]","\frac{2}{3} \sqrt[4]{-1+x^6}-\frac{\tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{\sqrt{-1+x^6}}{\sqrt{2}}}{\sqrt[4]{-1+x^6}}\right)}{3 \sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^6}}{1+\sqrt{-1+x^6}}\right)}{3 \sqrt{2}}","\frac{2}{3} \sqrt[4]{-1+x^6}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^6}}{-1+\sqrt{-1+x^6}}\right)}{3 \sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^6}}{1+\sqrt{-1+x^6}}\right)}{3 \sqrt{2}}",1,"(2*(-1 + x^6)^(1/4))/3 - ArcTan[(-(1/Sqrt[2]) + Sqrt[-1 + x^6]/Sqrt[2])/(-1 + x^6)^(1/4)]/(3*Sqrt[2]) - ArcTanh[(Sqrt[2]*(-1 + x^6)^(1/4))/(1 + Sqrt[-1 + x^6])]/(3*Sqrt[2])","A",1
1246,1,90,90,0.0858658,"\int \frac{1}{x^7 \sqrt[3]{1+x^6}} \, dx","IntegrateAlgebraic[1/(x^7*(1 + x^6)^(1/3)),x]","-\frac{\left(1+x^6\right)^{2/3}}{6 x^6}-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^6}}{\sqrt{3}}\right)}{6 \sqrt{3}}-\frac{1}{18} \log \left(-1+\sqrt[3]{1+x^6}\right)+\frac{1}{36} \log \left(1+\sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)","-\frac{\left(1+x^6\right)^{2/3}}{6 x^6}-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^6}}{\sqrt{3}}\right)}{6 \sqrt{3}}-\frac{1}{18} \log \left(-1+\sqrt[3]{1+x^6}\right)+\frac{1}{36} \log \left(1+\sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)",1,"-1/6*(1 + x^6)^(2/3)/x^6 - ArcTan[1/Sqrt[3] + (2*(1 + x^6)^(1/3))/Sqrt[3]]/(6*Sqrt[3]) - Log[-1 + (1 + x^6)^(1/3)]/18 + Log[1 + (1 + x^6)^(1/3) + (1 + x^6)^(2/3)]/36","A",1
1247,1,90,90,0.0603416,"\int \frac{\left(1+x^6\right)^{2/3}}{x^7} \, dx","IntegrateAlgebraic[(1 + x^6)^(2/3)/x^7,x]","-\frac{\left(1+x^6\right)^{2/3}}{6 x^6}+\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^6}}{\sqrt{3}}\right)}{3 \sqrt{3}}+\frac{1}{9} \log \left(-1+\sqrt[3]{1+x^6}\right)-\frac{1}{18} \log \left(1+\sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)","-\frac{\left(1+x^6\right)^{2/3}}{6 x^6}+\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^6}}{\sqrt{3}}\right)}{3 \sqrt{3}}+\frac{1}{9} \log \left(-1+\sqrt[3]{1+x^6}\right)-\frac{1}{18} \log \left(1+\sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)",1,"-1/6*(1 + x^6)^(2/3)/x^6 + ArcTan[1/Sqrt[3] + (2*(1 + x^6)^(1/3))/Sqrt[3]]/(3*Sqrt[3]) + Log[-1 + (1 + x^6)^(1/3)]/9 - Log[1 + (1 + x^6)^(1/3) + (1 + x^6)^(2/3)]/18","A",1
1248,1,90,90,12.9502373,"\int \frac{\left(-3+k^2\right) x+2 k^2 x^3}{\sqrt[4]{\left(1-x^2\right) \left(1-k^2 x^2\right)} \left(-1+d+\left(3-d k^2\right) x^2-3 x^4+x^6\right)} \, dx","IntegrateAlgebraic[((-3 + k^2)*x + 2*k^2*x^3)/(((1 - x^2)*(1 - k^2*x^2))^(1/4)*(-1 + d + (3 - d*k^2)*x^2 - 3*x^4 + x^6)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{-1+x^2}\right)}{d^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{-1+x^2}\right)}{d^{3/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{-1+x^2}\right)}{d^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{-1+x^2}\right)}{d^{3/4}}",1,"ArcTan[(d^(1/4)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/4))/(-1 + x^2)]/d^(3/4) - ArcTanh[(d^(1/4)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/4))/(-1 + x^2)]/d^(3/4)","A",1
1249,1,90,90,17.7912655,"\int \frac{\left(-1+x^7\right)^{2/3} \left(3+4 x^7\right)}{x^3 \left(-1+x^3+x^7\right)} \, dx","IntegrateAlgebraic[((-1 + x^7)^(2/3)*(3 + 4*x^7))/(x^3*(-1 + x^3 + x^7)),x]","\frac{3 \left(-1+x^7\right)^{2/3}}{2 x^2}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-1+x^7}}\right)+\log \left(x+\sqrt[3]{-1+x^7}\right)-\frac{1}{2} \log \left(x^2-x \sqrt[3]{-1+x^7}+\left(-1+x^7\right)^{2/3}\right)","\frac{3 \left(-1+x^7\right)^{2/3}}{2 x^2}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-1+x^7}}\right)+\log \left(x+\sqrt[3]{-1+x^7}\right)-\frac{1}{2} \log \left(x^2-x \sqrt[3]{-1+x^7}+\left(-1+x^7\right)^{2/3}\right)",1,"(3*(-1 + x^7)^(2/3))/(2*x^2) + Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(-1 + x^7)^(1/3))] + Log[x + (-1 + x^7)^(1/3)] - Log[x^2 - x*(-1 + x^7)^(1/3) + (-1 + x^7)^(2/3)]/2","A",1
1250,1,90,90,5.0458157,"\int \frac{x}{\sqrt{-b x+a^2 x^2} \left(a x^2+x \sqrt{-b x+a^2 x^2}\right)^{3/2}} \, dx","IntegrateAlgebraic[x/(Sqrt[-(b*x) + a^2*x^2]*(a*x^2 + x*Sqrt[-(b*x) + a^2*x^2])^(3/2)),x]","\frac{20 a \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{3 b^2 x}-\frac{4 \sqrt{-b x+a^2 x^2} \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{3 b^2 x^2}","\frac{20 a \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{3 b^2 x}-\frac{4 \sqrt{-b x+a^2 x^2} \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{3 b^2 x^2}",1,"(20*a*Sqrt[x*(a*x + Sqrt[-(b*x) + a^2*x^2])])/(3*b^2*x) - (4*Sqrt[-(b*x) + a^2*x^2]*Sqrt[x*(a*x + Sqrt[-(b*x) + a^2*x^2])])/(3*b^2*x^2)","A",1
1251,1,91,91,0.7768715,"\int \frac{-\sqrt{a b}+x}{\sqrt{x (a+x) (b+x)} \left(\sqrt{a b}+x\right)} \, dx","IntegrateAlgebraic[(-Sqrt[a*b] + x)/(Sqrt[x*(a + x)*(b + x)]*(Sqrt[a*b] + x)),x]","\frac{2 \tan ^{-1}\left(\frac{(b+x) \left(\frac{a}{\sqrt{-a-b+2 \sqrt{a b}}}+\frac{x}{\sqrt{-a-b+2 \sqrt{a b}}}\right)}{\sqrt{a b x+(a+b) x^2+x^3}}\right)}{\sqrt{-a-b+2 \sqrt{a b}}}","\frac{2 \tan ^{-1}\left(\frac{(b+x) \left(\frac{a}{\sqrt{-a-b+2 \sqrt{a b}}}+\frac{x}{\sqrt{-a-b+2 \sqrt{a b}}}\right)}{\sqrt{a b x+(a+b) x^2+x^3}}\right)}{\sqrt{-a-b+2 \sqrt{a b}}}",1,"(2*ArcTan[((b + x)*(a/Sqrt[-a - b + 2*Sqrt[a*b]] + x/Sqrt[-a - b + 2*Sqrt[a*b]]))/Sqrt[a*b*x + (a + b)*x^2 + x^3]])/Sqrt[-a - b + 2*Sqrt[a*b]]","A",1
1252,1,91,91,0.1588491,"\int \frac{1}{\sqrt[3]{-x^2+x^3}} \, dx","IntegrateAlgebraic[(-x^2 + x^3)^(-1/3),x]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{-x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{-x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)",1,"Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-x^2 + x^3)^(1/3))] - Log[-x + (-x^2 + x^3)^(1/3)] + Log[x^2 + x*(-x^2 + x^3)^(1/3) + (-x^2 + x^3)^(2/3)]/2","A",1
1253,1,91,91,5.1689439,"\int \frac{3-x^2}{\left(1-x^2\right) \sqrt[4]{1-6 x^2+x^4}} \, dx","IntegrateAlgebraic[(3 - x^2)/((1 - x^2)*(1 - 6*x^2 + x^4)^(1/4)),x]","\tan ^{-1}\left(\frac{i+x}{\sqrt[4]{1-6 x^2+x^4}}\right)-\tan ^{-1}\left(\frac{\sqrt[4]{1-6 x^2+x^4}}{-i+x}\right)+\tanh ^{-1}\left(\frac{i+x}{\sqrt[4]{1-6 x^2+x^4}}\right)+\tanh ^{-1}\left(\frac{\sqrt[4]{1-6 x^2+x^4}}{-i+x}\right)","\tan ^{-1}\left(\frac{i+x}{\sqrt[4]{1-6 x^2+x^4}}\right)-\tan ^{-1}\left(\frac{\sqrt[4]{1-6 x^2+x^4}}{-i+x}\right)+\tanh ^{-1}\left(\frac{i+x}{\sqrt[4]{1-6 x^2+x^4}}\right)+\tanh ^{-1}\left(\frac{\sqrt[4]{1-6 x^2+x^4}}{-i+x}\right)",1,"ArcTan[(I + x)/(1 - 6*x^2 + x^4)^(1/4)] - ArcTan[(1 - 6*x^2 + x^4)^(1/4)/(-I + x)] + ArcTanh[(I + x)/(1 - 6*x^2 + x^4)^(1/4)] + ArcTanh[(1 - 6*x^2 + x^4)^(1/4)/(-I + x)]","A",1
1254,1,91,91,0.1555793,"\int \frac{-3+x^4}{\left(1+x^4\right) \sqrt[4]{-3 x+4 x^4-3 x^5}} \, dx","IntegrateAlgebraic[(-3 + x^4)/((1 + x^4)*(-3*x + 4*x^4 - 3*x^5)^(1/4)),x]","\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \left(-3 x+4 x^4-3 x^5\right)^{3/4}}{3-4 x^3+3 x^4}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \left(-3 x+4 x^4-3 x^5\right)^{3/4}}{3-4 x^3+3 x^4}\right)","\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \left(-3 x+4 x^4-3 x^5\right)^{3/4}}{3-4 x^3+3 x^4}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \left(-3 x+4 x^4-3 x^5\right)^{3/4}}{3-4 x^3+3 x^4}\right)",1,"Sqrt[2]*ArcTan[(Sqrt[2]*(-3*x + 4*x^4 - 3*x^5)^(3/4))/(3 - 4*x^3 + 3*x^4)] + Sqrt[2]*ArcTanh[(Sqrt[2]*(-3*x + 4*x^4 - 3*x^5)^(3/4))/(3 - 4*x^3 + 3*x^4)]","A",1
1255,1,91,91,0.9973014,"\int \frac{\sqrt[3]{-1+x^6} \left(1+x^6\right)}{x^2 \left(-1+x^3+x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^6)^(1/3)*(1 + x^6))/(x^2*(-1 + x^3 + x^6)),x]","\frac{\sqrt[3]{-1+x^6}}{x}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-1+x^6}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(x+\sqrt[3]{-1+x^6}\right)+\frac{1}{6} \log \left(x^2-x \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)","\frac{\sqrt[3]{-1+x^6}}{x}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-1+x^6}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(x+\sqrt[3]{-1+x^6}\right)+\frac{1}{6} \log \left(x^2-x \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)",1,"(-1 + x^6)^(1/3)/x + ArcTan[(Sqrt[3]*x)/(-x + 2*(-1 + x^6)^(1/3))]/Sqrt[3] - Log[x + (-1 + x^6)^(1/3)]/3 + Log[x^2 - x*(-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)]/6","A",1
1256,1,65,91,0.595952,"\int \frac{\left(-1+x^4\right) \sqrt{1+x^4}}{1+x^2+3 x^4+x^6+x^8} \, dx","IntegrateAlgebraic[((-1 + x^4)*Sqrt[1 + x^4])/(1 + x^2 + 3*x^4 + x^6 + x^8),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x \sqrt{1+x^4}}{1-x^2+x^4}\right)}{2 \sqrt{3}}-\frac{1}{2} \tanh ^{-1}\left(\frac{x \sqrt{1+x^4}}{1+x^2+x^4}\right)","-\sqrt{\frac{1}{6} \left(-1-i \sqrt{3}\right)} \tan ^{-1}\left(\frac{2 x}{\left(-i+\sqrt{3}\right) \sqrt{1+x^4}}\right)-\sqrt{\frac{1}{6} \left(-1+i \sqrt{3}\right)} \tan ^{-1}\left(\frac{2 x}{\left(i+\sqrt{3}\right) \sqrt{1+x^4}}\right)",1,"-1/2*ArcTan[(Sqrt[3]*x*Sqrt[1 + x^4])/(1 - x^2 + x^4)]/Sqrt[3] - ArcTanh[(x*Sqrt[1 + x^4])/(1 + x^2 + x^4)]/2","A",1
1257,1,91,91,0.4036074,"\int \frac{-1+2 x^8}{\sqrt[4]{1+x^4} \left(-1+x^8\right)} \, dx","IntegrateAlgebraic[(-1 + 2*x^8)/((1 + x^4)^(1/4)*(-1 + x^8)),x]","-\frac{x}{2 \sqrt[4]{1+x^4}}+\tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{1+x^4}}\right)}{4 \sqrt[4]{2}}+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{1+x^4}}\right)}{4 \sqrt[4]{2}}","-\frac{x}{2 \sqrt[4]{1+x^4}}+\tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{1+x^4}}\right)}{4 \sqrt[4]{2}}+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{1+x^4}}\right)}{4 \sqrt[4]{2}}",1,"-1/2*x/(1 + x^4)^(1/4) + ArcTan[x/(1 + x^4)^(1/4)] - ArcTan[(2^(1/4)*x)/(1 + x^4)^(1/4)]/(4*2^(1/4)) + ArcTanh[x/(1 + x^4)^(1/4)] - ArcTanh[(2^(1/4)*x)/(1 + x^4)^(1/4)]/(4*2^(1/4))","A",1
1258,1,91,91,0.4569523,"\int \sqrt[4]{b x^5+a x^8} \, dx","IntegrateAlgebraic[(b*x^5 + a*x^8)^(1/4),x]","\frac{1}{3} x \sqrt[4]{b x^5+a x^8}+\frac{b \tan ^{-1}\left(\frac{\sqrt[4]{b x^5+a x^8}}{\sqrt[4]{a} x^2}\right)}{6 a^{3/4}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt[4]{b x^5+a x^8}}{\sqrt[4]{a} x^2}\right)}{6 a^{3/4}}","\frac{1}{3} x \sqrt[4]{b x^5+a x^8}+\frac{b \tan ^{-1}\left(\frac{\sqrt[4]{b x^5+a x^8}}{\sqrt[4]{a} x^2}\right)}{6 a^{3/4}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt[4]{b x^5+a x^8}}{\sqrt[4]{a} x^2}\right)}{6 a^{3/4}}",1,"(x*(b*x^5 + a*x^8)^(1/4))/3 + (b*ArcTan[(b*x^5 + a*x^8)^(1/4)/(a^(1/4)*x^2)])/(6*a^(3/4)) + (b*ArcTanh[(b*x^5 + a*x^8)^(1/4)/(a^(1/4)*x^2)])/(6*a^(3/4))","A",1
1259,1,91,91,0.1038082,"\int \frac{\sqrt{x-\sqrt{1+x^2}}}{1-\sqrt{1+x^2}} \, dx","IntegrateAlgebraic[Sqrt[x - Sqrt[1 + x^2]]/(1 - Sqrt[1 + x^2]),x]","\frac{2 (-2-x) \sqrt{x-\sqrt{1+x^2}}+2 \sqrt{1+x^2} \sqrt{x-\sqrt{1+x^2}}}{-1-x+\sqrt{1+x^2}}-2 \tan ^{-1}\left(\sqrt{x-\sqrt{1+x^2}}\right)","\frac{2 (-2-x) \sqrt{x-\sqrt{1+x^2}}+2 \sqrt{1+x^2} \sqrt{x-\sqrt{1+x^2}}}{-1-x+\sqrt{1+x^2}}-2 \tan ^{-1}\left(\sqrt{x-\sqrt{1+x^2}}\right)",1,"(2*(-2 - x)*Sqrt[x - Sqrt[1 + x^2]] + 2*Sqrt[1 + x^2]*Sqrt[x - Sqrt[1 + x^2]])/(-1 - x + Sqrt[1 + x^2]) - 2*ArcTan[Sqrt[x - Sqrt[1 + x^2]]]","A",1
1260,1,92,92,0.1454234,"\int \frac{(-1+x) (3+x)}{\left(-1+x^2\right)^{2/3} \left(2-x+x^2\right)} \, dx","IntegrateAlgebraic[((-1 + x)*(3 + x))/((-1 + x^2)^(2/3)*(2 - x + x^2)),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \left(-1+x^2\right)^{2/3}}{-2-2 x+\left(-1+x^2\right)^{2/3}}\right)-\log \left(1+x+\left(-1+x^2\right)^{2/3}\right)+\frac{1}{2} \log \left(1+2 x+x^2+(-1-x) \left(-1+x^2\right)^{2/3}+\left(-1+x^2\right)^{4/3}\right)","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \left(-1+x^2\right)^{2/3}}{-2-2 x+\left(-1+x^2\right)^{2/3}}\right)-\log \left(1+x+\left(-1+x^2\right)^{2/3}\right)+\frac{1}{2} \log \left(1+2 x+x^2+(-1-x) \left(-1+x^2\right)^{2/3}+\left(-1+x^2\right)^{4/3}\right)",1,"-(Sqrt[3]*ArcTan[(Sqrt[3]*(-1 + x^2)^(2/3))/(-2 - 2*x + (-1 + x^2)^(2/3))]) - Log[1 + x + (-1 + x^2)^(2/3)] + Log[1 + 2*x + x^2 + (-1 - x)*(-1 + x^2)^(2/3) + (-1 + x^2)^(4/3)]/2","A",1
1261,1,92,92,0.1435281,"\int \frac{\sqrt[3]{-1+x^3}}{x^4} \, dx","IntegrateAlgebraic[(-1 + x^3)^(1/3)/x^4,x]","-\frac{\sqrt[3]{-1+x^3}}{3 x^3}-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^3}}{\sqrt{3}}\right)}{3 \sqrt{3}}+\frac{1}{9} \log \left(1+\sqrt[3]{-1+x^3}\right)-\frac{1}{18} \log \left(1-\sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)","-\frac{\sqrt[3]{-1+x^3}}{3 x^3}-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^3}}{\sqrt{3}}\right)}{3 \sqrt{3}}+\frac{1}{9} \log \left(1+\sqrt[3]{-1+x^3}\right)-\frac{1}{18} \log \left(1-\sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)",1,"-1/3*(-1 + x^3)^(1/3)/x^3 - ArcTan[1/Sqrt[3] - (2*(-1 + x^3)^(1/3))/Sqrt[3]]/(3*Sqrt[3]) + Log[1 + (-1 + x^3)^(1/3)]/9 - Log[1 - (-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)]/18","A",1
1262,1,92,92,0.0725258,"\int \frac{\left(-1+x^3\right)^{2/3}}{x^4} \, dx","IntegrateAlgebraic[(-1 + x^3)^(2/3)/x^4,x]","-\frac{\left(-1+x^3\right)^{2/3}}{3 x^3}-\frac{2 \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^3}}{\sqrt{3}}\right)}{3 \sqrt{3}}-\frac{2}{9} \log \left(1+\sqrt[3]{-1+x^3}\right)+\frac{1}{9} \log \left(1-\sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)","-\frac{\left(-1+x^3\right)^{2/3}}{3 x^3}-\frac{2 \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^3}}{\sqrt{3}}\right)}{3 \sqrt{3}}-\frac{2}{9} \log \left(1+\sqrt[3]{-1+x^3}\right)+\frac{1}{9} \log \left(1-\sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)",1,"-1/3*(-1 + x^3)^(2/3)/x^3 - (2*ArcTan[1/Sqrt[3] - (2*(-1 + x^3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]) - (2*Log[1 + (-1 + x^3)^(1/3)])/9 + Log[1 - (-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)]/9","A",1
1263,1,92,92,0.1345316,"\int \frac{\sqrt[3]{1+x^3}}{x^2} \, dx","IntegrateAlgebraic[(1 + x^3)^(1/3)/x^2,x]","-\frac{\sqrt[3]{1+x^3}}{x}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(-x+\sqrt[3]{1+x^3}\right)+\frac{1}{6} \log \left(x^2+x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)","-\frac{\sqrt[3]{1+x^3}}{x}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(-x+\sqrt[3]{1+x^3}\right)+\frac{1}{6} \log \left(x^2+x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)",1,"-((1 + x^3)^(1/3)/x) - ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^3)^(1/3))]/Sqrt[3] - Log[-x + (1 + x^3)^(1/3)]/3 + Log[x^2 + x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/6","A",1
1264,1,92,92,0.8391027,"\int \frac{-1+x}{(1+x) \sqrt[3]{2+x^3}} \, dx","IntegrateAlgebraic[(-1 + x)/((1 + x)*(2 + x^3)^(1/3)),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{2+x^3}}{4+2 x+\sqrt[3]{2+x^3}}\right)-\log \left(-2-x+\sqrt[3]{2+x^3}\right)+\frac{1}{2} \log \left(4+4 x+x^2+(2+x) \sqrt[3]{2+x^3}+\left(2+x^3\right)^{2/3}\right)","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{2+x^3}}{4+2 x+\sqrt[3]{2+x^3}}\right)-\log \left(-2-x+\sqrt[3]{2+x^3}\right)+\frac{1}{2} \log \left(4+4 x+x^2+(2+x) \sqrt[3]{2+x^3}+\left(2+x^3\right)^{2/3}\right)",1,"-(Sqrt[3]*ArcTan[(Sqrt[3]*(2 + x^3)^(1/3))/(4 + 2*x + (2 + x^3)^(1/3))]) - Log[-2 - x + (2 + x^3)^(1/3)] + Log[4 + 4*x + x^2 + (2 + x)*(2 + x^3)^(1/3) + (2 + x^3)^(2/3)]/2","A",1
1265,1,92,92,0.148151,"\int \frac{1}{x^7 \left(b+a x^3\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x^7*(b + a*x^3)^(3/4)),x]","\frac{\sqrt[4]{b+a x^3} \left(-4 b+7 a x^3\right)}{24 b^2 x^6}-\frac{7 a^2 \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^3}}{\sqrt[4]{b}}\right)}{16 b^{11/4}}-\frac{7 a^2 \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^3}}{\sqrt[4]{b}}\right)}{16 b^{11/4}}","\frac{\sqrt[4]{b+a x^3} \left(-4 b+7 a x^3\right)}{24 b^2 x^6}-\frac{7 a^2 \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^3}}{\sqrt[4]{b}}\right)}{16 b^{11/4}}-\frac{7 a^2 \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^3}}{\sqrt[4]{b}}\right)}{16 b^{11/4}}",1,"((b + a*x^3)^(1/4)*(-4*b + 7*a*x^3))/(24*b^2*x^6) - (7*a^2*ArcTan[(b + a*x^3)^(1/4)/b^(1/4)])/(16*b^(11/4)) - (7*a^2*ArcTanh[(b + a*x^3)^(1/4)/b^(1/4)])/(16*b^(11/4))","A",1
1266,1,92,92,15.704923,"\int \frac{\left(1+2 x+x^2\right) \left(-2-(-1+k) (1+k) x+2 k^2 x^2\right)}{\left(\left(1-x^2\right) \left(1-k^2 x^2\right)\right)^{3/4} \left(1-d-(1+3 d) x-\left(3 d+k^2\right) x^2+\left(-d+k^2\right) x^3\right)} \, dx","IntegrateAlgebraic[((1 + 2*x + x^2)*(-2 - (-1 + k)*(1 + k)*x + 2*k^2*x^2))/(((1 - x^2)*(1 - k^2*x^2))^(3/4)*(1 - d - (1 + 3*d)*x - (3*d + k^2)*x^2 + (-d + k^2)*x^3)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{d}+\sqrt[4]{d} x}{\sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{d}+\sqrt[4]{d} x}{\sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{3/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{d}+\sqrt[4]{d} x}{\sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{d}+\sqrt[4]{d} x}{\sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{3/4}}",1,"ArcTan[(d^(1/4) + d^(1/4)*x)/(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/4)]/d^(3/4) - ArcTanh[(d^(1/4) + d^(1/4)*x)/(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/4)]/d^(3/4)","A",1
1267,1,92,92,0.0691145,"\int \frac{1}{x^5 \sqrt[3]{-1+x^4}} \, dx","IntegrateAlgebraic[1/(x^5*(-1 + x^4)^(1/3)),x]","\frac{\left(-1+x^4\right)^{2/3}}{4 x^4}-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^4}}{\sqrt{3}}\right)}{4 \sqrt{3}}-\frac{1}{12} \log \left(1+\sqrt[3]{-1+x^4}\right)+\frac{1}{24} \log \left(1-\sqrt[3]{-1+x^4}+\left(-1+x^4\right)^{2/3}\right)","\frac{\left(-1+x^4\right)^{2/3}}{4 x^4}-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^4}}{\sqrt{3}}\right)}{4 \sqrt{3}}-\frac{1}{12} \log \left(1+\sqrt[3]{-1+x^4}\right)+\frac{1}{24} \log \left(1-\sqrt[3]{-1+x^4}+\left(-1+x^4\right)^{2/3}\right)",1,"(-1 + x^4)^(2/3)/(4*x^4) - ArcTan[1/Sqrt[3] - (2*(-1 + x^4)^(1/3))/Sqrt[3]]/(4*Sqrt[3]) - Log[1 + (-1 + x^4)^(1/3)]/12 + Log[1 - (-1 + x^4)^(1/3) + (-1 + x^4)^(2/3)]/24","A",1
1268,1,92,92,1.0408277,"\int \frac{x (-b+x) \left(a b-2 a x+x^2\right)}{\sqrt{x (-a+x) (-b+x)} \left(-a^2 d+2 a d x+\left(b^2-d\right) x^2-2 b x^3+x^4\right)} \, dx","IntegrateAlgebraic[(x*(-b + x)*(a*b - 2*a*x + x^2))/(Sqrt[x*(-a + x)*(-b + x)]*(-(a^2*d) + 2*a*d*x + (b^2 - d)*x^2 - 2*b*x^3 + x^4)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{a b x+(-a-b) x^2+x^3}}{\sqrt[4]{d} (a-x)}\right)}{\sqrt[4]{d}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{a b x+(-a-b) x^2+x^3}}{x (-b+x)}\right)}{\sqrt[4]{d}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{a b x+(-a-b) x^2+x^3}}{\sqrt[4]{d} (a-x)}\right)}{\sqrt[4]{d}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{a b x+(-a-b) x^2+x^3}}{x (-b+x)}\right)}{\sqrt[4]{d}}",1,"-(ArcTan[Sqrt[a*b*x + (-a - b)*x^2 + x^3]/(d^(1/4)*(a - x))]/d^(1/4)) - ArcTanh[(d^(1/4)*Sqrt[a*b*x + (-a - b)*x^2 + x^3])/(x*(-b + x))]/d^(1/4)","A",1
1269,1,92,92,0.0785409,"\int \frac{1}{x^9 \left(b+a x^4\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x^9*(b + a*x^4)^(3/4)),x]","\frac{\sqrt[4]{b+a x^4} \left(-4 b+7 a x^4\right)}{32 b^2 x^8}-\frac{21 a^2 \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^4}}{\sqrt[4]{b}}\right)}{64 b^{11/4}}-\frac{21 a^2 \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^4}}{\sqrt[4]{b}}\right)}{64 b^{11/4}}","\frac{\sqrt[4]{b+a x^4} \left(-4 b+7 a x^4\right)}{32 b^2 x^8}-\frac{21 a^2 \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^4}}{\sqrt[4]{b}}\right)}{64 b^{11/4}}-\frac{21 a^2 \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^4}}{\sqrt[4]{b}}\right)}{64 b^{11/4}}",1,"((b + a*x^4)^(1/4)*(-4*b + 7*a*x^4))/(32*b^2*x^8) - (21*a^2*ArcTan[(b + a*x^4)^(1/4)/b^(1/4)])/(64*b^(11/4)) - (21*a^2*ArcTanh[(b + a*x^4)^(1/4)/b^(1/4)])/(64*b^(11/4))","A",1
1270,1,92,92,0.3656746,"\int x^4 \left(b+a x^4\right)^{3/4} \, dx","IntegrateAlgebraic[x^4*(b + a*x^4)^(3/4),x]","\frac{\left(b+a x^4\right)^{3/4} \left(3 b x+4 a x^5\right)}{32 a}-\frac{3 b^2 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{64 a^{5/4}}-\frac{3 b^2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{64 a^{5/4}}","\frac{\left(b+a x^4\right)^{3/4} \left(3 b x+4 a x^5\right)}{32 a}-\frac{3 b^2 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{64 a^{5/4}}-\frac{3 b^2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{64 a^{5/4}}",1,"((b + a*x^4)^(3/4)*(3*b*x + 4*a*x^5))/(32*a) - (3*b^2*ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)])/(64*a^(5/4)) - (3*b^2*ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)])/(64*a^(5/4))","A",1
1271,1,92,92,2.9377278,"\int \frac{\sqrt[4]{-b x^3+a x^4}}{x \left(d+c x+x^2\right)} \, dx","IntegrateAlgebraic[(-(b*x^3) + a*x^4)^(1/4)/(x*(d + c*x + x^2)),x]","b \text{RootSum}\left[b^2+a b c+a^2 d-b c \text{$\#$1}^4-2 a d \text{$\#$1}^4+d \text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-b c-2 a d+2 d \text{$\#$1}^4}\&\right]","b \text{RootSum}\left[b^2+a b c+a^2 d-b c \text{$\#$1}^4-2 a d \text{$\#$1}^4+d \text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-b c-2 a d+2 d \text{$\#$1}^4}\&\right]",1,"b*RootSum[b^2 + a*b*c + a^2*d - b*c*#1^4 - 2*a*d*#1^4 + d*#1^8 & , (-(Log[x]*#1) + Log[(-(b*x^3) + a*x^4)^(1/4) - x*#1]*#1)/(-(b*c) - 2*a*d + 2*d*#1^4) & ]","A",1
1272,1,92,92,0.0016171,"\int \frac{\sqrt[4]{-b x^3+a x^4}}{x \left(d+c x+x^2\right)} \, dx","IntegrateAlgebraic[(-(b*x^3) + a*x^4)^(1/4)/(x*(d + c*x + x^2)),x]","b \text{RootSum}\left[b^2+a b c+a^2 d-b c \text{$\#$1}^4-2 a d \text{$\#$1}^4+d \text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-b c-2 a d+2 d \text{$\#$1}^4}\&\right]","b \text{RootSum}\left[b^2+a b c+a^2 d-b c \text{$\#$1}^4-2 a d \text{$\#$1}^4+d \text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-b c-2 a d+2 d \text{$\#$1}^4}\&\right]",1,"b*RootSum[b^2 + a*b*c + a^2*d - b*c*#1^4 - 2*a*d*#1^4 + d*#1^8 & , (-(Log[x]*#1) + Log[(-(b*x^3) + a*x^4)^(1/4) - x*#1]*#1)/(-(b*c) - 2*a*d + 2*d*#1^4) & ]","A",1
1273,1,92,92,0.0619031,"\int \frac{\left(-1+x^5\right)^{2/3}}{x^6} \, dx","IntegrateAlgebraic[(-1 + x^5)^(2/3)/x^6,x]","-\frac{\left(-1+x^5\right)^{2/3}}{5 x^5}-\frac{2 \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^5}}{\sqrt{3}}\right)}{5 \sqrt{3}}-\frac{2}{15} \log \left(1+\sqrt[3]{-1+x^5}\right)+\frac{1}{15} \log \left(1-\sqrt[3]{-1+x^5}+\left(-1+x^5\right)^{2/3}\right)","-\frac{\left(-1+x^5\right)^{2/3}}{5 x^5}-\frac{2 \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^5}}{\sqrt{3}}\right)}{5 \sqrt{3}}-\frac{2}{15} \log \left(1+\sqrt[3]{-1+x^5}\right)+\frac{1}{15} \log \left(1-\sqrt[3]{-1+x^5}+\left(-1+x^5\right)^{2/3}\right)",1,"-1/5*(-1 + x^5)^(2/3)/x^5 - (2*ArcTan[1/Sqrt[3] - (2*(-1 + x^5)^(1/3))/Sqrt[3]])/(5*Sqrt[3]) - (2*Log[1 + (-1 + x^5)^(1/3)])/15 + Log[1 - (-1 + x^5)^(1/3) + (-1 + x^5)^(2/3)]/15","A",1
1274,1,92,92,0.0742266,"\int \frac{1}{x^{11} \left(b+a x^5\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x^11*(b + a*x^5)^(3/4)),x]","\frac{\sqrt[4]{b+a x^5} \left(-4 b+7 a x^5\right)}{40 b^2 x^{10}}-\frac{21 a^2 \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^5}}{\sqrt[4]{b}}\right)}{80 b^{11/4}}-\frac{21 a^2 \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^5}}{\sqrt[4]{b}}\right)}{80 b^{11/4}}","\frac{\sqrt[4]{b+a x^5} \left(-4 b+7 a x^5\right)}{40 b^2 x^{10}}-\frac{21 a^2 \tan ^{-1}\left(\frac{\sqrt[4]{b+a x^5}}{\sqrt[4]{b}}\right)}{80 b^{11/4}}-\frac{21 a^2 \tanh ^{-1}\left(\frac{\sqrt[4]{b+a x^5}}{\sqrt[4]{b}}\right)}{80 b^{11/4}}",1,"((b + a*x^5)^(1/4)*(-4*b + 7*a*x^5))/(40*b^2*x^10) - (21*a^2*ArcTan[(b + a*x^5)^(1/4)/b^(1/4)])/(80*b^(11/4)) - (21*a^2*ArcTanh[(b + a*x^5)^(1/4)/b^(1/4)])/(80*b^(11/4))","A",1
1275,1,92,92,0.0663381,"\int \frac{1}{x^7 \sqrt[3]{-1+x^6}} \, dx","IntegrateAlgebraic[1/(x^7*(-1 + x^6)^(1/3)),x]","\frac{\left(-1+x^6\right)^{2/3}}{6 x^6}-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^6}}{\sqrt{3}}\right)}{6 \sqrt{3}}-\frac{1}{18} \log \left(1+\sqrt[3]{-1+x^6}\right)+\frac{1}{36} \log \left(1-\sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)","\frac{\left(-1+x^6\right)^{2/3}}{6 x^6}-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^6}}{\sqrt{3}}\right)}{6 \sqrt{3}}-\frac{1}{18} \log \left(1+\sqrt[3]{-1+x^6}\right)+\frac{1}{36} \log \left(1-\sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)",1,"(-1 + x^6)^(2/3)/(6*x^6) - ArcTan[1/Sqrt[3] - (2*(-1 + x^6)^(1/3))/Sqrt[3]]/(6*Sqrt[3]) - Log[1 + (-1 + x^6)^(1/3)]/18 + Log[1 - (-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)]/36","A",1
1276,1,92,92,0.0615589,"\int \frac{\left(-1+x^6\right)^{2/3}}{x^7} \, dx","IntegrateAlgebraic[(-1 + x^6)^(2/3)/x^7,x]","-\frac{\left(-1+x^6\right)^{2/3}}{6 x^6}-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^6}}{\sqrt{3}}\right)}{3 \sqrt{3}}-\frac{1}{9} \log \left(1+\sqrt[3]{-1+x^6}\right)+\frac{1}{18} \log \left(1-\sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)","-\frac{\left(-1+x^6\right)^{2/3}}{6 x^6}-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^6}}{\sqrt{3}}\right)}{3 \sqrt{3}}-\frac{1}{9} \log \left(1+\sqrt[3]{-1+x^6}\right)+\frac{1}{18} \log \left(1-\sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)",1,"-1/6*(-1 + x^6)^(2/3)/x^6 - ArcTan[1/Sqrt[3] - (2*(-1 + x^6)^(1/3))/Sqrt[3]]/(3*Sqrt[3]) - Log[1 + (-1 + x^6)^(1/3)]/9 + Log[1 - (-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)]/18","A",1
1277,1,92,92,0.2058262,"\int \frac{\left(-1+x^4+2 x^6\right) \sqrt[3]{x+x^5+x^7}}{\left(1+x^4+x^6\right) \left(1-x^2+x^4+x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^4 + 2*x^6)*(x + x^5 + x^7)^(1/3))/((1 + x^4 + x^6)*(1 - x^2 + x^4 + x^6)),x]","\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^5+x^7}}\right)+\frac{1}{2} \log \left(-x+\sqrt[3]{x+x^5+x^7}\right)-\frac{1}{4} \log \left(x^2+x \sqrt[3]{x+x^5+x^7}+\left(x+x^5+x^7\right)^{2/3}\right)","\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^5+x^7}}\right)+\frac{1}{2} \log \left(-x+\sqrt[3]{x+x^5+x^7}\right)-\frac{1}{4} \log \left(x^2+x \sqrt[3]{x+x^5+x^7}+\left(x+x^5+x^7\right)^{2/3}\right)",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^5 + x^7)^(1/3))])/2 + Log[-x + (x + x^5 + x^7)^(1/3)]/2 - Log[x^2 + x*(x + x^5 + x^7)^(1/3) + (x + x^5 + x^7)^(2/3)]/4","A",1
1278,1,92,92,0.1518468,"\int \frac{\sqrt{1+\sqrt{1+x}}}{x-\sqrt{1+x}} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[1 + x]]/(x - Sqrt[1 + x]),x]","4 \sqrt{1+\sqrt{1+x}}-\frac{2}{5} \left(5+3 \sqrt{5}\right) \tanh ^{-1}\left(\frac{1}{2} \left(-1+\sqrt{5}\right) \sqrt{1+\sqrt{1+x}}\right)-\frac{2}{5} \left(-5+3 \sqrt{5}\right) \tanh ^{-1}\left(\frac{1}{2} \left(1+\sqrt{5}\right) \sqrt{1+\sqrt{1+x}}\right)","4 \sqrt{1+\sqrt{1+x}}-\frac{2}{5} \left(-5+3 \sqrt{5}\right) \tanh ^{-1}\left(\frac{2 \sqrt{1+\sqrt{1+x}}}{-1+\sqrt{5}}\right)-\frac{2}{5} \left(5+3 \sqrt{5}\right) \tanh ^{-1}\left(\frac{2 \sqrt{1+\sqrt{1+x}}}{1+\sqrt{5}}\right)",1,"4*Sqrt[1 + Sqrt[1 + x]] - (2*(5 + 3*Sqrt[5])*ArcTanh[((-1 + Sqrt[5])*Sqrt[1 + Sqrt[1 + x]])/2])/5 - (2*(-5 + 3*Sqrt[5])*ArcTanh[((1 + Sqrt[5])*Sqrt[1 + Sqrt[1 + x]])/2])/5","A",1
1279,1,98,93,0.1710206,"\int \frac{1}{x^3 \left(-1+x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x^3*(-1 + x^2)^(3/4)),x]","\frac{\sqrt[4]{-1+x^2}}{2 x^2}+\frac{3 \tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{\sqrt{-1+x^2}}{\sqrt{2}}}{\sqrt[4]{-1+x^2}}\right)}{4 \sqrt{2}}+\frac{3 \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^2}}{1+\sqrt{-1+x^2}}\right)}{4 \sqrt{2}}","\frac{\sqrt[4]{-1+x^2}}{2 x^2}-\frac{3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^2}}{-1+\sqrt{-1+x^2}}\right)}{4 \sqrt{2}}+\frac{3 \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^2}}{1+\sqrt{-1+x^2}}\right)}{4 \sqrt{2}}",1,"(-1 + x^2)^(1/4)/(2*x^2) + (3*ArcTan[(-(1/Sqrt[2]) + Sqrt[-1 + x^2]/Sqrt[2])/(-1 + x^2)^(1/4)])/(4*Sqrt[2]) + (3*ArcTanh[(Sqrt[2]*(-1 + x^2)^(1/4))/(1 + Sqrt[-1 + x^2])])/(4*Sqrt[2])","A",1
1280,1,93,93,0.1458278,"\int \frac{\left(-1+x^3\right)^{2/3}}{x^3} \, dx","IntegrateAlgebraic[(-1 + x^3)^(2/3)/x^3,x]","-\frac{\left(-1+x^3\right)^{2/3}}{2 x^2}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(-x+\sqrt[3]{-1+x^3}\right)+\frac{1}{6} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)","-\frac{\left(-1+x^3\right)^{2/3}}{2 x^2}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(-x+\sqrt[3]{-1+x^3}\right)+\frac{1}{6} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)",1,"-1/2*(-1 + x^3)^(2/3)/x^2 + ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^3)^(1/3))]/Sqrt[3] - Log[-x + (-1 + x^3)^(1/3)]/3 + Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)]/6","A",1
1281,1,93,93,1.5542811,"\int \frac{2+x^2}{\left(-2+2 x+x^2\right) \sqrt{1+x^3}} \, dx","IntegrateAlgebraic[(2 + x^2)/((-2 + 2*x + x^2)*Sqrt[1 + x^3]),x]","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} \sqrt{1+x^3}}{1-x+x^2}\right)}{3^{3/4}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{-3+2 \sqrt{3}} \sqrt{1+x^3}}{1-x+x^2}\right)}{3^{3/4}}","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} \sqrt{1+x^3}}{1-x+x^2}\right)}{3^{3/4}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{-3+2 \sqrt{3}} \sqrt{1+x^3}}{1-x+x^2}\right)}{3^{3/4}}",1,"-((Sqrt[2]*ArcTan[(Sqrt[3 + 2*Sqrt[3]]*Sqrt[1 + x^3])/(1 - x + x^2)])/3^(3/4)) - (Sqrt[2]*ArcTanh[(Sqrt[-3 + 2*Sqrt[3]]*Sqrt[1 + x^3])/(1 - x + x^2)])/3^(3/4)","A",1
1282,1,93,93,0.1420618,"\int \frac{\left(1+x^3\right)^{2/3}}{x^3} \, dx","IntegrateAlgebraic[(1 + x^3)^(2/3)/x^3,x]","-\frac{\left(1+x^3\right)^{2/3}}{2 x^2}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(-x+\sqrt[3]{1+x^3}\right)+\frac{1}{6} \log \left(x^2+x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)","-\frac{\left(1+x^3\right)^{2/3}}{2 x^2}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(-x+\sqrt[3]{1+x^3}\right)+\frac{1}{6} \log \left(x^2+x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)",1,"-1/2*(1 + x^3)^(2/3)/x^2 + ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^3)^(1/3))]/Sqrt[3] - Log[-x + (1 + x^3)^(1/3)]/3 + Log[x^2 + x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/6","A",1
1283,1,93,93,2.8633091,"\int \frac{2 a b x+(-3 a+b) x^2}{\sqrt[4]{x^2 (-a+x) (-b+x)} \left(a^3-3 a^2 x+(3 a-b d) x^2+(-1+d) x^3\right)} \, dx","IntegrateAlgebraic[(2*a*b*x + (-3*a + b)*x^2)/((x^2*(-a + x)*(-b + x))^(1/4)*(a^3 - 3*a^2*x + (3*a - b*d)*x^2 + (-1 + d)*x^3)),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b x^2+(-a-b) x^3+x^4}}{a-x}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b x^2+(-a-b) x^3+x^4}}{a-x}\right)}{d^{3/4}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b x^2+(-a-b) x^3+x^4}}{a-x}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b x^2+(-a-b) x^3+x^4}}{a-x}\right)}{d^{3/4}}",1,"(-2*ArcTan[(d^(1/4)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/4))/(a - x)])/d^(3/4) + (2*ArcTanh[(d^(1/4)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/4))/(a - x)])/d^(3/4)","A",1
1284,1,98,93,0.1205322,"\int \frac{\sqrt[4]{-1+x^4}}{x^5} \, dx","IntegrateAlgebraic[(-1 + x^4)^(1/4)/x^5,x]","-\frac{\sqrt[4]{-1+x^4}}{4 x^4}+\frac{\tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{\sqrt{-1+x^4}}{\sqrt{2}}}{\sqrt[4]{-1+x^4}}\right)}{8 \sqrt{2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^4}}{1+\sqrt{-1+x^4}}\right)}{8 \sqrt{2}}","-\frac{\sqrt[4]{-1+x^4}}{4 x^4}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^4}}{-1+\sqrt{-1+x^4}}\right)}{8 \sqrt{2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^4}}{1+\sqrt{-1+x^4}}\right)}{8 \sqrt{2}}",1,"-1/4*(-1 + x^4)^(1/4)/x^4 + ArcTan[(-(1/Sqrt[2]) + Sqrt[-1 + x^4]/Sqrt[2])/(-1 + x^4)^(1/4)]/(8*Sqrt[2]) + ArcTanh[(Sqrt[2]*(-1 + x^4)^(1/4))/(1 + Sqrt[-1 + x^4])]/(8*Sqrt[2])","A",1
1285,1,93,93,0.3517401,"\int \frac{1+x^2}{\left(-1+x+x^2\right) \sqrt[3]{-x^2+x^4}} \, dx","IntegrateAlgebraic[(1 + x^2)/((-1 + x + x^2)*(-x^2 + x^4)^(1/3)),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-x^2+x^4}}\right)-\log \left(x+\sqrt[3]{-x^2+x^4}\right)+\frac{1}{2} \log \left(x^2-x \sqrt[3]{-x^2+x^4}+\left(-x^2+x^4\right)^{2/3}\right)","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-x^2+x^4}}\right)-\log \left(x+\sqrt[3]{-x^2+x^4}\right)+\frac{1}{2} \log \left(x^2-x \sqrt[3]{-x^2+x^4}+\left(-x^2+x^4\right)^{2/3}\right)",1,"-(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(-x^2 + x^4)^(1/3))]) - Log[x + (-x^2 + x^4)^(1/3)] + Log[x^2 - x*(-x^2 + x^4)^(1/3) + (-x^2 + x^4)^(2/3)]/2","A",1
1286,1,93,93,0.2960899,"\int \frac{(-1+2 x) \left(2-x+x^2\right) \sqrt{-2+x^2-2 x^3+x^4}}{3-2 x+2 x^2} \, dx","IntegrateAlgebraic[((-1 + 2*x)*(2 - x + x^2)*Sqrt[-2 + x^2 - 2*x^3 + x^4])/(3 - 2*x + 2*x^2),x]","\frac{1}{4} \left(1-x+x^2\right) \sqrt{-2+x^2-2 x^3+x^4}-\frac{1}{4} \tanh ^{-1}\left(3-2 x+2 x^2+2 \sqrt{-2+x^2-2 x^3+x^4}\right)-\frac{7}{8} \log \left(-x+x^2+\sqrt{-2+x^2-2 x^3+x^4}\right)","\frac{1}{4} \left(1-x+x^2\right) \sqrt{-2+x^2-2 x^3+x^4}-\frac{1}{4} \tanh ^{-1}\left(3-2 x+2 x^2+2 \sqrt{-2+x^2-2 x^3+x^4}\right)-\frac{7}{8} \log \left(-x+x^2+\sqrt{-2+x^2-2 x^3+x^4}\right)",1,"((1 - x + x^2)*Sqrt[-2 + x^2 - 2*x^3 + x^4])/4 - ArcTanh[3 - 2*x + 2*x^2 + 2*Sqrt[-2 + x^2 - 2*x^3 + x^4]]/4 - (7*Log[-x + x^2 + Sqrt[-2 + x^2 - 2*x^3 + x^4]])/8","A",1
1287,1,93,93,0.3167456,"\int \frac{\sqrt[4]{-x^3+x^4}}{x (1+x)} \, dx","IntegrateAlgebraic[(-x^3 + x^4)^(1/4)/(x*(1 + x)),x]","-2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)+2 \sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)+2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)-2 \sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)","-2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)+2 \sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)+2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)-2 \sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)",1,"-2*ArcTan[x/(-x^3 + x^4)^(1/4)] + 2*2^(1/4)*ArcTan[(2^(1/4)*x)/(-x^3 + x^4)^(1/4)] + 2*ArcTanh[x/(-x^3 + x^4)^(1/4)] - 2*2^(1/4)*ArcTanh[(2^(1/4)*x)/(-x^3 + x^4)^(1/4)]","A",1
1288,1,93,93,0.9264989,"\int \frac{1+2 x^3}{\left(-1+x+x^3\right) \sqrt[3]{-x^2+x^5}} \, dx","IntegrateAlgebraic[(1 + 2*x^3)/((-1 + x + x^3)*(-x^2 + x^5)^(1/3)),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-x^2+x^5}}\right)-\log \left(x+\sqrt[3]{-x^2+x^5}\right)+\frac{1}{2} \log \left(x^2-x \sqrt[3]{-x^2+x^5}+\left(-x^2+x^5\right)^{2/3}\right)","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-x^2+x^5}}\right)-\log \left(x+\sqrt[3]{-x^2+x^5}\right)+\frac{1}{2} \log \left(x^2-x \sqrt[3]{-x^2+x^5}+\left(-x^2+x^5\right)^{2/3}\right)",1,"-(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(-x^2 + x^5)^(1/3))]) - Log[x + (-x^2 + x^5)^(1/3)] + Log[x^2 - x*(-x^2 + x^5)^(1/3) + (-x^2 + x^5)^(2/3)]/2","A",1
1289,1,98,93,0.1025881,"\int \frac{\sqrt[4]{-1+x^6}}{x^7} \, dx","IntegrateAlgebraic[(-1 + x^6)^(1/4)/x^7,x]","-\frac{\sqrt[4]{-1+x^6}}{6 x^6}+\frac{\tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{\sqrt{-1+x^6}}{\sqrt{2}}}{\sqrt[4]{-1+x^6}}\right)}{12 \sqrt{2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^6}}{1+\sqrt{-1+x^6}}\right)}{12 \sqrt{2}}","-\frac{\sqrt[4]{-1+x^6}}{6 x^6}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^6}}{-1+\sqrt{-1+x^6}}\right)}{12 \sqrt{2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^6}}{1+\sqrt{-1+x^6}}\right)}{12 \sqrt{2}}",1,"-1/6*(-1 + x^6)^(1/4)/x^6 + ArcTan[(-(1/Sqrt[2]) + Sqrt[-1 + x^6]/Sqrt[2])/(-1 + x^6)^(1/4)]/(12*Sqrt[2]) + ArcTanh[(Sqrt[2]*(-1 + x^6)^(1/4))/(1 + Sqrt[-1 + x^6])]/(12*Sqrt[2])","A",1
1290,1,93,93,2.6180254,"\int \frac{1+3 x^4}{\left(-1+x+x^4\right) \sqrt[3]{-x^2+x^6}} \, dx","IntegrateAlgebraic[(1 + 3*x^4)/((-1 + x + x^4)*(-x^2 + x^6)^(1/3)),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-x^2+x^6}}\right)-\log \left(x+\sqrt[3]{-x^2+x^6}\right)+\frac{1}{2} \log \left(x^2-x \sqrt[3]{-x^2+x^6}+\left(-x^2+x^6\right)^{2/3}\right)","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-x^2+x^6}}\right)-\log \left(x+\sqrt[3]{-x^2+x^6}\right)+\frac{1}{2} \log \left(x^2-x \sqrt[3]{-x^2+x^6}+\left(-x^2+x^6\right)^{2/3}\right)",1,"-(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(-x^2 + x^6)^(1/3))]) - Log[x + (-x^2 + x^6)^(1/3)] + Log[x^2 - x*(-x^2 + x^6)^(1/3) + (-x^2 + x^6)^(2/3)]/2","A",1
1291,1,81,93,0.4977373,"\int \frac{\left(2+x^3\right) \sqrt{-1-x^2+x^3}}{1-x^2-2 x^3-x^4+x^5+x^6} \, dx","IntegrateAlgebraic[((2 + x^3)*Sqrt[-1 - x^2 + x^3])/(1 - x^2 - 2*x^3 - x^4 + x^5 + x^6),x]","\frac{1}{5} \left(5-\sqrt{5}\right) \tan ^{-1}\left(\frac{\left(-1+\sqrt{5}\right) x}{2 \sqrt{-1-x^2+x^3}}\right)+\frac{1}{5} \left(-5-\sqrt{5}\right) \tan ^{-1}\left(\frac{\left(1+\sqrt{5}\right) x}{2 \sqrt{-1-x^2+x^3}}\right)","\frac{1}{5} \left(-5-\sqrt{5}\right) \tan ^{-1}\left(\frac{\sqrt{\frac{3}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt{-1-x^2+x^3}}\right)+\frac{1}{5} \left(5-\sqrt{5}\right) \tan ^{-1}\left(\frac{\sqrt{\frac{2}{3+\sqrt{5}}} x}{\sqrt{-1-x^2+x^3}}\right)",1,"((5 - Sqrt[5])*ArcTan[((-1 + Sqrt[5])*x)/(2*Sqrt[-1 - x^2 + x^3])])/5 + ((-5 - Sqrt[5])*ArcTan[((1 + Sqrt[5])*x)/(2*Sqrt[-1 - x^2 + x^3])])/5","A",1
1292,1,81,93,0.4039117,"\int \frac{\left(1+x^4\right) \sqrt{-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8} \, dx","IntegrateAlgebraic[((1 + x^4)*Sqrt[-1 - x^2 + x^4])/(1 - x^2 - 3*x^4 + x^6 + x^8),x]","\frac{1}{10} \left(5-\sqrt{5}\right) \tan ^{-1}\left(\frac{\left(-1+\sqrt{5}\right) x}{2 \sqrt{-1-x^2+x^4}}\right)+\frac{1}{10} \left(-5-\sqrt{5}\right) \tan ^{-1}\left(\frac{\left(1+\sqrt{5}\right) x}{2 \sqrt{-1-x^2+x^4}}\right)","\frac{1}{10} \left(-5-\sqrt{5}\right) \tan ^{-1}\left(\frac{\sqrt{\frac{3}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt{-1-x^2+x^4}}\right)+\frac{1}{10} \left(5-\sqrt{5}\right) \tan ^{-1}\left(\frac{\sqrt{\frac{2}{3+\sqrt{5}}} x}{\sqrt{-1-x^2+x^4}}\right)",1,"((5 - Sqrt[5])*ArcTan[((-1 + Sqrt[5])*x)/(2*Sqrt[-1 - x^2 + x^4])])/10 + ((-5 - Sqrt[5])*ArcTan[((1 + Sqrt[5])*x)/(2*Sqrt[-1 - x^2 + x^4])])/10","A",1
1293,1,90,93,0.2336994,"\int \frac{x^2 \sqrt{x+\sqrt{1+x}}}{\sqrt{1+x}} \, dx","IntegrateAlgebraic[(x^2*Sqrt[x + Sqrt[1 + x]])/Sqrt[1 + x],x]","\frac{\sqrt{x+\sqrt{1+x}} \left(363+4974 \sqrt{1+x}+72 (1+x)-4304 (1+x)^{3/2}+128 (1+x)^2+1280 (1+x)^{5/2}\right)}{3840}+\frac{385}{512} \log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}\right)","\frac{\left(563+328 x+128 x^2\right) \sqrt{x+\sqrt{1+x}}}{3840}+\frac{\sqrt{1+x} \left(975-872 x+640 x^2\right) \sqrt{x+\sqrt{1+x}}}{1920}+\frac{385}{512} \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}\right)",1,"(Sqrt[x + Sqrt[1 + x]]*(363 + 4974*Sqrt[1 + x] + 72*(1 + x) - 4304*(1 + x)^(3/2) + 128*(1 + x)^2 + 1280*(1 + x)^(5/2)))/3840 + (385*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]])/512","A",1
1294,1,94,94,0.1736464,"\int \left(-1+x^3\right)^{2/3} \, dx","IntegrateAlgebraic[(-1 + x^3)^(2/3),x]","\frac{1}{3} x \left(-1+x^3\right)^{2/3}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{3 \sqrt{3}}+\frac{2}{9} \log \left(-x+\sqrt[3]{-1+x^3}\right)-\frac{1}{9} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)","\frac{1}{3} x \left(-1+x^3\right)^{2/3}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{3 \sqrt{3}}+\frac{2}{9} \log \left(-x+\sqrt[3]{-1+x^3}\right)-\frac{1}{9} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)",1,"(x*(-1 + x^3)^(2/3))/3 - (2*ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^3)^(1/3))])/(3*Sqrt[3]) + (2*Log[-x + (-1 + x^3)^(1/3)])/9 - Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)]/9","A",1
1295,1,94,94,0.1766365,"\int \left(1+x^3\right)^{2/3} \, dx","IntegrateAlgebraic[(1 + x^3)^(2/3),x]","\frac{1}{3} x \left(1+x^3\right)^{2/3}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3}}\right)}{3 \sqrt{3}}-\frac{2}{9} \log \left(-x+\sqrt[3]{1+x^3}\right)+\frac{1}{9} \log \left(x^2+x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)","\frac{1}{3} x \left(1+x^3\right)^{2/3}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3}}\right)}{3 \sqrt{3}}-\frac{2}{9} \log \left(-x+\sqrt[3]{1+x^3}\right)+\frac{1}{9} \log \left(x^2+x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)",1,"(x*(1 + x^3)^(2/3))/3 + (2*ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^3)^(1/3))])/(3*Sqrt[3]) - (2*Log[-x + (1 + x^3)^(1/3)])/9 + Log[x^2 + x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/9","A",1
1296,1,94,94,0.2036661,"\int \frac{-1+x^3}{\sqrt[3]{2+x^3}} \, dx","IntegrateAlgebraic[(-1 + x^3)/(2 + x^3)^(1/3),x]","\frac{1}{3} x \left(2+x^3\right)^{2/3}-\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2+x^3}}\right)}{3 \sqrt{3}}+\frac{5}{9} \log \left(-x+\sqrt[3]{2+x^3}\right)-\frac{5}{18} \log \left(x^2+x \sqrt[3]{2+x^3}+\left(2+x^3\right)^{2/3}\right)","\frac{1}{3} x \left(2+x^3\right)^{2/3}-\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2+x^3}}\right)}{3 \sqrt{3}}+\frac{5}{9} \log \left(-x+\sqrt[3]{2+x^3}\right)-\frac{5}{18} \log \left(x^2+x \sqrt[3]{2+x^3}+\left(2+x^3\right)^{2/3}\right)",1,"(x*(2 + x^3)^(2/3))/3 - (5*ArcTan[(Sqrt[3]*x)/(x + 2*(2 + x^3)^(1/3))])/(3*Sqrt[3]) + (5*Log[-x + (2 + x^3)^(1/3)])/9 - (5*Log[x^2 + x*(2 + x^3)^(1/3) + (2 + x^3)^(2/3)])/18","A",1
1297,1,94,94,0.1709389,"\int \sqrt[3]{x+x^3} \, dx","IntegrateAlgebraic[(x + x^3)^(1/3),x]","\frac{1}{2} x \sqrt[3]{x+x^3}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)}{2 \sqrt{3}}-\frac{1}{6} \log \left(-x+\sqrt[3]{x+x^3}\right)+\frac{1}{12} \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)","\frac{1}{2} x \sqrt[3]{x+x^3}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)}{2 \sqrt{3}}-\frac{1}{6} \log \left(-x+\sqrt[3]{x+x^3}\right)+\frac{1}{12} \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)",1,"(x*(x + x^3)^(1/3))/2 - ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^3)^(1/3))]/(2*Sqrt[3]) - Log[-x + (x + x^3)^(1/3)]/6 + Log[x^2 + x*(x + x^3)^(1/3) + (x + x^3)^(2/3)]/12","A",1
1298,1,94,94,15.7625429,"\int \frac{\left(-2 k+(-1+k) (1+k) x+2 k x^2\right) \left(1+2 k x+k^2 x^2\right)}{\left(\left(1-x^2\right) \left(1-k^2 x^2\right)\right)^{3/4} \left(-1+d+(1+3 d) k x+\left(1+3 d k^2\right) x^2+k \left(-1+d k^2\right) x^3\right)} \, dx","IntegrateAlgebraic[((-2*k + (-1 + k)*(1 + k)*x + 2*k*x^2)*(1 + 2*k*x + k^2*x^2))/(((1 - x^2)*(1 - k^2*x^2))^(3/4)*(-1 + d + (1 + 3*d)*k*x + (1 + 3*d*k^2)*x^2 + k*(-1 + d*k^2)*x^3)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{d}+\sqrt[4]{d} k x}{\sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{d}+\sqrt[4]{d} k x}{\sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{3/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{d}+\sqrt[4]{d} k x}{\sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{d}+\sqrt[4]{d} k x}{\sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{3/4}}",1,"-(ArcTan[(d^(1/4) + d^(1/4)*k*x)/(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/4)]/d^(3/4)) + ArcTanh[(d^(1/4) + d^(1/4)*k*x)/(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/4)]/d^(3/4)","A",1
1299,1,94,94,0.478744,"\int \frac{\left(-1+x^4\right)^{3/4} \left(4+x^4\right)}{x^8 \left(-4+x^4\right)} \, dx","IntegrateAlgebraic[((-1 + x^4)^(3/4)*(4 + x^4))/(x^8*(-4 + x^4)),x]","\frac{\left(-1+x^4\right)^{3/4} \left(6+x^4\right)}{42 x^7}-\frac{3^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{3} x}{\sqrt{2} \sqrt[4]{-1+x^4}}\right)}{8 \sqrt{2}}-\frac{3^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{3} x}{\sqrt{2} \sqrt[4]{-1+x^4}}\right)}{8 \sqrt{2}}","\frac{\left(-1+x^4\right)^{3/4} \left(6+x^4\right)}{42 x^7}-\frac{3^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{3} x}{\sqrt{2} \sqrt[4]{-1+x^4}}\right)}{8 \sqrt{2}}-\frac{3^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{3} x}{\sqrt{2} \sqrt[4]{-1+x^4}}\right)}{8 \sqrt{2}}",1,"((-1 + x^4)^(3/4)*(6 + x^4))/(42*x^7) - (3^(3/4)*ArcTan[(3^(1/4)*x)/(Sqrt[2]*(-1 + x^4)^(1/4))])/(8*Sqrt[2]) - (3^(3/4)*ArcTanh[(3^(1/4)*x)/(Sqrt[2]*(-1 + x^4)^(1/4))])/(8*Sqrt[2])","A",1
1300,1,94,94,0.7705407,"\int \frac{4 b+a x^3}{\sqrt[4]{b+a x^3} \left(b+a x^3+x^4\right)} \, dx","IntegrateAlgebraic[(4*b + a*x^3)/((b + a*x^3)^(1/4)*(b + a*x^3 + x^4)),x]","-\sqrt{2} \tan ^{-1}\left(\frac{-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{b+a x^3}}{\sqrt{2}}}{x \sqrt[4]{b+a x^3}}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{b+a x^3}}{x^2+\sqrt{b+a x^3}}\right)","-\sqrt{2} \tan ^{-1}\left(\frac{-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{b+a x^3}}{\sqrt{2}}}{x \sqrt[4]{b+a x^3}}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{b+a x^3}}{x^2+\sqrt{b+a x^3}}\right)",1,"-(Sqrt[2]*ArcTan[(-(x^2/Sqrt[2]) + Sqrt[b + a*x^3]/Sqrt[2])/(x*(b + a*x^3)^(1/4))]) + Sqrt[2]*ArcTanh[(Sqrt[2]*x*(b + a*x^3)^(1/4))/(x^2 + Sqrt[b + a*x^3])]","A",1
1301,1,94,94,1.201208,"\int \frac{-a^2 b+a (2 a+b) x-3 a x^2+x^3}{\sqrt{x (-a+x) (-b+x)} \left(-a^2 d+2 a d x+\left(b^2-d\right) x^2-2 b x^3+x^4\right)} \, dx","IntegrateAlgebraic[(-(a^2*b) + a*(2*a + b)*x - 3*a*x^2 + x^3)/(Sqrt[x*(-a + x)*(-b + x)]*(-(a^2*d) + 2*a*d*x + (b^2 - d)*x^2 - 2*b*x^3 + x^4)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{a b x+(-a-b) x^2+x^3}}{x (-b+x)}\right)}{d^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{a b x+(-a-b) x^2+x^3}}{x (-b+x)}\right)}{d^{3/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{a b x+(-a-b) x^2+x^3}}{x (-b+x)}\right)}{d^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt{a b x+(-a-b) x^2+x^3}}{x (-b+x)}\right)}{d^{3/4}}",1,"ArcTan[(d^(1/4)*Sqrt[a*b*x + (-a - b)*x^2 + x^3])/(x*(-b + x))]/d^(3/4) - ArcTanh[(d^(1/4)*Sqrt[a*b*x + (-a - b)*x^2 + x^3])/(x*(-b + x))]/d^(3/4)","A",1
1302,1,94,94,0.5324637,"\int \frac{-2 b+a x^4}{x^4 \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[(-2*b + a*x^4)/(x^4*(b*x^2 + a*x^4)^(1/4)),x]","-\frac{4 \left(-3 b+4 a x^2\right) \left(b x^2+a x^4\right)^{3/4}}{21 b x^5}+a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)+a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)","-\frac{4 \left(-3 b+4 a x^2\right) \left(b x^2+a x^4\right)^{3/4}}{21 b x^5}+a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)+a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)",1,"(-4*(-3*b + 4*a*x^2)*(b*x^2 + a*x^4)^(3/4))/(21*b*x^5) + a^(3/4)*ArcTan[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)] + a^(3/4)*ArcTanh[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)]","A",1
1303,1,94,94,0.4884933,"\int \frac{2 b+a x^4}{x^4 \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[(2*b + a*x^4)/(x^4*(b*x^2 + a*x^4)^(1/4)),x]","\frac{4 \left(-3 b+4 a x^2\right) \left(b x^2+a x^4\right)^{3/4}}{21 b x^5}+a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)+a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)","\frac{4 \left(-3 b+4 a x^2\right) \left(b x^2+a x^4\right)^{3/4}}{21 b x^5}+a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)+a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)",1,"(4*(-3*b + 4*a*x^2)*(b*x^2 + a*x^4)^(3/4))/(21*b*x^5) + a^(3/4)*ArcTan[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)] + a^(3/4)*ArcTanh[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)]","A",1
1304,1,94,94,0.7578303,"\int \frac{-1+x}{(1+x) \sqrt{a+b x+c x^2+b x^3+a x^4}} \, dx","IntegrateAlgebraic[(-1 + x)/((1 + x)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]),x]","-\frac{2 \sqrt{-2 a+2 b-c} \tan ^{-1}\left(\frac{\sqrt{-2 a+2 b-c} x}{\sqrt{a}+2 \sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{2 a-2 b+c}","-\frac{2 \sqrt{-2 a+2 b-c} \tan ^{-1}\left(\frac{\sqrt{-2 a+2 b-c} x}{\sqrt{a}+2 \sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{2 a-2 b+c}",1,"(-2*Sqrt[-2*a + 2*b - c]*ArcTan[(Sqrt[-2*a + 2*b - c]*x)/(Sqrt[a] + 2*Sqrt[a]*x + Sqrt[a]*x^2 - Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])])/(2*a - 2*b + c)","A",1
1305,1,94,94,0.7646614,"\int \frac{1+x}{(-1+x) \sqrt{a+b x+c x^2+b x^3+a x^4}} \, dx","IntegrateAlgebraic[(1 + x)/((-1 + x)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]),x]","-\frac{2 \sqrt{-2 a-2 b-c} \tan ^{-1}\left(\frac{\sqrt{-2 a-2 b-c} x}{\sqrt{a}-2 \sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{2 a+2 b+c}","-\frac{2 \sqrt{-2 a-2 b-c} \tan ^{-1}\left(\frac{\sqrt{-2 a-2 b-c} x}{\sqrt{a}-2 \sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{2 a+2 b+c}",1,"(-2*Sqrt[-2*a - 2*b - c]*ArcTan[(Sqrt[-2*a - 2*b - c]*x)/(Sqrt[a] - 2*Sqrt[a]*x + Sqrt[a]*x^2 - Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])])/(2*a + 2*b + c)","A",1
1306,1,94,94,0.9771386,"\int \frac{x^2}{\sqrt{\left(1-x^2\right) \left(1-k^2 x^2\right)} \left(-1+k^2 x^4\right)} \, dx","IntegrateAlgebraic[x^2/(Sqrt[(1 - x^2)*(1 - k^2*x^2)]*(-1 + k^2*x^4)),x]","-\frac{\tan ^{-1}\left(\frac{(-1+k) x}{\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{4 (-1+k) k}+\frac{\tan ^{-1}\left(\frac{(1+k) x}{1+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{2 k (1+k)}","-\frac{\tan ^{-1}\left(\frac{(-1+k) x}{\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{4 (-1+k) k}+\frac{\tan ^{-1}\left(\frac{(1+k) x}{1+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{2 k (1+k)}",1,"-1/4*ArcTan[((-1 + k)*x)/Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4]]/((-1 + k)*k) + ArcTan[((1 + k)*x)/(1 + k*x^2 + Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4])]/(2*k*(1 + k))","A",1
1307,1,94,94,0.9499062,"\int \frac{\left(-1+x^6\right)^{2/3} \left(1+x^6\right)}{x^3 \left(-1-x^3+x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^6)^(2/3)*(1 + x^6))/(x^3*(-1 - x^3 + x^6)),x]","\frac{\left(-1+x^6\right)^{2/3}}{2 x^2}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^6}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(-x+\sqrt[3]{-1+x^6}\right)-\frac{1}{6} \log \left(x^2+x \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)","\frac{\left(-1+x^6\right)^{2/3}}{2 x^2}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^6}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(-x+\sqrt[3]{-1+x^6}\right)-\frac{1}{6} \log \left(x^2+x \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)",1,"(-1 + x^6)^(2/3)/(2*x^2) - ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^6)^(1/3))]/Sqrt[3] + Log[-x + (-1 + x^6)^(1/3)]/3 - Log[x^2 + x*(-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)]/6","A",1
1308,1,94,94,0.9241718,"\int \frac{\left(-1+x^6\right) \left(1+x^6\right)^{2/3}}{x^3 \left(1-x^3+x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^6)*(1 + x^6)^(2/3))/(x^3*(1 - x^3 + x^6)),x]","\frac{\left(1+x^6\right)^{2/3}}{2 x^2}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^6}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(-x+\sqrt[3]{1+x^6}\right)-\frac{1}{6} \log \left(x^2+x \sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)","\frac{\left(1+x^6\right)^{2/3}}{2 x^2}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^6}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(-x+\sqrt[3]{1+x^6}\right)-\frac{1}{6} \log \left(x^2+x \sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)",1,"(1 + x^6)^(2/3)/(2*x^2) - ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^6)^(1/3))]/Sqrt[3] + Log[-x + (1 + x^6)^(1/3)]/3 - Log[x^2 + x*(1 + x^6)^(1/3) + (1 + x^6)^(2/3)]/6","A",1
1309,1,94,94,0.4423377,"\int \frac{-1+x^8}{\sqrt[4]{-x^2+x^4} \left(1+x^8\right)} \, dx","IntegrateAlgebraic[(-1 + x^8)/((-x^2 + x^4)^(1/4)*(1 + x^8)),x]","\tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)+\frac{1}{4} \text{RootSum}\left[2-4 \text{$\#$1}^4+6 \text{$\#$1}^8-4 \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)+\frac{1}{4} \text{RootSum}\left[2-4 \text{$\#$1}^4+6 \text{$\#$1}^8-4 \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"ArcTan[x/(-x^2 + x^4)^(1/4)] + ArcTanh[x/(-x^2 + x^4)^(1/4)] + RootSum[2 - 4*#1^4 + 6*#1^8 - 4*#1^12 + #1^16 & , (-Log[x] + Log[(-x^2 + x^4)^(1/4) - x*#1])/#1 & ]/4","A",1
1310,1,94,94,0.0010818,"\int \frac{-1+x^8}{\sqrt[4]{-x^2+x^4} \left(1+x^8\right)} \, dx","IntegrateAlgebraic[(-1 + x^8)/((-x^2 + x^4)^(1/4)*(1 + x^8)),x]","\tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)+\frac{1}{4} \text{RootSum}\left[2-4 \text{$\#$1}^4+6 \text{$\#$1}^8-4 \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)+\frac{1}{4} \text{RootSum}\left[2-4 \text{$\#$1}^4+6 \text{$\#$1}^8-4 \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"ArcTan[x/(-x^2 + x^4)^(1/4)] + ArcTanh[x/(-x^2 + x^4)^(1/4)] + RootSum[2 - 4*#1^4 + 6*#1^8 - 4*#1^12 + #1^16 & , (-Log[x] + Log[(-x^2 + x^4)^(1/4) - x*#1])/#1 & ]/4","A",1
1311,1,115,94,0.5106332,"\int \frac{1+3 x^4+x^8}{x^2 \left(1+x^4\right)^{3/4} \left(1+3 x^4+3 x^8\right)} \, dx","IntegrateAlgebraic[(1 + 3*x^4 + x^8)/(x^2*(1 + x^4)^(3/4)*(1 + 3*x^4 + 3*x^8)),x]","-\frac{\sqrt[4]{1+x^4}}{x}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[4]{1+x^4}}\right)}{\sqrt{3}}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[4]{1+x^4}}\right)}{\sqrt{3}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{3} x \sqrt[4]{1+x^4}}{x^2+\sqrt{1+x^4}}\right)}{\sqrt{3}}","-\frac{\sqrt[4]{1+x^4}}{x}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x \sqrt[4]{1+x^4}}{-x^2+\sqrt{1+x^4}}\right)}{\sqrt{3}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{3} x \sqrt[4]{1+x^4}}{x^2+\sqrt{1+x^4}}\right)}{\sqrt{3}}",1,"-((1 + x^4)^(1/4)/x) - ArcTan[(Sqrt[3]*x)/(-x + 2*(1 + x^4)^(1/4))]/Sqrt[3] - ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^4)^(1/4))]/Sqrt[3] + ArcTanh[(Sqrt[3]*x*(1 + x^4)^(1/4))/(x^2 + Sqrt[1 + x^4])]/Sqrt[3]","A",1
1312,1,94,94,0.3575824,"\int \frac{\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{b+a^2 x^2} \, dx","IntegrateAlgebraic[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/(b + a^2*x^2),x]","\frac{\text{RootSum}\left[b+c^4-4 c^3 \text{$\#$1}^2+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{c^2-2 c \text{$\#$1}^2+\text{$\#$1}^4}\&\right]}{a}","\frac{\text{RootSum}\left[b+c^4-4 c^3 \text{$\#$1}^2+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{c^2-2 c \text{$\#$1}^2+\text{$\#$1}^4}\&\right]}{a}",1,"RootSum[b + c^4 - 4*c^3*#1^2 + 6*c^2*#1^4 - 4*c*#1^6 + #1^8 & , (Log[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] - #1]*#1)/(c^2 - 2*c*#1^2 + #1^4) & ]/a","A",1
1313,1,94,94,0.0023262,"\int \frac{\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{b+a^2 x^2} \, dx","IntegrateAlgebraic[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/(b + a^2*x^2),x]","\frac{\text{RootSum}\left[b+c^4-4 c^3 \text{$\#$1}^2+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{c^2-2 c \text{$\#$1}^2+\text{$\#$1}^4}\&\right]}{a}","\frac{\text{RootSum}\left[b+c^4-4 c^3 \text{$\#$1}^2+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{c^2-2 c \text{$\#$1}^2+\text{$\#$1}^4}\&\right]}{a}",1,"RootSum[b + c^4 - 4*c^3*#1^2 + 6*c^2*#1^4 - 4*c*#1^6 + #1^8 & , (Log[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] - #1]*#1)/(c^2 - 2*c*#1^2 + #1^4) & ]/a","A",1
1314,1,95,95,8.0192539,"\int \frac{-1+x}{x^4 \sqrt[3]{1+x^3}} \, dx","IntegrateAlgebraic[(-1 + x)/(x^4*(1 + x^3)^(1/3)),x]","\frac{(2-3 x) \left(1+x^3\right)^{2/3}}{6 x^3}+\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^3}}{\sqrt{3}}\right)}{3 \sqrt{3}}+\frac{1}{9} \log \left(-1+\sqrt[3]{1+x^3}\right)-\frac{1}{18} \log \left(1+\sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)","\frac{(2-3 x) \left(1+x^3\right)^{2/3}}{6 x^3}+\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^3}}{\sqrt{3}}\right)}{3 \sqrt{3}}+\frac{1}{9} \log \left(-1+\sqrt[3]{1+x^3}\right)-\frac{1}{18} \log \left(1+\sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)",1,"((2 - 3*x)*(1 + x^3)^(2/3))/(6*x^3) + ArcTan[1/Sqrt[3] + (2*(1 + x^3)^(1/3))/Sqrt[3]]/(3*Sqrt[3]) + Log[-1 + (1 + x^3)^(1/3)]/9 - Log[1 + (1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/18","A",1
1315,1,95,95,0.2561812,"\int \frac{(3+2 x) \sqrt[3]{1+x+x^3}}{x^2 (1+x)} \, dx","IntegrateAlgebraic[((3 + 2*x)*(1 + x + x^3)^(1/3))/(x^2*(1 + x)),x]","-\frac{3 \sqrt[3]{1+x+x^3}}{x}-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x+x^3}}\right)-\log \left(-x+\sqrt[3]{1+x+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{1+x+x^3}+\left(1+x+x^3\right)^{2/3}\right)","-\frac{3 \sqrt[3]{1+x+x^3}}{x}-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x+x^3}}\right)-\log \left(-x+\sqrt[3]{1+x+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{1+x+x^3}+\left(1+x+x^3\right)^{2/3}\right)",1,"(-3*(1 + x + x^3)^(1/3))/x - Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x + x^3)^(1/3))] - Log[-x + (1 + x + x^3)^(1/3)] + Log[x^2 + x*(1 + x + x^3)^(1/3) + (1 + x + x^3)^(2/3)]/2","A",1
1316,1,95,95,3.3727485,"\int \frac{\left(a^2-2 a x+x^2\right) \left(-a b+2 (a-b) x+x^2\right)}{(x (-a+x) (-b+x))^{3/4} \left(-a^3 d+\left(b+3 a^2 d\right) x-(1+3 a d) x^2+d x^3\right)} \, dx","IntegrateAlgebraic[((a^2 - 2*a*x + x^2)*(-(a*b) + 2*(a - b)*x + x^2))/((x*(-a + x)*(-b + x))^(3/4)*(-(a^3*d) + (b + 3*a^2*d)*x - (1 + 3*a*d)*x^2 + d*x^3)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(a b x+(-a-b) x^2+x^3\right)^{3/4}}{x (-b+x)}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \left(a b x+(-a-b) x^2+x^3\right)^{3/4}}{x (-b+x)}\right)}{d^{3/4}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(a b x+(-a-b) x^2+x^3\right)^{3/4}}{x (-b+x)}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \left(a b x+(-a-b) x^2+x^3\right)^{3/4}}{x (-b+x)}\right)}{d^{3/4}}",1,"(2*ArcTan[(d^(1/4)*(a*b*x + (-a - b)*x^2 + x^3)^(3/4))/(x*(-b + x))])/d^(3/4) - (2*ArcTanh[(d^(1/4)*(a*b*x + (-a - b)*x^2 + x^3)^(3/4))/(x*(-b + x))])/d^(3/4)","A",1
1317,1,95,95,5.3108616,"\int \frac{\left(1+x^2\right)^2}{\left(1-x^2\right) \left(1-6 x^2+x^4\right)^{3/4}} \, dx","IntegrateAlgebraic[(1 + x^2)^2/((1 - x^2)*(1 - 6*x^2 + x^4)^(3/4)),x]","\tan ^{-1}\left(\frac{i+x}{\sqrt[4]{1-6 x^2+x^4}}\right)-\tan ^{-1}\left(\frac{\sqrt[4]{1-6 x^2+x^4}}{-i+x}\right)-\tanh ^{-1}\left(\frac{i+x}{\sqrt[4]{1-6 x^2+x^4}}\right)-\tanh ^{-1}\left(\frac{\sqrt[4]{1-6 x^2+x^4}}{-i+x}\right)","\tan ^{-1}\left(\frac{i+x}{\sqrt[4]{1-6 x^2+x^4}}\right)-\tan ^{-1}\left(\frac{\sqrt[4]{1-6 x^2+x^4}}{-i+x}\right)-\tanh ^{-1}\left(\frac{i+x}{\sqrt[4]{1-6 x^2+x^4}}\right)-\tanh ^{-1}\left(\frac{\sqrt[4]{1-6 x^2+x^4}}{-i+x}\right)",1,"ArcTan[(I + x)/(1 - 6*x^2 + x^4)^(1/4)] - ArcTan[(1 - 6*x^2 + x^4)^(1/4)/(-I + x)] - ArcTanh[(I + x)/(1 - 6*x^2 + x^4)^(1/4)] - ArcTanh[(1 - 6*x^2 + x^4)^(1/4)/(-I + x)]","A",1
1318,1,95,95,4.2416022,"\int \frac{\left(-4+x^3\right) \left(1-x^3+x^4\right)}{x^2 \left(-1+x^3\right)^{3/4} \left(-1+x^3+x^4\right)} \, dx","IntegrateAlgebraic[((-4 + x^3)*(1 - x^3 + x^4))/(x^2*(-1 + x^3)^(3/4)*(-1 + x^3 + x^4)),x]","\frac{4 \sqrt[4]{-1+x^3}}{x}+2 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-1+x^3}}{-x^2+\sqrt{-1+x^3}}\right)-2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-1+x^3}}{x^2+\sqrt{-1+x^3}}\right)","\frac{4 \sqrt[4]{-1+x^3}}{x}+2 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-1+x^3}}{-x^2+\sqrt{-1+x^3}}\right)-2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-1+x^3}}{x^2+\sqrt{-1+x^3}}\right)",1,"(4*(-1 + x^3)^(1/4))/x + 2*Sqrt[2]*ArcTan[(Sqrt[2]*x*(-1 + x^3)^(1/4))/(-x^2 + Sqrt[-1 + x^3])] - 2*Sqrt[2]*ArcTanh[(Sqrt[2]*x*(-1 + x^3)^(1/4))/(x^2 + Sqrt[-1 + x^3])]","A",1
1319,1,95,95,2.7811204,"\int \frac{\left(-1+x^4\right) \left(3+x^4\right) \left(-1-x^3+x^4\right)}{x^6 \left(-1-2 x^3+x^4\right) \sqrt[4]{-x+x^5}} \, dx","IntegrateAlgebraic[((-1 + x^4)*(3 + x^4)*(-1 - x^3 + x^4))/(x^6*(-1 - 2*x^3 + x^4)*(-x + x^5)^(1/4)),x]","\frac{4 \left(-3+7 x^3+3 x^4\right) \left(-x+x^5\right)^{3/4}}{21 x^6}-2\ 2^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \left(-x+x^5\right)^{3/4}}{-1+x^4}\right)-2\ 2^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \left(-x+x^5\right)^{3/4}}{-1+x^4}\right)","\frac{4 \left(-3+7 x^3+3 x^4\right) \left(-x+x^5\right)^{3/4}}{21 x^6}-2\ 2^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \left(-x+x^5\right)^{3/4}}{-1+x^4}\right)-2\ 2^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \left(-x+x^5\right)^{3/4}}{-1+x^4}\right)",1,"(4*(-3 + 7*x^3 + 3*x^4)*(-x + x^5)^(3/4))/(21*x^6) - 2*2^(3/4)*ArcTan[(2^(1/4)*(-x + x^5)^(3/4))/(-1 + x^4)] - 2*2^(3/4)*ArcTanh[(2^(1/4)*(-x + x^5)^(3/4))/(-1 + x^4)]","A",1
1320,1,95,95,13.1244515,"\int \frac{\left(4 b+a x^5\right) \left(-b+c x^4+a x^5\right)}{x^2 \left(-b+a x^5\right)^{3/4} \left(-b-c x^4+a x^5\right)} \, dx","IntegrateAlgebraic[((4*b + a*x^5)*(-b + c*x^4 + a*x^5))/(x^2*(-b + a*x^5)^(3/4)*(-b - c*x^4 + a*x^5)),x]","\frac{4 \sqrt[4]{-b+a x^5}}{x}-4 \sqrt[4]{c} \tan ^{-1}\left(\frac{\sqrt[4]{c} x \left(-b+a x^5\right)^{3/4}}{b-a x^5}\right)+4 \sqrt[4]{c} \tanh ^{-1}\left(\frac{\sqrt[4]{c} x \left(-b+a x^5\right)^{3/4}}{b-a x^5}\right)","\frac{4 \sqrt[4]{-b+a x^5}}{x}-4 \sqrt[4]{c} \tan ^{-1}\left(\frac{\sqrt[4]{c} x \left(-b+a x^5\right)^{3/4}}{b-a x^5}\right)+4 \sqrt[4]{c} \tanh ^{-1}\left(\frac{\sqrt[4]{c} x \left(-b+a x^5\right)^{3/4}}{b-a x^5}\right)",1,"(4*(-b + a*x^5)^(1/4))/x - 4*c^(1/4)*ArcTan[(c^(1/4)*x*(-b + a*x^5)^(3/4))/(b - a*x^5)] + 4*c^(1/4)*ArcTanh[(c^(1/4)*x*(-b + a*x^5)^(3/4))/(b - a*x^5)]","A",1
1321,1,95,95,7.0340263,"\int \frac{\left(-2+x^6\right) \left(1-x^4+x^6\right)}{x^4 \sqrt[4]{1+x^6} \left(1+x^4+x^6\right)} \, dx","IntegrateAlgebraic[((-2 + x^6)*(1 - x^4 + x^6))/(x^4*(1 + x^6)^(1/4)*(1 + x^4 + x^6)),x]","\frac{2 \left(1+x^6\right)^{3/4}}{3 x^3}+\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{1+x^6}}{-x^2+\sqrt{1+x^6}}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{1+x^6}}{x^2+\sqrt{1+x^6}}\right)","\frac{2 \left(1+x^6\right)^{3/4}}{3 x^3}+\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{1+x^6}}{-x^2+\sqrt{1+x^6}}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{1+x^6}}{x^2+\sqrt{1+x^6}}\right)",1,"(2*(1 + x^6)^(3/4))/(3*x^3) + Sqrt[2]*ArcTan[(Sqrt[2]*x*(1 + x^6)^(1/4))/(-x^2 + Sqrt[1 + x^6])] + Sqrt[2]*ArcTanh[(Sqrt[2]*x*(1 + x^6)^(1/4))/(x^2 + Sqrt[1 + x^6])]","A",1
1322,1,95,95,15.5634578,"\int \frac{x^4 \left(2 b+a x^6\right)}{\sqrt[4]{-b+a x^6} \left(-b-x^4+a x^6\right)^2} \, dx","IntegrateAlgebraic[(x^4*(2*b + a*x^6))/((-b + a*x^6)^(1/4)*(-b - x^4 + a*x^6)^2),x]","-\frac{x \left(-b+a x^6\right)^{3/4}}{2 \left(-b-x^4+a x^6\right)}-\frac{1}{4} \tan ^{-1}\left(\frac{x \left(-b+a x^6\right)^{3/4}}{b-a x^6}\right)-\frac{1}{4} \tanh ^{-1}\left(\frac{x \left(-b+a x^6\right)^{3/4}}{b-a x^6}\right)","-\frac{x \left(-b+a x^6\right)^{3/4}}{2 \left(-b-x^4+a x^6\right)}-\frac{1}{4} \tan ^{-1}\left(\frac{x \left(-b+a x^6\right)^{3/4}}{b-a x^6}\right)-\frac{1}{4} \tanh ^{-1}\left(\frac{x \left(-b+a x^6\right)^{3/4}}{b-a x^6}\right)",1,"-1/2*(x*(-b + a*x^6)^(3/4))/(-b - x^4 + a*x^6) - ArcTan[(x*(-b + a*x^6)^(3/4))/(b - a*x^6)]/4 - ArcTanh[(x*(-b + a*x^6)^(3/4))/(b - a*x^6)]/4","A",1
1323,1,95,95,15.4394577,"\int \frac{\left(2 b+a x^6\right) \left(-b-x^4+a x^6\right)}{x^4 \sqrt[4]{-b+a x^6} \left(-b-2 x^4+a x^6\right)} \, dx","IntegrateAlgebraic[((2*b + a*x^6)*(-b - x^4 + a*x^6))/(x^4*(-b + a*x^6)^(1/4)*(-b - 2*x^4 + a*x^6)),x]","\frac{2 \left(-b+a x^6\right)^{3/4}}{3 x^3}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x \left(-b+a x^6\right)^{3/4}}{b-a x^6}\right)}{\sqrt[4]{2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x \left(-b+a x^6\right)^{3/4}}{b-a x^6}\right)}{\sqrt[4]{2}}","\frac{2 \left(-b+a x^6\right)^{3/4}}{3 x^3}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x \left(-b+a x^6\right)^{3/4}}{b-a x^6}\right)}{\sqrt[4]{2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x \left(-b+a x^6\right)^{3/4}}{b-a x^6}\right)}{\sqrt[4]{2}}",1,"(2*(-b + a*x^6)^(3/4))/(3*x^3) + ArcTan[(2^(1/4)*x*(-b + a*x^6)^(3/4))/(b - a*x^6)]/2^(1/4) + ArcTanh[(2^(1/4)*x*(-b + a*x^6)^(3/4))/(b - a*x^6)]/2^(1/4)","A",1
1324,1,95,95,0.5779133,"\int \frac{\sqrt{-1-2 x^2-2 x^3-x^8} \left(-1+x^3+3 x^8\right)}{\left(1+2 x^3+x^8\right) \left(1+x^2+2 x^3+x^8\right)} \, dx","IntegrateAlgebraic[(Sqrt[-1 - 2*x^2 - 2*x^3 - x^8]*(-1 + x^3 + 3*x^8))/((1 + 2*x^3 + x^8)*(1 + x^2 + 2*x^3 + x^8)),x]","\tan ^{-1}\left(\frac{x \sqrt{-1-2 x^2-2 x^3-x^8}}{1+2 x^2+2 x^3+x^8}\right)-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt{-1-2 x^2-2 x^3-x^8}}{1+2 x^2+2 x^3+x^8}\right)","\tan ^{-1}\left(\frac{x \sqrt{-1-2 x^2-2 x^3-x^8}}{1+2 x^2+2 x^3+x^8}\right)-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt{-1-2 x^2-2 x^3-x^8}}{1+2 x^2+2 x^3+x^8}\right)",1,"ArcTan[(x*Sqrt[-1 - 2*x^2 - 2*x^3 - x^8])/(1 + 2*x^2 + 2*x^3 + x^8)] - Sqrt[2]*ArcTan[(Sqrt[2]*x*Sqrt[-1 - 2*x^2 - 2*x^3 - x^8])/(1 + 2*x^2 + 2*x^3 + x^8)]","A",1
1325,1,95,95,2.3555242,"\int \frac{\left(1+x^6\right) \sqrt{-2-x^2+x^6}}{4-3 x^4-4 x^6+x^{12}} \, dx","IntegrateAlgebraic[((1 + x^6)*Sqrt[-2 - x^2 + x^6])/(4 - 3*x^4 - 4*x^6 + x^12),x]","-\frac{1}{4} \sqrt{\frac{1}{3} \left(1+\sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{1+\sqrt{3}} x}{\sqrt{-2-x^2+x^6}}\right)-\frac{1}{4} \sqrt{\frac{1}{3} \left(-1+\sqrt{3}\right)} \tanh ^{-1}\left(\frac{\sqrt{-1+\sqrt{3}} x}{\sqrt{-2-x^2+x^6}}\right)","-\frac{1}{4} \sqrt{\frac{1}{3} \left(1+\sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{1+\sqrt{3}} x}{\sqrt{-2-x^2+x^6}}\right)-\frac{1}{4} \sqrt{\frac{1}{3} \left(-1+\sqrt{3}\right)} \tanh ^{-1}\left(\frac{\sqrt{-1+\sqrt{3}} x}{\sqrt{-2-x^2+x^6}}\right)",1,"-1/4*(Sqrt[(1 + Sqrt[3])/3]*ArcTan[(Sqrt[1 + Sqrt[3]]*x)/Sqrt[-2 - x^2 + x^6]]) - (Sqrt[(-1 + Sqrt[3])/3]*ArcTanh[(Sqrt[-1 + Sqrt[3]]*x)/Sqrt[-2 - x^2 + x^6]])/4","A",1
1326,1,96,96,0.1784796,"\int x \sqrt[3]{-1+x^3} \, dx","IntegrateAlgebraic[x*(-1 + x^3)^(1/3),x]","\frac{1}{3} x^2 \sqrt[3]{-1+x^3}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{3 \sqrt{3}}+\frac{1}{9} \log \left(-x+\sqrt[3]{-1+x^3}\right)-\frac{1}{18} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)","\frac{1}{3} x^2 \sqrt[3]{-1+x^3}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{3 \sqrt{3}}+\frac{1}{9} \log \left(-x+\sqrt[3]{-1+x^3}\right)-\frac{1}{18} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)",1,"(x^2*(-1 + x^3)^(1/3))/3 + ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^3)^(1/3))]/(3*Sqrt[3]) + Log[-x + (-1 + x^3)^(1/3)]/9 - Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)]/18","A",1
1327,1,96,96,0.1703073,"\int x \sqrt[3]{1+x^3} \, dx","IntegrateAlgebraic[x*(1 + x^3)^(1/3),x]","\frac{1}{3} x^2 \sqrt[3]{1+x^3}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3}}\right)}{3 \sqrt{3}}-\frac{1}{9} \log \left(-x+\sqrt[3]{1+x^3}\right)+\frac{1}{18} \log \left(x^2+x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)","\frac{1}{3} x^2 \sqrt[3]{1+x^3}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3}}\right)}{3 \sqrt{3}}-\frac{1}{9} \log \left(-x+\sqrt[3]{1+x^3}\right)+\frac{1}{18} \log \left(x^2+x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)",1,"(x^2*(1 + x^3)^(1/3))/3 - ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^3)^(1/3))]/(3*Sqrt[3]) - Log[-x + (1 + x^3)^(1/3)]/9 + Log[x^2 + x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/18","A",1
1328,1,96,96,0.2321535,"\int \frac{\sqrt[3]{x+x^3}}{x^2} \, dx","IntegrateAlgebraic[(x + x^3)^(1/3)/x^2,x]","-\frac{3 \sqrt[3]{x+x^3}}{2 x}-\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)-\frac{1}{2} \log \left(-x+\sqrt[3]{x+x^3}\right)+\frac{1}{4} \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)","-\frac{3 \sqrt[3]{x+x^3}}{2 x}-\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)-\frac{1}{2} \log \left(-x+\sqrt[3]{x+x^3}\right)+\frac{1}{4} \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)",1,"(-3*(x + x^3)^(1/3))/(2*x) - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^3)^(1/3))])/2 - Log[-x + (x + x^3)^(1/3)]/2 + Log[x^2 + x*(x + x^3)^(1/3) + (x + x^3)^(2/3)]/4","A",1
1329,1,96,96,4.2475492,"\int \frac{(-1+x) (-1+k x) \left(3-2 (1+k) x+k x^2\right)}{x ((1-x) x (1-k x))^{3/4} \left(-1+(1+k) x-k x^2+d x^3\right)} \, dx","IntegrateAlgebraic[((-1 + x)*(-1 + k*x)*(3 - 2*(1 + k)*x + k*x^2))/(x*((1 - x)*x*(1 - k*x))^(3/4)*(-1 + (1 + k)*x - k*x^2 + d*x^3)),x]","\frac{4 \sqrt[4]{x+(-1-k) x^2+k x^3}}{x}+2 \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{x+(-1-k) x^2+k x^3}}\right)-2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{x+(-1-k) x^2+k x^3}}\right)","\frac{4 \sqrt[4]{x+(-1-k) x^2+k x^3}}{x}+2 \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{x+(-1-k) x^2+k x^3}}\right)-2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{x+(-1-k) x^2+k x^3}}\right)",1,"(4*(x + (-1 - k)*x^2 + k*x^3)^(1/4))/x + 2*d^(1/4)*ArcTan[(d^(1/4)*x)/(x + (-1 - k)*x^2 + k*x^3)^(1/4)] - 2*d^(1/4)*ArcTanh[(d^(1/4)*x)/(x + (-1 - k)*x^2 + k*x^3)^(1/4)]","A",1
1330,1,96,96,15.8102696,"\int \frac{\left(1-2 x+x^2\right) \left(-2+(-1+k) (1+k) x+2 k^2 x^2\right)}{\left(\left(1-x^2\right) \left(1-k^2 x^2\right)\right)^{3/4} \left(-1+d-(1+3 d) x+\left(3 d+k^2\right) x^2+\left(-d+k^2\right) x^3\right)} \, dx","IntegrateAlgebraic[((1 - 2*x + x^2)*(-2 + (-1 + k)*(1 + k)*x + 2*k^2*x^2))/(((1 - x^2)*(1 - k^2*x^2))^(3/4)*(-1 + d - (1 + 3*d)*x + (3*d + k^2)*x^2 + (-d + k^2)*x^3)),x]","-\frac{\tan ^{-1}\left(\frac{-\sqrt[4]{d}+\sqrt[4]{d} x}{\sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{3/4}}+\frac{\tanh ^{-1}\left(\frac{-\sqrt[4]{d}+\sqrt[4]{d} x}{\sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{3/4}}","-\frac{\tan ^{-1}\left(\frac{-\sqrt[4]{d}+\sqrt[4]{d} x}{\sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{3/4}}+\frac{\tanh ^{-1}\left(\frac{-\sqrt[4]{d}+\sqrt[4]{d} x}{\sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{3/4}}",1,"-(ArcTan[(-d^(1/4) + d^(1/4)*x)/(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/4)]/d^(3/4)) + ArcTanh[(-d^(1/4) + d^(1/4)*x)/(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/4)]/d^(3/4)","A",1
1331,1,96,96,0.2905984,"\int \frac{x^2}{\left(-b+a x^2\right) \sqrt[4]{-b x^2+a x^4}} \, dx","IntegrateAlgebraic[x^2/((-b + a*x^2)*(-(b*x^2) + a*x^4)^(1/4)),x]","-\frac{2 \left(-b x^2+a x^4\right)^{3/4}}{a x \left(-b+a x^2\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{a^{5/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{a^{5/4}}","-\frac{2 \left(-b x^2+a x^4\right)^{3/4}}{a x \left(-b+a x^2\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{a^{5/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{a^{5/4}}",1,"(-2*(-(b*x^2) + a*x^4)^(3/4))/(a*x*(-b + a*x^2)) + ArcTan[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)]/a^(5/4) + ArcTanh[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)]/a^(5/4)","A",1
1332,1,96,96,12.9108089,"\int \frac{\sqrt{q+p x^5} \left(-2 q+3 p x^5\right) \left(a q+b x^2+a p x^5\right)}{x^4 \left(c q+d x^2+c p x^5\right)} \, dx","IntegrateAlgebraic[(Sqrt[q + p*x^5]*(-2*q + 3*p*x^5)*(a*q + b*x^2 + a*p*x^5))/(x^4*(c*q + d*x^2 + c*p*x^5)),x]","\frac{2 \sqrt{q+p x^5} \left(a c q+3 b c x^2-3 a d x^2+a c p x^5\right)}{3 c^2 x^3}+\frac{2 \left(b c \sqrt{d}-a d^{3/2}\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c} \sqrt{q+p x^5}}\right)}{c^{5/2}}","\frac{2 \sqrt{q+p x^5} \left(a c q+3 b c x^2-3 a d x^2+a c p x^5\right)}{3 c^2 x^3}+\frac{2 \left(b c \sqrt{d}-a d^{3/2}\right) \tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c} \sqrt{q+p x^5}}\right)}{c^{5/2}}",1,"(2*Sqrt[q + p*x^5]*(a*c*q + 3*b*c*x^2 - 3*a*d*x^2 + a*c*p*x^5))/(3*c^2*x^3) + (2*(b*c*Sqrt[d] - a*d^(3/2))*ArcTan[(Sqrt[d]*x)/(Sqrt[c]*Sqrt[q + p*x^5])])/c^(5/2)","A",1
1333,1,96,96,0.3172373,"\int \frac{x}{\sqrt[3]{x^2+x^6}} \, dx","IntegrateAlgebraic[x/(x^2 + x^6)^(1/3),x]","\frac{1}{4} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{x^2+x^6}}\right)-\frac{1}{4} \log \left(-x^2+\sqrt[3]{x^2+x^6}\right)+\frac{1}{8} \log \left(x^4+x^2 \sqrt[3]{x^2+x^6}+\left(x^2+x^6\right)^{2/3}\right)","\frac{1}{4} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{x^2+x^6}}\right)-\frac{1}{4} \log \left(-x^2+\sqrt[3]{x^2+x^6}\right)+\frac{1}{8} \log \left(x^4+x^2 \sqrt[3]{x^2+x^6}+\left(x^2+x^6\right)^{2/3}\right)",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(x^2 + x^6)^(1/3))])/4 - Log[-x^2 + (x^2 + x^6)^(1/3)]/4 + Log[x^4 + x^2*(x^2 + x^6)^(1/3) + (x^2 + x^6)^(2/3)]/8","A",1
1334,1,96,96,0.6487479,"\int \frac{-1+x^4}{\left(1+x^2+x^4\right) \sqrt[4]{x^2+x^6}} \, dx","IntegrateAlgebraic[(-1 + x^4)/((1 + x^2 + x^4)*(x^2 + x^6)^(1/4)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{x^2+x^6}}{-x^2+\sqrt{x^2+x^6}}\right)}{\sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{x^2+x^6}}{\sqrt{2}}}{x \sqrt[4]{x^2+x^6}}\right)}{\sqrt{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{x^2+x^6}}{-x^2+\sqrt{x^2+x^6}}\right)}{\sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{x^2+x^6}}{\sqrt{2}}}{x \sqrt[4]{x^2+x^6}}\right)}{\sqrt{2}}",1,"-(ArcTan[(Sqrt[2]*x*(x^2 + x^6)^(1/4))/(-x^2 + Sqrt[x^2 + x^6])]/Sqrt[2]) - ArcTanh[(x^2/Sqrt[2] + Sqrt[x^2 + x^6]/Sqrt[2])/(x*(x^2 + x^6)^(1/4))]/Sqrt[2]","A",1
1335,1,96,96,0.0013553,"\int \frac{-1+x^4}{\left(1+x^2+x^4\right) \sqrt[4]{x^2+x^6}} \, dx","IntegrateAlgebraic[(-1 + x^4)/((1 + x^2 + x^4)*(x^2 + x^6)^(1/4)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{x^2+x^6}}{-x^2+\sqrt{x^2+x^6}}\right)}{\sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{x^2+x^6}}{\sqrt{2}}}{x \sqrt[4]{x^2+x^6}}\right)}{\sqrt{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{x^2+x^6}}{-x^2+\sqrt{x^2+x^6}}\right)}{\sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{x^2+x^6}}{\sqrt{2}}}{x \sqrt[4]{x^2+x^6}}\right)}{\sqrt{2}}",1,"-(ArcTan[(Sqrt[2]*x*(x^2 + x^6)^(1/4))/(-x^2 + Sqrt[x^2 + x^6])]/Sqrt[2]) - ArcTanh[(x^2/Sqrt[2] + Sqrt[x^2 + x^6]/Sqrt[2])/(x*(x^2 + x^6)^(1/4))]/Sqrt[2]","A",1
1336,1,96,96,15.7235336,"\int \frac{x^2 \left(2 b+a x^6\right)}{\left(-b+a x^6\right)^{3/4} \left(-b-2 c x^4+a x^6\right)} \, dx","IntegrateAlgebraic[(x^2*(2*b + a*x^6))/((-b + a*x^6)^(3/4)*(-b - 2*c*x^4 + a*x^6)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x \left(-b+a x^6\right)^{3/4}}{b-a x^6}\right)}{2^{3/4} c^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x \left(-b+a x^6\right)^{3/4}}{b-a x^6}\right)}{2^{3/4} c^{3/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x \left(-b+a x^6\right)^{3/4}}{b-a x^6}\right)}{2^{3/4} c^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x \left(-b+a x^6\right)^{3/4}}{b-a x^6}\right)}{2^{3/4} c^{3/4}}",1,"-(ArcTan[(2^(1/4)*c^(1/4)*x*(-b + a*x^6)^(3/4))/(b - a*x^6)]/(2^(3/4)*c^(3/4))) + ArcTanh[(2^(1/4)*c^(1/4)*x*(-b + a*x^6)^(3/4))/(b - a*x^6)]/(2^(3/4)*c^(3/4))","A",1
1337,1,97,96,0.6449308,"\int \frac{1}{\sqrt[4]{b+a x^4} \left(-2 b-2 a x^4+x^8\right)} \, dx","IntegrateAlgebraic[1/((b + a*x^4)^(1/4)*(-2*b - 2*a*x^4 + x^8)),x]","-\frac{\text{RootSum}\left[b+2 a \text{$\#$1}^4-2 \text{$\#$1}^8\&,\frac{-a \log (x)+a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]}{8 b}","-\frac{\text{RootSum}\left[b+2 a \text{$\#$1}^4-2 \text{$\#$1}^8\&,\frac{a \log (x)-a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}-2 \text{$\#$1}^5}\&\right]}{8 b}",1,"-1/8*RootSum[b + 2*a*#1^4 - 2*#1^8 & , (-(a*Log[x]) + a*Log[(b + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(b + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1) + 2*#1^5) & ]/b","A",1
1338,1,97,96,0.0014718,"\int \frac{1}{\sqrt[4]{b+a x^4} \left(-2 b-2 a x^4+x^8\right)} \, dx","IntegrateAlgebraic[1/((b + a*x^4)^(1/4)*(-2*b - 2*a*x^4 + x^8)),x]","-\frac{\text{RootSum}\left[b+2 a \text{$\#$1}^4-2 \text{$\#$1}^8\&,\frac{-a \log (x)+a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]}{8 b}","-\frac{\text{RootSum}\left[b+2 a \text{$\#$1}^4-2 \text{$\#$1}^8\&,\frac{a \log (x)-a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}-2 \text{$\#$1}^5}\&\right]}{8 b}",1,"-1/8*RootSum[b + 2*a*#1^4 - 2*#1^8 & , (-(a*Log[x]) + a*Log[(b + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(b + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1) + 2*#1^5) & ]/b","A",1
1339,1,97,96,0.8698829,"\int \frac{-2 b+a x^4}{\sqrt[4]{b+a x^4} \left(-b-a x^4+2 x^8\right)} \, dx","IntegrateAlgebraic[(-2*b + a*x^4)/((b + a*x^4)^(1/4)*(-b - a*x^4 + 2*x^8)),x]","\frac{1}{4} \text{RootSum}\left[2 b+a \text{$\#$1}^4-\text{$\#$1}^8\&,\frac{-3 a \log (x)+3 a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)+2 \log (x) \text{$\#$1}^4-2 \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]","\frac{1}{4} \text{RootSum}\left[2 b+a \text{$\#$1}^4-\text{$\#$1}^8\&,\frac{3 a \log (x)-3 a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)-2 \log (x) \text{$\#$1}^4+2 \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}-2 \text{$\#$1}^5}\&\right]",1,"RootSum[2*b + a*#1^4 - #1^8 & , (-3*a*Log[x] + 3*a*Log[(b + a*x^4)^(1/4) - x*#1] + 2*Log[x]*#1^4 - 2*Log[(b + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1) + 2*#1^5) & ]/4","A",1
1340,1,97,97,0.2697044,"\int \frac{1+b x+k^2 x^2}{\sqrt{(1-x) x \left(1-k^2 x\right)} \left(-1+k^2 x^2\right)} \, dx","IntegrateAlgebraic[(1 + b*x + k^2*x^2)/(Sqrt[(1 - x)*x*(1 - k^2*x)]*(-1 + k^2*x^2)),x]","\frac{(-b-2 k) \tan ^{-1}\left(\frac{(-1+k) x}{\sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}\right)}{2 (-1+k) k}+\frac{(b-2 k) \tan ^{-1}\left(\frac{(1+k) x}{\sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}\right)}{2 k (1+k)}","\frac{(-b-2 k) \tan ^{-1}\left(\frac{(-1+k) x}{\sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}\right)}{2 (-1+k) k}+\frac{(b-2 k) \tan ^{-1}\left(\frac{(1+k) x}{\sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}\right)}{2 k (1+k)}",1,"((-b - 2*k)*ArcTan[((-1 + k)*x)/Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3]])/(2*(-1 + k)*k) + ((b - 2*k)*ArcTan[((1 + k)*x)/Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3]])/(2*k*(1 + k))","A",1
1341,1,97,97,0.1359002,"\int \frac{\sqrt[3]{-1+x^3}}{x^7} \, dx","IntegrateAlgebraic[(-1 + x^3)^(1/3)/x^7,x]","\frac{\left(-3+x^3\right) \sqrt[3]{-1+x^3}}{18 x^6}-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^3}}{\sqrt{3}}\right)}{9 \sqrt{3}}+\frac{1}{27} \log \left(1+\sqrt[3]{-1+x^3}\right)-\frac{1}{54} \log \left(1-\sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)","\frac{\left(-3+x^3\right) \sqrt[3]{-1+x^3}}{18 x^6}-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^3}}{\sqrt{3}}\right)}{9 \sqrt{3}}+\frac{1}{27} \log \left(1+\sqrt[3]{-1+x^3}\right)-\frac{1}{54} \log \left(1-\sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)",1,"((-3 + x^3)*(-1 + x^3)^(1/3))/(18*x^6) - ArcTan[1/Sqrt[3] - (2*(-1 + x^3)^(1/3))/Sqrt[3]]/(9*Sqrt[3]) + Log[1 + (-1 + x^3)^(1/3)]/27 - Log[1 - (-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)]/54","A",1
1342,1,97,97,0.1271667,"\int \frac{\sqrt[3]{1+x^3}}{x^7} \, dx","IntegrateAlgebraic[(1 + x^3)^(1/3)/x^7,x]","\frac{\left(-3-x^3\right) \sqrt[3]{1+x^3}}{18 x^6}+\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^3}}{\sqrt{3}}\right)}{9 \sqrt{3}}-\frac{1}{27} \log \left(-1+\sqrt[3]{1+x^3}\right)+\frac{1}{54} \log \left(1+\sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)","\frac{\left(-3-x^3\right) \sqrt[3]{1+x^3}}{18 x^6}+\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^3}}{\sqrt{3}}\right)}{9 \sqrt{3}}-\frac{1}{27} \log \left(-1+\sqrt[3]{1+x^3}\right)+\frac{1}{54} \log \left(1+\sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)",1,"((-3 - x^3)*(1 + x^3)^(1/3))/(18*x^6) + ArcTan[1/Sqrt[3] + (2*(1 + x^3)^(1/3))/Sqrt[3]]/(9*Sqrt[3]) - Log[-1 + (1 + x^3)^(1/3)]/27 + Log[1 + (1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/54","A",1
1343,1,97,97,0.0783055,"\int \frac{\left(-1+x^3\right) \sqrt[3]{1+x^3}}{x^4} \, dx","IntegrateAlgebraic[((-1 + x^3)*(1 + x^3)^(1/3))/x^4,x]","\frac{\sqrt[3]{1+x^3} \left(1+3 x^3\right)}{3 x^3}-\frac{2 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^3}}{\sqrt{3}}\right)}{3 \sqrt{3}}+\frac{2}{9} \log \left(-1+\sqrt[3]{1+x^3}\right)-\frac{1}{9} \log \left(1+\sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)","\frac{\sqrt[3]{1+x^3} \left(1+3 x^3\right)}{3 x^3}-\frac{2 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^3}}{\sqrt{3}}\right)}{3 \sqrt{3}}+\frac{2}{9} \log \left(-1+\sqrt[3]{1+x^3}\right)-\frac{1}{9} \log \left(1+\sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)",1,"((1 + x^3)^(1/3)*(1 + 3*x^3))/(3*x^3) - (2*ArcTan[1/Sqrt[3] + (2*(1 + x^3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]) + (2*Log[-1 + (1 + x^3)^(1/3)])/9 - Log[1 + (1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/9","A",1
1344,1,97,97,0.3518159,"\int \frac{1+x}{\left(-1-x+x^3\right) \sqrt[3]{-x^2+x^3}} \, dx","IntegrateAlgebraic[(1 + x)/((-1 - x + x^3)*(-x^2 + x^3)^(1/3)),x]","-\text{RootSum}\left[-1+5 \text{$\#$1}^3-4 \text{$\#$1}^6+\text{$\#$1}^9\&,\frac{-2 \log (x)+2 \log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-5 \text{$\#$1}+3 \text{$\#$1}^4}\&\right]","-\text{RootSum}\left[-1+5 \text{$\#$1}^3-4 \text{$\#$1}^6+\text{$\#$1}^9\&,\frac{-2 \log (x)+2 \log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-5 \text{$\#$1}+3 \text{$\#$1}^4}\&\right]",1,"-RootSum[-1 + 5*#1^3 - 4*#1^6 + #1^9 & , (-2*Log[x] + 2*Log[(-x^2 + x^3)^(1/3) - x*#1] + Log[x]*#1^3 - Log[(-x^2 + x^3)^(1/3) - x*#1]*#1^3)/(-5*#1 + 3*#1^4) & ]","A",1
1345,1,97,97,0.0011366,"\int \frac{1+x}{\left(-1-x+x^3\right) \sqrt[3]{-x^2+x^3}} \, dx","IntegrateAlgebraic[(1 + x)/((-1 - x + x^3)*(-x^2 + x^3)^(1/3)),x]","-\text{RootSum}\left[-1+5 \text{$\#$1}^3-4 \text{$\#$1}^6+\text{$\#$1}^9\&,\frac{-2 \log (x)+2 \log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-5 \text{$\#$1}+3 \text{$\#$1}^4}\&\right]","-\text{RootSum}\left[-1+5 \text{$\#$1}^3-4 \text{$\#$1}^6+\text{$\#$1}^9\&,\frac{-2 \log (x)+2 \log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-5 \text{$\#$1}+3 \text{$\#$1}^4}\&\right]",1,"-RootSum[-1 + 5*#1^3 - 4*#1^6 + #1^9 & , (-2*Log[x] + 2*Log[(-x^2 + x^3)^(1/3) - x*#1] + Log[x]*#1^3 - Log[(-x^2 + x^3)^(1/3) - x*#1]*#1^3)/(-5*#1 + 3*#1^4) & ]","A",1
1346,1,97,97,0.2912492,"\int \frac{-1+x^2}{\left(1+x^2\right) \sqrt{-x-x^2+x^3}} \, dx","IntegrateAlgebraic[(-1 + x^2)/((1 + x^2)*Sqrt[-x - x^2 + x^3]),x]","-\sqrt{\frac{1}{5}+\frac{2 i}{5}} \tan ^{-1}\left(\frac{\sqrt{1-2 i} \sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)-\sqrt{\frac{1}{5}-\frac{2 i}{5}} \tan ^{-1}\left(\frac{\sqrt{1+2 i} \sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)","-\sqrt{\frac{1}{5}+\frac{2 i}{5}} \tan ^{-1}\left(\frac{\sqrt{1-2 i} \sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)-\sqrt{\frac{1}{5}-\frac{2 i}{5}} \tan ^{-1}\left(\frac{\sqrt{1+2 i} \sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)",1,"-(Sqrt[1/5 + (2*I)/5]*ArcTan[(Sqrt[1 - 2*I]*Sqrt[-x - x^2 + x^3])/(-1 - x + x^2)]) - Sqrt[1/5 - (2*I)/5]*ArcTan[(Sqrt[1 + 2*I]*Sqrt[-x - x^2 + x^3])/(-1 - x + x^2)]","A",1
1347,1,97,97,0.2279744,"\int \frac{\sqrt[3]{x^2+x^3}}{x} \, dx","IntegrateAlgebraic[(x^2 + x^3)^(1/3)/x,x]","\sqrt[3]{x^2+x^3}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(-x+\sqrt[3]{x^2+x^3}\right)+\frac{1}{6} \log \left(x^2+x \sqrt[3]{x^2+x^3}+\left(x^2+x^3\right)^{2/3}\right)","\sqrt[3]{x^2+x^3}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(-x+\sqrt[3]{x^2+x^3}\right)+\frac{1}{6} \log \left(x^2+x \sqrt[3]{x^2+x^3}+\left(x^2+x^3\right)^{2/3}\right)",1,"(x^2 + x^3)^(1/3) - ArcTan[(Sqrt[3]*x)/(x + 2*(x^2 + x^3)^(1/3))]/Sqrt[3] - Log[-x + (x^2 + x^3)^(1/3)]/3 + Log[x^2 + x*(x^2 + x^3)^(1/3) + (x^2 + x^3)^(2/3)]/6","A",1
1348,1,97,97,0.193483,"\int \frac{\left(1+x^3\right)^{2/3} \left(2+x^3\right)}{x^6 \left(1+2 x^3\right)} \, dx","IntegrateAlgebraic[((1 + x^3)^(2/3)*(2 + x^3))/(x^6*(1 + 2*x^3)),x]","\frac{\left(1+x^3\right)^{2/3} \left(-4+11 x^3\right)}{10 x^5}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{1+x^3}}\right)+\log \left(x+\sqrt[3]{1+x^3}\right)-\frac{1}{2} \log \left(x^2-x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)","\frac{\left(1+x^3\right)^{2/3} \left(-4+11 x^3\right)}{10 x^5}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{1+x^3}}\right)+\log \left(x+\sqrt[3]{1+x^3}\right)-\frac{1}{2} \log \left(x^2-x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)",1,"((1 + x^3)^(2/3)*(-4 + 11*x^3))/(10*x^5) + Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(1 + x^3)^(1/3))] + Log[x + (1 + x^3)^(1/3)] - Log[x^2 - x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/2","A",1
1349,1,97,97,0.0991242,"\int \frac{\left(-3+x^4\right) \sqrt[3]{1+x^4}}{x^9} \, dx","IntegrateAlgebraic[((-3 + x^4)*(1 + x^4)^(1/3))/x^9,x]","\frac{\left(3-x^4\right) \sqrt[3]{1+x^4}}{8 x^8}-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^4}}{\sqrt{3}}\right)}{2 \sqrt{3}}+\frac{1}{6} \log \left(-1+\sqrt[3]{1+x^4}\right)-\frac{1}{12} \log \left(1+\sqrt[3]{1+x^4}+\left(1+x^4\right)^{2/3}\right)","\frac{\left(3-x^4\right) \sqrt[3]{1+x^4}}{8 x^8}-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^4}}{\sqrt{3}}\right)}{2 \sqrt{3}}+\frac{1}{6} \log \left(-1+\sqrt[3]{1+x^4}\right)-\frac{1}{12} \log \left(1+\sqrt[3]{1+x^4}+\left(1+x^4\right)^{2/3}\right)",1,"((3 - x^4)*(1 + x^4)^(1/3))/(8*x^8) - ArcTan[1/Sqrt[3] + (2*(1 + x^4)^(1/3))/Sqrt[3]]/(2*Sqrt[3]) + Log[-1 + (1 + x^4)^(1/3)]/6 - Log[1 + (1 + x^4)^(1/3) + (1 + x^4)^(2/3)]/12","A",1
1350,1,97,97,0.3419131,"\int \frac{-b+a x^3}{x^3 \sqrt[4]{b x+a x^4}} \, dx","IntegrateAlgebraic[(-b + a*x^3)/(x^3*(b*x + a*x^4)^(1/4)),x]","\frac{4 \left(b x+a x^4\right)^{3/4}}{9 x^3}+\frac{2}{3} a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)+\frac{2}{3} a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)","\frac{4 \left(b x+a x^4\right)^{3/4}}{9 x^3}+\frac{2}{3} a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)+\frac{2}{3} a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)",1,"(4*(b*x + a*x^4)^(3/4))/(9*x^3) + (2*a^(3/4)*ArcTan[(a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/3 + (2*a^(3/4)*ArcTanh[(a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/3","A",1
1351,1,97,97,0.3315164,"\int \frac{b+a x^3}{x^3 \sqrt[4]{b x+a x^4}} \, dx","IntegrateAlgebraic[(b + a*x^3)/(x^3*(b*x + a*x^4)^(1/4)),x]","-\frac{4 \left(b x+a x^4\right)^{3/4}}{9 x^3}+\frac{2}{3} a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)+\frac{2}{3} a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)","-\frac{4 \left(b x+a x^4\right)^{3/4}}{9 x^3}+\frac{2}{3} a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)+\frac{2}{3} a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)",1,"(-4*(b*x + a*x^4)^(3/4))/(9*x^3) + (2*a^(3/4)*ArcTan[(a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/3 + (2*a^(3/4)*ArcTanh[(a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/3","A",1
1352,1,97,97,0.1028478,"\int \frac{1}{x^{13} \sqrt[3]{1+x^6}} \, dx","IntegrateAlgebraic[1/(x^13*(1 + x^6)^(1/3)),x]","\frac{\left(1+x^6\right)^{2/3} \left(-3+4 x^6\right)}{36 x^{12}}+\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^6}}{\sqrt{3}}\right)}{9 \sqrt{3}}+\frac{1}{27} \log \left(-1+\sqrt[3]{1+x^6}\right)-\frac{1}{54} \log \left(1+\sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)","\frac{\left(1+x^6\right)^{2/3} \left(-3+4 x^6\right)}{36 x^{12}}+\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^6}}{\sqrt{3}}\right)}{9 \sqrt{3}}+\frac{1}{27} \log \left(-1+\sqrt[3]{1+x^6}\right)-\frac{1}{54} \log \left(1+\sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)",1,"((1 + x^6)^(2/3)*(-3 + 4*x^6))/(36*x^12) + ArcTan[1/Sqrt[3] + (2*(1 + x^6)^(1/3))/Sqrt[3]]/(9*Sqrt[3]) + Log[-1 + (1 + x^6)^(1/3)]/27 - Log[1 + (1 + x^6)^(1/3) + (1 + x^6)^(2/3)]/54","A",1
1353,1,97,97,2.8889467,"\int \frac{-3+5 x^8}{\left(1+x^8\right) \sqrt[3]{1-x^3+x^8}} \, dx","IntegrateAlgebraic[(-3 + 5*x^8)/((1 + x^8)*(1 - x^3 + x^8)^(1/3)),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{1-x^3+x^8}}\right)-\log \left(x+\sqrt[3]{1-x^3+x^8}\right)+\frac{1}{2} \log \left(x^2-x \sqrt[3]{1-x^3+x^8}+\left(1-x^3+x^8\right)^{2/3}\right)","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{1-x^3+x^8}}\right)-\log \left(x+\sqrt[3]{1-x^3+x^8}\right)+\frac{1}{2} \log \left(x^2-x \sqrt[3]{1-x^3+x^8}+\left(1-x^3+x^8\right)^{2/3}\right)",1,"-(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(1 - x^3 + x^8)^(1/3))]) - Log[x + (1 - x^3 + x^8)^(1/3)] + Log[x^2 - x*(1 - x^3 + x^8)^(1/3) + (1 - x^3 + x^8)^(2/3)]/2","A",1
1354,1,87,97,0.2158863,"\int \frac{\sqrt{x+\sqrt{1+x}}}{1+\sqrt{1+x}} \, dx","IntegrateAlgebraic[Sqrt[x + Sqrt[1 + x]]/(1 + Sqrt[1 + x]),x]","\frac{1}{2} \sqrt{x+\sqrt{1+x}} \left(-3+2 \sqrt{1+x}\right)-4 \tan ^{-1}\left(1+\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}\right)+\frac{1}{4} \log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}\right)","-\frac{3}{2} \sqrt{x+\sqrt{1+x}}+\sqrt{1+x} \sqrt{x+\sqrt{1+x}}-4 \tan ^{-1}\left(1+\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}\right)+\frac{1}{4} \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}\right)",1,"(Sqrt[x + Sqrt[1 + x]]*(-3 + 2*Sqrt[1 + x]))/2 - 4*ArcTan[1 + Sqrt[1 + x] - Sqrt[x + Sqrt[1 + x]]] + Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]]/4","A",1
1355,1,97,97,0.1456817,"\int \frac{x+\sqrt{1+x^2}}{1+\sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[(x + Sqrt[1 + x^2])/(1 + Sqrt[x + Sqrt[1 + x^2]]),x]","-\frac{x}{2}+\frac{1}{3} (3+x) \sqrt{x+\sqrt{1+x^2}}+\sqrt{1+x^2} \left(-\frac{1}{2}+\frac{1}{3} \sqrt{x+\sqrt{1+x^2}}\right)+\frac{1}{2} \log \left(x+\sqrt{1+x^2}\right)-2 \log \left(1+\sqrt{x+\sqrt{1+x^2}}\right)","-\frac{x}{2}+\frac{1}{3} (3+x) \sqrt{x+\sqrt{1+x^2}}+\sqrt{1+x^2} \left(-\frac{1}{2}+\frac{1}{3} \sqrt{x+\sqrt{1+x^2}}\right)+\frac{1}{2} \log \left(x+\sqrt{1+x^2}\right)-2 \log \left(1+\sqrt{x+\sqrt{1+x^2}}\right)",1,"-1/2*x + ((3 + x)*Sqrt[x + Sqrt[1 + x^2]])/3 + Sqrt[1 + x^2]*(-1/2 + Sqrt[x + Sqrt[1 + x^2]]/3) + Log[x + Sqrt[1 + x^2]]/2 - 2*Log[1 + Sqrt[x + Sqrt[1 + x^2]]]","A",1
1356,1,98,98,2.4713481,"\int \frac{x^2}{\left(-2 b+a x^2\right) \left(-b+a x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[x^2/((-2*b + a*x^2)*(-b + a*x^2)^(3/4)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^2}}\right)}{\sqrt{2} a^{3/2} \sqrt[4]{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^2}}{\sqrt{a} x}\right)}{\sqrt{2} a^{3/2} \sqrt[4]{b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^2}}\right)}{\sqrt{2} a^{3/2} \sqrt[4]{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^2}}{\sqrt{a} x}\right)}{\sqrt{2} a^{3/2} \sqrt[4]{b}}",1,"ArcTan[(Sqrt[a]*x)/(Sqrt[2]*b^(1/4)*(-b + a*x^2)^(1/4))]/(Sqrt[2]*a^(3/2)*b^(1/4)) - ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^2)^(1/4))/(Sqrt[a]*x)]/(Sqrt[2]*a^(3/2)*b^(1/4))","A",1
1357,1,98,98,15.9221947,"\int \frac{\left(-2 k-(-1+k) (1+k) x+2 k x^2\right) \left(1-2 k x+k^2 x^2\right)}{\left(\left(1-x^2\right) \left(1-k^2 x^2\right)\right)^{3/4} \left(1-d+(1+3 d) k x-\left(1+3 d k^2\right) x^2+k \left(-1+d k^2\right) x^3\right)} \, dx","IntegrateAlgebraic[((-2*k - (-1 + k)*(1 + k)*x + 2*k*x^2)*(1 - 2*k*x + k^2*x^2))/(((1 - x^2)*(1 - k^2*x^2))^(3/4)*(1 - d + (1 + 3*d)*k*x - (1 + 3*d*k^2)*x^2 + k*(-1 + d*k^2)*x^3)),x]","\frac{\tan ^{-1}\left(\frac{-\sqrt[4]{d}+\sqrt[4]{d} k x}{\sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{3/4}}-\frac{\tanh ^{-1}\left(\frac{-\sqrt[4]{d}+\sqrt[4]{d} k x}{\sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{3/4}}","\frac{\tan ^{-1}\left(\frac{-\sqrt[4]{d}+\sqrt[4]{d} k x}{\sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{3/4}}-\frac{\tanh ^{-1}\left(\frac{-\sqrt[4]{d}+\sqrt[4]{d} k x}{\sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{3/4}}",1,"ArcTan[(-d^(1/4) + d^(1/4)*k*x)/(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/4)]/d^(3/4) - ArcTanh[(-d^(1/4) + d^(1/4)*k*x)/(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/4)]/d^(3/4)","A",1
1358,1,98,98,1.0725373,"\int \frac{\left(-1+x^2\right) \left(1+x^2\right) \sqrt{1+3 x^2+x^4}}{x^2 \left(1+x+x^2\right)^2} \, dx","IntegrateAlgebraic[((-1 + x^2)*(1 + x^2)*Sqrt[1 + 3*x^2 + x^4])/(x^2*(1 + x + x^2)^2),x]","\frac{\left(1+2 x+x^2\right) \sqrt{1+3 x^2+x^4}}{x \left(1+x+x^2\right)}-3 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{1+x+x^2+\sqrt{1+3 x^2+x^4}}\right)+2 \log (x)-2 \log \left(1+x^2+\sqrt{1+3 x^2+x^4}\right)","\frac{\left(1+2 x+x^2\right) \sqrt{1+3 x^2+x^4}}{x \left(1+x+x^2\right)}-3 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{1+x+x^2+\sqrt{1+3 x^2+x^4}}\right)+2 \log (x)-2 \log \left(1+x^2+\sqrt{1+3 x^2+x^4}\right)",1,"((1 + 2*x + x^2)*Sqrt[1 + 3*x^2 + x^4])/(x*(1 + x + x^2)) - 3*Sqrt[2]*ArcTanh[(Sqrt[2]*x)/(1 + x + x^2 + Sqrt[1 + 3*x^2 + x^4])] + 2*Log[x] - 2*Log[1 + x^2 + Sqrt[1 + 3*x^2 + x^4]]","A",1
1359,1,98,98,0.6965378,"\int \frac{\left(2+x^2\right) \left(-4+x+2 x^2\right) \sqrt{8-7 x^2+2 x^4}}{x^4} \, dx","IntegrateAlgebraic[((2 + x^2)*(-4 + x + 2*x^2)*Sqrt[8 - 7*x^2 + 2*x^4])/x^4,x]","\frac{\sqrt{8-7 x^2+2 x^4} \left(16-6 x-14 x^2+3 x^3+4 x^4\right)}{6 x^3}-\frac{\log (x)}{2 \sqrt{2}}+\frac{\log \left(-2 \sqrt{2}+\sqrt{2} x^2+\sqrt{8-7 x^2+2 x^4}\right)}{2 \sqrt{2}}","\frac{\sqrt{8-7 x^2+2 x^4} \left(16-6 x-14 x^2+3 x^3+4 x^4\right)}{6 x^3}-\frac{\log (x)}{2 \sqrt{2}}+\frac{\log \left(-2 \sqrt{2}+\sqrt{2} x^2+\sqrt{8-7 x^2+2 x^4}\right)}{2 \sqrt{2}}",1,"(Sqrt[8 - 7*x^2 + 2*x^4]*(16 - 6*x - 14*x^2 + 3*x^3 + 4*x^4))/(6*x^3) - Log[x]/(2*Sqrt[2]) + Log[-2*Sqrt[2] + Sqrt[2]*x^2 + Sqrt[8 - 7*x^2 + 2*x^4]]/(2*Sqrt[2])","A",1
1360,1,98,98,0.4086594,"\int x^4 \left(-b+a x^4\right)^{3/4} \, dx","IntegrateAlgebraic[x^4*(-b + a*x^4)^(3/4),x]","\frac{\left(-b+a x^4\right)^{3/4} \left(-3 b x+4 a x^5\right)}{32 a}-\frac{3 b^2 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{64 a^{5/4}}-\frac{3 b^2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{64 a^{5/4}}","\frac{\left(-b+a x^4\right)^{3/4} \left(-3 b x+4 a x^5\right)}{32 a}-\frac{3 b^2 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{64 a^{5/4}}-\frac{3 b^2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{64 a^{5/4}}",1,"((-b + a*x^4)^(3/4)*(-3*b*x + 4*a*x^5))/(32*a) - (3*b^2*ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)])/(64*a^(5/4)) - (3*b^2*ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)])/(64*a^(5/4))","A",1
1361,1,98,98,0.3607077,"\int \frac{\left(-2 b+a x^2\right) \sqrt[4]{b x^2+a x^4}}{x^2} \, dx","IntegrateAlgebraic[((-2*b + a*x^2)*(b*x^2 + a*x^4)^(1/4))/x^2,x]","\frac{\left(8 b+a x^2\right) \sqrt[4]{b x^2+a x^4}}{2 x}+\frac{7}{4} \sqrt[4]{a} b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)-\frac{7}{4} \sqrt[4]{a} b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)","\frac{\left(8 b+a x^2\right) \sqrt[4]{b x^2+a x^4}}{2 x}+\frac{7}{4} \sqrt[4]{a} b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)-\frac{7}{4} \sqrt[4]{a} b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)",1,"((8*b + a*x^2)*(b*x^2 + a*x^4)^(1/4))/(2*x) + (7*a^(1/4)*b*ArcTan[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/4 - (7*a^(1/4)*b*ArcTanh[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/4","A",1
1362,1,98,98,0.3804812,"\int \frac{\left(2 b+a x^2\right) \sqrt[4]{b x^2+a x^4}}{x^2} \, dx","IntegrateAlgebraic[((2*b + a*x^2)*(b*x^2 + a*x^4)^(1/4))/x^2,x]","\frac{\left(-8 b+a x^2\right) \sqrt[4]{b x^2+a x^4}}{2 x}-\frac{9}{4} \sqrt[4]{a} b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)+\frac{9}{4} \sqrt[4]{a} b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)","\frac{\left(-8 b+a x^2\right) \sqrt[4]{b x^2+a x^4}}{2 x}-\frac{9}{4} \sqrt[4]{a} b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)+\frac{9}{4} \sqrt[4]{a} b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)",1,"((-8*b + a*x^2)*(b*x^2 + a*x^4)^(1/4))/(2*x) - (9*a^(1/4)*b*ArcTan[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/4 + (9*a^(1/4)*b*ArcTanh[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/4","A",1
1363,1,98,98,12.105979,"\int \frac{\left(1-3 k^2\right) x+2 k^2 x^3}{\sqrt[4]{\left(1-x^2\right) \left(1-k^2 x^2\right)} \left(-1+d+\left(-d+3 k^2\right) x^2-3 k^4 x^4+k^6 x^6\right)} \, dx","IntegrateAlgebraic[((1 - 3*k^2)*x + 2*k^2*x^3)/(((1 - x^2)*(1 - k^2*x^2))^(1/4)*(-1 + d + (-d + 3*k^2)*x^2 - 3*k^4*x^4 + k^6*x^6)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{-1+k^2 x^2}\right)}{d^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{-1+k^2 x^2}\right)}{d^{3/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{-1+k^2 x^2}\right)}{d^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{-1+k^2 x^2}\right)}{d^{3/4}}",1,"ArcTan[(d^(1/4)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/4))/(-1 + k^2*x^2)]/d^(3/4) - ArcTanh[(d^(1/4)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/4))/(-1 + k^2*x^2)]/d^(3/4)","A",1
1364,1,98,98,0.897519,"\int \frac{-b+a x^4}{\sqrt[4]{b+a x^4} \left(b-a x^4+x^8\right)} \, dx","IntegrateAlgebraic[(-b + a*x^4)/((b + a*x^4)^(1/4)*(b - a*x^4 + x^8)),x]","-\frac{1}{4} \text{RootSum}\left[2 a^2+b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 a \log (x)+2 a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]","-\frac{1}{4} \text{RootSum}\left[2 a^2+b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 a \log (x)-2 a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{3 a \text{$\#$1}-2 \text{$\#$1}^5}\&\right]",1,"-1/4*RootSum[2*a^2 + b - 3*a*#1^4 + #1^8 & , (-2*a*Log[x] + 2*a*Log[(b + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(b + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1 + 2*#1^5) & ]","A",1
1365,1,98,98,0.001523,"\int \frac{-b+a x^4}{\sqrt[4]{b+a x^4} \left(b-a x^4+x^8\right)} \, dx","IntegrateAlgebraic[(-b + a*x^4)/((b + a*x^4)^(1/4)*(b - a*x^4 + x^8)),x]","-\frac{1}{4} \text{RootSum}\left[2 a^2+b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 a \log (x)+2 a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]","-\frac{1}{4} \text{RootSum}\left[2 a^2+b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 a \log (x)-2 a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{3 a \text{$\#$1}-2 \text{$\#$1}^5}\&\right]",1,"-1/4*RootSum[2*a^2 + b - 3*a*#1^4 + #1^8 & , (-2*a*Log[x] + 2*a*Log[(b + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(b + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1 + 2*#1^5) & ]","A",1
1366,1,98,98,12.8705735,"\int \frac{\left(4+x^5\right) \left(1-x^4-2 x^5+x^8+x^9+x^{10}\right)}{x^2 \left(-1+x^5\right)^{3/4} \left(1+x^4-2 x^5-x^8-x^9+x^{10}\right)} \, dx","IntegrateAlgebraic[((4 + x^5)*(1 - x^4 - 2*x^5 + x^8 + x^9 + x^10))/(x^2*(-1 + x^5)^(3/4)*(1 + x^4 - 2*x^5 - x^8 - x^9 + x^10)),x]","\frac{4 \sqrt[4]{-1+x^5}}{x}-2 \text{RootSum}\left[-1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{\log (x)-\log \left(\sqrt[4]{-1+x^5}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-1+x^5}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]","\frac{4 \sqrt[4]{-1+x^5}}{x}-2 \text{RootSum}\left[-1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{\log (x)-\log \left(\sqrt[4]{-1+x^5}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-1+x^5}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]",1,"(4*(-1 + x^5)^(1/4))/x - 2*RootSum[-1 - #1^4 + #1^8 & , (Log[x] - Log[(-1 + x^5)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-1 + x^5)^(1/4) - x*#1]*#1^4)/(-#1^3 + 2*#1^7) & ]","A",1
1367,1,98,98,0.0019262,"\int \frac{\left(4+x^5\right) \left(1-x^4-2 x^5+x^8+x^9+x^{10}\right)}{x^2 \left(-1+x^5\right)^{3/4} \left(1+x^4-2 x^5-x^8-x^9+x^{10}\right)} \, dx","IntegrateAlgebraic[((4 + x^5)*(1 - x^4 - 2*x^5 + x^8 + x^9 + x^10))/(x^2*(-1 + x^5)^(3/4)*(1 + x^4 - 2*x^5 - x^8 - x^9 + x^10)),x]","\frac{4 \sqrt[4]{-1+x^5}}{x}-2 \text{RootSum}\left[-1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{\log (x)-\log \left(\sqrt[4]{-1+x^5}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-1+x^5}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]","\frac{4 \sqrt[4]{-1+x^5}}{x}-2 \text{RootSum}\left[-1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{\log (x)-\log \left(\sqrt[4]{-1+x^5}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-1+x^5}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]",1,"(4*(-1 + x^5)^(1/4))/x - 2*RootSum[-1 - #1^4 + #1^8 & , (Log[x] - Log[(-1 + x^5)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-1 + x^5)^(1/4) - x*#1]*#1^4)/(-#1^3 + 2*#1^7) & ]","A",1
1368,1,98,98,0.5639466,"\int \frac{1}{\sqrt{a x^2+\sqrt{b+a^2 x^4}}} \, dx","IntegrateAlgebraic[1/Sqrt[a*x^2 + Sqrt[b + a^2*x^4]],x]","\frac{x}{2 \sqrt{a x^2+\sqrt{b+a^2 x^4}}}+\frac{\log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{2 \sqrt{2} \sqrt{a}}","\frac{x}{2 \sqrt{a x^2+\sqrt{b+a^2 x^4}}}+\frac{\log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{2 \sqrt{2} \sqrt{a}}",1,"x/(2*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]) + Log[a*x^2 + Sqrt[b + a^2*x^4] + Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]]/(2*Sqrt[2]*Sqrt[a])","A",1
1369,1,99,99,0.0806289,"\int \frac{\left(-3 b+2 a x^2\right) \left(b^2+a^2 x^2\right)^{3/4}}{x} \, dx","IntegrateAlgebraic[((-3*b + 2*a*x^2)*(b^2 + a^2*x^2)^(3/4))/x,x]","\frac{2 \left(b^2+a^2 x^2\right)^{3/4} \left(-7 a b+2 b^2+2 a^2 x^2\right)}{7 a}-3 b^{5/2} \tan ^{-1}\left(\frac{\sqrt[4]{b^2+a^2 x^2}}{\sqrt{b}}\right)+3 b^{5/2} \tanh ^{-1}\left(\frac{\sqrt[4]{b^2+a^2 x^2}}{\sqrt{b}}\right)","\frac{2 \left(b^2+a^2 x^2\right)^{3/4} \left(-7 a b+2 b^2+2 a^2 x^2\right)}{7 a}-3 b^{5/2} \tan ^{-1}\left(\frac{\sqrt[4]{b^2+a^2 x^2}}{\sqrt{b}}\right)+3 b^{5/2} \tanh ^{-1}\left(\frac{\sqrt[4]{b^2+a^2 x^2}}{\sqrt{b}}\right)",1,"(2*(b^2 + a^2*x^2)^(3/4)*(-7*a*b + 2*b^2 + 2*a^2*x^2))/(7*a) - 3*b^(5/2)*ArcTan[(b^2 + a^2*x^2)^(1/4)/Sqrt[b]] + 3*b^(5/2)*ArcTanh[(b^2 + a^2*x^2)^(1/4)/Sqrt[b]]","A",1
1370,1,99,99,1.28465,"\int \frac{-1+x}{\left(-2-2 x+x^2\right) \sqrt{-1+x^3}} \, dx","IntegrateAlgebraic[(-1 + x)/((-2 - 2*x + x^2)*Sqrt[-1 + x^3]),x]","\frac{1}{6} \sqrt{3+2 \sqrt{3}} \tan ^{-1}\left(\frac{\sqrt{-3+2 \sqrt{3}} \sqrt{-1+x^3}}{1+x+x^2}\right)-\frac{1}{6} \sqrt{-3+2 \sqrt{3}} \tanh ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} \sqrt{-1+x^3}}{1+x+x^2}\right)","\frac{1}{6} \sqrt{3+2 \sqrt{3}} \tan ^{-1}\left(\frac{\sqrt{-3+2 \sqrt{3}} \sqrt{-1+x^3}}{1+x+x^2}\right)-\frac{1}{6} \sqrt{-3+2 \sqrt{3}} \tanh ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} \sqrt{-1+x^3}}{1+x+x^2}\right)",1,"(Sqrt[3 + 2*Sqrt[3]]*ArcTan[(Sqrt[-3 + 2*Sqrt[3]]*Sqrt[-1 + x^3])/(1 + x + x^2)])/6 - (Sqrt[-3 + 2*Sqrt[3]]*ArcTanh[(Sqrt[3 + 2*Sqrt[3]]*Sqrt[-1 + x^3])/(1 + x + x^2)])/6","A",1
1371,1,99,99,1.6127791,"\int \frac{3-x+x^2}{\left(-2-2 x+x^2\right) \sqrt{-1+x^3}} \, dx","IntegrateAlgebraic[(3 - x + x^2)/((-2 - 2*x + x^2)*Sqrt[-1 + x^3]),x]","-\frac{1}{6} \sqrt{15+14 \sqrt{3}} \tan ^{-1}\left(\frac{\sqrt{-3+2 \sqrt{3}} \sqrt{-1+x^3}}{1+x+x^2}\right)-\frac{1}{6} \sqrt{-15+14 \sqrt{3}} \tanh ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} \sqrt{-1+x^3}}{1+x+x^2}\right)","-\frac{1}{6} \sqrt{15+14 \sqrt{3}} \tan ^{-1}\left(\frac{\sqrt{-3+2 \sqrt{3}} \sqrt{-1+x^3}}{1+x+x^2}\right)-\frac{1}{6} \sqrt{-15+14 \sqrt{3}} \tanh ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} \sqrt{-1+x^3}}{1+x+x^2}\right)",1,"-1/6*(Sqrt[15 + 14*Sqrt[3]]*ArcTan[(Sqrt[-3 + 2*Sqrt[3]]*Sqrt[-1 + x^3])/(1 + x + x^2)]) - (Sqrt[-15 + 14*Sqrt[3]]*ArcTanh[(Sqrt[3 + 2*Sqrt[3]]*Sqrt[-1 + x^3])/(1 + x + x^2)])/6","A",1
1372,1,99,99,1.3540722,"\int \frac{2 x+x^2}{\left(-2-2 x+x^2\right) \sqrt{-1+x^3}} \, dx","IntegrateAlgebraic[(2*x + x^2)/((-2 - 2*x + x^2)*Sqrt[-1 + x^3]),x]","\frac{1}{3} \sqrt{-3+2 \sqrt{3}} \tan ^{-1}\left(\frac{\sqrt{-3+2 \sqrt{3}} \sqrt{-1+x^3}}{1+x+x^2}\right)-\frac{1}{3} \sqrt{3+2 \sqrt{3}} \tanh ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} \sqrt{-1+x^3}}{1+x+x^2}\right)","\frac{1}{3} \sqrt{-3+2 \sqrt{3}} \tan ^{-1}\left(\frac{\sqrt{-3+2 \sqrt{3}} \sqrt{-1+x^3}}{1+x+x^2}\right)-\frac{1}{3} \sqrt{3+2 \sqrt{3}} \tanh ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} \sqrt{-1+x^3}}{1+x+x^2}\right)",1,"(Sqrt[-3 + 2*Sqrt[3]]*ArcTan[(Sqrt[-3 + 2*Sqrt[3]]*Sqrt[-1 + x^3])/(1 + x + x^2)])/3 - (Sqrt[3 + 2*Sqrt[3]]*ArcTanh[(Sqrt[3 + 2*Sqrt[3]]*Sqrt[-1 + x^3])/(1 + x + x^2)])/3","A",1
1373,1,99,99,0.1119154,"\int \frac{\sqrt[3]{-1+x^3} \left(1+x^3\right)}{x^7} \, dx","IntegrateAlgebraic[((-1 + x^3)^(1/3)*(1 + x^3))/x^7,x]","\frac{\left(-3-5 x^3\right) \sqrt[3]{-1+x^3}}{18 x^6}-\frac{4 \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^3}}{\sqrt{3}}\right)}{9 \sqrt{3}}+\frac{4}{27} \log \left(1+\sqrt[3]{-1+x^3}\right)-\frac{2}{27} \log \left(1-\sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)","\frac{\left(-3-5 x^3\right) \sqrt[3]{-1+x^3}}{18 x^6}-\frac{4 \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^3}}{\sqrt{3}}\right)}{9 \sqrt{3}}+\frac{4}{27} \log \left(1+\sqrt[3]{-1+x^3}\right)-\frac{2}{27} \log \left(1-\sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)",1,"((-3 - 5*x^3)*(-1 + x^3)^(1/3))/(18*x^6) - (4*ArcTan[1/Sqrt[3] - (2*(-1 + x^3)^(1/3))/Sqrt[3]])/(9*Sqrt[3]) + (4*Log[1 + (-1 + x^3)^(1/3)])/27 - (2*Log[1 - (-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)])/27","A",1
1374,1,99,99,0.1123201,"\int \frac{\sqrt[3]{-1+x^3} \left(-1+2 x^3\right)}{x^7} \, dx","IntegrateAlgebraic[((-1 + x^3)^(1/3)*(-1 + 2*x^3))/x^7,x]","\frac{\left(3-13 x^3\right) \sqrt[3]{-1+x^3}}{18 x^6}-\frac{5 \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^3}}{\sqrt{3}}\right)}{9 \sqrt{3}}+\frac{5}{27} \log \left(1+\sqrt[3]{-1+x^3}\right)-\frac{5}{54} \log \left(1-\sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)","\frac{\left(3-13 x^3\right) \sqrt[3]{-1+x^3}}{18 x^6}-\frac{5 \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^3}}{\sqrt{3}}\right)}{9 \sqrt{3}}+\frac{5}{27} \log \left(1+\sqrt[3]{-1+x^3}\right)-\frac{5}{54} \log \left(1-\sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)",1,"((3 - 13*x^3)*(-1 + x^3)^(1/3))/(18*x^6) - (5*ArcTan[1/Sqrt[3] - (2*(-1 + x^3)^(1/3))/Sqrt[3]])/(9*Sqrt[3]) + (5*Log[1 + (-1 + x^3)^(1/3)])/27 - (5*Log[1 - (-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)])/54","A",1
1375,1,99,99,5.7936531,"\int \frac{\left(a^2-2 a x+x^2\right) \left(-2 a b x+(3 a-b) x^2\right)}{\left(x^2 (-a+x) (-b+x)\right)^{3/4} \left(a^3 d-3 a^2 d x+(-b+3 a d) x^2+(1-d) x^3\right)} \, dx","IntegrateAlgebraic[((a^2 - 2*a*x + x^2)*(-2*a*b*x + (3*a - b)*x^2))/((x^2*(-a + x)*(-b + x))^(3/4)*(a^3*d - 3*a^2*d*x + (-b + 3*a*d)*x^2 + (1 - d)*x^3)),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(a b x^2+(-a-b) x^3+x^4\right)^{3/4}}{x^2 (-b+x)}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \left(a b x^2+(-a-b) x^3+x^4\right)^{3/4}}{x^2 (-b+x)}\right)}{d^{3/4}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(a b x^2+(-a-b) x^3+x^4\right)^{3/4}}{x^2 (-b+x)}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \left(a b x^2+(-a-b) x^3+x^4\right)^{3/4}}{x^2 (-b+x)}\right)}{d^{3/4}}",1,"(-2*ArcTan[(d^(1/4)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(3/4))/(x^2*(-b + x))])/d^(3/4) + (2*ArcTanh[(d^(1/4)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(3/4))/(x^2*(-b + x))])/d^(3/4)","A",1
1376,1,97,99,0.4920826,"\int \frac{1+x^2}{\sqrt{\frac{-2-x+2 x^2}{-1+x+x^2}} \left(1-x^2+x^4\right)} \, dx","IntegrateAlgebraic[(1 + x^2)/(Sqrt[(-2 - x + 2*x^2)/(-1 + x + x^2)]*(1 - x^2 + x^4)),x]","\sqrt{\frac{1}{5}+\frac{3 i}{5}} \tan ^{-1}\left(\sqrt{-\frac{1}{5}-\frac{3 i}{5}} \sqrt{\frac{-2-x+2 x^2}{-1+x+x^2}}\right)+\sqrt{\frac{1}{5}-\frac{3 i}{5}} \tan ^{-1}\left(\sqrt{-\frac{1}{5}+\frac{3 i}{5}} \sqrt{\frac{-2-x+2 x^2}{-1+x+x^2}}\right)","\sqrt{\frac{1}{5}-\frac{3 i}{5}} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{-2-x+2 x^2}{-1+x+x^2}}}{\sqrt{-1-3 i}}\right)+\sqrt{\frac{1}{5}+\frac{3 i}{5}} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{-2-x+2 x^2}{-1+x+x^2}}}{\sqrt{-1+3 i}}\right)",1,"Sqrt[1/5 + (3*I)/5]*ArcTan[Sqrt[-1/5 - (3*I)/5]*Sqrt[(-2 - x + 2*x^2)/(-1 + x + x^2)]] + Sqrt[1/5 - (3*I)/5]*ArcTan[Sqrt[-1/5 + (3*I)/5]*Sqrt[(-2 - x + 2*x^2)/(-1 + x + x^2)]]","A",1
1377,1,99,99,1.3639343,"\int \frac{-2 b+a x^4}{x^4 \left(b+a x^4\right) \sqrt[4]{-b x^2+a x^4}} \, dx","IntegrateAlgebraic[(-2*b + a*x^4)/(x^4*(b + a*x^4)*(-(b*x^2) + a*x^4)^(1/4)),x]","-\frac{4 \left(3 b+4 a x^2\right) \left(-b x^2+a x^4\right)^{3/4}}{21 b^2 x^5}-\frac{3 a \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{4 b}","-\frac{4 \left(3 b+4 a x^2\right) \left(-b x^2+a x^4\right)^{3/4}}{21 b^2 x^5}-\frac{3 a \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{4 b}",1,"(-4*(3*b + 4*a*x^2)*(-(b*x^2) + a*x^4)^(3/4))/(21*b^2*x^5) - (3*a*RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (-Log[x] + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1])/#1 & ])/(4*b)","A",1
1378,1,99,99,0.0022201,"\int \frac{-2 b+a x^4}{x^4 \left(b+a x^4\right) \sqrt[4]{-b x^2+a x^4}} \, dx","IntegrateAlgebraic[(-2*b + a*x^4)/(x^4*(b + a*x^4)*(-(b*x^2) + a*x^4)^(1/4)),x]","-\frac{4 \left(3 b+4 a x^2\right) \left(-b x^2+a x^4\right)^{3/4}}{21 b^2 x^5}-\frac{3 a \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{4 b}","-\frac{4 \left(3 b+4 a x^2\right) \left(-b x^2+a x^4\right)^{3/4}}{21 b^2 x^5}-\frac{3 a \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{4 b}",1,"(-4*(3*b + 4*a*x^2)*(-(b*x^2) + a*x^4)^(3/4))/(21*b^2*x^5) - (3*a*RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (-Log[x] + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1])/#1 & ])/(4*b)","A",1
1379,1,99,99,0.4244338,"\int \sqrt[4]{b x^3+a x^4} \, dx","IntegrateAlgebraic[(b*x^3 + a*x^4)^(1/4),x]","\frac{(b+4 a x) \sqrt[4]{b x^3+a x^4}}{8 a}+\frac{3 b^2 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{16 a^{7/4}}-\frac{3 b^2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{16 a^{7/4}}","\frac{(b+4 a x) \sqrt[4]{b x^3+a x^4}}{8 a}+\frac{3 b^2 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{16 a^{7/4}}-\frac{3 b^2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{16 a^{7/4}}",1,"((b + 4*a*x)*(b*x^3 + a*x^4)^(1/4))/(8*a) + (3*b^2*ArcTan[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(16*a^(7/4)) - (3*b^2*ArcTanh[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(16*a^(7/4))","A",1
1380,1,99,99,0.1014136,"\int \frac{1}{x^{13} \sqrt[3]{-1+x^6}} \, dx","IntegrateAlgebraic[1/(x^13*(-1 + x^6)^(1/3)),x]","\frac{\left(-1+x^6\right)^{2/3} \left(3+4 x^6\right)}{36 x^{12}}-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^6}}{\sqrt{3}}\right)}{9 \sqrt{3}}-\frac{1}{27} \log \left(1+\sqrt[3]{-1+x^6}\right)+\frac{1}{54} \log \left(1-\sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)","\frac{\left(-1+x^6\right)^{2/3} \left(3+4 x^6\right)}{36 x^{12}}-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^6}}{\sqrt{3}}\right)}{9 \sqrt{3}}-\frac{1}{27} \log \left(1+\sqrt[3]{-1+x^6}\right)+\frac{1}{54} \log \left(1-\sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)",1,"((-1 + x^6)^(2/3)*(3 + 4*x^6))/(36*x^12) - ArcTan[1/Sqrt[3] - (2*(-1 + x^6)^(1/3))/Sqrt[3]]/(9*Sqrt[3]) - Log[1 + (-1 + x^6)^(1/3)]/27 + Log[1 - (-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)]/54","A",1
1381,1,99,99,0.9435678,"\int \frac{b+2 a x^4}{\sqrt[4]{b+a x^4} \left(-2 b-a x^4+x^8\right)} \, dx","IntegrateAlgebraic[(b + 2*a*x^4)/((b + a*x^4)^(1/4)*(-2*b - a*x^4 + x^8)),x]","-\frac{1}{4} \text{RootSum}\left[a^2-b-3 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{a \log (x)-a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 a \text{$\#$1}+4 \text{$\#$1}^5}\&\right]","-\frac{1}{4} \text{RootSum}\left[a^2-b-3 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-a \log (x)+a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{3 a \text{$\#$1}-4 \text{$\#$1}^5}\&\right]",1,"-1/4*RootSum[a^2 - b - 3*a*#1^4 + 2*#1^8 & , (a*Log[x] - a*Log[(b + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(b + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1 + 4*#1^5) & ]","A",1
1382,1,99,99,11.8034154,"\int \frac{2 b+a x^4}{\sqrt[4]{b x^2+a x^4} \left(-b-a x^4+2 x^8\right)} \, dx","IntegrateAlgebraic[(2*b + a*x^4)/((b*x^2 + a*x^4)^(1/4)*(-b - a*x^4 + 2*x^8)),x]","\frac{1}{4} \text{RootSum}\left[a^4+a^3 b-2 b^3-4 a^3 \text{$\#$1}^4-2 a^2 b \text{$\#$1}^4+6 a^2 \text{$\#$1}^8+a b \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{1}{4} \text{RootSum}\left[a^4+a^3 b-2 b^3-4 a^3 \text{$\#$1}^4-2 a^2 b \text{$\#$1}^4+6 a^2 \text{$\#$1}^8+a b \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"RootSum[a^4 + a^3*b - 2*b^3 - 4*a^3*#1^4 - 2*a^2*b*#1^4 + 6*a^2*#1^8 + a*b*#1^8 - 4*a*#1^12 + #1^16 & , (-Log[x] + Log[(b*x^2 + a*x^4)^(1/4) - x*#1])/#1 & ]/4","A",1
1383,1,99,99,0.002094,"\int \frac{2 b+a x^4}{\sqrt[4]{b x^2+a x^4} \left(-b-a x^4+2 x^8\right)} \, dx","IntegrateAlgebraic[(2*b + a*x^4)/((b*x^2 + a*x^4)^(1/4)*(-b - a*x^4 + 2*x^8)),x]","\frac{1}{4} \text{RootSum}\left[a^4+a^3 b-2 b^3-4 a^3 \text{$\#$1}^4-2 a^2 b \text{$\#$1}^4+6 a^2 \text{$\#$1}^8+a b \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{1}{4} \text{RootSum}\left[a^4+a^3 b-2 b^3-4 a^3 \text{$\#$1}^4-2 a^2 b \text{$\#$1}^4+6 a^2 \text{$\#$1}^8+a b \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"RootSum[a^4 + a^3*b - 2*b^3 - 4*a^3*#1^4 - 2*a^2*b*#1^4 + 6*a^2*#1^8 + a*b*#1^8 - 4*a*#1^12 + #1^16 & , (-Log[x] + Log[(b*x^2 + a*x^4)^(1/4) - x*#1])/#1 & ]/4","A",1
1384,1,99,99,0.8864885,"\int \frac{\left(-1+2 x^2\right) \left(-1+4 x-4 x^2+4 x^4\right)}{\sqrt{\frac{1-2 x^2}{1+2 x^2}} \left(1+2 x^2\right) \left(-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right)} \, dx","IntegrateAlgebraic[((-1 + 2*x^2)*(-1 + 4*x - 4*x^2 + 4*x^4))/(Sqrt[(1 - 2*x^2)/(1 + 2*x^2)]*(1 + 2*x^2)*(-1 - 8*x + 32*x^2 - 40*x^3 + 46*x^4 - 64*x^5 + 56*x^6 - 32*x^7 + 8*x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{\frac{3}{2}} \sqrt{\frac{1-2 x^2}{1+2 x^2}}}{-1+x}\right)}{2 \sqrt[4]{2} 3^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{\frac{3}{2}} \sqrt{\frac{1-2 x^2}{1+2 x^2}}}{-1+x}\right)}{2 \sqrt[4]{2} 3^{3/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{\frac{3}{2}} \sqrt{\frac{1-2 x^2}{1+2 x^2}}}{-1+x}\right)}{2 \sqrt[4]{2} 3^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{\frac{3}{2}} \sqrt{\frac{1-2 x^2}{1+2 x^2}}}{-1+x}\right)}{2 \sqrt[4]{2} 3^{3/4}}",1,"ArcTan[((3/2)^(1/4)*Sqrt[(1 - 2*x^2)/(1 + 2*x^2)])/(-1 + x)]/(2*2^(1/4)*3^(3/4)) - ArcTanh[((3/2)^(1/4)*Sqrt[(1 - 2*x^2)/(1 + 2*x^2)])/(-1 + x)]/(2*2^(1/4)*3^(3/4))","A",1
1385,1,99,99,0.6345787,"\int \frac{\sqrt{1+x^2}}{x^2+\sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[Sqrt[1 + x^2]/(x^2 + Sqrt[x + Sqrt[1 + x^2]]),x]","\log \left(x+\sqrt{1+x^2}\right)-2 \text{RootSum}\left[1-2 \text{$\#$1}^4+4 \text{$\#$1}^5+\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right)+\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}}{-2+5 \text{$\#$1}+2 \text{$\#$1}^4}\&\right]","\log \left(x+\sqrt{1+x^2}\right)-2 \text{RootSum}\left[1-2 \text{$\#$1}^4+4 \text{$\#$1}^5+\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right)+\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}}{-2+5 \text{$\#$1}+2 \text{$\#$1}^4}\&\right]",1,"Log[x + Sqrt[1 + x^2]] - 2*RootSum[1 - 2*#1^4 + 4*#1^5 + #1^8 & , (-Log[Sqrt[x + Sqrt[1 + x^2]] - #1] + Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1)/(-2 + 5*#1 + 2*#1^4) & ]","A",1
1386,1,99,99,0.0010571,"\int \frac{\sqrt{1+x^2}}{x^2+\sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[Sqrt[1 + x^2]/(x^2 + Sqrt[x + Sqrt[1 + x^2]]),x]","\log \left(x+\sqrt{1+x^2}\right)-2 \text{RootSum}\left[1-2 \text{$\#$1}^4+4 \text{$\#$1}^5+\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right)+\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}}{-2+5 \text{$\#$1}+2 \text{$\#$1}^4}\&\right]","\log \left(x+\sqrt{1+x^2}\right)-2 \text{RootSum}\left[1-2 \text{$\#$1}^4+4 \text{$\#$1}^5+\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right)+\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}}{-2+5 \text{$\#$1}+2 \text{$\#$1}^4}\&\right]",1,"Log[x + Sqrt[1 + x^2]] - 2*RootSum[1 - 2*#1^4 + 4*#1^5 + #1^8 & , (-Log[Sqrt[x + Sqrt[1 + x^2]] - #1] + Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1)/(-2 + 5*#1 + 2*#1^4) & ]","A",1
1387,1,100,100,4.0717308,"\int \frac{1+k x^2}{\left(-1+c k x+k x^2\right) \sqrt{\left(1-x^2\right) \left(1-k^2 x^2\right)}} \, dx","IntegrateAlgebraic[(1 + k*x^2)/((-1 + c*k*x + k*x^2)*Sqrt[(1 - x^2)*(1 - k^2*x^2)]),x]","\frac{2 \sqrt{1-2 k+k^2-c^2 k^2} \tan ^{-1}\left(\frac{\sqrt{1-2 k+k^2-c^2 k^2} x}{-1+c k x+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{(1-k+c k) (-1+k+c k)}","\frac{2 \sqrt{1-2 k+k^2-c^2 k^2} \tan ^{-1}\left(\frac{\sqrt{1-2 k+k^2-c^2 k^2} x}{-1+c k x+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{(1-k+c k) (-1+k+c k)}",1,"(2*Sqrt[1 - 2*k + k^2 - c^2*k^2]*ArcTan[(Sqrt[1 - 2*k + k^2 - c^2*k^2]*x)/(-1 + c*k*x + k*x^2 + Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4])])/((1 - k + c*k)*(-1 + k + c*k))","A",1
1388,1,100,100,3.719061,"\int \frac{-1+k x^2}{\left(1+c k x+k x^2\right) \sqrt{\left(1-x^2\right) \left(1-k^2 x^2\right)}} \, dx","IntegrateAlgebraic[(-1 + k*x^2)/((1 + c*k*x + k*x^2)*Sqrt[(1 - x^2)*(1 - k^2*x^2)]),x]","\frac{2 \sqrt{1+2 k+k^2-c^2 k^2} \tan ^{-1}\left(\frac{\sqrt{1+2 k+k^2-c^2 k^2} x}{1+c k x+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{(-1-k+c k) (1+k+c k)}","\frac{2 \sqrt{1+2 k+k^2-c^2 k^2} \tan ^{-1}\left(\frac{\sqrt{1+2 k+k^2-c^2 k^2} x}{1+c k x+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{(-1-k+c k) (1+k+c k)}",1,"(2*Sqrt[1 + 2*k + k^2 - c^2*k^2]*ArcTan[(Sqrt[1 + 2*k + k^2 - c^2*k^2]*x)/(1 + c*k*x + k*x^2 + Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4])])/((-1 - k + c*k)*(1 + k + c*k))","A",1
1389,1,100,100,0.2603938,"\int x^2 \sqrt[3]{x+x^3} \, dx","IntegrateAlgebraic[x^2*(x + x^3)^(1/3),x]","\frac{1}{12} \sqrt[3]{x+x^3} \left(x+3 x^3\right)+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)}{6 \sqrt{3}}+\frac{1}{18} \log \left(-x+\sqrt[3]{x+x^3}\right)-\frac{1}{36} \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)","\frac{1}{12} \sqrt[3]{x+x^3} \left(x+3 x^3\right)+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)}{6 \sqrt{3}}+\frac{1}{18} \log \left(-x+\sqrt[3]{x+x^3}\right)-\frac{1}{36} \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)",1,"((x + x^3)^(1/3)*(x + 3*x^3))/12 + ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^3)^(1/3))]/(6*Sqrt[3]) + Log[-x + (x + x^3)^(1/3)]/18 - Log[x^2 + x*(x + x^3)^(1/3) + (x + x^3)^(2/3)]/36","A",1
1390,1,100,100,0.1380806,"\int \frac{6+2 x+x^2}{\sqrt[3]{2+x^2} \left(1+3 x-2 x^2+x^3\right)} \, dx","IntegrateAlgebraic[(6 + 2*x + x^2)/((2 + x^2)^(1/3)*(1 + 3*x - 2*x^2 + x^3)),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\frac{2}{\sqrt{3}}-\frac{2 x}{\sqrt{3}}+\frac{\sqrt[3]{2+x^2}}{\sqrt{3}}}{\sqrt[3]{2+x^2}}\right)+\log \left(-1+x+\sqrt[3]{2+x^2}\right)-\frac{1}{2} \log \left(1-2 x+x^2+(1-x) \sqrt[3]{2+x^2}+\left(2+x^2\right)^{2/3}\right)","-\sqrt{3} \tan ^{-1}\left(\frac{\frac{2}{\sqrt{3}}-\frac{2 x}{\sqrt{3}}+\frac{\sqrt[3]{2+x^2}}{\sqrt{3}}}{\sqrt[3]{2+x^2}}\right)+\log \left(-1+x+\sqrt[3]{2+x^2}\right)-\frac{1}{2} \log \left(1-2 x+x^2+(1-x) \sqrt[3]{2+x^2}+\left(2+x^2\right)^{2/3}\right)",1,"-(Sqrt[3]*ArcTan[(2/Sqrt[3] - (2*x)/Sqrt[3] + (2 + x^2)^(1/3)/Sqrt[3])/(2 + x^2)^(1/3)]) + Log[-1 + x + (2 + x^2)^(1/3)] - Log[1 - 2*x + x^2 + (1 - x)*(2 + x^2)^(1/3) + (2 + x^2)^(2/3)]/2","A",1
1391,1,100,100,0.164758,"\int \frac{\sqrt[3]{x^2+x^3}}{x^2} \, dx","IntegrateAlgebraic[(x^2 + x^3)^(1/3)/x^2,x]","-\frac{3 \sqrt[3]{x^2+x^3}}{x}-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{x^2+x^3}+\left(x^2+x^3\right)^{2/3}\right)","-\frac{3 \sqrt[3]{x^2+x^3}}{x}-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{x^2+x^3}+\left(x^2+x^3\right)^{2/3}\right)",1,"(-3*(x^2 + x^3)^(1/3))/x - Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(x^2 + x^3)^(1/3))] - Log[-x + (x^2 + x^3)^(1/3)] + Log[x^2 + x*(x^2 + x^3)^(1/3) + (x^2 + x^3)^(2/3)]/2","A",1
1392,1,100,100,0.1546722,"\int \frac{\sqrt[3]{1+x^4} \left(3+x^4\right)}{x^{13}} \, dx","IntegrateAlgebraic[((1 + x^4)^(1/3)*(3 + x^4))/x^13,x]","\frac{\sqrt[3]{1+x^4} \left(-9-6 x^4+x^8\right)}{36 x^{12}}-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^4}}{\sqrt{3}}\right)}{18 \sqrt{3}}+\frac{1}{54} \log \left(-1+\sqrt[3]{1+x^4}\right)-\frac{1}{108} \log \left(1+\sqrt[3]{1+x^4}+\left(1+x^4\right)^{2/3}\right)","\frac{\sqrt[3]{1+x^4} \left(-9-6 x^4+x^8\right)}{36 x^{12}}-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^4}}{\sqrt{3}}\right)}{18 \sqrt{3}}+\frac{1}{54} \log \left(-1+\sqrt[3]{1+x^4}\right)-\frac{1}{108} \log \left(1+\sqrt[3]{1+x^4}+\left(1+x^4\right)^{2/3}\right)",1,"((1 + x^4)^(1/3)*(-9 - 6*x^4 + x^8))/(36*x^12) - ArcTan[1/Sqrt[3] + (2*(1 + x^4)^(1/3))/Sqrt[3]]/(18*Sqrt[3]) + Log[-1 + (1 + x^4)^(1/3)]/54 - Log[1 + (1 + x^4)^(1/3) + (1 + x^4)^(2/3)]/108","A",1
1393,1,100,100,3.3883239,"\int \frac{\left(-1+x^4\right)^{2/3} \left(3+x^4\right) \left(-1+x^3+x^4\right)}{x^6 \left(-1-x^3+x^4\right)} \, dx","IntegrateAlgebraic[((-1 + x^4)^(2/3)*(3 + x^4)*(-1 + x^3 + x^4))/(x^6*(-1 - x^3 + x^4)),x]","\frac{3 \left(-1+x^4\right)^{2/3} \left(-1+5 x^3+x^4\right)}{5 x^5}-2 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^4}}\right)+2 \log \left(-x+\sqrt[3]{-1+x^4}\right)-\log \left(x^2+x \sqrt[3]{-1+x^4}+\left(-1+x^4\right)^{2/3}\right)","\frac{3 \left(-1+x^4\right)^{2/3} \left(-1+5 x^3+x^4\right)}{5 x^5}-2 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^4}}\right)+2 \log \left(-x+\sqrt[3]{-1+x^4}\right)-\log \left(x^2+x \sqrt[3]{-1+x^4}+\left(-1+x^4\right)^{2/3}\right)",1,"(3*(-1 + x^4)^(2/3)*(-1 + 5*x^3 + x^4))/(5*x^5) - 2*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^4)^(1/3))] + 2*Log[-x + (-1 + x^4)^(1/3)] - Log[x^2 + x*(-1 + x^4)^(1/3) + (-1 + x^4)^(2/3)]","A",1
1394,1,100,100,3.2767593,"\int \frac{\left(-3+x^4\right) \left(1+x^4\right)^{2/3} \left(1+x^3+x^4\right)}{x^6 \left(1-x^3+x^4\right)} \, dx","IntegrateAlgebraic[((-3 + x^4)*(1 + x^4)^(2/3)*(1 + x^3 + x^4))/(x^6*(1 - x^3 + x^4)),x]","\frac{3 \left(1+x^4\right)^{2/3} \left(1+5 x^3+x^4\right)}{5 x^5}-2 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^4}}\right)+2 \log \left(-x+\sqrt[3]{1+x^4}\right)-\log \left(x^2+x \sqrt[3]{1+x^4}+\left(1+x^4\right)^{2/3}\right)","\frac{3 \left(1+x^4\right)^{2/3} \left(1+5 x^3+x^4\right)}{5 x^5}-2 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^4}}\right)+2 \log \left(-x+\sqrt[3]{1+x^4}\right)-\log \left(x^2+x \sqrt[3]{1+x^4}+\left(1+x^4\right)^{2/3}\right)",1,"(3*(1 + x^4)^(2/3)*(1 + 5*x^3 + x^4))/(5*x^5) - 2*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^4)^(1/3))] + 2*Log[-x + (1 + x^4)^(1/3)] - Log[x^2 + x*(1 + x^4)^(1/3) + (1 + x^4)^(2/3)]","A",1
1395,1,101,100,0.6159379,"\int \frac{-b+a x^2}{\left(b+a x^2+x^4\right) \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[(-b + a*x^2)/((b + a*x^2 + x^4)*(b*x^2 + a*x^4)^(1/4)),x]","-\frac{1}{2} \text{RootSum}\left[b-a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 a \log (x)+2 a \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]","-\frac{1}{2} \text{RootSum}\left[b-a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 a \log (x)-2 a \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}-2 \text{$\#$1}^5}\&\right]",1,"-1/2*RootSum[b - a*#1^4 + #1^8 & , (-2*a*Log[x] + 2*a*Log[(b*x^2 + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1) + 2*#1^5) & ]","A",1
1396,1,101,100,0.0019365,"\int \frac{-b+a x^2}{\left(b+a x^2+x^4\right) \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[(-b + a*x^2)/((b + a*x^2 + x^4)*(b*x^2 + a*x^4)^(1/4)),x]","-\frac{1}{2} \text{RootSum}\left[b-a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 a \log (x)+2 a \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]","-\frac{1}{2} \text{RootSum}\left[b-a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 a \log (x)-2 a \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}-2 \text{$\#$1}^5}\&\right]",1,"-1/2*RootSum[b - a*#1^4 + #1^8 & , (-2*a*Log[x] + 2*a*Log[(b*x^2 + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1) + 2*#1^5) & ]","A",1
1397,1,100,100,0.2011079,"\int \frac{\left(-1+x^3\right) \left(1+x^3\right)^{2/3}}{x^3 \left(-1-x^3+x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)*(1 + x^3)^(2/3))/(x^3*(-1 - x^3 + x^6)),x]","-\frac{\left(1+x^3\right)^{2/3}}{2 x^2}-\frac{1}{3} \text{RootSum}\left[-1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-\text{$\#$1}+2 \text{$\#$1}^4}\&\right]","-\frac{\left(1+x^3\right)^{2/3}}{2 x^2}-\frac{1}{3} \text{RootSum}\left[-1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-\text{$\#$1}+2 \text{$\#$1}^4}\&\right]",1,"-1/2*(1 + x^3)^(2/3)/x^2 - RootSum[-1 - #1^3 + #1^6 & , (-Log[x] + Log[(1 + x^3)^(1/3) - x*#1] + Log[x]*#1^3 - Log[(1 + x^3)^(1/3) - x*#1]*#1^3)/(-#1 + 2*#1^4) & ]/3","A",1
1398,1,100,100,0.0011462,"\int \frac{\left(-1+x^3\right) \left(1+x^3\right)^{2/3}}{x^3 \left(-1-x^3+x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)*(1 + x^3)^(2/3))/(x^3*(-1 - x^3 + x^6)),x]","-\frac{\left(1+x^3\right)^{2/3}}{2 x^2}-\frac{1}{3} \text{RootSum}\left[-1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-\text{$\#$1}+2 \text{$\#$1}^4}\&\right]","-\frac{\left(1+x^3\right)^{2/3}}{2 x^2}-\frac{1}{3} \text{RootSum}\left[-1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-\text{$\#$1}+2 \text{$\#$1}^4}\&\right]",1,"-1/2*(1 + x^3)^(2/3)/x^2 - RootSum[-1 - #1^3 + #1^6 & , (-Log[x] + Log[(1 + x^3)^(1/3) - x*#1] + Log[x]*#1^3 - Log[(1 + x^3)^(1/3) - x*#1]*#1^3)/(-#1 + 2*#1^4) & ]/3","A",1
1399,1,100,100,17.7925559,"\int \frac{\sqrt[3]{1+2 x^7} \left(-3+8 x^7\right)}{x^2 \left(1+x^3+2 x^7\right)} \, dx","IntegrateAlgebraic[((1 + 2*x^7)^(1/3)*(-3 + 8*x^7))/(x^2*(1 + x^3 + 2*x^7)),x]","\frac{3 \sqrt[3]{1+2 x^7}}{x}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{1+2 x^7}}\right)-\log \left(x+\sqrt[3]{1+2 x^7}\right)+\frac{1}{2} \log \left(x^2-x \sqrt[3]{1+2 x^7}+\left(1+2 x^7\right)^{2/3}\right)","\frac{3 \sqrt[3]{1+2 x^7}}{x}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{1+2 x^7}}\right)-\log \left(x+\sqrt[3]{1+2 x^7}\right)+\frac{1}{2} \log \left(x^2-x \sqrt[3]{1+2 x^7}+\left(1+2 x^7\right)^{2/3}\right)",1,"(3*(1 + 2*x^7)^(1/3))/x + Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(1 + 2*x^7)^(1/3))] - Log[x + (1 + 2*x^7)^(1/3)] + Log[x^2 - x*(1 + 2*x^7)^(1/3) + (1 + 2*x^7)^(2/3)]/2","A",1
1400,1,100,100,0.9231065,"\int \frac{-b+2 a x^4}{\sqrt[4]{b+a x^4} \left(-b+a x^4+x^8\right)} \, dx","IntegrateAlgebraic[(-b + 2*a*x^4)/((b + a*x^4)^(1/4)*(-b + a*x^4 + x^8)),x]","\frac{1}{4} \text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-3 a \log (x)+3 a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]","\frac{1}{4} \text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{3 a \log (x)-3 a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{3 a \text{$\#$1}-2 \text{$\#$1}^5}\&\right]",1,"RootSum[2*a^2 - b - 3*a*#1^4 + #1^8 & , (-3*a*Log[x] + 3*a*Log[(b + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(b + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1 + 2*#1^5) & ]/4","A",1
1401,1,100,100,0.0014979,"\int \frac{-b+2 a x^4}{\sqrt[4]{b+a x^4} \left(-b+a x^4+x^8\right)} \, dx","IntegrateAlgebraic[(-b + 2*a*x^4)/((b + a*x^4)^(1/4)*(-b + a*x^4 + x^8)),x]","\frac{1}{4} \text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-3 a \log (x)+3 a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]","\frac{1}{4} \text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{3 a \log (x)-3 a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{3 a \text{$\#$1}-2 \text{$\#$1}^5}\&\right]",1,"RootSum[2*a^2 - b - 3*a*#1^4 + #1^8 & , (-3*a*Log[x] + 3*a*Log[(b + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(b + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1 + 2*#1^5) & ]/4","A",1
1402,1,100,100,0.3574485,"\int \frac{\left(-1+3 x^4\right) \sqrt{1+x+2 x^4+x^5+x^8}}{x^2 \left(4+x+4 x^4\right)} \, dx","IntegrateAlgebraic[((-1 + 3*x^4)*Sqrt[1 + x + 2*x^4 + x^5 + x^8])/(x^2*(4 + x + 4*x^4)),x]","\frac{\sqrt{1+x+2 x^4+x^5+x^8}}{4 x}-\frac{1}{8} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt{1+x+2 x^4+x^5+x^8}}{1+x+x^4}\right)+\frac{1}{8} \tanh ^{-1}\left(\frac{\sqrt{1+x+2 x^4+x^5+x^8}}{1+x+x^4}\right)","\frac{\sqrt{1+x+2 x^4+x^5+x^8}}{4 x}-\frac{1}{8} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt{1+x+2 x^4+x^5+x^8}}{1+x+x^4}\right)+\frac{1}{8} \tanh ^{-1}\left(\frac{\sqrt{1+x+2 x^4+x^5+x^8}}{1+x+x^4}\right)",1,"Sqrt[1 + x + 2*x^4 + x^5 + x^8]/(4*x) - (Sqrt[3]*ArcTan[(Sqrt[3]*Sqrt[1 + x + 2*x^4 + x^5 + x^8])/(1 + x + x^4)])/8 + ArcTanh[Sqrt[1 + x + 2*x^4 + x^5 + x^8]/(1 + x + x^4)]/8","A",1
1403,1,100,100,1.047348,"\int \frac{-b+c x^4+a x^8}{x^2 \left(-b+a x^4\right)^{3/4}} \, dx","IntegrateAlgebraic[(-b + c*x^4 + a*x^8)/(x^2*(-b + a*x^4)^(3/4)),x]","\frac{\left(-4+x^4\right) \sqrt[4]{-b+a x^4}}{4 x}+\frac{(-3 b-4 c) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{8 a^{3/4}}+\frac{(3 b+4 c) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{8 a^{3/4}}","\frac{\left(-4+x^4\right) \sqrt[4]{-b+a x^4}}{4 x}+\frac{(-3 b-4 c) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{8 a^{3/4}}+\frac{(3 b+4 c) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{8 a^{3/4}}",1,"((-4 + x^4)*(-b + a*x^4)^(1/4))/(4*x) + ((-3*b - 4*c)*ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)])/(8*a^(3/4)) + ((3*b + 4*c)*ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)])/(8*a^(3/4))","A",1
1404,1,100,100,0.1959623,"\int \frac{1}{(1+x) \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[1/((1 + x)*Sqrt[x + Sqrt[1 + x^2]]),x]","\frac{2}{\sqrt{x+\sqrt{1+x^2}}}+2 \sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\sqrt{-1+\sqrt{2}} \sqrt{x+\sqrt{1+x^2}}\right)-2 \sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\sqrt{1+\sqrt{2}} \sqrt{x+\sqrt{1+x^2}}\right)","\frac{2}{\sqrt{x+\sqrt{1+x^2}}}+2 \sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{x+\sqrt{1+x^2}}}{\sqrt{1+\sqrt{2}}}\right)-2 \sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{x+\sqrt{1+x^2}}}{\sqrt{-1+\sqrt{2}}}\right)",1,"2/Sqrt[x + Sqrt[1 + x^2]] + 2*Sqrt[-1 + Sqrt[2]]*ArcTan[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] - 2*Sqrt[1 + Sqrt[2]]*ArcTanh[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]]","A",1
1405,1,100,100,0.1209124,"\int \frac{1}{x \sqrt{a x+\sqrt{b^2+a^2 x^2}}} \, dx","IntegrateAlgebraic[1/(x*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]),x]","\frac{2}{\sqrt{a x+\sqrt{b^2+a^2 x^2}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{b}}\right)}{\sqrt{b}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{b}}\right)}{\sqrt{b}}","\frac{2}{\sqrt{a x+\sqrt{b^2+a^2 x^2}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{b}}\right)}{\sqrt{b}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{b}}\right)}{\sqrt{b}}",1,"2/Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] + (2*ArcTan[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]/Sqrt[b]])/Sqrt[b] - (2*ArcTanh[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]/Sqrt[b]])/Sqrt[b]","A",1
1406,1,101,101,0.5229255,"\int \frac{(-3+x)^6 \left(-1-x+x^2\right)^{3/2}}{-1+x} \, dx","IntegrateAlgebraic[((-3 + x)^6*(-1 - x + x^2)^(3/2))/(-1 + x),x]","\frac{\sqrt{-1-x+x^2} \left(-1245336401+4423205098 x-508033624 x^2-2700564848 x^3+2304529024 x^4-910869760 x^5+199009280 x^6-23296000 x^7+1146880 x^8\right)}{10321920}+128 \tan ^{-1}\left(1-x+\sqrt{-1-x+x^2}\right)+\frac{19451047 \log \left(1-2 x+2 \sqrt{-1-x+x^2}\right)}{65536}","\frac{\sqrt{-1-x+x^2} \left(-1245336401+4423205098 x-508033624 x^2-2700564848 x^3+2304529024 x^4-910869760 x^5+199009280 x^6-23296000 x^7+1146880 x^8\right)}{10321920}+128 \tan ^{-1}\left(1-x+\sqrt{-1-x+x^2}\right)+\frac{19451047 \log \left(1-2 x+2 \sqrt{-1-x+x^2}\right)}{65536}",1,"(Sqrt[-1 - x + x^2]*(-1245336401 + 4423205098*x - 508033624*x^2 - 2700564848*x^3 + 2304529024*x^4 - 910869760*x^5 + 199009280*x^6 - 23296000*x^7 + 1146880*x^8))/10321920 + 128*ArcTan[1 - x + Sqrt[-1 - x + x^2]] + (19451047*Log[1 - 2*x + 2*Sqrt[-1 - x + x^2]])/65536","A",1
1407,1,101,101,0.1499961,"\int \frac{\left(-1+x^3\right) \sqrt[3]{1+x^3}}{x^5} \, dx","IntegrateAlgebraic[((-1 + x^3)*(1 + x^3)^(1/3))/x^5,x]","\frac{\left(1-3 x^3\right) \sqrt[3]{1+x^3}}{4 x^4}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(-x+\sqrt[3]{1+x^3}\right)+\frac{1}{6} \log \left(x^2+x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)","\frac{\left(1-3 x^3\right) \sqrt[3]{1+x^3}}{4 x^4}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(-x+\sqrt[3]{1+x^3}\right)+\frac{1}{6} \log \left(x^2+x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)",1,"((1 - 3*x^3)*(1 + x^3)^(1/3))/(4*x^4) - ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^3)^(1/3))]/Sqrt[3] - Log[-x + (1 + x^3)^(1/3)]/3 + Log[x^2 + x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/6","A",1
1408,1,101,101,0.1791236,"\int \frac{\left(-1+x^3\right) \sqrt[3]{1+x^3}}{x^2} \, dx","IntegrateAlgebraic[((-1 + x^3)*(1 + x^3)^(1/3))/x^2,x]","\frac{\sqrt[3]{1+x^3} \left(3+x^3\right)}{3 x}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3}}\right)}{3 \sqrt{3}}+\frac{2}{9} \log \left(-x+\sqrt[3]{1+x^3}\right)-\frac{1}{9} \log \left(x^2+x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)","\frac{\sqrt[3]{1+x^3} \left(3+x^3\right)}{3 x}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3}}\right)}{3 \sqrt{3}}+\frac{2}{9} \log \left(-x+\sqrt[3]{1+x^3}\right)-\frac{1}{9} \log \left(x^2+x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)",1,"((1 + x^3)^(1/3)*(3 + x^3))/(3*x) + (2*ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^3)^(1/3))])/(3*Sqrt[3]) + (2*Log[-x + (1 + x^3)^(1/3)])/9 - Log[x^2 + x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/9","A",1
1409,1,101,101,0.1542566,"\int \frac{\sqrt[3]{-1+x^3} \left(1+x^3\right)}{x^5} \, dx","IntegrateAlgebraic[((-1 + x^3)^(1/3)*(1 + x^3))/x^5,x]","\frac{\left(-1-3 x^3\right) \sqrt[3]{-1+x^3}}{4 x^4}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(-x+\sqrt[3]{-1+x^3}\right)+\frac{1}{6} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)","\frac{\left(-1-3 x^3\right) \sqrt[3]{-1+x^3}}{4 x^4}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(-x+\sqrt[3]{-1+x^3}\right)+\frac{1}{6} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)",1,"((-1 - 3*x^3)*(-1 + x^3)^(1/3))/(4*x^4) - ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^3)^(1/3))]/Sqrt[3] - Log[-x + (-1 + x^3)^(1/3)]/3 + Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)]/6","A",1
1410,1,101,101,0.3098338,"\int \frac{x}{\left(1+x^2\right) \sqrt{-x-x^2+x^3}} \, dx","IntegrateAlgebraic[x/((1 + x^2)*Sqrt[-x - x^2 + x^3]),x]","-\frac{1}{2} \sqrt{-\frac{1}{5}-\frac{2 i}{5}} \tan ^{-1}\left(\frac{\sqrt{1-2 i} \sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)-\frac{1}{2} \sqrt{-\frac{1}{5}+\frac{2 i}{5}} \tan ^{-1}\left(\frac{\sqrt{1+2 i} \sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)","-\frac{1}{2} \sqrt{-\frac{1}{5}-\frac{2 i}{5}} \tan ^{-1}\left(\frac{\sqrt{1-2 i} \sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)-\frac{1}{2} \sqrt{-\frac{1}{5}+\frac{2 i}{5}} \tan ^{-1}\left(\frac{\sqrt{1+2 i} \sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)",1,"-1/2*(Sqrt[-1/5 - (2*I)/5]*ArcTan[(Sqrt[1 - 2*I]*Sqrt[-x - x^2 + x^3])/(-1 - x + x^2)]) - (Sqrt[-1/5 + (2*I)/5]*ArcTan[(Sqrt[1 + 2*I]*Sqrt[-x - x^2 + x^3])/(-1 - x + x^2)])/2","A",1
1411,1,101,101,0.3161286,"\int \frac{-1+x+x^2}{\left(1+x^2\right) \sqrt{-x-x^2+x^3}} \, dx","IntegrateAlgebraic[(-1 + x + x^2)/((1 + x^2)*Sqrt[-x - x^2 + x^3]),x]","-\frac{1}{2} \sqrt{\frac{11}{5}+\frac{2 i}{5}} \tan ^{-1}\left(\frac{\sqrt{1-2 i} \sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)-\frac{1}{2} \sqrt{\frac{11}{5}-\frac{2 i}{5}} \tan ^{-1}\left(\frac{\sqrt{1+2 i} \sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)","-\frac{1}{2} \sqrt{\frac{11}{5}+\frac{2 i}{5}} \tan ^{-1}\left(\frac{\sqrt{1-2 i} \sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)-\frac{1}{2} \sqrt{\frac{11}{5}-\frac{2 i}{5}} \tan ^{-1}\left(\frac{\sqrt{1+2 i} \sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)",1,"-1/2*(Sqrt[11/5 + (2*I)/5]*ArcTan[(Sqrt[1 - 2*I]*Sqrt[-x - x^2 + x^3])/(-1 - x + x^2)]) - (Sqrt[11/5 - (2*I)/5]*ArcTan[(Sqrt[1 + 2*I]*Sqrt[-x - x^2 + x^3])/(-1 - x + x^2)])/2","A",1
1412,1,101,101,0.1707856,"\int \frac{x}{\sqrt[3]{x^2+x^3}} \, dx","IntegrateAlgebraic[x/(x^2 + x^3)^(1/3),x]","\frac{\left(x^2+x^3\right)^{2/3}}{x}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^3}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(-x+\sqrt[3]{x^2+x^3}\right)-\frac{1}{6} \log \left(x^2+x \sqrt[3]{x^2+x^3}+\left(x^2+x^3\right)^{2/3}\right)","\frac{\left(x^2+x^3\right)^{2/3}}{x}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^3}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(-x+\sqrt[3]{x^2+x^3}\right)-\frac{1}{6} \log \left(x^2+x \sqrt[3]{x^2+x^3}+\left(x^2+x^3\right)^{2/3}\right)",1,"(x^2 + x^3)^(2/3)/x - ArcTan[(Sqrt[3]*x)/(x + 2*(x^2 + x^3)^(1/3))]/Sqrt[3] + Log[-x + (x^2 + x^3)^(1/3)]/3 - Log[x^2 + x*(x^2 + x^3)^(1/3) + (x^2 + x^3)^(2/3)]/6","A",1
1413,1,101,101,0.3486247,"\int \frac{1}{x^6 \left(1+x^3\right) \sqrt[3]{x^2+x^3}} \, dx","IntegrateAlgebraic[1/(x^6*(1 + x^3)*(x^2 + x^3)^(1/3)),x]","\frac{\left(x^2+x^3\right)^{2/3} \left(-9240+660 x-900 x^2+20985 x^3-6357 x^4+19071 x^5+109573 x^6\right)}{52360 x^7 (1+x)}-\frac{1}{3} \text{RootSum}\left[3-3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{\left(x^2+x^3\right)^{2/3} \left(-9240+660 x-900 x^2+20985 x^3-6357 x^4+19071 x^5+109573 x^6\right)}{52360 x^7 (1+x)}-\frac{1}{3} \text{RootSum}\left[3-3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"((x^2 + x^3)^(2/3)*(-9240 + 660*x - 900*x^2 + 20985*x^3 - 6357*x^4 + 19071*x^5 + 109573*x^6))/(52360*x^7*(1 + x)) - RootSum[3 - 3*#1^3 + #1^6 & , (-Log[x] + Log[(x^2 + x^3)^(1/3) - x*#1])/#1 & ]/3","A",1
1414,1,101,101,0.0013874,"\int \frac{1}{x^6 \left(1+x^3\right) \sqrt[3]{x^2+x^3}} \, dx","IntegrateAlgebraic[1/(x^6*(1 + x^3)*(x^2 + x^3)^(1/3)),x]","\frac{\left(x^2+x^3\right)^{2/3} \left(-9240+660 x-900 x^2+20985 x^3-6357 x^4+19071 x^5+109573 x^6\right)}{52360 x^7 (1+x)}-\frac{1}{3} \text{RootSum}\left[3-3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{\left(x^2+x^3\right)^{2/3} \left(-9240+660 x-900 x^2+20985 x^3-6357 x^4+19071 x^5+109573 x^6\right)}{52360 x^7 (1+x)}-\frac{1}{3} \text{RootSum}\left[3-3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"((x^2 + x^3)^(2/3)*(-9240 + 660*x - 900*x^2 + 20985*x^3 - 6357*x^4 + 19071*x^5 + 109573*x^6))/(52360*x^7*(1 + x)) - RootSum[3 - 3*#1^3 + #1^6 & , (-Log[x] + Log[(x^2 + x^3)^(1/3) - x*#1])/#1 & ]/3","A",1
1415,1,101,101,0.1505865,"\int \frac{\sqrt[3]{-1+x^3} \left(-1+2 x^3\right)}{x^5} \, dx","IntegrateAlgebraic[((-1 + x^3)^(1/3)*(-1 + 2*x^3))/x^5,x]","\frac{\left(1-9 x^3\right) \sqrt[3]{-1+x^3}}{4 x^4}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{\sqrt{3}}-\frac{2}{3} \log \left(-x+\sqrt[3]{-1+x^3}\right)+\frac{1}{3} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)","\frac{\left(1-9 x^3\right) \sqrt[3]{-1+x^3}}{4 x^4}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{\sqrt{3}}-\frac{2}{3} \log \left(-x+\sqrt[3]{-1+x^3}\right)+\frac{1}{3} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)",1,"((1 - 9*x^3)*(-1 + x^3)^(1/3))/(4*x^4) - (2*ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^3)^(1/3))])/Sqrt[3] - (2*Log[-x + (-1 + x^3)^(1/3)])/3 + Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)]/3","A",1
1416,1,101,101,0.1850355,"\int \frac{\left(-1+x^3\right)^{2/3} \left(-1+3 x^3\right)}{x^6 \left(-1+2 x^3\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(-1 + 3*x^3))/(x^6*(-1 + 2*x^3)),x]","\frac{\left(-1+x^3\right)^{2/3} \left(-2+7 x^3\right)}{10 x^5}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-1+x^3}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(x+\sqrt[3]{-1+x^3}\right)-\frac{1}{6} \log \left(x^2-x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)","\frac{\left(-1+x^3\right)^{2/3} \left(-2+7 x^3\right)}{10 x^5}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-1+x^3}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(x+\sqrt[3]{-1+x^3}\right)-\frac{1}{6} \log \left(x^2-x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)",1,"((-1 + x^3)^(2/3)*(-2 + 7*x^3))/(10*x^5) + ArcTan[(Sqrt[3]*x)/(-x + 2*(-1 + x^3)^(1/3))]/Sqrt[3] + Log[x + (-1 + x^3)^(1/3)]/3 - Log[x^2 - x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)]/6","A",1
1417,1,101,101,0.4252658,"\int \frac{x^4}{\left(-1+x^4\right)^2 \sqrt[4]{x^2+x^4}} \, dx","IntegrateAlgebraic[x^4/((-1 + x^4)^2*(x^2 + x^4)^(1/4)),x]","\frac{\left(-5-2 x^2-13 x^4\right) \left(x^2+x^4\right)^{3/4}}{80 x \left(-1+x^2\right) \left(1+x^2\right)^2}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{32 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{32 \sqrt[4]{2}}","\frac{\left(-5-2 x^2-13 x^4\right) \left(x^2+x^4\right)^{3/4}}{80 x \left(-1+x^2\right) \left(1+x^2\right)^2}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{32 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{32 \sqrt[4]{2}}",1,"((-5 - 2*x^2 - 13*x^4)*(x^2 + x^4)^(3/4))/(80*x*(-1 + x^2)*(1 + x^2)^2) - ArcTan[(2^(1/4)*x)/(x^2 + x^4)^(1/4)]/(32*2^(1/4)) - ArcTanh[(2^(1/4)*x)/(x^2 + x^4)^(1/4)]/(32*2^(1/4))","A",1
1418,1,101,101,0.3847337,"\int \frac{\left(1+x^2\right) \sqrt[4]{x^3+x^4}}{x^2 \left(-1+x^2\right)} \, dx","IntegrateAlgebraic[((1 + x^2)*(x^3 + x^4)^(1/4))/(x^2*(-1 + x^2)),x]","\frac{4 \sqrt[4]{x^3+x^4}}{x}-2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)+2 \sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^4}}\right)+2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)-2 \sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^4}}\right)","\frac{4 \sqrt[4]{x^3+x^4}}{x}-2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)+2 \sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^4}}\right)+2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)-2 \sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^4}}\right)",1,"(4*(x^3 + x^4)^(1/4))/x - 2*ArcTan[x/(x^3 + x^4)^(1/4)] + 2*2^(1/4)*ArcTan[(2^(1/4)*x)/(x^3 + x^4)^(1/4)] + 2*ArcTanh[x/(x^3 + x^4)^(1/4)] - 2*2^(1/4)*ArcTanh[(2^(1/4)*x)/(x^3 + x^4)^(1/4)]","A",1
1419,1,101,101,4.1234992,"\int \frac{\left(4+x^3\right) \left(-1-x^3+x^4\right)}{x^2 \left(1+x^3\right)^{3/4} \left(1+x^3+x^4\right)} \, dx","IntegrateAlgebraic[((4 + x^3)*(-1 - x^3 + x^4))/(x^2*(1 + x^3)^(3/4)*(1 + x^3 + x^4)),x]","\frac{4 \sqrt[4]{1+x^3}}{x}-2 \sqrt{2} \tan ^{-1}\left(\frac{-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^3}}{\sqrt{2}}}{x \sqrt[4]{1+x^3}}\right)-2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{1+x^3}}{x^2+\sqrt{1+x^3}}\right)","\frac{4 \sqrt[4]{1+x^3}}{x}-2 \sqrt{2} \tan ^{-1}\left(\frac{-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^3}}{\sqrt{2}}}{x \sqrt[4]{1+x^3}}\right)-2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{1+x^3}}{x^2+\sqrt{1+x^3}}\right)",1,"(4*(1 + x^3)^(1/4))/x - 2*Sqrt[2]*ArcTan[(-(x^2/Sqrt[2]) + Sqrt[1 + x^3]/Sqrt[2])/(x*(1 + x^3)^(1/4))] - 2*Sqrt[2]*ArcTanh[(Sqrt[2]*x*(1 + x^3)^(1/4))/(x^2 + Sqrt[1 + x^3])]","A",1
1420,1,101,101,0.3885857,"\int \frac{1+x^2+2 x^4}{\sqrt[4]{1+x^2} \left(2+3 x^2+x^4\right)} \, dx","IntegrateAlgebraic[(1 + x^2 + 2*x^4)/((1 + x^2)^(1/4)*(2 + 3*x^2 + x^4)),x]","\frac{4 x}{\sqrt[4]{1+x^2}}+\frac{7}{4} \tan ^{-1}\left(\frac{-x+\sqrt[4]{1+x^2}}{\sqrt[4]{1+x^2}}\right)-\frac{7}{4} \tan ^{-1}\left(\frac{x+\sqrt[4]{1+x^2}}{\sqrt[4]{1+x^2}}\right)-\frac{7}{4} \tanh ^{-1}\left(\frac{2 x \sqrt[4]{1+x^2}}{x^2+2 \sqrt{1+x^2}}\right)","\frac{4 x}{\sqrt[4]{1+x^2}}+\frac{7}{4} \tan ^{-1}\left(\frac{-x+\sqrt[4]{1+x^2}}{\sqrt[4]{1+x^2}}\right)-\frac{7}{4} \tan ^{-1}\left(\frac{x+\sqrt[4]{1+x^2}}{\sqrt[4]{1+x^2}}\right)-\frac{7}{4} \tanh ^{-1}\left(\frac{2 x \sqrt[4]{1+x^2}}{x^2+2 \sqrt{1+x^2}}\right)",1,"(4*x)/(1 + x^2)^(1/4) + (7*ArcTan[(-x + (1 + x^2)^(1/4))/(1 + x^2)^(1/4)])/4 - (7*ArcTan[(x + (1 + x^2)^(1/4))/(1 + x^2)^(1/4)])/4 - (7*ArcTanh[(2*x*(1 + x^2)^(1/4))/(x^2 + 2*Sqrt[1 + x^2])])/4","A",1
1421,1,101,101,0.3344216,"\int \frac{\left(-b+a x^2\right) \sqrt[4]{-b x^2+a x^4}}{x^2} \, dx","IntegrateAlgebraic[((-b + a*x^2)*(-(b*x^2) + a*x^4)^(1/4))/x^2,x]","\frac{\left(4 b+a x^2\right) \sqrt[4]{-b x^2+a x^4}}{2 x}+\frac{5}{4} \sqrt[4]{a} b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)-\frac{5}{4} \sqrt[4]{a} b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)","\frac{\left(4 b+a x^2\right) \sqrt[4]{-b x^2+a x^4}}{2 x}+\frac{5}{4} \sqrt[4]{a} b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)-\frac{5}{4} \sqrt[4]{a} b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)",1,"((4*b + a*x^2)*(-(b*x^2) + a*x^4)^(1/4))/(2*x) + (5*a^(1/4)*b*ArcTan[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/4 - (5*a^(1/4)*b*ArcTanh[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/4","A",1
1422,1,101,101,1.4107033,"\int \frac{b+a x^4}{x^4 \left(2 b+a x^4\right) \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[(b + a*x^4)/(x^4*(2*b + a*x^4)*(b*x^2 + a*x^4)^(1/4)),x]","\frac{\left(-3 b+4 a x^2\right) \left(b x^2+a x^4\right)^{3/4}}{21 b^2 x^5}-\frac{a \text{RootSum}\left[2 a^2+a b-4 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{16 b}","\frac{\left(-3 b+4 a x^2\right) \left(b x^2+a x^4\right)^{3/4}}{21 b^2 x^5}-\frac{a \text{RootSum}\left[2 a^2+a b-4 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{16 b}",1,"((-3*b + 4*a*x^2)*(b*x^2 + a*x^4)^(3/4))/(21*b^2*x^5) - (a*RootSum[2*a^2 + a*b - 4*a*#1^4 + 2*#1^8 & , (-Log[x] + Log[(b*x^2 + a*x^4)^(1/4) - x*#1])/#1 & ])/(16*b)","A",1
1423,1,101,101,0.0022868,"\int \frac{b+a x^4}{x^4 \left(2 b+a x^4\right) \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[(b + a*x^4)/(x^4*(2*b + a*x^4)*(b*x^2 + a*x^4)^(1/4)),x]","\frac{\left(-3 b+4 a x^2\right) \left(b x^2+a x^4\right)^{3/4}}{21 b^2 x^5}-\frac{a \text{RootSum}\left[2 a^2+a b-4 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{16 b}","\frac{\left(-3 b+4 a x^2\right) \left(b x^2+a x^4\right)^{3/4}}{21 b^2 x^5}-\frac{a \text{RootSum}\left[2 a^2+a b-4 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{16 b}",1,"((-3*b + 4*a*x^2)*(b*x^2 + a*x^4)^(3/4))/(21*b^2*x^5) - (a*RootSum[2*a^2 + a*b - 4*a*#1^4 + 2*#1^8 & , (-Log[x] + Log[(b*x^2 + a*x^4)^(1/4) - x*#1])/#1 & ])/(16*b)","A",1
1424,1,105,101,0.3209057,"\int \frac{1-x^4}{x^2 \sqrt{a+b x+c x^2+b x^3+a x^4}} \, dx","IntegrateAlgebraic[(1 - x^4)/(x^2*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]),x]","-\frac{\sqrt{a+b x+c x^2+b x^3+a x^4}}{a x}+\frac{b \log (x)}{2 a^{3/2}}-\frac{b \log \left(2 a^2+a b x+2 a^2 x^2-2 a^{3/2} \sqrt{a+b x+c x^2+b x^3+a x^4}\right)}{2 a^{3/2}}","-\frac{\sqrt{a+b x+c x^2+b x^3+a x^4}}{a x}+\frac{b \log (x)}{2 a^{3/2}}-\frac{b \log \left(-2 a-b x-2 a x^2+2 \sqrt{a} \sqrt{a+b x+c x^2+b x^3+a x^4}\right)}{2 a^{3/2}}",1,"-(Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]/(a*x)) + (b*Log[x])/(2*a^(3/2)) - (b*Log[2*a^2 + a*b*x + 2*a^2*x^2 - 2*a^(3/2)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]])/(2*a^(3/2))","A",1
1425,1,101,101,0.4820935,"\int \frac{-1+x^2}{\left(1+x^2\right) \sqrt[4]{x^2+x^6}} \, dx","IntegrateAlgebraic[(-1 + x^2)/((1 + x^2)*(x^2 + x^6)^(1/4)),x]","\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{2^{3/4}}","\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{2^{3/4}}",1,"ArcTan[(2^(3/4)*x*(x^2 + x^6)^(1/4))/(Sqrt[2]*x^2 - Sqrt[x^2 + x^6])]/2^(3/4) - ArcTanh[(x^2/2^(1/4) + Sqrt[x^2 + x^6]/2^(3/4))/(x*(x^2 + x^6)^(1/4))]/2^(3/4)","A",1
1426,1,101,101,0.0011691,"\int \frac{-1+x^2}{\left(1+x^2\right) \sqrt[4]{x^2+x^6}} \, dx","IntegrateAlgebraic[(-1 + x^2)/((1 + x^2)*(x^2 + x^6)^(1/4)),x]","\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{2^{3/4}}","\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{2^{3/4}}",1,"ArcTan[(2^(3/4)*x*(x^2 + x^6)^(1/4))/(Sqrt[2]*x^2 - Sqrt[x^2 + x^6])]/2^(3/4) - ArcTanh[(x^2/2^(1/4) + Sqrt[x^2 + x^6]/2^(3/4))/(x*(x^2 + x^6)^(1/4))]/2^(3/4)","A",1
1427,1,101,101,0.5764356,"\int \frac{\sqrt[4]{-x^3+x^4} \left(-1+x^8\right)}{x^4} \, dx","IntegrateAlgebraic[((-x^3 + x^4)^(1/4)*(-1 + x^8))/x^4,x]","\frac{\sqrt[4]{-x^3+x^4} \left(327680-65536 x-262144 x^2-21945 x^3-12540 x^4-9120 x^5-7296 x^6-6144 x^7+122880 x^8\right)}{737280 x^3}+\frac{1463 \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)}{32768}-\frac{1463 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)}{32768}","\frac{\sqrt[4]{-x^3+x^4} \left(327680-65536 x-262144 x^2-21945 x^3-12540 x^4-9120 x^5-7296 x^6-6144 x^7+122880 x^8\right)}{737280 x^3}+\frac{1463 \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)}{32768}-\frac{1463 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)}{32768}",1,"((-x^3 + x^4)^(1/4)*(327680 - 65536*x - 262144*x^2 - 21945*x^3 - 12540*x^4 - 9120*x^5 - 7296*x^6 - 6144*x^7 + 122880*x^8))/(737280*x^3) + (1463*ArcTan[x/(-x^3 + x^4)^(1/4)])/32768 - (1463*ArcTanh[x/(-x^3 + x^4)^(1/4)])/32768","A",1
1428,1,101,101,0.224341,"\int \frac{\sqrt[4]{1+x^4} \left(2+x^4\right)}{x^2 \left(-1+2 x^4+x^8\right)} \, dx","IntegrateAlgebraic[((1 + x^4)^(1/4)*(2 + x^4))/(x^2*(-1 + 2*x^4 + x^8)),x]","\frac{2 \sqrt[4]{1+x^4}}{x}-\frac{1}{8} \text{RootSum}\left[2-4 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-4 \log (x)+4 \log \left(\sqrt[4]{1+x^4}-x \text{$\#$1}\right)+7 \log (x) \text{$\#$1}^4-7 \log \left(\sqrt[4]{1+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-2 \text{$\#$1}^3+\text{$\#$1}^7}\&\right]","\frac{2 \sqrt[4]{1+x^4}}{x}-\frac{1}{8} \text{RootSum}\left[2-4 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-4 \log (x)+4 \log \left(\sqrt[4]{1+x^4}-x \text{$\#$1}\right)+7 \log (x) \text{$\#$1}^4-7 \log \left(\sqrt[4]{1+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-2 \text{$\#$1}^3+\text{$\#$1}^7}\&\right]",1,"(2*(1 + x^4)^(1/4))/x - RootSum[2 - 4*#1^4 + #1^8 & , (-4*Log[x] + 4*Log[(1 + x^4)^(1/4) - x*#1] + 7*Log[x]*#1^4 - 7*Log[(1 + x^4)^(1/4) - x*#1]*#1^4)/(-2*#1^3 + #1^7) & ]/8","A",1
1429,1,101,101,0.0008757,"\int \frac{\sqrt[4]{1+x^4} \left(2+x^4\right)}{x^2 \left(-1+2 x^4+x^8\right)} \, dx","IntegrateAlgebraic[((1 + x^4)^(1/4)*(2 + x^4))/(x^2*(-1 + 2*x^4 + x^8)),x]","\frac{2 \sqrt[4]{1+x^4}}{x}-\frac{1}{8} \text{RootSum}\left[2-4 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-4 \log (x)+4 \log \left(\sqrt[4]{1+x^4}-x \text{$\#$1}\right)+7 \log (x) \text{$\#$1}^4-7 \log \left(\sqrt[4]{1+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-2 \text{$\#$1}^3+\text{$\#$1}^7}\&\right]","\frac{2 \sqrt[4]{1+x^4}}{x}-\frac{1}{8} \text{RootSum}\left[2-4 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-4 \log (x)+4 \log \left(\sqrt[4]{1+x^4}-x \text{$\#$1}\right)+7 \log (x) \text{$\#$1}^4-7 \log \left(\sqrt[4]{1+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-2 \text{$\#$1}^3+\text{$\#$1}^7}\&\right]",1,"(2*(1 + x^4)^(1/4))/x - RootSum[2 - 4*#1^4 + #1^8 & , (-4*Log[x] + 4*Log[(1 + x^4)^(1/4) - x*#1] + 7*Log[x]*#1^4 - 7*Log[(1 + x^4)^(1/4) - x*#1]*#1^4)/(-2*#1^3 + #1^7) & ]/8","A",1
1430,1,100,101,1.5014887,"\int \frac{-b+2 a x^4}{\sqrt[4]{-b+a x^4} \left(-b+a x^8\right)} \, dx","IntegrateAlgebraic[(-b + 2*a*x^4)/((-b + a*x^4)^(1/4)*(-b + a*x^8)),x]","\frac{1}{8} \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a \log (x)-a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}+\text{$\#$1}^5}\&\right]","\frac{1}{8} \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a \log (x)+a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}-\text{$\#$1}^5}\&\right]",1,"RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (a*Log[x] - a*Log[(-b + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1) + #1^5) & ]/8","A",1
1431,1,101,101,5.0272048,"\int \frac{\sqrt{1+2 x^6} \left(-1+4 x^6\right)}{2+x^4+8 x^6+8 x^{12}} \, dx","IntegrateAlgebraic[(Sqrt[1 + 2*x^6]*(-1 + 4*x^6))/(2 + x^4 + 8*x^6 + 8*x^12),x]","-\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt{1+2 x^6}}{\sqrt{2}-x^2+2 \sqrt{2} x^6}\right)}{4 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{2^{3/4} x \sqrt{1+2 x^6}}{\sqrt{2}+x^2+2 \sqrt{2} x^6}\right)}{4 \sqrt[4]{2}}","-\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt{1+2 x^6}}{\sqrt{2}-x^2+2 \sqrt{2} x^6}\right)}{4 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{2^{3/4} x \sqrt{1+2 x^6}}{\sqrt{2}+x^2+2 \sqrt{2} x^6}\right)}{4 \sqrt[4]{2}}",1,"-1/4*ArcTan[(2^(3/4)*x*Sqrt[1 + 2*x^6])/(Sqrt[2] - x^2 + 2*Sqrt[2]*x^6)]/2^(1/4) - ArcTanh[(2^(3/4)*x*Sqrt[1 + 2*x^6])/(Sqrt[2] + x^2 + 2*Sqrt[2]*x^6)]/(4*2^(1/4))","A",1
1432,1,86,101,0.0913632,"\int \frac{\left(-1+x^2\right) \sqrt{1+\sqrt{1+x}}}{1+x^2} \, dx","IntegrateAlgebraic[((-1 + x^2)*Sqrt[1 + Sqrt[1 + x]])/(1 + x^2),x]","\frac{4}{15} \sqrt{1+\sqrt{1+x}} \left(-2+\sqrt{1+x}+3 (1+x)\right)-\text{RootSum}\left[1+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)}{-2 \text{$\#$1}+\text{$\#$1}^3}\&\right]","\frac{4}{15} \sqrt{1+x} \sqrt{1+\sqrt{1+x}}+\frac{4}{15} (1+3 x) \sqrt{1+\sqrt{1+x}}-\text{RootSum}\left[1+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)}{-2 \text{$\#$1}+\text{$\#$1}^3}\&\right]",1,"(4*Sqrt[1 + Sqrt[1 + x]]*(-2 + Sqrt[1 + x] + 3*(1 + x)))/15 - RootSum[1 + 4*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[1 + x]] - #1]/(-2*#1 + #1^3) & ]","A",1
1433,1,86,101,0.0010784,"\int \frac{\left(-1+x^2\right) \sqrt{1+\sqrt{1+x}}}{1+x^2} \, dx","IntegrateAlgebraic[((-1 + x^2)*Sqrt[1 + Sqrt[1 + x]])/(1 + x^2),x]","\frac{4}{15} \sqrt{1+\sqrt{1+x}} \left(-2+\sqrt{1+x}+3 (1+x)\right)-\text{RootSum}\left[1+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)}{-2 \text{$\#$1}+\text{$\#$1}^3}\&\right]","\frac{4}{15} \sqrt{1+x} \sqrt{1+\sqrt{1+x}}+\frac{4}{15} (1+3 x) \sqrt{1+\sqrt{1+x}}-\text{RootSum}\left[1+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)}{-2 \text{$\#$1}+\text{$\#$1}^3}\&\right]",1,"(4*Sqrt[1 + Sqrt[1 + x]]*(-2 + Sqrt[1 + x] + 3*(1 + x)))/15 - RootSum[1 + 4*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[1 + x]] - #1]/(-2*#1 + #1^3) & ]","A",1
1434,1,101,101,0.3622699,"\int x^2 \sqrt{x^2+\sqrt{1+x^4}} \, dx","IntegrateAlgebraic[x^2*Sqrt[x^2 + Sqrt[1 + x^4]],x]","\frac{2 x^4 \sqrt{1+x^4}+x^2 \left(-1+2 x^4\right)}{8 x \sqrt{x^2+\sqrt{1+x^4}}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)}{4 \sqrt{2}}","\frac{2 x^4 \sqrt{1+x^4}+x^2 \left(-1+2 x^4\right)}{8 x \sqrt{x^2+\sqrt{1+x^4}}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)}{4 \sqrt{2}}",1,"(2*x^4*Sqrt[1 + x^4] + x^2*(-1 + 2*x^4))/(8*x*Sqrt[x^2 + Sqrt[1 + x^4]]) + ArcTanh[(Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2 + Sqrt[1 + x^4])]/(4*Sqrt[2])","A",1
1435,1,114,101,0.3124394,"\int \frac{\left(-1+x^4\right) \sqrt{x^2+\sqrt{1+x^4}}}{\sqrt{1+x^4}} \, dx","IntegrateAlgebraic[((-1 + x^4)*Sqrt[x^2 + Sqrt[1 + x^4]])/Sqrt[1 + x^4],x]","\frac{2 x^4 \sqrt{1+x^4}+x^2 \left(3+2 x^4\right)}{8 x \sqrt{x^2+\sqrt{1+x^4}}}-\frac{11 \tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^4}}{\sqrt{2}}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)}{4 \sqrt{2}}","\frac{2 x^4 \sqrt{1+x^4}+x^2 \left(3+2 x^4\right)}{8 x \sqrt{x^2+\sqrt{1+x^4}}}-\frac{11 \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)}{4 \sqrt{2}}",1,"(2*x^4*Sqrt[1 + x^4] + x^2*(3 + 2*x^4))/(8*x*Sqrt[x^2 + Sqrt[1 + x^4]]) - (11*ArcTanh[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])])/(4*Sqrt[2])","A",1
1436,1,101,101,0.4238376,"\int \sqrt{1+x^4} \sqrt{x^2+\sqrt{1+x^4}} \, dx","IntegrateAlgebraic[Sqrt[1 + x^4]*Sqrt[x^2 + Sqrt[1 + x^4]],x]","\frac{2 x^4 \sqrt{1+x^4}+x^2 \left(3+2 x^4\right)}{8 x \sqrt{x^2+\sqrt{1+x^4}}}+\frac{5 \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)}{4 \sqrt{2}}","\frac{2 x^4 \sqrt{1+x^4}+x^2 \left(3+2 x^4\right)}{8 x \sqrt{x^2+\sqrt{1+x^4}}}+\frac{5 \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)}{4 \sqrt{2}}",1,"(2*x^4*Sqrt[1 + x^4] + x^2*(3 + 2*x^4))/(8*x*Sqrt[x^2 + Sqrt[1 + x^4]]) + (5*ArcTanh[(Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2 + Sqrt[1 + x^4])])/(4*Sqrt[2])","A",1
1437,1,102,102,0.2007669,"\int x^3 \left(-1+x^3\right)^{2/3} \, dx","IntegrateAlgebraic[x^3*(-1 + x^3)^(2/3),x]","\frac{1}{18} \left(-1+x^3\right)^{2/3} \left(-2 x+3 x^4\right)-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{9 \sqrt{3}}+\frac{1}{27} \log \left(-x+\sqrt[3]{-1+x^3}\right)-\frac{1}{54} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)","\frac{1}{18} \left(-1+x^3\right)^{2/3} \left(-2 x+3 x^4\right)-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{9 \sqrt{3}}+\frac{1}{27} \log \left(-x+\sqrt[3]{-1+x^3}\right)-\frac{1}{54} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)",1,"((-1 + x^3)^(2/3)*(-2*x + 3*x^4))/18 - ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^3)^(1/3))]/(9*Sqrt[3]) + Log[-x + (-1 + x^3)^(1/3)]/27 - Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)]/54","A",1
1438,1,102,102,0.1809281,"\int x^4 \sqrt[3]{1+x^3} \, dx","IntegrateAlgebraic[x^4*(1 + x^3)^(1/3),x]","\frac{1}{18} \sqrt[3]{1+x^3} \left(x^2+3 x^5\right)+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3}}\right)}{9 \sqrt{3}}+\frac{1}{27} \log \left(-x+\sqrt[3]{1+x^3}\right)-\frac{1}{54} \log \left(x^2+x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)","\frac{1}{18} \sqrt[3]{1+x^3} \left(x^2+3 x^5\right)+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3}}\right)}{9 \sqrt{3}}+\frac{1}{27} \log \left(-x+\sqrt[3]{1+x^3}\right)-\frac{1}{54} \log \left(x^2+x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)",1,"((1 + x^3)^(1/3)*(x^2 + 3*x^5))/18 + ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^3)^(1/3))]/(9*Sqrt[3]) + Log[-x + (1 + x^3)^(1/3)]/27 - Log[x^2 + x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/54","A",1
1439,1,102,102,0.2064353,"\int x^3 \left(1+x^3\right)^{2/3} \, dx","IntegrateAlgebraic[x^3*(1 + x^3)^(2/3),x]","\frac{1}{18} \left(1+x^3\right)^{2/3} \left(2 x+3 x^4\right)-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3}}\right)}{9 \sqrt{3}}+\frac{1}{27} \log \left(-x+\sqrt[3]{1+x^3}\right)-\frac{1}{54} \log \left(x^2+x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)","\frac{1}{18} \left(1+x^3\right)^{2/3} \left(2 x+3 x^4\right)-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3}}\right)}{9 \sqrt{3}}+\frac{1}{27} \log \left(-x+\sqrt[3]{1+x^3}\right)-\frac{1}{54} \log \left(x^2+x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)",1,"((1 + x^3)^(2/3)*(2*x + 3*x^4))/18 - ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^3)^(1/3))]/(9*Sqrt[3]) + Log[-x + (1 + x^3)^(1/3)]/27 - Log[x^2 + x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/54","A",1
1440,1,102,102,0.1889073,"\int \frac{\left(-2+x^3\right)^{2/3} \left(4+x^3\right)}{x^6 \left(-1+x^3\right)} \, dx","IntegrateAlgebraic[((-2 + x^3)^(2/3)*(4 + x^3))/(x^6*(-1 + x^3)),x]","\frac{\left(-2+x^3\right)^{2/3} \left(8+21 x^3\right)}{10 x^5}+\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-2+x^3}}\right)}{\sqrt{3}}+\frac{5}{3} \log \left(x+\sqrt[3]{-2+x^3}\right)-\frac{5}{6} \log \left(x^2-x \sqrt[3]{-2+x^3}+\left(-2+x^3\right)^{2/3}\right)","\frac{\left(-2+x^3\right)^{2/3} \left(8+21 x^3\right)}{10 x^5}+\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-2+x^3}}\right)}{\sqrt{3}}+\frac{5}{3} \log \left(x+\sqrt[3]{-2+x^3}\right)-\frac{5}{6} \log \left(x^2-x \sqrt[3]{-2+x^3}+\left(-2+x^3\right)^{2/3}\right)",1,"((-2 + x^3)^(2/3)*(8 + 21*x^3))/(10*x^5) + (5*ArcTan[(Sqrt[3]*x)/(-x + 2*(-2 + x^3)^(1/3))])/Sqrt[3] + (5*Log[x + (-2 + x^3)^(1/3)])/3 - (5*Log[x^2 - x*(-2 + x^3)^(1/3) + (-2 + x^3)^(2/3)])/6","A",1
1441,1,102,102,0.5418447,"\int \frac{3 b+a x^2}{\left(b+a x^2+x^3\right) \sqrt[4]{b x+a x^3}} \, dx","IntegrateAlgebraic[(3*b + a*x^2)/((b + a*x^2 + x^3)*(b*x + a*x^3)^(1/4)),x]","-\sqrt{2} \tan ^{-1}\left(\frac{-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{b x+a x^3}}{\sqrt{2}}}{x \sqrt[4]{b x+a x^3}}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{b x+a x^3}}{x^2+\sqrt{b x+a x^3}}\right)","-\sqrt{2} \tan ^{-1}\left(\frac{-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{b x+a x^3}}{\sqrt{2}}}{x \sqrt[4]{b x+a x^3}}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{b x+a x^3}}{x^2+\sqrt{b x+a x^3}}\right)",1,"-(Sqrt[2]*ArcTan[(-(x^2/Sqrt[2]) + Sqrt[b*x + a*x^3]/Sqrt[2])/(x*(b*x + a*x^3)^(1/4))]) + Sqrt[2]*ArcTanh[(Sqrt[2]*x*(b*x + a*x^3)^(1/4))/(x^2 + Sqrt[b*x + a*x^3])]","A",1
1442,1,102,102,0.1236069,"\int \frac{\left(-3+x^4\right) \sqrt[3]{1+x^4}}{x^{13}} \, dx","IntegrateAlgebraic[((-3 + x^4)*(1 + x^4)^(1/3))/x^13,x]","\frac{\sqrt[3]{1+x^4} \left(9-3 x^4-4 x^8\right)}{36 x^{12}}+\frac{2 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^4}}{\sqrt{3}}\right)}{9 \sqrt{3}}-\frac{2}{27} \log \left(-1+\sqrt[3]{1+x^4}\right)+\frac{1}{27} \log \left(1+\sqrt[3]{1+x^4}+\left(1+x^4\right)^{2/3}\right)","\frac{\sqrt[3]{1+x^4} \left(9-3 x^4-4 x^8\right)}{36 x^{12}}+\frac{2 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^4}}{\sqrt{3}}\right)}{9 \sqrt{3}}-\frac{2}{27} \log \left(-1+\sqrt[3]{1+x^4}\right)+\frac{1}{27} \log \left(1+\sqrt[3]{1+x^4}+\left(1+x^4\right)^{2/3}\right)",1,"((1 + x^4)^(1/3)*(9 - 3*x^4 - 4*x^8))/(36*x^12) + (2*ArcTan[1/Sqrt[3] + (2*(1 + x^4)^(1/3))/Sqrt[3]])/(9*Sqrt[3]) - (2*Log[-1 + (1 + x^4)^(1/3)])/27 + Log[1 + (1 + x^4)^(1/3) + (1 + x^4)^(2/3)]/27","A",1
1443,1,102,102,0.29875,"\int \frac{-2 b+a x^2}{\sqrt[4]{-b+a x^2} \left(-b+a x^2+x^4\right)} \, dx","IntegrateAlgebraic[(-2*b + a*x^2)/((-b + a*x^2)^(1/4)*(-b + a*x^2 + x^4)),x]","-\frac{\tan ^{-1}\left(\frac{-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{-b+a x^2}}{\sqrt{2}}}{x \sqrt[4]{-b+a x^2}}\right)}{\sqrt{2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-b+a x^2}}{x^2+\sqrt{-b+a x^2}}\right)}{\sqrt{2}}","-\frac{\tan ^{-1}\left(\frac{-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{-b+a x^2}}{\sqrt{2}}}{x \sqrt[4]{-b+a x^2}}\right)}{\sqrt{2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-b+a x^2}}{x^2+\sqrt{-b+a x^2}}\right)}{\sqrt{2}}",1,"-(ArcTan[(-(x^2/Sqrt[2]) + Sqrt[-b + a*x^2]/Sqrt[2])/(x*(-b + a*x^2)^(1/4))]/Sqrt[2]) + ArcTanh[(Sqrt[2]*x*(-b + a*x^2)^(1/4))/(x^2 + Sqrt[-b + a*x^2])]/Sqrt[2]","A",1
1444,1,102,102,0.7563335,"\int \frac{-4 b+a x^3}{\sqrt[4]{-b+a x^3} \left(-b+a x^3+x^4\right)} \, dx","IntegrateAlgebraic[(-4*b + a*x^3)/((-b + a*x^3)^(1/4)*(-b + a*x^3 + x^4)),x]","-\sqrt{2} \tan ^{-1}\left(\frac{-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{-b+a x^3}}{\sqrt{2}}}{x \sqrt[4]{-b+a x^3}}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-b+a x^3}}{x^2+\sqrt{-b+a x^3}}\right)","-\sqrt{2} \tan ^{-1}\left(\frac{-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{-b+a x^3}}{\sqrt{2}}}{x \sqrt[4]{-b+a x^3}}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-b+a x^3}}{x^2+\sqrt{-b+a x^3}}\right)",1,"-(Sqrt[2]*ArcTan[(-(x^2/Sqrt[2]) + Sqrt[-b + a*x^3]/Sqrt[2])/(x*(-b + a*x^3)^(1/4))]) + Sqrt[2]*ArcTanh[(Sqrt[2]*x*(-b + a*x^3)^(1/4))/(x^2 + Sqrt[-b + a*x^3])]","A",1
1445,1,103,102,0.6148263,"\int \frac{b+a x^2}{\left(b-a x^2+x^4\right) \sqrt[4]{-b x^2+a x^4}} \, dx","IntegrateAlgebraic[(b + a*x^2)/((b - a*x^2 + x^4)*(-(b*x^2) + a*x^4)^(1/4)),x]","\frac{1}{2} \text{RootSum}\left[b-a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 a \log (x)+2 a \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]","\frac{1}{2} \text{RootSum}\left[b-a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 a \log (x)-2 a \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}-2 \text{$\#$1}^5}\&\right]",1,"RootSum[b - a*#1^4 + #1^8 & , (-2*a*Log[x] + 2*a*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1) + 2*#1^5) & ]/2","A",1
1446,1,103,102,0.0019189,"\int \frac{b+a x^2}{\left(b-a x^2+x^4\right) \sqrt[4]{-b x^2+a x^4}} \, dx","IntegrateAlgebraic[(b + a*x^2)/((b - a*x^2 + x^4)*(-(b*x^2) + a*x^4)^(1/4)),x]","\frac{1}{2} \text{RootSum}\left[b-a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 a \log (x)+2 a \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]","\frac{1}{2} \text{RootSum}\left[b-a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 a \log (x)-2 a \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}-2 \text{$\#$1}^5}\&\right]",1,"RootSum[b - a*#1^4 + #1^8 & , (-2*a*Log[x] + 2*a*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1) + 2*#1^5) & ]/2","A",1
1447,1,102,102,1.5267598,"\int \frac{b+a x^4}{x^4 \left(-2 b+a x^4\right) \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[(b + a*x^4)/(x^4*(-2*b + a*x^4)*(b*x^2 + a*x^4)^(1/4)),x]","\frac{\left(3 b-4 a x^2\right) \left(b x^2+a x^4\right)^{3/4}}{21 b^2 x^5}+\frac{3 a \text{RootSum}\left[2 a^2-a b-4 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{16 b}","\frac{\left(3 b-4 a x^2\right) \left(b x^2+a x^4\right)^{3/4}}{21 b^2 x^5}+\frac{3 a \text{RootSum}\left[2 a^2-a b-4 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{16 b}",1,"((3*b - 4*a*x^2)*(b*x^2 + a*x^4)^(3/4))/(21*b^2*x^5) + (3*a*RootSum[2*a^2 - a*b - 4*a*#1^4 + 2*#1^8 & , (-Log[x] + Log[(b*x^2 + a*x^4)^(1/4) - x*#1])/#1 & ])/(16*b)","A",1
1448,1,102,102,0.0022969,"\int \frac{b+a x^4}{x^4 \left(-2 b+a x^4\right) \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[(b + a*x^4)/(x^4*(-2*b + a*x^4)*(b*x^2 + a*x^4)^(1/4)),x]","\frac{\left(3 b-4 a x^2\right) \left(b x^2+a x^4\right)^{3/4}}{21 b^2 x^5}+\frac{3 a \text{RootSum}\left[2 a^2-a b-4 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{16 b}","\frac{\left(3 b-4 a x^2\right) \left(b x^2+a x^4\right)^{3/4}}{21 b^2 x^5}+\frac{3 a \text{RootSum}\left[2 a^2-a b-4 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{16 b}",1,"((3*b - 4*a*x^2)*(b*x^2 + a*x^4)^(3/4))/(21*b^2*x^5) + (3*a*RootSum[2*a^2 - a*b - 4*a*#1^4 + 2*#1^8 & , (-Log[x] + Log[(b*x^2 + a*x^4)^(1/4) - x*#1])/#1 & ])/(16*b)","A",1
1449,1,102,102,2.2658386,"\int \frac{\left(-1+x^3\right) \left(1+x^3\right)^3 \left(1+x^6\right)^{2/3}}{x^6 \left(1-x^3+x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)*(1 + x^3)^3*(1 + x^6)^(2/3))/(x^6*(1 - x^3 + x^6)),x]","\frac{\left(1+x^6\right)^{2/3} \left(2+15 x^3+2 x^6\right)}{10 x^5}-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^6}}\right)+\log \left(-x+\sqrt[3]{1+x^6}\right)-\frac{1}{2} \log \left(x^2+x \sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)","\frac{\left(1+x^6\right)^{2/3} \left(2+15 x^3+2 x^6\right)}{10 x^5}-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^6}}\right)+\log \left(-x+\sqrt[3]{1+x^6}\right)-\frac{1}{2} \log \left(x^2+x \sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)",1,"((1 + x^6)^(2/3)*(2 + 15*x^3 + 2*x^6))/(10*x^5) - Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^6)^(1/3))] + Log[-x + (1 + x^6)^(1/3)] - Log[x^2 + x*(1 + x^6)^(1/3) + (1 + x^6)^(2/3)]/2","A",1
1450,1,102,102,2.273632,"\int \frac{\left(-1+x^6\right)^{2/3} \left(1+x^6\right) \left(-2+x^3+2 x^6\right)}{x^6 \left(-1-x^3+x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^6)^(2/3)*(1 + x^6)*(-2 + x^3 + 2*x^6))/(x^6*(-1 - x^3 + x^6)),x]","\frac{\left(-1+x^6\right)^{2/3} \left(-4+15 x^3+4 x^6\right)}{10 x^5}-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^6}}\right)+\log \left(-x+\sqrt[3]{-1+x^6}\right)-\frac{1}{2} \log \left(x^2+x \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)","\frac{\left(-1+x^6\right)^{2/3} \left(-4+15 x^3+4 x^6\right)}{10 x^5}-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^6}}\right)+\log \left(-x+\sqrt[3]{-1+x^6}\right)-\frac{1}{2} \log \left(x^2+x \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)",1,"((-1 + x^6)^(2/3)*(-4 + 15*x^3 + 4*x^6))/(10*x^5) - Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^6)^(1/3))] + Log[-x + (-1 + x^6)^(1/3)] - Log[x^2 + x*(-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)]/2","A",1
1451,1,102,102,0.6179492,"\int \frac{1}{\sqrt[4]{b+a x^4} \left(-b+a x^4+x^8\right)} \, dx","IntegrateAlgebraic[1/((b + a*x^4)^(1/4)*(-b + a*x^4 + x^8)),x]","-\frac{\text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a \log (x)+a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]}{4 b}","-\frac{\text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a \log (x)-a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{3 a \text{$\#$1}-2 \text{$\#$1}^5}\&\right]}{4 b}",1,"-1/4*RootSum[2*a^2 - b - 3*a*#1^4 + #1^8 & , (-(a*Log[x]) + a*Log[(b + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(b + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1 + 2*#1^5) & ]/b","A",1
1452,1,102,102,0.0013834,"\int \frac{1}{\sqrt[4]{b+a x^4} \left(-b+a x^4+x^8\right)} \, dx","IntegrateAlgebraic[1/((b + a*x^4)^(1/4)*(-b + a*x^4 + x^8)),x]","-\frac{\text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a \log (x)+a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]}{4 b}","-\frac{\text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a \log (x)-a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{3 a \text{$\#$1}-2 \text{$\#$1}^5}\&\right]}{4 b}",1,"-1/4*RootSum[2*a^2 - b - 3*a*#1^4 + #1^8 & , (-(a*Log[x]) + a*Log[(b + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(b + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1 + 2*#1^5) & ]/b","A",1
1453,1,102,102,11.6527532,"\int \frac{-b+a x^4}{\sqrt[4]{-b x^2+a x^4} \left(-b+2 a x^4+2 x^8\right)} \, dx","IntegrateAlgebraic[(-b + a*x^4)/((-(b*x^2) + a*x^4)^(1/4)*(-b + 2*a*x^4 + 2*x^8)),x]","-\frac{1}{8} \text{RootSum}\left[a^4-2 a^3 b-2 b^3-4 a^3 \text{$\#$1}^4+4 a^2 b \text{$\#$1}^4+6 a^2 \text{$\#$1}^8-2 a b \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","-\frac{1}{8} \text{RootSum}\left[a^4-2 a^3 b-2 b^3-4 a^3 \text{$\#$1}^4+4 a^2 b \text{$\#$1}^4+6 a^2 \text{$\#$1}^8-2 a b \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"-1/8*RootSum[a^4 - 2*a^3*b - 2*b^3 - 4*a^3*#1^4 + 4*a^2*b*#1^4 + 6*a^2*#1^8 - 2*a*b*#1^8 - 4*a*#1^12 + #1^16 & , (-Log[x] + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1])/#1 & ]","A",1
1454,1,102,102,0.0024796,"\int \frac{-b+a x^4}{\sqrt[4]{-b x^2+a x^4} \left(-b+2 a x^4+2 x^8\right)} \, dx","IntegrateAlgebraic[(-b + a*x^4)/((-(b*x^2) + a*x^4)^(1/4)*(-b + 2*a*x^4 + 2*x^8)),x]","-\frac{1}{8} \text{RootSum}\left[a^4-2 a^3 b-2 b^3-4 a^3 \text{$\#$1}^4+4 a^2 b \text{$\#$1}^4+6 a^2 \text{$\#$1}^8-2 a b \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","-\frac{1}{8} \text{RootSum}\left[a^4-2 a^3 b-2 b^3-4 a^3 \text{$\#$1}^4+4 a^2 b \text{$\#$1}^4+6 a^2 \text{$\#$1}^8-2 a b \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"-1/8*RootSum[a^4 - 2*a^3*b - 2*b^3 - 4*a^3*#1^4 + 4*a^2*b*#1^4 + 6*a^2*#1^8 - 2*a*b*#1^8 - 4*a*#1^12 + #1^16 & , (-Log[x] + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1])/#1 & ]","A",1
1455,1,102,102,7.0715431,"\int \frac{\sqrt{c x^2-x \sqrt{-b x+a x^2}}}{x^3} \, dx","IntegrateAlgebraic[Sqrt[c*x^2 - x*Sqrt[-(b*x) + a*x^2]]/x^3,x]","-\frac{4 \left(3 b-3 a x+2 c^2 x\right) \sqrt{-x \left(-c x+\sqrt{-b x+a x^2}\right)}}{15 b x^2}-\frac{4 c \sqrt{-b x+a x^2} \sqrt{-x \left(-c x+\sqrt{-b x+a x^2}\right)}}{15 b x^2}","-\frac{4 \left(3 b-3 a x+2 c^2 x\right) \sqrt{-x \left(-c x+\sqrt{-b x+a x^2}\right)}}{15 b x^2}-\frac{4 c \sqrt{-b x+a x^2} \sqrt{-x \left(-c x+\sqrt{-b x+a x^2}\right)}}{15 b x^2}",1,"(-4*(3*b - 3*a*x + 2*c^2*x)*Sqrt[-(x*(-(c*x) + Sqrt[-(b*x) + a*x^2]))])/(15*b*x^2) - (4*c*Sqrt[-(b*x) + a*x^2]*Sqrt[-(x*(-(c*x) + Sqrt[-(b*x) + a*x^2]))])/(15*b*x^2)","A",1
1456,1,102,102,0.6106141,"\int \frac{x}{x+\sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[x/(x + Sqrt[x + Sqrt[1 + x^2]]),x]","x-2 \sqrt{x+\sqrt{1+x^2}}+2 \text{RootSum}\left[-1+2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{-\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right)+\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^3}{3 \text{$\#$1}^2+2 \text{$\#$1}^3}\&\right]","x-2 \sqrt{x+\sqrt{1+x^2}}+2 \text{RootSum}\left[-1+2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{-\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right)+\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^3}{3 \text{$\#$1}^2+2 \text{$\#$1}^3}\&\right]",1,"x - 2*Sqrt[x + Sqrt[1 + x^2]] + 2*RootSum[-1 + 2*#1^3 + #1^4 & , (-Log[Sqrt[x + Sqrt[1 + x^2]] - #1] + Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1^3)/(3*#1^2 + 2*#1^3) & ]","A",1
1457,1,102,102,0.0008871,"\int \frac{x}{x+\sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[x/(x + Sqrt[x + Sqrt[1 + x^2]]),x]","x-2 \sqrt{x+\sqrt{1+x^2}}+2 \text{RootSum}\left[-1+2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{-\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right)+\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^3}{3 \text{$\#$1}^2+2 \text{$\#$1}^3}\&\right]","x-2 \sqrt{x+\sqrt{1+x^2}}+2 \text{RootSum}\left[-1+2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{-\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right)+\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^3}{3 \text{$\#$1}^2+2 \text{$\#$1}^3}\&\right]",1,"x - 2*Sqrt[x + Sqrt[1 + x^2]] + 2*RootSum[-1 + 2*#1^3 + #1^4 & , (-Log[Sqrt[x + Sqrt[1 + x^2]] - #1] + Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1^3)/(3*#1^2 + 2*#1^3) & ]","A",1
1458,1,103,103,1.2521302,"\int \frac{1+x}{\left(-2+2 x+x^2\right) \sqrt{1+x^3}} \, dx","IntegrateAlgebraic[(1 + x)/((-2 + 2*x + x^2)*Sqrt[1 + x^3]),x]","\frac{1}{6} \sqrt{-3+2 \sqrt{3}} \tan ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} \sqrt{1+x^3}}{1-x+x^2}\right)-\frac{1}{6} \sqrt{3+2 \sqrt{3}} \tanh ^{-1}\left(\frac{\sqrt{-3+2 \sqrt{3}} \sqrt{1+x^3}}{1-x+x^2}\right)","\frac{1}{6} \sqrt{-3+2 \sqrt{3}} \tan ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} \sqrt{1+x^3}}{1-x+x^2}\right)-\frac{1}{6} \sqrt{3+2 \sqrt{3}} \tanh ^{-1}\left(\frac{\sqrt{-3+2 \sqrt{3}} \sqrt{1+x^3}}{1-x+x^2}\right)",1,"(Sqrt[-3 + 2*Sqrt[3]]*ArcTan[(Sqrt[3 + 2*Sqrt[3]]*Sqrt[1 + x^3])/(1 - x + x^2)])/6 - (Sqrt[3 + 2*Sqrt[3]]*ArcTanh[(Sqrt[-3 + 2*Sqrt[3]]*Sqrt[1 + x^3])/(1 - x + x^2)])/6","A",1
1459,1,103,103,1.5481011,"\int \frac{3+x+x^2}{\left(-2+2 x+x^2\right) \sqrt{1+x^3}} \, dx","IntegrateAlgebraic[(3 + x + x^2)/((-2 + 2*x + x^2)*Sqrt[1 + x^3]),x]","-\frac{1}{6} \sqrt{-15+14 \sqrt{3}} \tan ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} \sqrt{1+x^3}}{1-x+x^2}\right)-\frac{1}{6} \sqrt{15+14 \sqrt{3}} \tanh ^{-1}\left(\frac{\sqrt{-3+2 \sqrt{3}} \sqrt{1+x^3}}{1-x+x^2}\right)","-\frac{1}{6} \sqrt{-15+14 \sqrt{3}} \tan ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} \sqrt{1+x^3}}{1-x+x^2}\right)-\frac{1}{6} \sqrt{15+14 \sqrt{3}} \tanh ^{-1}\left(\frac{\sqrt{-3+2 \sqrt{3}} \sqrt{1+x^3}}{1-x+x^2}\right)",1,"-1/6*(Sqrt[-15 + 14*Sqrt[3]]*ArcTan[(Sqrt[3 + 2*Sqrt[3]]*Sqrt[1 + x^3])/(1 - x + x^2)]) - (Sqrt[15 + 14*Sqrt[3]]*ArcTanh[(Sqrt[-3 + 2*Sqrt[3]]*Sqrt[1 + x^3])/(1 - x + x^2)])/6","A",1
1460,1,103,103,1.6026048,"\int \frac{3-x+2 x^2}{\left(-2+2 x+x^2\right) \sqrt{1+x^3}} \, dx","IntegrateAlgebraic[(3 - x + 2*x^2)/((-2 + 2*x + x^2)*Sqrt[1 + x^3]),x]","-\frac{1}{6} \sqrt{21+26 \sqrt{3}} \tan ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} \sqrt{1+x^3}}{1-x+x^2}\right)-\frac{1}{6} \sqrt{-21+26 \sqrt{3}} \tanh ^{-1}\left(\frac{\sqrt{-3+2 \sqrt{3}} \sqrt{1+x^3}}{1-x+x^2}\right)","-\frac{1}{6} \sqrt{21+26 \sqrt{3}} \tan ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} \sqrt{1+x^3}}{1-x+x^2}\right)-\frac{1}{6} \sqrt{-21+26 \sqrt{3}} \tanh ^{-1}\left(\frac{\sqrt{-3+2 \sqrt{3}} \sqrt{1+x^3}}{1-x+x^2}\right)",1,"-1/6*(Sqrt[21 + 26*Sqrt[3]]*ArcTan[(Sqrt[3 + 2*Sqrt[3]]*Sqrt[1 + x^3])/(1 - x + x^2)]) - (Sqrt[-21 + 26*Sqrt[3]]*ArcTanh[(Sqrt[-3 + 2*Sqrt[3]]*Sqrt[1 + x^3])/(1 - x + x^2)])/6","A",1
1461,1,103,103,0.2048719,"\int \frac{\left(-1+x^3\right)^{2/3} \left(1+x^3\right)}{x^3} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(1 + x^3))/x^3,x]","\frac{\left(-1+x^3\right)^{2/3} \left(-3+2 x^3\right)}{6 x^2}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{3 \sqrt{3}}-\frac{1}{9} \log \left(-x+\sqrt[3]{-1+x^3}\right)+\frac{1}{18} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)","\frac{\left(-1+x^3\right)^{2/3} \left(-3+2 x^3\right)}{6 x^2}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{3 \sqrt{3}}-\frac{1}{9} \log \left(-x+\sqrt[3]{-1+x^3}\right)+\frac{1}{18} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)",1,"((-1 + x^3)^(2/3)*(-3 + 2*x^3))/(6*x^2) + ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^3)^(1/3))]/(3*Sqrt[3]) - Log[-x + (-1 + x^3)^(1/3)]/9 + Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)]/18","A",1
1462,1,103,103,0.41641,"\int x^8 \left(b+a x^4\right)^{3/4} \, dx","IntegrateAlgebraic[x^8*(b + a*x^4)^(3/4),x]","\frac{\left(b+a x^4\right)^{3/4} \left(-15 b^2 x+12 a b x^5+32 a^2 x^9\right)}{384 a^2}+\frac{5 b^3 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{256 a^{9/4}}+\frac{5 b^3 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{256 a^{9/4}}","\frac{\left(b+a x^4\right)^{3/4} \left(-15 b^2 x+12 a b x^5+32 a^2 x^9\right)}{384 a^2}+\frac{5 b^3 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{256 a^{9/4}}+\frac{5 b^3 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{256 a^{9/4}}",1,"((b + a*x^4)^(3/4)*(-15*b^2*x + 12*a*b*x^5 + 32*a^2*x^9))/(384*a^2) + (5*b^3*ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)])/(256*a^(9/4)) + (5*b^3*ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)])/(256*a^(9/4))","A",1
1463,1,103,103,0.3895476,"\int \frac{1}{\left(-b+a x^3\right) \sqrt[4]{b x+a x^4}} \, dx","IntegrateAlgebraic[1/((-b + a*x^3)*(b*x + a*x^4)^(1/4)),x]","-\frac{2^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 \sqrt[4]{a} b}-\frac{2^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 \sqrt[4]{a} b}","-\frac{2^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 \sqrt[4]{a} b}-\frac{2^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 \sqrt[4]{a} b}",1,"-1/3*(2^(3/4)*ArcTan[(2^(1/4)*a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/(a^(1/4)*b) - (2^(3/4)*ArcTanh[(2^(1/4)*a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/(3*a^(1/4)*b)","A",1
1464,1,103,103,0.1980298,"\int \frac{\left(-2+x^3\right) \left(1+x^3\right)^{2/3}}{x^3 \left(-2+x^3+2 x^6\right)} \, dx","IntegrateAlgebraic[((-2 + x^3)*(1 + x^3)^(2/3))/(x^3*(-2 + x^3 + 2*x^6)),x]","-\frac{\left(1+x^3\right)^{2/3}}{2 x^2}+\frac{1}{3} \text{RootSum}\left[1-5 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)+2 \log (x) \text{$\#$1}^3-2 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-5 \text{$\#$1}+4 \text{$\#$1}^4}\&\right]","-\frac{\left(1+x^3\right)^{2/3}}{2 x^2}+\frac{1}{3} \text{RootSum}\left[1-5 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)+2 \log (x) \text{$\#$1}^3-2 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-5 \text{$\#$1}+4 \text{$\#$1}^4}\&\right]",1,"-1/2*(1 + x^3)^(2/3)/x^2 + RootSum[1 - 5*#1^3 + 2*#1^6 & , (-Log[x] + Log[(1 + x^3)^(1/3) - x*#1] + 2*Log[x]*#1^3 - 2*Log[(1 + x^3)^(1/3) - x*#1]*#1^3)/(-5*#1 + 4*#1^4) & ]/3","A",1
1465,1,103,103,0.0009694,"\int \frac{\left(-2+x^3\right) \left(1+x^3\right)^{2/3}}{x^3 \left(-2+x^3+2 x^6\right)} \, dx","IntegrateAlgebraic[((-2 + x^3)*(1 + x^3)^(2/3))/(x^3*(-2 + x^3 + 2*x^6)),x]","-\frac{\left(1+x^3\right)^{2/3}}{2 x^2}+\frac{1}{3} \text{RootSum}\left[1-5 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)+2 \log (x) \text{$\#$1}^3-2 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-5 \text{$\#$1}+4 \text{$\#$1}^4}\&\right]","-\frac{\left(1+x^3\right)^{2/3}}{2 x^2}+\frac{1}{3} \text{RootSum}\left[1-5 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)+2 \log (x) \text{$\#$1}^3-2 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-5 \text{$\#$1}+4 \text{$\#$1}^4}\&\right]",1,"-1/2*(1 + x^3)^(2/3)/x^2 + RootSum[1 - 5*#1^3 + 2*#1^6 & , (-Log[x] + Log[(1 + x^3)^(1/3) - x*#1] + 2*Log[x]*#1^3 - 2*Log[(1 + x^3)^(1/3) - x*#1]*#1^3)/(-5*#1 + 4*#1^4) & ]/3","A",1
1466,1,103,103,0.5116855,"\int \frac{x^2 \left(-2+x^4\right)}{\sqrt[3]{-x+x^5} \left(-1+x^4+x^8\right)} \, dx","IntegrateAlgebraic[(x^2*(-2 + x^4))/((-x + x^5)^(1/3)*(-1 + x^4 + x^8)),x]","\frac{1}{4} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-x+x^5}}{-2 x^3+\sqrt[3]{-x+x^5}}\right)+\frac{1}{4} \log \left(x^3+\sqrt[3]{-x+x^5}\right)-\frac{1}{8} \log \left(x^6-x^3 \sqrt[3]{-x+x^5}+\left(-x+x^5\right)^{2/3}\right)","\frac{1}{4} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-x+x^5}}{-2 x^3+\sqrt[3]{-x+x^5}}\right)+\frac{1}{4} \log \left(x^3+\sqrt[3]{-x+x^5}\right)-\frac{1}{8} \log \left(x^6-x^3 \sqrt[3]{-x+x^5}+\left(-x+x^5\right)^{2/3}\right)",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(-x + x^5)^(1/3))/(-2*x^3 + (-x + x^5)^(1/3))])/4 + Log[x^3 + (-x + x^5)^(1/3)]/4 - Log[x^6 - x^3*(-x + x^5)^(1/3) + (-x + x^5)^(2/3)]/8","A",1
1467,1,103,103,0.8743403,"\int \frac{-b+2 a x^4}{\sqrt[4]{-b+a x^4} \left(-b-a x^4+x^8\right)} \, dx","IntegrateAlgebraic[(-b + 2*a*x^4)/((-b + a*x^4)^(1/4)*(-b - a*x^4 + x^8)),x]","\frac{1}{4} \text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a \log (x)-a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]","\frac{1}{4} \text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a \log (x)+a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{3 a \text{$\#$1}-2 \text{$\#$1}^5}\&\right]",1,"RootSum[2*a^2 - b - 3*a*#1^4 + #1^8 & , (a*Log[x] - a*Log[(-b + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1 + 2*#1^5) & ]/4","A",1
1468,1,103,103,0.858265,"\int \frac{-2 b+a x^4}{\sqrt[4]{b+a x^4} \left(-2 b+a x^4+x^8\right)} \, dx","IntegrateAlgebraic[(-2*b + a*x^4)/((b + a*x^4)^(1/4)*(-2*b + a*x^4 + x^8)),x]","\frac{1}{4} \text{RootSum}\left[3 a^2-b-5 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-3 a \log (x)+3 a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)+2 \log (x) \text{$\#$1}^4-2 \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-5 a \text{$\#$1}+4 \text{$\#$1}^5}\&\right]","\frac{1}{4} \text{RootSum}\left[3 a^2-b-5 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{3 a \log (x)-3 a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)-2 \log (x) \text{$\#$1}^4+2 \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{5 a \text{$\#$1}-4 \text{$\#$1}^5}\&\right]",1,"RootSum[3*a^2 - b - 5*a*#1^4 + 2*#1^8 & , (-3*a*Log[x] + 3*a*Log[(b + a*x^4)^(1/4) - x*#1] + 2*Log[x]*#1^4 - 2*Log[(b + a*x^4)^(1/4) - x*#1]*#1^4)/(-5*a*#1 + 4*#1^5) & ]/4","A",1
1469,1,103,103,0.1824656,"\int \frac{1}{x^2 \sqrt{a x+\sqrt{b^2+a^2 x^2}}} \, dx","IntegrateAlgebraic[1/(x^2*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]),x]","-\frac{1}{x \sqrt{a x+\sqrt{b^2+a^2 x^2}}}+\frac{a \tan ^{-1}\left(\frac{\sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{b}}\right)}{b^{3/2}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{b}}\right)}{b^{3/2}}","-\frac{1}{x \sqrt{a x+\sqrt{b^2+a^2 x^2}}}+\frac{a \tan ^{-1}\left(\frac{\sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{b}}\right)}{b^{3/2}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{b}}\right)}{b^{3/2}}",1,"-(1/(x*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])) + (a*ArcTan[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]/Sqrt[b]])/b^(3/2) + (a*ArcTanh[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]/Sqrt[b]])/b^(3/2)","A",1
1470,1,104,104,0.1731537,"\int \frac{\sqrt[3]{-1+x^3}}{x^{10}} \, dx","IntegrateAlgebraic[(-1 + x^3)^(1/3)/x^10,x]","\frac{\sqrt[3]{-1+x^3} \left(-18+3 x^3+5 x^6\right)}{162 x^9}-\frac{5 \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^3}}{\sqrt{3}}\right)}{81 \sqrt{3}}+\frac{5}{243} \log \left(1+\sqrt[3]{-1+x^3}\right)-\frac{5}{486} \log \left(1-\sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)","\frac{\sqrt[3]{-1+x^3} \left(-18+3 x^3+5 x^6\right)}{162 x^9}-\frac{5 \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^3}}{\sqrt{3}}\right)}{81 \sqrt{3}}+\frac{5}{243} \log \left(1+\sqrt[3]{-1+x^3}\right)-\frac{5}{486} \log \left(1-\sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)",1,"((-1 + x^3)^(1/3)*(-18 + 3*x^3 + 5*x^6))/(162*x^9) - (5*ArcTan[1/Sqrt[3] - (2*(-1 + x^3)^(1/3))/Sqrt[3]])/(81*Sqrt[3]) + (5*Log[1 + (-1 + x^3)^(1/3)])/243 - (5*Log[1 - (-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)])/486","A",1
1471,1,104,104,0.1798929,"\int x^4 \sqrt[3]{-1+x^3} \, dx","IntegrateAlgebraic[x^4*(-1 + x^3)^(1/3),x]","\frac{1}{18} \sqrt[3]{-1+x^3} \left(-x^2+3 x^5\right)+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{9 \sqrt{3}}+\frac{1}{27} \log \left(-x+\sqrt[3]{-1+x^3}\right)-\frac{1}{54} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)","\frac{1}{18} \sqrt[3]{-1+x^3} \left(-x^2+3 x^5\right)+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{9 \sqrt{3}}+\frac{1}{27} \log \left(-x+\sqrt[3]{-1+x^3}\right)-\frac{1}{54} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)",1,"((-1 + x^3)^(1/3)*(-x^2 + 3*x^5))/18 + ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^3)^(1/3))]/(9*Sqrt[3]) + Log[-x + (-1 + x^3)^(1/3)]/27 - Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)]/54","A",1
1472,1,104,104,0.1778214,"\int \sqrt[3]{-x+x^3} \, dx","IntegrateAlgebraic[(-x + x^3)^(1/3),x]","\frac{1}{2} x \sqrt[3]{-x+x^3}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)}{2 \sqrt{3}}+\frac{1}{6} \log \left(-x+\sqrt[3]{-x+x^3}\right)-\frac{1}{12} \log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)","\frac{1}{2} x \sqrt[3]{-x+x^3}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)}{2 \sqrt{3}}+\frac{1}{6} \log \left(-x+\sqrt[3]{-x+x^3}\right)-\frac{1}{12} \log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)",1,"(x*(-x + x^3)^(1/3))/2 + ArcTan[(Sqrt[3]*x)/(x + 2*(-x + x^3)^(1/3))]/(2*Sqrt[3]) + Log[-x + (-x + x^3)^(1/3)]/6 - Log[x^2 + x*(-x + x^3)^(1/3) + (-x + x^3)^(2/3)]/12","A",1
1473,1,104,104,0.1573012,"\int \frac{\sqrt[3]{-1+x^3} \left(-1+2 x^3\right)}{x^{10}} \, dx","IntegrateAlgebraic[((-1 + x^3)^(1/3)*(-1 + 2*x^3))/x^10,x]","\frac{\sqrt[3]{-1+x^3} \left(18-57 x^3+13 x^6\right)}{162 x^9}-\frac{13 \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^3}}{\sqrt{3}}\right)}{81 \sqrt{3}}+\frac{13}{243} \log \left(1+\sqrt[3]{-1+x^3}\right)-\frac{13}{486} \log \left(1-\sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)","\frac{\sqrt[3]{-1+x^3} \left(18-57 x^3+13 x^6\right)}{162 x^9}-\frac{13 \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^3}}{\sqrt{3}}\right)}{81 \sqrt{3}}+\frac{13}{243} \log \left(1+\sqrt[3]{-1+x^3}\right)-\frac{13}{486} \log \left(1-\sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)",1,"((-1 + x^3)^(1/3)*(18 - 57*x^3 + 13*x^6))/(162*x^9) - (13*ArcTan[1/Sqrt[3] - (2*(-1 + x^3)^(1/3))/Sqrt[3]])/(81*Sqrt[3]) + (13*Log[1 + (-1 + x^3)^(1/3)])/243 - (13*Log[1 - (-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)])/486","A",1
1474,1,104,104,0.3611028,"\int \frac{1+k^3 x^3}{\sqrt{(1-x) x \left(1-k^2 x\right)} \left(-1+k^3 x^3\right)} \, dx","IntegrateAlgebraic[(1 + k^3*x^3)/(Sqrt[(1 - x)*x*(1 - k^2*x)]*(-1 + k^3*x^3)),x]","-\frac{2 \tan ^{-1}\left(\frac{(-1+k) x}{\sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}\right)}{3 (-1+k)}-\frac{4 \tan ^{-1}\left(\frac{\sqrt{1+k+k^2} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{(-1+x) \left(-1+k^2 x\right)}\right)}{3 \sqrt{1+k+k^2}}","-\frac{2 \tan ^{-1}\left(\frac{(-1+k) x}{\sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}\right)}{3 (-1+k)}-\frac{4 \tan ^{-1}\left(\frac{\sqrt{1+k+k^2} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{(-1+x) \left(-1+k^2 x\right)}\right)}{3 \sqrt{1+k+k^2}}",1,"(-2*ArcTan[((-1 + k)*x)/Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3]])/(3*(-1 + k)) - (4*ArcTan[(Sqrt[1 + k + k^2]*Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3])/((-1 + x)*(-1 + k^2*x))])/(3*Sqrt[1 + k + k^2])","A",1
1475,1,104,104,3.5010617,"\int \frac{\left(-3+x^4\right) \left(1+x^4\right)^{2/3} \left(1+2 x^3+x^4\right)}{x^6 \left(1-x^3+x^4\right)} \, dx","IntegrateAlgebraic[((-3 + x^4)*(1 + x^4)^(2/3)*(1 + 2*x^3 + x^4))/(x^6*(1 - x^3 + x^4)),x]","\frac{3 \left(1+x^4\right)^{2/3} \left(2+15 x^3+2 x^4\right)}{10 x^5}-3 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^4}}\right)+3 \log \left(-x+\sqrt[3]{1+x^4}\right)-\frac{3}{2} \log \left(x^2+x \sqrt[3]{1+x^4}+\left(1+x^4\right)^{2/3}\right)","\frac{3 \left(1+x^4\right)^{2/3} \left(2+15 x^3+2 x^4\right)}{10 x^5}-3 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^4}}\right)+3 \log \left(-x+\sqrt[3]{1+x^4}\right)-\frac{3}{2} \log \left(x^2+x \sqrt[3]{1+x^4}+\left(1+x^4\right)^{2/3}\right)",1,"(3*(1 + x^4)^(2/3)*(2 + 15*x^3 + 2*x^4))/(10*x^5) - 3*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^4)^(1/3))] + 3*Log[-x + (1 + x^4)^(1/3)] - (3*Log[x^2 + x*(1 + x^4)^(1/3) + (1 + x^4)^(2/3)])/2","A",1
1476,1,104,104,0.3539696,"\int \frac{\left(b+a x^4\right)^{3/4}}{x^4 \left(2 b+a x^4\right)} \, dx","IntegrateAlgebraic[(b + a*x^4)^(3/4)/(x^4*(2*b + a*x^4)),x]","-\frac{\left(b+a x^4\right)^{3/4}}{6 b x^3}+\frac{a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{2} \sqrt[4]{b+a x^4}}\right)}{4\ 2^{3/4} b}+\frac{a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{2} \sqrt[4]{b+a x^4}}\right)}{4\ 2^{3/4} b}","-\frac{\left(b+a x^4\right)^{3/4}}{6 b x^3}+\frac{a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{2} \sqrt[4]{b+a x^4}}\right)}{4\ 2^{3/4} b}+\frac{a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{2} \sqrt[4]{b+a x^4}}\right)}{4\ 2^{3/4} b}",1,"-1/6*(b + a*x^4)^(3/4)/(b*x^3) + (a^(3/4)*ArcTan[(a^(1/4)*x)/(2^(1/4)*(b + a*x^4)^(1/4))])/(4*2^(3/4)*b) + (a^(3/4)*ArcTanh[(a^(1/4)*x)/(2^(1/4)*(b + a*x^4)^(1/4))])/(4*2^(3/4)*b)","A",1
1477,1,104,104,0.3228334,"\int \frac{-b+a x^3}{x^3 \sqrt[4]{-b x+a x^4}} \, dx","IntegrateAlgebraic[(-b + a*x^3)/(x^3*(-(b*x) + a*x^4)^(1/4)),x]","-\frac{4 \left(-b x+a x^4\right)^{3/4}}{9 x^3}+\frac{2}{3} a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)+\frac{2}{3} a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)","-\frac{4 \left(-b x+a x^4\right)^{3/4}}{9 x^3}+\frac{2}{3} a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)+\frac{2}{3} a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)",1,"(-4*(-(b*x) + a*x^4)^(3/4))/(9*x^3) + (2*a^(3/4)*ArcTan[(a^(1/4)*(-(b*x) + a*x^4)^(3/4))/(-b + a*x^3)])/3 + (2*a^(3/4)*ArcTanh[(a^(1/4)*(-(b*x) + a*x^4)^(3/4))/(-b + a*x^3)])/3","A",1
1478,1,104,104,0.3501285,"\int \frac{b+a x^3}{x^3 \sqrt[4]{-b x+a x^4}} \, dx","IntegrateAlgebraic[(b + a*x^3)/(x^3*(-(b*x) + a*x^4)^(1/4)),x]","\frac{4 \left(-b x+a x^4\right)^{3/4}}{9 x^3}+\frac{2}{3} a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)+\frac{2}{3} a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)","\frac{4 \left(-b x+a x^4\right)^{3/4}}{9 x^3}+\frac{2}{3} a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)+\frac{2}{3} a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)",1,"(4*(-(b*x) + a*x^4)^(3/4))/(9*x^3) + (2*a^(3/4)*ArcTan[(a^(1/4)*(-(b*x) + a*x^4)^(3/4))/(-b + a*x^3)])/3 + (2*a^(3/4)*ArcTanh[(a^(1/4)*(-(b*x) + a*x^4)^(3/4))/(-b + a*x^3)])/3","A",1
1479,1,104,104,0.308158,"\int \frac{\sqrt[4]{-b x+a x^4}}{x^2} \, dx","IntegrateAlgebraic[(-(b*x) + a*x^4)^(1/4)/x^2,x]","-\frac{4 \sqrt[4]{-b x+a x^4}}{3 x}-\frac{2}{3} \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)+\frac{2}{3} \sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)","-\frac{4 \sqrt[4]{-b x+a x^4}}{3 x}-\frac{2}{3} \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)+\frac{2}{3} \sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)",1,"(-4*(-(b*x) + a*x^4)^(1/4))/(3*x) - (2*a^(1/4)*ArcTan[(a^(1/4)*(-(b*x) + a*x^4)^(3/4))/(-b + a*x^3)])/3 + (2*a^(1/4)*ArcTanh[(a^(1/4)*(-(b*x) + a*x^4)^(3/4))/(-b + a*x^3)])/3","A",1
1480,1,105,104,0.4790997,"\int \frac{1}{\left(b-a x^2+x^4\right) \sqrt[4]{-b x^2+a x^4}} \, dx","IntegrateAlgebraic[1/((b - a*x^2 + x^4)*(-(b*x^2) + a*x^4)^(1/4)),x]","\frac{\text{RootSum}\left[b-a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a \log (x)+a \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]}{2 b}","\frac{\text{RootSum}\left[b-a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a \log (x)-a \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}-2 \text{$\#$1}^5}\&\right]}{2 b}",1,"RootSum[b - a*#1^4 + #1^8 & , (-(a*Log[x]) + a*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1) + 2*#1^5) & ]/(2*b)","A",1
1481,1,105,104,0.0018305,"\int \frac{1}{\left(b-a x^2+x^4\right) \sqrt[4]{-b x^2+a x^4}} \, dx","IntegrateAlgebraic[1/((b - a*x^2 + x^4)*(-(b*x^2) + a*x^4)^(1/4)),x]","\frac{\text{RootSum}\left[b-a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a \log (x)+a \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]}{2 b}","\frac{\text{RootSum}\left[b-a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a \log (x)-a \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}-2 \text{$\#$1}^5}\&\right]}{2 b}",1,"RootSum[b - a*#1^4 + #1^8 & , (-(a*Log[x]) + a*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1) + 2*#1^5) & ]/(2*b)","A",1
1482,1,104,104,0.3710824,"\int \frac{-b+a x^4}{x^4 \sqrt[4]{-b+2 a x^4}} \, dx","IntegrateAlgebraic[(-b + a*x^4)/(x^4*(-b + 2*a*x^4)^(1/4)),x]","-\frac{\left(-b+2 a x^4\right)^{3/4}}{3 x^3}+\frac{a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{-b+2 a x^4}}\right)}{2 \sqrt[4]{2}}+\frac{a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{-b+2 a x^4}}\right)}{2 \sqrt[4]{2}}","-\frac{\left(-b+2 a x^4\right)^{3/4}}{3 x^3}+\frac{a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{-b+2 a x^4}}\right)}{2 \sqrt[4]{2}}+\frac{a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{-b+2 a x^4}}\right)}{2 \sqrt[4]{2}}",1,"-1/3*(-b + 2*a*x^4)^(3/4)/x^3 + (a^(3/4)*ArcTan[(2^(1/4)*a^(1/4)*x)/(-b + 2*a*x^4)^(1/4)])/(2*2^(1/4)) + (a^(3/4)*ArcTanh[(2^(1/4)*a^(1/4)*x)/(-b + 2*a*x^4)^(1/4)])/(2*2^(1/4))","A",1
1483,1,104,104,3.0976307,"\int \frac{x^4 \left(4 b+a x^5\right)}{\left(-b+a x^5\right)^2 \sqrt[4]{-b+c x^4+a x^5}} \, dx","IntegrateAlgebraic[(x^4*(4*b + a*x^5))/((-b + a*x^5)^2*(-b + c*x^4 + a*x^5)^(1/4)),x]","\frac{x \left(-b+c x^4+a x^5\right)^{3/4}}{c \left(b-a x^5\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{-b+c x^4+a x^5}}\right)}{2 c^{5/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{-b+c x^4+a x^5}}\right)}{2 c^{5/4}}","\frac{x \left(-b+c x^4+a x^5\right)^{3/4}}{c \left(b-a x^5\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{-b+c x^4+a x^5}}\right)}{2 c^{5/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{-b+c x^4+a x^5}}\right)}{2 c^{5/4}}",1,"(x*(-b + c*x^4 + a*x^5)^(3/4))/(c*(b - a*x^5)) + ArcTan[(c^(1/4)*x)/(-b + c*x^4 + a*x^5)^(1/4)]/(2*c^(5/4)) + ArcTanh[(c^(1/4)*x)/(-b + c*x^4 + a*x^5)^(1/4)]/(2*c^(5/4))","A",1
1484,1,104,104,0.1751143,"\int \frac{1}{x^{19} \sqrt[3]{-1+x^6}} \, dx","IntegrateAlgebraic[1/(x^19*(-1 + x^6)^(1/3)),x]","\frac{\left(-1+x^6\right)^{2/3} \left(18+21 x^6+28 x^{12}\right)}{324 x^{18}}-\frac{7 \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^6}}{\sqrt{3}}\right)}{81 \sqrt{3}}-\frac{7}{243} \log \left(1+\sqrt[3]{-1+x^6}\right)+\frac{7}{486} \log \left(1-\sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)","\frac{\left(-1+x^6\right)^{2/3} \left(18+21 x^6+28 x^{12}\right)}{324 x^{18}}-\frac{7 \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^6}}{\sqrt{3}}\right)}{81 \sqrt{3}}-\frac{7}{243} \log \left(1+\sqrt[3]{-1+x^6}\right)+\frac{7}{486} \log \left(1-\sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)",1,"((-1 + x^6)^(2/3)*(18 + 21*x^6 + 28*x^12))/(324*x^18) - (7*ArcTan[1/Sqrt[3] - (2*(-1 + x^6)^(1/3))/Sqrt[3]])/(81*Sqrt[3]) - (7*Log[1 + (-1 + x^6)^(1/3)])/243 + (7*Log[1 - (-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)])/486","A",1
1485,1,104,104,1.0088094,"\int \frac{x^7}{\sqrt[3]{-1+x^6}} \, dx","IntegrateAlgebraic[x^7/(-1 + x^6)^(1/3),x]","\frac{1}{6} x^2 \left(-1+x^6\right)^{2/3}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{-1+x^6}}\right)}{6 \sqrt{3}}-\frac{1}{18} \log \left(-x^2+\sqrt[3]{-1+x^6}\right)+\frac{1}{36} \log \left(x^4+x^2 \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)","\frac{1}{6} x^2 \left(-1+x^6\right)^{2/3}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{-1+x^6}}\right)}{6 \sqrt{3}}-\frac{1}{18} \log \left(-x^2+\sqrt[3]{-1+x^6}\right)+\frac{1}{36} \log \left(x^4+x^2 \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)",1,"(x^2*(-1 + x^6)^(2/3))/6 + ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(-1 + x^6)^(1/3))]/(6*Sqrt[3]) - Log[-x^2 + (-1 + x^6)^(1/3)]/18 + Log[x^4 + x^2*(-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)]/36","A",1
1486,1,104,104,0.6319324,"\int x^3 \sqrt[3]{-1+x^6} \, dx","IntegrateAlgebraic[x^3*(-1 + x^6)^(1/3),x]","\frac{1}{6} x^4 \sqrt[3]{-1+x^6}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{-1+x^6}}\right)}{6 \sqrt{3}}+\frac{1}{18} \log \left(-x^2+\sqrt[3]{-1+x^6}\right)-\frac{1}{36} \log \left(x^4+x^2 \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)","\frac{1}{6} x^4 \sqrt[3]{-1+x^6}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{-1+x^6}}\right)}{6 \sqrt{3}}+\frac{1}{18} \log \left(-x^2+\sqrt[3]{-1+x^6}\right)-\frac{1}{36} \log \left(x^4+x^2 \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)",1,"(x^4*(-1 + x^6)^(1/3))/6 + ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(-1 + x^6)^(1/3))]/(6*Sqrt[3]) + Log[-x^2 + (-1 + x^6)^(1/3)]/18 - Log[x^4 + x^2*(-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)]/36","A",1
1487,1,104,104,0.5953824,"\int x \left(-1+x^6\right)^{2/3} \, dx","IntegrateAlgebraic[x*(-1 + x^6)^(2/3),x]","\frac{1}{6} x^2 \left(-1+x^6\right)^{2/3}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{-1+x^6}}\right)}{3 \sqrt{3}}+\frac{1}{9} \log \left(-x^2+\sqrt[3]{-1+x^6}\right)-\frac{1}{18} \log \left(x^4+x^2 \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)","\frac{1}{6} x^2 \left(-1+x^6\right)^{2/3}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{-1+x^6}}\right)}{3 \sqrt{3}}+\frac{1}{9} \log \left(-x^2+\sqrt[3]{-1+x^6}\right)-\frac{1}{18} \log \left(x^4+x^2 \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)",1,"(x^2*(-1 + x^6)^(2/3))/6 - ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(-1 + x^6)^(1/3))]/(3*Sqrt[3]) + Log[-x^2 + (-1 + x^6)^(1/3)]/9 - Log[x^4 + x^2*(-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)]/18","A",1
1488,1,104,104,0.9721244,"\int \frac{x^7}{\sqrt[3]{1+x^6}} \, dx","IntegrateAlgebraic[x^7/(1 + x^6)^(1/3),x]","\frac{1}{6} x^2 \left(1+x^6\right)^{2/3}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{1+x^6}}\right)}{6 \sqrt{3}}+\frac{1}{18} \log \left(-x^2+\sqrt[3]{1+x^6}\right)-\frac{1}{36} \log \left(x^4+x^2 \sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)","\frac{1}{6} x^2 \left(1+x^6\right)^{2/3}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{1+x^6}}\right)}{6 \sqrt{3}}+\frac{1}{18} \log \left(-x^2+\sqrt[3]{1+x^6}\right)-\frac{1}{36} \log \left(x^4+x^2 \sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)",1,"(x^2*(1 + x^6)^(2/3))/6 - ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(1 + x^6)^(1/3))]/(6*Sqrt[3]) + Log[-x^2 + (1 + x^6)^(1/3)]/18 - Log[x^4 + x^2*(1 + x^6)^(1/3) + (1 + x^6)^(2/3)]/36","A",1
1489,1,104,104,0.6167204,"\int x^3 \sqrt[3]{1+x^6} \, dx","IntegrateAlgebraic[x^3*(1 + x^6)^(1/3),x]","\frac{1}{6} x^4 \sqrt[3]{1+x^6}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{1+x^6}}\right)}{6 \sqrt{3}}-\frac{1}{18} \log \left(-x^2+\sqrt[3]{1+x^6}\right)+\frac{1}{36} \log \left(x^4+x^2 \sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)","\frac{1}{6} x^4 \sqrt[3]{1+x^6}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{1+x^6}}\right)}{6 \sqrt{3}}-\frac{1}{18} \log \left(-x^2+\sqrt[3]{1+x^6}\right)+\frac{1}{36} \log \left(x^4+x^2 \sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)",1,"(x^4*(1 + x^6)^(1/3))/6 - ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(1 + x^6)^(1/3))]/(6*Sqrt[3]) - Log[-x^2 + (1 + x^6)^(1/3)]/18 + Log[x^4 + x^2*(1 + x^6)^(1/3) + (1 + x^6)^(2/3)]/36","A",1
1490,1,104,104,0.6021476,"\int x \left(1+x^6\right)^{2/3} \, dx","IntegrateAlgebraic[x*(1 + x^6)^(2/3),x]","\frac{1}{6} x^2 \left(1+x^6\right)^{2/3}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{1+x^6}}\right)}{3 \sqrt{3}}-\frac{1}{9} \log \left(-x^2+\sqrt[3]{1+x^6}\right)+\frac{1}{18} \log \left(x^4+x^2 \sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)","\frac{1}{6} x^2 \left(1+x^6\right)^{2/3}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{1+x^6}}\right)}{3 \sqrt{3}}-\frac{1}{9} \log \left(-x^2+\sqrt[3]{1+x^6}\right)+\frac{1}{18} \log \left(x^4+x^2 \sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)",1,"(x^2*(1 + x^6)^(2/3))/6 + ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(1 + x^6)^(1/3))]/(3*Sqrt[3]) - Log[-x^2 + (1 + x^6)^(1/3)]/9 + Log[x^4 + x^2*(1 + x^6)^(1/3) + (1 + x^6)^(2/3)]/18","A",1
1491,1,104,104,2.8147025,"\int \frac{\left(1-x^3+x^5\right) \left(-3+2 x^5\right)}{x^3 \left(1+x^3+x^5\right) \sqrt[4]{x+x^6}} \, dx","IntegrateAlgebraic[((1 - x^3 + x^5)*(-3 + 2*x^5))/(x^3*(1 + x^3 + x^5)*(x + x^6)^(1/4)),x]","\frac{4 \left(x+x^6\right)^{3/4}}{3 x^3}+2 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{x+x^6}}{-x^2+\sqrt{x+x^6}}\right)+2 \sqrt{2} \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{x+x^6}}{\sqrt{2}}}{x \sqrt[4]{x+x^6}}\right)","\frac{4 \left(x+x^6\right)^{3/4}}{3 x^3}+2 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{x+x^6}}{-x^2+\sqrt{x+x^6}}\right)+2 \sqrt{2} \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{x+x^6}}{\sqrt{2}}}{x \sqrt[4]{x+x^6}}\right)",1,"(4*(x + x^6)^(3/4))/(3*x^3) + 2*Sqrt[2]*ArcTan[(Sqrt[2]*x*(x + x^6)^(1/4))/(-x^2 + Sqrt[x + x^6])] + 2*Sqrt[2]*ArcTanh[(x^2/Sqrt[2] + Sqrt[x + x^6]/Sqrt[2])/(x*(x + x^6)^(1/4))]","A",1
1492,1,104,104,0.3248498,"\int \frac{x}{\sqrt[3]{-x^2+x^6}} \, dx","IntegrateAlgebraic[x/(-x^2 + x^6)^(1/3),x]","\frac{1}{4} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{-x^2+x^6}}\right)-\frac{1}{4} \log \left(-x^2+\sqrt[3]{-x^2+x^6}\right)+\frac{1}{8} \log \left(x^4+x^2 \sqrt[3]{-x^2+x^6}+\left(-x^2+x^6\right)^{2/3}\right)","\frac{1}{4} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{-x^2+x^6}}\right)-\frac{1}{4} \log \left(-x^2+\sqrt[3]{-x^2+x^6}\right)+\frac{1}{8} \log \left(x^4+x^2 \sqrt[3]{-x^2+x^6}+\left(-x^2+x^6\right)^{2/3}\right)",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(-x^2 + x^6)^(1/3))])/4 - Log[-x^2 + (-x^2 + x^6)^(1/3)]/4 + Log[x^4 + x^2*(-x^2 + x^6)^(1/3) + (-x^2 + x^6)^(2/3)]/8","A",1
1493,1,104,104,0.201293,"\int \frac{\left(-1+x^3\right)^{2/3} \left(2+x^3\right)}{x^3 \left(-4+x^3+x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(2 + x^3))/(x^3*(-4 + x^3 + x^6)),x]","\frac{\left(-1+x^3\right)^{2/3}}{4 x^2}-\frac{1}{6} \text{RootSum}\left[2-7 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-2 \log (x)+2 \log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-7 \text{$\#$1}+8 \text{$\#$1}^4}\&\right]","\frac{\left(-1+x^3\right)^{2/3}}{4 x^2}-\frac{1}{6} \text{RootSum}\left[2-7 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-2 \log (x)+2 \log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-7 \text{$\#$1}+8 \text{$\#$1}^4}\&\right]",1,"(-1 + x^3)^(2/3)/(4*x^2) - RootSum[2 - 7*#1^3 + 4*#1^6 & , (-2*Log[x] + 2*Log[(-1 + x^3)^(1/3) - x*#1] + Log[x]*#1^3 - Log[(-1 + x^3)^(1/3) - x*#1]*#1^3)/(-7*#1 + 8*#1^4) & ]/6","A",1
1494,1,104,104,0.0010586,"\int \frac{\left(-1+x^3\right)^{2/3} \left(2+x^3\right)}{x^3 \left(-4+x^3+x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(2 + x^3))/(x^3*(-4 + x^3 + x^6)),x]","\frac{\left(-1+x^3\right)^{2/3}}{4 x^2}-\frac{1}{6} \text{RootSum}\left[2-7 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-2 \log (x)+2 \log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-7 \text{$\#$1}+8 \text{$\#$1}^4}\&\right]","\frac{\left(-1+x^3\right)^{2/3}}{4 x^2}-\frac{1}{6} \text{RootSum}\left[2-7 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-2 \log (x)+2 \log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-7 \text{$\#$1}+8 \text{$\#$1}^4}\&\right]",1,"(-1 + x^3)^(2/3)/(4*x^2) - RootSum[2 - 7*#1^3 + 4*#1^6 & , (-2*Log[x] + 2*Log[(-1 + x^3)^(1/3) - x*#1] + Log[x]*#1^3 - Log[(-1 + x^3)^(1/3) - x*#1]*#1^3)/(-7*#1 + 8*#1^4) & ]/6","A",1
1495,1,106,104,0.4302843,"\int \frac{(-3+2 x) \left(1-x+x^3\right)^{2/3}}{1-2 x+x^2+2 x^3-2 x^4+2 x^6} \, dx","IntegrateAlgebraic[((-3 + 2*x)*(1 - x + x^3)^(2/3))/(1 - 2*x + x^2 + 2*x^3 - 2*x^4 + 2*x^6),x]","-\tan ^{-1}\left(\frac{x}{\sqrt[3]{1-x+x^3}}\right)-\frac{1}{2} i \left(-i+\sqrt{3}\right) \tan ^{-1}\left(\frac{\left(1-i \sqrt{3}\right) x}{2 \sqrt[3]{1-x+x^3}}\right)+\frac{1}{2} i \left(i+\sqrt{3}\right) \tan ^{-1}\left(\frac{\left(1+i \sqrt{3}\right) x}{2 \sqrt[3]{1-x+x^3}}\right)","-\tan ^{-1}\left(\frac{x}{\sqrt[3]{1-x+x^3}}\right)-\frac{1}{2} \tan ^{-1}\left(\frac{x \sqrt[3]{1-x+x^3}}{-x^2+\left(1-x+x^3\right)^{2/3}}\right)-\frac{1}{2} \sqrt{3} \tanh ^{-1}\left(\frac{\sqrt{3} x \sqrt[3]{1-x+x^3}}{x^2+\left(1-x+x^3\right)^{2/3}}\right)",1,"-ArcTan[x/(1 - x + x^3)^(1/3)] - (I/2)*(-I + Sqrt[3])*ArcTan[((1 - I*Sqrt[3])*x)/(2*(1 - x + x^3)^(1/3))] + (I/2)*(I + Sqrt[3])*ArcTan[((1 + I*Sqrt[3])*x)/(2*(1 - x + x^3)^(1/3))]","C",1
1496,1,104,104,2.7975748,"\int \frac{\left(-2+x^6\right) \left(4+x^6\right) \sqrt[4]{-2+2 x^4+x^6}}{x^6 \left(-4-x^4+2 x^6\right)} \, dx","IntegrateAlgebraic[((-2 + x^6)*(4 + x^6)*(-2 + 2*x^4 + x^6)^(1/4))/(x^6*(-4 - x^4 + 2*x^6)),x]","\frac{\sqrt[4]{-2+2 x^4+x^6} \left(-4+9 x^4+2 x^6\right)}{10 x^5}+\frac{1}{4} \sqrt[4]{\frac{5}{2}} \tan ^{-1}\left(\frac{\sqrt[4]{\frac{5}{2}} x}{\sqrt[4]{-2+2 x^4+x^6}}\right)-\frac{1}{4} \sqrt[4]{\frac{5}{2}} \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{5}{2}} x}{\sqrt[4]{-2+2 x^4+x^6}}\right)","\frac{\sqrt[4]{-2+2 x^4+x^6} \left(-4+9 x^4+2 x^6\right)}{10 x^5}+\frac{1}{4} \sqrt[4]{\frac{5}{2}} \tan ^{-1}\left(\frac{\sqrt[4]{\frac{5}{2}} x}{\sqrt[4]{-2+2 x^4+x^6}}\right)-\frac{1}{4} \sqrt[4]{\frac{5}{2}} \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{5}{2}} x}{\sqrt[4]{-2+2 x^4+x^6}}\right)",1,"((-2 + 2*x^4 + x^6)^(1/4)*(-4 + 9*x^4 + 2*x^6))/(10*x^5) + ((5/2)^(1/4)*ArcTan[((5/2)^(1/4)*x)/(-2 + 2*x^4 + x^6)^(1/4)])/4 - ((5/2)^(1/4)*ArcTanh[((5/2)^(1/4)*x)/(-2 + 2*x^4 + x^6)^(1/4)])/4","A",1
1497,1,104,104,0.1989116,"\int \frac{\left(1+x^3\right)^{2/3} \left(-1+3 x^6\right)}{x^9 \left(1+2 x^3\right)} \, dx","IntegrateAlgebraic[((1 + x^3)^(2/3)*(-1 + 3*x^6))/(x^9*(1 + 2*x^3)),x]","\frac{\left(1+x^3\right)^{2/3} \left(5-14 x^3+x^6\right)}{40 x^8}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{1+x^3}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(x+\sqrt[3]{1+x^3}\right)-\frac{1}{6} \log \left(x^2-x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)","\frac{\left(1+x^3\right)^{2/3} \left(5-14 x^3+x^6\right)}{40 x^8}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{1+x^3}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(x+\sqrt[3]{1+x^3}\right)-\frac{1}{6} \log \left(x^2-x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)",1,"((1 + x^3)^(2/3)*(5 - 14*x^3 + x^6))/(40*x^8) + ArcTan[(Sqrt[3]*x)/(-x + 2*(1 + x^3)^(1/3))]/Sqrt[3] + Log[x + (1 + x^3)^(1/3)]/3 - Log[x^2 - x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/6","A",1
1498,1,71,104,0.3374694,"\int \frac{\sqrt{1-x^4} \left(1+x^4\right)}{1-x^4+x^8} \, dx","IntegrateAlgebraic[(Sqrt[1 - x^4]*(1 + x^4))/(1 - x^4 + x^8),x]","\frac{1}{2} (-1)^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{-1} x \sqrt{1-x^4}}{-1+x^4}\right)+\frac{1}{2} \sqrt[4]{-1} \tanh ^{-1}\left(\frac{(-1)^{3/4} x \sqrt{1-x^4}}{-1+x^4}\right)","\frac{\tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{x^4}{\sqrt{2}}}{x \sqrt{1-x^4}}\right)}{2 \sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}-\frac{x^2}{\sqrt{2}}+\frac{x^4}{\sqrt{2}}}{x \sqrt{1-x^4}}\right)}{2 \sqrt{2}}",1,"((-1)^(3/4)*ArcTanh[((-1)^(1/4)*x*Sqrt[1 - x^4])/(-1 + x^4)])/2 + ((-1)^(1/4)*ArcTanh[((-1)^(3/4)*x*Sqrt[1 - x^4])/(-1 + x^4)])/2","A",1
1499,1,104,104,0.5191621,"\int \frac{x^2}{\left(b+a x^4\right)^{3/4} \left(-b^2+a^2 x^8\right)} \, dx","IntegrateAlgebraic[x^2/((b + a*x^4)^(3/4)*(-b^2 + a^2*x^8)),x]","-\frac{x^3}{6 b^2 \left(b+a x^4\right)^{3/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{4\ 2^{3/4} a^{3/4} b^2}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{4\ 2^{3/4} a^{3/4} b^2}","-\frac{x^3}{6 b^2 \left(b+a x^4\right)^{3/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{4\ 2^{3/4} a^{3/4} b^2}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{4\ 2^{3/4} a^{3/4} b^2}",1,"-1/6*x^3/(b^2*(b + a*x^4)^(3/4)) + ArcTan[(2^(1/4)*a^(1/4)*x)/(b + a*x^4)^(1/4)]/(4*2^(3/4)*a^(3/4)*b^2) - ArcTanh[(2^(1/4)*a^(1/4)*x)/(b + a*x^4)^(1/4)]/(4*2^(3/4)*a^(3/4)*b^2)","A",1
1500,1,104,104,0.4026018,"\int \frac{\sqrt{1+\sqrt{1+x^2}}}{x+\sqrt{1+x^2}} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[1 + x^2]]/(x + Sqrt[1 + x^2]),x]","\frac{2 x \left(11+3 x^2\right)}{15 \sqrt{1+\sqrt{1+x^2}}}-\frac{2}{15} \left(1+3 x^2\right) \sqrt{1+\sqrt{1+x^2}}+\sqrt{1+x^2} \left(\frac{8 x}{15 \sqrt{1+\sqrt{1+x^2}}}-\frac{2}{15} \sqrt{1+\sqrt{1+x^2}}\right)","\frac{2 x \left(11+3 x^2\right)}{15 \sqrt{1+\sqrt{1+x^2}}}-\frac{2}{15} \left(1+3 x^2\right) \sqrt{1+\sqrt{1+x^2}}+\sqrt{1+x^2} \left(\frac{8 x}{15 \sqrt{1+\sqrt{1+x^2}}}-\frac{2}{15} \sqrt{1+\sqrt{1+x^2}}\right)",1,"(2*x*(11 + 3*x^2))/(15*Sqrt[1 + Sqrt[1 + x^2]]) - (2*(1 + 3*x^2)*Sqrt[1 + Sqrt[1 + x^2]])/15 + Sqrt[1 + x^2]*((8*x)/(15*Sqrt[1 + Sqrt[1 + x^2]]) - (2*Sqrt[1 + Sqrt[1 + x^2]])/15)","A",1
1501,1,105,105,0.1129368,"\int \frac{(2+x)^2}{x \left(4-2 x+x^2\right) \sqrt[3]{1+x+x^2}} \, dx","IntegrateAlgebraic[(2 + x)^2/(x*(4 - 2*x + x^2)*(1 + x + x^2)^(1/3)),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\frac{2}{\sqrt{3}}-\frac{2 x}{\sqrt{3}}+\frac{\sqrt[3]{1+x+x^2}}{\sqrt{3}}}{\sqrt[3]{1+x+x^2}}\right)+\log \left(-1+x+\sqrt[3]{1+x+x^2}\right)-\frac{1}{2} \log \left(1-2 x+x^2+(1-x) \sqrt[3]{1+x+x^2}+\left(1+x+x^2\right)^{2/3}\right)","-\sqrt{3} \tan ^{-1}\left(\frac{\frac{2}{\sqrt{3}}-\frac{2 x}{\sqrt{3}}+\frac{\sqrt[3]{1+x+x^2}}{\sqrt{3}}}{\sqrt[3]{1+x+x^2}}\right)+\log \left(-1+x+\sqrt[3]{1+x+x^2}\right)-\frac{1}{2} \log \left(1-2 x+x^2+(1-x) \sqrt[3]{1+x+x^2}+\left(1+x+x^2\right)^{2/3}\right)",1,"-(Sqrt[3]*ArcTan[(2/Sqrt[3] - (2*x)/Sqrt[3] + (1 + x + x^2)^(1/3)/Sqrt[3])/(1 + x + x^2)^(1/3)]) + Log[-1 + x + (1 + x + x^2)^(1/3)] - Log[1 - 2*x + x^2 + (1 - x)*(1 + x + x^2)^(1/3) + (1 + x + x^2)^(2/3)]/2","A",1
1502,1,105,105,0.1850517,"\int \frac{1}{\left(-2 b+a x^2\right) \sqrt[4]{-b+a x^2}} \, dx","IntegrateAlgebraic[1/((-2*b + a*x^2)*(-b + a*x^2)^(1/4)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^2}}{\sqrt{a} x}\right)}{2 \sqrt{2} \sqrt{a} b^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^2}}{\sqrt{a} x}\right)}{2 \sqrt{2} \sqrt{a} b^{3/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^2}}{\sqrt{a} x}\right)}{2 \sqrt{2} \sqrt{a} b^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^2}}{\sqrt{a} x}\right)}{2 \sqrt{2} \sqrt{a} b^{3/4}}",1,"ArcTan[(Sqrt[2]*b^(1/4)*(-b + a*x^2)^(1/4))/(Sqrt[a]*x)]/(2*Sqrt[2]*Sqrt[a]*b^(3/4)) - ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^2)^(1/4))/(Sqrt[a]*x)]/(2*Sqrt[2]*Sqrt[a]*b^(3/4))","A",1
1503,1,105,105,15.7086014,"\int \frac{-1+x}{x^7 \sqrt[3]{1+x^3}} \, dx","IntegrateAlgebraic[(-1 + x)/(x^7*(1 + x^3)^(1/3)),x]","\frac{\left(1+x^3\right)^{2/3} \left(15-18 x-20 x^3+27 x^4\right)}{90 x^6}-\frac{2 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^3}}{\sqrt{3}}\right)}{9 \sqrt{3}}-\frac{2}{27} \log \left(-1+\sqrt[3]{1+x^3}\right)+\frac{1}{27} \log \left(1+\sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)","\frac{\left(1+x^3\right)^{2/3} \left(15-18 x-20 x^3+27 x^4\right)}{90 x^6}-\frac{2 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^3}}{\sqrt{3}}\right)}{9 \sqrt{3}}-\frac{2}{27} \log \left(-1+\sqrt[3]{1+x^3}\right)+\frac{1}{27} \log \left(1+\sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)",1,"((1 + x^3)^(2/3)*(15 - 18*x - 20*x^3 + 27*x^4))/(90*x^6) - (2*ArcTan[1/Sqrt[3] + (2*(1 + x^3)^(1/3))/Sqrt[3]])/(9*Sqrt[3]) - (2*Log[-1 + (1 + x^3)^(1/3)])/27 + Log[1 + (1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/27","A",1
1504,1,105,105,0.5458769,"\int \frac{3 c+2 b x+a x^2}{\sqrt[3]{c+b x+a x^2} \left(c+b x+a x^2+x^3\right)} \, dx","IntegrateAlgebraic[(3*c + 2*b*x + a*x^2)/((c + b*x + a*x^2)^(1/3)*(c + b*x + a*x^2 + x^3)),x]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{c+b x+a x^2}}{-2 x+\sqrt[3]{c+b x+a x^2}}\right)+\log \left(x+\sqrt[3]{c+b x+a x^2}\right)-\frac{1}{2} \log \left(x^2-x \sqrt[3]{c+b x+a x^2}+\left(c+b x+a x^2\right)^{2/3}\right)","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{c+b x+a x^2}}{-2 x+\sqrt[3]{c+b x+a x^2}}\right)+\log \left(x+\sqrt[3]{c+b x+a x^2}\right)-\frac{1}{2} \log \left(x^2-x \sqrt[3]{c+b x+a x^2}+\left(c+b x+a x^2\right)^{2/3}\right)",1,"Sqrt[3]*ArcTan[(Sqrt[3]*(c + b*x + a*x^2)^(1/3))/(-2*x + (c + b*x + a*x^2)^(1/3))] + Log[x + (c + b*x + a*x^2)^(1/3)] - Log[x^2 - x*(c + b*x + a*x^2)^(1/3) + (c + b*x + a*x^2)^(2/3)]/2","A",1
1505,1,104,105,0.5485679,"\int \frac{-b+a x^2}{\left(b+a x^4\right) \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[(-b + a*x^2)/((b + a*x^4)*(b*x^2 + a*x^4)^(1/4)),x]","-\frac{1}{4} \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 a \log (x)+2 a \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}+\text{$\#$1}^5}\&\right]","-\frac{1}{4} \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 a \log (x)-2 a \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}-\text{$\#$1}^5}\&\right]",1,"-1/4*RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (-2*a*Log[x] + 2*a*Log[(b*x^2 + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1) + #1^5) & ]","A",1
1506,1,104,105,0.0017294,"\int \frac{-b+a x^2}{\left(b+a x^4\right) \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[(-b + a*x^2)/((b + a*x^4)*(b*x^2 + a*x^4)^(1/4)),x]","-\frac{1}{4} \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 a \log (x)+2 a \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}+\text{$\#$1}^5}\&\right]","-\frac{1}{4} \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 a \log (x)-2 a \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}-\text{$\#$1}^5}\&\right]",1,"-1/4*RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (-2*a*Log[x] + 2*a*Log[(b*x^2 + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1) + #1^5) & ]","A",1
1507,1,105,105,0.6071102,"\int \frac{\sqrt[4]{b x^2+a x^4}}{x^4 \left(b+a x^4\right)} \, dx","IntegrateAlgebraic[(b*x^2 + a*x^4)^(1/4)/(x^4*(b + a*x^4)),x]","-\frac{2 \left(b+a x^2\right) \sqrt[4]{b x^2+a x^4}}{5 b^2 x^3}+\frac{a \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]}{4 b}","-\frac{2 \left(b+a x^2\right) \sqrt[4]{b x^2+a x^4}}{5 b^2 x^3}+\frac{a \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]}{4 b}",1,"(-2*(b + a*x^2)*(b*x^2 + a*x^4)^(1/4))/(5*b^2*x^3) + (a*RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (-(Log[x]*#1) + Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1)/(-a + #1^4) & ])/(4*b)","A",1
1508,1,105,105,0.0015374,"\int \frac{\sqrt[4]{b x^2+a x^4}}{x^4 \left(b+a x^4\right)} \, dx","IntegrateAlgebraic[(b*x^2 + a*x^4)^(1/4)/(x^4*(b + a*x^4)),x]","-\frac{2 \left(b+a x^2\right) \sqrt[4]{b x^2+a x^4}}{5 b^2 x^3}+\frac{a \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]}{4 b}","-\frac{2 \left(b+a x^2\right) \sqrt[4]{b x^2+a x^4}}{5 b^2 x^3}+\frac{a \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]}{4 b}",1,"(-2*(b + a*x^2)*(b*x^2 + a*x^4)^(1/4))/(5*b^2*x^3) + (a*RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (-(Log[x]*#1) + Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1)/(-a + #1^4) & ])/(4*b)","A",1
1509,1,105,105,0.0015551,"\int \frac{\sqrt[4]{b x^2+a x^4}}{x^4 \left(b+a x^4\right)} \, dx","IntegrateAlgebraic[(b*x^2 + a*x^4)^(1/4)/(x^4*(b + a*x^4)),x]","-\frac{2 \left(b+a x^2\right) \sqrt[4]{b x^2+a x^4}}{5 b^2 x^3}+\frac{a \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]}{4 b}","-\frac{2 \left(b+a x^2\right) \sqrt[4]{b x^2+a x^4}}{5 b^2 x^3}+\frac{a \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]}{4 b}",1,"(-2*(b + a*x^2)*(b*x^2 + a*x^4)^(1/4))/(5*b^2*x^3) + (a*RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (-(Log[x]*#1) + Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1)/(-a + #1^4) & ])/(4*b)","A",1
1510,1,105,105,2.5659679,"\int \frac{x \left(3+5 x^2\right)}{\sqrt[3]{1+x^2} \left(-1+x^3+x^5\right)} \, dx","IntegrateAlgebraic[(x*(3 + 5*x^2))/((1 + x^2)^(1/3)*(-1 + x^3 + x^5)),x]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x \sqrt[3]{1+x^2}}{2+x \sqrt[3]{1+x^2}}\right)+2 \tanh ^{-1}\left(1-2 x \sqrt[3]{1+x^2}\right)+\frac{1}{2} \log \left(x^2 \left(1+x^2\right)^{2/3}\right)-\frac{1}{2} \log \left(1+x \sqrt[3]{1+x^2}+x^2 \left(1+x^2\right)^{2/3}\right)","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x \sqrt[3]{1+x^2}}{2+x \sqrt[3]{1+x^2}}\right)+2 \tanh ^{-1}\left(1-2 x \sqrt[3]{1+x^2}\right)+\frac{1}{2} \log \left(x^2 \left(1+x^2\right)^{2/3}\right)-\frac{1}{2} \log \left(1+x \sqrt[3]{1+x^2}+x^2 \left(1+x^2\right)^{2/3}\right)",1,"Sqrt[3]*ArcTan[(Sqrt[3]*x*(1 + x^2)^(1/3))/(2 + x*(1 + x^2)^(1/3))] + 2*ArcTanh[1 - 2*x*(1 + x^2)^(1/3)] + Log[x^2*(1 + x^2)^(2/3)]/2 - Log[1 + x*(1 + x^2)^(1/3) + x^2*(1 + x^2)^(2/3)]/2","A",1
1511,1,105,105,3.3845681,"\int \frac{\left(-1+x^5\right)^{2/3} \left(3+2 x^5\right) \left(-2+x^3+2 x^5\right)}{x^6 \left(-1+x^3+x^5\right)} \, dx","IntegrateAlgebraic[((-1 + x^5)^(2/3)*(3 + 2*x^5)*(-2 + x^3 + 2*x^5))/(x^6*(-1 + x^3 + x^5)),x]","\frac{3 \left(-1+x^5\right)^{2/3} \left(-4-5 x^3+4 x^5\right)}{10 x^5}-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-1+x^5}}\right)-\log \left(x+\sqrt[3]{-1+x^5}\right)+\frac{1}{2} \log \left(x^2-x \sqrt[3]{-1+x^5}+\left(-1+x^5\right)^{2/3}\right)","\frac{3 \left(-1+x^5\right)^{2/3} \left(-4-5 x^3+4 x^5\right)}{10 x^5}-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-1+x^5}}\right)-\log \left(x+\sqrt[3]{-1+x^5}\right)+\frac{1}{2} \log \left(x^2-x \sqrt[3]{-1+x^5}+\left(-1+x^5\right)^{2/3}\right)",1,"(3*(-1 + x^5)^(2/3)*(-4 - 5*x^3 + 4*x^5))/(10*x^5) - Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(-1 + x^5)^(1/3))] - Log[x + (-1 + x^5)^(1/3)] + Log[x^2 - x*(-1 + x^5)^(1/3) + (-1 + x^5)^(2/3)]/2","A",1
1512,1,105,105,0.1751769,"\int \frac{\left(1+x^3\right)^{2/3} \left(-2+x^3+x^6\right)}{x^9} \, dx","IntegrateAlgebraic[((1 + x^3)^(2/3)*(-2 + x^3 + x^6))/x^9,x]","\frac{\left(1+x^3\right)^{2/3} \left(5-2 x^3-17 x^6\right)}{20 x^8}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(-x+\sqrt[3]{1+x^3}\right)+\frac{1}{6} \log \left(x^2+x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)","\frac{\left(1+x^3\right)^{2/3} \left(5-2 x^3-17 x^6\right)}{20 x^8}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(-x+\sqrt[3]{1+x^3}\right)+\frac{1}{6} \log \left(x^2+x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)",1,"((1 + x^3)^(2/3)*(5 - 2*x^3 - 17*x^6))/(20*x^8) + ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^3)^(1/3))]/Sqrt[3] - Log[-x + (1 + x^3)^(1/3)]/3 + Log[x^2 + x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/6","A",1
1513,1,105,105,2.2567688,"\int \frac{\left(-1+x^6\right)^{2/3} \left(1+x^6\right) \left(-1-x^3+x^6\right)}{x^6 \left(-1+x^3+x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^6)^(2/3)*(1 + x^6)*(-1 - x^3 + x^6))/(x^6*(-1 + x^3 + x^6)),x]","\frac{\left(-1+x^6\right)^{2/3} \left(-1-5 x^3+x^6\right)}{5 x^5}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-1+x^6}}\right)}{\sqrt{3}}-\frac{2}{3} \log \left(x+\sqrt[3]{-1+x^6}\right)+\frac{1}{3} \log \left(x^2-x \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)","\frac{\left(-1+x^6\right)^{2/3} \left(-1-5 x^3+x^6\right)}{5 x^5}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-1+x^6}}\right)}{\sqrt{3}}-\frac{2}{3} \log \left(x+\sqrt[3]{-1+x^6}\right)+\frac{1}{3} \log \left(x^2-x \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)",1,"((-1 + x^6)^(2/3)*(-1 - 5*x^3 + x^6))/(5*x^5) - (2*ArcTan[(Sqrt[3]*x)/(-x + 2*(-1 + x^6)^(1/3))])/Sqrt[3] - (2*Log[x + (-1 + x^6)^(1/3)])/3 + Log[x^2 - x*(-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)]/3","A",1
1514,1,105,105,0.1661301,"\int \frac{\left(-1+x^3\right)^{2/3} \left(-2+2 x^3+x^6\right)}{x^9} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(-2 + 2*x^3 + x^6))/x^9,x]","\frac{\left(-1+x^3\right)^{2/3} \left(1-2 x^3-x^6\right)}{4 x^8}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(-x+\sqrt[3]{-1+x^3}\right)+\frac{1}{6} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)","\frac{\left(-1+x^3\right)^{2/3} \left(1-2 x^3-x^6\right)}{4 x^8}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(-x+\sqrt[3]{-1+x^3}\right)+\frac{1}{6} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)",1,"((-1 + x^3)^(2/3)*(1 - 2*x^3 - x^6))/(4*x^8) + ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^3)^(1/3))]/Sqrt[3] - Log[-x + (-1 + x^3)^(1/3)]/3 + Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)]/6","A",1
1515,1,105,105,0.387513,"\int \frac{-2+x^4}{\sqrt[4]{1+x^4} \left(-2+x^4+x^8\right)} \, dx","IntegrateAlgebraic[(-2 + x^4)/((1 + x^4)^(1/4)*(-2 + x^4 + x^8)),x]","\frac{1}{3} \sqrt[4]{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{2} \sqrt[4]{1+x^4}}\right)+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{1+x^4}}\right)}{6 \sqrt[4]{2}}+\frac{1}{3} \sqrt[4]{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{2} \sqrt[4]{1+x^4}}\right)+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{1+x^4}}\right)}{6 \sqrt[4]{2}}","\frac{1}{3} \sqrt[4]{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{2} \sqrt[4]{1+x^4}}\right)+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{1+x^4}}\right)}{6 \sqrt[4]{2}}+\frac{1}{3} \sqrt[4]{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{2} \sqrt[4]{1+x^4}}\right)+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{1+x^4}}\right)}{6 \sqrt[4]{2}}",1,"(2^(1/4)*ArcTan[x/(2^(1/4)*(1 + x^4)^(1/4))])/3 + ArcTan[(2^(1/4)*x)/(1 + x^4)^(1/4)]/(6*2^(1/4)) + (2^(1/4)*ArcTanh[x/(2^(1/4)*(1 + x^4)^(1/4))])/3 + ArcTanh[(2^(1/4)*x)/(1 + x^4)^(1/4)]/(6*2^(1/4))","A",1
1516,1,105,105,0.2310118,"\int \frac{\sqrt[4]{-1+x^4} \left(2+x^4\right)}{x^2 \left(2+2 x^4+x^8\right)} \, dx","IntegrateAlgebraic[((-1 + x^4)^(1/4)*(2 + x^4))/(x^2*(2 + 2*x^4 + x^8)),x]","-\frac{\sqrt[4]{-1+x^4}}{x}+\frac{1}{8} \text{RootSum}\left[5-6 \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-5 \log (x)+5 \log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right)+3 \log (x) \text{$\#$1}^4-3 \log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 \text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]","-\frac{\sqrt[4]{-1+x^4}}{x}+\frac{1}{8} \text{RootSum}\left[5-6 \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-5 \log (x)+5 \log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right)+3 \log (x) \text{$\#$1}^4-3 \log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 \text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]",1,"-((-1 + x^4)^(1/4)/x) + RootSum[5 - 6*#1^4 + 2*#1^8 & , (-5*Log[x] + 5*Log[(-1 + x^4)^(1/4) - x*#1] + 3*Log[x]*#1^4 - 3*Log[(-1 + x^4)^(1/4) - x*#1]*#1^4)/(-3*#1^3 + 2*#1^7) & ]/8","A",1
1517,1,105,105,0.0009495,"\int \frac{\sqrt[4]{-1+x^4} \left(2+x^4\right)}{x^2 \left(2+2 x^4+x^8\right)} \, dx","IntegrateAlgebraic[((-1 + x^4)^(1/4)*(2 + x^4))/(x^2*(2 + 2*x^4 + x^8)),x]","-\frac{\sqrt[4]{-1+x^4}}{x}+\frac{1}{8} \text{RootSum}\left[5-6 \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-5 \log (x)+5 \log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right)+3 \log (x) \text{$\#$1}^4-3 \log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 \text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]","-\frac{\sqrt[4]{-1+x^4}}{x}+\frac{1}{8} \text{RootSum}\left[5-6 \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-5 \log (x)+5 \log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right)+3 \log (x) \text{$\#$1}^4-3 \log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 \text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]",1,"-((-1 + x^4)^(1/4)/x) + RootSum[5 - 6*#1^4 + 2*#1^8 & , (-5*Log[x] + 5*Log[(-1 + x^4)^(1/4) - x*#1] + 3*Log[x]*#1^4 - 3*Log[(-1 + x^4)^(1/4) - x*#1]*#1^4)/(-3*#1^3 + 2*#1^7) & ]/8","A",1
1518,1,105,105,0.5019248,"\int \frac{-2 b-2 a x^4+x^8}{x^4 \sqrt[4]{b+a x^4}} \, dx","IntegrateAlgebraic[(-2*b - 2*a*x^4 + x^8)/(x^4*(b + a*x^4)^(1/4)),x]","\frac{\left(8 a+3 x^4\right) \left(b+a x^4\right)^{3/4}}{12 a x^3}+\frac{\left(-8 a^2-b\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{8 a^{5/4}}+\frac{\left(-8 a^2-b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{8 a^{5/4}}","\frac{\left(8 a+3 x^4\right) \left(b+a x^4\right)^{3/4}}{12 a x^3}+\frac{\left(-8 a^2-b\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{8 a^{5/4}}+\frac{\left(-8 a^2-b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{8 a^{5/4}}",1,"((8*a + 3*x^4)*(b + a*x^4)^(3/4))/(12*a*x^3) + ((-8*a^2 - b)*ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)])/(8*a^(5/4)) + ((-8*a^2 - b)*ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)])/(8*a^(5/4))","A",1
1519,1,105,105,0.4748691,"\int \frac{1-2 x^4+2 x^8}{\sqrt[4]{1+x^4} \left(-2-x^4+x^8\right)} \, dx","IntegrateAlgebraic[(1 - 2*x^4 + 2*x^8)/((1 + x^4)^(1/4)*(-2 - x^4 + x^8)),x]","-\frac{5 x}{3 \sqrt[4]{1+x^4}}+\tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\frac{5 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{3}{2}} x}{\sqrt[4]{1+x^4}}\right)}{6\ 2^{3/4} \sqrt[4]{3}}+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\frac{5 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{3}{2}} x}{\sqrt[4]{1+x^4}}\right)}{6\ 2^{3/4} \sqrt[4]{3}}","-\frac{5 x}{3 \sqrt[4]{1+x^4}}+\tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\frac{5 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{3}{2}} x}{\sqrt[4]{1+x^4}}\right)}{6\ 2^{3/4} \sqrt[4]{3}}+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\frac{5 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{3}{2}} x}{\sqrt[4]{1+x^4}}\right)}{6\ 2^{3/4} \sqrt[4]{3}}",1,"(-5*x)/(3*(1 + x^4)^(1/4)) + ArcTan[x/(1 + x^4)^(1/4)] - (5*ArcTan[((3/2)^(1/4)*x)/(1 + x^4)^(1/4)])/(6*2^(3/4)*3^(1/4)) + ArcTanh[x/(1 + x^4)^(1/4)] - (5*ArcTanh[((3/2)^(1/4)*x)/(1 + x^4)^(1/4)])/(6*2^(3/4)*3^(1/4))","A",1
1520,1,105,105,2.6775624,"\int \frac{\left(1+x^3+x^8\right)^{2/3} \left(-3+5 x^8\right)}{x^3 \left(1+x^8\right)} \, dx","IntegrateAlgebraic[((1 + x^3 + x^8)^(2/3)*(-3 + 5*x^8))/(x^3*(1 + x^8)),x]","\frac{3 \left(1+x^3+x^8\right)^{2/3}}{2 x^2}-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3+x^8}}\right)+\log \left(-x+\sqrt[3]{1+x^3+x^8}\right)-\frac{1}{2} \log \left(x^2+x \sqrt[3]{1+x^3+x^8}+\left(1+x^3+x^8\right)^{2/3}\right)","\frac{3 \left(1+x^3+x^8\right)^{2/3}}{2 x^2}-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3+x^8}}\right)+\log \left(-x+\sqrt[3]{1+x^3+x^8}\right)-\frac{1}{2} \log \left(x^2+x \sqrt[3]{1+x^3+x^8}+\left(1+x^3+x^8\right)^{2/3}\right)",1,"(3*(1 + x^3 + x^8)^(2/3))/(2*x^2) - Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^3 + x^8)^(1/3))] + Log[-x + (1 + x^3 + x^8)^(1/3)] - Log[x^2 + x*(1 + x^3 + x^8)^(1/3) + (1 + x^3 + x^8)^(2/3)]/2","A",1
1521,1,105,105,1.2721716,"\int \frac{-b+2 a x^8}{\sqrt[4]{b+a x^4} \left(-b+a x^8\right)} \, dx","IntegrateAlgebraic[(-b + 2*a*x^8)/((b + a*x^4)^(1/4)*(-b + a*x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{1}{8} \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{1}{8} \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(1/4) + ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(1/4) + RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (-Log[x] + Log[(b + a*x^4)^(1/4) - x*#1])/#1 & ]/8","A",1
1522,1,105,105,2.1803409,"\int \frac{\sqrt{-1-x^2+x^6} \left(1+2 x^6\right)}{8-x^4-16 x^6+8 x^{12}} \, dx","IntegrateAlgebraic[(Sqrt[-1 - x^2 + x^6]*(1 + 2*x^6))/(8 - x^4 - 16*x^6 + 8*x^12),x]","\frac{1}{8} \sqrt{\frac{1}{2} \left(4-\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{4-\sqrt{2}} x}{2 \sqrt{-1-x^2+x^6}}\right)-\frac{1}{8} \sqrt{\frac{1}{2} \left(4+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{4+\sqrt{2}} x}{2 \sqrt{-1-x^2+x^6}}\right)","\frac{1}{8} \sqrt{\frac{1}{2} \left(4-\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{4-\sqrt{2}} x}{2 \sqrt{-1-x^2+x^6}}\right)-\frac{1}{8} \sqrt{\frac{1}{2} \left(4+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{4+\sqrt{2}} x}{2 \sqrt{-1-x^2+x^6}}\right)",1,"(Sqrt[(4 - Sqrt[2])/2]*ArcTan[(Sqrt[4 - Sqrt[2]]*x)/(2*Sqrt[-1 - x^2 + x^6])])/8 - (Sqrt[(4 + Sqrt[2])/2]*ArcTan[(Sqrt[4 + Sqrt[2]]*x)/(2*Sqrt[-1 - x^2 + x^6])])/8","A",1
1523,1,105,105,0.1443953,"\int \frac{1}{\left(1+x^2\right) \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[1/((1 + x^2)*Sqrt[x + Sqrt[1 + x^2]]),x]","\sqrt{2} \tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x}{\sqrt{2}}+\frac{\sqrt{1+x^2}}{\sqrt{2}}}{\sqrt{x+\sqrt{1+x^2}}}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{\frac{1}{\sqrt{2}}+\frac{x}{\sqrt{2}}+\frac{\sqrt{1+x^2}}{\sqrt{2}}}{\sqrt{x+\sqrt{1+x^2}}}\right)","\sqrt{2} \tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x}{\sqrt{2}}+\frac{\sqrt{1+x^2}}{\sqrt{2}}}{\sqrt{x+\sqrt{1+x^2}}}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{\frac{1}{\sqrt{2}}+\frac{x}{\sqrt{2}}+\frac{\sqrt{1+x^2}}{\sqrt{2}}}{\sqrt{x+\sqrt{1+x^2}}}\right)",1,"Sqrt[2]*ArcTan[(-(1/Sqrt[2]) + x/Sqrt[2] + Sqrt[1 + x^2]/Sqrt[2])/Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[2]*ArcTanh[(1/Sqrt[2] + x/Sqrt[2] + Sqrt[1 + x^2]/Sqrt[2])/Sqrt[x + Sqrt[1 + x^2]]]","A",1
1524,1,105,105,0.1359273,"\int \frac{\sqrt{x-\sqrt{1+x^2}}}{x^2+\sqrt{1+x^2}} \, dx","IntegrateAlgebraic[Sqrt[x - Sqrt[1 + x^2]]/(x^2 + Sqrt[1 + x^2]),x]","\text{RootSum}\left[1-2 \text{$\#$1}^2-2 \text{$\#$1}^4-2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{x-\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}+\log \left(\sqrt{x-\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^5}{-1-2 \text{$\#$1}^2-3 \text{$\#$1}^4+2 \text{$\#$1}^6}\&\right]","\text{RootSum}\left[1-2 \text{$\#$1}^2-2 \text{$\#$1}^4-2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{x-\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}+\log \left(\sqrt{x-\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^5}{-1-2 \text{$\#$1}^2-3 \text{$\#$1}^4+2 \text{$\#$1}^6}\&\right]",1,"RootSum[1 - 2*#1^2 - 2*#1^4 - 2*#1^6 + #1^8 & , (Log[Sqrt[x - Sqrt[1 + x^2]] - #1]*#1 + Log[Sqrt[x - Sqrt[1 + x^2]] - #1]*#1^5)/(-1 - 2*#1^2 - 3*#1^4 + 2*#1^6) & ]","A",1
1525,1,105,105,0.5421673,"\int \frac{\sqrt{a x^2+\sqrt{b+a^2 x^4}}}{x^2} \, dx","IntegrateAlgebraic[Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/x^2,x]","-\frac{\sqrt{a x^2+\sqrt{b+a^2 x^4}}}{x}+\frac{\sqrt{a} \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{\sqrt{2}}","-\frac{\sqrt{a x^2+\sqrt{b+a^2 x^4}}}{x}+\frac{\sqrt{a} \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{\sqrt{2}}",1,"-(Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/x) + (Sqrt[a]*Log[I*a*x^2 + I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]])/Sqrt[2]","A",1
1526,1,102,105,0.1433394,"\int \frac{-1+x}{(1+x) \sqrt{1+\sqrt{1+\sqrt{1+x}}}} \, dx","IntegrateAlgebraic[(-1 + x)/((1 + x)*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]),x]","\frac{-8 \sqrt{1+\sqrt{1+x}} \left(-8+3 \sqrt{1+x}\right)-8 \left(-113-\sqrt{1+x}-15 (1+x)\right)}{105 \sqrt{1+\sqrt{1+\sqrt{1+x}}}}+4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{1+\sqrt{1+\sqrt{1+x}}}}{\sqrt{2}}\right)","\frac{-8 (-128-15 x)+64 \sqrt{1+\sqrt{1+x}}+\sqrt{1+x} \left(8-24 \sqrt{1+\sqrt{1+x}}\right)}{105 \sqrt{1+\sqrt{1+\sqrt{1+x}}}}+4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{1+\sqrt{1+\sqrt{1+x}}}}{\sqrt{2}}\right)",1,"(-8*Sqrt[1 + Sqrt[1 + x]]*(-8 + 3*Sqrt[1 + x]) - 8*(-113 - Sqrt[1 + x] - 15*(1 + x)))/(105*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]) + 4*Sqrt[2]*ArcTanh[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]/Sqrt[2]]","A",1
1527,1,153,106,0.2186783,"\int \frac{1}{(-2+x) \sqrt[3]{1+2 x+x^2}} \, dx","IntegrateAlgebraic[1/((-2 + x)*(1 + 2*x + x^2)^(1/3)),x]","\frac{\tan ^{-1}\left(\frac{3^{5/6} \sqrt[3]{1+2 x+x^2}}{2+2 x+\sqrt[3]{3} \sqrt[3]{1+2 x+x^2}}\right)}{\sqrt[6]{3}}-\frac{\log (1+x)}{3\ 3^{2/3}}+\frac{\log \left(3^{2/3}+3^{2/3} x-3 \sqrt[3]{1+2 x+x^2}\right)}{3^{2/3}}-\frac{\log \left(3^{2/3}+3^{2/3} x+3 \sqrt[3]{1+2 x+x^2}+\sqrt[3]{3} \left(1+2 x+x^2\right)^{2/3}\right)}{2\ 3^{2/3}}","\frac{\left((1+x)^2\right)^{2/3} \left(-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x}}{3^{5/6}}\right)}{\sqrt[6]{3}}+\frac{\log \left(-3+3^{2/3} \sqrt[3]{1+x}\right)}{3^{2/3}}-\frac{\log \left(3+3^{2/3} \sqrt[3]{1+x}+\sqrt[3]{3} (1+x)^{2/3}\right)}{2\ 3^{2/3}}\right)}{(1+x)^{4/3}}",1,"ArcTan[(3^(5/6)*(1 + 2*x + x^2)^(1/3))/(2 + 2*x + 3^(1/3)*(1 + 2*x + x^2)^(1/3))]/3^(1/6) - Log[1 + x]/(3*3^(2/3)) + Log[3^(2/3) + 3^(2/3)*x - 3*(1 + 2*x + x^2)^(1/3)]/3^(2/3) - Log[3^(2/3) + 3^(2/3)*x + 3*(1 + 2*x + x^2)^(1/3) + 3^(1/3)*(1 + 2*x + x^2)^(2/3)]/(2*3^(2/3))","A",1
1528,1,106,106,0.205298,"\int \frac{\sqrt[3]{-x+x^3}}{x^2} \, dx","IntegrateAlgebraic[(-x + x^3)^(1/3)/x^2,x]","-\frac{3 \sqrt[3]{-x+x^3}}{2 x}-\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)-\frac{1}{2} \log \left(-x+\sqrt[3]{-x+x^3}\right)+\frac{1}{4} \log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)","-\frac{3 \sqrt[3]{-x+x^3}}{2 x}-\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)-\frac{1}{2} \log \left(-x+\sqrt[3]{-x+x^3}\right)+\frac{1}{4} \log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)",1,"(-3*(-x + x^3)^(1/3))/(2*x) - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-x + x^3)^(1/3))])/2 - Log[-x + (-x + x^3)^(1/3)]/2 + Log[x^2 + x*(-x + x^3)^(1/3) + (-x + x^3)^(2/3)]/4","A",1
1529,1,106,106,0.2207449,"\int \frac{\sqrt[3]{-x^2+x^3}}{x} \, dx","IntegrateAlgebraic[(-x^2 + x^3)^(1/3)/x,x]","\sqrt[3]{-x^2+x^3}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(-x+\sqrt[3]{-x^2+x^3}\right)-\frac{1}{6} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)","\sqrt[3]{-x^2+x^3}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(-x+\sqrt[3]{-x^2+x^3}\right)-\frac{1}{6} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)",1,"(-x^2 + x^3)^(1/3) + ArcTan[(Sqrt[3]*x)/(x + 2*(-x^2 + x^3)^(1/3))]/Sqrt[3] + Log[-x + (-x^2 + x^3)^(1/3)]/3 - Log[x^2 + x*(-x^2 + x^3)^(1/3) + (-x^2 + x^3)^(2/3)]/6","A",1
1530,1,106,106,0.5409219,"\int \frac{-3 b+a x^2}{\left(-b+a x^2+x^3\right) \sqrt[4]{-b x+a x^3}} \, dx","IntegrateAlgebraic[(-3*b + a*x^2)/((-b + a*x^2 + x^3)*(-(b*x) + a*x^3)^(1/4)),x]","-\sqrt{2} \tan ^{-1}\left(\frac{-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{-b x+a x^3}}{\sqrt{2}}}{x \sqrt[4]{-b x+a x^3}}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-b x+a x^3}}{x^2+\sqrt{-b x+a x^3}}\right)","-\sqrt{2} \tan ^{-1}\left(\frac{-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{-b x+a x^3}}{\sqrt{2}}}{x \sqrt[4]{-b x+a x^3}}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-b x+a x^3}}{x^2+\sqrt{-b x+a x^3}}\right)",1,"-(Sqrt[2]*ArcTan[(-(x^2/Sqrt[2]) + Sqrt[-(b*x) + a*x^3]/Sqrt[2])/(x*(-(b*x) + a*x^3)^(1/4))]) + Sqrt[2]*ArcTanh[(Sqrt[2]*x*(-(b*x) + a*x^3)^(1/4))/(x^2 + Sqrt[-(b*x) + a*x^3])]","A",1
1531,1,106,106,0.3375407,"\int \frac{1}{(-2 b+a x) \sqrt[4]{-b x^2+a x^3}} \, dx","IntegrateAlgebraic[1/((-2*b + a*x)*(-(b*x^2) + a*x^3)^(1/4)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b x^2+a x^3}}{\sqrt{a} x}\right)}{\sqrt{2} \sqrt{a} b^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b x^2+a x^3}}{\sqrt{a} x}\right)}{\sqrt{2} \sqrt{a} b^{3/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b x^2+a x^3}}{\sqrt{a} x}\right)}{\sqrt{2} \sqrt{a} b^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b x^2+a x^3}}{\sqrt{a} x}\right)}{\sqrt{2} \sqrt{a} b^{3/4}}",1,"ArcTan[(Sqrt[2]*b^(1/4)*(-(b*x^2) + a*x^3)^(1/4))/(Sqrt[a]*x)]/(Sqrt[2]*Sqrt[a]*b^(3/4)) - ArcTanh[(Sqrt[2]*b^(1/4)*(-(b*x^2) + a*x^3)^(1/4))/(Sqrt[a]*x)]/(Sqrt[2]*Sqrt[a]*b^(3/4))","A",1
1532,1,106,106,1.7884816,"\int \frac{\left(-b+a x^4\right) \sqrt[4]{-b x^2+a x^4}}{x^4 \left(b+a x^4\right)} \, dx","IntegrateAlgebraic[((-b + a*x^4)*(-(b*x^2) + a*x^4)^(1/4))/(x^4*(b + a*x^4)),x]","-\frac{2 \left(-b+a x^2\right) \sqrt[4]{-b x^2+a x^4}}{5 b x^3}+\frac{1}{2} a \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]","-\frac{2 \left(-b+a x^2\right) \sqrt[4]{-b x^2+a x^4}}{5 b x^3}+\frac{1}{2} a \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]",1,"(-2*(-b + a*x^2)*(-(b*x^2) + a*x^4)^(1/4))/(5*b*x^3) + (a*RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (-(Log[x]*#1) + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1)/(-a + #1^4) & ])/2","A",1
1533,1,106,106,0.0020872,"\int \frac{\left(-b+a x^4\right) \sqrt[4]{-b x^2+a x^4}}{x^4 \left(b+a x^4\right)} \, dx","IntegrateAlgebraic[((-b + a*x^4)*(-(b*x^2) + a*x^4)^(1/4))/(x^4*(b + a*x^4)),x]","-\frac{2 \left(-b+a x^2\right) \sqrt[4]{-b x^2+a x^4}}{5 b x^3}+\frac{1}{2} a \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]","-\frac{2 \left(-b+a x^2\right) \sqrt[4]{-b x^2+a x^4}}{5 b x^3}+\frac{1}{2} a \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]",1,"(-2*(-b + a*x^2)*(-(b*x^2) + a*x^4)^(1/4))/(5*b*x^3) + (a*RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (-(Log[x]*#1) + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1)/(-a + #1^4) & ])/2","A",1
1534,1,106,106,0.61065,"\int \frac{-b+a x^2}{\left(b-a x^2+x^4\right) \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[(-b + a*x^2)/((b - a*x^2 + x^4)*(b*x^2 + a*x^4)^(1/4)),x]","-\frac{1}{2} \text{RootSum}\left[2 a^2+b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 a \log (x)+2 a \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]","-\frac{1}{2} \text{RootSum}\left[2 a^2+b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 a \log (x)-2 a \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{3 a \text{$\#$1}-2 \text{$\#$1}^5}\&\right]",1,"-1/2*RootSum[2*a^2 + b - 3*a*#1^4 + #1^8 & , (-2*a*Log[x] + 2*a*Log[(b*x^2 + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1 + 2*#1^5) & ]","A",1
1535,1,106,106,0.0019351,"\int \frac{-b+a x^2}{\left(b-a x^2+x^4\right) \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[(-b + a*x^2)/((b - a*x^2 + x^4)*(b*x^2 + a*x^4)^(1/4)),x]","-\frac{1}{2} \text{RootSum}\left[2 a^2+b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 a \log (x)+2 a \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]","-\frac{1}{2} \text{RootSum}\left[2 a^2+b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 a \log (x)-2 a \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{3 a \text{$\#$1}-2 \text{$\#$1}^5}\&\right]",1,"-1/2*RootSum[2*a^2 + b - 3*a*#1^4 + #1^8 & , (-2*a*Log[x] + 2*a*Log[(b*x^2 + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1 + 2*#1^5) & ]","A",1
1536,1,106,106,0.6590515,"\int \frac{\sqrt[4]{b x^2+a x^4}}{x^4 \left(-b+a x^4\right)} \, dx","IntegrateAlgebraic[(b*x^2 + a*x^4)^(1/4)/(x^4*(-b + a*x^4)),x]","\frac{2 \left(b+a x^2\right) \sqrt[4]{b x^2+a x^4}}{5 b^2 x^3}+\frac{a \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]}{4 b}","\frac{2 \left(b+a x^2\right) \sqrt[4]{b x^2+a x^4}}{5 b^2 x^3}+\frac{a \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]}{4 b}",1,"(2*(b + a*x^2)*(b*x^2 + a*x^4)^(1/4))/(5*b^2*x^3) + (a*RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (-(Log[x]*#1) + Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1)/(-a + #1^4) & ])/(4*b)","A",1
1537,1,106,106,0.0022817,"\int \frac{\sqrt[4]{b x^2+a x^4}}{x^4 \left(-b+a x^4\right)} \, dx","IntegrateAlgebraic[(b*x^2 + a*x^4)^(1/4)/(x^4*(-b + a*x^4)),x]","\frac{2 \left(b+a x^2\right) \sqrt[4]{b x^2+a x^4}}{5 b^2 x^3}+\frac{a \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]}{4 b}","\frac{2 \left(b+a x^2\right) \sqrt[4]{b x^2+a x^4}}{5 b^2 x^3}+\frac{a \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]}{4 b}",1,"(2*(b + a*x^2)*(b*x^2 + a*x^4)^(1/4))/(5*b^2*x^3) + (a*RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (-(Log[x]*#1) + Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1)/(-a + #1^4) & ])/(4*b)","A",1
1538,1,106,106,2.3436325,"\int \frac{b^3+a^3 x^3}{\left(-b^3+a^3 x^3\right) \sqrt{b^4+a^4 x^4}} \, dx","IntegrateAlgebraic[(b^3 + a^3*x^3)/((-b^3 + a^3*x^3)*Sqrt[b^4 + a^4*x^4]),x]","-\frac{4 \tan ^{-1}\left(\frac{a b x}{b^2+a b x+a^2 x^2+\sqrt{b^4+a^4 x^4}}\right)}{3 a b}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} a b x}{b^2-2 a b x+a^2 x^2+\sqrt{b^4+a^4 x^4}}\right)}{3 a b}","-\frac{4 \tan ^{-1}\left(\frac{a b x}{b^2+a b x+a^2 x^2+\sqrt{b^4+a^4 x^4}}\right)}{3 a b}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} a b x}{b^2-2 a b x+a^2 x^2+\sqrt{b^4+a^4 x^4}}\right)}{3 a b}",1,"(-4*ArcTan[(a*b*x)/(b^2 + a*b*x + a^2*x^2 + Sqrt[b^4 + a^4*x^4])])/(3*a*b) - (Sqrt[2]*ArcTanh[(Sqrt[2]*a*b*x)/(b^2 - 2*a*b*x + a^2*x^2 + Sqrt[b^4 + a^4*x^4])])/(3*a*b)","A",1
1539,1,106,106,0.2157337,"\int \frac{\left(1+x^3\right)^{2/3} \left(1+x^3+2 x^6\right)}{x^6 \left(-1+2 x^6\right)} \, dx","IntegrateAlgebraic[((1 + x^3)^(2/3)*(1 + x^3 + 2*x^6))/(x^6*(-1 + 2*x^6)),x]","\frac{\left(1+x^3\right)^{2/3} \left(2+7 x^3\right)}{10 x^5}-\frac{1}{6} \text{RootSum}\left[-1-2 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{\log (x)-\log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)+5 \log (x) \text{$\#$1}^3-5 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-\text{$\#$1}+\text{$\#$1}^4}\&\right]","\frac{\left(1+x^3\right)^{2/3} \left(2+7 x^3\right)}{10 x^5}-\frac{1}{6} \text{RootSum}\left[-1-2 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{\log (x)-\log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)+5 \log (x) \text{$\#$1}^3-5 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-\text{$\#$1}+\text{$\#$1}^4}\&\right]",1,"((1 + x^3)^(2/3)*(2 + 7*x^3))/(10*x^5) - RootSum[-1 - 2*#1^3 + #1^6 & , (Log[x] - Log[(1 + x^3)^(1/3) - x*#1] + 5*Log[x]*#1^3 - 5*Log[(1 + x^3)^(1/3) - x*#1]*#1^3)/(-#1 + #1^4) & ]/6","A",1
1540,1,106,106,0.0011294,"\int \frac{\left(1+x^3\right)^{2/3} \left(1+x^3+2 x^6\right)}{x^6 \left(-1+2 x^6\right)} \, dx","IntegrateAlgebraic[((1 + x^3)^(2/3)*(1 + x^3 + 2*x^6))/(x^6*(-1 + 2*x^6)),x]","\frac{\left(1+x^3\right)^{2/3} \left(2+7 x^3\right)}{10 x^5}-\frac{1}{6} \text{RootSum}\left[-1-2 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{\log (x)-\log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)+5 \log (x) \text{$\#$1}^3-5 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-\text{$\#$1}+\text{$\#$1}^4}\&\right]","\frac{\left(1+x^3\right)^{2/3} \left(2+7 x^3\right)}{10 x^5}-\frac{1}{6} \text{RootSum}\left[-1-2 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{\log (x)-\log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)+5 \log (x) \text{$\#$1}^3-5 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-\text{$\#$1}+\text{$\#$1}^4}\&\right]",1,"((1 + x^3)^(2/3)*(2 + 7*x^3))/(10*x^5) - RootSum[-1 - 2*#1^3 + #1^6 & , (Log[x] - Log[(1 + x^3)^(1/3) - x*#1] + 5*Log[x]*#1^3 - 5*Log[(1 + x^3)^(1/3) - x*#1]*#1^3)/(-#1 + #1^4) & ]/6","A",1
1541,1,106,106,2.750061,"\int \frac{\left(1+x^6\right) \left(-1+2 x^6\right) \left(-1+x^4+2 x^6\right)^{5/4}}{x^{10} \left(-1-x^4+2 x^6\right)} \, dx","IntegrateAlgebraic[((1 + x^6)*(-1 + 2*x^6)*(-1 + x^4 + 2*x^6)^(5/4))/(x^10*(-1 - x^4 + 2*x^6)),x]","\frac{\sqrt[4]{-1+x^4+2 x^6} \left(5-19 x^4-20 x^6+104 x^8+38 x^{10}+20 x^{12}\right)}{45 x^9}+\sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-1+x^4+2 x^6}}\right)-\sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-1+x^4+2 x^6}}\right)","\frac{\sqrt[4]{-1+x^4+2 x^6} \left(5-19 x^4-20 x^6+104 x^8+38 x^{10}+20 x^{12}\right)}{45 x^9}+\sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-1+x^4+2 x^6}}\right)-\sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-1+x^4+2 x^6}}\right)",1,"((-1 + x^4 + 2*x^6)^(1/4)*(5 - 19*x^4 - 20*x^6 + 104*x^8 + 38*x^10 + 20*x^12))/(45*x^9) + 2^(1/4)*ArcTan[(2^(1/4)*x)/(-1 + x^4 + 2*x^6)^(1/4)] - 2^(1/4)*ArcTanh[(2^(1/4)*x)/(-1 + x^4 + 2*x^6)^(1/4)]","A",1
1542,1,106,106,13.1446086,"\int \frac{1}{\sqrt[3]{b^2 x^2+a^3 x^3} \left(b+a^6 x^6\right)} \, dx","IntegrateAlgebraic[1/((b^2*x^2 + a^3*x^3)^(1/3)*(b + a^6*x^6)),x]","-\frac{\text{RootSum}\left[a^{18}+a^6 b^{11}-6 a^{15} \text{$\#$1}^3+15 a^{12} \text{$\#$1}^6-20 a^9 \text{$\#$1}^9+15 a^6 \text{$\#$1}^{12}-6 a^3 \text{$\#$1}^{15}+\text{$\#$1}^{18}\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 b}","-\frac{\text{RootSum}\left[a^{18}+a^6 b^{11}-6 a^{15} \text{$\#$1}^3+15 a^{12} \text{$\#$1}^6-20 a^9 \text{$\#$1}^9+15 a^6 \text{$\#$1}^{12}-6 a^3 \text{$\#$1}^{15}+\text{$\#$1}^{18}\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 b}",1,"-1/6*RootSum[a^18 + a^6*b^11 - 6*a^15*#1^3 + 15*a^12*#1^6 - 20*a^9*#1^9 + 15*a^6*#1^12 - 6*a^3*#1^15 + #1^18 & , (-Log[x] + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1])/#1 & ]/b","A",1
1543,1,106,106,0.0020243,"\int \frac{1}{\sqrt[3]{b^2 x^2+a^3 x^3} \left(b+a^6 x^6\right)} \, dx","IntegrateAlgebraic[1/((b^2*x^2 + a^3*x^3)^(1/3)*(b + a^6*x^6)),x]","-\frac{\text{RootSum}\left[a^{18}+a^6 b^{11}-6 a^{15} \text{$\#$1}^3+15 a^{12} \text{$\#$1}^6-20 a^9 \text{$\#$1}^9+15 a^6 \text{$\#$1}^{12}-6 a^3 \text{$\#$1}^{15}+\text{$\#$1}^{18}\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 b}","-\frac{\text{RootSum}\left[a^{18}+a^6 b^{11}-6 a^{15} \text{$\#$1}^3+15 a^{12} \text{$\#$1}^6-20 a^9 \text{$\#$1}^9+15 a^6 \text{$\#$1}^{12}-6 a^3 \text{$\#$1}^{15}+\text{$\#$1}^{18}\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 b}",1,"-1/6*RootSum[a^18 + a^6*b^11 - 6*a^15*#1^3 + 15*a^12*#1^6 - 20*a^9*#1^9 + 15*a^6*#1^12 - 6*a^3*#1^15 + #1^18 & , (-Log[x] + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1])/#1 & ]/b","A",1
1544,1,106,106,15.9134979,"\int \frac{\left(\left(1-3 k^2\right) x+2 k^2 x^3\right) \left(1-2 k^2 x^2+k^4 x^4\right)}{\left(\left(1-x^2\right) \left(1-k^2 x^2\right)\right)^{3/4} \left(1-d+\left(-1+3 d k^2\right) x^2-3 d k^4 x^4+d k^6 x^6\right)} \, dx","IntegrateAlgebraic[(((1 - 3*k^2)*x + 2*k^2*x^3)*(1 - 2*k^2*x^2 + k^4*x^4))/(((1 - x^2)*(1 - k^2*x^2))^(3/4)*(1 - d + (-1 + 3*d*k^2)*x^2 - 3*d*k^4*x^4 + d*k^6*x^6)),x]","\frac{\tan ^{-1}\left(\frac{-\sqrt[4]{d}+\sqrt[4]{d} k^2 x^2}{\sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{3/4}}-\frac{\tanh ^{-1}\left(\frac{-\sqrt[4]{d}+\sqrt[4]{d} k^2 x^2}{\sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{3/4}}","\frac{\tan ^{-1}\left(\frac{-\sqrt[4]{d}+\sqrt[4]{d} k^2 x^2}{\sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{3/4}}-\frac{\tanh ^{-1}\left(\frac{-\sqrt[4]{d}+\sqrt[4]{d} k^2 x^2}{\sqrt[4]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{3/4}}",1,"ArcTan[(-d^(1/4) + d^(1/4)*k^2*x^2)/(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/4)]/d^(3/4) - ArcTanh[(-d^(1/4) + d^(1/4)*k^2*x^2)/(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/4)]/d^(3/4)","A",1
1545,1,106,106,0.3210068,"\int \frac{\sqrt[4]{-1+2 x^4} \left(-1+x^4+x^8\right)}{x^6 \left(-1+x^4\right)} \, dx","IntegrateAlgebraic[((-1 + 2*x^4)^(1/4)*(-1 + x^4 + x^8))/(x^6*(-1 + x^4)),x]","\frac{\left(-1+2 x^4\right)^{5/4}}{5 x^5}+\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-1+2 x^4}}\right)-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-1+2 x^4}}\right)}{2^{3/4}}-\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-1+2 x^4}}\right)+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-1+2 x^4}}\right)}{2^{3/4}}","\frac{\left(-1+2 x^4\right)^{5/4}}{5 x^5}+\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-1+2 x^4}}\right)-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-1+2 x^4}}\right)}{2^{3/4}}-\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-1+2 x^4}}\right)+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-1+2 x^4}}\right)}{2^{3/4}}",1,"(-1 + 2*x^4)^(5/4)/(5*x^5) + ArcTan[x/(-1 + 2*x^4)^(1/4)]/2 - ArcTan[(2^(1/4)*x)/(-1 + 2*x^4)^(1/4)]/2^(3/4) - ArcTanh[x/(-1 + 2*x^4)^(1/4)]/2 + ArcTanh[(2^(1/4)*x)/(-1 + 2*x^4)^(1/4)]/2^(3/4)","A",1
1546,1,106,106,0.5933226,"\int \frac{1}{\sqrt[4]{-b+a x^4} \left(-b-a x^4+x^8\right)} \, dx","IntegrateAlgebraic[1/((-b + a*x^4)^(1/4)*(-b - a*x^4 + x^8)),x]","-\frac{\text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a \log (x)+a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]}{4 b}","-\frac{\text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a \log (x)-a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{3 a \text{$\#$1}-2 \text{$\#$1}^5}\&\right]}{4 b}",1,"-1/4*RootSum[2*a^2 - b - 3*a*#1^4 + #1^8 & , (-(a*Log[x]) + a*Log[(-b + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1 + 2*#1^5) & ]/b","A",1
1547,1,106,106,0.0017223,"\int \frac{1}{\sqrt[4]{-b+a x^4} \left(-b-a x^4+x^8\right)} \, dx","IntegrateAlgebraic[1/((-b + a*x^4)^(1/4)*(-b - a*x^4 + x^8)),x]","-\frac{\text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a \log (x)+a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]}{4 b}","-\frac{\text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a \log (x)-a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{3 a \text{$\#$1}-2 \text{$\#$1}^5}\&\right]}{4 b}",1,"-1/4*RootSum[2*a^2 - b - 3*a*#1^4 + #1^8 & , (-(a*Log[x]) + a*Log[(-b + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1 + 2*#1^5) & ]/b","A",1
1548,1,107,107,0.2563101,"\int x^6 \left(-1+x^3\right)^{2/3} \, dx","IntegrateAlgebraic[x^6*(-1 + x^3)^(2/3),x]","\frac{1}{81} \left(-1+x^3\right)^{2/3} \left(-4 x-3 x^4+9 x^7\right)-\frac{4 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{81 \sqrt{3}}+\frac{4}{243} \log \left(-x+\sqrt[3]{-1+x^3}\right)-\frac{2}{243} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)","\frac{1}{81} \left(-1+x^3\right)^{2/3} \left(-4 x-3 x^4+9 x^7\right)-\frac{4 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{81 \sqrt{3}}+\frac{4}{243} \log \left(-x+\sqrt[3]{-1+x^3}\right)-\frac{2}{243} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)",1,"((-1 + x^3)^(2/3)*(-4*x - 3*x^4 + 9*x^7))/81 - (4*ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^3)^(1/3))])/(81*Sqrt[3]) + (4*Log[-x + (-1 + x^3)^(1/3)])/243 - (2*Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)])/243","A",1
1549,1,107,107,0.2410855,"\int x^6 \left(1+x^3\right)^{2/3} \, dx","IntegrateAlgebraic[x^6*(1 + x^3)^(2/3),x]","\frac{1}{81} \left(1+x^3\right)^{2/3} \left(-4 x+3 x^4+9 x^7\right)+\frac{4 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3}}\right)}{81 \sqrt{3}}-\frac{4}{243} \log \left(-x+\sqrt[3]{1+x^3}\right)+\frac{2}{243} \log \left(x^2+x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)","\frac{1}{81} \left(1+x^3\right)^{2/3} \left(-4 x+3 x^4+9 x^7\right)+\frac{4 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3}}\right)}{81 \sqrt{3}}-\frac{4}{243} \log \left(-x+\sqrt[3]{1+x^3}\right)+\frac{2}{243} \log \left(x^2+x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)",1,"((1 + x^3)^(2/3)*(-4*x + 3*x^4 + 9*x^7))/81 + (4*ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^3)^(1/3))])/(81*Sqrt[3]) - (4*Log[-x + (1 + x^3)^(1/3)])/243 + (2*Log[x^2 + x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)])/243","A",1
1550,1,107,107,0.3145707,"\int x^4 \sqrt[3]{x+x^3} \, dx","IntegrateAlgebraic[x^4*(x + x^3)^(1/3),x]","\frac{1}{108} \sqrt[3]{x+x^3} \left(-5 x+3 x^3+18 x^5\right)-\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)}{54 \sqrt{3}}-\frac{5}{162} \log \left(-x+\sqrt[3]{x+x^3}\right)+\frac{5}{324} \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)","\frac{1}{108} \sqrt[3]{x+x^3} \left(-5 x+3 x^3+18 x^5\right)-\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)}{54 \sqrt{3}}-\frac{5}{162} \log \left(-x+\sqrt[3]{x+x^3}\right)+\frac{5}{324} \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)",1,"((x + x^3)^(1/3)*(-5*x + 3*x^3 + 18*x^5))/108 - (5*ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^3)^(1/3))])/(54*Sqrt[3]) - (5*Log[-x + (x + x^3)^(1/3)])/162 + (5*Log[x^2 + x*(x + x^3)^(1/3) + (x + x^3)^(2/3)])/324","A",1
1551,1,107,107,3.6382033,"\int \frac{x^2 \left(3 a b^2-2 b (2 a+b) x+(a+2 b) x^2\right)}{\left(x (-a+x) (-b+x)^2\right)^{3/4} \left(a b^2-b (2 a+b) x+(a+2 b) x^2+(-1+d) x^3\right)} \, dx","IntegrateAlgebraic[(x^2*(3*a*b^2 - 2*b*(2*a + b)*x + (a + 2*b)*x^2))/((x*(-a + x)*(-b + x)^2)^(3/4)*(a*b^2 - b*(2*a + b)*x + (a + 2*b)*x^2 + (-1 + d)*x^3)),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)}{d^{3/4}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)}{d^{3/4}}",1,"(-2*ArcTan[(d^(1/4)*x)/(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/4)])/d^(3/4) + (2*ArcTanh[(d^(1/4)*x)/(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/4)])/d^(3/4)","A",1
1552,1,107,107,0.2253083,"\int \frac{\sqrt[3]{1+x^4} \left(3+x^4\right)}{x^{17}} \, dx","IntegrateAlgebraic[((1 + x^4)^(1/3)*(3 + x^4))/x^17,x]","\frac{\sqrt[3]{1+x^4} \left(-81-45 x^4+6 x^8-10 x^{12}\right)}{432 x^{16}}+\frac{5 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^4}}{\sqrt{3}}\right)}{108 \sqrt{3}}-\frac{5}{324} \log \left(-1+\sqrt[3]{1+x^4}\right)+\frac{5}{648} \log \left(1+\sqrt[3]{1+x^4}+\left(1+x^4\right)^{2/3}\right)","\frac{\sqrt[3]{1+x^4} \left(-81-45 x^4+6 x^8-10 x^{12}\right)}{432 x^{16}}+\frac{5 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^4}}{\sqrt{3}}\right)}{108 \sqrt{3}}-\frac{5}{324} \log \left(-1+\sqrt[3]{1+x^4}\right)+\frac{5}{648} \log \left(1+\sqrt[3]{1+x^4}+\left(1+x^4\right)^{2/3}\right)",1,"((1 + x^4)^(1/3)*(-81 - 45*x^4 + 6*x^8 - 10*x^12))/(432*x^16) + (5*ArcTan[1/Sqrt[3] + (2*(1 + x^4)^(1/3))/Sqrt[3]])/(108*Sqrt[3]) - (5*Log[-1 + (1 + x^4)^(1/3)])/324 + (5*Log[1 + (1 + x^4)^(1/3) + (1 + x^4)^(2/3)])/648","A",1
1553,1,107,107,0.2604591,"\int \frac{\left(-2+x^2\right) \sqrt[3]{x+x^3}}{x^2 \left(4-2 x^2+x^4\right)} \, dx","IntegrateAlgebraic[((-2 + x^2)*(x + x^3)^(1/3))/(x^2*(4 - 2*x^2 + x^4)),x]","\frac{3 \sqrt[3]{x+x^3}}{4 x}-\frac{1}{8} \text{RootSum}\left[7-10 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-7 \log (x)+7 \log \left(\sqrt[3]{x+x^3}-x \text{$\#$1}\right)+4 \log (x) \text{$\#$1}^3-4 \log \left(\sqrt[3]{x+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-5 \text{$\#$1}^2+4 \text{$\#$1}^5}\&\right]","\frac{3 \sqrt[3]{x+x^3}}{4 x}-\frac{1}{8} \text{RootSum}\left[7-10 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-7 \log (x)+7 \log \left(\sqrt[3]{x+x^3}-x \text{$\#$1}\right)+4 \log (x) \text{$\#$1}^3-4 \log \left(\sqrt[3]{x+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-5 \text{$\#$1}^2+4 \text{$\#$1}^5}\&\right]",1,"(3*(x + x^3)^(1/3))/(4*x) - RootSum[7 - 10*#1^3 + 4*#1^6 & , (-7*Log[x] + 7*Log[(x + x^3)^(1/3) - x*#1] + 4*Log[x]*#1^3 - 4*Log[(x + x^3)^(1/3) - x*#1]*#1^3)/(-5*#1^2 + 4*#1^5) & ]/8","A",1
1554,1,107,107,0.0010452,"\int \frac{\left(-2+x^2\right) \sqrt[3]{x+x^3}}{x^2 \left(4-2 x^2+x^4\right)} \, dx","IntegrateAlgebraic[((-2 + x^2)*(x + x^3)^(1/3))/(x^2*(4 - 2*x^2 + x^4)),x]","\frac{3 \sqrt[3]{x+x^3}}{4 x}-\frac{1}{8} \text{RootSum}\left[7-10 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-7 \log (x)+7 \log \left(\sqrt[3]{x+x^3}-x \text{$\#$1}\right)+4 \log (x) \text{$\#$1}^3-4 \log \left(\sqrt[3]{x+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-5 \text{$\#$1}^2+4 \text{$\#$1}^5}\&\right]","\frac{3 \sqrt[3]{x+x^3}}{4 x}-\frac{1}{8} \text{RootSum}\left[7-10 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-7 \log (x)+7 \log \left(\sqrt[3]{x+x^3}-x \text{$\#$1}\right)+4 \log (x) \text{$\#$1}^3-4 \log \left(\sqrt[3]{x+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-5 \text{$\#$1}^2+4 \text{$\#$1}^5}\&\right]",1,"(3*(x + x^3)^(1/3))/(4*x) - RootSum[7 - 10*#1^3 + 4*#1^6 & , (-7*Log[x] + 7*Log[(x + x^3)^(1/3) - x*#1] + 4*Log[x]*#1^3 - 4*Log[(x + x^3)^(1/3) - x*#1]*#1^3)/(-5*#1^2 + 4*#1^5) & ]/8","A",1
1555,1,107,107,0.37636,"\int \frac{1}{\left(-1+x^4\right)^2 \sqrt[4]{-x^2+x^4}} \, dx","IntegrateAlgebraic[1/((-1 + x^4)^2*(-x^2 + x^4)^(1/4)),x]","\frac{\left(-x^2+x^4\right)^{3/4} \left(-85+2 x^2+67 x^4\right)}{80 x \left(-1+x^2\right)^2 \left(1+x^2\right)}+\frac{15 \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{32 \sqrt[4]{2}}+\frac{15 \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{32 \sqrt[4]{2}}","\frac{\left(-x^2+x^4\right)^{3/4} \left(-85+2 x^2+67 x^4\right)}{80 x \left(-1+x^2\right)^2 \left(1+x^2\right)}+\frac{15 \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{32 \sqrt[4]{2}}+\frac{15 \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{32 \sqrt[4]{2}}",1,"((-x^2 + x^4)^(3/4)*(-85 + 2*x^2 + 67*x^4))/(80*x*(-1 + x^2)^2*(1 + x^2)) + (15*ArcTan[(2^(1/4)*x)/(-x^2 + x^4)^(1/4)])/(32*2^(1/4)) + (15*ArcTanh[(2^(1/4)*x)/(-x^2 + x^4)^(1/4)])/(32*2^(1/4))","A",1
1556,1,107,107,2.1027657,"\int \frac{\left(-1+x^2\right) \left(1-x+x^2-x^3+x^4\right)}{\left(1-x+x^2\right)^2 \left(1+x+x^2\right) \sqrt{1+3 x^2+x^4}} \, dx","IntegrateAlgebraic[((-1 + x^2)*(1 - x + x^2 - x^3 + x^4))/((1 - x + x^2)^2*(1 + x + x^2)*Sqrt[1 + 3*x^2 + x^4]),x]","\frac{\sqrt{1+3 x^2+x^4}}{4 \left(1-x+x^2\right)}-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{1-x+x^2+\sqrt{1+3 x^2+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x}{1+x+x^2+\sqrt{1+3 x^2+x^4}}\right)}{2 \sqrt{2}}","\frac{\sqrt{1+3 x^2+x^4}}{4 \left(1-x+x^2\right)}-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{1-x+x^2+\sqrt{1+3 x^2+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x}{1+x+x^2+\sqrt{1+3 x^2+x^4}}\right)}{2 \sqrt{2}}",1,"Sqrt[1 + 3*x^2 + x^4]/(4*(1 - x + x^2)) - Sqrt[2]*ArcTanh[(Sqrt[2]*x)/(1 - x + x^2 + Sqrt[1 + 3*x^2 + x^4])] - ArcTanh[(Sqrt[2]*x)/(1 + x + x^2 + Sqrt[1 + 3*x^2 + x^4])]/(2*Sqrt[2])","A",1
1557,1,107,107,0.4057415,"\int \frac{\left(b+a x^3\right) \left(b+2 a x^3\right)}{x^6 \sqrt[4]{b x+a x^4}} \, dx","IntegrateAlgebraic[((b + a*x^3)*(b + 2*a*x^3))/(x^6*(b*x + a*x^4)^(1/4)),x]","-\frac{4 \left(3 b+17 a x^3\right) \left(b x+a x^4\right)^{3/4}}{63 x^6}+\frac{4}{3} a^{7/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)+\frac{4}{3} a^{7/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)","-\frac{4 \left(3 b+17 a x^3\right) \left(b x+a x^4\right)^{3/4}}{63 x^6}+\frac{4}{3} a^{7/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)+\frac{4}{3} a^{7/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)",1,"(-4*(3*b + 17*a*x^3)*(b*x + a*x^4)^(3/4))/(63*x^6) + (4*a^(7/4)*ArcTan[(a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/3 + (4*a^(7/4)*ArcTanh[(a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/3","A",1
1558,1,107,107,2.3113708,"\int \frac{-b^3+a^3 x^3}{\left(b^3+a^3 x^3\right) \sqrt{b^4+a^4 x^4}} \, dx","IntegrateAlgebraic[(-b^3 + a^3*x^3)/((b^3 + a^3*x^3)*Sqrt[b^4 + a^4*x^4]),x]","-\frac{4 \tan ^{-1}\left(\frac{a b x}{b^2-a b x+a^2 x^2+\sqrt{b^4+a^4 x^4}}\right)}{3 a b}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} a b x}{b^2+2 a b x+a^2 x^2+\sqrt{b^4+a^4 x^4}}\right)}{3 a b}","-\frac{4 \tan ^{-1}\left(\frac{a b x}{b^2-a b x+a^2 x^2+\sqrt{b^4+a^4 x^4}}\right)}{3 a b}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} a b x}{b^2+2 a b x+a^2 x^2+\sqrt{b^4+a^4 x^4}}\right)}{3 a b}",1,"(-4*ArcTan[(a*b*x)/(b^2 - a*b*x + a^2*x^2 + Sqrt[b^4 + a^4*x^4])])/(3*a*b) - (Sqrt[2]*ArcTanh[(Sqrt[2]*a*b*x)/(b^2 + 2*a*b*x + a^2*x^2 + Sqrt[b^4 + a^4*x^4])])/(3*a*b)","A",1
1559,1,107,107,0.9898094,"\int \frac{x^4 \left(-2 b+a x^2\right)}{\left(-b+a x^2\right)^2 \sqrt[4]{-b+a x^2+c x^4}} \, dx","IntegrateAlgebraic[(x^4*(-2*b + a*x^2))/((-b + a*x^2)^2*(-b + a*x^2 + c*x^4)^(1/4)),x]","-\frac{x \left(-b+a x^2+c x^4\right)^{3/4}}{2 c \left(b-a x^2\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{-b+a x^2+c x^4}}\right)}{4 c^{5/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{-b+a x^2+c x^4}}\right)}{4 c^{5/4}}","-\frac{x \left(-b+a x^2+c x^4\right)^{3/4}}{2 c \left(b-a x^2\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{-b+a x^2+c x^4}}\right)}{4 c^{5/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{-b+a x^2+c x^4}}\right)}{4 c^{5/4}}",1,"-1/2*(x*(-b + a*x^2 + c*x^4)^(3/4))/(c*(b - a*x^2)) - ArcTan[(c^(1/4)*x)/(-b + a*x^2 + c*x^4)^(1/4)]/(4*c^(5/4)) - ArcTanh[(c^(1/4)*x)/(-b + a*x^2 + c*x^4)^(1/4)]/(4*c^(5/4))","A",1
1560,1,107,107,2.1758524,"\int \frac{\left(-1-2 x+x^2\right) \left(-1+2 x+x^2\right)}{\left(1-x+2 x^2+x^3+x^4\right) \sqrt[3]{-x+x^5}} \, dx","IntegrateAlgebraic[((-1 - 2*x + x^2)*(-1 + 2*x + x^2))/((1 - x + 2*x^2 + x^3 + x^4)*(-x + x^5)^(1/3)),x]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-x+x^5}}{-2-2 x^2+\sqrt[3]{-x+x^5}}\right)+\log \left(1+x^2+\sqrt[3]{-x+x^5}\right)-\frac{1}{2} \log \left(1+2 x^2+x^4+\left(-1-x^2\right) \sqrt[3]{-x+x^5}+\left(-x+x^5\right)^{2/3}\right)","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-x+x^5}}{-2-2 x^2+\sqrt[3]{-x+x^5}}\right)+\log \left(1+x^2+\sqrt[3]{-x+x^5}\right)-\frac{1}{2} \log \left(1+2 x^2+x^4+\left(-1-x^2\right) \sqrt[3]{-x+x^5}+\left(-x+x^5\right)^{2/3}\right)",1,"Sqrt[3]*ArcTan[(Sqrt[3]*(-x + x^5)^(1/3))/(-2 - 2*x^2 + (-x + x^5)^(1/3))] + Log[1 + x^2 + (-x + x^5)^(1/3)] - Log[1 + 2*x^2 + x^4 + (-1 - x^2)*(-x + x^5)^(1/3) + (-x + x^5)^(2/3)]/2","A",1
1561,1,107,107,0.405937,"\int \frac{-b-a x^3+x^6}{x^6 \sqrt[4]{b x+a x^4}} \, dx","IntegrateAlgebraic[(-b - a*x^3 + x^6)/(x^6*(b*x + a*x^4)^(1/4)),x]","\frac{4 \left(b+a x^3\right) \left(b x+a x^4\right)^{3/4}}{21 b x^6}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 \sqrt[4]{a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 \sqrt[4]{a}}","\frac{4 \left(b+a x^3\right) \left(b x+a x^4\right)^{3/4}}{21 b x^6}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 \sqrt[4]{a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 \sqrt[4]{a}}",1,"(4*(b + a*x^3)*(b*x + a*x^4)^(3/4))/(21*b*x^6) + (2*ArcTan[(a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/(3*a^(1/4)) + (2*ArcTanh[(a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/(3*a^(1/4))","A",1
1562,1,107,107,13.3977598,"\int \frac{1}{\sqrt[3]{b^2 x^2+a^3 x^3} \left(-b+a^6 x^6\right)} \, dx","IntegrateAlgebraic[1/((b^2*x^2 + a^3*x^3)^(1/3)*(-b + a^6*x^6)),x]","\frac{\text{RootSum}\left[a^{18}-a^6 b^{11}-6 a^{15} \text{$\#$1}^3+15 a^{12} \text{$\#$1}^6-20 a^9 \text{$\#$1}^9+15 a^6 \text{$\#$1}^{12}-6 a^3 \text{$\#$1}^{15}+\text{$\#$1}^{18}\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 b}","\frac{\text{RootSum}\left[a^{18}-a^6 b^{11}-6 a^{15} \text{$\#$1}^3+15 a^{12} \text{$\#$1}^6-20 a^9 \text{$\#$1}^9+15 a^6 \text{$\#$1}^{12}-6 a^3 \text{$\#$1}^{15}+\text{$\#$1}^{18}\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 b}",1,"RootSum[a^18 - a^6*b^11 - 6*a^15*#1^3 + 15*a^12*#1^6 - 20*a^9*#1^9 + 15*a^6*#1^12 - 6*a^3*#1^15 + #1^18 & , (-Log[x] + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1])/#1 & ]/(6*b)","A",1
1563,1,107,107,0.0020267,"\int \frac{1}{\sqrt[3]{b^2 x^2+a^3 x^3} \left(-b+a^6 x^6\right)} \, dx","IntegrateAlgebraic[1/((b^2*x^2 + a^3*x^3)^(1/3)*(-b + a^6*x^6)),x]","\frac{\text{RootSum}\left[a^{18}-a^6 b^{11}-6 a^{15} \text{$\#$1}^3+15 a^{12} \text{$\#$1}^6-20 a^9 \text{$\#$1}^9+15 a^6 \text{$\#$1}^{12}-6 a^3 \text{$\#$1}^{15}+\text{$\#$1}^{18}\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 b}","\frac{\text{RootSum}\left[a^{18}-a^6 b^{11}-6 a^{15} \text{$\#$1}^3+15 a^{12} \text{$\#$1}^6-20 a^9 \text{$\#$1}^9+15 a^6 \text{$\#$1}^{12}-6 a^3 \text{$\#$1}^{15}+\text{$\#$1}^{18}\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 b}",1,"RootSum[a^18 - a^6*b^11 - 6*a^15*#1^3 + 15*a^12*#1^6 - 20*a^9*#1^9 + 15*a^6*#1^12 - 6*a^3*#1^15 + #1^18 & , (-Log[x] + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1])/#1 & ]/(6*b)","A",1
1564,1,107,107,13.9304467,"\int \frac{1}{\sqrt[3]{b^2 x^2+a^3 x^3} \left(-b+2 a^6 x^6\right)} \, dx","IntegrateAlgebraic[1/((b^2*x^2 + a^3*x^3)^(1/3)*(-b + 2*a^6*x^6)),x]","\frac{\text{RootSum}\left[a^{18}-2 a^6 b^{11}-6 a^{15} \text{$\#$1}^3+15 a^{12} \text{$\#$1}^6-20 a^9 \text{$\#$1}^9+15 a^6 \text{$\#$1}^{12}-6 a^3 \text{$\#$1}^{15}+\text{$\#$1}^{18}\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 b}","\frac{\text{RootSum}\left[a^{18}-2 a^6 b^{11}-6 a^{15} \text{$\#$1}^3+15 a^{12} \text{$\#$1}^6-20 a^9 \text{$\#$1}^9+15 a^6 \text{$\#$1}^{12}-6 a^3 \text{$\#$1}^{15}+\text{$\#$1}^{18}\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 b}",1,"RootSum[a^18 - 2*a^6*b^11 - 6*a^15*#1^3 + 15*a^12*#1^6 - 20*a^9*#1^9 + 15*a^6*#1^12 - 6*a^3*#1^15 + #1^18 & , (-Log[x] + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1])/#1 & ]/(6*b)","A",1
1565,1,107,107,0.0022809,"\int \frac{1}{\sqrt[3]{b^2 x^2+a^3 x^3} \left(-b+2 a^6 x^6\right)} \, dx","IntegrateAlgebraic[1/((b^2*x^2 + a^3*x^3)^(1/3)*(-b + 2*a^6*x^6)),x]","\frac{\text{RootSum}\left[a^{18}-2 a^6 b^{11}-6 a^{15} \text{$\#$1}^3+15 a^{12} \text{$\#$1}^6-20 a^9 \text{$\#$1}^9+15 a^6 \text{$\#$1}^{12}-6 a^3 \text{$\#$1}^{15}+\text{$\#$1}^{18}\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 b}","\frac{\text{RootSum}\left[a^{18}-2 a^6 b^{11}-6 a^{15} \text{$\#$1}^3+15 a^{12} \text{$\#$1}^6-20 a^9 \text{$\#$1}^9+15 a^6 \text{$\#$1}^{12}-6 a^3 \text{$\#$1}^{15}+\text{$\#$1}^{18}\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 b}",1,"RootSum[a^18 - 2*a^6*b^11 - 6*a^15*#1^3 + 15*a^12*#1^6 - 20*a^9*#1^9 + 15*a^6*#1^12 - 6*a^3*#1^15 + #1^18 & , (-Log[x] + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1])/#1 & ]/(6*b)","A",1
1566,1,107,107,13.7373723,"\int \frac{1}{\sqrt[3]{b^2 x^2+a^3 x^3} \left(b+2 a^6 x^6\right)} \, dx","IntegrateAlgebraic[1/((b^2*x^2 + a^3*x^3)^(1/3)*(b + 2*a^6*x^6)),x]","-\frac{\text{RootSum}\left[a^{18}+2 a^6 b^{11}-6 a^{15} \text{$\#$1}^3+15 a^{12} \text{$\#$1}^6-20 a^9 \text{$\#$1}^9+15 a^6 \text{$\#$1}^{12}-6 a^3 \text{$\#$1}^{15}+\text{$\#$1}^{18}\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 b}","-\frac{\text{RootSum}\left[a^{18}+2 a^6 b^{11}-6 a^{15} \text{$\#$1}^3+15 a^{12} \text{$\#$1}^6-20 a^9 \text{$\#$1}^9+15 a^6 \text{$\#$1}^{12}-6 a^3 \text{$\#$1}^{15}+\text{$\#$1}^{18}\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 b}",1,"-1/6*RootSum[a^18 + 2*a^6*b^11 - 6*a^15*#1^3 + 15*a^12*#1^6 - 20*a^9*#1^9 + 15*a^6*#1^12 - 6*a^3*#1^15 + #1^18 & , (-Log[x] + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1])/#1 & ]/b","A",1
1567,1,107,107,0.0020342,"\int \frac{1}{\sqrt[3]{b^2 x^2+a^3 x^3} \left(b+2 a^6 x^6\right)} \, dx","IntegrateAlgebraic[1/((b^2*x^2 + a^3*x^3)^(1/3)*(b + 2*a^6*x^6)),x]","-\frac{\text{RootSum}\left[a^{18}+2 a^6 b^{11}-6 a^{15} \text{$\#$1}^3+15 a^{12} \text{$\#$1}^6-20 a^9 \text{$\#$1}^9+15 a^6 \text{$\#$1}^{12}-6 a^3 \text{$\#$1}^{15}+\text{$\#$1}^{18}\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 b}","-\frac{\text{RootSum}\left[a^{18}+2 a^6 b^{11}-6 a^{15} \text{$\#$1}^3+15 a^{12} \text{$\#$1}^6-20 a^9 \text{$\#$1}^9+15 a^6 \text{$\#$1}^{12}-6 a^3 \text{$\#$1}^{15}+\text{$\#$1}^{18}\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 b}",1,"-1/6*RootSum[a^18 + 2*a^6*b^11 - 6*a^15*#1^3 + 15*a^12*#1^6 - 20*a^9*#1^9 + 15*a^6*#1^12 - 6*a^3*#1^15 + #1^18 & , (-Log[x] + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1])/#1 & ]/b","A",1
1568,1,107,107,1.1408958,"\int \frac{x^4 \left(-4+x^3\right)}{\sqrt[4]{-1+x^3} \left(-1+2 x^3-x^6+x^8\right)} \, dx","IntegrateAlgebraic[(x^4*(-4 + x^3))/((-1 + x^3)^(1/4)*(-1 + 2*x^3 - x^6 + x^8)),x]","\tan ^{-1}\left(\frac{\sqrt[4]{-1+x^3}}{x}\right)+\frac{\tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-1+x^3}}{-x^2+\sqrt{-1+x^3}}\right)}{\sqrt{2}}-\tanh ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^3}}\right)+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-1+x^3}}{x^2+\sqrt{-1+x^3}}\right)}{\sqrt{2}}","\tan ^{-1}\left(\frac{\sqrt[4]{-1+x^3}}{x}\right)+\frac{\tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-1+x^3}}{-x^2+\sqrt{-1+x^3}}\right)}{\sqrt{2}}-\tanh ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^3}}\right)+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-1+x^3}}{x^2+\sqrt{-1+x^3}}\right)}{\sqrt{2}}",1,"ArcTan[(-1 + x^3)^(1/4)/x] + ArcTan[(Sqrt[2]*x*(-1 + x^3)^(1/4))/(-x^2 + Sqrt[-1 + x^3])]/Sqrt[2] - ArcTanh[x/(-1 + x^3)^(1/4)] + ArcTanh[(Sqrt[2]*x*(-1 + x^3)^(1/4))/(x^2 + Sqrt[-1 + x^3])]/Sqrt[2]","A",1
1569,1,107,107,0.6468948,"\int \frac{x^6}{\left(b+a x^4\right)^{3/4} \left(-b^2+a^2 x^8\right)} \, dx","IntegrateAlgebraic[x^6/((b + a*x^4)^(3/4)*(-b^2 + a^2*x^8)),x]","\frac{x^3}{6 a b \left(b+a x^4\right)^{3/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{4\ 2^{3/4} a^{7/4} b}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{4\ 2^{3/4} a^{7/4} b}","\frac{x^3}{6 a b \left(b+a x^4\right)^{3/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{4\ 2^{3/4} a^{7/4} b}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{4\ 2^{3/4} a^{7/4} b}",1,"x^3/(6*a*b*(b + a*x^4)^(3/4)) + ArcTan[(2^(1/4)*a^(1/4)*x)/(b + a*x^4)^(1/4)]/(4*2^(3/4)*a^(7/4)*b) - ArcTanh[(2^(1/4)*a^(1/4)*x)/(b + a*x^4)^(1/4)]/(4*2^(3/4)*a^(7/4)*b)","A",1
1570,1,107,107,0.2379854,"\int \frac{\left(-b^2+a x^2\right) \sqrt{b+\sqrt{b^2+a x^2}}}{b^2+a x^2} \, dx","IntegrateAlgebraic[((-b^2 + a*x^2)*Sqrt[b + Sqrt[b^2 + a*x^2]])/(b^2 + a*x^2),x]","\frac{4 b x}{3 \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{2 x \sqrt{b^2+a x^2}}{3 \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{4 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{\sqrt{a}}","\frac{4 b x}{3 \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{2 x \sqrt{b^2+a x^2}}{3 \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{4 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{\sqrt{a}}",1,"(4*b*x)/(3*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (2*x*Sqrt[b^2 + a*x^2])/(3*Sqrt[b + Sqrt[b^2 + a*x^2]]) - (4*b^(3/2)*ArcTan[(Sqrt[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/Sqrt[a]","A",1
1571,1,108,108,0.1924558,"\int \sqrt[3]{x^2+x^3} \, dx","IntegrateAlgebraic[(x^2 + x^3)^(1/3),x]","\frac{1}{6} (1+3 x) \sqrt[3]{x^2+x^3}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^3}}\right)}{3 \sqrt{3}}+\frac{1}{9} \log \left(-x+\sqrt[3]{x^2+x^3}\right)-\frac{1}{18} \log \left(x^2+x \sqrt[3]{x^2+x^3}+\left(x^2+x^3\right)^{2/3}\right)","\frac{1}{6} (1+3 x) \sqrt[3]{x^2+x^3}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^3}}\right)}{3 \sqrt{3}}+\frac{1}{9} \log \left(-x+\sqrt[3]{x^2+x^3}\right)-\frac{1}{18} \log \left(x^2+x \sqrt[3]{x^2+x^3}+\left(x^2+x^3\right)^{2/3}\right)",1,"((1 + 3*x)*(x^2 + x^3)^(1/3))/6 + ArcTan[(Sqrt[3]*x)/(x + 2*(x^2 + x^3)^(1/3))]/(3*Sqrt[3]) + Log[-x + (x^2 + x^3)^(1/3)]/9 - Log[x^2 + x*(x^2 + x^3)^(1/3) + (x^2 + x^3)^(2/3)]/18","A",1
1572,1,108,108,0.3389276,"\int \frac{-1+k^3 x^3}{\sqrt{(1-x) x \left(1-k^2 x\right)} \left(1+k^3 x^3\right)} \, dx","IntegrateAlgebraic[(-1 + k^3*x^3)/(Sqrt[(1 - x)*x*(1 - k^2*x)]*(1 + k^3*x^3)),x]","-\frac{2 \tan ^{-1}\left(\frac{(1+k) x}{\sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}\right)}{3 (1+k)}-\frac{4 \tan ^{-1}\left(\frac{\sqrt{1-k+k^2} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{(-1+x) \left(-1+k^2 x\right)}\right)}{3 \sqrt{1-k+k^2}}","-\frac{2 \tan ^{-1}\left(\frac{(1+k) x}{\sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}\right)}{3 (1+k)}-\frac{4 \tan ^{-1}\left(\frac{\sqrt{1-k+k^2} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{(-1+x) \left(-1+k^2 x\right)}\right)}{3 \sqrt{1-k+k^2}}",1,"(-2*ArcTan[((1 + k)*x)/Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3]])/(3*(1 + k)) - (4*ArcTan[(Sqrt[1 - k + k^2]*Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3])/((-1 + x)*(-1 + k^2*x))])/(3*Sqrt[1 - k + k^2])","A",1
1573,1,108,108,0.5810255,"\int \frac{b+a x^2}{\left(b+a x^2+x^4\right) \sqrt[4]{-b x^2+a x^4}} \, dx","IntegrateAlgebraic[(b + a*x^2)/((b + a*x^2 + x^4)*(-(b*x^2) + a*x^4)^(1/4)),x]","\frac{1}{2} \text{RootSum}\left[2 a^2+b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 a \log (x)+2 a \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]","\frac{1}{2} \text{RootSum}\left[2 a^2+b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 a \log (x)-2 a \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{3 a \text{$\#$1}-2 \text{$\#$1}^5}\&\right]",1,"RootSum[2*a^2 + b - 3*a*#1^4 + #1^8 & , (-2*a*Log[x] + 2*a*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1 + 2*#1^5) & ]/2","A",1
1574,1,108,108,0.0036703,"\int \frac{b+a x^2}{\left(b+a x^2+x^4\right) \sqrt[4]{-b x^2+a x^4}} \, dx","IntegrateAlgebraic[(b + a*x^2)/((b + a*x^2 + x^4)*(-(b*x^2) + a*x^4)^(1/4)),x]","\frac{1}{2} \text{RootSum}\left[2 a^2+b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 a \log (x)+2 a \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]","\frac{1}{2} \text{RootSum}\left[2 a^2+b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 a \log (x)-2 a \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{3 a \text{$\#$1}-2 \text{$\#$1}^5}\&\right]",1,"RootSum[2*a^2 + b - 3*a*#1^4 + #1^8 & , (-2*a*Log[x] + 2*a*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1 + 2*#1^5) & ]/2","A",1
1575,1,108,108,3.2741042,"\int \frac{c+b x^2+c k^2 x^4}{\sqrt{\left(1-x^2\right) \left(1-k^2 x^2\right)} \left(-1+k^2 x^4\right)} \, dx","IntegrateAlgebraic[(c + b*x^2 + c*k^2*x^4)/(Sqrt[(1 - x^2)*(1 - k^2*x^2)]*(-1 + k^2*x^4)),x]","\frac{(-b-2 c k) \tan ^{-1}\left(\frac{(-1+k) x}{\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{4 (-1+k) k}+\frac{(b-2 c k) \tan ^{-1}\left(\frac{(1+k) x}{1+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{2 k (1+k)}","\frac{(-b-2 c k) \tan ^{-1}\left(\frac{(-1+k) x}{\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{4 (-1+k) k}+\frac{(b-2 c k) \tan ^{-1}\left(\frac{(1+k) x}{1+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{2 k (1+k)}",1,"((-b - 2*c*k)*ArcTan[((-1 + k)*x)/Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4]])/(4*(-1 + k)*k) + ((b - 2*c*k)*ArcTan[((1 + k)*x)/(1 + k*x^2 + Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4])])/(2*k*(1 + k))","A",1
1576,1,108,108,1.2329738,"\int \frac{3 b+a x^4}{\left(-b+x^3+a x^4\right) \sqrt[4]{-b x+a x^5}} \, dx","IntegrateAlgebraic[(3*b + a*x^4)/((-b + x^3 + a*x^4)*(-(b*x) + a*x^5)^(1/4)),x]","-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-b x+a x^5}}{-x^2+\sqrt{-b x+a x^5}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{-b x+a x^5}}{\sqrt{2}}}{x \sqrt[4]{-b x+a x^5}}\right)","-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-b x+a x^5}}{-x^2+\sqrt{-b x+a x^5}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{-b x+a x^5}}{\sqrt{2}}}{x \sqrt[4]{-b x+a x^5}}\right)",1,"-(Sqrt[2]*ArcTan[(Sqrt[2]*x*(-(b*x) + a*x^5)^(1/4))/(-x^2 + Sqrt[-(b*x) + a*x^5])]) - Sqrt[2]*ArcTanh[(x^2/Sqrt[2] + Sqrt[-(b*x) + a*x^5]/Sqrt[2])/(x*(-(b*x) + a*x^5)^(1/4))]","A",1
1577,1,108,108,0.345883,"\int \frac{-1+x^6}{x^6 \sqrt[3]{x+x^3}} \, dx","IntegrateAlgebraic[(-1 + x^6)/(x^6*(x + x^3)^(1/3)),x]","\frac{3 \left(x+x^3\right)^{2/3} \left(5-6 x^2+9 x^4\right)}{80 x^6}+\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)-\frac{1}{2} \log \left(-x+\sqrt[3]{x+x^3}\right)+\frac{1}{4} \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)","\frac{3 \left(x+x^3\right)^{2/3} \left(5-6 x^2+9 x^4\right)}{80 x^6}+\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)-\frac{1}{2} \log \left(-x+\sqrt[3]{x+x^3}\right)+\frac{1}{4} \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)",1,"(3*(x + x^3)^(2/3)*(5 - 6*x^2 + 9*x^4))/(80*x^6) + (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^3)^(1/3))])/2 - Log[-x + (x + x^3)^(1/3)]/2 + Log[x^2 + x*(x + x^3)^(1/3) + (x + x^3)^(2/3)]/4","A",1
1578,1,108,108,0.203018,"\int \frac{\left(-1+x^3\right)^{2/3} \left(2+x^6\right)}{x^6} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(2 + x^6))/x^6,x]","\frac{\left(-1+x^3\right)^{2/3} \left(-6+6 x^3+5 x^6\right)}{15 x^5}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{3 \sqrt{3}}+\frac{2}{9} \log \left(-x+\sqrt[3]{-1+x^3}\right)-\frac{1}{9} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)","\frac{\left(-1+x^3\right)^{2/3} \left(-6+6 x^3+5 x^6\right)}{15 x^5}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{3 \sqrt{3}}+\frac{2}{9} \log \left(-x+\sqrt[3]{-1+x^3}\right)-\frac{1}{9} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)",1,"((-1 + x^3)^(2/3)*(-6 + 6*x^3 + 5*x^6))/(15*x^5) - (2*ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^3)^(1/3))])/(3*Sqrt[3]) + (2*Log[-x + (-1 + x^3)^(1/3)])/9 - Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)]/9","A",1
1579,1,108,108,0.2215767,"\int \frac{\left(-1+x^3\right)^{2/3} \left(-2+x^3+x^6\right)}{x^6} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(-2 + x^3 + x^6))/x^6,x]","\frac{\left(-1+x^3\right)^{2/3} \left(12-27 x^3+10 x^6\right)}{30 x^5}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{3 \sqrt{3}}-\frac{1}{9} \log \left(-x+\sqrt[3]{-1+x^3}\right)+\frac{1}{18} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)","\frac{\left(-1+x^3\right)^{2/3} \left(12-27 x^3+10 x^6\right)}{30 x^5}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{3 \sqrt{3}}-\frac{1}{9} \log \left(-x+\sqrt[3]{-1+x^3}\right)+\frac{1}{18} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)",1,"((-1 + x^3)^(2/3)*(12 - 27*x^3 + 10*x^6))/(30*x^5) + ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^3)^(1/3))]/(3*Sqrt[3]) - Log[-x + (-1 + x^3)^(1/3)]/9 + Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)]/18","A",1
1580,1,108,108,0.2055665,"\int \frac{\left(1+x^3\right)^{2/3} \left(1+2 x^6\right)}{x^6} \, dx","IntegrateAlgebraic[((1 + x^3)^(2/3)*(1 + 2*x^6))/x^6,x]","\frac{\left(1+x^3\right)^{2/3} \left(-3-3 x^3+10 x^6\right)}{15 x^5}+\frac{4 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3}}\right)}{3 \sqrt{3}}-\frac{4}{9} \log \left(-x+\sqrt[3]{1+x^3}\right)+\frac{2}{9} \log \left(x^2+x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)","\frac{\left(1+x^3\right)^{2/3} \left(-3-3 x^3+10 x^6\right)}{15 x^5}+\frac{4 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3}}\right)}{3 \sqrt{3}}-\frac{4}{9} \log \left(-x+\sqrt[3]{1+x^3}\right)+\frac{2}{9} \log \left(x^2+x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)",1,"((1 + x^3)^(2/3)*(-3 - 3*x^3 + 10*x^6))/(15*x^5) + (4*ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^3)^(1/3))])/(3*Sqrt[3]) - (4*Log[-x + (1 + x^3)^(1/3)])/9 + (2*Log[x^2 + x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)])/9","A",1
1581,1,108,108,2.9381208,"\int \frac{3 b+2 a x^5}{\left(-b+x^3+a x^5\right) \sqrt[4]{-b x+a x^6}} \, dx","IntegrateAlgebraic[(3*b + 2*a*x^5)/((-b + x^3 + a*x^5)*(-(b*x) + a*x^6)^(1/4)),x]","-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-b x+a x^6}}{-x^2+\sqrt{-b x+a x^6}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{-b x+a x^6}}{\sqrt{2}}}{x \sqrt[4]{-b x+a x^6}}\right)","-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-b x+a x^6}}{-x^2+\sqrt{-b x+a x^6}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{-b x+a x^6}}{\sqrt{2}}}{x \sqrt[4]{-b x+a x^6}}\right)",1,"-(Sqrt[2]*ArcTan[(Sqrt[2]*x*(-(b*x) + a*x^6)^(1/4))/(-x^2 + Sqrt[-(b*x) + a*x^6])]) - Sqrt[2]*ArcTanh[(x^2/Sqrt[2] + Sqrt[-(b*x) + a*x^6]/Sqrt[2])/(x*(-(b*x) + a*x^6)^(1/4))]","A",1
1582,1,108,108,0.2573474,"\int \frac{\sqrt[4]{-1+x^4} \left(1+x^4+x^8\right)}{x^6 \left(-1+2 x^8\right)} \, dx","IntegrateAlgebraic[((-1 + x^4)^(1/4)*(1 + x^4 + x^8))/(x^6*(-1 + 2*x^8)),x]","\frac{\sqrt[4]{-1+x^4} \left(1+4 x^4\right)}{5 x^5}+\frac{1}{8} \text{RootSum}\left[-1-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right)+2 \log (x) \text{$\#$1}^4-2 \log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+\text{$\#$1}^7}\&\right]","\frac{\sqrt[4]{-1+x^4} \left(1+4 x^4\right)}{5 x^5}+\frac{1}{8} \text{RootSum}\left[-1-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right)+2 \log (x) \text{$\#$1}^4-2 \log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+\text{$\#$1}^7}\&\right]",1,"((-1 + x^4)^(1/4)*(1 + 4*x^4))/(5*x^5) + RootSum[-1 - 2*#1^4 + #1^8 & , (-Log[x] + Log[(-1 + x^4)^(1/4) - x*#1] + 2*Log[x]*#1^4 - 2*Log[(-1 + x^4)^(1/4) - x*#1]*#1^4)/(-#1^3 + #1^7) & ]/8","A",1
1583,1,108,108,0.0011355,"\int \frac{\sqrt[4]{-1+x^4} \left(1+x^4+x^8\right)}{x^6 \left(-1+2 x^8\right)} \, dx","IntegrateAlgebraic[((-1 + x^4)^(1/4)*(1 + x^4 + x^8))/(x^6*(-1 + 2*x^8)),x]","\frac{\sqrt[4]{-1+x^4} \left(1+4 x^4\right)}{5 x^5}+\frac{1}{8} \text{RootSum}\left[-1-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right)+2 \log (x) \text{$\#$1}^4-2 \log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+\text{$\#$1}^7}\&\right]","\frac{\sqrt[4]{-1+x^4} \left(1+4 x^4\right)}{5 x^5}+\frac{1}{8} \text{RootSum}\left[-1-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right)+2 \log (x) \text{$\#$1}^4-2 \log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+\text{$\#$1}^7}\&\right]",1,"((-1 + x^4)^(1/4)*(1 + 4*x^4))/(5*x^5) + RootSum[-1 - 2*#1^4 + #1^8 & , (-Log[x] + Log[(-1 + x^4)^(1/4) - x*#1] + 2*Log[x]*#1^4 - 2*Log[(-1 + x^4)^(1/4) - x*#1]*#1^4)/(-#1^3 + #1^7) & ]/8","A",1
1584,1,108,108,4.9156773,"\int \frac{\sqrt{-b x+a^2 x^2}}{\left(a x^2+x \sqrt{-b x+a^2 x^2}\right)^{3/2}} \, dx","IntegrateAlgebraic[Sqrt[-(b*x) + a^2*x^2]/(a*x^2 + x*Sqrt[-(b*x) + a^2*x^2])^(3/2),x]","-\frac{4 \left(5 a b+3 a^3 x\right) \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{3 b^2 x}+\frac{4 \left(b+3 a^2 x\right) \sqrt{-b x+a^2 x^2} \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{3 b^2 x^2}","-\frac{4 \left(5 a b+3 a^3 x\right) \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{3 b^2 x}+\frac{4 \left(b+3 a^2 x\right) \sqrt{-b x+a^2 x^2} \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{3 b^2 x^2}",1,"(-4*(5*a*b + 3*a^3*x)*Sqrt[x*(a*x + Sqrt[-(b*x) + a^2*x^2])])/(3*b^2*x) + (4*(b + 3*a^2*x)*Sqrt[-(b*x) + a^2*x^2]*Sqrt[x*(a*x + Sqrt[-(b*x) + a^2*x^2])])/(3*b^2*x^2)","A",1
1585,1,109,109,0.245128,"\int x^7 \sqrt[3]{-1+x^3} \, dx","IntegrateAlgebraic[x^7*(-1 + x^3)^(1/3),x]","\frac{1}{162} \sqrt[3]{-1+x^3} \left(-5 x^2-3 x^5+18 x^8\right)+\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{81 \sqrt{3}}+\frac{5}{243} \log \left(-x+\sqrt[3]{-1+x^3}\right)-\frac{5}{486} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)","\frac{1}{162} \sqrt[3]{-1+x^3} \left(-5 x^2-3 x^5+18 x^8\right)+\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{81 \sqrt{3}}+\frac{5}{243} \log \left(-x+\sqrt[3]{-1+x^3}\right)-\frac{5}{486} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)",1,"((-1 + x^3)^(1/3)*(-5*x^2 - 3*x^5 + 18*x^8))/162 + (5*ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^3)^(1/3))])/(81*Sqrt[3]) + (5*Log[-x + (-1 + x^3)^(1/3)])/243 - (5*Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)])/486","A",1
1586,1,109,109,0.2445995,"\int x^7 \sqrt[3]{1+x^3} \, dx","IntegrateAlgebraic[x^7*(1 + x^3)^(1/3),x]","\frac{1}{162} \sqrt[3]{1+x^3} \left(-5 x^2+3 x^5+18 x^8\right)-\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3}}\right)}{81 \sqrt{3}}-\frac{5}{243} \log \left(-x+\sqrt[3]{1+x^3}\right)+\frac{5}{486} \log \left(x^2+x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)","\frac{1}{162} \sqrt[3]{1+x^3} \left(-5 x^2+3 x^5+18 x^8\right)-\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3}}\right)}{81 \sqrt{3}}-\frac{5}{243} \log \left(-x+\sqrt[3]{1+x^3}\right)+\frac{5}{486} \log \left(x^2+x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)",1,"((1 + x^3)^(1/3)*(-5*x^2 + 3*x^5 + 18*x^8))/162 - (5*ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^3)^(1/3))])/(81*Sqrt[3]) - (5*Log[-x + (1 + x^3)^(1/3)])/243 + (5*Log[x^2 + x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)])/486","A",1
1587,1,109,109,0.1815402,"\int \frac{\sqrt[3]{-1+x^3} \left(1+x^3\right)}{x^{13}} \, dx","IntegrateAlgebraic[((-1 + x^3)^(1/3)*(1 + x^3))/x^13,x]","\frac{\sqrt[3]{-1+x^3} \left(-81-99 x^3+30 x^6+50 x^9\right)}{972 x^{12}}-\frac{25 \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^3}}{\sqrt{3}}\right)}{243 \sqrt{3}}+\frac{25}{729} \log \left(1+\sqrt[3]{-1+x^3}\right)-\frac{25 \log \left(1-\sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)}{1458}","\frac{\sqrt[3]{-1+x^3} \left(-81-99 x^3+30 x^6+50 x^9\right)}{972 x^{12}}-\frac{25 \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^3}}{\sqrt{3}}\right)}{243 \sqrt{3}}+\frac{25}{729} \log \left(1+\sqrt[3]{-1+x^3}\right)-\frac{25 \log \left(1-\sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)}{1458}",1,"((-1 + x^3)^(1/3)*(-81 - 99*x^3 + 30*x^6 + 50*x^9))/(972*x^12) - (25*ArcTan[1/Sqrt[3] - (2*(-1 + x^3)^(1/3))/Sqrt[3]])/(243*Sqrt[3]) + (25*Log[1 + (-1 + x^3)^(1/3)])/729 - (25*Log[1 - (-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)])/1458","A",1
1588,1,109,109,7.2657294,"\int \frac{(-b+x) (a-3 b+2 x)}{(-a+x) ((-a+x) (-b+x))^{3/4} \left(b-a^3 d-\left(1-3 a^2 d\right) x-3 a d x^2+d x^3\right)} \, dx","IntegrateAlgebraic[((-b + x)*(a - 3*b + 2*x))/((-a + x)*((-a + x)*(-b + x))^(3/4)*(b - a^3*d - (1 - 3*a^2*d)*x - 3*a*d*x^2 + d*x^3)),x]","-\frac{4 (b-x)}{\left(a b-a x-b x+x^2\right)^{3/4}}-2 \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(a b+(-a-b) x+x^2\right)^{3/4}}{b-x}\right)+2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt[4]{a b+(-a-b) x+x^2}}{\sqrt[4]{d} (a-x)}\right)","-\frac{4 (b-x)}{\left(a b-a x-b x+x^2\right)^{3/4}}-2 \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(a b+(-a-b) x+x^2\right)^{3/4}}{b-x}\right)+2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt[4]{a b+(-a-b) x+x^2}}{\sqrt[4]{d} (a-x)}\right)",1,"(-4*(b - x))/(a*b - a*x - b*x + x^2)^(3/4) - 2*d^(1/4)*ArcTan[(d^(1/4)*(a*b + (-a - b)*x + x^2)^(3/4))/(b - x)] + 2*d^(1/4)*ArcTanh[(a*b + (-a - b)*x + x^2)^(1/4)/(d^(1/4)*(a - x))]","A",1
1589,1,109,109,0.4258781,"\int x^8 \left(-b+a x^4\right)^{3/4} \, dx","IntegrateAlgebraic[x^8*(-b + a*x^4)^(3/4),x]","\frac{\left(-b+a x^4\right)^{3/4} \left(-15 b^2 x-12 a b x^5+32 a^2 x^9\right)}{384 a^2}-\frac{5 b^3 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{256 a^{9/4}}-\frac{5 b^3 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{256 a^{9/4}}","\frac{\left(-b+a x^4\right)^{3/4} \left(-15 b^2 x-12 a b x^5+32 a^2 x^9\right)}{384 a^2}-\frac{5 b^3 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{256 a^{9/4}}-\frac{5 b^3 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{256 a^{9/4}}",1,"((-b + a*x^4)^(3/4)*(-15*b^2*x - 12*a*b*x^5 + 32*a^2*x^9))/(384*a^2) - (5*b^3*ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)])/(256*a^(9/4)) - (5*b^3*ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)])/(256*a^(9/4))","A",1
1590,1,109,109,0.3965771,"\int \frac{1}{\left(b+a x^3\right) \sqrt[4]{-b x+a x^4}} \, dx","IntegrateAlgebraic[1/((b + a*x^3)*(-(b*x) + a*x^4)^(1/4)),x]","\frac{2^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)}{3 \sqrt[4]{a} b}+\frac{2^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)}{3 \sqrt[4]{a} b}","\frac{2^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)}{3 \sqrt[4]{a} b}+\frac{2^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)}{3 \sqrt[4]{a} b}",1,"(2^(3/4)*ArcTan[(2^(1/4)*a^(1/4)*(-(b*x) + a*x^4)^(3/4))/(-b + a*x^3)])/(3*a^(1/4)*b) + (2^(3/4)*ArcTanh[(2^(1/4)*a^(1/4)*(-(b*x) + a*x^4)^(3/4))/(-b + a*x^3)])/(3*a^(1/4)*b)","A",1
1591,1,109,109,0.657653,"\int \frac{\sqrt[4]{-b x^2+a x^4}}{x^4 \left(-b+a x^4\right)} \, dx","IntegrateAlgebraic[(-(b*x^2) + a*x^4)^(1/4)/(x^4*(-b + a*x^4)),x]","\frac{2 \left(b-a x^2\right) \sqrt[4]{-b x^2+a x^4}}{5 b^2 x^3}-\frac{a \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]}{4 b}","\frac{2 \left(b-a x^2\right) \sqrt[4]{-b x^2+a x^4}}{5 b^2 x^3}-\frac{a \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]}{4 b}",1,"(2*(b - a*x^2)*(-(b*x^2) + a*x^4)^(1/4))/(5*b^2*x^3) - (a*RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (-(Log[x]*#1) + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1)/(-a + #1^4) & ])/(4*b)","A",1
1592,1,109,109,0.0017394,"\int \frac{\sqrt[4]{-b x^2+a x^4}}{x^4 \left(-b+a x^4\right)} \, dx","IntegrateAlgebraic[(-(b*x^2) + a*x^4)^(1/4)/(x^4*(-b + a*x^4)),x]","\frac{2 \left(b-a x^2\right) \sqrt[4]{-b x^2+a x^4}}{5 b^2 x^3}-\frac{a \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]}{4 b}","\frac{2 \left(b-a x^2\right) \sqrt[4]{-b x^2+a x^4}}{5 b^2 x^3}-\frac{a \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]}{4 b}",1,"(2*(b - a*x^2)*(-(b*x^2) + a*x^4)^(1/4))/(5*b^2*x^3) - (a*RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (-(Log[x]*#1) + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1)/(-a + #1^4) & ])/(4*b)","A",1
1593,1,109,109,0.9673778,"\int \frac{(-2+(1+k) x) \left(1-(1+k) x+(a+k) x^2\right)}{\sqrt[3]{(1-x) x (1-k x)} \left(1-2 (1+k) x+\left(1+c+4 k+k^2\right) x^2-\left(c+2 k+c k+2 k^2\right) x^3+\left(-b+c k+k^2\right) x^4\right)} \, dx","IntegrateAlgebraic[((-2 + (1 + k)*x)*(1 - (1 + k)*x + (a + k)*x^2))/(((1 - x)*x*(1 - k*x))^(1/3)*(1 - 2*(1 + k)*x + (1 + c + 4*k + k^2)*x^2 - (c + 2*k + c*k + 2*k^2)*x^3 + (-b + c*k + k^2)*x^4)),x]","-\text{RootSum}\left[b-c \text{$\#$1}^3-\text{$\#$1}^6\&,\frac{a \log (x)-a \log \left(\sqrt[3]{x+(-1-k) x^2+k x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{x+(-1-k) x^2+k x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{c \text{$\#$1}+2 \text{$\#$1}^4}\&\right]","-\text{RootSum}\left[b-c \text{$\#$1}^3-\text{$\#$1}^6\&,\frac{a \log (x)-a \log \left(\sqrt[3]{x+(-1-k) x^2+k x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{x+(-1-k) x^2+k x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{c \text{$\#$1}+2 \text{$\#$1}^4}\&\right]",1,"-RootSum[b - c*#1^3 - #1^6 & , (a*Log[x] - a*Log[(x + (-1 - k)*x^2 + k*x^3)^(1/3) - x*#1] + Log[x]*#1^3 - Log[(x + (-1 - k)*x^2 + k*x^3)^(1/3) - x*#1]*#1^3)/(c*#1 + 2*#1^4) & ]","A",1
1594,1,109,109,0.4736734,"\int \frac{\sqrt[3]{-1+x^6} \left(1+x^6\right)}{x^3} \, dx","IntegrateAlgebraic[((-1 + x^6)^(1/3)*(1 + x^6))/x^3,x]","\frac{\left(-3+x^6\right) \sqrt[3]{-1+x^6}}{6 x^2}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{-1+x^6}}\right)}{3 \sqrt{3}}-\frac{1}{9} \log \left(-x^2+\sqrt[3]{-1+x^6}\right)+\frac{1}{18} \log \left(x^4+x^2 \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)","\frac{\left(-3+x^6\right) \sqrt[3]{-1+x^6}}{6 x^2}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{-1+x^6}}\right)}{3 \sqrt{3}}-\frac{1}{9} \log \left(-x^2+\sqrt[3]{-1+x^6}\right)+\frac{1}{18} \log \left(x^4+x^2 \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)",1,"((-3 + x^6)*(-1 + x^6)^(1/3))/(6*x^2) - ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(-1 + x^6)^(1/3))]/(3*Sqrt[3]) - Log[-x^2 + (-1 + x^6)^(1/3)]/9 + Log[x^4 + x^2*(-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)]/18","A",1
1595,1,109,109,0.2182157,"\int \frac{\left(1+x^3\right)^{2/3} \left(1-2 x^3+x^6\right)}{x^6 \left(-2+x^6\right)} \, dx","IntegrateAlgebraic[((1 + x^3)^(2/3)*(1 - 2*x^3 + x^6))/(x^6*(-2 + x^6)),x]","\frac{\left(1-4 x^3\right) \left(1+x^3\right)^{2/3}}{10 x^5}+\frac{1}{24} \text{RootSum}\left[1-4 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{-2 \log (x)+2 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-\text{$\#$1}+\text{$\#$1}^4}\&\right]","\frac{\left(1-4 x^3\right) \left(1+x^3\right)^{2/3}}{10 x^5}+\frac{1}{24} \text{RootSum}\left[1-4 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{-2 \log (x)+2 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-\text{$\#$1}+\text{$\#$1}^4}\&\right]",1,"((1 - 4*x^3)*(1 + x^3)^(2/3))/(10*x^5) + RootSum[1 - 4*#1^3 + 2*#1^6 & , (-2*Log[x] + 2*Log[(1 + x^3)^(1/3) - x*#1] + Log[x]*#1^3 - Log[(1 + x^3)^(1/3) - x*#1]*#1^3)/(-#1 + #1^4) & ]/24","A",1
1596,1,109,109,0.0011898,"\int \frac{\left(1+x^3\right)^{2/3} \left(1-2 x^3+x^6\right)}{x^6 \left(-2+x^6\right)} \, dx","IntegrateAlgebraic[((1 + x^3)^(2/3)*(1 - 2*x^3 + x^6))/(x^6*(-2 + x^6)),x]","\frac{\left(1-4 x^3\right) \left(1+x^3\right)^{2/3}}{10 x^5}+\frac{1}{24} \text{RootSum}\left[1-4 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{-2 \log (x)+2 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-\text{$\#$1}+\text{$\#$1}^4}\&\right]","\frac{\left(1-4 x^3\right) \left(1+x^3\right)^{2/3}}{10 x^5}+\frac{1}{24} \text{RootSum}\left[1-4 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{-2 \log (x)+2 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-\text{$\#$1}+\text{$\#$1}^4}\&\right]",1,"((1 - 4*x^3)*(1 + x^3)^(2/3))/(10*x^5) + RootSum[1 - 4*#1^3 + 2*#1^6 & , (-2*Log[x] + 2*Log[(1 + x^3)^(1/3) - x*#1] + Log[x]*#1^3 - Log[(1 + x^3)^(1/3) - x*#1]*#1^3)/(-#1 + #1^4) & ]/24","A",1
1597,1,109,109,0.7107892,"\int \frac{\sqrt[3]{-1+x^6} \left(-1+2 x^6\right)}{x^9} \, dx","IntegrateAlgebraic[((-1 + x^6)^(1/3)*(-1 + 2*x^6))/x^9,x]","\frac{\left(1-9 x^6\right) \sqrt[3]{-1+x^6}}{8 x^8}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{-1+x^6}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(-x^2+\sqrt[3]{-1+x^6}\right)+\frac{1}{6} \log \left(x^4+x^2 \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)","\frac{\left(1-9 x^6\right) \sqrt[3]{-1+x^6}}{8 x^8}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{-1+x^6}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(-x^2+\sqrt[3]{-1+x^6}\right)+\frac{1}{6} \log \left(x^4+x^2 \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)",1,"((1 - 9*x^6)*(-1 + x^6)^(1/3))/(8*x^8) - ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(-1 + x^6)^(1/3))]/Sqrt[3] - Log[-x^2 + (-1 + x^6)^(1/3)]/3 + Log[x^4 + x^2*(-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)]/6","A",1
1598,1,109,109,0.2225861,"\int \frac{\left(-1+x^3\right)^{2/3} \left(-1+x^6\right)}{x^6 \left(-2+x^3+2 x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(-1 + x^6))/(x^6*(-2 + x^3 + 2*x^6)),x]","\frac{\left(-4-x^3\right) \left(-1+x^3\right)^{2/3}}{40 x^5}+\frac{1}{12} \text{RootSum}\left[-1-3 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{\log (x)-\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-3 \text{$\#$1}+4 \text{$\#$1}^4}\&\right]","\frac{\left(-4-x^3\right) \left(-1+x^3\right)^{2/3}}{40 x^5}+\frac{1}{12} \text{RootSum}\left[-1-3 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{\log (x)-\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-3 \text{$\#$1}+4 \text{$\#$1}^4}\&\right]",1,"((-4 - x^3)*(-1 + x^3)^(2/3))/(40*x^5) + RootSum[-1 - 3*#1^3 + 2*#1^6 & , (Log[x] - Log[(-1 + x^3)^(1/3) - x*#1] + Log[x]*#1^3 - Log[(-1 + x^3)^(1/3) - x*#1]*#1^3)/(-3*#1 + 4*#1^4) & ]/12","A",1
1599,1,109,109,0.0012032,"\int \frac{\left(-1+x^3\right)^{2/3} \left(-1+x^6\right)}{x^6 \left(-2+x^3+2 x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(-1 + x^6))/(x^6*(-2 + x^3 + 2*x^6)),x]","\frac{\left(-4-x^3\right) \left(-1+x^3\right)^{2/3}}{40 x^5}+\frac{1}{12} \text{RootSum}\left[-1-3 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{\log (x)-\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-3 \text{$\#$1}+4 \text{$\#$1}^4}\&\right]","\frac{\left(-4-x^3\right) \left(-1+x^3\right)^{2/3}}{40 x^5}+\frac{1}{12} \text{RootSum}\left[-1-3 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{\log (x)-\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-3 \text{$\#$1}+4 \text{$\#$1}^4}\&\right]",1,"((-4 - x^3)*(-1 + x^3)^(2/3))/(40*x^5) + RootSum[-1 - 3*#1^3 + 2*#1^6 & , (Log[x] - Log[(-1 + x^3)^(1/3) - x*#1] + Log[x]*#1^3 - Log[(-1 + x^3)^(1/3) - x*#1]*#1^3)/(-3*#1 + 4*#1^4) & ]/12","A",1
1600,1,109,109,0.2962689,"\int \frac{\sqrt[4]{-1+x^4} \left(2-x^4+2 x^8\right)}{x^{10} \left(-1+2 x^4\right)} \, dx","IntegrateAlgebraic[((-1 + x^4)^(1/4)*(2 - x^4 + 2*x^8))/(x^10*(-1 + 2*x^4)),x]","\frac{\sqrt[4]{-1+x^4} \left(2+5 x^4+65 x^8\right)}{9 x^9}+2 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-1+x^4}}{-x^2+\sqrt{-1+x^4}}\right)-2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-1+x^4}}{x^2+\sqrt{-1+x^4}}\right)","\frac{\sqrt[4]{-1+x^4} \left(2+5 x^4+65 x^8\right)}{9 x^9}+2 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-1+x^4}}{-x^2+\sqrt{-1+x^4}}\right)-2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-1+x^4}}{x^2+\sqrt{-1+x^4}}\right)",1,"((-1 + x^4)^(1/4)*(2 + 5*x^4 + 65*x^8))/(9*x^9) + 2*Sqrt[2]*ArcTan[(Sqrt[2]*x*(-1 + x^4)^(1/4))/(-x^2 + Sqrt[-1 + x^4])] - 2*Sqrt[2]*ArcTanh[(Sqrt[2]*x*(-1 + x^4)^(1/4))/(x^2 + Sqrt[-1 + x^4])]","A",1
1601,1,109,109,15.3357235,"\int \frac{\left(2+x^6\right) \left(1-2 x^6+x^8+x^{12}\right)}{x^8 \sqrt[4]{-1+x^6} \left(-1+x^4+x^6\right)} \, dx","IntegrateAlgebraic[((2 + x^6)*(1 - 2*x^6 + x^8 + x^12))/(x^8*(-1 + x^6)^(1/4)*(-1 + x^4 + x^6)),x]","\frac{2 \left(-1+x^6\right)^{3/4} \left(-3-7 x^4+3 x^6\right)}{21 x^7}-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-1+x^6}}{-x^2+\sqrt{-1+x^6}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-1+x^6}}{x^2+\sqrt{-1+x^6}}\right)","\frac{2 \left(-1+x^6\right)^{3/4} \left(-3-7 x^4+3 x^6\right)}{21 x^7}-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-1+x^6}}{-x^2+\sqrt{-1+x^6}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-1+x^6}}{x^2+\sqrt{-1+x^6}}\right)",1,"(2*(-1 + x^6)^(3/4)*(-3 - 7*x^4 + 3*x^6))/(21*x^7) - Sqrt[2]*ArcTan[(Sqrt[2]*x*(-1 + x^6)^(1/4))/(-x^2 + Sqrt[-1 + x^6])] - Sqrt[2]*ArcTanh[(Sqrt[2]*x*(-1 + x^6)^(1/4))/(x^2 + Sqrt[-1 + x^6])]","A",1
1602,1,109,109,4.0942449,"\int \frac{\sqrt{x+x^2}}{x \sqrt{x^2+x \sqrt{x+x^2}}} \, dx","IntegrateAlgebraic[Sqrt[x + x^2]/(x*Sqrt[x^2 + x*Sqrt[x + x^2]]),x]","-\frac{2 \sqrt{x+x^2} \sqrt{x \left(x+\sqrt{x+x^2}\right)}}{x}+\sqrt{x \left(x+\sqrt{x+x^2}\right)} \left(\frac{2 (-2+x)}{x}+\frac{\sqrt{2} \sqrt{-x+\sqrt{x+x^2}} \tanh ^{-1}\left(\sqrt{2} \sqrt{-x+\sqrt{x+x^2}}\right)}{x}\right)","-\frac{2 \sqrt{x+x^2} \sqrt{x \left(x+\sqrt{x+x^2}\right)}}{x}+\sqrt{x \left(x+\sqrt{x+x^2}\right)} \left(\frac{2 (-2+x)}{x}+\frac{\sqrt{2} \sqrt{-x+\sqrt{x+x^2}} \tanh ^{-1}\left(\sqrt{2} \sqrt{-x+\sqrt{x+x^2}}\right)}{x}\right)",1,"(-2*Sqrt[x + x^2]*Sqrt[x*(x + Sqrt[x + x^2])])/x + Sqrt[x*(x + Sqrt[x + x^2])]*((2*(-2 + x))/x + (Sqrt[2]*Sqrt[-x + Sqrt[x + x^2]]*ArcTanh[Sqrt[2]*Sqrt[-x + Sqrt[x + x^2]]])/x)","A",1
1603,1,109,109,4.3801105,"\int \frac{\sqrt{x^2+x \sqrt{x+x^2}}}{\sqrt{x+x^2}} \, dx","IntegrateAlgebraic[Sqrt[x^2 + x*Sqrt[x + x^2]]/Sqrt[x + x^2],x]","\frac{3 \sqrt{x+x^2} \sqrt{x \left(x+\sqrt{x+x^2}\right)}}{2 x}+\sqrt{x \left(x+\sqrt{x+x^2}\right)} \left(-\frac{1}{2}-\frac{3 \sqrt{-x+\sqrt{x+x^2}} \tanh ^{-1}\left(\sqrt{2} \sqrt{-x+\sqrt{x+x^2}}\right)}{2 \sqrt{2} x}\right)","\frac{3 \sqrt{x+x^2} \sqrt{x \left(x+\sqrt{x+x^2}\right)}}{2 x}+\sqrt{x \left(x+\sqrt{x+x^2}\right)} \left(-\frac{1}{2}-\frac{3 \sqrt{-x+\sqrt{x+x^2}} \tanh ^{-1}\left(\sqrt{2} \sqrt{-x+\sqrt{x+x^2}}\right)}{2 \sqrt{2} x}\right)",1,"(3*Sqrt[x + x^2]*Sqrt[x*(x + Sqrt[x + x^2])])/(2*x) + Sqrt[x*(x + Sqrt[x + x^2])]*(-1/2 - (3*Sqrt[-x + Sqrt[x + x^2]]*ArcTanh[Sqrt[2]*Sqrt[-x + Sqrt[x + x^2]]])/(2*Sqrt[2]*x))","A",1
1604,1,77,109,0.1542088,"\int \frac{1}{\sqrt{b+\sqrt{b^2+a x^2}}} \, dx","IntegrateAlgebraic[1/Sqrt[b + Sqrt[b^2 + a*x^2]],x]","\frac{2 x}{\sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{2} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{\sqrt{a}}","\frac{2 x}{\sqrt{b+\sqrt{b^2+a x^2}}}-\frac{2 \sqrt{2} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2} \sqrt{b}}\right)}{\sqrt{a}}",1,"(2*x)/Sqrt[b + Sqrt[b^2 + a*x^2]] - (Sqrt[2]*Sqrt[b]*ArcTan[(Sqrt[a]*x)/(Sqrt[2]*Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/Sqrt[a]","A",1
1605,1,109,109,0.1759541,"\int \left(b^2+a x^2\right)^2 \sqrt{b+\sqrt{b^2+a x^2}} \, dx","IntegrateAlgebraic[(b^2 + a*x^2)^2*Sqrt[b + Sqrt[b^2 + a*x^2]],x]","\frac{2 x \sqrt{b^2+a x^2} \left(271 b^4+206 a b^2 x^2+63 a^2 x^4\right)}{693 \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{4 x \left(211 b^5+118 a b^3 x^2+35 a^2 b x^4\right)}{693 \sqrt{b+\sqrt{b^2+a x^2}}}","\frac{2 x \sqrt{b^2+a x^2} \left(271 b^4+206 a b^2 x^2+63 a^2 x^4\right)}{693 \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{4 x \left(211 b^5+118 a b^3 x^2+35 a^2 b x^4\right)}{693 \sqrt{b+\sqrt{b^2+a x^2}}}",1,"(2*x*Sqrt[b^2 + a*x^2]*(271*b^4 + 206*a*b^2*x^2 + 63*a^2*x^4))/(693*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (4*x*(211*b^5 + 118*a*b^3*x^2 + 35*a^2*b*x^4))/(693*Sqrt[b + Sqrt[b^2 + a*x^2]])","A",1
1606,1,110,110,0.2886657,"\int \frac{1}{(1-3 x) \sqrt[3]{-x+x^3}} \, dx","IntegrateAlgebraic[1/((1 - 3*x)*(-x + x^3)^(1/3)),x]","\frac{1}{4} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-x+x^3}}{-1-x+\sqrt[3]{-x+x^3}}\right)+\frac{1}{4} \log \left(1+x+2 \sqrt[3]{-x+x^3}\right)-\frac{1}{8} \log \left(1+2 x+x^2+(-2-2 x) \sqrt[3]{-x+x^3}+4 \left(-x+x^3\right)^{2/3}\right)","\frac{1}{4} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-x+x^3}}{-1-x+\sqrt[3]{-x+x^3}}\right)+\frac{1}{4} \log \left(1+x+2 \sqrt[3]{-x+x^3}\right)-\frac{1}{8} \log \left(1+2 x+x^2+(-2-2 x) \sqrt[3]{-x+x^3}+4 \left(-x+x^3\right)^{2/3}\right)",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(-x + x^3)^(1/3))/(-1 - x + (-x + x^3)^(1/3))])/4 + Log[1 + x + 2*(-x + x^3)^(1/3)]/4 - Log[1 + 2*x + x^2 + (-2 - 2*x)*(-x + x^3)^(1/3) + 4*(-x + x^3)^(2/3)]/8","A",1
1607,1,110,110,0.2871182,"\int \frac{1}{(1+3 x) \sqrt[3]{-x+x^3}} \, dx","IntegrateAlgebraic[1/((1 + 3*x)*(-x + x^3)^(1/3)),x]","\frac{1}{4} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-x+x^3}}{1-x+\sqrt[3]{-x+x^3}}\right)+\frac{1}{4} \log \left(-1+x+2 \sqrt[3]{-x+x^3}\right)-\frac{1}{8} \log \left(1-2 x+x^2+(2-2 x) \sqrt[3]{-x+x^3}+4 \left(-x+x^3\right)^{2/3}\right)","\frac{1}{4} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-x+x^3}}{1-x+\sqrt[3]{-x+x^3}}\right)+\frac{1}{4} \log \left(-1+x+2 \sqrt[3]{-x+x^3}\right)-\frac{1}{8} \log \left(1-2 x+x^2+(2-2 x) \sqrt[3]{-x+x^3}+4 \left(-x+x^3\right)^{2/3}\right)",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(-x + x^3)^(1/3))/(1 - x + (-x + x^3)^(1/3))])/4 + Log[-1 + x + 2*(-x + x^3)^(1/3)]/4 - Log[1 - 2*x + x^2 + (2 - 2*x)*(-x + x^3)^(1/3) + 4*(-x + x^3)^(2/3)]/8","A",1
1608,1,110,110,0.1978368,"\int \frac{x}{\sqrt[3]{-x^2+x^3}} \, dx","IntegrateAlgebraic[x/(-x^2 + x^3)^(1/3),x]","\frac{\left(-x^2+x^3\right)^{2/3}}{x}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(-x+\sqrt[3]{-x^2+x^3}\right)+\frac{1}{6} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)","\frac{\left(-x^2+x^3\right)^{2/3}}{x}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(-x+\sqrt[3]{-x^2+x^3}\right)+\frac{1}{6} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)",1,"(-x^2 + x^3)^(2/3)/x + ArcTan[(Sqrt[3]*x)/(x + 2*(-x^2 + x^3)^(1/3))]/Sqrt[3] - Log[-x + (-x^2 + x^3)^(1/3)]/3 + Log[x^2 + x*(-x^2 + x^3)^(1/3) + (-x^2 + x^3)^(2/3)]/6","A",1
1609,1,110,110,0.202031,"\int \frac{\left(1-x^3\right)^{2/3} \left(-1+x^3\right)}{x^6 \left(-1+2 x^3\right)} \, dx","IntegrateAlgebraic[((1 - x^3)^(2/3)*(-1 + x^3))/(x^6*(-1 + 2*x^3)),x]","\frac{\left(-2-3 x^3\right) \left(1-x^3\right)^{2/3}}{10 x^5}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1-x^3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(-x+\sqrt[3]{1-x^3}\right)+\frac{1}{6} \log \left(x^2+x \sqrt[3]{1-x^3}+\left(1-x^3\right)^{2/3}\right)","\frac{\left(-2-3 x^3\right) \left(1-x^3\right)^{2/3}}{10 x^5}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1-x^3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(-x+\sqrt[3]{1-x^3}\right)+\frac{1}{6} \log \left(x^2+x \sqrt[3]{1-x^3}+\left(1-x^3\right)^{2/3}\right)",1,"((-2 - 3*x^3)*(1 - x^3)^(2/3))/(10*x^5) + ArcTan[(Sqrt[3]*x)/(x + 2*(1 - x^3)^(1/3))]/Sqrt[3] - Log[-x + (1 - x^3)^(1/3)]/3 + Log[x^2 + x*(1 - x^3)^(1/3) + (1 - x^3)^(2/3)]/6","A",1
1610,1,110,110,0.5144583,"\int \frac{2 b+a x}{\left(b+a x+x^2\right) \sqrt[4]{b x^2+a x^3}} \, dx","IntegrateAlgebraic[(2*b + a*x)/((b + a*x + x^2)*(b*x^2 + a*x^3)^(1/4)),x]","-\sqrt{2} \tan ^{-1}\left(\frac{-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{b x^2+a x^3}}{\sqrt{2}}}{x \sqrt[4]{b x^2+a x^3}}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{b x^2+a x^3}}{x^2+\sqrt{b x^2+a x^3}}\right)","-\sqrt{2} \tan ^{-1}\left(\frac{-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{b x^2+a x^3}}{\sqrt{2}}}{x \sqrt[4]{b x^2+a x^3}}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{b x^2+a x^3}}{x^2+\sqrt{b x^2+a x^3}}\right)",1,"-(Sqrt[2]*ArcTan[(-(x^2/Sqrt[2]) + Sqrt[b*x^2 + a*x^3]/Sqrt[2])/(x*(b*x^2 + a*x^3)^(1/4))]) + Sqrt[2]*ArcTanh[(Sqrt[2]*x*(b*x^2 + a*x^3)^(1/4))/(x^2 + Sqrt[b*x^2 + a*x^3])]","A",1
1611,1,110,110,0.5912812,"\int \frac{x^3 \sqrt{x+x^4}}{-b+a x^3} \, dx","IntegrateAlgebraic[(x^3*Sqrt[x + x^4])/(-b + a*x^3),x]","\frac{x \sqrt{x+x^4}}{3 a}+\frac{2 \sqrt{-a-b} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{-a-b} x \sqrt{x+x^4}}{\sqrt{b} (1+x) \left(1-x+x^2\right)}\right)}{3 a^2}+\frac{(a+2 b) \tanh ^{-1}\left(\frac{x^2}{\sqrt{x+x^4}}\right)}{3 a^2}","\frac{x \sqrt{x+x^4}}{3 a}+\frac{2 \sqrt{-a-b} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{-a-b} x \sqrt{x+x^4}}{\sqrt{b} (1+x) \left(1-x+x^2\right)}\right)}{3 a^2}+\frac{(a+2 b) \tanh ^{-1}\left(\frac{x^2}{\sqrt{x+x^4}}\right)}{3 a^2}",1,"(x*Sqrt[x + x^4])/(3*a) + (2*Sqrt[-a - b]*Sqrt[b]*ArcTan[(Sqrt[-a - b]*x*Sqrt[x + x^4])/(Sqrt[b]*(1 + x)*(1 - x + x^2))])/(3*a^2) + ((a + 2*b)*ArcTanh[x^2/Sqrt[x + x^4]])/(3*a^2)","A",1
1612,1,110,110,0.817362,"\int x^9 \sqrt[3]{1+x^6} \, dx","IntegrateAlgebraic[x^9*(1 + x^6)^(1/3),x]","\frac{1}{36} \sqrt[3]{1+x^6} \left(x^4+3 x^{10}\right)+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{1+x^6}}\right)}{18 \sqrt{3}}+\frac{1}{54} \log \left(-x^2+\sqrt[3]{1+x^6}\right)-\frac{1}{108} \log \left(x^4+x^2 \sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)","\frac{1}{36} \sqrt[3]{1+x^6} \left(x^4+3 x^{10}\right)+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{1+x^6}}\right)}{18 \sqrt{3}}+\frac{1}{54} \log \left(-x^2+\sqrt[3]{1+x^6}\right)-\frac{1}{108} \log \left(x^4+x^2 \sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)",1,"((1 + x^6)^(1/3)*(x^4 + 3*x^10))/36 + ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(1 + x^6)^(1/3))]/(18*Sqrt[3]) + Log[-x^2 + (1 + x^6)^(1/3)]/54 - Log[x^4 + x^2*(1 + x^6)^(1/3) + (1 + x^6)^(2/3)]/108","A",1
1613,1,110,110,0.2036806,"\int \frac{\left(-1+x^3\right)^{2/3} \left(2+x^3\right)}{x^6 \left(4+x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(2 + x^3))/(x^6*(4 + x^6)),x]","\frac{\left(-4-x^3\right) \left(-1+x^3\right)^{2/3}}{40 x^5}+\frac{1}{96} \text{RootSum}\left[5-8 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-5 \log (x)+5 \log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right)+6 \log (x) \text{$\#$1}^3-6 \log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-\text{$\#$1}+\text{$\#$1}^4}\&\right]","\frac{\left(-4-x^3\right) \left(-1+x^3\right)^{2/3}}{40 x^5}+\frac{1}{96} \text{RootSum}\left[5-8 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-5 \log (x)+5 \log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right)+6 \log (x) \text{$\#$1}^3-6 \log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-\text{$\#$1}+\text{$\#$1}^4}\&\right]",1,"((-4 - x^3)*(-1 + x^3)^(2/3))/(40*x^5) + RootSum[5 - 8*#1^3 + 4*#1^6 & , (-5*Log[x] + 5*Log[(-1 + x^3)^(1/3) - x*#1] + 6*Log[x]*#1^3 - 6*Log[(-1 + x^3)^(1/3) - x*#1]*#1^3)/(-#1 + #1^4) & ]/96","A",1
1614,1,110,110,0.0010248,"\int \frac{\left(-1+x^3\right)^{2/3} \left(2+x^3\right)}{x^6 \left(4+x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(2 + x^3))/(x^6*(4 + x^6)),x]","\frac{\left(-4-x^3\right) \left(-1+x^3\right)^{2/3}}{40 x^5}+\frac{1}{96} \text{RootSum}\left[5-8 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-5 \log (x)+5 \log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right)+6 \log (x) \text{$\#$1}^3-6 \log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-\text{$\#$1}+\text{$\#$1}^4}\&\right]","\frac{\left(-4-x^3\right) \left(-1+x^3\right)^{2/3}}{40 x^5}+\frac{1}{96} \text{RootSum}\left[5-8 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-5 \log (x)+5 \log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right)+6 \log (x) \text{$\#$1}^3-6 \log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-\text{$\#$1}+\text{$\#$1}^4}\&\right]",1,"((-4 - x^3)*(-1 + x^3)^(2/3))/(40*x^5) + RootSum[5 - 8*#1^3 + 4*#1^6 & , (-5*Log[x] + 5*Log[(-1 + x^3)^(1/3) - x*#1] + 6*Log[x]*#1^3 - 6*Log[(-1 + x^3)^(1/3) - x*#1]*#1^3)/(-#1 + #1^4) & ]/96","A",1
1615,1,110,110,0.528458,"\int \frac{b-a x^3+x^6}{x^6 \sqrt[4]{b x+a x^4}} \, dx","IntegrateAlgebraic[(b - a*x^3 + x^6)/(x^6*(b*x + a*x^4)^(1/4)),x]","\frac{4 \left(-3 b+11 a x^3\right) \left(b x+a x^4\right)^{3/4}}{63 b x^6}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 \sqrt[4]{a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 \sqrt[4]{a}}","\frac{4 \left(-3 b+11 a x^3\right) \left(b x+a x^4\right)^{3/4}}{63 b x^6}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 \sqrt[4]{a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 \sqrt[4]{a}}",1,"(4*(-3*b + 11*a*x^3)*(b*x + a*x^4)^(3/4))/(63*b*x^6) + (2*ArcTan[(a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/(3*a^(1/4)) + (2*ArcTanh[(a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/(3*a^(1/4))","A",1
1616,1,110,110,0.669615,"\int \frac{-b+a x^6}{x^6 \sqrt[4]{b x+a x^4}} \, dx","IntegrateAlgebraic[(-b + a*x^6)/(x^6*(b*x + a*x^4)^(1/4)),x]","-\frac{4 \left(-3 b+4 a x^3\right) \left(b x+a x^4\right)^{3/4}}{63 b x^6}+\frac{2}{3} a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)+\frac{2}{3} a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)","-\frac{4 \left(-3 b+4 a x^3\right) \left(b x+a x^4\right)^{3/4}}{63 b x^6}+\frac{2}{3} a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)+\frac{2}{3} a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)",1,"(-4*(-3*b + 4*a*x^3)*(b*x + a*x^4)^(3/4))/(63*b*x^6) + (2*a^(3/4)*ArcTan[(a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/3 + (2*a^(3/4)*ArcTanh[(a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/3","A",1
1617,1,110,110,0.6957812,"\int \frac{b+a x^6}{x^6 \sqrt[4]{b x+a x^4}} \, dx","IntegrateAlgebraic[(b + a*x^6)/(x^6*(b*x + a*x^4)^(1/4)),x]","\frac{4 \left(-3 b+4 a x^3\right) \left(b x+a x^4\right)^{3/4}}{63 b x^6}+\frac{2}{3} a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)+\frac{2}{3} a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)","\frac{4 \left(-3 b+4 a x^3\right) \left(b x+a x^4\right)^{3/4}}{63 b x^6}+\frac{2}{3} a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)+\frac{2}{3} a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)",1,"(4*(-3*b + 4*a*x^3)*(b*x + a*x^4)^(3/4))/(63*b*x^6) + (2*a^(3/4)*ArcTan[(a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/3 + (2*a^(3/4)*ArcTanh[(a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/3","A",1
1618,1,110,110,3.0015221,"\int \frac{-3 b+2 a x^5}{\left(2 b+x^3+2 a x^5\right) \sqrt[4]{b x+a x^6}} \, dx","IntegrateAlgebraic[(-3*b + 2*a*x^5)/((2*b + x^3 + 2*a*x^5)*(b*x + a*x^6)^(1/4)),x]","-\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{b x+a x^6}}{-x^2+\sqrt{2} \sqrt{b x+a x^6}}\right)}{\sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{2^{3/4}}+\frac{\sqrt{b x+a x^6}}{\sqrt[4]{2}}}{x \sqrt[4]{b x+a x^6}}\right)}{\sqrt[4]{2}}","-\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{b x+a x^6}}{-x^2+\sqrt{2} \sqrt{b x+a x^6}}\right)}{\sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{2^{3/4}}+\frac{\sqrt{b x+a x^6}}{\sqrt[4]{2}}}{x \sqrt[4]{b x+a x^6}}\right)}{\sqrt[4]{2}}",1,"-(ArcTan[(2^(3/4)*x*(b*x + a*x^6)^(1/4))/(-x^2 + Sqrt[2]*Sqrt[b*x + a*x^6])]/2^(1/4)) - ArcTanh[(x^2/2^(3/4) + Sqrt[b*x + a*x^6]/2^(1/4))/(x*(b*x + a*x^6)^(1/4))]/2^(1/4)","A",1
1619,1,110,110,14.6794926,"\int \frac{1}{\sqrt[3]{b^2 x^2+a^3 x^3} \left(2 b+a^6 x^6\right)} \, dx","IntegrateAlgebraic[1/((b^2*x^2 + a^3*x^3)^(1/3)*(2*b + a^6*x^6)),x]","-\frac{\text{RootSum}\left[2 a^{18}+a^6 b^{11}-12 a^{15} \text{$\#$1}^3+30 a^{12} \text{$\#$1}^6-40 a^9 \text{$\#$1}^9+30 a^6 \text{$\#$1}^{12}-12 a^3 \text{$\#$1}^{15}+2 \text{$\#$1}^{18}\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{12 b}","-\frac{\text{RootSum}\left[2 a^{18}+a^6 b^{11}-12 a^{15} \text{$\#$1}^3+30 a^{12} \text{$\#$1}^6-40 a^9 \text{$\#$1}^9+30 a^6 \text{$\#$1}^{12}-12 a^3 \text{$\#$1}^{15}+2 \text{$\#$1}^{18}\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{12 b}",1,"-1/12*RootSum[2*a^18 + a^6*b^11 - 12*a^15*#1^3 + 30*a^12*#1^6 - 40*a^9*#1^9 + 30*a^6*#1^12 - 12*a^3*#1^15 + 2*#1^18 & , (-Log[x] + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1])/#1 & ]/b","A",1
1620,1,110,110,0.002215,"\int \frac{1}{\sqrt[3]{b^2 x^2+a^3 x^3} \left(2 b+a^6 x^6\right)} \, dx","IntegrateAlgebraic[1/((b^2*x^2 + a^3*x^3)^(1/3)*(2*b + a^6*x^6)),x]","-\frac{\text{RootSum}\left[2 a^{18}+a^6 b^{11}-12 a^{15} \text{$\#$1}^3+30 a^{12} \text{$\#$1}^6-40 a^9 \text{$\#$1}^9+30 a^6 \text{$\#$1}^{12}-12 a^3 \text{$\#$1}^{15}+2 \text{$\#$1}^{18}\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{12 b}","-\frac{\text{RootSum}\left[2 a^{18}+a^6 b^{11}-12 a^{15} \text{$\#$1}^3+30 a^{12} \text{$\#$1}^6-40 a^9 \text{$\#$1}^9+30 a^6 \text{$\#$1}^{12}-12 a^3 \text{$\#$1}^{15}+2 \text{$\#$1}^{18}\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{12 b}",1,"-1/12*RootSum[2*a^18 + a^6*b^11 - 12*a^15*#1^3 + 30*a^12*#1^6 - 40*a^9*#1^9 + 30*a^6*#1^12 - 12*a^3*#1^15 + 2*#1^18 & , (-Log[x] + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1])/#1 & ]/b","A",1
1621,1,73,110,1.1478252,"\int \frac{\left(-1+x^4\right) \left(1+x^2+3 x^4+x^6+x^8\right)}{\left(1+x^2+x^4\right)^{3/2} \left(1+3 x^2+5 x^4+3 x^6+x^8\right)} \, dx","IntegrateAlgebraic[((-1 + x^4)*(1 + x^2 + 3*x^4 + x^6 + x^8))/((1 + x^2 + x^4)^(3/2)*(1 + 3*x^2 + 5*x^4 + 3*x^6 + x^8)),x]","-\frac{x}{\sqrt{1+x^2+x^4}}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x \sqrt{1+x^2+x^4}}{1+x^4}\right)}{\sqrt{3}}-\tanh ^{-1}\left(\frac{x \sqrt{1+x^2+x^4}}{\left(1+x^2\right)^2}\right)","-\frac{x}{\sqrt{1+x^2+x^4}}+\sqrt{\frac{1}{3} \left(-2+2 i \sqrt{3}\right)} \tan ^{-1}\left(\frac{2 x}{\left(-i+\sqrt{3}\right) \sqrt{1+x^2+x^4}}\right)+\sqrt{\frac{1}{3} \left(-2-2 i \sqrt{3}\right)} \tan ^{-1}\left(\frac{2 x}{\left(i+\sqrt{3}\right) \sqrt{1+x^2+x^4}}\right)",1,"-(x/Sqrt[1 + x^2 + x^4]) + ArcTan[(Sqrt[3]*x*Sqrt[1 + x^2 + x^4])/(1 + x^4)]/Sqrt[3] - ArcTanh[(x*Sqrt[1 + x^2 + x^4])/(1 + x^2)^2]","A",1
1622,1,110,110,1.2615192,"\int \frac{-b+a x^8}{\sqrt[4]{b+a x^4} \left(b+a x^8\right)} \, dx","IntegrateAlgebraic[(-b + a*x^8)/((b + a*x^4)^(1/4)*(b + a*x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2 \sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2 \sqrt[4]{a}}+\frac{1}{4} \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2 \sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2 \sqrt[4]{a}}+\frac{1}{4} \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/(2*a^(1/4)) + ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/(2*a^(1/4)) + RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (-Log[x] + Log[(b + a*x^4)^(1/4) - x*#1])/#1 & ]/4","A",1
1623,1,110,110,0.0013361,"\int \frac{-b+a x^8}{\sqrt[4]{b+a x^4} \left(b+a x^8\right)} \, dx","IntegrateAlgebraic[(-b + a*x^8)/((b + a*x^4)^(1/4)*(b + a*x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2 \sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2 \sqrt[4]{a}}+\frac{1}{4} \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2 \sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2 \sqrt[4]{a}}+\frac{1}{4} \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/(2*a^(1/4)) + ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/(2*a^(1/4)) + RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (-Log[x] + Log[(b + a*x^4)^(1/4) - x*#1])/#1 & ]/4","A",1
1624,1,110,110,9.4560454,"\int \frac{\left(-3+2 x+2 x^5\right) \sqrt{x-x^2+x^6}}{1-2 x+x^2-x^3+x^4+2 x^5-3 x^6-x^8+x^{10}} \, dx","IntegrateAlgebraic[((-3 + 2*x + 2*x^5)*Sqrt[x - x^2 + x^6])/(1 - 2*x + x^2 - x^3 + x^4 + 2*x^5 - 3*x^6 - x^8 + x^10),x]","\sqrt{\frac{1}{5} \left(-2+2 \sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} \sqrt{x-x^2+x^6}}{x^2}\right)-\sqrt{\frac{1}{5} \left(2+2 \sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} \sqrt{x-x^2+x^6}}{x^2}\right)","\sqrt{\frac{1}{5} \left(-2+2 \sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} \sqrt{x-x^2+x^6}}{x^2}\right)-\sqrt{\frac{1}{5} \left(2+2 \sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} \sqrt{x-x^2+x^6}}{x^2}\right)",1,"Sqrt[(-2 + 2*Sqrt[5])/5]*ArcTan[(Sqrt[1/2 + Sqrt[5]/2]*Sqrt[x - x^2 + x^6])/x^2] - Sqrt[(2 + 2*Sqrt[5])/5]*ArcTanh[(Sqrt[-1/2 + Sqrt[5]/2]*Sqrt[x - x^2 + x^6])/x^2]","A",1
1625,1,84,110,0.2822604,"\int \frac{\sqrt{x+\sqrt{1+x}}}{x^2 \sqrt{1+x}} \, dx","IntegrateAlgebraic[Sqrt[x + Sqrt[1 + x]]/(x^2*Sqrt[1 + x]),x]","-\frac{\sqrt{1+x} \sqrt{x+\sqrt{1+x}}}{x}+\frac{3}{2} \tan ^{-1}\left(1+\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}\right)-\frac{1}{2} \tanh ^{-1}\left(1-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}\right)","-\frac{\sqrt{1+x} \sqrt{x+\sqrt{1+x}}}{x}+\frac{3}{2} \tan ^{-1}\left(\frac{\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}}{2+\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}}\right)-\frac{1}{2} \tanh ^{-1}\left(1-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}\right)",1,"-((Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]])/x) + (3*ArcTan[1 + Sqrt[1 + x] - Sqrt[x + Sqrt[1 + x]]])/2 - ArcTanh[1 - Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]]]/2","A",1
1626,1,84,110,0.1875937,"\int \frac{\sqrt{x+\sqrt{1+x}}}{x \sqrt{1+x}} \, dx","IntegrateAlgebraic[Sqrt[x + Sqrt[1 + x]]/(x*Sqrt[1 + x]),x]","-2 \tan ^{-1}\left(1+\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}\right)-2 \tanh ^{-1}\left(1-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}\right)-2 \log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}\right)","-2 \tan ^{-1}\left(\frac{\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}}{2+\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}}\right)-2 \tanh ^{-1}\left(1-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}\right)-2 \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}\right)",1,"-2*ArcTan[1 + Sqrt[1 + x] - Sqrt[x + Sqrt[1 + x]]] - 2*ArcTanh[1 - Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]]] - 2*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]]","A",1
1627,1,95,110,0.2027922,"\int \frac{\sqrt{a x+\sqrt{-b+a x}}}{\sqrt{-b+a x}} \, dx","IntegrateAlgebraic[Sqrt[a*x + Sqrt[-b + a*x]]/Sqrt[-b + a*x],x]","\frac{\sqrt{a x+\sqrt{-b+a x}} \left(1+2 \sqrt{-b+a x}\right)}{2 a}+\frac{(1-4 b) \log \left(a \left(-1-2 \sqrt{-b+a x}\right)+2 a \sqrt{a x+\sqrt{-b+a x}}\right)}{4 a}","\frac{\sqrt{a x+\sqrt{-b+a x}}}{2 a}+\frac{\sqrt{-b+a x} \sqrt{a x+\sqrt{-b+a x}}}{a}+\frac{(1-4 b) \log \left(1+2 \sqrt{-b+a x}-2 \sqrt{a x+\sqrt{-b+a x}}\right)}{4 a}",1,"(Sqrt[a*x + Sqrt[-b + a*x]]*(1 + 2*Sqrt[-b + a*x]))/(2*a) + ((1 - 4*b)*Log[a*(-1 - 2*Sqrt[-b + a*x]) + 2*a*Sqrt[a*x + Sqrt[-b + a*x]]])/(4*a)","A",1
1628,1,124,110,0.1556941,"\int \frac{1-a x+b \sqrt{a+b x}}{\sqrt{a+b x} \left(x+a b \sqrt{a+b x}\right)} \, dx","IntegrateAlgebraic[(1 - a*x + b*Sqrt[a + b*x])/(Sqrt[a + b*x]*(x + a*b*Sqrt[a + b*x])),x]","-\frac{2 a \sqrt{a+b x}}{b}+\frac{2 \left(-2+a b^3+a^3 b^3\right) \tanh ^{-1}\left(\frac{\sqrt{a} b^2}{\sqrt{4+a b^4}}+\frac{2 \sqrt{a+b x}}{\sqrt{a} \sqrt{4+a b^4}}\right)}{\sqrt{a} \sqrt{4+a b^4}}+\left(b+a^2 b\right) \log \left(b x+a b^2 \sqrt{a+b x}\right)","-\frac{2 a \sqrt{a+b x}}{b}+\frac{2 \left(-2+a b^3+a^3 b^3\right) \tanh ^{-1}\left(\frac{a b^2+2 \sqrt{a+b x}}{\sqrt{a} \sqrt{4+a b^4}}\right)}{\sqrt{a} \sqrt{4+a b^4}}+\left(b+a^2 b\right) \log \left(b x+a b^2 \sqrt{a+b x}\right)",1,"(-2*a*Sqrt[a + b*x])/b + (2*(-2 + a*b^3 + a^3*b^3)*ArcTanh[(Sqrt[a]*b^2)/Sqrt[4 + a*b^4] + (2*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[4 + a*b^4])])/(Sqrt[a]*Sqrt[4 + a*b^4]) + (b + a^2*b)*Log[b*x + a*b^2*Sqrt[a + b*x]]","A",1
1629,1,78,110,0.1499211,"\int \frac{\sqrt{b+\sqrt{b^2+a x^2}}}{x^2} \, dx","IntegrateAlgebraic[Sqrt[b + Sqrt[b^2 + a*x^2]]/x^2,x]","-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{x}+\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{\sqrt{2} \sqrt{b}}","-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{x}+\frac{\sqrt{2} \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2} \sqrt{b}}\right)}{\sqrt{b}}",1,"-(Sqrt[b + Sqrt[b^2 + a*x^2]]/x) + (Sqrt[a]*ArcTan[(Sqrt[a]*x)/(Sqrt[2]*Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/(Sqrt[2]*Sqrt[b])","A",1
1630,1,110,110,5.3113119,"\int \frac{1}{\sqrt{-b x+a^2 x^2} \left(a x^2+x \sqrt{-b x+a^2 x^2}\right)^{3/2}} \, dx","IntegrateAlgebraic[1/(Sqrt[-(b*x) + a^2*x^2]*(a*x^2 + x*Sqrt[-(b*x) + a^2*x^2])^(3/2)),x]","\frac{4 \left(39 a b+32 a^3 x\right) \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{105 b^3 x^2}-\frac{4 \left(15 b-32 a^2 x\right) \sqrt{-b x+a^2 x^2} \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{105 b^3 x^3}","\frac{4 \left(39 a b+32 a^3 x\right) \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{105 b^3 x^2}-\frac{4 \left(15 b-32 a^2 x\right) \sqrt{-b x+a^2 x^2} \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{105 b^3 x^3}",1,"(4*(39*a*b + 32*a^3*x)*Sqrt[x*(a*x + Sqrt[-(b*x) + a^2*x^2])])/(105*b^3*x^2) - (4*(15*b - 32*a^2*x)*Sqrt[-(b*x) + a^2*x^2]*Sqrt[x*(a*x + Sqrt[-(b*x) + a^2*x^2])])/(105*b^3*x^3)","A",1
1631,1,110,110,0.1289358,"\int \frac{1}{1+\sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[(1 + Sqrt[x + Sqrt[1 + x^2]])^(-1),x]","\frac{-1+5 x+(2+2 x) \sqrt{x+\sqrt{1+x^2}}+\sqrt{1+x^2} \left(5+2 \sqrt{x+\sqrt{1+x^2}}\right)}{2 x+2 \sqrt{1+x^2}}+\frac{1}{2} \log \left(x+\sqrt{1+x^2}\right)-2 \log \left(1+\sqrt{x+\sqrt{1+x^2}}\right)","\frac{-1+5 x+(2+2 x) \sqrt{x+\sqrt{1+x^2}}+\sqrt{1+x^2} \left(5+2 \sqrt{x+\sqrt{1+x^2}}\right)}{2 x+2 \sqrt{1+x^2}}+\frac{1}{2} \log \left(x+\sqrt{1+x^2}\right)-2 \log \left(1+\sqrt{x+\sqrt{1+x^2}}\right)",1,"(-1 + 5*x + (2 + 2*x)*Sqrt[x + Sqrt[1 + x^2]] + Sqrt[1 + x^2]*(5 + 2*Sqrt[x + Sqrt[1 + x^2]]))/(2*x + 2*Sqrt[1 + x^2]) + Log[x + Sqrt[1 + x^2]]/2 - 2*Log[1 + Sqrt[x + Sqrt[1 + x^2]]]","A",1
1632,1,111,111,0.1229102,"\int \frac{-2+x}{\left(2+x^2\right) \sqrt[3]{-1+x+2 x^2}} \, dx","IntegrateAlgebraic[(-2 + x)/((2 + x^2)*(-1 + x + 2*x^2)^(1/3)),x]","\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-1+x+2 x^2}}{2+2 x+\sqrt[3]{-1+x+2 x^2}}\right)+\frac{1}{2} \log \left(-1-x+\sqrt[3]{-1+x+2 x^2}\right)-\frac{1}{4} \log \left(1+2 x+x^2+(1+x) \sqrt[3]{-1+x+2 x^2}+\left(-1+x+2 x^2\right)^{2/3}\right)","\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-1+x+2 x^2}}{2+2 x+\sqrt[3]{-1+x+2 x^2}}\right)+\frac{1}{2} \log \left(-1-x+\sqrt[3]{-1+x+2 x^2}\right)-\frac{1}{4} \log \left(1+2 x+x^2+(1+x) \sqrt[3]{-1+x+2 x^2}+\left(-1+x+2 x^2\right)^{2/3}\right)",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(-1 + x + 2*x^2)^(1/3))/(2 + 2*x + (-1 + x + 2*x^2)^(1/3))])/2 + Log[-1 - x + (-1 + x + 2*x^2)^(1/3)]/2 - Log[1 + 2*x + x^2 + (1 + x)*(-1 + x + 2*x^2)^(1/3) + (-1 + x + 2*x^2)^(2/3)]/4","A",1
1633,1,111,111,0.8407967,"\int \frac{(-a+x) (-b+x) (-2 a b+(a+b) x)}{\left(x^2 (-a+x) (-b+x)\right)^{3/4} \left(-a b+(a+b) x+(-1+d) x^2\right)} \, dx","IntegrateAlgebraic[((-a + x)*(-b + x)*(-2*a*b + (a + b)*x))/((x^2*(-a + x)*(-b + x))^(3/4)*(-(a*b) + (a + b)*x + (-1 + d)*x^2)),x]","-\frac{4 \sqrt[4]{a b x^2+(-a-b) x^3+x^4}}{x}-2 \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{a b x^2+(-a-b) x^3+x^4}}\right)+2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{a b x^2+(-a-b) x^3+x^4}}\right)","-\frac{4 \sqrt[4]{a b x^2+(-a-b) x^3+x^4}}{x}-2 \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{a b x^2+(-a-b) x^3+x^4}}\right)+2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{a b x^2+(-a-b) x^3+x^4}}\right)",1,"(-4*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/4))/x - 2*d^(1/4)*ArcTan[(d^(1/4)*x)/(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/4)] + 2*d^(1/4)*ArcTanh[(d^(1/4)*x)/(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/4)]","A",1
1634,1,111,111,1.0451921,"\int \frac{1-x+x^2}{\left(-2+2 x+x^2\right) \sqrt{1+x^3}} \, dx","IntegrateAlgebraic[(1 - x + x^2)/((-2 + 2*x + x^2)*Sqrt[1 + x^3]),x]","-\frac{1}{2} \sqrt{\frac{1}{3} \left(3+2 \sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} \sqrt{1+x^3}}{1-x+x^2}\right)-\frac{1}{2} \sqrt{\frac{1}{3} \left(-3+2 \sqrt{3}\right)} \tanh ^{-1}\left(\frac{\sqrt{-3+2 \sqrt{3}} \sqrt{1+x^3}}{1-x+x^2}\right)","-\frac{1}{2} \sqrt{\frac{1}{3} \left(3+2 \sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} \sqrt{1+x^3}}{1-x+x^2}\right)-\frac{1}{2} \sqrt{\frac{1}{3} \left(-3+2 \sqrt{3}\right)} \tanh ^{-1}\left(\frac{\sqrt{-3+2 \sqrt{3}} \sqrt{1+x^3}}{1-x+x^2}\right)",1,"-1/2*(Sqrt[(3 + 2*Sqrt[3])/3]*ArcTan[(Sqrt[3 + 2*Sqrt[3]]*Sqrt[1 + x^3])/(1 - x + x^2)]) - (Sqrt[(-3 + 2*Sqrt[3])/3]*ArcTanh[(Sqrt[-3 + 2*Sqrt[3]]*Sqrt[1 + x^3])/(1 - x + x^2)])/2","A",1
1635,1,111,111,0.2397941,"\int \frac{-1+x}{x \sqrt[3]{-x^2+x^3}} \, dx","IntegrateAlgebraic[(-1 + x)/(x*(-x^2 + x^3)^(1/3)),x]","-\frac{3 \left(-x^2+x^3\right)^{2/3}}{2 x^2}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{-x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)","-\frac{3 \left(-x^2+x^3\right)^{2/3}}{2 x^2}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{-x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)",1,"(-3*(-x^2 + x^3)^(2/3))/(2*x^2) + Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-x^2 + x^3)^(1/3))] - Log[-x + (-x^2 + x^3)^(1/3)] + Log[x^2 + x*(-x^2 + x^3)^(1/3) + (-x^2 + x^3)^(2/3)]/2","A",1
1636,1,111,111,0.242906,"\int \frac{1+x}{x \sqrt[3]{-x^2+x^3}} \, dx","IntegrateAlgebraic[(1 + x)/(x*(-x^2 + x^3)^(1/3)),x]","\frac{3 \left(-x^2+x^3\right)^{2/3}}{2 x^2}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{-x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)","\frac{3 \left(-x^2+x^3\right)^{2/3}}{2 x^2}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{-x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)",1,"(3*(-x^2 + x^3)^(2/3))/(2*x^2) + Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-x^2 + x^3)^(1/3))] - Log[-x + (-x^2 + x^3)^(1/3)] + Log[x^2 + x*(-x^2 + x^3)^(1/3) + (-x^2 + x^3)^(2/3)]/2","A",1
1637,1,111,111,0.2905006,"\int \frac{x^2}{\sqrt[3]{x^2+x^3}} \, dx","IntegrateAlgebraic[x^2/(x^2 + x^3)^(1/3),x]","\frac{(-4+3 x) \left(x^2+x^3\right)^{2/3}}{6 x}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^3}}\right)}{3 \sqrt{3}}-\frac{2}{9} \log \left(-x+\sqrt[3]{x^2+x^3}\right)+\frac{1}{9} \log \left(x^2+x \sqrt[3]{x^2+x^3}+\left(x^2+x^3\right)^{2/3}\right)","\frac{(-4+3 x) \left(x^2+x^3\right)^{2/3}}{6 x}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^3}}\right)}{3 \sqrt{3}}-\frac{2}{9} \log \left(-x+\sqrt[3]{x^2+x^3}\right)+\frac{1}{9} \log \left(x^2+x \sqrt[3]{x^2+x^3}+\left(x^2+x^3\right)^{2/3}\right)",1,"((-4 + 3*x)*(x^2 + x^3)^(2/3))/(6*x) + (2*ArcTan[(Sqrt[3]*x)/(x + 2*(x^2 + x^3)^(1/3))])/(3*Sqrt[3]) - (2*Log[-x + (x^2 + x^3)^(1/3)])/9 + Log[x^2 + x*(x^2 + x^3)^(1/3) + (x^2 + x^3)^(2/3)]/9","A",1
1638,1,111,111,1.0702745,"\int \frac{(a-3 b+2 x) \left(-a^3+3 a^2 x-3 a x^2+x^3\right)}{(-b+x) \sqrt[4]{(-a+x) (-b+x)} \left(-a^3+b d-\left(-3 a^2+d\right) x-3 a x^2+x^3\right)} \, dx","IntegrateAlgebraic[((a - 3*b + 2*x)*(-a^3 + 3*a^2*x - 3*a*x^2 + x^3))/((-b + x)*((-a + x)*(-b + x))^(1/4)*(-a^3 + b*d - (-3*a^2 + d)*x - 3*a*x^2 + x^3)),x]","-\frac{4 \left(a b-a x-b x+x^2\right)^{3/4}}{b-x}-2 \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b+(-a-b) x+x^2}}{a-x}\right)+2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b+(-a-b) x+x^2}}{a-x}\right)","-\frac{4 \left(a b-a x-b x+x^2\right)^{3/4}}{b-x}-2 \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b+(-a-b) x+x^2}}{a-x}\right)+2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b+(-a-b) x+x^2}}{a-x}\right)",1,"(-4*(a*b - a*x - b*x + x^2)^(3/4))/(b - x) - 2*d^(1/4)*ArcTan[(d^(1/4)*(a*b + (-a - b)*x + x^2)^(1/4))/(a - x)] + 2*d^(1/4)*ArcTanh[(d^(1/4)*(a*b + (-a - b)*x + x^2)^(1/4))/(a - x)]","A",1
1639,1,111,111,0.9366382,"\int \frac{x^3 \left(3-2 (1+k) x+k x^2\right)}{(-1+x) \sqrt[4]{(1-x) x (1-k x)} (-1+k x) \left(-d+d (1+k) x-d k x^2+x^3\right)} \, dx","IntegrateAlgebraic[(x^3*(3 - 2*(1 + k)*x + k*x^2))/((-1 + x)*((1 - x)*x*(1 - k*x))^(1/4)*(-1 + k*x)*(-d + d*(1 + k)*x - d*k*x^2 + x^3)),x]","\frac{4 \left(x-x^2-k x^2+k x^3\right)^{3/4}}{(-1+x) (-1+k x)}+2 \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{x+(-1-k) x^2+k x^3}}{x}\right)-2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{x+(-1-k) x^2+k x^3}}{x}\right)","\frac{4 \left(x-x^2-k x^2+k x^3\right)^{3/4}}{(-1+x) (-1+k x)}+2 \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{x+(-1-k) x^2+k x^3}}{x}\right)-2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{x+(-1-k) x^2+k x^3}}{x}\right)",1,"(4*(x - x^2 - k*x^2 + k*x^3)^(3/4))/((-1 + x)*(-1 + k*x)) + 2*d^(1/4)*ArcTan[(d^(1/4)*(x + (-1 - k)*x^2 + k*x^3)^(1/4))/x] - 2*d^(1/4)*ArcTanh[(d^(1/4)*(x + (-1 - k)*x^2 + k*x^3)^(1/4))/x]","A",1
1640,1,111,111,0.2352553,"\int \frac{\left(2+x^3\right) \left(1+2 x^3\right)^{2/3}}{x^6 \left(1+x^3\right)} \, dx","IntegrateAlgebraic[((2 + x^3)*(1 + 2*x^3)^(2/3))/(x^6*(1 + x^3)),x]","\frac{\left(-4-3 x^3\right) \left(1+2 x^3\right)^{2/3}}{10 x^5}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+2 x^3}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(-x+\sqrt[3]{1+2 x^3}\right)-\frac{1}{6} \log \left(x^2+x \sqrt[3]{1+2 x^3}+\left(1+2 x^3\right)^{2/3}\right)","\frac{\left(-4-3 x^3\right) \left(1+2 x^3\right)^{2/3}}{10 x^5}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+2 x^3}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(-x+\sqrt[3]{1+2 x^3}\right)-\frac{1}{6} \log \left(x^2+x \sqrt[3]{1+2 x^3}+\left(1+2 x^3\right)^{2/3}\right)",1,"((-4 - 3*x^3)*(1 + 2*x^3)^(2/3))/(10*x^5) - ArcTan[(Sqrt[3]*x)/(x + 2*(1 + 2*x^3)^(1/3))]/Sqrt[3] + Log[-x + (1 + 2*x^3)^(1/3)]/3 - Log[x^2 + x*(1 + 2*x^3)^(1/3) + (1 + 2*x^3)^(2/3)]/6","A",1
1641,1,111,111,0.4928027,"\int \frac{3 a b^2-2 b (2 a+b) x+(a+2 b) x^2}{\sqrt[4]{x (-a+x) (-b+x)^2} \left(-a b^2 d+b (2 a+b) d x-(a+2 b) d x^2+(-1+d) x^3\right)} \, dx","IntegrateAlgebraic[(3*a*b^2 - 2*b*(2*a + b)*x + (a + 2*b)*x^2)/((x*(-a + x)*(-b + x)^2)^(1/4)*(-(a*b^2*d) + b*(2*a + b)*d*x - (a + 2*b)*d*x^2 + (-1 + d)*x^3)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{x}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{x}\right)}{d^{3/4}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{x}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{x}\right)}{d^{3/4}}",1,"(2*ArcTan[(d^(1/4)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/4))/x])/d^(3/4) - (2*ArcTanh[(d^(1/4)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/4))/x])/d^(3/4)","A",1
1642,1,111,111,0.6608086,"\int \frac{(d+c x) \sqrt[4]{-b x^3+a x^4}}{x^2} \, dx","IntegrateAlgebraic[((d + c*x)*(-(b*x^3) + a*x^4)^(1/4))/x^2,x]","\frac{(-4 d+c x) \sqrt[4]{-b x^3+a x^4}}{x}+\frac{(b c-4 a d) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)}{2 a^{3/4}}+\frac{(-b c+4 a d) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)}{2 a^{3/4}}","\frac{(-4 d+c x) \sqrt[4]{-b x^3+a x^4}}{x}+\frac{(b c-4 a d) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)}{2 a^{3/4}}+\frac{(-b c+4 a d) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)}{2 a^{3/4}}",1,"((-4*d + c*x)*(-(b*x^3) + a*x^4)^(1/4))/x + ((b*c - 4*a*d)*ArcTan[(a^(1/4)*x)/(-(b*x^3) + a*x^4)^(1/4)])/(2*a^(3/4)) + ((-(b*c) + 4*a*d)*ArcTanh[(a^(1/4)*x)/(-(b*x^3) + a*x^4)^(1/4)])/(2*a^(3/4))","A",1
1643,1,111,111,2.330147,"\int \frac{\left(3+x^5\right) \sqrt[3]{-2+x^3+x^5}}{x^2 \left(-2+x^5\right)} \, dx","IntegrateAlgebraic[((3 + x^5)*(-2 + x^3 + x^5)^(1/3))/(x^2*(-2 + x^5)),x]","\frac{3 \sqrt[3]{-2+x^3+x^5}}{2 x}+\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-2+x^3+x^5}}\right)+\frac{1}{2} \log \left(-x+\sqrt[3]{-2+x^3+x^5}\right)-\frac{1}{4} \log \left(x^2+x \sqrt[3]{-2+x^3+x^5}+\left(-2+x^3+x^5\right)^{2/3}\right)","\frac{3 \sqrt[3]{-2+x^3+x^5}}{2 x}+\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-2+x^3+x^5}}\right)+\frac{1}{2} \log \left(-x+\sqrt[3]{-2+x^3+x^5}\right)-\frac{1}{4} \log \left(x^2+x \sqrt[3]{-2+x^3+x^5}+\left(-2+x^3+x^5\right)^{2/3}\right)",1,"(3*(-2 + x^3 + x^5)^(1/3))/(2*x) + (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-2 + x^3 + x^5)^(1/3))])/2 + Log[-x + (-2 + x^3 + x^5)^(1/3)]/2 - Log[x^2 + x*(-2 + x^3 + x^5)^(1/3) + (-2 + x^3 + x^5)^(2/3)]/4","A",1
1644,1,111,111,2.6710141,"\int \frac{\left(-3+x^5\right) \left(2+x^3+x^5\right)^{2/3}}{x^3 \left(2+x^5\right)} \, dx","IntegrateAlgebraic[((-3 + x^5)*(2 + x^3 + x^5)^(2/3))/(x^3*(2 + x^5)),x]","\frac{3 \left(2+x^3+x^5\right)^{2/3}}{4 x^2}-\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2+x^3+x^5}}\right)+\frac{1}{2} \log \left(-x+\sqrt[3]{2+x^3+x^5}\right)-\frac{1}{4} \log \left(x^2+x \sqrt[3]{2+x^3+x^5}+\left(2+x^3+x^5\right)^{2/3}\right)","\frac{3 \left(2+x^3+x^5\right)^{2/3}}{4 x^2}-\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2+x^3+x^5}}\right)+\frac{1}{2} \log \left(-x+\sqrt[3]{2+x^3+x^5}\right)-\frac{1}{4} \log \left(x^2+x \sqrt[3]{2+x^3+x^5}+\left(2+x^3+x^5\right)^{2/3}\right)",1,"(3*(2 + x^3 + x^5)^(2/3))/(4*x^2) - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(2 + x^3 + x^5)^(1/3))])/2 + Log[-x + (2 + x^3 + x^5)^(1/3)]/2 - Log[x^2 + x*(2 + x^3 + x^5)^(1/3) + (2 + x^3 + x^5)^(2/3)]/4","A",1
1645,1,111,111,0.5038709,"\int \frac{\sqrt[3]{-1+x^6} \left(1+x^6\right)}{x^9} \, dx","IntegrateAlgebraic[((-1 + x^6)^(1/3)*(1 + x^6))/x^9,x]","\frac{\left(-1-3 x^6\right) \sqrt[3]{-1+x^6}}{8 x^8}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{-1+x^6}}\right)}{2 \sqrt{3}}-\frac{1}{6} \log \left(-x^2+\sqrt[3]{-1+x^6}\right)+\frac{1}{12} \log \left(x^4+x^2 \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)","\frac{\left(-1-3 x^6\right) \sqrt[3]{-1+x^6}}{8 x^8}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{-1+x^6}}\right)}{2 \sqrt{3}}-\frac{1}{6} \log \left(-x^2+\sqrt[3]{-1+x^6}\right)+\frac{1}{12} \log \left(x^4+x^2 \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)",1,"((-1 - 3*x^6)*(-1 + x^6)^(1/3))/(8*x^8) - ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(-1 + x^6)^(1/3))]/(2*Sqrt[3]) - Log[-x^2 + (-1 + x^6)^(1/3)]/6 + Log[x^4 + x^2*(-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)]/12","A",1
1646,1,111,111,15.5628383,"\int \frac{1}{\sqrt[3]{b^2 x^2+a^3 x^3} \left(-2 b+a^6 x^6\right)} \, dx","IntegrateAlgebraic[1/((b^2*x^2 + a^3*x^3)^(1/3)*(-2*b + a^6*x^6)),x]","\frac{\text{RootSum}\left[2 a^{18}-a^6 b^{11}-12 a^{15} \text{$\#$1}^3+30 a^{12} \text{$\#$1}^6-40 a^9 \text{$\#$1}^9+30 a^6 \text{$\#$1}^{12}-12 a^3 \text{$\#$1}^{15}+2 \text{$\#$1}^{18}\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{12 b}","\frac{\text{RootSum}\left[2 a^{18}-a^6 b^{11}-12 a^{15} \text{$\#$1}^3+30 a^{12} \text{$\#$1}^6-40 a^9 \text{$\#$1}^9+30 a^6 \text{$\#$1}^{12}-12 a^3 \text{$\#$1}^{15}+2 \text{$\#$1}^{18}\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{12 b}",1,"RootSum[2*a^18 - a^6*b^11 - 12*a^15*#1^3 + 30*a^12*#1^6 - 40*a^9*#1^9 + 30*a^6*#1^12 - 12*a^3*#1^15 + 2*#1^18 & , (-Log[x] + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1])/#1 & ]/(12*b)","A",1
1647,1,111,111,0.0022885,"\int \frac{1}{\sqrt[3]{b^2 x^2+a^3 x^3} \left(-2 b+a^6 x^6\right)} \, dx","IntegrateAlgebraic[1/((b^2*x^2 + a^3*x^3)^(1/3)*(-2*b + a^6*x^6)),x]","\frac{\text{RootSum}\left[2 a^{18}-a^6 b^{11}-12 a^{15} \text{$\#$1}^3+30 a^{12} \text{$\#$1}^6-40 a^9 \text{$\#$1}^9+30 a^6 \text{$\#$1}^{12}-12 a^3 \text{$\#$1}^{15}+2 \text{$\#$1}^{18}\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{12 b}","\frac{\text{RootSum}\left[2 a^{18}-a^6 b^{11}-12 a^{15} \text{$\#$1}^3+30 a^{12} \text{$\#$1}^6-40 a^9 \text{$\#$1}^9+30 a^6 \text{$\#$1}^{12}-12 a^3 \text{$\#$1}^{15}+2 \text{$\#$1}^{18}\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{12 b}",1,"RootSum[2*a^18 - a^6*b^11 - 12*a^15*#1^3 + 30*a^12*#1^6 - 40*a^9*#1^9 + 30*a^6*#1^12 - 12*a^3*#1^15 + 2*#1^18 & , (-Log[x] + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1])/#1 & ]/(12*b)","A",1
1648,1,111,111,0.4174773,"\int \frac{1}{\sqrt[4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}} \, dx","IntegrateAlgebraic[(-1 - 3*x^4 - 2*x^8 + 2*x^12 + 3*x^16 + x^20)^(-1/4),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}}{\sqrt[4]{2} x \left(1+x^4\right)}\right)}{2 \sqrt[4]{2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}}{\sqrt[4]{2} x \left(1+x^4\right)}\right)}{2 \sqrt[4]{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}}{\sqrt[4]{2} x \left(1+x^4\right)}\right)}{2 \sqrt[4]{2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}}{\sqrt[4]{2} x \left(1+x^4\right)}\right)}{2 \sqrt[4]{2}}",1,"-1/2*ArcTan[(-1 - 3*x^4 - 2*x^8 + 2*x^12 + 3*x^16 + x^20)^(1/4)/(2^(1/4)*x*(1 + x^4))]/2^(1/4) + ArcTanh[(-1 - 3*x^4 - 2*x^8 + 2*x^12 + 3*x^16 + x^20)^(1/4)/(2^(1/4)*x*(1 + x^4))]/(2*2^(1/4))","A",1
1649,1,110,111,0.0852677,"\int \frac{1+x^2}{\sqrt{b+a x} \sqrt{c+\sqrt{b+a x}}} \, dx","IntegrateAlgebraic[(1 + x^2)/(Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]]),x]","\frac{4 \sqrt{c+\sqrt{b+a x}} \left(315 a^2+315 b^2-336 b c^2+128 c^4+168 b c \sqrt{b+a x}-64 c^3 \sqrt{b+a x}-126 b (b+a x)+48 c^2 (b+a x)-40 c (b+a x)^{3/2}+35 (b+a x)^2\right)}{315 a^3}","-\frac{32 \sqrt{b+a x} \left(-16 b c+8 c^3+5 a c x\right) \sqrt{c+\sqrt{b+a x}}}{315 a^3}+\frac{4 \left(315 a^2+224 b^2-288 b c^2+128 c^4-56 a b x+48 a c^2 x+35 a^2 x^2\right) \sqrt{c+\sqrt{b+a x}}}{315 a^3}",1,"(4*Sqrt[c + Sqrt[b + a*x]]*(315*a^2 + 315*b^2 - 336*b*c^2 + 128*c^4 + 168*b*c*Sqrt[b + a*x] - 64*c^3*Sqrt[b + a*x] - 126*b*(b + a*x) + 48*c^2*(b + a*x) - 40*c*(b + a*x)^(3/2) + 35*(b + a*x)^2))/(315*a^3)","A",1
1650,1,111,111,0.1684765,"\int \frac{x^2}{\sqrt{-b+a^2 x^2} \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}} \, dx","IntegrateAlgebraic[x^2/(Sqrt[-b + a^2*x^2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4)),x]","\frac{8 \sqrt{-b+a^2 x^2} \left(-4 b x+a^2 x^3\right)}{7 a^2 \left(a x+\sqrt{-b+a^2 x^2}\right)^{9/4}}+\frac{4 \left(32 b^2-81 a^2 b x^2+18 a^4 x^4\right)}{63 a^3 \left(a x+\sqrt{-b+a^2 x^2}\right)^{9/4}}","\frac{8 \sqrt{-b+a^2 x^2} \left(-4 b x+a^2 x^3\right)}{7 a^2 \left(a x+\sqrt{-b+a^2 x^2}\right)^{9/4}}+\frac{4 \left(32 b^2-81 a^2 b x^2+18 a^4 x^4\right)}{63 a^3 \left(a x+\sqrt{-b+a^2 x^2}\right)^{9/4}}",1,"(8*Sqrt[-b + a^2*x^2]*(-4*b*x + a^2*x^3))/(7*a^2*(a*x + Sqrt[-b + a^2*x^2])^(9/4)) + (4*(32*b^2 - 81*a^2*b*x^2 + 18*a^4*x^4))/(63*a^3*(a*x + Sqrt[-b + a^2*x^2])^(9/4))","A",1
1651,1,112,112,0.243775,"\int x^2 \sqrt[3]{-x+x^3} \, dx","IntegrateAlgebraic[x^2*(-x + x^3)^(1/3),x]","\frac{1}{12} \sqrt[3]{-x+x^3} \left(-x+3 x^3\right)+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)}{6 \sqrt{3}}+\frac{1}{18} \log \left(-x+\sqrt[3]{-x+x^3}\right)-\frac{1}{36} \log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)","\frac{1}{12} \sqrt[3]{-x+x^3} \left(-x+3 x^3\right)+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)}{6 \sqrt{3}}+\frac{1}{18} \log \left(-x+\sqrt[3]{-x+x^3}\right)-\frac{1}{36} \log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)",1,"((-x + x^3)^(1/3)*(-x + 3*x^3))/12 + ArcTan[(Sqrt[3]*x)/(x + 2*(-x + x^3)^(1/3))]/(6*Sqrt[3]) + Log[-x + (-x + x^3)^(1/3)]/18 - Log[x^2 + x*(-x + x^3)^(1/3) + (-x + x^3)^(2/3)]/36","A",1
1652,1,112,112,0.4046709,"\int x^6 \sqrt[3]{x+x^3} \, dx","IntegrateAlgebraic[x^6*(x + x^3)^(1/3),x]","\frac{1}{648} \sqrt[3]{x+x^3} \left(20 x-12 x^3+9 x^5+81 x^7\right)+\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)}{81 \sqrt{3}}+\frac{5}{243} \log \left(-x+\sqrt[3]{x+x^3}\right)-\frac{5}{486} \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)","\frac{1}{648} \sqrt[3]{x+x^3} \left(20 x-12 x^3+9 x^5+81 x^7\right)+\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)}{81 \sqrt{3}}+\frac{5}{243} \log \left(-x+\sqrt[3]{x+x^3}\right)-\frac{5}{486} \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)",1,"((x + x^3)^(1/3)*(20*x - 12*x^3 + 9*x^5 + 81*x^7))/648 + (5*ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^3)^(1/3))])/(81*Sqrt[3]) + (5*Log[-x + (x + x^3)^(1/3)])/243 - (5*Log[x^2 + x*(x + x^3)^(1/3) + (x + x^3)^(2/3)])/486","A",1
1653,1,107,112,4.5670536,"\int \frac{-3+2 x}{x \sqrt[4]{-1+x^4}} \, dx","IntegrateAlgebraic[(-3 + 2*x)/(x*(-1 + x^4)^(1/4)),x]","-\tan ^{-1}\left(\frac{\sqrt[4]{-1+x^4}}{x}\right)+\frac{3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^4}}{-1+\sqrt{-1+x^4}}\right)}{2 \sqrt{2}}+\tanh ^{-1}\left(\frac{\sqrt[4]{-1+x^4}}{x}\right)+\frac{3 \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^4}}{1+\sqrt{-1+x^4}}\right)}{2 \sqrt{2}}","-\tan ^{-1}\left(\frac{\sqrt[4]{-1+x^4}}{x}\right)-\frac{3 \tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{\sqrt{-1+x^4}}{\sqrt{2}}}{\sqrt[4]{-1+x^4}}\right)}{2 \sqrt{2}}+\tanh ^{-1}\left(\frac{\sqrt[4]{-1+x^4}}{x}\right)+\frac{3 \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^4}}{1+\sqrt{-1+x^4}}\right)}{2 \sqrt{2}}",1,"-ArcTan[(-1 + x^4)^(1/4)/x] + (3*ArcTan[(Sqrt[2]*(-1 + x^4)^(1/4))/(-1 + Sqrt[-1 + x^4])])/(2*Sqrt[2]) + ArcTanh[(-1 + x^4)^(1/4)/x] + (3*ArcTanh[(Sqrt[2]*(-1 + x^4)^(1/4))/(1 + Sqrt[-1 + x^4])])/(2*Sqrt[2])","A",1
1654,1,112,112,0.6314921,"\int \frac{\sqrt[3]{-1+x^3-x^4} \left(-3+x^4\right)}{x^2 \left(1+x^4\right)} \, dx","IntegrateAlgebraic[((-1 + x^3 - x^4)^(1/3)*(-3 + x^4))/(x^2*(1 + x^4)),x]","\frac{3 \sqrt[3]{-1+x^3-x^4}}{x}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3-x^4}}\right)+\log \left(-x+\sqrt[3]{-1+x^3-x^4}\right)-\frac{1}{2} \log \left(x^2+x \sqrt[3]{-1+x^3-x^4}+\left(-1+x^3-x^4\right)^{2/3}\right)","\frac{3 \sqrt[3]{-1+x^3-x^4}}{x}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3-x^4}}\right)+\log \left(-x+\sqrt[3]{-1+x^3-x^4}\right)-\frac{1}{2} \log \left(x^2+x \sqrt[3]{-1+x^3-x^4}+\left(-1+x^3-x^4\right)^{2/3}\right)",1,"(3*(-1 + x^3 - x^4)^(1/3))/x + Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^3 - x^4)^(1/3))] + Log[-x + (-1 + x^3 - x^4)^(1/3)] - Log[x^2 + x*(-1 + x^3 - x^4)^(1/3) + (-1 + x^3 - x^4)^(2/3)]/2","A",1
1655,1,112,112,0.6151109,"\int \frac{x^3 \sqrt{-x+x^4}}{-b+a x^3} \, dx","IntegrateAlgebraic[(x^3*Sqrt[-x + x^4])/(-b + a*x^3),x]","\frac{x \sqrt{-x+x^4}}{3 a}+\frac{2 \sqrt{a-b} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a-b} x \sqrt{-x+x^4}}{\sqrt{b} (-1+x) \left(1+x+x^2\right)}\right)}{3 a^2}+\frac{(-a+2 b) \tanh ^{-1}\left(\frac{x^2}{\sqrt{-x+x^4}}\right)}{3 a^2}","\frac{x \sqrt{-x+x^4}}{3 a}+\frac{2 \sqrt{a-b} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a-b} x \sqrt{-x+x^4}}{\sqrt{b} (-1+x) \left(1+x+x^2\right)}\right)}{3 a^2}+\frac{(-a+2 b) \tanh ^{-1}\left(\frac{x^2}{\sqrt{-x+x^4}}\right)}{3 a^2}",1,"(x*Sqrt[-x + x^4])/(3*a) + (2*Sqrt[a - b]*Sqrt[b]*ArcTan[(Sqrt[a - b]*x*Sqrt[-x + x^4])/(Sqrt[b]*(-1 + x)*(1 + x + x^2))])/(3*a^2) + ((-a + 2*b)*ArcTanh[x^2/Sqrt[-x + x^4]])/(3*a^2)","A",1
1656,1,112,112,0.3219292,"\int \frac{1+2 x^4}{\left(-1+x^4\right) \sqrt[4]{x^2+x^4}} \, dx","IntegrateAlgebraic[(1 + 2*x^4)/((-1 + x^4)*(x^2 + x^4)^(1/4)),x]","-\frac{3 \left(x^2+x^4\right)^{3/4}}{x \left(1+x^2\right)}+2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^2+x^4}}\right)-\frac{3 \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2 \sqrt[4]{2}}+2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^2+x^4}}\right)-\frac{3 \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2 \sqrt[4]{2}}","-\frac{3 \left(x^2+x^4\right)^{3/4}}{x \left(1+x^2\right)}+2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^2+x^4}}\right)-\frac{3 \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2 \sqrt[4]{2}}+2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^2+x^4}}\right)-\frac{3 \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2 \sqrt[4]{2}}",1,"(-3*(x^2 + x^4)^(3/4))/(x*(1 + x^2)) + 2*ArcTan[x/(x^2 + x^4)^(1/4)] - (3*ArcTan[(2^(1/4)*x)/(x^2 + x^4)^(1/4)])/(2*2^(1/4)) + 2*ArcTanh[x/(x^2 + x^4)^(1/4)] - (3*ArcTanh[(2^(1/4)*x)/(x^2 + x^4)^(1/4)])/(2*2^(1/4))","A",1
1657,1,112,112,0.393022,"\int \frac{b+a x^2}{x^2 \left(-b+a x^2\right) \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[(b + a*x^2)/(x^2*(-b + a*x^2)*(b*x^2 + a*x^4)^(1/4)),x]","\frac{2 \left(b x^2+a x^4\right)^{3/4}}{3 b x^3}-\frac{2^{3/4} a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{b}-\frac{2^{3/4} a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{b}","\frac{2 \left(b x^2+a x^4\right)^{3/4}}{3 b x^3}-\frac{2^{3/4} a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{b}-\frac{2^{3/4} a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{b}",1,"(2*(b*x^2 + a*x^4)^(3/4))/(3*b*x^3) - (2^(3/4)*a^(3/4)*ArcTan[(2^(1/4)*a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/b - (2^(3/4)*a^(3/4)*ArcTanh[(2^(1/4)*a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/b","A",1
1658,1,112,112,0.4883226,"\int x \sqrt[4]{b x^3+a x^4} \, dx","IntegrateAlgebraic[x*(b*x^3 + a*x^4)^(1/4),x]","\frac{\left(-7 b^2+4 a b x+32 a^2 x^2\right) \sqrt[4]{b x^3+a x^4}}{96 a^2}-\frac{7 b^3 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{64 a^{11/4}}+\frac{7 b^3 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{64 a^{11/4}}","\frac{\left(-7 b^2+4 a b x+32 a^2 x^2\right) \sqrt[4]{b x^3+a x^4}}{96 a^2}-\frac{7 b^3 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{64 a^{11/4}}+\frac{7 b^3 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{64 a^{11/4}}",1,"((-7*b^2 + 4*a*b*x + 32*a^2*x^2)*(b*x^3 + a*x^4)^(1/4))/(96*a^2) - (7*b^3*ArcTan[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(64*a^(11/4)) + (7*b^3*ArcTanh[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(64*a^(11/4))","A",1
1659,1,112,112,2.9497944,"\int \frac{-2 b+a x^3}{\left(b+x^2+a x^3\right) \sqrt[4]{b x^2+a x^5}} \, dx","IntegrateAlgebraic[(-2*b + a*x^3)/((b + x^2 + a*x^3)*(b*x^2 + a*x^5)^(1/4)),x]","-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{b x^2+a x^5}}{-x^2+\sqrt{b x^2+a x^5}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{b x^2+a x^5}}{\sqrt{2}}}{x \sqrt[4]{b x^2+a x^5}}\right)","-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{b x^2+a x^5}}{-x^2+\sqrt{b x^2+a x^5}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{b x^2+a x^5}}{\sqrt{2}}}{x \sqrt[4]{b x^2+a x^5}}\right)",1,"-(Sqrt[2]*ArcTan[(Sqrt[2]*x*(b*x^2 + a*x^5)^(1/4))/(-x^2 + Sqrt[b*x^2 + a*x^5])]) - Sqrt[2]*ArcTanh[(x^2/Sqrt[2] + Sqrt[b*x^2 + a*x^5]/Sqrt[2])/(x*(b*x^2 + a*x^5)^(1/4))]","A",1
1660,1,112,112,2.9033897,"\int \frac{-b+a x^2}{\left(b+x+a x^2\right) \sqrt[4]{b x^3+a x^5}} \, dx","IntegrateAlgebraic[(-b + a*x^2)/((b + x + a*x^2)*(b*x^3 + a*x^5)^(1/4)),x]","-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{b x^3+a x^5}}{-x^2+\sqrt{b x^3+a x^5}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{b x^3+a x^5}}{\sqrt{2}}}{x \sqrt[4]{b x^3+a x^5}}\right)","-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{b x^3+a x^5}}{-x^2+\sqrt{b x^3+a x^5}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{b x^3+a x^5}}{\sqrt{2}}}{x \sqrt[4]{b x^3+a x^5}}\right)",1,"-(Sqrt[2]*ArcTan[(Sqrt[2]*x*(b*x^3 + a*x^5)^(1/4))/(-x^2 + Sqrt[b*x^3 + a*x^5])]) - Sqrt[2]*ArcTanh[(x^2/Sqrt[2] + Sqrt[b*x^3 + a*x^5]/Sqrt[2])/(x*(b*x^3 + a*x^5)^(1/4))]","A",1
1661,1,112,112,2.8293632,"\int \frac{x^{13}}{\sqrt[3]{-1+x^6}} \, dx","IntegrateAlgebraic[x^13/(-1 + x^6)^(1/3),x]","\frac{1}{36} \left(-1+x^6\right)^{2/3} \left(4 x^2+3 x^8\right)+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{-1+x^6}}\right)}{9 \sqrt{3}}-\frac{1}{27} \log \left(-x^2+\sqrt[3]{-1+x^6}\right)+\frac{1}{54} \log \left(x^4+x^2 \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)","\frac{1}{36} \left(-1+x^6\right)^{2/3} \left(4 x^2+3 x^8\right)+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{-1+x^6}}\right)}{9 \sqrt{3}}-\frac{1}{27} \log \left(-x^2+\sqrt[3]{-1+x^6}\right)+\frac{1}{54} \log \left(x^4+x^2 \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)",1,"((-1 + x^6)^(2/3)*(4*x^2 + 3*x^8))/36 + ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(-1 + x^6)^(1/3))]/(9*Sqrt[3]) - Log[-x^2 + (-1 + x^6)^(1/3)]/27 + Log[x^4 + x^2*(-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)]/54","A",1
1662,1,112,112,0.8478711,"\int x^9 \sqrt[3]{-1+x^6} \, dx","IntegrateAlgebraic[x^9*(-1 + x^6)^(1/3),x]","\frac{1}{36} \sqrt[3]{-1+x^6} \left(-x^4+3 x^{10}\right)+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{-1+x^6}}\right)}{18 \sqrt{3}}+\frac{1}{54} \log \left(-x^2+\sqrt[3]{-1+x^6}\right)-\frac{1}{108} \log \left(x^4+x^2 \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)","\frac{1}{36} \sqrt[3]{-1+x^6} \left(-x^4+3 x^{10}\right)+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{-1+x^6}}\right)}{18 \sqrt{3}}+\frac{1}{54} \log \left(-x^2+\sqrt[3]{-1+x^6}\right)-\frac{1}{108} \log \left(x^4+x^2 \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)",1,"((-1 + x^6)^(1/3)*(-x^4 + 3*x^10))/36 + ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(-1 + x^6)^(1/3))]/(18*Sqrt[3]) + Log[-x^2 + (-1 + x^6)^(1/3)]/54 - Log[x^4 + x^2*(-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)]/108","A",1
1663,1,112,112,1.0876319,"\int x^7 \left(-1+x^6\right)^{2/3} \, dx","IntegrateAlgebraic[x^7*(-1 + x^6)^(2/3),x]","\frac{1}{36} \left(-1+x^6\right)^{2/3} \left(-2 x^2+3 x^8\right)-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{-1+x^6}}\right)}{18 \sqrt{3}}+\frac{1}{54} \log \left(-x^2+\sqrt[3]{-1+x^6}\right)-\frac{1}{108} \log \left(x^4+x^2 \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)","\frac{1}{36} \left(-1+x^6\right)^{2/3} \left(-2 x^2+3 x^8\right)-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{-1+x^6}}\right)}{18 \sqrt{3}}+\frac{1}{54} \log \left(-x^2+\sqrt[3]{-1+x^6}\right)-\frac{1}{108} \log \left(x^4+x^2 \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)",1,"((-1 + x^6)^(2/3)*(-2*x^2 + 3*x^8))/36 - ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(-1 + x^6)^(1/3))]/(18*Sqrt[3]) + Log[-x^2 + (-1 + x^6)^(1/3)]/54 - Log[x^4 + x^2*(-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)]/108","A",1
1664,1,112,112,2.6923034,"\int \frac{x^{13}}{\sqrt[3]{1+x^6}} \, dx","IntegrateAlgebraic[x^13/(1 + x^6)^(1/3),x]","\frac{1}{36} \left(1+x^6\right)^{2/3} \left(-4 x^2+3 x^8\right)+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{1+x^6}}\right)}{9 \sqrt{3}}-\frac{1}{27} \log \left(-x^2+\sqrt[3]{1+x^6}\right)+\frac{1}{54} \log \left(x^4+x^2 \sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)","\frac{1}{36} \left(1+x^6\right)^{2/3} \left(-4 x^2+3 x^8\right)+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{1+x^6}}\right)}{9 \sqrt{3}}-\frac{1}{27} \log \left(-x^2+\sqrt[3]{1+x^6}\right)+\frac{1}{54} \log \left(x^4+x^2 \sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)",1,"((1 + x^6)^(2/3)*(-4*x^2 + 3*x^8))/36 + ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(1 + x^6)^(1/3))]/(9*Sqrt[3]) - Log[-x^2 + (1 + x^6)^(1/3)]/27 + Log[x^4 + x^2*(1 + x^6)^(1/3) + (1 + x^6)^(2/3)]/54","A",1
1665,1,112,112,1.0574399,"\int x^7 \left(1+x^6\right)^{2/3} \, dx","IntegrateAlgebraic[x^7*(1 + x^6)^(2/3),x]","\frac{1}{36} \left(1+x^6\right)^{2/3} \left(2 x^2+3 x^8\right)-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{1+x^6}}\right)}{18 \sqrt{3}}+\frac{1}{54} \log \left(-x^2+\sqrt[3]{1+x^6}\right)-\frac{1}{108} \log \left(x^4+x^2 \sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)","\frac{1}{36} \left(1+x^6\right)^{2/3} \left(2 x^2+3 x^8\right)-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{1+x^6}}\right)}{18 \sqrt{3}}+\frac{1}{54} \log \left(-x^2+\sqrt[3]{1+x^6}\right)-\frac{1}{108} \log \left(x^4+x^2 \sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)",1,"((1 + x^6)^(2/3)*(2*x^2 + 3*x^8))/36 - ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(1 + x^6)^(1/3))]/(18*Sqrt[3]) + Log[-x^2 + (1 + x^6)^(1/3)]/54 - Log[x^4 + x^2*(1 + x^6)^(1/3) + (1 + x^6)^(2/3)]/108","A",1
1666,1,112,112,0.2400488,"\int \frac{\left(1+x^3\right)^{2/3} \left(4+x^3\right)}{x^6 \left(8-4 x^3+x^6\right)} \, dx","IntegrateAlgebraic[((1 + x^3)^(2/3)*(4 + x^3))/(x^6*(8 - 4*x^3 + x^6)),x]","\frac{\left(-8-23 x^3\right) \left(1+x^3\right)^{2/3}}{80 x^5}+\frac{1}{96} \text{RootSum}\left[13-20 \text{$\#$1}^3+8 \text{$\#$1}^6\&,\frac{-39 \log (x)+39 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)+32 \log (x) \text{$\#$1}^3-32 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-5 \text{$\#$1}+4 \text{$\#$1}^4}\&\right]","\frac{\left(-8-23 x^3\right) \left(1+x^3\right)^{2/3}}{80 x^5}+\frac{1}{96} \text{RootSum}\left[13-20 \text{$\#$1}^3+8 \text{$\#$1}^6\&,\frac{-39 \log (x)+39 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)+32 \log (x) \text{$\#$1}^3-32 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-5 \text{$\#$1}+4 \text{$\#$1}^4}\&\right]",1,"((-8 - 23*x^3)*(1 + x^3)^(2/3))/(80*x^5) + RootSum[13 - 20*#1^3 + 8*#1^6 & , (-39*Log[x] + 39*Log[(1 + x^3)^(1/3) - x*#1] + 32*Log[x]*#1^3 - 32*Log[(1 + x^3)^(1/3) - x*#1]*#1^3)/(-5*#1 + 4*#1^4) & ]/96","A",1
1667,1,112,112,0.001408,"\int \frac{\left(1+x^3\right)^{2/3} \left(4+x^3\right)}{x^6 \left(8-4 x^3+x^6\right)} \, dx","IntegrateAlgebraic[((1 + x^3)^(2/3)*(4 + x^3))/(x^6*(8 - 4*x^3 + x^6)),x]","\frac{\left(-8-23 x^3\right) \left(1+x^3\right)^{2/3}}{80 x^5}+\frac{1}{96} \text{RootSum}\left[13-20 \text{$\#$1}^3+8 \text{$\#$1}^6\&,\frac{-39 \log (x)+39 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)+32 \log (x) \text{$\#$1}^3-32 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-5 \text{$\#$1}+4 \text{$\#$1}^4}\&\right]","\frac{\left(-8-23 x^3\right) \left(1+x^3\right)^{2/3}}{80 x^5}+\frac{1}{96} \text{RootSum}\left[13-20 \text{$\#$1}^3+8 \text{$\#$1}^6\&,\frac{-39 \log (x)+39 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)+32 \log (x) \text{$\#$1}^3-32 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-5 \text{$\#$1}+4 \text{$\#$1}^4}\&\right]",1,"((-8 - 23*x^3)*(1 + x^3)^(2/3))/(80*x^5) + RootSum[13 - 20*#1^3 + 8*#1^6 & , (-39*Log[x] + 39*Log[(1 + x^3)^(1/3) - x*#1] + 32*Log[x]*#1^3 - 32*Log[(1 + x^3)^(1/3) - x*#1]*#1^3)/(-5*#1 + 4*#1^4) & ]/96","A",1
1668,1,112,112,0.2413315,"\int \frac{\left(1+x^3\right)^{2/3} \left(2+x^3\right) \left(4+3 x^3\right)}{x^6 \left(4+2 x^3+x^6\right)} \, dx","IntegrateAlgebraic[((1 + x^3)^(2/3)*(2 + x^3)*(4 + 3*x^3))/(x^6*(4 + 2*x^3 + x^6)),x]","\frac{\left(-8-23 x^3\right) \left(1+x^3\right)^{2/3}}{20 x^5}+\frac{1}{12} \text{RootSum}\left[3-6 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-9 \log (x)+9 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)+8 \log (x) \text{$\#$1}^3-8 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-3 \text{$\#$1}+4 \text{$\#$1}^4}\&\right]","\frac{\left(-8-23 x^3\right) \left(1+x^3\right)^{2/3}}{20 x^5}+\frac{1}{12} \text{RootSum}\left[3-6 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-9 \log (x)+9 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)+8 \log (x) \text{$\#$1}^3-8 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-3 \text{$\#$1}+4 \text{$\#$1}^4}\&\right]",1,"((-8 - 23*x^3)*(1 + x^3)^(2/3))/(20*x^5) + RootSum[3 - 6*#1^3 + 4*#1^6 & , (-9*Log[x] + 9*Log[(1 + x^3)^(1/3) - x*#1] + 8*Log[x]*#1^3 - 8*Log[(1 + x^3)^(1/3) - x*#1]*#1^3)/(-3*#1 + 4*#1^4) & ]/12","A",1
1669,1,112,112,0.0016898,"\int \frac{\left(1+x^3\right)^{2/3} \left(2+x^3\right) \left(4+3 x^3\right)}{x^6 \left(4+2 x^3+x^6\right)} \, dx","IntegrateAlgebraic[((1 + x^3)^(2/3)*(2 + x^3)*(4 + 3*x^3))/(x^6*(4 + 2*x^3 + x^6)),x]","\frac{\left(-8-23 x^3\right) \left(1+x^3\right)^{2/3}}{20 x^5}+\frac{1}{12} \text{RootSum}\left[3-6 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-9 \log (x)+9 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)+8 \log (x) \text{$\#$1}^3-8 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-3 \text{$\#$1}+4 \text{$\#$1}^4}\&\right]","\frac{\left(-8-23 x^3\right) \left(1+x^3\right)^{2/3}}{20 x^5}+\frac{1}{12} \text{RootSum}\left[3-6 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-9 \log (x)+9 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)+8 \log (x) \text{$\#$1}^3-8 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-3 \text{$\#$1}+4 \text{$\#$1}^4}\&\right]",1,"((-8 - 23*x^3)*(1 + x^3)^(2/3))/(20*x^5) + RootSum[3 - 6*#1^3 + 4*#1^6 & , (-9*Log[x] + 9*Log[(1 + x^3)^(1/3) - x*#1] + 8*Log[x]*#1^3 - 8*Log[(1 + x^3)^(1/3) - x*#1]*#1^3)/(-3*#1 + 4*#1^4) & ]/12","A",1
1670,1,112,112,0.2327144,"\int \frac{\left(-1+x^3\right)^{2/3} \left(4+x^6\right)}{x^6 \left(4+2 x^3+x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(4 + x^6))/(x^6*(4 + 2*x^3 + x^6)),x]","\frac{\left(-1+x^3\right)^{2/3} \left(-4+9 x^3\right)}{20 x^5}-\frac{1}{12} \text{RootSum}\left[7-10 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-7 \log (x)+7 \log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right)+6 \log (x) \text{$\#$1}^3-6 \log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-5 \text{$\#$1}+4 \text{$\#$1}^4}\&\right]","\frac{\left(-1+x^3\right)^{2/3} \left(-4+9 x^3\right)}{20 x^5}-\frac{1}{12} \text{RootSum}\left[7-10 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-7 \log (x)+7 \log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right)+6 \log (x) \text{$\#$1}^3-6 \log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-5 \text{$\#$1}+4 \text{$\#$1}^4}\&\right]",1,"((-1 + x^3)^(2/3)*(-4 + 9*x^3))/(20*x^5) - RootSum[7 - 10*#1^3 + 4*#1^6 & , (-7*Log[x] + 7*Log[(-1 + x^3)^(1/3) - x*#1] + 6*Log[x]*#1^3 - 6*Log[(-1 + x^3)^(1/3) - x*#1]*#1^3)/(-5*#1 + 4*#1^4) & ]/12","A",1
1671,1,112,112,0.0012507,"\int \frac{\left(-1+x^3\right)^{2/3} \left(4+x^6\right)}{x^6 \left(4+2 x^3+x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(4 + x^6))/(x^6*(4 + 2*x^3 + x^6)),x]","\frac{\left(-1+x^3\right)^{2/3} \left(-4+9 x^3\right)}{20 x^5}-\frac{1}{12} \text{RootSum}\left[7-10 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-7 \log (x)+7 \log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right)+6 \log (x) \text{$\#$1}^3-6 \log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-5 \text{$\#$1}+4 \text{$\#$1}^4}\&\right]","\frac{\left(-1+x^3\right)^{2/3} \left(-4+9 x^3\right)}{20 x^5}-\frac{1}{12} \text{RootSum}\left[7-10 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-7 \log (x)+7 \log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right)+6 \log (x) \text{$\#$1}^3-6 \log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-5 \text{$\#$1}+4 \text{$\#$1}^4}\&\right]",1,"((-1 + x^3)^(2/3)*(-4 + 9*x^3))/(20*x^5) - RootSum[7 - 10*#1^3 + 4*#1^6 & , (-7*Log[x] + 7*Log[(-1 + x^3)^(1/3) - x*#1] + 6*Log[x]*#1^3 - 6*Log[(-1 + x^3)^(1/3) - x*#1]*#1^3)/(-5*#1 + 4*#1^4) & ]/12","A",1
1672,1,112,112,0.2346559,"\int \frac{\left(-1+x^3\right)^{2/3} \left(1-x^3+x^6\right)}{x^6 \left(-2-x^3+2 x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(1 - x^3 + x^6))/(x^6*(-2 - x^3 + 2*x^6)),x]","\frac{\left(4-19 x^3\right) \left(-1+x^3\right)^{2/3}}{40 x^5}+\frac{1}{12} \text{RootSum}\left[1-5 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{-3 \log (x)+3 \log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right)+17 \log (x) \text{$\#$1}^3-17 \log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-5 \text{$\#$1}+4 \text{$\#$1}^4}\&\right]","\frac{\left(4-19 x^3\right) \left(-1+x^3\right)^{2/3}}{40 x^5}+\frac{1}{12} \text{RootSum}\left[1-5 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{-3 \log (x)+3 \log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right)+17 \log (x) \text{$\#$1}^3-17 \log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-5 \text{$\#$1}+4 \text{$\#$1}^4}\&\right]",1,"((4 - 19*x^3)*(-1 + x^3)^(2/3))/(40*x^5) + RootSum[1 - 5*#1^3 + 2*#1^6 & , (-3*Log[x] + 3*Log[(-1 + x^3)^(1/3) - x*#1] + 17*Log[x]*#1^3 - 17*Log[(-1 + x^3)^(1/3) - x*#1]*#1^3)/(-5*#1 + 4*#1^4) & ]/12","A",1
1673,1,112,112,0.0012034,"\int \frac{\left(-1+x^3\right)^{2/3} \left(1-x^3+x^6\right)}{x^6 \left(-2-x^3+2 x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(1 - x^3 + x^6))/(x^6*(-2 - x^3 + 2*x^6)),x]","\frac{\left(4-19 x^3\right) \left(-1+x^3\right)^{2/3}}{40 x^5}+\frac{1}{12} \text{RootSum}\left[1-5 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{-3 \log (x)+3 \log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right)+17 \log (x) \text{$\#$1}^3-17 \log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-5 \text{$\#$1}+4 \text{$\#$1}^4}\&\right]","\frac{\left(4-19 x^3\right) \left(-1+x^3\right)^{2/3}}{40 x^5}+\frac{1}{12} \text{RootSum}\left[1-5 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{-3 \log (x)+3 \log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right)+17 \log (x) \text{$\#$1}^3-17 \log \left(\sqrt[3]{-1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-5 \text{$\#$1}+4 \text{$\#$1}^4}\&\right]",1,"((4 - 19*x^3)*(-1 + x^3)^(2/3))/(40*x^5) + RootSum[1 - 5*#1^3 + 2*#1^6 & , (-3*Log[x] + 3*Log[(-1 + x^3)^(1/3) - x*#1] + 17*Log[x]*#1^3 - 17*Log[(-1 + x^3)^(1/3) - x*#1]*#1^3)/(-5*#1 + 4*#1^4) & ]/12","A",1
1674,1,112,112,0.2342795,"\int \frac{\left(1+x^3\right)^{2/3} \left(4+6 x^3+3 x^6\right)}{x^6 \left(8+6 x^3+3 x^6\right)} \, dx","IntegrateAlgebraic[((1 + x^3)^(2/3)*(4 + 6*x^3 + 3*x^6))/(x^6*(8 + 6*x^3 + 3*x^6)),x]","\frac{\left(-8-23 x^3\right) \left(1+x^3\right)^{2/3}}{80 x^5}+\frac{1}{16} \text{RootSum}\left[5-10 \text{$\#$1}^3+8 \text{$\#$1}^6\&,\frac{-5 \log (x)+5 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)+6 \log (x) \text{$\#$1}^3-6 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-5 \text{$\#$1}+8 \text{$\#$1}^4}\&\right]","\frac{\left(-8-23 x^3\right) \left(1+x^3\right)^{2/3}}{80 x^5}+\frac{1}{16} \text{RootSum}\left[5-10 \text{$\#$1}^3+8 \text{$\#$1}^6\&,\frac{-5 \log (x)+5 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)+6 \log (x) \text{$\#$1}^3-6 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-5 \text{$\#$1}+8 \text{$\#$1}^4}\&\right]",1,"((-8 - 23*x^3)*(1 + x^3)^(2/3))/(80*x^5) + RootSum[5 - 10*#1^3 + 8*#1^6 & , (-5*Log[x] + 5*Log[(1 + x^3)^(1/3) - x*#1] + 6*Log[x]*#1^3 - 6*Log[(1 + x^3)^(1/3) - x*#1]*#1^3)/(-5*#1 + 8*#1^4) & ]/16","A",1
1675,1,112,112,0.0014077,"\int \frac{\left(1+x^3\right)^{2/3} \left(4+6 x^3+3 x^6\right)}{x^6 \left(8+6 x^3+3 x^6\right)} \, dx","IntegrateAlgebraic[((1 + x^3)^(2/3)*(4 + 6*x^3 + 3*x^6))/(x^6*(8 + 6*x^3 + 3*x^6)),x]","\frac{\left(-8-23 x^3\right) \left(1+x^3\right)^{2/3}}{80 x^5}+\frac{1}{16} \text{RootSum}\left[5-10 \text{$\#$1}^3+8 \text{$\#$1}^6\&,\frac{-5 \log (x)+5 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)+6 \log (x) \text{$\#$1}^3-6 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-5 \text{$\#$1}+8 \text{$\#$1}^4}\&\right]","\frac{\left(-8-23 x^3\right) \left(1+x^3\right)^{2/3}}{80 x^5}+\frac{1}{16} \text{RootSum}\left[5-10 \text{$\#$1}^3+8 \text{$\#$1}^6\&,\frac{-5 \log (x)+5 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right)+6 \log (x) \text{$\#$1}^3-6 \log \left(\sqrt[3]{1+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-5 \text{$\#$1}+8 \text{$\#$1}^4}\&\right]",1,"((-8 - 23*x^3)*(1 + x^3)^(2/3))/(80*x^5) + RootSum[5 - 10*#1^3 + 8*#1^6 & , (-5*Log[x] + 5*Log[(1 + x^3)^(1/3) - x*#1] + 6*Log[x]*#1^3 - 6*Log[(1 + x^3)^(1/3) - x*#1]*#1^3)/(-5*#1 + 8*#1^4) & ]/16","A",1
1676,1,112,112,2.7285981,"\int \frac{\sqrt[3]{-1+2 x^3+x^8} \left(3+5 x^8\right)}{x^2 \left(-1+x^3+x^8\right)} \, dx","IntegrateAlgebraic[((-1 + 2*x^3 + x^8)^(1/3)*(3 + 5*x^8))/(x^2*(-1 + x^3 + x^8)),x]","\frac{3 \sqrt[3]{-1+2 x^3+x^8}}{x}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+2 x^3+x^8}}\right)+\log \left(-x+\sqrt[3]{-1+2 x^3+x^8}\right)-\frac{1}{2} \log \left(x^2+x \sqrt[3]{-1+2 x^3+x^8}+\left(-1+2 x^3+x^8\right)^{2/3}\right)","\frac{3 \sqrt[3]{-1+2 x^3+x^8}}{x}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+2 x^3+x^8}}\right)+\log \left(-x+\sqrt[3]{-1+2 x^3+x^8}\right)-\frac{1}{2} \log \left(x^2+x \sqrt[3]{-1+2 x^3+x^8}+\left(-1+2 x^3+x^8\right)^{2/3}\right)",1,"(3*(-1 + 2*x^3 + x^8)^(1/3))/x + Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + 2*x^3 + x^8)^(1/3))] + Log[-x + (-1 + 2*x^3 + x^8)^(1/3)] - Log[x^2 + x*(-1 + 2*x^3 + x^8)^(1/3) + (-1 + 2*x^3 + x^8)^(2/3)]/2","A",1
1677,1,112,112,0.2956913,"\int \frac{\left(-1+x^4\right) \sqrt{1+\sqrt{1+x^2}}}{1+x^4} \, dx","IntegrateAlgebraic[((-1 + x^4)*Sqrt[1 + Sqrt[1 + x^2]])/(1 + x^4),x]","\frac{4 x}{3 \sqrt{1+\sqrt{1+x^2}}}+\frac{2 x \sqrt{1+x^2}}{3 \sqrt{1+\sqrt{1+x^2}}}-\frac{1}{2} \text{RootSum}\left[1+4 \text{$\#$1}^4+4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\frac{x}{\sqrt{1+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{2 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]","\frac{4 x}{3 \sqrt{1+\sqrt{1+x^2}}}+\frac{2 x \sqrt{1+x^2}}{3 \sqrt{1+\sqrt{1+x^2}}}-\frac{1}{2} \text{RootSum}\left[1+4 \text{$\#$1}^4+4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\frac{x}{\sqrt{1+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{2 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]",1,"(4*x)/(3*Sqrt[1 + Sqrt[1 + x^2]]) + (2*x*Sqrt[1 + x^2])/(3*Sqrt[1 + Sqrt[1 + x^2]]) - RootSum[1 + 4*#1^4 + 4*#1^6 + #1^8 & , Log[x/Sqrt[1 + Sqrt[1 + x^2]] - #1]/(2*#1^3 + #1^5) & ]/2","A",1
1678,1,112,112,0.0012527,"\int \frac{\left(-1+x^4\right) \sqrt{1+\sqrt{1+x^2}}}{1+x^4} \, dx","IntegrateAlgebraic[((-1 + x^4)*Sqrt[1 + Sqrt[1 + x^2]])/(1 + x^4),x]","\frac{4 x}{3 \sqrt{1+\sqrt{1+x^2}}}+\frac{2 x \sqrt{1+x^2}}{3 \sqrt{1+\sqrt{1+x^2}}}-\frac{1}{2} \text{RootSum}\left[1+4 \text{$\#$1}^4+4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\frac{x}{\sqrt{1+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{2 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]","\frac{4 x}{3 \sqrt{1+\sqrt{1+x^2}}}+\frac{2 x \sqrt{1+x^2}}{3 \sqrt{1+\sqrt{1+x^2}}}-\frac{1}{2} \text{RootSum}\left[1+4 \text{$\#$1}^4+4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\frac{x}{\sqrt{1+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{2 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]",1,"(4*x)/(3*Sqrt[1 + Sqrt[1 + x^2]]) + (2*x*Sqrt[1 + x^2])/(3*Sqrt[1 + Sqrt[1 + x^2]]) - RootSum[1 + 4*#1^4 + 4*#1^6 + #1^8 & , Log[x/Sqrt[1 + Sqrt[1 + x^2]] - #1]/(2*#1^3 + #1^5) & ]/2","A",1
1679,1,112,112,0.1585679,"\int \frac{\sqrt{b^2+a^2 x^2}}{\sqrt{a x+\sqrt{b^2+a^2 x^2}}} \, dx","IntegrateAlgebraic[Sqrt[b^2 + a^2*x^2]/Sqrt[a*x + Sqrt[b^2 + a^2*x^2]],x]","\frac{4 \sqrt{b^2+a^2 x^2} \left(-b^2 x+a^2 x^3\right)}{3 \left(a x+\sqrt{b^2+a^2 x^2}\right)^{5/2}}+\frac{2 \left(-7 b^4-5 a^2 b^2 x^2+10 a^4 x^4\right)}{15 a \left(a x+\sqrt{b^2+a^2 x^2}\right)^{5/2}}","\frac{4 \sqrt{b^2+a^2 x^2} \left(-b^2 x+a^2 x^3\right)}{3 \left(a x+\sqrt{b^2+a^2 x^2}\right)^{5/2}}+\frac{2 \left(-7 b^4-5 a^2 b^2 x^2+10 a^4 x^4\right)}{15 a \left(a x+\sqrt{b^2+a^2 x^2}\right)^{5/2}}",1,"(4*Sqrt[b^2 + a^2*x^2]*(-(b^2*x) + a^2*x^3))/(3*(a*x + Sqrt[b^2 + a^2*x^2])^(5/2)) + (2*(-7*b^4 - 5*a^2*b^2*x^2 + 10*a^4*x^4))/(15*a*(a*x + Sqrt[b^2 + a^2*x^2])^(5/2))","A",1
1680,1,113,113,0.2903409,"\int \frac{-2 a+b+x}{\sqrt[4]{(-a+x) (-b+x)^2} \left(b^2+a d-(2 b+d) x+x^2\right)} \, dx","IntegrateAlgebraic[(-2*a + b + x)/(((-a + x)*(-b + x)^2)^(1/4)*(b^2 + a*d - (2*b + d)*x + x^2)),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{b-x}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{b-x}\right)}{d^{3/4}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{b-x}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{b-x}\right)}{d^{3/4}}",1,"(-2*ArcTan[(d^(1/4)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/4))/(b - x)])/d^(3/4) + (2*ArcTanh[(d^(1/4)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/4))/(b - x)])/d^(3/4)","A",1
1681,1,113,113,0.2683431,"\int x \sqrt[3]{x^2+x^3} \, dx","IntegrateAlgebraic[x*(x^2 + x^3)^(1/3),x]","\frac{1}{54} \left(-5+3 x+18 x^2\right) \sqrt[3]{x^2+x^3}-\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^3}}\right)}{27 \sqrt{3}}-\frac{5}{81} \log \left(-x+\sqrt[3]{x^2+x^3}\right)+\frac{5}{162} \log \left(x^2+x \sqrt[3]{x^2+x^3}+\left(x^2+x^3\right)^{2/3}\right)","\frac{1}{54} \left(-5+3 x+18 x^2\right) \sqrt[3]{x^2+x^3}-\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^3}}\right)}{27 \sqrt{3}}-\frac{5}{81} \log \left(-x+\sqrt[3]{x^2+x^3}\right)+\frac{5}{162} \log \left(x^2+x \sqrt[3]{x^2+x^3}+\left(x^2+x^3\right)^{2/3}\right)",1,"((-5 + 3*x + 18*x^2)*(x^2 + x^3)^(1/3))/54 - (5*ArcTan[(Sqrt[3]*x)/(x + 2*(x^2 + x^3)^(1/3))])/(27*Sqrt[3]) - (5*Log[-x + (x^2 + x^3)^(1/3)])/81 + (5*Log[x^2 + x*(x^2 + x^3)^(1/3) + (x^2 + x^3)^(2/3)])/162","A",1
1682,1,113,113,0.3443554,"\int \frac{-((2 a-3 b) b)+2 (a-2 b) x+x^2}{\sqrt[4]{(-a+x) (-b+x)^2} \left(-a^3-b^2 d+\left(3 a^2+2 b d\right) x-(3 a+d) x^2+x^3\right)} \, dx","IntegrateAlgebraic[(-((2*a - 3*b)*b) + 2*(a - 2*b)*x + x^2)/(((-a + x)*(-b + x)^2)^(1/4)*(-a^3 - b^2*d + (3*a^2 + 2*b*d)*x - (3*a + d)*x^2 + x^3)),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{a-x}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{a-x}\right)}{d^{3/4}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{a-x}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{a-x}\right)}{d^{3/4}}",1,"(-2*ArcTan[(d^(1/4)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/4))/(a - x)])/d^(3/4) + (2*ArcTanh[(d^(1/4)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/4))/(a - x)])/d^(3/4)","A",1
1683,-1,0,113,180.0270086,"\int \frac{(-1+x)^3 \left(-1+2 (-1+k) x+k x^2\right)}{x \sqrt[4]{(1-x) x (1-k x)} (-1+k x) \left(-1+(3+d) x-(3+d k) x^2+x^3\right)} \, dx","IntegrateAlgebraic[((-1 + x)^3*(-1 + 2*(-1 + k)*x + k*x^2))/(x*((1 - x)*x*(1 - k*x))^(1/4)*(-1 + k*x)*(-1 + (3 + d)*x - (3 + d*k)*x^2 + x^3)),x]","\text{\$Aborted}","\frac{4 \left(x-x^2-k x^2+k x^3\right)^{3/4}}{x (-1+k x)}+2 \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{x+(-1-k) x^2+k x^3}}{-1+x}\right)-2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{x+(-1-k) x^2+k x^3}}{-1+x}\right)",1,"$Aborted","F",-1
1684,1,113,113,0.3562256,"\int \frac{b+a x^2}{\left(-b+a x^2\right) \sqrt{b x+a x^3}} \, dx","IntegrateAlgebraic[(b + a*x^2)/((-b + a*x^2)*Sqrt[b*x + a*x^3]),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b}}",1,"-(ArcTan[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[b*x + a*x^3])/(b + a*x^2)]/(Sqrt[2]*a^(1/4)*b^(1/4))) - ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[b*x + a*x^3])/(b + a*x^2)]/(Sqrt[2]*a^(1/4)*b^(1/4))","A",1
1685,1,113,113,3.6449094,"\int \frac{\left(a^2-2 a x+x^2\right) \left(-((2 a-3 b) b)+2 (a-2 b) x+x^2\right)}{\left((-a+x) (-b+x)^2\right)^{3/4} \left(-b^2-a^3 d+\left(2 b+3 a^2 d\right) x-(1+3 a d) x^2+d x^3\right)} \, dx","IntegrateAlgebraic[((a^2 - 2*a*x + x^2)*(-((2*a - 3*b)*b) + 2*(a - 2*b)*x + x^2))/(((-a + x)*(-b + x)^2)^(3/4)*(-b^2 - a^3*d + (2*b + 3*a^2*d)*x - (1 + 3*a*d)*x^2 + d*x^3)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{3/4}}{(b-x)^2}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{3/4}}{(b-x)^2}\right)}{d^{3/4}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{3/4}}{(b-x)^2}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{3/4}}{(b-x)^2}\right)}{d^{3/4}}",1,"(2*ArcTan[(d^(1/4)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(3/4))/(b - x)^2])/d^(3/4) - (2*ArcTanh[(d^(1/4)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(3/4))/(b - x)^2])/d^(3/4)","A",1
1686,1,113,113,0.4101802,"\int \frac{\left(-1+x^3\right) \sqrt[4]{x^3+x^4}}{1+x^3} \, dx","IntegrateAlgebraic[((-1 + x^3)*(x^3 + x^4)^(1/4))/(1 + x^3),x]","\frac{1}{8} (1+4 x) \sqrt[4]{x^3+x^4}+\frac{3}{16} \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)-\frac{3}{16} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)+2 \text{RootSum}\left[3-3 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right)}{-3 \text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]","\frac{1}{8} (1+4 x) \sqrt[4]{x^3+x^4}+\frac{3}{16} \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)-\frac{3}{16} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)+2 \text{RootSum}\left[3-3 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right)}{-3 \text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]",1,"((1 + 4*x)*(x^3 + x^4)^(1/4))/8 + (3*ArcTan[x/(x^3 + x^4)^(1/4)])/16 - (3*ArcTanh[x/(x^3 + x^4)^(1/4)])/16 + 2*RootSum[3 - 3*#1^4 + #1^8 & , (-Log[x] + Log[(x^3 + x^4)^(1/4) - x*#1])/(-3*#1^3 + 2*#1^7) & ]","A",1
1687,1,113,113,0.3966626,"\int \frac{-b+a x^2}{x^2 \left(b+a x^2\right) \sqrt[4]{-b x^2+a x^4}} \, dx","IntegrateAlgebraic[(-b + a*x^2)/(x^2*(b + a*x^2)*(-(b*x^2) + a*x^4)^(1/4)),x]","-\frac{2 \left(-b x^2+a x^4\right)^{3/4}}{3 b x^3}+\frac{2^{3/4} a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{b}+\frac{2^{3/4} a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{b}","-\frac{2 \left(-b x^2+a x^4\right)^{3/4}}{3 b x^3}+\frac{2^{3/4} a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{b}+\frac{2^{3/4} a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{b}",1,"(-2*(-(b*x^2) + a*x^4)^(3/4))/(3*b*x^3) + (2^(3/4)*a^(3/4)*ArcTan[(2^(1/4)*a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/b + (2^(3/4)*a^(3/4)*ArcTanh[(2^(1/4)*a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/b","A",1
1688,1,113,113,0.4487216,"\int \frac{(-b+x) (-6 a+b+5 x)}{\sqrt[4]{(-a+x) (-b+x)^2} \left(b^6+a d-\left(6 b^5+d\right) x+15 b^4 x^2-20 b^3 x^3+15 b^2 x^4-6 b x^5+x^6\right)} \, dx","IntegrateAlgebraic[((-b + x)*(-6*a + b + 5*x))/(((-a + x)*(-b + x)^2)^(1/4)*(b^6 + a*d - (6*b^5 + d)*x + 15*b^4*x^2 - 20*b^3*x^3 + 15*b^2*x^4 - 6*b*x^5 + x^6)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{(b-x)^2}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{(b-x)^2}\right)}{d^{3/4}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{(b-x)^2}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{(b-x)^2}\right)}{d^{3/4}}",1,"(2*ArcTan[(d^(1/4)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/4))/(b - x)^2])/d^(3/4) - (2*ArcTanh[(d^(1/4)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/4))/(b - x)^2])/d^(3/4)","A",1
1689,1,113,113,0.4694448,"\int \frac{(-6 a+b+5 x) \left(-b^5+5 b^4 x-10 b^3 x^2+10 b^2 x^3-5 b x^4+x^5\right)}{\left((-a+x) (-b+x)^2\right)^{3/4} \left(a+b^6 d-\left(1+6 b^5 d\right) x+15 b^4 d x^2-20 b^3 d x^3+15 b^2 d x^4-6 b d x^5+d x^6\right)} \, dx","IntegrateAlgebraic[((-6*a + b + 5*x)*(-b^5 + 5*b^4*x - 10*b^3*x^2 + 10*b^2*x^3 - 5*b*x^4 + x^5))/(((-a + x)*(-b + x)^2)^(3/4)*(a + b^6*d - (1 + 6*b^5*d)*x + 15*b^4*d*x^2 - 20*b^3*d*x^3 + 15*b^2*d*x^4 - 6*b*d*x^5 + d*x^6)),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{3/4}}{a-x}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{3/4}}{a-x}\right)}{d^{3/4}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{3/4}}{a-x}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{3/4}}{a-x}\right)}{d^{3/4}}",1,"(-2*ArcTan[(d^(1/4)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(3/4))/(a - x)])/d^(3/4) + (2*ArcTanh[(d^(1/4)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(3/4))/(a - x)])/d^(3/4)","A",1
1690,1,113,113,0.3323371,"\int \frac{x^4 \sqrt[4]{x^2+x^4}}{-1+x^8} \, dx","IntegrateAlgebraic[(x^4*(x^2 + x^4)^(1/4))/(-1 + x^8),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2\ 2^{3/4}}-\frac{1}{8} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^4}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2\ 2^{3/4}}-\frac{1}{8} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^4}\&\right]",1,"ArcTan[(2^(1/4)*x)/(x^2 + x^4)^(1/4)]/(2*2^(3/4)) - ArcTanh[(2^(1/4)*x)/(x^2 + x^4)^(1/4)]/(2*2^(3/4)) - RootSum[2 - 2*#1^4 + #1^8 & , (-(Log[x]*#1) + Log[(x^2 + x^4)^(1/4) - x*#1]*#1)/(-1 + #1^4) & ]/8","A",1
1691,1,113,113,0.0011296,"\int \frac{x^4 \sqrt[4]{x^2+x^4}}{-1+x^8} \, dx","IntegrateAlgebraic[(x^4*(x^2 + x^4)^(1/4))/(-1 + x^8),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2\ 2^{3/4}}-\frac{1}{8} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^4}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2\ 2^{3/4}}-\frac{1}{8} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^4}\&\right]",1,"ArcTan[(2^(1/4)*x)/(x^2 + x^4)^(1/4)]/(2*2^(3/4)) - ArcTanh[(2^(1/4)*x)/(x^2 + x^4)^(1/4)]/(2*2^(3/4)) - RootSum[2 - 2*#1^4 + #1^8 & , (-(Log[x]*#1) + Log[(x^2 + x^4)^(1/4) - x*#1]*#1)/(-1 + #1^4) & ]/8","A",1
1692,1,113,113,0.3070625,"\int \frac{x^4}{\sqrt[4]{x^2+x^4} \left(1+x^4+x^8\right)} \, dx","IntegrateAlgebraic[x^4/((x^2 + x^4)^(1/4)*(1 + x^4 + x^8)),x]","-\frac{1}{4} \text{RootSum}\left[3-3 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{x^2+x^4}-x \text{$\#$1}\right)}{-3 \text{$\#$1}+2 \text{$\#$1}^5}\&\right]+\frac{1}{4} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{x^2+x^4}-x \text{$\#$1}\right)}{-\text{$\#$1}+2 \text{$\#$1}^5}\&\right]","-\frac{1}{4} \text{RootSum}\left[3-3 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{x^2+x^4}-x \text{$\#$1}\right)}{-3 \text{$\#$1}+2 \text{$\#$1}^5}\&\right]+\frac{1}{4} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{x^2+x^4}-x \text{$\#$1}\right)}{-\text{$\#$1}+2 \text{$\#$1}^5}\&\right]",1,"-1/4*RootSum[3 - 3*#1^4 + #1^8 & , (-Log[x] + Log[(x^2 + x^4)^(1/4) - x*#1])/(-3*#1 + 2*#1^5) & ] + RootSum[1 - #1^4 + #1^8 & , (-Log[x] + Log[(x^2 + x^4)^(1/4) - x*#1])/(-#1 + 2*#1^5) & ]/4","A",1
1693,1,113,113,0.0015039,"\int \frac{x^4}{\sqrt[4]{x^2+x^4} \left(1+x^4+x^8\right)} \, dx","IntegrateAlgebraic[x^4/((x^2 + x^4)^(1/4)*(1 + x^4 + x^8)),x]","-\frac{1}{4} \text{RootSum}\left[3-3 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{x^2+x^4}-x \text{$\#$1}\right)}{-3 \text{$\#$1}+2 \text{$\#$1}^5}\&\right]+\frac{1}{4} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{x^2+x^4}-x \text{$\#$1}\right)}{-\text{$\#$1}+2 \text{$\#$1}^5}\&\right]","-\frac{1}{4} \text{RootSum}\left[3-3 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{x^2+x^4}-x \text{$\#$1}\right)}{-3 \text{$\#$1}+2 \text{$\#$1}^5}\&\right]+\frac{1}{4} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{x^2+x^4}-x \text{$\#$1}\right)}{-\text{$\#$1}+2 \text{$\#$1}^5}\&\right]",1,"-1/4*RootSum[3 - 3*#1^4 + #1^8 & , (-Log[x] + Log[(x^2 + x^4)^(1/4) - x*#1])/(-3*#1 + 2*#1^5) & ] + RootSum[1 - #1^4 + #1^8 & , (-Log[x] + Log[(x^2 + x^4)^(1/4) - x*#1])/(-#1 + 2*#1^5) & ]/4","A",1
1694,1,113,113,3.7217455,"\int \frac{x^6 \left(4+x^3\right)}{\left(1+x^3\right)^{3/4} \left(-1-2 x^3-x^6+x^8\right)} \, dx","IntegrateAlgebraic[(x^6*(4 + x^3))/((1 + x^3)^(3/4)*(-1 - 2*x^3 - x^6 + x^8)),x]","\tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^3}}\right)-\frac{\tan ^{-1}\left(\frac{-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^3}}{\sqrt{2}}}{x \sqrt[4]{1+x^3}}\right)}{\sqrt{2}}-\tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^3}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{1+x^3}}{x^2+\sqrt{1+x^3}}\right)}{\sqrt{2}}","\tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^3}}\right)-\frac{\tan ^{-1}\left(\frac{-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^3}}{\sqrt{2}}}{x \sqrt[4]{1+x^3}}\right)}{\sqrt{2}}-\tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^3}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{1+x^3}}{x^2+\sqrt{1+x^3}}\right)}{\sqrt{2}}",1,"ArcTan[x/(1 + x^3)^(1/4)] - ArcTan[(-(x^2/Sqrt[2]) + Sqrt[1 + x^3]/Sqrt[2])/(x*(1 + x^3)^(1/4))]/Sqrt[2] - ArcTanh[x/(1 + x^3)^(1/4)] - ArcTanh[(Sqrt[2]*x*(1 + x^3)^(1/4))/(x^2 + Sqrt[1 + x^3])]/Sqrt[2]","A",1
1695,1,114,114,0.3207178,"\int x^{10} \sqrt[3]{-1+x^3} \, dx","IntegrateAlgebraic[x^10*(-1 + x^3)^(1/3),x]","\frac{1}{972} \sqrt[3]{-1+x^3} \left(-20 x^2-12 x^5-9 x^8+81 x^{11}\right)+\frac{10 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{243 \sqrt{3}}+\frac{10}{729} \log \left(-x+\sqrt[3]{-1+x^3}\right)-\frac{5}{729} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)","\frac{1}{972} \sqrt[3]{-1+x^3} \left(-20 x^2-12 x^5-9 x^8+81 x^{11}\right)+\frac{10 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{243 \sqrt{3}}+\frac{10}{729} \log \left(-x+\sqrt[3]{-1+x^3}\right)-\frac{5}{729} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)",1,"((-1 + x^3)^(1/3)*(-20*x^2 - 12*x^5 - 9*x^8 + 81*x^11))/972 + (10*ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^3)^(1/3))])/(243*Sqrt[3]) + (10*Log[-x + (-1 + x^3)^(1/3)])/729 - (5*Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)])/729","A",1
1696,1,114,114,0.2341353,"\int \frac{x}{(-1+x) \sqrt[3]{-x^2+x^3}} \, dx","IntegrateAlgebraic[x/((-1 + x)*(-x^2 + x^3)^(1/3)),x]","-\frac{3 \left(-x^2+x^3\right)^{2/3}}{(-1+x) x}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{-x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)","-\frac{3 \left(-x^2+x^3\right)^{2/3}}{(-1+x) x}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{-x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)",1,"(-3*(-x^2 + x^3)^(2/3))/((-1 + x)*x) + Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-x^2 + x^3)^(1/3))] - Log[-x + (-x^2 + x^3)^(1/3)] + Log[x^2 + x*(-x^2 + x^3)^(1/3) + (-x^2 + x^3)^(2/3)]/2","A",1
1697,1,114,114,0.2392763,"\int \frac{1+x}{(-1+x) \sqrt[3]{-x^2+x^3}} \, dx","IntegrateAlgebraic[(1 + x)/((-1 + x)*(-x^2 + x^3)^(1/3)),x]","-\frac{6 \left(-x^2+x^3\right)^{2/3}}{(-1+x) x}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{-x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)","-\frac{6 \left(-x^2+x^3\right)^{2/3}}{(-1+x) x}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{-x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)",1,"(-6*(-x^2 + x^3)^(2/3))/((-1 + x)*x) + Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-x^2 + x^3)^(1/3))] - Log[-x + (-x^2 + x^3)^(1/3)] + Log[x^2 + x*(-x^2 + x^3)^(1/3) + (-x^2 + x^3)^(2/3)]/2","A",1
1698,1,114,114,0.5455896,"\int \frac{-2 b+a x}{\left(-b+a x+x^2\right) \sqrt[4]{-b x^2+a x^3}} \, dx","IntegrateAlgebraic[(-2*b + a*x)/((-b + a*x + x^2)*(-(b*x^2) + a*x^3)^(1/4)),x]","-\sqrt{2} \tan ^{-1}\left(\frac{-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{-b x^2+a x^3}}{\sqrt{2}}}{x \sqrt[4]{-b x^2+a x^3}}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-b x^2+a x^3}}{x^2+\sqrt{-b x^2+a x^3}}\right)","-\sqrt{2} \tan ^{-1}\left(\frac{-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{-b x^2+a x^3}}{\sqrt{2}}}{x \sqrt[4]{-b x^2+a x^3}}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-b x^2+a x^3}}{x^2+\sqrt{-b x^2+a x^3}}\right)",1,"-(Sqrt[2]*ArcTan[(-(x^2/Sqrt[2]) + Sqrt[-(b*x^2) + a*x^3]/Sqrt[2])/(x*(-(b*x^2) + a*x^3)^(1/4))]) + Sqrt[2]*ArcTanh[(Sqrt[2]*x*(-(b*x^2) + a*x^3)^(1/4))/(x^2 + Sqrt[-(b*x^2) + a*x^3])]","A",1
1699,1,114,114,0.2570236,"\int \frac{\left(1+x^2\right) \sqrt[3]{-x+2 x^3}}{x^2 \left(1+x^4\right)} \, dx","IntegrateAlgebraic[((1 + x^2)*(-x + 2*x^3)^(1/3))/(x^2*(1 + x^4)),x]","-\frac{3 \sqrt[3]{-x+2 x^3}}{2 x}+\frac{1}{4} \text{RootSum}\left[5-4 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-5 \log (x)+5 \log \left(\sqrt[3]{-x+2 x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{-x+2 x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-2 \text{$\#$1}^2+\text{$\#$1}^5}\&\right]","-\frac{3 \sqrt[3]{-x+2 x^3}}{2 x}+\frac{1}{4} \text{RootSum}\left[5-4 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-5 \log (x)+5 \log \left(\sqrt[3]{-x+2 x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{-x+2 x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-2 \text{$\#$1}^2+\text{$\#$1}^5}\&\right]",1,"(-3*(-x + 2*x^3)^(1/3))/(2*x) + RootSum[5 - 4*#1^3 + #1^6 & , (-5*Log[x] + 5*Log[(-x + 2*x^3)^(1/3) - x*#1] + Log[x]*#1^3 - Log[(-x + 2*x^3)^(1/3) - x*#1]*#1^3)/(-2*#1^2 + #1^5) & ]/4","A",1
1700,1,114,114,1.537516,"\int \frac{-1+x}{\left(1+x+x^2\right) \sqrt[4]{1+x^4}} \, dx","IntegrateAlgebraic[(-1 + x)/((1 + x + x^2)*(1 + x^4)^(1/4)),x]","\frac{\tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}-\sqrt{2} x-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^4}}{\sqrt{2}}}{(1+x) \sqrt[4]{1+x^4}}\right)}{\sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\left(\sqrt{2}+\sqrt{2} x\right) \sqrt[4]{1+x^4}}{1+2 x+x^2+\sqrt{1+x^4}}\right)}{\sqrt{2}}","\frac{\tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}-\sqrt{2} x-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^4}}{\sqrt{2}}}{(1+x) \sqrt[4]{1+x^4}}\right)}{\sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\left(\sqrt{2}+\sqrt{2} x\right) \sqrt[4]{1+x^4}}{1+2 x+x^2+\sqrt{1+x^4}}\right)}{\sqrt{2}}",1,"ArcTan[(-(1/Sqrt[2]) - Sqrt[2]*x - x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/((1 + x)*(1 + x^4)^(1/4))]/Sqrt[2] - ArcTanh[((Sqrt[2] + Sqrt[2]*x)*(1 + x^4)^(1/4))/(1 + 2*x + x^2 + Sqrt[1 + x^4])]/Sqrt[2]","A",1
1701,1,114,114,0.5700406,"\int \frac{x^2 \sqrt[4]{x^3+x^4}}{-1+x} \, dx","IntegrateAlgebraic[(x^2*(x^3 + x^4)^(1/4))/(-1 + x),x]","\frac{1}{96} \left(101+52 x+32 x^2\right) \sqrt[4]{x^3+x^4}-\frac{155}{64} \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)+2 \sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^4}}\right)+\frac{155}{64} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)-2 \sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^4}}\right)","\frac{1}{96} \left(101+52 x+32 x^2\right) \sqrt[4]{x^3+x^4}-\frac{155}{64} \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)+2 \sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^4}}\right)+\frac{155}{64} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)-2 \sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^4}}\right)",1,"((101 + 52*x + 32*x^2)*(x^3 + x^4)^(1/4))/96 - (155*ArcTan[x/(x^3 + x^4)^(1/4)])/64 + 2*2^(1/4)*ArcTan[(2^(1/4)*x)/(x^3 + x^4)^(1/4)] + (155*ArcTanh[x/(x^3 + x^4)^(1/4)])/64 - 2*2^(1/4)*ArcTanh[(2^(1/4)*x)/(x^3 + x^4)^(1/4)]","A",1
1702,1,114,114,0.4336899,"\int \frac{\left(-b+a x^3\right) \left(-b+2 a x^3\right)}{x^6 \sqrt[4]{-b x+a x^4}} \, dx","IntegrateAlgebraic[((-b + a*x^3)*(-b + 2*a*x^3))/(x^6*(-(b*x) + a*x^4)^(1/4)),x]","\frac{4 \left(3 b-17 a x^3\right) \left(-b x+a x^4\right)^{3/4}}{63 x^6}+\frac{4}{3} a^{7/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)+\frac{4}{3} a^{7/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)","\frac{4 \left(3 b-17 a x^3\right) \left(-b x+a x^4\right)^{3/4}}{63 x^6}+\frac{4}{3} a^{7/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)+\frac{4}{3} a^{7/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)",1,"(4*(3*b - 17*a*x^3)*(-(b*x) + a*x^4)^(3/4))/(63*x^6) + (4*a^(7/4)*ArcTan[(a^(1/4)*(-(b*x) + a*x^4)^(3/4))/(-b + a*x^3)])/3 + (4*a^(7/4)*ArcTanh[(a^(1/4)*(-(b*x) + a*x^4)^(3/4))/(-b + a*x^3)])/3","A",1
1703,1,114,114,0.8225207,"\int \frac{-b+a x^4+x^8}{x^8 \left(-b+a x^4\right) \sqrt[4]{b+a x^4}} \, dx","IntegrateAlgebraic[(-b + a*x^4 + x^8)/(x^8*(-b + a*x^4)*(b + a*x^4)^(1/4)),x]","\frac{\left(b+a x^4\right)^{3/4} \left(-3 b+4 a x^4\right)}{21 b^2 x^7}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2 \sqrt[4]{2} \sqrt[4]{a} b}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2 \sqrt[4]{2} \sqrt[4]{a} b}","\frac{\left(b+a x^4\right)^{3/4} \left(-3 b+4 a x^4\right)}{21 b^2 x^7}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2 \sqrt[4]{2} \sqrt[4]{a} b}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2 \sqrt[4]{2} \sqrt[4]{a} b}",1,"((b + a*x^4)^(3/4)*(-3*b + 4*a*x^4))/(21*b^2*x^7) - ArcTan[(2^(1/4)*a^(1/4)*x)/(b + a*x^4)^(1/4)]/(2*2^(1/4)*a^(1/4)*b) - ArcTanh[(2^(1/4)*a^(1/4)*x)/(b + a*x^4)^(1/4)]/(2*2^(1/4)*a^(1/4)*b)","A",1
1704,1,114,114,0.1981576,"\int \frac{1+x^4}{x^4 \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[(1 + x^4)/(x^4*Sqrt[x + Sqrt[1 + x^2]]),x]","\frac{-8-57 x^2+48 x^4+456 x^6+384 x^8+\sqrt{1+x^2} \left(-30 x-36 x^3+264 x^5+384 x^7\right)}{24 x^3 \left(x+\sqrt{1+x^2}\right)^{9/2}}-\frac{1}{8} \tan ^{-1}\left(\sqrt{x+\sqrt{1+x^2}}\right)-\frac{1}{8} \tanh ^{-1}\left(\sqrt{x+\sqrt{1+x^2}}\right)","\frac{-8-57 x^2+48 x^4+456 x^6+384 x^8+\sqrt{1+x^2} \left(-30 x-36 x^3+264 x^5+384 x^7\right)}{24 x^3 \left(x+\sqrt{1+x^2}\right)^{9/2}}-\frac{1}{8} \tan ^{-1}\left(\sqrt{x+\sqrt{1+x^2}}\right)-\frac{1}{8} \tanh ^{-1}\left(\sqrt{x+\sqrt{1+x^2}}\right)",1,"(-8 - 57*x^2 + 48*x^4 + 456*x^6 + 384*x^8 + Sqrt[1 + x^2]*(-30*x - 36*x^3 + 264*x^5 + 384*x^7))/(24*x^3*(x + Sqrt[1 + x^2])^(9/2)) - ArcTan[Sqrt[x + Sqrt[1 + x^2]]]/8 - ArcTanh[Sqrt[x + Sqrt[1 + x^2]]]/8","A",1
1705,1,82,114,0.1581997,"\int \frac{\sqrt{b+\sqrt{b^2+a x^2}}}{x^2 \sqrt{b^2+a x^2}} \, dx","IntegrateAlgebraic[Sqrt[b + Sqrt[b^2 + a*x^2]]/(x^2*Sqrt[b^2 + a*x^2]),x]","-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{b x}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{\sqrt{2} b^{3/2}}","-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{b x}-\frac{\sqrt{2} \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2} \sqrt{b}}\right)}{b^{3/2}}",1,"-(Sqrt[b + Sqrt[b^2 + a*x^2]]/(b*x)) - (Sqrt[a]*ArcTan[(Sqrt[a]*x)/(Sqrt[2]*Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/(Sqrt[2]*b^(3/2))","A",1
1706,1,115,115,0.124582,"\int \frac{2+x^2}{x \left(2-2 x+x^2\right) \sqrt[3]{1-x+x^2}} \, dx","IntegrateAlgebraic[(2 + x^2)/(x*(2 - 2*x + x^2)*(1 - x + x^2)^(1/3)),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\frac{2}{\sqrt{3}}-\frac{2 x}{\sqrt{3}}+\frac{\sqrt[3]{1-x+x^2}}{\sqrt{3}}}{\sqrt[3]{1-x+x^2}}\right)+\log \left(-1+x+\sqrt[3]{1-x+x^2}\right)-\frac{1}{2} \log \left(1-2 x+x^2+(1-x) \sqrt[3]{1-x+x^2}+\left(1-x+x^2\right)^{2/3}\right)","-\sqrt{3} \tan ^{-1}\left(\frac{\frac{2}{\sqrt{3}}-\frac{2 x}{\sqrt{3}}+\frac{\sqrt[3]{1-x+x^2}}{\sqrt{3}}}{\sqrt[3]{1-x+x^2}}\right)+\log \left(-1+x+\sqrt[3]{1-x+x^2}\right)-\frac{1}{2} \log \left(1-2 x+x^2+(1-x) \sqrt[3]{1-x+x^2}+\left(1-x+x^2\right)^{2/3}\right)",1,"-(Sqrt[3]*ArcTan[(2/Sqrt[3] - (2*x)/Sqrt[3] + (1 - x + x^2)^(1/3)/Sqrt[3])/(1 - x + x^2)^(1/3)]) + Log[-1 + x + (1 - x + x^2)^(1/3)] - Log[1 - 2*x + x^2 + (1 - x)*(1 - x + x^2)^(1/3) + (1 - x + x^2)^(2/3)]/2","A",1
1707,1,115,115,17.2914202,"\int \frac{-1+x}{x^{10} \sqrt[3]{1+x^3}} \, dx","IntegrateAlgebraic[(-1 + x)/(x^10*(1 + x^3)^(1/3)),x]","\frac{\left(1+x^3\right)^{2/3} \left(360-405 x-420 x^3+486 x^4+560 x^6-729 x^7\right)}{3240 x^9}+\frac{14 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^3}}{\sqrt{3}}\right)}{81 \sqrt{3}}+\frac{14}{243} \log \left(-1+\sqrt[3]{1+x^3}\right)-\frac{7}{243} \log \left(1+\sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)","\frac{\left(1+x^3\right)^{2/3} \left(360-405 x-420 x^3+486 x^4+560 x^6-729 x^7\right)}{3240 x^9}+\frac{14 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x^3}}{\sqrt{3}}\right)}{81 \sqrt{3}}+\frac{14}{243} \log \left(-1+\sqrt[3]{1+x^3}\right)-\frac{7}{243} \log \left(1+\sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)",1,"((1 + x^3)^(2/3)*(360 - 405*x - 420*x^3 + 486*x^4 + 560*x^6 - 729*x^7))/(3240*x^9) + (14*ArcTan[1/Sqrt[3] + (2*(1 + x^3)^(1/3))/Sqrt[3]])/(81*Sqrt[3]) + (14*Log[-1 + (1 + x^3)^(1/3)])/243 - (7*Log[1 + (1 + x^3)^(1/3) + (1 + x^3)^(2/3)])/243","A",1
1708,1,115,115,0.0958936,"\int \frac{1}{x \sqrt[3]{b+a x^3}} \, dx","IntegrateAlgebraic[1/(x*(b + a*x^3)^(1/3)),x]","\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{b+a x^3}}{\sqrt{3} \sqrt[3]{b}}\right)}{\sqrt{3} \sqrt[3]{b}}+\frac{\log \left(-\sqrt[3]{b}+\sqrt[3]{b+a x^3}\right)}{3 \sqrt[3]{b}}-\frac{\log \left(b^{2/3}+\sqrt[3]{b} \sqrt[3]{b+a x^3}+\left(b+a x^3\right)^{2/3}\right)}{6 \sqrt[3]{b}}","\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{b+a x^3}}{\sqrt{3} \sqrt[3]{b}}\right)}{\sqrt{3} \sqrt[3]{b}}+\frac{\log \left(-\sqrt[3]{b}+\sqrt[3]{b+a x^3}\right)}{3 \sqrt[3]{b}}-\frac{\log \left(b^{2/3}+\sqrt[3]{b} \sqrt[3]{b+a x^3}+\left(b+a x^3\right)^{2/3}\right)}{6 \sqrt[3]{b}}",1,"ArcTan[1/Sqrt[3] + (2*(b + a*x^3)^(1/3))/(Sqrt[3]*b^(1/3))]/(Sqrt[3]*b^(1/3)) + Log[-b^(1/3) + (b + a*x^3)^(1/3)]/(3*b^(1/3)) - Log[b^(2/3) + b^(1/3)*(b + a*x^3)^(1/3) + (b + a*x^3)^(2/3)]/(6*b^(1/3))","A",1
1709,1,115,115,0.8952205,"\int \frac{\left(-1-2 (-1+k) x+k x^2\right) \left(-1+3 k x-3 k^2 x^2+k^3 x^3\right)}{(-1+x) x \sqrt[4]{(1-x) x (1-k x)} \left(-1+(d+3 k) x-\left(d+3 k^2\right) x^2+k^3 x^3\right)} \, dx","IntegrateAlgebraic[((-1 - 2*(-1 + k)*x + k*x^2)*(-1 + 3*k*x - 3*k^2*x^2 + k^3*x^3))/((-1 + x)*x*((1 - x)*x*(1 - k*x))^(1/4)*(-1 + (d + 3*k)*x - (d + 3*k^2)*x^2 + k^3*x^3)),x]","\frac{4 \left(x-x^2-k x^2+k x^3\right)^{3/4}}{(-1+x) x}+2 \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{x+(-1-k) x^2+k x^3}}{-1+k x}\right)-2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{x+(-1-k) x^2+k x^3}}{-1+k x}\right)","\frac{4 \left(x-x^2-k x^2+k x^3\right)^{3/4}}{(-1+x) x}+2 \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{x+(-1-k) x^2+k x^3}}{-1+k x}\right)-2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{x+(-1-k) x^2+k x^3}}{-1+k x}\right)",1,"(4*(x - x^2 - k*x^2 + k*x^3)^(3/4))/((-1 + x)*x) + 2*d^(1/4)*ArcTan[(d^(1/4)*(x + (-1 - k)*x^2 + k*x^3)^(1/4))/(-1 + k*x)] - 2*d^(1/4)*ArcTanh[(d^(1/4)*(x + (-1 - k)*x^2 + k*x^3)^(1/4))/(-1 + k*x)]","A",1
1710,1,115,115,0.5435165,"\int \frac{\left(3+x^4\right) \left(-1-x^3+x^4\right)^{2/3}}{x^3 \left(-1+x^4\right)} \, dx","IntegrateAlgebraic[((3 + x^4)*(-1 - x^3 + x^4)^(2/3))/(x^3*(-1 + x^4)),x]","\frac{3 \left(-1-x^3+x^4\right)^{2/3}}{2 x^2}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-1-x^3+x^4}}\right)+\log \left(x+\sqrt[3]{-1-x^3+x^4}\right)-\frac{1}{2} \log \left(x^2-x \sqrt[3]{-1-x^3+x^4}+\left(-1-x^3+x^4\right)^{2/3}\right)","\frac{3 \left(-1-x^3+x^4\right)^{2/3}}{2 x^2}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-1-x^3+x^4}}\right)+\log \left(x+\sqrt[3]{-1-x^3+x^4}\right)-\frac{1}{2} \log \left(x^2-x \sqrt[3]{-1-x^3+x^4}+\left(-1-x^3+x^4\right)^{2/3}\right)",1,"(3*(-1 - x^3 + x^4)^(2/3))/(2*x^2) + Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(-1 - x^3 + x^4)^(1/3))] + Log[x + (-1 - x^3 + x^4)^(1/3)] - Log[x^2 - x*(-1 - x^3 + x^4)^(1/3) + (-1 - x^3 + x^4)^(2/3)]/2","A",1
1711,1,115,115,0.3047603,"\int \frac{\left(1+2 x^2\right) \sqrt[3]{x+2 x^3}}{x^4 \left(1+2 x^4\right)} \, dx","IntegrateAlgebraic[((1 + 2*x^2)*(x + 2*x^3)^(1/3))/(x^4*(1 + 2*x^4)),x]","-\frac{3 \left(1+10 x^2\right) \sqrt[3]{x+2 x^3}}{8 x^3}+\frac{1}{2} \text{RootSum}\left[6-4 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-6 \log (x)+6 \log \left(\sqrt[3]{x+2 x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{x+2 x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-2 \text{$\#$1}^2+\text{$\#$1}^5}\&\right]","-\frac{3 \left(1+10 x^2\right) \sqrt[3]{x+2 x^3}}{8 x^3}+\frac{1}{2} \text{RootSum}\left[6-4 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-6 \log (x)+6 \log \left(\sqrt[3]{x+2 x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{x+2 x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-2 \text{$\#$1}^2+\text{$\#$1}^5}\&\right]",1,"(-3*(1 + 10*x^2)*(x + 2*x^3)^(1/3))/(8*x^3) + RootSum[6 - 4*#1^3 + #1^6 & , (-6*Log[x] + 6*Log[(x + 2*x^3)^(1/3) - x*#1] + Log[x]*#1^3 - Log[(x + 2*x^3)^(1/3) - x*#1]*#1^3)/(-2*#1^2 + #1^5) & ]/2","A",1
1712,1,115,115,0.124616,"\int \frac{1}{x-\sqrt{b+a x} \sqrt{c+\sqrt{b+a x}}} \, dx","IntegrateAlgebraic[(x - Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]])^(-1),x]","4 \text{RootSum}\left[b-c^2-a c \text{$\#$1}+2 c \text{$\#$1}^2+a \text{$\#$1}^3-\text{$\#$1}^4\&,\frac{-c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}+\log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3}{a c-4 c \text{$\#$1}-3 a \text{$\#$1}^2+4 \text{$\#$1}^3}\&\right]","4 \text{RootSum}\left[b-c^2-a c \text{$\#$1}+2 c \text{$\#$1}^2+a \text{$\#$1}^3-\text{$\#$1}^4\&,\frac{-c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}+\log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3}{a c-4 c \text{$\#$1}-3 a \text{$\#$1}^2+4 \text{$\#$1}^3}\&\right]",1,"4*RootSum[b - c^2 - a*c*#1 + 2*c*#1^2 + a*#1^3 - #1^4 & , (-(c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1) + Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^3)/(a*c - 4*c*#1 - 3*a*#1^2 + 4*#1^3) & ]","A",1
1713,1,115,115,0.0019426,"\int \frac{1}{x-\sqrt{b+a x} \sqrt{c+\sqrt{b+a x}}} \, dx","IntegrateAlgebraic[(x - Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]])^(-1),x]","4 \text{RootSum}\left[b-c^2-a c \text{$\#$1}+2 c \text{$\#$1}^2+a \text{$\#$1}^3-\text{$\#$1}^4\&,\frac{-c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}+\log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3}{a c-4 c \text{$\#$1}-3 a \text{$\#$1}^2+4 \text{$\#$1}^3}\&\right]","4 \text{RootSum}\left[b-c^2-a c \text{$\#$1}+2 c \text{$\#$1}^2+a \text{$\#$1}^3-\text{$\#$1}^4\&,\frac{-c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}+\log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3}{a c-4 c \text{$\#$1}-3 a \text{$\#$1}^2+4 \text{$\#$1}^3}\&\right]",1,"4*RootSum[b - c^2 - a*c*#1 + 2*c*#1^2 + a*#1^3 - #1^4 & , (-(c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1) + Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^3)/(a*c - 4*c*#1 - 3*a*#1^2 + 4*#1^3) & ]","A",1
1714,1,116,116,0.3580271,"\int \frac{2-x+x^2}{\sqrt[3]{-1+x^2} \left(3+4 x+x^2\right)} \, dx","IntegrateAlgebraic[(2 - x + x^2)/((-1 + x^2)^(1/3)*(3 + 4*x + x^2)),x]","\frac{3 \left(-1+x^2\right)^{2/3}}{1+x}-\frac{7}{4} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-1+x^2}}{1-x+\sqrt[3]{-1+x^2}}\right)-\frac{7}{4} \log \left(-1+x+2 \sqrt[3]{-1+x^2}\right)+\frac{7}{8} \log \left(1-2 x+x^2+(2-2 x) \sqrt[3]{-1+x^2}+4 \left(-1+x^2\right)^{2/3}\right)","\frac{3 \left(-1+x^2\right)^{2/3}}{1+x}-\frac{7}{4} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-1+x^2}}{1-x+\sqrt[3]{-1+x^2}}\right)-\frac{7}{4} \log \left(-1+x+2 \sqrt[3]{-1+x^2}\right)+\frac{7}{8} \log \left(1-2 x+x^2+(2-2 x) \sqrt[3]{-1+x^2}+4 \left(-1+x^2\right)^{2/3}\right)",1,"(3*(-1 + x^2)^(2/3))/(1 + x) - (7*Sqrt[3]*ArcTan[(Sqrt[3]*(-1 + x^2)^(1/3))/(1 - x + (-1 + x^2)^(1/3))])/4 - (7*Log[-1 + x + 2*(-1 + x^2)^(1/3)])/4 + (7*Log[1 - 2*x + x^2 + (2 - 2*x)*(-1 + x^2)^(1/3) + 4*(-1 + x^2)^(2/3)])/8","A",1
1715,1,116,116,4.2303372,"\int \frac{x (-1+k x) \left(-1+2 (-1+k) x+k x^2\right)}{(-1+x) ((1-x) x (1-k x))^{3/4} \left(-d+(1+3 d) x-(3 d+k) x^2+d x^3\right)} \, dx","IntegrateAlgebraic[(x*(-1 + k*x)*(-1 + 2*(-1 + k)*x + k*x^2))/((-1 + x)*((1 - x)*x*(1 - k*x))^(3/4)*(-d + (1 + 3*d)*x - (3*d + k)*x^2 + d*x^3)),x]","\frac{4 \sqrt[4]{x+(-1-k) x^2+k x^3}}{-1+x}+2 \sqrt[4]{d} \tan ^{-1}\left(\frac{-\sqrt[4]{d}+\sqrt[4]{d} x}{\sqrt[4]{x+(-1-k) x^2+k x^3}}\right)-2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{-\sqrt[4]{d}+\sqrt[4]{d} x}{\sqrt[4]{x+(-1-k) x^2+k x^3}}\right)","\frac{4 \sqrt[4]{x+(-1-k) x^2+k x^3}}{-1+x}+2 \sqrt[4]{d} \tan ^{-1}\left(\frac{-\sqrt[4]{d}+\sqrt[4]{d} x}{\sqrt[4]{x+(-1-k) x^2+k x^3}}\right)-2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{-\sqrt[4]{d}+\sqrt[4]{d} x}{\sqrt[4]{x+(-1-k) x^2+k x^3}}\right)",1,"(4*(x + (-1 - k)*x^2 + k*x^3)^(1/4))/(-1 + x) + 2*d^(1/4)*ArcTan[(-d^(1/4) + d^(1/4)*x)/(x + (-1 - k)*x^2 + k*x^3)^(1/4)] - 2*d^(1/4)*ArcTanh[(-d^(1/4) + d^(1/4)*x)/(x + (-1 - k)*x^2 + k*x^3)^(1/4)]","A",1
1716,1,116,116,4.303923,"\int \frac{(-1+x) x \left(-1-2 (-1+k) x+k x^2\right)}{((1-x) x (1-k x))^{3/4} (-1+k x) \left(-d+(1+3 d k) x-\left(1+3 d k^2\right) x^2+d k^3 x^3\right)} \, dx","IntegrateAlgebraic[((-1 + x)*x*(-1 - 2*(-1 + k)*x + k*x^2))/(((1 - x)*x*(1 - k*x))^(3/4)*(-1 + k*x)*(-d + (1 + 3*d*k)*x - (1 + 3*d*k^2)*x^2 + d*k^3*x^3)),x]","\frac{4 \left(-x+x^2\right)}{\left(x-x^2-k x^2+k x^3\right)^{3/4}}+2 \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(x+(-1-k) x^2+k x^3\right)^{3/4}}{(-1+x) x}\right)-2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt[4]{d} \left(x+(-1-k) x^2+k x^3\right)^{3/4}}{(-1+x) x}\right)","\frac{4 \left(-x+x^2\right)}{\left(x-x^2-k x^2+k x^3\right)^{3/4}}+2 \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(x+(-1-k) x^2+k x^3\right)^{3/4}}{(-1+x) x}\right)-2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt[4]{d} \left(x+(-1-k) x^2+k x^3\right)^{3/4}}{(-1+x) x}\right)",1,"(4*(-x + x^2))/(x - x^2 - k*x^2 + k*x^3)^(3/4) + 2*d^(1/4)*ArcTan[(d^(1/4)*(x + (-1 - k)*x^2 + k*x^3)^(3/4))/((-1 + x)*x)] - 2*d^(1/4)*ArcTanh[(d^(1/4)*(x + (-1 - k)*x^2 + k*x^3)^(3/4))/((-1 + x)*x)]","A",1
1717,1,116,116,0.2853523,"\int \frac{\left(4+x^2\right) \sqrt[3]{-2 x+x^3}}{x^4 \left(-4-4 x^2+x^4\right)} \, dx","IntegrateAlgebraic[((4 + x^2)*(-2*x + x^3)^(1/3))/(x^4*(-4 - 4*x^2 + x^4)),x]","-\frac{3 \left(-2+7 x^2\right) \sqrt[3]{-2 x+x^3}}{16 x^3}+\frac{1}{16} \text{RootSum}\left[2-4 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-6 \log (x)+6 \log \left(\sqrt[3]{-2 x+x^3}-x \text{$\#$1}\right)+11 \log (x) \text{$\#$1}^3-11 \log \left(\sqrt[3]{-2 x+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-2 \text{$\#$1}^2+\text{$\#$1}^5}\&\right]","-\frac{3 \left(-2+7 x^2\right) \sqrt[3]{-2 x+x^3}}{16 x^3}+\frac{1}{16} \text{RootSum}\left[2-4 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-6 \log (x)+6 \log \left(\sqrt[3]{-2 x+x^3}-x \text{$\#$1}\right)+11 \log (x) \text{$\#$1}^3-11 \log \left(\sqrt[3]{-2 x+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-2 \text{$\#$1}^2+\text{$\#$1}^5}\&\right]",1,"(-3*(-2 + 7*x^2)*(-2*x + x^3)^(1/3))/(16*x^3) + RootSum[2 - 4*#1^3 + #1^6 & , (-6*Log[x] + 6*Log[(-2*x + x^3)^(1/3) - x*#1] + 11*Log[x]*#1^3 - 11*Log[(-2*x + x^3)^(1/3) - x*#1]*#1^3)/(-2*#1^2 + #1^5) & ]/16","A",1
1718,1,116,116,0.0011362,"\int \frac{\left(4+x^2\right) \sqrt[3]{-2 x+x^3}}{x^4 \left(-4-4 x^2+x^4\right)} \, dx","IntegrateAlgebraic[((4 + x^2)*(-2*x + x^3)^(1/3))/(x^4*(-4 - 4*x^2 + x^4)),x]","-\frac{3 \left(-2+7 x^2\right) \sqrt[3]{-2 x+x^3}}{16 x^3}+\frac{1}{16} \text{RootSum}\left[2-4 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-6 \log (x)+6 \log \left(\sqrt[3]{-2 x+x^3}-x \text{$\#$1}\right)+11 \log (x) \text{$\#$1}^3-11 \log \left(\sqrt[3]{-2 x+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-2 \text{$\#$1}^2+\text{$\#$1}^5}\&\right]","-\frac{3 \left(-2+7 x^2\right) \sqrt[3]{-2 x+x^3}}{16 x^3}+\frac{1}{16} \text{RootSum}\left[2-4 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-6 \log (x)+6 \log \left(\sqrt[3]{-2 x+x^3}-x \text{$\#$1}\right)+11 \log (x) \text{$\#$1}^3-11 \log \left(\sqrt[3]{-2 x+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-2 \text{$\#$1}^2+\text{$\#$1}^5}\&\right]",1,"(-3*(-2 + 7*x^2)*(-2*x + x^3)^(1/3))/(16*x^3) + RootSum[2 - 4*#1^3 + #1^6 & , (-6*Log[x] + 6*Log[(-2*x + x^3)^(1/3) - x*#1] + 11*Log[x]*#1^3 - 11*Log[(-2*x + x^3)^(1/3) - x*#1]*#1^3)/(-2*#1^2 + #1^5) & ]/16","A",1
1719,1,116,116,2.9427377,"\int \frac{2 b+a x^3}{\left(-b+x^2+a x^3\right) \sqrt[4]{-b x^2+a x^5}} \, dx","IntegrateAlgebraic[(2*b + a*x^3)/((-b + x^2 + a*x^3)*(-(b*x^2) + a*x^5)^(1/4)),x]","-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-b x^2+a x^5}}{-x^2+\sqrt{-b x^2+a x^5}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{-b x^2+a x^5}}{\sqrt{2}}}{x \sqrt[4]{-b x^2+a x^5}}\right)","-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-b x^2+a x^5}}{-x^2+\sqrt{-b x^2+a x^5}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{-b x^2+a x^5}}{\sqrt{2}}}{x \sqrt[4]{-b x^2+a x^5}}\right)",1,"-(Sqrt[2]*ArcTan[(Sqrt[2]*x*(-(b*x^2) + a*x^5)^(1/4))/(-x^2 + Sqrt[-(b*x^2) + a*x^5])]) - Sqrt[2]*ArcTanh[(x^2/Sqrt[2] + Sqrt[-(b*x^2) + a*x^5]/Sqrt[2])/(x*(-(b*x^2) + a*x^5)^(1/4))]","A",1
1720,1,116,116,2.9003273,"\int \frac{b+a x^2}{\left(-b+x+a x^2\right) \sqrt[4]{-b x^3+a x^5}} \, dx","IntegrateAlgebraic[(b + a*x^2)/((-b + x + a*x^2)*(-(b*x^3) + a*x^5)^(1/4)),x]","-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-b x^3+a x^5}}{-x^2+\sqrt{-b x^3+a x^5}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{-b x^3+a x^5}}{\sqrt{2}}}{x \sqrt[4]{-b x^3+a x^5}}\right)","-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-b x^3+a x^5}}{-x^2+\sqrt{-b x^3+a x^5}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{-b x^3+a x^5}}{\sqrt{2}}}{x \sqrt[4]{-b x^3+a x^5}}\right)",1,"-(Sqrt[2]*ArcTan[(Sqrt[2]*x*(-(b*x^3) + a*x^5)^(1/4))/(-x^2 + Sqrt[-(b*x^3) + a*x^5])]) - Sqrt[2]*ArcTanh[(x^2/Sqrt[2] + Sqrt[-(b*x^3) + a*x^5]/Sqrt[2])/(x*(-(b*x^3) + a*x^5)^(1/4))]","A",1
1721,1,116,116,1.4910553,"\int \frac{b+a x^4}{\left(-b+x^2+a x^4\right) \sqrt[4]{-b x^2+a x^6}} \, dx","IntegrateAlgebraic[(b + a*x^4)/((-b + x^2 + a*x^4)*(-(b*x^2) + a*x^6)^(1/4)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-b x^2+a x^6}}{-x^2+\sqrt{-b x^2+a x^6}}\right)}{\sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{-b x^2+a x^6}}{\sqrt{2}}}{x \sqrt[4]{-b x^2+a x^6}}\right)}{\sqrt{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-b x^2+a x^6}}{-x^2+\sqrt{-b x^2+a x^6}}\right)}{\sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{-b x^2+a x^6}}{\sqrt{2}}}{x \sqrt[4]{-b x^2+a x^6}}\right)}{\sqrt{2}}",1,"-(ArcTan[(Sqrt[2]*x*(-(b*x^2) + a*x^6)^(1/4))/(-x^2 + Sqrt[-(b*x^2) + a*x^6])]/Sqrt[2]) - ArcTanh[(x^2/Sqrt[2] + Sqrt[-(b*x^2) + a*x^6]/Sqrt[2])/(x*(-(b*x^2) + a*x^6)^(1/4))]/Sqrt[2]","A",1
1722,1,116,116,2.9245811,"\int \frac{b+2 a x^3}{\left(-b+x+a x^3\right) \sqrt[4]{-b x^3+a x^6}} \, dx","IntegrateAlgebraic[(b + 2*a*x^3)/((-b + x + a*x^3)*(-(b*x^3) + a*x^6)^(1/4)),x]","-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-b x^3+a x^6}}{-x^2+\sqrt{-b x^3+a x^6}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{-b x^3+a x^6}}{\sqrt{2}}}{x \sqrt[4]{-b x^3+a x^6}}\right)","-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-b x^3+a x^6}}{-x^2+\sqrt{-b x^3+a x^6}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{-b x^3+a x^6}}{\sqrt{2}}}{x \sqrt[4]{-b x^3+a x^6}}\right)",1,"-(Sqrt[2]*ArcTan[(Sqrt[2]*x*(-(b*x^3) + a*x^6)^(1/4))/(-x^2 + Sqrt[-(b*x^3) + a*x^6])]) - Sqrt[2]*ArcTanh[(x^2/Sqrt[2] + Sqrt[-(b*x^3) + a*x^6]/Sqrt[2])/(x*(-(b*x^3) + a*x^6)^(1/4))]","A",1
1723,1,116,116,0.798867,"\int \frac{1}{\sqrt[4]{-b+a x^4} \left(-b-x^4+a x^8\right)} \, dx","IntegrateAlgebraic[1/((-b + a*x^4)^(1/4)*(-b - x^4 + a*x^8)),x]","-\frac{\text{RootSum}\left[a+a^2-a b-\text{$\#$1}^4-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a \log (x)+a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}-2 a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]}{4 b}","\frac{\text{RootSum}\left[a+a^2-a b-\text{$\#$1}^4-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a \log (x)-a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}-2 a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]}{4 b}",1,"-1/4*RootSum[a + a^2 - a*b - #1^4 - 2*a*#1^4 + #1^8 & , (-(a*Log[x]) + a*Log[(-b + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4)/(-#1 - 2*a*#1 + 2*#1^5) & ]/b","A",1
1724,1,116,116,0.0015569,"\int \frac{1}{\sqrt[4]{-b+a x^4} \left(-b-x^4+a x^8\right)} \, dx","IntegrateAlgebraic[1/((-b + a*x^4)^(1/4)*(-b - x^4 + a*x^8)),x]","-\frac{\text{RootSum}\left[a+a^2-a b-\text{$\#$1}^4-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a \log (x)+a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}-2 a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]}{4 b}","\frac{\text{RootSum}\left[a+a^2-a b-\text{$\#$1}^4-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a \log (x)-a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}-2 a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]}{4 b}",1,"-1/4*RootSum[a + a^2 - a*b - #1^4 - 2*a*#1^4 + #1^8 & , (-(a*Log[x]) + a*Log[(-b + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4)/(-#1 - 2*a*#1 + 2*#1^5) & ]/b","A",1
1725,1,117,116,1.7379112,"\int \frac{b+a x^4}{\sqrt[4]{-b+a x^4} \left(b+c x^4+a x^8\right)} \, dx","IntegrateAlgebraic[(b + a*x^4)/((-b + a*x^4)^(1/4)*(b + c*x^4 + a*x^8)),x]","\frac{1}{4} \text{RootSum}\left[a^2+a b+a c-2 a \text{$\#$1}^4-c \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 a \log (x)+2 a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-2 a \text{$\#$1}-c \text{$\#$1}+2 \text{$\#$1}^5}\&\right]","\frac{1}{4} \text{RootSum}\left[a^2+a b+a c-2 a \text{$\#$1}^4-c \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 a \log (x)-2 a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{2 a \text{$\#$1}+c \text{$\#$1}-2 \text{$\#$1}^5}\&\right]",1,"RootSum[a^2 + a*b + a*c - 2*a*#1^4 - c*#1^4 + #1^8 & , (-2*a*Log[x] + 2*a*Log[(-b + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4)/(-2*a*#1 - c*#1 + 2*#1^5) & ]/4","A",1
1726,1,117,116,0.0015861,"\int \frac{b+a x^4}{\sqrt[4]{-b+a x^4} \left(b+c x^4+a x^8\right)} \, dx","IntegrateAlgebraic[(b + a*x^4)/((-b + a*x^4)^(1/4)*(b + c*x^4 + a*x^8)),x]","\frac{1}{4} \text{RootSum}\left[a^2+a b+a c-2 a \text{$\#$1}^4-c \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 a \log (x)+2 a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-2 a \text{$\#$1}-c \text{$\#$1}+2 \text{$\#$1}^5}\&\right]","\frac{1}{4} \text{RootSum}\left[a^2+a b+a c-2 a \text{$\#$1}^4-c \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 a \log (x)-2 a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{2 a \text{$\#$1}+c \text{$\#$1}-2 \text{$\#$1}^5}\&\right]",1,"RootSum[a^2 + a*b + a*c - 2*a*#1^4 - c*#1^4 + #1^8 & , (-2*a*Log[x] + 2*a*Log[(-b + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4)/(-2*a*#1 - c*#1 + 2*#1^5) & ]/4","A",1
1727,1,116,116,2.7731528,"\int \frac{\left(1+x^5\right) \sqrt[3]{1+x^3+x^5} \left(-3+2 x^5\right)}{x^2 \left(2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}\right)} \, dx","IntegrateAlgebraic[((1 + x^5)*(1 + x^3 + x^5)^(1/3)*(-3 + 2*x^5))/(x^2*(2 - 2*x^3 + 4*x^5 - x^6 - 2*x^8 + 2*x^10)),x]","\frac{3 \sqrt[3]{1+x^3+x^5}}{2 x}-\frac{1}{4} \text{RootSum}\left[3-6 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{-3 \log (x)+3 \log \left(\sqrt[3]{1+x^3+x^5}-x \text{$\#$1}\right)+4 \log (x) \text{$\#$1}^3-4 \log \left(\sqrt[3]{1+x^3+x^5}-x \text{$\#$1}\right) \text{$\#$1}^3}{-3 \text{$\#$1}^2+2 \text{$\#$1}^5}\&\right]","\frac{3 \sqrt[3]{1+x^3+x^5}}{2 x}-\frac{1}{4} \text{RootSum}\left[3-6 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{-3 \log (x)+3 \log \left(\sqrt[3]{1+x^3+x^5}-x \text{$\#$1}\right)+4 \log (x) \text{$\#$1}^3-4 \log \left(\sqrt[3]{1+x^3+x^5}-x \text{$\#$1}\right) \text{$\#$1}^3}{-3 \text{$\#$1}^2+2 \text{$\#$1}^5}\&\right]",1,"(3*(1 + x^3 + x^5)^(1/3))/(2*x) - RootSum[3 - 6*#1^3 + 2*#1^6 & , (-3*Log[x] + 3*Log[(1 + x^3 + x^5)^(1/3) - x*#1] + 4*Log[x]*#1^3 - 4*Log[(1 + x^3 + x^5)^(1/3) - x*#1]*#1^3)/(-3*#1^2 + 2*#1^5) & ]/4","A",1
1728,1,116,116,0.0014359,"\int \frac{\left(1+x^5\right) \sqrt[3]{1+x^3+x^5} \left(-3+2 x^5\right)}{x^2 \left(2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}\right)} \, dx","IntegrateAlgebraic[((1 + x^5)*(1 + x^3 + x^5)^(1/3)*(-3 + 2*x^5))/(x^2*(2 - 2*x^3 + 4*x^5 - x^6 - 2*x^8 + 2*x^10)),x]","\frac{3 \sqrt[3]{1+x^3+x^5}}{2 x}-\frac{1}{4} \text{RootSum}\left[3-6 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{-3 \log (x)+3 \log \left(\sqrt[3]{1+x^3+x^5}-x \text{$\#$1}\right)+4 \log (x) \text{$\#$1}^3-4 \log \left(\sqrt[3]{1+x^3+x^5}-x \text{$\#$1}\right) \text{$\#$1}^3}{-3 \text{$\#$1}^2+2 \text{$\#$1}^5}\&\right]","\frac{3 \sqrt[3]{1+x^3+x^5}}{2 x}-\frac{1}{4} \text{RootSum}\left[3-6 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{-3 \log (x)+3 \log \left(\sqrt[3]{1+x^3+x^5}-x \text{$\#$1}\right)+4 \log (x) \text{$\#$1}^3-4 \log \left(\sqrt[3]{1+x^3+x^5}-x \text{$\#$1}\right) \text{$\#$1}^3}{-3 \text{$\#$1}^2+2 \text{$\#$1}^5}\&\right]",1,"(3*(1 + x^3 + x^5)^(1/3))/(2*x) - RootSum[3 - 6*#1^3 + 2*#1^6 & , (-3*Log[x] + 3*Log[(1 + x^3 + x^5)^(1/3) - x*#1] + 4*Log[x]*#1^3 - 4*Log[(1 + x^3 + x^5)^(1/3) - x*#1]*#1^3)/(-3*#1^2 + 2*#1^5) & ]/4","A",1
1729,1,117,117,0.3273008,"\int x^4 \sqrt[3]{-x+x^3} \, dx","IntegrateAlgebraic[x^4*(-x + x^3)^(1/3),x]","\frac{1}{108} \sqrt[3]{-x+x^3} \left(-5 x-3 x^3+18 x^5\right)+\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)}{54 \sqrt{3}}+\frac{5}{162} \log \left(-x+\sqrt[3]{-x+x^3}\right)-\frac{5}{324} \log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)","\frac{1}{108} \sqrt[3]{-x+x^3} \left(-5 x-3 x^3+18 x^5\right)+\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)}{54 \sqrt{3}}+\frac{5}{162} \log \left(-x+\sqrt[3]{-x+x^3}\right)-\frac{5}{324} \log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)",1,"((-x + x^3)^(1/3)*(-5*x - 3*x^3 + 18*x^5))/108 + (5*ArcTan[(Sqrt[3]*x)/(x + 2*(-x + x^3)^(1/3))])/(54*Sqrt[3]) + (5*Log[-x + (-x + x^3)^(1/3)])/162 - (5*Log[x^2 + x*(-x + x^3)^(1/3) + (-x + x^3)^(2/3)])/324","A",1
1730,1,117,117,0.3675313,"\int \frac{x}{\left(-b+a x^2\right) \sqrt{b x+a x^3}} \, dx","IntegrateAlgebraic[x/((-b + a*x^2)*Sqrt[b*x + a*x^3]),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)}{2 \sqrt{2} a^{3/4} b^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)}{2 \sqrt{2} a^{3/4} b^{3/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)}{2 \sqrt{2} a^{3/4} b^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)}{2 \sqrt{2} a^{3/4} b^{3/4}}",1,"ArcTan[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[b*x + a*x^3])/(b + a*x^2)]/(2*Sqrt[2]*a^(3/4)*b^(3/4)) - ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[b*x + a*x^3])/(b + a*x^2)]/(2*Sqrt[2]*a^(3/4)*b^(3/4))","A",1
1731,1,117,117,0.3526343,"\int \frac{\sqrt{b x+a x^3}}{-b^2+a^2 x^4} \, dx","IntegrateAlgebraic[Sqrt[b*x + a*x^3]/(-b^2 + a^2*x^4),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)}{2 \sqrt{2} a^{3/4} b^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)}{2 \sqrt{2} a^{3/4} b^{3/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)}{2 \sqrt{2} a^{3/4} b^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)}{2 \sqrt{2} a^{3/4} b^{3/4}}",1,"ArcTan[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[b*x + a*x^3])/(b + a*x^2)]/(2*Sqrt[2]*a^(3/4)*b^(3/4)) - ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[b*x + a*x^3])/(b + a*x^2)]/(2*Sqrt[2]*a^(3/4)*b^(3/4))","A",1
1732,1,117,117,7.5018651,"\int \frac{x^{19}}{\sqrt[3]{-1+x^6}} \, dx","IntegrateAlgebraic[x^19/(-1 + x^6)^(1/3),x]","\frac{1}{324} \left(-1+x^6\right)^{2/3} \left(28 x^2+21 x^8+18 x^{14}\right)+\frac{7 \tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{-1+x^6}}\right)}{81 \sqrt{3}}-\frac{7}{243} \log \left(-x^2+\sqrt[3]{-1+x^6}\right)+\frac{7}{486} \log \left(x^4+x^2 \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)","\frac{1}{324} \left(-1+x^6\right)^{2/3} \left(28 x^2+21 x^8+18 x^{14}\right)+\frac{7 \tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{-1+x^6}}\right)}{81 \sqrt{3}}-\frac{7}{243} \log \left(-x^2+\sqrt[3]{-1+x^6}\right)+\frac{7}{486} \log \left(x^4+x^2 \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)",1,"((-1 + x^6)^(2/3)*(28*x^2 + 21*x^8 + 18*x^14))/324 + (7*ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(-1 + x^6)^(1/3))])/(81*Sqrt[3]) - (7*Log[-x^2 + (-1 + x^6)^(1/3)])/243 + (7*Log[x^4 + x^2*(-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)])/486","A",1
1733,1,117,117,1.5818114,"\int x^{15} \sqrt[3]{-1+x^6} \, dx","IntegrateAlgebraic[x^15*(-1 + x^6)^(1/3),x]","\frac{1}{324} \sqrt[3]{-1+x^6} \left(-5 x^4-3 x^{10}+18 x^{16}\right)+\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{-1+x^6}}\right)}{162 \sqrt{3}}+\frac{5}{486} \log \left(-x^2+\sqrt[3]{-1+x^6}\right)-\frac{5}{972} \log \left(x^4+x^2 \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)","\frac{1}{324} \sqrt[3]{-1+x^6} \left(-5 x^4-3 x^{10}+18 x^{16}\right)+\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{-1+x^6}}\right)}{162 \sqrt{3}}+\frac{5}{486} \log \left(-x^2+\sqrt[3]{-1+x^6}\right)-\frac{5}{972} \log \left(x^4+x^2 \sqrt[3]{-1+x^6}+\left(-1+x^6\right)^{2/3}\right)",1,"((-1 + x^6)^(1/3)*(-5*x^4 - 3*x^10 + 18*x^16))/324 + (5*ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(-1 + x^6)^(1/3))])/(162*Sqrt[3]) + (5*Log[-x^2 + (-1 + x^6)^(1/3)])/486 - (5*Log[x^4 + x^2*(-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)])/972","A",1
1734,1,117,117,7.5272565,"\int \frac{x^{19}}{\sqrt[3]{1+x^6}} \, dx","IntegrateAlgebraic[x^19/(1 + x^6)^(1/3),x]","\frac{1}{324} \left(1+x^6\right)^{2/3} \left(28 x^2-21 x^8+18 x^{14}\right)-\frac{7 \tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{1+x^6}}\right)}{81 \sqrt{3}}+\frac{7}{243} \log \left(-x^2+\sqrt[3]{1+x^6}\right)-\frac{7}{486} \log \left(x^4+x^2 \sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)","\frac{1}{324} \left(1+x^6\right)^{2/3} \left(28 x^2-21 x^8+18 x^{14}\right)-\frac{7 \tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{1+x^6}}\right)}{81 \sqrt{3}}+\frac{7}{243} \log \left(-x^2+\sqrt[3]{1+x^6}\right)-\frac{7}{486} \log \left(x^4+x^2 \sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)",1,"((1 + x^6)^(2/3)*(28*x^2 - 21*x^8 + 18*x^14))/324 - (7*ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(1 + x^6)^(1/3))])/(81*Sqrt[3]) + (7*Log[-x^2 + (1 + x^6)^(1/3)])/243 - (7*Log[x^4 + x^2*(1 + x^6)^(1/3) + (1 + x^6)^(2/3)])/486","A",1
1735,1,117,117,1.5411641,"\int x^{15} \sqrt[3]{1+x^6} \, dx","IntegrateAlgebraic[x^15*(1 + x^6)^(1/3),x]","\frac{1}{324} \sqrt[3]{1+x^6} \left(-5 x^4+3 x^{10}+18 x^{16}\right)-\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{1+x^6}}\right)}{162 \sqrt{3}}-\frac{5}{486} \log \left(-x^2+\sqrt[3]{1+x^6}\right)+\frac{5}{972} \log \left(x^4+x^2 \sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)","\frac{1}{324} \sqrt[3]{1+x^6} \left(-5 x^4+3 x^{10}+18 x^{16}\right)-\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{1+x^6}}\right)}{162 \sqrt{3}}-\frac{5}{486} \log \left(-x^2+\sqrt[3]{1+x^6}\right)+\frac{5}{972} \log \left(x^4+x^2 \sqrt[3]{1+x^6}+\left(1+x^6\right)^{2/3}\right)",1,"((1 + x^6)^(1/3)*(-5*x^4 + 3*x^10 + 18*x^16))/324 - (5*ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(1 + x^6)^(1/3))])/(162*Sqrt[3]) - (5*Log[-x^2 + (1 + x^6)^(1/3)])/486 + (5*Log[x^4 + x^2*(1 + x^6)^(1/3) + (1 + x^6)^(2/3)])/972","A",1
1736,1,117,117,0.5653406,"\int \frac{-b-a x^3+x^6}{x^6 \sqrt[4]{-b x+a x^4}} \, dx","IntegrateAlgebraic[(-b - a*x^3 + x^6)/(x^6*(-(b*x) + a*x^4)^(1/4)),x]","-\frac{4 \left(3 b+11 a x^3\right) \left(-b x+a x^4\right)^{3/4}}{63 b x^6}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)}{3 \sqrt[4]{a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)}{3 \sqrt[4]{a}}","-\frac{4 \left(3 b+11 a x^3\right) \left(-b x+a x^4\right)^{3/4}}{63 b x^6}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)}{3 \sqrt[4]{a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)}{3 \sqrt[4]{a}}",1,"(-4*(3*b + 11*a*x^3)*(-(b*x) + a*x^4)^(3/4))/(63*b*x^6) + (2*ArcTan[(a^(1/4)*(-(b*x) + a*x^4)^(3/4))/(-b + a*x^3)])/(3*a^(1/4)) + (2*ArcTanh[(a^(1/4)*(-(b*x) + a*x^4)^(3/4))/(-b + a*x^3)])/(3*a^(1/4))","A",1
1737,1,117,117,0.4198807,"\int \frac{-3-x^4+3 x^6}{\left(1-x^4+x^6\right) \sqrt[3]{1-x^3-x^4+x^6}} \, dx","IntegrateAlgebraic[(-3 - x^4 + 3*x^6)/((1 - x^4 + x^6)*(1 - x^3 - x^4 + x^6)^(1/3)),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{1-x^3-x^4+x^6}}\right)-\log \left(x+\sqrt[3]{1-x^3-x^4+x^6}\right)+\frac{1}{2} \log \left(x^2-x \sqrt[3]{1-x^3-x^4+x^6}+\left(1-x^3-x^4+x^6\right)^{2/3}\right)","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{1-x^3-x^4+x^6}}\right)-\log \left(x+\sqrt[3]{1-x^3-x^4+x^6}\right)+\frac{1}{2} \log \left(x^2-x \sqrt[3]{1-x^3-x^4+x^6}+\left(1-x^3-x^4+x^6\right)^{2/3}\right)",1,"-(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(1 - x^3 - x^4 + x^6)^(1/3))]) - Log[x + (1 - x^3 - x^4 + x^6)^(1/3)] + Log[x^2 - x*(1 - x^3 - x^4 + x^6)^(1/3) + (1 - x^3 - x^4 + x^6)^(2/3)]/2","A",1
1738,1,117,117,0.7092756,"\int \frac{-b+a x^6}{x^6 \sqrt[4]{-b x+a x^4}} \, dx","IntegrateAlgebraic[(-b + a*x^6)/(x^6*(-(b*x) + a*x^4)^(1/4)),x]","-\frac{4 \left(3 b+4 a x^3\right) \left(-b x+a x^4\right)^{3/4}}{63 b x^6}+\frac{2}{3} a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)+\frac{2}{3} a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)","-\frac{4 \left(3 b+4 a x^3\right) \left(-b x+a x^4\right)^{3/4}}{63 b x^6}+\frac{2}{3} a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)+\frac{2}{3} a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)",1,"(-4*(3*b + 4*a*x^3)*(-(b*x) + a*x^4)^(3/4))/(63*b*x^6) + (2*a^(3/4)*ArcTan[(a^(1/4)*(-(b*x) + a*x^4)^(3/4))/(-b + a*x^3)])/3 + (2*a^(3/4)*ArcTanh[(a^(1/4)*(-(b*x) + a*x^4)^(3/4))/(-b + a*x^3)])/3","A",1
1739,1,117,117,0.7318574,"\int \frac{b+a x^6}{x^6 \sqrt[4]{-b x+a x^4}} \, dx","IntegrateAlgebraic[(b + a*x^6)/(x^6*(-(b*x) + a*x^4)^(1/4)),x]","\frac{4 \left(3 b+4 a x^3\right) \left(-b x+a x^4\right)^{3/4}}{63 b x^6}+\frac{2}{3} a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)+\frac{2}{3} a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)","\frac{4 \left(3 b+4 a x^3\right) \left(-b x+a x^4\right)^{3/4}}{63 b x^6}+\frac{2}{3} a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)+\frac{2}{3} a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)",1,"(4*(3*b + 4*a*x^3)*(-(b*x) + a*x^4)^(3/4))/(63*b*x^6) + (2*a^(3/4)*ArcTan[(a^(1/4)*(-(b*x) + a*x^4)^(3/4))/(-b + a*x^3)])/3 + (2*a^(3/4)*ArcTanh[(a^(1/4)*(-(b*x) + a*x^4)^(3/4))/(-b + a*x^3)])/3","A",1
1740,1,117,117,0.4960376,"\int \frac{\sqrt[4]{x^2+x^4} \left(-1-x^4+x^8\right)}{-1+x^4} \, dx","IntegrateAlgebraic[((x^2 + x^4)^(1/4)*(-1 - x^4 + x^8))/(-1 + x^4),x]","\frac{1}{192} \sqrt[4]{x^2+x^4} \left(-7 x+4 x^3+32 x^5\right)-\frac{7}{128} \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^2+x^4}}\right)-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2^{3/4}}+\frac{7}{128} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^2+x^4}}\right)+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2^{3/4}}","\frac{1}{192} \sqrt[4]{x^2+x^4} \left(-7 x+4 x^3+32 x^5\right)-\frac{7}{128} \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^2+x^4}}\right)-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2^{3/4}}+\frac{7}{128} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^2+x^4}}\right)+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2^{3/4}}",1,"((x^2 + x^4)^(1/4)*(-7*x + 4*x^3 + 32*x^5))/192 - (7*ArcTan[x/(x^2 + x^4)^(1/4)])/128 - ArcTan[(2^(1/4)*x)/(x^2 + x^4)^(1/4)]/2^(3/4) + (7*ArcTanh[x/(x^2 + x^4)^(1/4)])/128 + ArcTanh[(2^(1/4)*x)/(x^2 + x^4)^(1/4)]/2^(3/4)","A",1
1741,1,117,117,0.8131022,"\int \frac{-b-a x^4+2 x^8}{\sqrt[4]{b+a x^4} \left(-b-a x^4+x^8\right)} \, dx","IntegrateAlgebraic[(-b - a*x^4 + 2*x^8)/((b + a*x^4)^(1/4)*(-b - a*x^4 + x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{1}{4} \text{RootSum}\left[b+a \text{$\#$1}^4-\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}^3+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^3}{-a+2 \text{$\#$1}^4}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{1}{4} \text{RootSum}\left[b+a \text{$\#$1}^4-\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}^3+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^3}{-a+2 \text{$\#$1}^4}\&\right]",1,"ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(1/4) + ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(1/4) + RootSum[b + a*#1^4 - #1^8 & , (-(Log[x]*#1^3) + Log[(b + a*x^4)^(1/4) - x*#1]*#1^3)/(-a + 2*#1^4) & ]/4","A",1
1742,1,117,117,0.0017607,"\int \frac{-b-a x^4+2 x^8}{\sqrt[4]{b+a x^4} \left(-b-a x^4+x^8\right)} \, dx","IntegrateAlgebraic[(-b - a*x^4 + 2*x^8)/((b + a*x^4)^(1/4)*(-b - a*x^4 + x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{1}{4} \text{RootSum}\left[b+a \text{$\#$1}^4-\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}^3+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^3}{-a+2 \text{$\#$1}^4}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{1}{4} \text{RootSum}\left[b+a \text{$\#$1}^4-\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}^3+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^3}{-a+2 \text{$\#$1}^4}\&\right]",1,"ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(1/4) + ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(1/4) + RootSum[b + a*#1^4 - #1^8 & , (-(Log[x]*#1^3) + Log[(b + a*x^4)^(1/4) - x*#1]*#1^3)/(-a + 2*#1^4) & ]/4","A",1
1743,-1,0,117,180.0180789,"\int \frac{\sqrt{c x^2-x \sqrt{-b x+a x^2}}}{x^3 \sqrt{-b x+a x^2}} \, dx","IntegrateAlgebraic[Sqrt[c*x^2 - x*Sqrt[-(b*x) + a*x^2]]/(x^3*Sqrt[-(b*x) + a*x^2]),x]","\text{\$Aborted}","-\frac{4 \left(3 b c+32 a c x-8 c^3 x\right) \sqrt{-x \left(-c x+\sqrt{-b x+a x^2}\right)}}{105 b^2 x^2}+\frac{4 \left(15 b+20 a x+4 c^2 x\right) \sqrt{-b x+a x^2} \sqrt{-x \left(-c x+\sqrt{-b x+a x^2}\right)}}{105 b^2 x^3}",1,"$Aborted","F",-1
1744,1,118,118,0.2809057,"\int \frac{b+x^3}{\sqrt[3]{a+x^3}} \, dx","IntegrateAlgebraic[(b + x^3)/(a + x^3)^(1/3),x]","\frac{1}{3} x \left(a+x^3\right)^{2/3}+\frac{1}{9} \left(-\sqrt{3} a+3 \sqrt{3} b\right) \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{a+x^3}}\right)+\frac{1}{9} (a-3 b) \log \left(-x+\sqrt[3]{a+x^3}\right)+\frac{1}{18} (-a+3 b) \log \left(x^2+x \sqrt[3]{a+x^3}+\left(a+x^3\right)^{2/3}\right)","\frac{1}{3} x \left(a+x^3\right)^{2/3}+\frac{1}{9} \left(-\sqrt{3} a+3 \sqrt{3} b\right) \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{a+x^3}}\right)+\frac{1}{9} (a-3 b) \log \left(-x+\sqrt[3]{a+x^3}\right)+\frac{1}{18} (-a+3 b) \log \left(x^2+x \sqrt[3]{a+x^3}+\left(a+x^3\right)^{2/3}\right)",1,"(x*(a + x^3)^(2/3))/3 + ((-(Sqrt[3]*a) + 3*Sqrt[3]*b)*ArcTan[(Sqrt[3]*x)/(x + 2*(a + x^3)^(1/3))])/9 + ((a - 3*b)*Log[-x + (a + x^3)^(1/3)])/9 + ((-a + 3*b)*Log[x^2 + x*(a + x^3)^(1/3) + (a + x^3)^(2/3)])/18","A",1
1745,1,118,118,0.3373811,"\int \frac{b+a x^2}{\sqrt[3]{x+x^3}} \, dx","IntegrateAlgebraic[(b + a*x^2)/(x + x^3)^(1/3),x]","\frac{1}{2} a \left(x+x^3\right)^{2/3}+\frac{1}{6} \left(-\sqrt{3} a+3 \sqrt{3} b\right) \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)+\frac{1}{6} (a-3 b) \log \left(-x+\sqrt[3]{x+x^3}\right)+\frac{1}{12} (-a+3 b) \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)","\frac{1}{2} a \left(x+x^3\right)^{2/3}+\frac{1}{6} \left(-\sqrt{3} a+3 \sqrt{3} b\right) \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)+\frac{1}{6} (a-3 b) \log \left(-x+\sqrt[3]{x+x^3}\right)+\frac{1}{12} (-a+3 b) \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)",1,"(a*(x + x^3)^(2/3))/2 + ((-(Sqrt[3]*a) + 3*Sqrt[3]*b)*ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^3)^(1/3))])/6 + ((a - 3*b)*Log[-x + (x + x^3)^(1/3)])/6 + ((-a + 3*b)*Log[x^2 + x*(x + x^3)^(1/3) + (x + x^3)^(2/3)])/12","A",1
1746,1,118,118,0.1964573,"\int \sqrt[3]{-x^2+x^3} \, dx","IntegrateAlgebraic[(-x^2 + x^3)^(1/3),x]","\frac{1}{6} (-1+3 x) \sqrt[3]{-x^2+x^3}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)}{3 \sqrt{3}}+\frac{1}{9} \log \left(-x+\sqrt[3]{-x^2+x^3}\right)-\frac{1}{18} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)","\frac{1}{6} (-1+3 x) \sqrt[3]{-x^2+x^3}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)}{3 \sqrt{3}}+\frac{1}{9} \log \left(-x+\sqrt[3]{-x^2+x^3}\right)-\frac{1}{18} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)",1,"((-1 + 3*x)*(-x^2 + x^3)^(1/3))/6 + ArcTan[(Sqrt[3]*x)/(x + 2*(-x^2 + x^3)^(1/3))]/(3*Sqrt[3]) + Log[-x + (-x^2 + x^3)^(1/3)]/9 - Log[x^2 + x*(-x^2 + x^3)^(1/3) + (-x^2 + x^3)^(2/3)]/18","A",1
1747,1,118,118,0.3520605,"\int x^2 \sqrt[3]{x^2+x^3} \, dx","IntegrateAlgebraic[x^2*(x^2 + x^3)^(1/3),x]","\frac{1}{324} \sqrt[3]{x^2+x^3} \left(20-12 x+9 x^2+81 x^3\right)+\frac{10 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^3}}\right)}{81 \sqrt{3}}+\frac{10}{243} \log \left(-x+\sqrt[3]{x^2+x^3}\right)-\frac{5}{243} \log \left(x^2+x \sqrt[3]{x^2+x^3}+\left(x^2+x^3\right)^{2/3}\right)","\frac{1}{324} \sqrt[3]{x^2+x^3} \left(20-12 x+9 x^2+81 x^3\right)+\frac{10 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^3}}\right)}{81 \sqrt{3}}+\frac{10}{243} \log \left(-x+\sqrt[3]{x^2+x^3}\right)-\frac{5}{243} \log \left(x^2+x \sqrt[3]{x^2+x^3}+\left(x^2+x^3\right)^{2/3}\right)",1,"((x^2 + x^3)^(1/3)*(20 - 12*x + 9*x^2 + 81*x^3))/324 + (10*ArcTan[(Sqrt[3]*x)/(x + 2*(x^2 + x^3)^(1/3))])/(81*Sqrt[3]) + (10*Log[-x + (x^2 + x^3)^(1/3)])/243 - (5*Log[x^2 + x*(x^2 + x^3)^(1/3) + (x^2 + x^3)^(2/3)])/243","A",1
1748,1,118,118,0.3160502,"\int \frac{x^4}{\left(1+x^4\right)^2 \sqrt[4]{x^2+x^4}} \, dx","IntegrateAlgebraic[x^4/((1 + x^4)^2*(x^2 + x^4)^(1/4)),x]","\frac{\left(-1+x^2\right) \left(x^2+x^4\right)^{3/4}}{8 x \left(1+x^4\right)}+\frac{1}{64} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 \log (x)+2 \log \left(\sqrt[4]{x^2+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}+\text{$\#$1}^5}\&\right]","\frac{\left(-1+x^2\right) \left(x^2+x^4\right)^{3/4}}{8 x \left(1+x^4\right)}+\frac{1}{64} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 \log (x)+2 \log \left(\sqrt[4]{x^2+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}+\text{$\#$1}^5}\&\right]",1,"((-1 + x^2)*(x^2 + x^4)^(3/4))/(8*x*(1 + x^4)) + RootSum[2 - 2*#1^4 + #1^8 & , (-2*Log[x] + 2*Log[(x^2 + x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(x^2 + x^4)^(1/4) - x*#1]*#1^4)/(-#1 + #1^5) & ]/64","A",1
1749,1,118,118,0.0016115,"\int \frac{x^4}{\left(1+x^4\right)^2 \sqrt[4]{x^2+x^4}} \, dx","IntegrateAlgebraic[x^4/((1 + x^4)^2*(x^2 + x^4)^(1/4)),x]","\frac{\left(-1+x^2\right) \left(x^2+x^4\right)^{3/4}}{8 x \left(1+x^4\right)}+\frac{1}{64} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 \log (x)+2 \log \left(\sqrt[4]{x^2+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}+\text{$\#$1}^5}\&\right]","\frac{\left(-1+x^2\right) \left(x^2+x^4\right)^{3/4}}{8 x \left(1+x^4\right)}+\frac{1}{64} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 \log (x)+2 \log \left(\sqrt[4]{x^2+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}+\text{$\#$1}^5}\&\right]",1,"((-1 + x^2)*(x^2 + x^4)^(3/4))/(8*x*(1 + x^4)) + RootSum[2 - 2*#1^4 + #1^8 & , (-2*Log[x] + 2*Log[(x^2 + x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(x^2 + x^4)^(1/4) - x*#1]*#1^4)/(-#1 + #1^5) & ]/64","A",1
1750,1,118,118,2.0804501,"\int \frac{\left(-3+x^4\right) \left(1-x^3+x^4\right) \left(1+x^3+x^4\right)^{2/3}}{x^6 \left(1+x^4\right)} \, dx","IntegrateAlgebraic[((-3 + x^4)*(1 - x^3 + x^4)*(1 + x^3 + x^4)^(2/3))/(x^6*(1 + x^4)),x]","\frac{3 \left(1+x^3+x^4\right)^{2/3} \left(2-3 x^3+2 x^4\right)}{10 x^5}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3+x^4}}\right)-\log \left(-x+\sqrt[3]{1+x^3+x^4}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{1+x^3+x^4}+\left(1+x^3+x^4\right)^{2/3}\right)","\frac{3 \left(1+x^3+x^4\right)^{2/3} \left(2-3 x^3+2 x^4\right)}{10 x^5}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3+x^4}}\right)-\log \left(-x+\sqrt[3]{1+x^3+x^4}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{1+x^3+x^4}+\left(1+x^3+x^4\right)^{2/3}\right)",1,"(3*(1 + x^3 + x^4)^(2/3)*(2 - 3*x^3 + 2*x^4))/(10*x^5) + Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^3 + x^4)^(1/3))] - Log[-x + (1 + x^3 + x^4)^(1/3)] + Log[x^2 + x*(1 + x^3 + x^4)^(1/3) + (1 + x^3 + x^4)^(2/3)]/2","A",1
1751,1,118,118,0.5256644,"\int \frac{-b+2 a x^4}{\left(b+a x^4\right) \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[(-b + 2*a*x^4)/((b + a*x^4)*(b*x^2 + a*x^4)^(1/4)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{3}{4} \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{3}{4} \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"(2*ArcTan[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/a^(1/4) + (2*ArcTanh[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/a^(1/4) + (3*RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (-Log[x] + Log[(b*x^2 + a*x^4)^(1/4) - x*#1])/#1 & ])/4","A",1
1752,1,118,118,0.00184,"\int \frac{-b+2 a x^4}{\left(b+a x^4\right) \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[(-b + 2*a*x^4)/((b + a*x^4)*(b*x^2 + a*x^4)^(1/4)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{3}{4} \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{3}{4} \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"(2*ArcTan[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/a^(1/4) + (2*ArcTanh[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/a^(1/4) + (3*RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (-Log[x] + Log[(b*x^2 + a*x^4)^(1/4) - x*#1])/#1 & ])/4","A",1
1753,1,118,118,0.4040536,"\int \frac{1+x^2}{\left(-1+x^2\right) \sqrt[3]{x+x^5}} \, dx","IntegrateAlgebraic[(1 + x^2)/((-1 + x^2)*(x + x^5)^(1/3)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x+x^5}}\right)}{2 \sqrt[3]{2}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x+x^5}\right)}{2 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x+x^5}+\sqrt[3]{2} \left(x+x^5\right)^{2/3}\right)}{4 \sqrt[3]{2}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x+x^5}}\right)}{2 \sqrt[3]{2}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x+x^5}\right)}{2 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x+x^5}+\sqrt[3]{2} \left(x+x^5\right)^{2/3}\right)}{4 \sqrt[3]{2}}",1,"-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(x + x^5)^(1/3))])/2^(1/3) + Log[-2*x + 2^(2/3)*(x + x^5)^(1/3)]/(2*2^(1/3)) - Log[2*x^2 + 2^(2/3)*x*(x + x^5)^(1/3) + 2^(1/3)*(x + x^5)^(2/3)]/(4*2^(1/3))","A",1
1754,1,118,118,0.4825168,"\int \frac{\left(-1+x^2\right) \sqrt[4]{x^2+x^6}}{x^2 \left(1+x^2\right)} \, dx","IntegrateAlgebraic[((-1 + x^2)*(x^2 + x^6)^(1/4))/(x^2*(1 + x^2)),x]","\frac{2 \sqrt[4]{x^2+x^6}}{x}-\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{\sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{\sqrt[4]{2}}","\frac{2 \sqrt[4]{x^2+x^6}}{x}-\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{\sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{\sqrt[4]{2}}",1,"(2*(x^2 + x^6)^(1/4))/x - ArcTan[(2^(3/4)*x*(x^2 + x^6)^(1/4))/(Sqrt[2]*x^2 - Sqrt[x^2 + x^6])]/2^(1/4) - ArcTanh[(x^2/2^(1/4) + Sqrt[x^2 + x^6]/2^(3/4))/(x*(x^2 + x^6)^(1/4))]/2^(1/4)","A",1
1755,1,118,118,0.0009428,"\int \frac{\left(-1+x^2\right) \sqrt[4]{x^2+x^6}}{x^2 \left(1+x^2\right)} \, dx","IntegrateAlgebraic[((-1 + x^2)*(x^2 + x^6)^(1/4))/(x^2*(1 + x^2)),x]","\frac{2 \sqrt[4]{x^2+x^6}}{x}-\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{\sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{\sqrt[4]{2}}","\frac{2 \sqrt[4]{x^2+x^6}}{x}-\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{\sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{\sqrt[4]{2}}",1,"(2*(x^2 + x^6)^(1/4))/x - ArcTan[(2^(3/4)*x*(x^2 + x^6)^(1/4))/(Sqrt[2]*x^2 - Sqrt[x^2 + x^6])]/2^(1/4) - ArcTanh[(x^2/2^(1/4) + Sqrt[x^2 + x^6]/2^(3/4))/(x*(x^2 + x^6)^(1/4))]/2^(1/4)","A",1
1756,1,118,118,0.258131,"\int \frac{\left(-1+x^3\right)^{2/3} \left(1-5 x^3+4 x^6\right)}{x^6 \left(-1+2 x^3\right)^2} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(1 - 5*x^3 + 4*x^6))/(x^6*(-1 + 2*x^3)^2),x]","\frac{\left(-1+x^3\right)^{2/3} \left(6-33 x^3+62 x^6\right)}{30 x^5 \left(-1+2 x^3\right)}+\frac{7 \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-1+x^3}}\right)}{3 \sqrt{3}}+\frac{7}{9} \log \left(x+\sqrt[3]{-1+x^3}\right)-\frac{7}{18} \log \left(x^2-x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)","\frac{\left(-1+x^3\right)^{2/3} \left(6-33 x^3+62 x^6\right)}{30 x^5 \left(-1+2 x^3\right)}+\frac{7 \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-1+x^3}}\right)}{3 \sqrt{3}}+\frac{7}{9} \log \left(x+\sqrt[3]{-1+x^3}\right)-\frac{7}{18} \log \left(x^2-x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)",1,"((-1 + x^3)^(2/3)*(6 - 33*x^3 + 62*x^6))/(30*x^5*(-1 + 2*x^3)) + (7*ArcTan[(Sqrt[3]*x)/(-x + 2*(-1 + x^3)^(1/3))])/(3*Sqrt[3]) + (7*Log[x + (-1 + x^3)^(1/3)])/9 - (7*Log[x^2 - x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)])/18","A",1
1757,1,103,118,0.1840253,"\int \frac{\sqrt{1+x} \sqrt{1+\sqrt{1+x}}}{x-\sqrt{1+x}} \, dx","IntegrateAlgebraic[(Sqrt[1 + x]*Sqrt[1 + Sqrt[1 + x]])/(x - Sqrt[1 + x]),x]","\frac{4}{3} \sqrt{1+\sqrt{1+x}} \left(4+\sqrt{1+x}\right)-\frac{4}{5} \left(5+2 \sqrt{5}\right) \tanh ^{-1}\left(\frac{1}{2} \left(-1+\sqrt{5}\right) \sqrt{1+\sqrt{1+x}}\right)-\frac{4}{5} \left(-5+2 \sqrt{5}\right) \tanh ^{-1}\left(\frac{1}{2} \left(1+\sqrt{5}\right) \sqrt{1+\sqrt{1+x}}\right)","\frac{16}{3} \sqrt{1+\sqrt{1+x}}+\frac{4}{3} \sqrt{1+x} \sqrt{1+\sqrt{1+x}}-\frac{4}{5} \left(-5+2 \sqrt{5}\right) \tanh ^{-1}\left(\frac{2 \sqrt{1+\sqrt{1+x}}}{-1+\sqrt{5}}\right)-\frac{4}{5} \left(5+2 \sqrt{5}\right) \tanh ^{-1}\left(\frac{2 \sqrt{1+\sqrt{1+x}}}{1+\sqrt{5}}\right)",1,"(4*Sqrt[1 + Sqrt[1 + x]]*(4 + Sqrt[1 + x]))/3 - (4*(5 + 2*Sqrt[5])*ArcTanh[((-1 + Sqrt[5])*Sqrt[1 + Sqrt[1 + x]])/2])/5 - (4*(-5 + 2*Sqrt[5])*ArcTanh[((1 + Sqrt[5])*Sqrt[1 + Sqrt[1 + x]])/2])/5","A",1
1758,1,126,118,0.271834,"\int \frac{\sqrt{b+a x}}{\sqrt{a b x+\sqrt{b+a x}}} \, dx","IntegrateAlgebraic[Sqrt[b + a*x]/Sqrt[a*b*x + Sqrt[b + a*x]],x]","\frac{\left(-3+2 b \sqrt{b+a x}\right) \sqrt{-b^2+\sqrt{b+a x}+b (b+a x)}}{2 a b^2}+\frac{\left(-3-4 b^3\right) \log \left(a b^2+2 a b^3 \sqrt{b+a x}-2 a b^{5/2} \sqrt{-b^2+\sqrt{b+a x}+b (b+a x)}\right)}{4 a b^{5/2}}","\frac{\left(-3+2 b \sqrt{b+a x}\right) \sqrt{-b^2+\sqrt{b+a x}+b (b+a x)}}{2 a b^2}+\frac{\left(-3-4 b^3\right) \log \left(-1-2 b \sqrt{b+a x}+2 \sqrt{b} \sqrt{-b^2+\sqrt{b+a x}+b (b+a x)}\right)}{4 a b^{5/2}}",1,"((-3 + 2*b*Sqrt[b + a*x])*Sqrt[-b^2 + Sqrt[b + a*x] + b*(b + a*x)])/(2*a*b^2) + ((-3 - 4*b^3)*Log[a*b^2 + 2*a*b^3*Sqrt[b + a*x] - 2*a*b^(5/2)*Sqrt[-b^2 + Sqrt[b + a*x] + b*(b + a*x)]])/(4*a*b^(5/2))","A",1
1759,1,118,118,0.2836285,"\int \frac{\sqrt{b+a^2 x^4}}{\sqrt{a x^2+\sqrt{b+a^2 x^4}}} \, dx","IntegrateAlgebraic[Sqrt[b + a^2*x^4]/Sqrt[a*x^2 + Sqrt[b + a^2*x^4]],x]","\frac{b x}{8 \left(a x^2+\sqrt{b+a^2 x^4}\right)^{3/2}}+\frac{1}{4} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\frac{5 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{b}}\right)}{8 \sqrt{2} \sqrt{a}}","\frac{b x}{8 \left(a x^2+\sqrt{b+a^2 x^4}\right)^{3/2}}+\frac{1}{4} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\frac{5 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{b}}\right)}{8 \sqrt{2} \sqrt{a}}",1,"(b*x)/(8*(a*x^2 + Sqrt[b + a^2*x^4])^(3/2)) + (x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/4 + (5*Sqrt[b]*ArcTan[(Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[b]])/(8*Sqrt[2]*Sqrt[a])","A",1
1760,1,118,118,0.4017841,"\int \frac{1}{(f+e x) \sqrt{d+\sqrt{c+\sqrt{b+a x}}}} \, dx","IntegrateAlgebraic[1/((f + e*x)*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]]),x]","\frac{\text{RootSum}\left[b e-c^2 e+2 c d^2 e-d^4 e-a f-4 c d e \text{$\#$1}^2+4 d^3 e \text{$\#$1}^2+2 c e \text{$\#$1}^4-6 d^2 e \text{$\#$1}^4+4 d e \text{$\#$1}^6-e \text{$\#$1}^8\&,\frac{\log \left(\sqrt{d+\sqrt{c+\sqrt{b+a x}}}-\text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{e}","\frac{\text{RootSum}\left[b e-c^2 e+2 c d^2 e-d^4 e-a f-4 c d e \text{$\#$1}^2+4 d^3 e \text{$\#$1}^2+2 c e \text{$\#$1}^4-6 d^2 e \text{$\#$1}^4+4 d e \text{$\#$1}^6-e \text{$\#$1}^8\&,\frac{\log \left(\sqrt{d+\sqrt{c+\sqrt{b+a x}}}-\text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{e}",1,"RootSum[b*e - c^2*e + 2*c*d^2*e - d^4*e - a*f - 4*c*d*e*#1^2 + 4*d^3*e*#1^2 + 2*c*e*#1^4 - 6*d^2*e*#1^4 + 4*d*e*#1^6 - e*#1^8 & , Log[Sqrt[d + Sqrt[c + Sqrt[b + a*x]]] - #1]/#1 & ]/e","A",1
1761,1,119,119,0.3914318,"\int \frac{-a b+(-a+2 b) x}{\sqrt[4]{x (-a+x) (-b+x)^2} \left(-b^2+(2 b-a d) x+(-1+d) x^2\right)} \, dx","IntegrateAlgebraic[(-(a*b) + (-a + 2*b)*x)/((x*(-a + x)*(-b + x)^2)^(1/4)*(-b^2 + (2*b - a*d)*x + (-1 + d)*x^2)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{b-x}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{b-x}\right)}{d^{3/4}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{b-x}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{b-x}\right)}{d^{3/4}}",1,"(2*ArcTan[(d^(1/4)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/4))/(b - x)])/d^(3/4) - (2*ArcTanh[(d^(1/4)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/4))/(b - x)])/d^(3/4)","A",1
1762,1,119,119,0.3613187,"\int x^{13} \sqrt[3]{-1+x^3} \, dx","IntegrateAlgebraic[x^13*(-1 + x^3)^(1/3),x]","\frac{\sqrt[3]{-1+x^3} \left(-220 x^2-132 x^5-99 x^8-81 x^{11}+972 x^{14}\right)}{14580}+\frac{22 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{729 \sqrt{3}}+\frac{22 \log \left(-x+\sqrt[3]{-1+x^3}\right)}{2187}-\frac{11 \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)}{2187}","\frac{\sqrt[3]{-1+x^3} \left(-220 x^2-132 x^5-99 x^8-81 x^{11}+972 x^{14}\right)}{14580}+\frac{22 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{729 \sqrt{3}}+\frac{22 \log \left(-x+\sqrt[3]{-1+x^3}\right)}{2187}-\frac{11 \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)}{2187}",1,"((-1 + x^3)^(1/3)*(-220*x^2 - 132*x^5 - 99*x^8 - 81*x^11 + 972*x^14))/14580 + (22*ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^3)^(1/3))])/(729*Sqrt[3]) + (22*Log[-x + (-1 + x^3)^(1/3)])/2187 - (11*Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)])/2187","A",1
1763,1,119,119,0.4207872,"\int \frac{a b^2-2 (2 a-b) b x+(3 a-2 b) x^2}{\sqrt[4]{x (-a+x) (-b+x)^2} \left(a^3+\left(-3 a^2+b^2 d\right) x+(3 a-2 b d) x^2+(-1+d) x^3\right)} \, dx","IntegrateAlgebraic[(a*b^2 - 2*(2*a - b)*b*x + (3*a - 2*b)*x^2)/((x*(-a + x)*(-b + x)^2)^(1/4)*(a^3 + (-3*a^2 + b^2*d)*x + (3*a - 2*b*d)*x^2 + (-1 + d)*x^3)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{a-x}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{a-x}\right)}{d^{3/4}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{a-x}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{a-x}\right)}{d^{3/4}}",1,"(2*ArcTan[(d^(1/4)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/4))/(a - x)])/d^(3/4) - (2*ArcTanh[(d^(1/4)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/4))/(a - x)])/d^(3/4)","A",1
1764,1,119,119,0.2849829,"\int \frac{\sqrt[3]{x+2 x^3} \left(-1+x^4\right)}{x^4 \left(2-x^2+x^4\right)} \, dx","IntegrateAlgebraic[((x + 2*x^3)^(1/3)*(-1 + x^4))/(x^4*(2 - x^2 + x^4)),x]","\frac{3 \left(1+4 x^2\right) \sqrt[3]{x+2 x^3}}{16 x^3}+\frac{1}{8} \text{RootSum}\left[11-9 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{11 \log (x)-11 \log \left(\sqrt[3]{x+2 x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{x+2 x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-9 \text{$\#$1}^2+4 \text{$\#$1}^5}\&\right]","\frac{3 \left(1+4 x^2\right) \sqrt[3]{x+2 x^3}}{16 x^3}+\frac{1}{8} \text{RootSum}\left[11-9 \text{$\#$1}^3+2 \text{$\#$1}^6\&,\frac{11 \log (x)-11 \log \left(\sqrt[3]{x+2 x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{x+2 x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-9 \text{$\#$1}^2+4 \text{$\#$1}^5}\&\right]",1,"(3*(1 + 4*x^2)*(x + 2*x^3)^(1/3))/(16*x^3) + RootSum[11 - 9*#1^3 + 2*#1^6 & , (11*Log[x] - 11*Log[(x + 2*x^3)^(1/3) - x*#1] + Log[x]*#1^3 - Log[(x + 2*x^3)^(1/3) - x*#1]*#1^3)/(-9*#1^2 + 4*#1^5) & ]/8","A",1
1765,1,119,119,0.5119633,"\int \frac{b+2 a x^4}{\left(-b+a x^4\right) \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[(b + 2*a*x^4)/((-b + a*x^4)*(b*x^2 + a*x^4)^(1/4)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{3}{4} \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{3}{4} \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"(2*ArcTan[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/a^(1/4) + (2*ArcTanh[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/a^(1/4) + (3*RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (-Log[x] + Log[(b*x^2 + a*x^4)^(1/4) - x*#1])/#1 & ])/4","A",1
1766,1,119,119,0.0020607,"\int \frac{b+2 a x^4}{\left(-b+a x^4\right) \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[(b + 2*a*x^4)/((-b + a*x^4)*(b*x^2 + a*x^4)^(1/4)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{3}{4} \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{3}{4} \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"(2*ArcTan[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/a^(1/4) + (2*ArcTanh[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/a^(1/4) + (3*RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (-Log[x] + Log[(b*x^2 + a*x^4)^(1/4) - x*#1])/#1 & ])/4","A",1
1767,1,119,119,0.3292489,"\int \frac{\left(1+x^6\right) \left(-1-x^3+x^6\right)^{2/3}}{x^3 \left(-1+x^6\right)} \, dx","IntegrateAlgebraic[((1 + x^6)*(-1 - x^3 + x^6)^(2/3))/(x^3*(-1 + x^6)),x]","\frac{\left(-1-x^3+x^6\right)^{2/3}}{2 x^2}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-1-x^3+x^6}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(x+\sqrt[3]{-1-x^3+x^6}\right)-\frac{1}{6} \log \left(x^2-x \sqrt[3]{-1-x^3+x^6}+\left(-1-x^3+x^6\right)^{2/3}\right)","\frac{\left(-1-x^3+x^6\right)^{2/3}}{2 x^2}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-1-x^3+x^6}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(x+\sqrt[3]{-1-x^3+x^6}\right)-\frac{1}{6} \log \left(x^2-x \sqrt[3]{-1-x^3+x^6}+\left(-1-x^3+x^6\right)^{2/3}\right)",1,"(-1 - x^3 + x^6)^(2/3)/(2*x^2) + ArcTan[(Sqrt[3]*x)/(-x + 2*(-1 - x^3 + x^6)^(1/3))]/Sqrt[3] + Log[x + (-1 - x^3 + x^6)^(1/3)]/3 - Log[x^2 - x*(-1 - x^3 + x^6)^(1/3) + (-1 - x^3 + x^6)^(2/3)]/6","A",1
1768,1,119,119,1.0344935,"\int \frac{\left(4+x^6\right) \sqrt[4]{-2-x^4+x^6}}{x^2 \left(-2+x^6\right)} \, dx","IntegrateAlgebraic[((4 + x^6)*(-2 - x^4 + x^6)^(1/4))/(x^2*(-2 + x^6)),x]","\frac{2 \sqrt[4]{-2-x^4+x^6}}{x}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-2-x^4+x^6}}{-x^2+\sqrt{-2-x^4+x^6}}\right)}{\sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-2-x^4+x^6}}{x^2+\sqrt{-2-x^4+x^6}}\right)}{\sqrt{2}}","\frac{2 \sqrt[4]{-2-x^4+x^6}}{x}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-2-x^4+x^6}}{-x^2+\sqrt{-2-x^4+x^6}}\right)}{\sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-2-x^4+x^6}}{x^2+\sqrt{-2-x^4+x^6}}\right)}{\sqrt{2}}",1,"(2*(-2 - x^4 + x^6)^(1/4))/x + ArcTan[(Sqrt[2]*x*(-2 - x^4 + x^6)^(1/4))/(-x^2 + Sqrt[-2 - x^4 + x^6])]/Sqrt[2] - ArcTanh[(Sqrt[2]*x*(-2 - x^4 + x^6)^(1/4))/(x^2 + Sqrt[-2 - x^4 + x^6])]/Sqrt[2]","A",1
1769,1,119,119,0.5759657,"\int \sqrt[4]{b x^7+a x^8} \, dx","IntegrateAlgebraic[(b*x^7 + a*x^8)^(1/4),x]","\frac{\left(-7 b^2+4 a b x+32 a^2 x^2\right) \sqrt[4]{b x^7+a x^8}}{96 a^2 x}+\frac{7 b^3 \tan ^{-1}\left(\frac{\sqrt[4]{b x^7+a x^8}}{\sqrt[4]{a} x^2}\right)}{64 a^{11/4}}+\frac{7 b^3 \tanh ^{-1}\left(\frac{\sqrt[4]{b x^7+a x^8}}{\sqrt[4]{a} x^2}\right)}{64 a^{11/4}}","\frac{\left(-7 b^2+4 a b x+32 a^2 x^2\right) \sqrt[4]{b x^7+a x^8}}{96 a^2 x}+\frac{7 b^3 \tan ^{-1}\left(\frac{\sqrt[4]{b x^7+a x^8}}{\sqrt[4]{a} x^2}\right)}{64 a^{11/4}}+\frac{7 b^3 \tanh ^{-1}\left(\frac{\sqrt[4]{b x^7+a x^8}}{\sqrt[4]{a} x^2}\right)}{64 a^{11/4}}",1,"((-7*b^2 + 4*a*b*x + 32*a^2*x^2)*(b*x^7 + a*x^8)^(1/4))/(96*a^2*x) + (7*b^3*ArcTan[(b*x^7 + a*x^8)^(1/4)/(a^(1/4)*x^2)])/(64*a^(11/4)) + (7*b^3*ArcTanh[(b*x^7 + a*x^8)^(1/4)/(a^(1/4)*x^2)])/(64*a^(11/4))","A",1
1770,1,119,119,2.4515561,"\int \frac{-1+x^{10}}{\sqrt{1+x^4} \left(1+x^{10}\right)} \, dx","IntegrateAlgebraic[(-1 + x^10)/(Sqrt[1 + x^4]*(1 + x^10)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{5 \sqrt{2}}-\frac{1}{5} \sqrt{2 \left(1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt{1+x^4}}\right)-\frac{1}{5} \sqrt{2 \left(-1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt{1+x^4}}\right)","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{5 \sqrt{2}}-\frac{1}{5} \sqrt{2 \left(1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt{1+x^4}}\right)-\frac{1}{5} \sqrt{2 \left(-1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt{1+x^4}}\right)",1,"-1/5*ArcTan[(Sqrt[2]*x)/Sqrt[1 + x^4]]/Sqrt[2] - (Sqrt[2*(1 + Sqrt[5])]*ArcTan[(Sqrt[-1/2 + Sqrt[5]/2]*x)/Sqrt[1 + x^4]])/5 - (Sqrt[2*(-1 + Sqrt[5])]*ArcTanh[(Sqrt[1/2 + Sqrt[5]/2]*x)/Sqrt[1 + x^4]])/5","A",1
1771,1,119,119,2.3080388,"\int \frac{1+x^{10}}{\sqrt{1+x^4} \left(-1+x^{10}\right)} \, dx","IntegrateAlgebraic[(1 + x^10)/(Sqrt[1 + x^4]*(-1 + x^10)),x]","-\frac{1}{5} \sqrt{2 \left(-1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt{1+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{5 \sqrt{2}}-\frac{1}{5} \sqrt{2 \left(1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt{1+x^4}}\right)","-\frac{1}{5} \sqrt{2 \left(-1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt{1+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{5 \sqrt{2}}-\frac{1}{5} \sqrt{2 \left(1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt{1+x^4}}\right)",1,"-1/5*(Sqrt[2*(-1 + Sqrt[5])]*ArcTan[(Sqrt[1/2 + Sqrt[5]/2]*x)/Sqrt[1 + x^4]]) - ArcTanh[(Sqrt[2]*x)/Sqrt[1 + x^4]]/(5*Sqrt[2]) - (Sqrt[2*(1 + Sqrt[5])]*ArcTanh[(Sqrt[-1/2 + Sqrt[5]/2]*x)/Sqrt[1 + x^4]])/5","A",1
1772,1,119,119,0.6848647,"\int \frac{1+x^{16}}{\sqrt{1+x^4} \left(-1+x^{16}\right)} \, dx","IntegrateAlgebraic[(1 + x^16)/(Sqrt[1 + x^4]*(-1 + x^16)),x]","-\frac{x}{4 \sqrt{1+x^4}}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{1+x^4}}\right)}{4 \sqrt[4]{2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{8 \sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{1+x^4}}\right)}{4 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{8 \sqrt{2}}","-\frac{x}{4 \sqrt{1+x^4}}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{1+x^4}}\right)}{4 \sqrt[4]{2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{8 \sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{1+x^4}}\right)}{4 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+x^4}}\right)}{8 \sqrt{2}}",1,"-1/4*x/Sqrt[1 + x^4] - ArcTan[(2^(1/4)*x)/Sqrt[1 + x^4]]/(4*2^(1/4)) - ArcTan[(Sqrt[2]*x)/Sqrt[1 + x^4]]/(8*Sqrt[2]) - ArcTanh[(2^(1/4)*x)/Sqrt[1 + x^4]]/(4*2^(1/4)) - ArcTanh[(Sqrt[2]*x)/Sqrt[1 + x^4]]/(8*Sqrt[2])","A",1
1773,1,108,119,0.1590789,"\int \frac{\sqrt{b+a x} \sqrt{c+\sqrt{b+a x}}}{1-\sqrt{b+a x}} \, dx","IntegrateAlgebraic[(Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]])/(1 - Sqrt[b + a*x]),x]","\frac{4 \sqrt{c+\sqrt{b+a x}} \left(-15-5 c+2 c^2-5 \sqrt{b+a x}-c \sqrt{b+a x}-3 (b+a x)\right)}{15 a}-\frac{4 \sqrt{-1-c} \tan ^{-1}\left(\frac{\sqrt{-1-c} \sqrt{c+\sqrt{b+a x}}}{1+c}\right)}{a}","-\frac{4 (5+c) \sqrt{b+a x} \sqrt{c+\sqrt{b+a x}}}{15 a}-\frac{4 \left(15+3 b+5 c-2 c^2+3 a x\right) \sqrt{c+\sqrt{b+a x}}}{15 a}-\frac{4 \sqrt{-1-c} \tan ^{-1}\left(\frac{\sqrt{-1-c} \sqrt{c+\sqrt{b+a x}}}{1+c}\right)}{a}",1,"(4*Sqrt[c + Sqrt[b + a*x]]*(-15 - 5*c + 2*c^2 - 5*Sqrt[b + a*x] - c*Sqrt[b + a*x] - 3*(b + a*x)))/(15*a) - (4*Sqrt[-1 - c]*ArcTan[(Sqrt[-1 - c]*Sqrt[c + Sqrt[b + a*x]])/(1 + c)])/a","A",1
1774,1,119,119,1.575643,"\int x \sqrt{-1+x^2} \sqrt{x^2+x \sqrt{-1+x^2}} \, dx","IntegrateAlgebraic[x*Sqrt[-1 + x^2]*Sqrt[x^2 + x*Sqrt[-1 + x^2]],x]","\frac{1}{192} \sqrt{-1+x^2} \left(-39+56 x^2\right) \sqrt{x^2+x \sqrt{-1+x^2}}+\frac{1}{192} \left(13 x-8 x^3\right) \sqrt{x^2+x \sqrt{-1+x^2}}+\frac{13 \log \left(x+\sqrt{-1+x^2}-\sqrt{2} \sqrt{x^2+x \sqrt{-1+x^2}}\right)}{64 \sqrt{2}}","\frac{1}{192} \sqrt{-1+x^2} \left(-39+56 x^2\right) \sqrt{x^2+x \sqrt{-1+x^2}}+\frac{1}{192} \left(13 x-8 x^3\right) \sqrt{x^2+x \sqrt{-1+x^2}}+\frac{13 \log \left(x+\sqrt{-1+x^2}-\sqrt{2} \sqrt{x^2+x \sqrt{-1+x^2}}\right)}{64 \sqrt{2}}",1,"(Sqrt[-1 + x^2]*(-39 + 56*x^2)*Sqrt[x^2 + x*Sqrt[-1 + x^2]])/192 + ((13*x - 8*x^3)*Sqrt[x^2 + x*Sqrt[-1 + x^2]])/192 + (13*Log[x + Sqrt[-1 + x^2] - Sqrt[2]*Sqrt[x^2 + x*Sqrt[-1 + x^2]]])/(64*Sqrt[2])","A",1
1775,1,119,119,3.578374,"\int \frac{\left(-q+p x^2\right) \sqrt{q^2+p^2 x^4} \left(b x^3+a \left(q+p x^2\right)^3\right)}{x^6} \, dx","IntegrateAlgebraic[((-q + p*x^2)*Sqrt[q^2 + p^2*x^4]*(b*x^3 + a*(q + p*x^2)^3))/x^6,x]","\frac{\sqrt{q^2+p^2 x^4} \left(6 a q^4+20 a p q^3 x^2+15 b q x^3+12 a p^2 q^2 x^4+15 b p x^5+20 a p^3 q x^6+6 a p^4 x^8\right)}{30 x^5}+b p q \log (x)-b p q \log \left(q+p x^2+\sqrt{q^2+p^2 x^4}\right)","\frac{\sqrt{q^2+p^2 x^4} \left(6 a q^4+20 a p q^3 x^2+15 b q x^3+12 a p^2 q^2 x^4+15 b p x^5+20 a p^3 q x^6+6 a p^4 x^8\right)}{30 x^5}+b p q \log (x)-b p q \log \left(q+p x^2+\sqrt{q^2+p^2 x^4}\right)",1,"(Sqrt[q^2 + p^2*x^4]*(6*a*q^4 + 20*a*p*q^3*x^2 + 15*b*q*x^3 + 12*a*p^2*q^2*x^4 + 15*b*p*x^5 + 20*a*p^3*q*x^6 + 6*a*p^4*x^8))/(30*x^5) + b*p*q*Log[x] - b*p*q*Log[q + p*x^2 + Sqrt[q^2 + p^2*x^4]]","A",1
1776,1,145,119,0.4413611,"\int \frac{\left(-1+x^2\right) \sqrt{x^2+\sqrt{1+x^4}}}{\sqrt{1+x^4}} \, dx","IntegrateAlgebraic[((-1 + x^2)*Sqrt[x^2 + Sqrt[1 + x^4]])/Sqrt[1 + x^4],x]","\frac{1}{2} x \sqrt{x^2+\sqrt{1+x^4}}-\frac{\tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^4}}{\sqrt{2}}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)}{\sqrt{2}}-\sqrt{2} \tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^4}}{\sqrt{2}}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)","\frac{1}{2} x \sqrt{x^2+\sqrt{1+x^4}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)}{\sqrt{2}}-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)",1,"(x*Sqrt[x^2 + Sqrt[1 + x^4]])/2 - ArcTan[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])]/Sqrt[2] - Sqrt[2]*ArcTanh[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])]","A",1
1777,1,99,119,0.140713,"\int \frac{x \sqrt{1+x}}{x+\sqrt{1+\sqrt{1+x}}} \, dx","IntegrateAlgebraic[(x*Sqrt[1 + x])/(x + Sqrt[1 + Sqrt[1 + x]]),x]","-\frac{4}{3} \left(1+\sqrt{1+x}\right)^{3/2}+\frac{2}{3} \left(1+(1+x)^{3/2}\right)-\frac{2}{5} \left(-5+\sqrt{5}\right) \log \left(-1+\sqrt{5}-2 \sqrt{1+\sqrt{1+x}}\right)+\frac{2}{5} \left(5+\sqrt{5}\right) \log \left(1+\sqrt{5}+2 \sqrt{1+\sqrt{1+x}}\right)","-\frac{4}{3} \sqrt{1+\sqrt{1+x}}+\sqrt{1+x} \left(\frac{2 (1+x)}{3}-\frac{4}{3} \sqrt{1+\sqrt{1+x}}\right)-\frac{2}{5} \left(-5+\sqrt{5}\right) \log \left(-1+\sqrt{5}-2 \sqrt{1+\sqrt{1+x}}\right)+\frac{2}{5} \left(5+\sqrt{5}\right) \log \left(1+\sqrt{5}+2 \sqrt{1+\sqrt{1+x}}\right)",1,"(-4*(1 + Sqrt[1 + x])^(3/2))/3 + (2*(1 + (1 + x)^(3/2)))/3 - (2*(-5 + Sqrt[5])*Log[-1 + Sqrt[5] - 2*Sqrt[1 + Sqrt[1 + x]]])/5 + (2*(5 + Sqrt[5])*Log[1 + Sqrt[5] + 2*Sqrt[1 + Sqrt[1 + x]]])/5","A",1
1778,1,120,120,0.3115811,"\int \frac{\sqrt[3]{-x+x^3} \left(8-10 x^2+x^4\right)}{x^4 \left(4-2 x^2+x^4\right)} \, dx","IntegrateAlgebraic[((-x + x^3)^(1/3)*(8 - 10*x^2 + x^4))/(x^4*(4 - 2*x^2 + x^4)),x]","\frac{3 \left(-1+4 x^2\right) \sqrt[3]{-x+x^3}}{4 x^3}-\frac{1}{8} \text{RootSum}\left[3-6 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-9 \log (x)+9 \log \left(\sqrt[3]{-x+x^3}-x \text{$\#$1}\right)+4 \log (x) \text{$\#$1}^3-4 \log \left(\sqrt[3]{-x+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-3 \text{$\#$1}^2+4 \text{$\#$1}^5}\&\right]","\frac{3 \left(-1+4 x^2\right) \sqrt[3]{-x+x^3}}{4 x^3}-\frac{1}{8} \text{RootSum}\left[3-6 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-9 \log (x)+9 \log \left(\sqrt[3]{-x+x^3}-x \text{$\#$1}\right)+4 \log (x) \text{$\#$1}^3-4 \log \left(\sqrt[3]{-x+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-3 \text{$\#$1}^2+4 \text{$\#$1}^5}\&\right]",1,"(3*(-1 + 4*x^2)*(-x + x^3)^(1/3))/(4*x^3) - RootSum[3 - 6*#1^3 + 4*#1^6 & , (-9*Log[x] + 9*Log[(-x + x^3)^(1/3) - x*#1] + 4*Log[x]*#1^3 - 4*Log[(-x + x^3)^(1/3) - x*#1]*#1^3)/(-3*#1^2 + 4*#1^5) & ]/8","A",1
1779,1,120,120,0.0014618,"\int \frac{\sqrt[3]{-x+x^3} \left(8-10 x^2+x^4\right)}{x^4 \left(4-2 x^2+x^4\right)} \, dx","IntegrateAlgebraic[((-x + x^3)^(1/3)*(8 - 10*x^2 + x^4))/(x^4*(4 - 2*x^2 + x^4)),x]","\frac{3 \left(-1+4 x^2\right) \sqrt[3]{-x+x^3}}{4 x^3}-\frac{1}{8} \text{RootSum}\left[3-6 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-9 \log (x)+9 \log \left(\sqrt[3]{-x+x^3}-x \text{$\#$1}\right)+4 \log (x) \text{$\#$1}^3-4 \log \left(\sqrt[3]{-x+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-3 \text{$\#$1}^2+4 \text{$\#$1}^5}\&\right]","\frac{3 \left(-1+4 x^2\right) \sqrt[3]{-x+x^3}}{4 x^3}-\frac{1}{8} \text{RootSum}\left[3-6 \text{$\#$1}^3+4 \text{$\#$1}^6\&,\frac{-9 \log (x)+9 \log \left(\sqrt[3]{-x+x^3}-x \text{$\#$1}\right)+4 \log (x) \text{$\#$1}^3-4 \log \left(\sqrt[3]{-x+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-3 \text{$\#$1}^2+4 \text{$\#$1}^5}\&\right]",1,"(3*(-1 + 4*x^2)*(-x + x^3)^(1/3))/(4*x^3) - RootSum[3 - 6*#1^3 + 4*#1^6 & , (-9*Log[x] + 9*Log[(-x + x^3)^(1/3) - x*#1] + 4*Log[x]*#1^3 - 4*Log[(-x + x^3)^(1/3) - x*#1]*#1^3)/(-3*#1^2 + 4*#1^5) & ]/8","A",1
1780,1,120,120,0.2942886,"\int \frac{-1+x^2}{\left(1+x^2\right) \sqrt{1-x-x^2+x^3+x^4}} \, dx","IntegrateAlgebraic[(-1 + x^2)/((1 + x^2)*Sqrt[1 - x - x^2 + x^3 + x^4]),x]","\frac{1}{2} \text{RootSum}\left[5-16 \text{$\#$1}+14 \text{$\#$1}^2+\text{$\#$1}^4\&,\frac{\log (x)-\log \left(1-x^2+\sqrt{1-x-x^2+x^3+x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^2+\log \left(1-x^2+\sqrt{1-x-x^2+x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^2}{-4+7 \text{$\#$1}+\text{$\#$1}^3}\&\right]","\frac{1}{2} \text{RootSum}\left[5-16 \text{$\#$1}+14 \text{$\#$1}^2+\text{$\#$1}^4\&,\frac{\log (x)-\log \left(1-x^2+\sqrt{1-x-x^2+x^3+x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^2+\log \left(1-x^2+\sqrt{1-x-x^2+x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^2}{-4+7 \text{$\#$1}+\text{$\#$1}^3}\&\right]",1,"RootSum[5 - 16*#1 + 14*#1^2 + #1^4 & , (Log[x] - Log[1 - x^2 + Sqrt[1 - x - x^2 + x^3 + x^4] - x*#1] - Log[x]*#1^2 + Log[1 - x^2 + Sqrt[1 - x - x^2 + x^3 + x^4] - x*#1]*#1^2)/(-4 + 7*#1 + #1^3) & ]/2","A",1
1781,1,120,120,0.0014698,"\int \frac{-1+x^2}{\left(1+x^2\right) \sqrt{1-x-x^2+x^3+x^4}} \, dx","IntegrateAlgebraic[(-1 + x^2)/((1 + x^2)*Sqrt[1 - x - x^2 + x^3 + x^4]),x]","\frac{1}{2} \text{RootSum}\left[5-16 \text{$\#$1}+14 \text{$\#$1}^2+\text{$\#$1}^4\&,\frac{\log (x)-\log \left(1-x^2+\sqrt{1-x-x^2+x^3+x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^2+\log \left(1-x^2+\sqrt{1-x-x^2+x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^2}{-4+7 \text{$\#$1}+\text{$\#$1}^3}\&\right]","\frac{1}{2} \text{RootSum}\left[5-16 \text{$\#$1}+14 \text{$\#$1}^2+\text{$\#$1}^4\&,\frac{\log (x)-\log \left(1-x^2+\sqrt{1-x-x^2+x^3+x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^2+\log \left(1-x^2+\sqrt{1-x-x^2+x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^2}{-4+7 \text{$\#$1}+\text{$\#$1}^3}\&\right]",1,"RootSum[5 - 16*#1 + 14*#1^2 + #1^4 & , (Log[x] - Log[1 - x^2 + Sqrt[1 - x - x^2 + x^3 + x^4] - x*#1] - Log[x]*#1^2 + Log[1 - x^2 + Sqrt[1 - x - x^2 + x^3 + x^4] - x*#1]*#1^2)/(-4 + 7*#1 + #1^3) & ]/2","A",1
1782,1,120,120,0.2840905,"\int \frac{-1+x^4}{\left(1+x^4\right) \sqrt{1-x-x^2+x^3+x^4}} \, dx","IntegrateAlgebraic[(-1 + x^4)/((1 + x^4)*Sqrt[1 - x - x^2 + x^3 + x^4]),x]","\frac{1}{2} \text{RootSum}\left[3-8 \text{$\#$1}+6 \text{$\#$1}^2+\text{$\#$1}^4\&,\frac{\log (x)-\log \left(1-x^2+\sqrt{1-x-x^2+x^3+x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^2+\log \left(1-x^2+\sqrt{1-x-x^2+x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^2}{-2+3 \text{$\#$1}+\text{$\#$1}^3}\&\right]","\frac{1}{2} \text{RootSum}\left[3-8 \text{$\#$1}+6 \text{$\#$1}^2+\text{$\#$1}^4\&,\frac{\log (x)-\log \left(1-x^2+\sqrt{1-x-x^2+x^3+x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^2+\log \left(1-x^2+\sqrt{1-x-x^2+x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^2}{-2+3 \text{$\#$1}+\text{$\#$1}^3}\&\right]",1,"RootSum[3 - 8*#1 + 6*#1^2 + #1^4 & , (Log[x] - Log[1 - x^2 + Sqrt[1 - x - x^2 + x^3 + x^4] - x*#1] - Log[x]*#1^2 + Log[1 - x^2 + Sqrt[1 - x - x^2 + x^3 + x^4] - x*#1]*#1^2)/(-2 + 3*#1 + #1^3) & ]/2","A",1
1783,1,120,120,0.0014612,"\int \frac{-1+x^4}{\left(1+x^4\right) \sqrt{1-x-x^2+x^3+x^4}} \, dx","IntegrateAlgebraic[(-1 + x^4)/((1 + x^4)*Sqrt[1 - x - x^2 + x^3 + x^4]),x]","\frac{1}{2} \text{RootSum}\left[3-8 \text{$\#$1}+6 \text{$\#$1}^2+\text{$\#$1}^4\&,\frac{\log (x)-\log \left(1-x^2+\sqrt{1-x-x^2+x^3+x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^2+\log \left(1-x^2+\sqrt{1-x-x^2+x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^2}{-2+3 \text{$\#$1}+\text{$\#$1}^3}\&\right]","\frac{1}{2} \text{RootSum}\left[3-8 \text{$\#$1}+6 \text{$\#$1}^2+\text{$\#$1}^4\&,\frac{\log (x)-\log \left(1-x^2+\sqrt{1-x-x^2+x^3+x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^2+\log \left(1-x^2+\sqrt{1-x-x^2+x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^2}{-2+3 \text{$\#$1}+\text{$\#$1}^3}\&\right]",1,"RootSum[3 - 8*#1 + 6*#1^2 + #1^4 & , (Log[x] - Log[1 - x^2 + Sqrt[1 - x - x^2 + x^3 + x^4] - x*#1] - Log[x]*#1^2 + Log[1 - x^2 + Sqrt[1 - x - x^2 + x^3 + x^4] - x*#1]*#1^2)/(-2 + 3*#1 + #1^3) & ]/2","A",1
1784,1,120,120,1.1651673,"\int \frac{b+a x^8}{x^6 \left(-b+a x^4\right) \left(b+a x^4\right)^{3/4}} \, dx","IntegrateAlgebraic[(b + a*x^8)/(x^6*(-b + a*x^4)*(b + a*x^4)^(3/4)),x]","\frac{\left(b+a x^4\right)^{5/4}}{5 b^2 x^5}+\frac{\left(a^{5/4}+\sqrt[4]{a} b\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2\ 2^{3/4} b^2}-\frac{\left(a^{5/4}+\sqrt[4]{a} b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2\ 2^{3/4} b^2}","\frac{\left(b+a x^4\right)^{5/4}}{5 b^2 x^5}+\frac{\left(a^{5/4}+\sqrt[4]{a} b\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2\ 2^{3/4} b^2}-\frac{\left(a^{5/4}+\sqrt[4]{a} b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2\ 2^{3/4} b^2}",1,"(b + a*x^4)^(5/4)/(5*b^2*x^5) + ((a^(5/4) + a^(1/4)*b)*ArcTan[(2^(1/4)*a^(1/4)*x)/(b + a*x^4)^(1/4)])/(2*2^(3/4)*b^2) - ((a^(5/4) + a^(1/4)*b)*ArcTanh[(2^(1/4)*a^(1/4)*x)/(b + a*x^4)^(1/4)])/(2*2^(3/4)*b^2)","A",1
1785,1,121,120,1.8973883,"\int \frac{-b+2 a x^4}{\sqrt[4]{-b+a x^4} \left(-2 b-c x^4+2 a x^8\right)} \, dx","IntegrateAlgebraic[(-b + 2*a*x^4)/((-b + a*x^4)^(1/4)*(-2*b - c*x^4 + 2*a*x^8)),x]","\frac{1}{4} \text{RootSum}\left[2 a^2-2 a b+a c-4 a \text{$\#$1}^4-c \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{a \log (x)-a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-4 a \text{$\#$1}-c \text{$\#$1}+4 \text{$\#$1}^5}\&\right]","\frac{1}{4} \text{RootSum}\left[2 a^2-2 a b+a c-4 a \text{$\#$1}^4-c \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-a \log (x)+a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{4 a \text{$\#$1}+c \text{$\#$1}-4 \text{$\#$1}^5}\&\right]",1,"RootSum[2*a^2 - 2*a*b + a*c - 4*a*#1^4 - c*#1^4 + 2*#1^8 & , (a*Log[x] - a*Log[(-b + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4)/(-4*a*#1 - c*#1 + 4*#1^5) & ]/4","A",1
1786,1,121,120,0.0021438,"\int \frac{-b+2 a x^4}{\sqrt[4]{-b+a x^4} \left(-2 b-c x^4+2 a x^8\right)} \, dx","IntegrateAlgebraic[(-b + 2*a*x^4)/((-b + a*x^4)^(1/4)*(-2*b - c*x^4 + 2*a*x^8)),x]","\frac{1}{4} \text{RootSum}\left[2 a^2-2 a b+a c-4 a \text{$\#$1}^4-c \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{a \log (x)-a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-4 a \text{$\#$1}-c \text{$\#$1}+4 \text{$\#$1}^5}\&\right]","\frac{1}{4} \text{RootSum}\left[2 a^2-2 a b+a c-4 a \text{$\#$1}^4-c \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-a \log (x)+a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{4 a \text{$\#$1}+c \text{$\#$1}-4 \text{$\#$1}^5}\&\right]",1,"RootSum[2*a^2 - 2*a*b + a*c - 4*a*#1^4 - c*#1^4 + 2*#1^8 & , (a*Log[x] - a*Log[(-b + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4)/(-4*a*#1 - c*#1 + 4*#1^5) & ]/4","A",1
1787,1,121,121,0.5110442,"\int \frac{-a b-a c+3 b c+2 (a-b-c) x+x^2}{\sqrt[4]{(-a+x) (-b+x) (-c+x)} \left(-a^3-b c d+\left(3 a^2+b d+c d\right) x-(3 a+d) x^2+x^3\right)} \, dx","IntegrateAlgebraic[(-(a*b) - a*c + 3*b*c + 2*(a - b - c)*x + x^2)/(((-a + x)*(-b + x)*(-c + x))^(1/4)*(-a^3 - b*c*d + (3*a^2 + b*d + c*d)*x - (3*a + d)*x^2 + x^3)),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{a-x}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{a-x}\right)}{d^{3/4}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{a-x}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{a-x}\right)}{d^{3/4}}",1,"(-2*ArcTan[(d^(1/4)*(-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3)^(1/4))/(a - x)])/d^(3/4) + (2*ArcTanh[(d^(1/4)*(-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3)^(1/4))/(a - x)])/d^(3/4)","A",1
1788,1,121,121,0.4249957,"\int \frac{1}{(-b+a x) \sqrt[4]{b^2 x+a^2 x^3}} \, dx","IntegrateAlgebraic[1/((-b + a*x)*(b^2*x + a^2*x^3)^(1/4)),x]","\frac{\tan ^{-1}\left(\frac{2^{3/4} \sqrt[4]{a} \sqrt[4]{b} \sqrt[4]{b^2 x+a^2 x^3}}{b+a x}\right)}{2 \sqrt[4]{2} a^{3/4} b^{3/4}}-\frac{\tanh ^{-1}\left(\frac{2^{3/4} \sqrt[4]{a} \sqrt[4]{b} \sqrt[4]{b^2 x+a^2 x^3}}{b+a x}\right)}{2 \sqrt[4]{2} a^{3/4} b^{3/4}}","\frac{\tan ^{-1}\left(\frac{2^{3/4} \sqrt[4]{a} \sqrt[4]{b} \sqrt[4]{b^2 x+a^2 x^3}}{b+a x}\right)}{2 \sqrt[4]{2} a^{3/4} b^{3/4}}-\frac{\tanh ^{-1}\left(\frac{2^{3/4} \sqrt[4]{a} \sqrt[4]{b} \sqrt[4]{b^2 x+a^2 x^3}}{b+a x}\right)}{2 \sqrt[4]{2} a^{3/4} b^{3/4}}",1,"ArcTan[(2^(3/4)*a^(1/4)*b^(1/4)*(b^2*x + a^2*x^3)^(1/4))/(b + a*x)]/(2*2^(1/4)*a^(3/4)*b^(3/4)) - ArcTanh[(2^(3/4)*a^(1/4)*b^(1/4)*(b^2*x + a^2*x^3)^(1/4))/(b + a*x)]/(2*2^(1/4)*a^(3/4)*b^(3/4))","A",1
1789,1,121,121,1.2387351,"\int \frac{\left(2+x^2\right) \left(-2-2 x+x^2\right) \sqrt{4-3 x^2+x^4}}{x^2 \left(-2+x^2\right) \left(-4+x+2 x^2\right)} \, dx","IntegrateAlgebraic[((2 + x^2)*(-2 - 2*x + x^2)*Sqrt[4 - 3*x^2 + x^4])/(x^2*(-2 + x^2)*(-4 + x + 2*x^2)),x]","\frac{\sqrt{4-3 x^2+x^4}}{2 x}+4 \tanh ^{-1}\left(\frac{x}{-2+x^2+\sqrt{4-3 x^2+x^4}}\right)-\frac{5}{2} \sqrt{5} \tanh ^{-1}\left(\frac{\sqrt{5} x}{-4+x+2 x^2+2 \sqrt{4-3 x^2+x^4}}\right)+\frac{5 \log (x)}{4}-\frac{5}{4} \log \left(-2+x^2+\sqrt{4-3 x^2+x^4}\right)","\frac{\sqrt{4-3 x^2+x^4}}{2 x}+4 \tanh ^{-1}\left(\frac{x}{-2+x^2+\sqrt{4-3 x^2+x^4}}\right)-\frac{5}{2} \sqrt{5} \tanh ^{-1}\left(\frac{\sqrt{5} x}{-4+x+2 x^2+2 \sqrt{4-3 x^2+x^4}}\right)+\frac{5 \log (x)}{4}-\frac{5}{4} \log \left(-2+x^2+\sqrt{4-3 x^2+x^4}\right)",1,"Sqrt[4 - 3*x^2 + x^4]/(2*x) + 4*ArcTanh[x/(-2 + x^2 + Sqrt[4 - 3*x^2 + x^4])] - (5*Sqrt[5]*ArcTanh[(Sqrt[5]*x)/(-4 + x + 2*x^2 + 2*Sqrt[4 - 3*x^2 + x^4])])/2 + (5*Log[x])/4 - (5*Log[-2 + x^2 + Sqrt[4 - 3*x^2 + x^4]])/4","A",1
1790,1,121,121,0.4320841,"\int \frac{1+x}{(-1+x) \sqrt[3]{x^2+x^4}} \, dx","IntegrateAlgebraic[(1 + x)/((-1 + x)*(x^2 + x^4)^(1/3)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x^2+x^4}}\right)}{\sqrt[3]{2}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x^2+x^4}\right)}{\sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x^2+x^4}+\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}\right)}{2 \sqrt[3]{2}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x^2+x^4}}\right)}{\sqrt[3]{2}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x^2+x^4}\right)}{\sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x^2+x^4}+\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}\right)}{2 \sqrt[3]{2}}",1,"-((Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(x^2 + x^4)^(1/3))])/2^(1/3)) + Log[-2*x + 2^(2/3)*(x^2 + x^4)^(1/3)]/2^(1/3) - Log[2*x^2 + 2^(2/3)*x*(x^2 + x^4)^(1/3) + 2^(1/3)*(x^2 + x^4)^(2/3)]/(2*2^(1/3))","A",1
1791,1,121,121,0.429043,"\int \frac{-b^2+a^4 x^4}{\sqrt{b x+a^2 x^3} \left(b^2+a^4 x^4\right)} \, dx","IntegrateAlgebraic[(-b^2 + a^4*x^4)/(Sqrt[b*x + a^2*x^3]*(b^2 + a^4*x^4)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt{a} \sqrt[4]{b} \sqrt{b x+a^2 x^3}}{b+a^2 x^2}\right)}{\sqrt[4]{2} \sqrt{a} \sqrt[4]{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt{a} \sqrt[4]{b} \sqrt{b x+a^2 x^3}}{b+a^2 x^2}\right)}{\sqrt[4]{2} \sqrt{a} \sqrt[4]{b}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt{a} \sqrt[4]{b} \sqrt{b x+a^2 x^3}}{b+a^2 x^2}\right)}{\sqrt[4]{2} \sqrt{a} \sqrt[4]{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt{a} \sqrt[4]{b} \sqrt{b x+a^2 x^3}}{b+a^2 x^2}\right)}{\sqrt[4]{2} \sqrt{a} \sqrt[4]{b}}",1,"-(ArcTan[(2^(1/4)*Sqrt[a]*b^(1/4)*Sqrt[b*x + a^2*x^3])/(b + a^2*x^2)]/(2^(1/4)*Sqrt[a]*b^(1/4))) - ArcTanh[(2^(1/4)*Sqrt[a]*b^(1/4)*Sqrt[b*x + a^2*x^3])/(b + a^2*x^2)]/(2^(1/4)*Sqrt[a]*b^(1/4))","A",1
1792,1,121,121,0.4234128,"\int \frac{-1+x^2}{\left(1+x^2\right) \sqrt[3]{x+x^5}} \, dx","IntegrateAlgebraic[(-1 + x^2)/((1 + x^2)*(x + x^5)^(1/3)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2^{2/3} \sqrt[3]{x+x^5}}\right)}{2 \sqrt[3]{2}}-\frac{\log \left(2 x+2^{2/3} \sqrt[3]{x+x^5}\right)}{2 \sqrt[3]{2}}+\frac{\log \left(-2 x^2+2^{2/3} x \sqrt[3]{x+x^5}-\sqrt[3]{2} \left(x+x^5\right)^{2/3}\right)}{4 \sqrt[3]{2}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2^{2/3} \sqrt[3]{x+x^5}}\right)}{2 \sqrt[3]{2}}-\frac{\log \left(2 x+2^{2/3} \sqrt[3]{x+x^5}\right)}{2 \sqrt[3]{2}}+\frac{\log \left(-2 x^2+2^{2/3} x \sqrt[3]{x+x^5}-\sqrt[3]{2} \left(x+x^5\right)^{2/3}\right)}{4 \sqrt[3]{2}}",1,"-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2^(2/3)*(x + x^5)^(1/3))])/2^(1/3) - Log[2*x + 2^(2/3)*(x + x^5)^(1/3)]/(2*2^(1/3)) + Log[-2*x^2 + 2^(2/3)*x*(x + x^5)^(1/3) - 2^(1/3)*(x + x^5)^(2/3)]/(4*2^(1/3))","A",1
1793,1,121,121,0.3211815,"\int \frac{\left(1+x^4\right) \sqrt[4]{-x^2+x^4}}{1+x^4+x^8} \, dx","IntegrateAlgebraic[((1 + x^4)*(-x^2 + x^4)^(1/4))/(1 + x^4 + x^8),x]","\frac{1}{4} \text{RootSum}\left[3-3 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-3+2 \text{$\#$1}^4}\&\right]+\frac{1}{4} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-1+2 \text{$\#$1}^4}\&\right]","\frac{1}{4} \text{RootSum}\left[3-3 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-3+2 \text{$\#$1}^4}\&\right]+\frac{1}{4} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-1+2 \text{$\#$1}^4}\&\right]",1,"RootSum[3 - 3*#1^4 + #1^8 & , (-(Log[x]*#1) + Log[(-x^2 + x^4)^(1/4) - x*#1]*#1)/(-3 + 2*#1^4) & ]/4 + RootSum[1 - #1^4 + #1^8 & , (-(Log[x]*#1) + Log[(-x^2 + x^4)^(1/4) - x*#1]*#1)/(-1 + 2*#1^4) & ]/4","A",1
1794,1,121,121,0.0012271,"\int \frac{\left(1+x^4\right) \sqrt[4]{-x^2+x^4}}{1+x^4+x^8} \, dx","IntegrateAlgebraic[((1 + x^4)*(-x^2 + x^4)^(1/4))/(1 + x^4 + x^8),x]","\frac{1}{4} \text{RootSum}\left[3-3 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-3+2 \text{$\#$1}^4}\&\right]+\frac{1}{4} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-1+2 \text{$\#$1}^4}\&\right]","\frac{1}{4} \text{RootSum}\left[3-3 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-3+2 \text{$\#$1}^4}\&\right]+\frac{1}{4} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-1+2 \text{$\#$1}^4}\&\right]",1,"RootSum[3 - 3*#1^4 + #1^8 & , (-(Log[x]*#1) + Log[(-x^2 + x^4)^(1/4) - x*#1]*#1)/(-3 + 2*#1^4) & ]/4 + RootSum[1 - #1^4 + #1^8 & , (-(Log[x]*#1) + Log[(-x^2 + x^4)^(1/4) - x*#1]*#1)/(-1 + 2*#1^4) & ]/4","A",1
1795,1,121,121,0.3526253,"\int \frac{1-2 x^4+2 x^8}{\sqrt[4]{-1+x^4} \left(-2-x^4+2 x^8\right)} \, dx","IntegrateAlgebraic[(1 - 2*x^4 + 2*x^8)/((-1 + x^4)^(1/4)*(-2 - x^4 + 2*x^8)),x]","\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)+\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)-\frac{1}{4} \text{RootSum}\left[1-5 \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-2 \log (x)+2 \log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right)+3 \log (x) \text{$\#$1}^4-3 \log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-5 \text{$\#$1}+4 \text{$\#$1}^5}\&\right]","\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)+\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)-\frac{1}{4} \text{RootSum}\left[1-5 \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-2 \log (x)+2 \log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right)+3 \log (x) \text{$\#$1}^4-3 \log \left(\sqrt[4]{-1+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-5 \text{$\#$1}+4 \text{$\#$1}^5}\&\right]",1,"ArcTan[x/(-1 + x^4)^(1/4)]/2 + ArcTanh[x/(-1 + x^4)^(1/4)]/2 - RootSum[1 - 5*#1^4 + 2*#1^8 & , (-2*Log[x] + 2*Log[(-1 + x^4)^(1/4) - x*#1] + 3*Log[x]*#1^4 - 3*Log[(-1 + x^4)^(1/4) - x*#1]*#1^4)/(-5*#1 + 4*#1^5) & ]/4","A",1
1796,1,121,121,4.2231263,"\int \frac{\sqrt{-x+x^2} \sqrt{x^2-x \sqrt{-x+x^2}}}{x^3} \, dx","IntegrateAlgebraic[(Sqrt[-x + x^2]*Sqrt[x^2 - x*Sqrt[-x + x^2]])/x^3,x]","-\frac{4 \sqrt{-x+x^2} \sqrt{x \left(x-\sqrt{-x+x^2}\right)}}{3 x^2}+\sqrt{x \left(x-\sqrt{-x+x^2}\right)} \left(\frac{4}{3 x}-\frac{2 \sqrt{2} \sqrt{x+\sqrt{-x+x^2}} \tanh ^{-1}\left(\sqrt{2} \sqrt{x+\sqrt{-x+x^2}}\right)}{x}\right)","-\frac{4 \sqrt{-x+x^2} \sqrt{x \left(x-\sqrt{-x+x^2}\right)}}{3 x^2}+\sqrt{x \left(x-\sqrt{-x+x^2}\right)} \left(\frac{4}{3 x}-\frac{2 \sqrt{2} \sqrt{x+\sqrt{-x+x^2}} \tanh ^{-1}\left(\sqrt{2} \sqrt{x+\sqrt{-x+x^2}}\right)}{x}\right)",1,"(-4*Sqrt[-x + x^2]*Sqrt[x*(x - Sqrt[-x + x^2])])/(3*x^2) + Sqrt[x*(x - Sqrt[-x + x^2])]*(4/(3*x) - (2*Sqrt[2]*Sqrt[x + Sqrt[-x + x^2]]*ArcTanh[Sqrt[2]*Sqrt[x + Sqrt[-x + x^2]]])/x)","A",1
1797,1,122,122,0.3017827,"\int \frac{b+2 a x}{\sqrt[4]{c+b x+a x^2} \left(5 c+4 b x+4 a x^2\right)} \, dx","IntegrateAlgebraic[(b + 2*a*x)/((c + b*x + a*x^2)^(1/4)*(5*c + 4*b*x + 4*a*x^2)),x]","-\frac{\tan ^{-1}\left(1-\frac{2 \sqrt[4]{c+b x+a x^2}}{\sqrt[4]{c}}\right)}{2 \sqrt[4]{c}}+\frac{\tan ^{-1}\left(1+\frac{2 \sqrt[4]{c+b x+a x^2}}{\sqrt[4]{c}}\right)}{2 \sqrt[4]{c}}-\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[4]{c}}{2}+\frac{\sqrt{c+b x+a x^2}}{\sqrt[4]{c}}}{\sqrt[4]{c+b x+a x^2}}\right)}{2 \sqrt[4]{c}}","-\frac{\tan ^{-1}\left(1-\frac{2 \sqrt[4]{c+b x+a x^2}}{\sqrt[4]{c}}\right)}{2 \sqrt[4]{c}}+\frac{\tan ^{-1}\left(1+\frac{2 \sqrt[4]{c+b x+a x^2}}{\sqrt[4]{c}}\right)}{2 \sqrt[4]{c}}-\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[4]{c}}{2}+\frac{\sqrt{c+b x+a x^2}}{\sqrt[4]{c}}}{\sqrt[4]{c+b x+a x^2}}\right)}{2 \sqrt[4]{c}}",1,"-1/2*ArcTan[1 - (2*(c + b*x + a*x^2)^(1/4))/c^(1/4)]/c^(1/4) + ArcTan[1 + (2*(c + b*x + a*x^2)^(1/4))/c^(1/4)]/(2*c^(1/4)) - ArcTanh[(c^(1/4)/2 + Sqrt[c + b*x + a*x^2]/c^(1/4))/(c + b*x + a*x^2)^(1/4)]/(2*c^(1/4))","A",1
1798,1,122,122,0.4196732,"\int x^6 \sqrt[3]{-x+x^3} \, dx","IntegrateAlgebraic[x^6*(-x + x^3)^(1/3),x]","\frac{1}{648} \sqrt[3]{-x+x^3} \left(-20 x-12 x^3-9 x^5+81 x^7\right)+\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)}{81 \sqrt{3}}+\frac{5}{243} \log \left(-x+\sqrt[3]{-x+x^3}\right)-\frac{5}{486} \log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)","\frac{1}{648} \sqrt[3]{-x+x^3} \left(-20 x-12 x^3-9 x^5+81 x^7\right)+\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)}{81 \sqrt{3}}+\frac{5}{243} \log \left(-x+\sqrt[3]{-x+x^3}\right)-\frac{5}{486} \log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)",1,"((-x + x^3)^(1/3)*(-20*x - 12*x^3 - 9*x^5 + 81*x^7))/648 + (5*ArcTan[(Sqrt[3]*x)/(x + 2*(-x + x^3)^(1/3))])/(81*Sqrt[3]) + (5*Log[-x + (-x + x^3)^(1/3)])/243 - (5*Log[x^2 + x*(-x + x^3)^(1/3) + (-x + x^3)^(2/3)])/486","A",1
1799,1,122,122,0.2798952,"\int \frac{1}{\sqrt[3]{-1-x+x^2+x^3}} \, dx","IntegrateAlgebraic[(-1 - x + x^2 + x^3)^(-1/3),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-1-x+x^2+x^3}}{2+2 x+\sqrt[3]{-1-x+x^2+x^3}}\right)-\log \left(-1-x+\sqrt[3]{-1-x+x^2+x^3}\right)+\frac{1}{2} \log \left(1+2 x+x^2+(1+x) \sqrt[3]{-1-x+x^2+x^3}+\left(-1-x+x^2+x^3\right)^{2/3}\right)","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-1-x+x^2+x^3}}{2+2 x+\sqrt[3]{-1-x+x^2+x^3}}\right)-\log \left(-1-x+\sqrt[3]{-1-x+x^2+x^3}\right)+\frac{1}{2} \log \left(1+2 x+x^2+(1+x) \sqrt[3]{-1-x+x^2+x^3}+\left(-1-x+x^2+x^3\right)^{2/3}\right)",1,"-(Sqrt[3]*ArcTan[(Sqrt[3]*(-1 - x + x^2 + x^3)^(1/3))/(2 + 2*x + (-1 - x + x^2 + x^3)^(1/3))]) - Log[-1 - x + (-1 - x + x^2 + x^3)^(1/3)] + Log[1 + 2*x + x^2 + (1 + x)*(-1 - x + x^2 + x^3)^(1/3) + (-1 - x + x^2 + x^3)^(2/3)]/2","A",1
1800,1,122,122,0.347615,"\int \frac{1+2 x^4}{\left(-1+x^4\right) \sqrt[4]{-x^2+x^4}} \, dx","IntegrateAlgebraic[(1 + 2*x^4)/((-1 + x^4)*(-x^2 + x^4)^(1/4)),x]","-\frac{3 \left(-x^2+x^4\right)^{3/4}}{x \left(-1+x^2\right)}+2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)-\frac{3 \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{2 \sqrt[4]{2}}+2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)-\frac{3 \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{2 \sqrt[4]{2}}","-\frac{3 \left(-x^2+x^4\right)^{3/4}}{x \left(-1+x^2\right)}+2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)-\frac{3 \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{2 \sqrt[4]{2}}+2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)-\frac{3 \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{2 \sqrt[4]{2}}",1,"(-3*(-x^2 + x^4)^(3/4))/(x*(-1 + x^2)) + 2*ArcTan[x/(-x^2 + x^4)^(1/4)] - (3*ArcTan[(2^(1/4)*x)/(-x^2 + x^4)^(1/4)])/(2*2^(1/4)) + 2*ArcTanh[x/(-x^2 + x^4)^(1/4)] - (3*ArcTanh[(2^(1/4)*x)/(-x^2 + x^4)^(1/4)])/(2*2^(1/4))","A",1
1801,1,122,122,0.537751,"\int \frac{\left(-b+a x^4\right) \sqrt[4]{b x^2+a x^4}}{x^2} \, dx","IntegrateAlgebraic[((-b + a*x^4)*(b*x^2 + a*x^4)^(1/4))/x^2,x]","\frac{\sqrt[4]{b x^2+a x^4} \left(32 b+b x^2+4 a x^4\right)}{16 x}+\frac{\left(32 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{32 a^{3/4}}+\frac{\left(-32 a b-3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{32 a^{3/4}}","\frac{\sqrt[4]{b x^2+a x^4} \left(32 b+b x^2+4 a x^4\right)}{16 x}+\frac{\left(32 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{32 a^{3/4}}+\frac{\left(-32 a b-3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{32 a^{3/4}}",1,"((b*x^2 + a*x^4)^(1/4)*(32*b + b*x^2 + 4*a*x^4))/(16*x) + ((32*a*b + 3*b^2)*ArcTan[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/(32*a^(3/4)) + ((-32*a*b - 3*b^2)*ArcTanh[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/(32*a^(3/4))","A",1
1802,1,122,122,0.5517168,"\int \frac{\left(b+a x^4\right) \sqrt[4]{b x^2+a x^4}}{x^2} \, dx","IntegrateAlgebraic[((b + a*x^4)*(b*x^2 + a*x^4)^(1/4))/x^2,x]","\frac{\sqrt[4]{b x^2+a x^4} \left(-32 b+b x^2+4 a x^4\right)}{16 x}+\frac{\left(-32 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{32 a^{3/4}}+\frac{\left(32 a b-3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{32 a^{3/4}}","\frac{\sqrt[4]{b x^2+a x^4} \left(-32 b+b x^2+4 a x^4\right)}{16 x}+\frac{\left(-32 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{32 a^{3/4}}+\frac{\left(32 a b-3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{32 a^{3/4}}",1,"((b*x^2 + a*x^4)^(1/4)*(-32*b + b*x^2 + 4*a*x^4))/(16*x) + ((-32*a*b + 3*b^2)*ArcTan[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/(32*a^(3/4)) + ((32*a*b - 3*b^2)*ArcTanh[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/(32*a^(3/4))","A",1
1803,1,122,122,0.5918414,"\int \frac{b+2 a x^4}{\left(-b+a x^4\right) \sqrt[4]{-b x^2+a x^4}} \, dx","IntegrateAlgebraic[(b + 2*a*x^4)/((-b + a*x^4)*(-(b*x^2) + a*x^4)^(1/4)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{3}{4} \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{3}{4} \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"(2*ArcTan[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/a^(1/4) + (2*ArcTanh[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/a^(1/4) + (3*RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (-Log[x] + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1])/#1 & ])/4","A",1
1804,1,122,122,0.0021214,"\int \frac{b+2 a x^4}{\left(-b+a x^4\right) \sqrt[4]{-b x^2+a x^4}} \, dx","IntegrateAlgebraic[(b + 2*a*x^4)/((-b + a*x^4)*(-(b*x^2) + a*x^4)^(1/4)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{3}{4} \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{3}{4} \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"(2*ArcTan[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/a^(1/4) + (2*ArcTanh[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/a^(1/4) + (3*RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (-Log[x] + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1])/#1 & ])/4","A",1
1805,1,122,122,0.8226489,"\int \frac{-b-2 a x^4+2 x^8}{\sqrt[4]{-b+a x^4}} \, dx","IntegrateAlgebraic[(-b - 2*a*x^4 + 2*x^8)/(-b + a*x^4)^(1/4),x]","\frac{\left(-b+a x^4\right)^{3/4} \left(-8 a^2 x+5 b x+4 a x^5\right)}{16 a^2}+\frac{\left(-24 a^2 b+5 b^2\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{32 a^{9/4}}+\frac{\left(-24 a^2 b+5 b^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{32 a^{9/4}}","\frac{\left(-b+a x^4\right)^{3/4} \left(-8 a^2 x+5 b x+4 a x^5\right)}{16 a^2}+\frac{\left(-24 a^2 b+5 b^2\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{32 a^{9/4}}+\frac{\left(-24 a^2 b+5 b^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{32 a^{9/4}}",1,"((-b + a*x^4)^(3/4)*(-8*a^2*x + 5*b*x + 4*a*x^5))/(16*a^2) + ((-24*a^2*b + 5*b^2)*ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)])/(32*a^(9/4)) + ((-24*a^2*b + 5*b^2)*ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)])/(32*a^(9/4))","A",1
1806,1,123,123,0.2554813,"\int \frac{-1+x}{\left(-4-2 x+x^2\right) \sqrt[3]{-2-2 x+x^2}} \, dx","IntegrateAlgebraic[(-1 + x)/((-4 - 2*x + x^2)*(-2 - 2*x + x^2)^(1/3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2^{2/3} \sqrt[3]{-2-2 x+x^2}}{\sqrt{3}}\right)}{2 \sqrt[3]{2}}+\frac{\log \left(-2+2^{2/3} \sqrt[3]{-2-2 x+x^2}\right)}{2 \sqrt[3]{2}}-\frac{\log \left(2+2^{2/3} \sqrt[3]{-2-2 x+x^2}+\sqrt[3]{2} \left(-2-2 x+x^2\right)^{2/3}\right)}{4 \sqrt[3]{2}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2^{2/3} \sqrt[3]{-2-2 x+x^2}}{\sqrt{3}}\right)}{2 \sqrt[3]{2}}+\frac{\log \left(-2+2^{2/3} \sqrt[3]{-2-2 x+x^2}\right)}{2 \sqrt[3]{2}}-\frac{\log \left(2+2^{2/3} \sqrt[3]{-2-2 x+x^2}+\sqrt[3]{2} \left(-2-2 x+x^2\right)^{2/3}\right)}{4 \sqrt[3]{2}}",1,"(Sqrt[3]*ArcTan[1/Sqrt[3] + (2^(2/3)*(-2 - 2*x + x^2)^(1/3))/Sqrt[3]])/(2*2^(1/3)) + Log[-2 + 2^(2/3)*(-2 - 2*x + x^2)^(1/3)]/(2*2^(1/3)) - Log[2 + 2^(2/3)*(-2 - 2*x + x^2)^(1/3) + 2^(1/3)*(-2 - 2*x + x^2)^(2/3)]/(4*2^(1/3))","A",1
1807,1,123,123,0.2353677,"\int \frac{1}{(1+x) \sqrt[3]{3+2 x+x^2}} \, dx","IntegrateAlgebraic[1/((1 + x)*(3 + 2*x + x^2)^(1/3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2^{2/3} \sqrt[3]{3+2 x+x^2}}{\sqrt{3}}\right)}{2 \sqrt[3]{2}}+\frac{\log \left(-2+2^{2/3} \sqrt[3]{3+2 x+x^2}\right)}{2 \sqrt[3]{2}}-\frac{\log \left(2+2^{2/3} \sqrt[3]{3+2 x+x^2}+\sqrt[3]{2} \left(3+2 x+x^2\right)^{2/3}\right)}{4 \sqrt[3]{2}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2^{2/3} \sqrt[3]{3+2 x+x^2}}{\sqrt{3}}\right)}{2 \sqrt[3]{2}}+\frac{\log \left(-2+2^{2/3} \sqrt[3]{3+2 x+x^2}\right)}{2 \sqrt[3]{2}}-\frac{\log \left(2+2^{2/3} \sqrt[3]{3+2 x+x^2}+\sqrt[3]{2} \left(3+2 x+x^2\right)^{2/3}\right)}{4 \sqrt[3]{2}}",1,"(Sqrt[3]*ArcTan[1/Sqrt[3] + (2^(2/3)*(3 + 2*x + x^2)^(1/3))/Sqrt[3]])/(2*2^(1/3)) + Log[-2 + 2^(2/3)*(3 + 2*x + x^2)^(1/3)]/(2*2^(1/3)) - Log[2 + 2^(2/3)*(3 + 2*x + x^2)^(1/3) + 2^(1/3)*(3 + 2*x + x^2)^(2/3)]/(4*2^(1/3))","A",1
1808,1,121,123,0.2312676,"\int \frac{1}{x \left(-b+a x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x*(-b + a*x^2)^(3/4)),x]","\frac{\tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^2}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^2}}\right)}{\sqrt{2} b^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^2}}{\sqrt{b}+\sqrt{-b+a x^2}}\right)}{\sqrt{2} b^{3/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^2}}{-\sqrt{b}+\sqrt{-b+a x^2}}\right)}{\sqrt{2} b^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^2}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^2}}\right)}{\sqrt{2} b^{3/4}}",1,"ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^2]/(Sqrt[2]*b^(1/4)))/(-b + a*x^2)^(1/4)]/(Sqrt[2]*b^(3/4)) + ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^2)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^2])]/(Sqrt[2]*b^(3/4))","A",1
1809,1,123,123,0.1186407,"\int \frac{1}{x \sqrt[3]{-b+a x^3}} \, dx","IntegrateAlgebraic[1/(x*(-b + a*x^3)^(1/3)),x]","-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-b+a x^3}}{\sqrt{3} \sqrt[3]{b}}\right)}{\sqrt{3} \sqrt[3]{b}}-\frac{\log \left(\sqrt[3]{b}+\sqrt[3]{-b+a x^3}\right)}{3 \sqrt[3]{b}}+\frac{\log \left(b^{2/3}-\sqrt[3]{b} \sqrt[3]{-b+a x^3}+\left(-b+a x^3\right)^{2/3}\right)}{6 \sqrt[3]{b}}","-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-b+a x^3}}{\sqrt{3} \sqrt[3]{b}}\right)}{\sqrt{3} \sqrt[3]{b}}-\frac{\log \left(\sqrt[3]{b}+\sqrt[3]{-b+a x^3}\right)}{3 \sqrt[3]{b}}+\frac{\log \left(b^{2/3}-\sqrt[3]{b} \sqrt[3]{-b+a x^3}+\left(-b+a x^3\right)^{2/3}\right)}{6 \sqrt[3]{b}}",1,"-(ArcTan[1/Sqrt[3] - (2*(-b + a*x^3)^(1/3))/(Sqrt[3]*b^(1/3))]/(Sqrt[3]*b^(1/3))) - Log[b^(1/3) + (-b + a*x^3)^(1/3)]/(3*b^(1/3)) + Log[b^(2/3) - b^(1/3)*(-b + a*x^3)^(1/3) + (-b + a*x^3)^(2/3)]/(6*b^(1/3))","A",1
1810,1,123,123,0.5147734,"\int \frac{-b^3+a^3 x^3}{\sqrt{b^2 x+a^2 x^3} \left(b^3+a^3 x^3\right)} \, dx","IntegrateAlgebraic[(-b^3 + a^3*x^3)/(Sqrt[b^2*x + a^2*x^3]*(b^3 + a^3*x^3)),x]","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{3 \sqrt{a} \sqrt{b}}-\frac{4 \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{3 \sqrt{a} \sqrt{b}}","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{3 \sqrt{a} \sqrt{b}}-\frac{4 \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{3 \sqrt{a} \sqrt{b}}",1,"-1/3*(Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[b]*Sqrt[b^2*x + a^2*x^3])/(b^2 + a^2*x^2)])/(Sqrt[a]*Sqrt[b]) - (4*ArcTanh[(Sqrt[a]*Sqrt[b]*Sqrt[b^2*x + a^2*x^3])/(b^2 + a^2*x^2)])/(3*Sqrt[a]*Sqrt[b])","A",1
1811,1,123,123,0.5065299,"\int \frac{b^3+a^3 x^3}{\sqrt{b^2 x+a^2 x^3} \left(-b^3+a^3 x^3\right)} \, dx","IntegrateAlgebraic[(b^3 + a^3*x^3)/(Sqrt[b^2*x + a^2*x^3]*(-b^3 + a^3*x^3)),x]","-\frac{4 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{3 \sqrt{a} \sqrt{b}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{3 \sqrt{a} \sqrt{b}}","-\frac{4 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{3 \sqrt{a} \sqrt{b}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{3 \sqrt{a} \sqrt{b}}",1,"(-4*ArcTan[(Sqrt[a]*Sqrt[b]*Sqrt[b^2*x + a^2*x^3])/(b^2 + a^2*x^2)])/(3*Sqrt[a]*Sqrt[b]) - (Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[b]*Sqrt[b^2*x + a^2*x^3])/(b^2 + a^2*x^2)])/(3*Sqrt[a]*Sqrt[b])","A",1
1812,1,123,123,0.2986518,"\int \frac{\sqrt[4]{x^3+x^4}}{-2+x+2 x^2} \, dx","IntegrateAlgebraic[(x^3 + x^4)^(1/4)/(-2 + x + 2*x^2),x]","-\tan ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)-\frac{1}{2} \text{RootSum}\left[1-5 \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right)+2 \log (x) \text{$\#$1}^4-2 \log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-5 \text{$\#$1}^3+4 \text{$\#$1}^7}\&\right]","-\tan ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)-\frac{1}{2} \text{RootSum}\left[1-5 \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right)+2 \log (x) \text{$\#$1}^4-2 \log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-5 \text{$\#$1}^3+4 \text{$\#$1}^7}\&\right]",1,"-ArcTan[x/(x^3 + x^4)^(1/4)] + ArcTanh[x/(x^3 + x^4)^(1/4)] - RootSum[1 - 5*#1^4 + 2*#1^8 & , (-Log[x] + Log[(x^3 + x^4)^(1/4) - x*#1] + 2*Log[x]*#1^4 - 2*Log[(x^3 + x^4)^(1/4) - x*#1]*#1^4)/(-5*#1^3 + 4*#1^7) & ]/2","A",1
1813,1,123,123,1.8028873,"\int \frac{\left(-4 b+a x^3\right) \left(b-a x^3+x^4\right)}{x^4 \sqrt[4]{-b+a x^3} \left(-b+a x^3+x^4\right)} \, dx","IntegrateAlgebraic[((-4*b + a*x^3)*(b - a*x^3 + x^4))/(x^4*(-b + a*x^3)^(1/4)*(-b + a*x^3 + x^4)),x]","\frac{4 \left(-b+a x^3\right)^{3/4}}{3 x^3}-2 \sqrt{2} \tan ^{-1}\left(\frac{-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{-b+a x^3}}{\sqrt{2}}}{x \sqrt[4]{-b+a x^3}}\right)+2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-b+a x^3}}{x^2+\sqrt{-b+a x^3}}\right)","\frac{4 \left(-b+a x^3\right)^{3/4}}{3 x^3}-2 \sqrt{2} \tan ^{-1}\left(\frac{-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{-b+a x^3}}{\sqrt{2}}}{x \sqrt[4]{-b+a x^3}}\right)+2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-b+a x^3}}{x^2+\sqrt{-b+a x^3}}\right)",1,"(4*(-b + a*x^3)^(3/4))/(3*x^3) - 2*Sqrt[2]*ArcTan[(-(x^2/Sqrt[2]) + Sqrt[-b + a*x^3]/Sqrt[2])/(x*(-b + a*x^3)^(1/4))] + 2*Sqrt[2]*ArcTanh[(Sqrt[2]*x*(-b + a*x^3)^(1/4))/(x^2 + Sqrt[-b + a*x^3])]","A",1
1814,1,123,123,0.379608,"\int \frac{\left(-2+x^2\right) \left(-1+x^2\right) \sqrt[4]{-1+x^2+x^4}}{x^6 \left(-1+x^2+2 x^4\right)} \, dx","IntegrateAlgebraic[((-2 + x^2)*(-1 + x^2)*(-1 + x^2 + x^4)^(1/4))/(x^6*(-1 + x^2 + 2*x^4)),x]","\frac{2 \sqrt[4]{-1+x^2+x^4} \left(1-x^2+9 x^4\right)}{5 x^5}+\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-1+x^2+x^4}}{-x^2+\sqrt{-1+x^2+x^4}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-1+x^2+x^4}}{x^2+\sqrt{-1+x^2+x^4}}\right)","\frac{2 \sqrt[4]{-1+x^2+x^4} \left(1-x^2+9 x^4\right)}{5 x^5}+\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-1+x^2+x^4}}{-x^2+\sqrt{-1+x^2+x^4}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-1+x^2+x^4}}{x^2+\sqrt{-1+x^2+x^4}}\right)",1,"(2*(-1 + x^2 + x^4)^(1/4)*(1 - x^2 + 9*x^4))/(5*x^5) + Sqrt[2]*ArcTan[(Sqrt[2]*x*(-1 + x^2 + x^4)^(1/4))/(-x^2 + Sqrt[-1 + x^2 + x^4])] - Sqrt[2]*ArcTanh[(Sqrt[2]*x*(-1 + x^2 + x^4)^(1/4))/(x^2 + Sqrt[-1 + x^2 + x^4])]","A",1
1815,1,123,123,0.5212036,"\int x^2 \sqrt[4]{b x^3+a x^4} \, dx","IntegrateAlgebraic[x^2*(b*x^3 + a*x^4)^(1/4),x]","\frac{\left(77 b^3-44 a b^2 x+32 a^2 b x^2+384 a^3 x^3\right) \sqrt[4]{b x^3+a x^4}}{1536 a^3}+\frac{77 b^4 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{1024 a^{15/4}}-\frac{77 b^4 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{1024 a^{15/4}}","\frac{\left(77 b^3-44 a b^2 x+32 a^2 b x^2+384 a^3 x^3\right) \sqrt[4]{b x^3+a x^4}}{1536 a^3}+\frac{77 b^4 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{1024 a^{15/4}}-\frac{77 b^4 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{1024 a^{15/4}}",1,"((77*b^3 - 44*a*b^2*x + 32*a^2*b*x^2 + 384*a^3*x^3)*(b*x^3 + a*x^4)^(1/4))/(1536*a^3) + (77*b^4*ArcTan[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(1024*a^(15/4)) - (77*b^4*ArcTanh[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(1024*a^(15/4))","A",1
1816,1,123,123,0.2264028,"\int \frac{1+x^6}{\sqrt{x+x^2+x^3} \left(1-x^6\right)} \, dx","IntegrateAlgebraic[(1 + x^6)/(Sqrt[x + x^2 + x^3]*(1 - x^6)),x]","\frac{2 \sqrt{x+x^2+x^3}}{3 \left(1+x+x^2\right)}+\frac{1}{3} \tan ^{-1}\left(\frac{\sqrt{x+x^2+x^3}}{1+x+x^2}\right)+\frac{1}{3} \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{x+x^2+x^3}}{1+x+x^2}\right)+\frac{\tanh ^{-1}\left(\frac{\sqrt{3} \sqrt{x+x^2+x^3}}{1+x+x^2}\right)}{3 \sqrt{3}}","\frac{2 \sqrt{x+x^2+x^3}}{3 \left(1+x+x^2\right)}+\frac{1}{3} \tan ^{-1}\left(\frac{\sqrt{x+x^2+x^3}}{1+x+x^2}\right)+\frac{1}{3} \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{x+x^2+x^3}}{1+x+x^2}\right)+\frac{\tanh ^{-1}\left(\frac{\sqrt{3} \sqrt{x+x^2+x^3}}{1+x+x^2}\right)}{3 \sqrt{3}}",1,"(2*Sqrt[x + x^2 + x^3])/(3*(1 + x + x^2)) + ArcTan[Sqrt[x + x^2 + x^3]/(1 + x + x^2)]/3 + (Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[x + x^2 + x^3])/(1 + x + x^2)])/3 + ArcTanh[(Sqrt[3]*Sqrt[x + x^2 + x^3])/(1 + x + x^2)]/(3*Sqrt[3])","A",1
1817,1,123,123,0.6916989,"\int \frac{1-3 x^3+3 x^6}{x^6 \left(-1+2 x^3\right) \sqrt[4]{-x+x^4}} \, dx","IntegrateAlgebraic[(1 - 3*x^3 + 3*x^6)/(x^6*(-1 + 2*x^3)*(-x + x^4)^(1/4)),x]","\frac{4 \left(-1+x^3\right) \left(-x+x^4\right)^{3/4}}{21 x^6}-\frac{1}{3} \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-x+x^4}}{-x^2+\sqrt{-x+x^4}}\right)-\frac{1}{3} \sqrt{2} \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{-x+x^4}}{\sqrt{2}}}{x \sqrt[4]{-x+x^4}}\right)","\frac{4 \left(-1+x^3\right) \left(-x+x^4\right)^{3/4}}{21 x^6}-\frac{1}{3} \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-x+x^4}}{-x^2+\sqrt{-x+x^4}}\right)-\frac{1}{3} \sqrt{2} \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{-x+x^4}}{\sqrt{2}}}{x \sqrt[4]{-x+x^4}}\right)",1,"(4*(-1 + x^3)*(-x + x^4)^(3/4))/(21*x^6) - (Sqrt[2]*ArcTan[(Sqrt[2]*x*(-x + x^4)^(1/4))/(-x^2 + Sqrt[-x + x^4])])/3 - (Sqrt[2]*ArcTanh[(x^2/Sqrt[2] + Sqrt[-x + x^4]/Sqrt[2])/(x*(-x + x^4)^(1/4))])/3","A",1
1818,1,123,123,0.4943826,"\int \frac{b+a x^6}{x^3 \left(b+a x^3\right) \sqrt[4]{b x+a x^4}} \, dx","IntegrateAlgebraic[(b + a*x^6)/(x^3*(b + a*x^3)*(b*x + a*x^4)^(1/4)),x]","-\frac{4 \left(b+4 a x^3+3 b x^3\right) \left(b x+a x^4\right)^{3/4}}{9 b x^3 \left(b+a x^3\right)}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 \sqrt[4]{a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 \sqrt[4]{a}}","-\frac{4 \left(b+4 a x^3+3 b x^3\right) \left(b x+a x^4\right)^{3/4}}{9 b x^3 \left(b+a x^3\right)}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 \sqrt[4]{a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 \sqrt[4]{a}}",1,"(-4*(b + 4*a*x^3 + 3*b*x^3)*(b*x + a*x^4)^(3/4))/(9*b*x^3*(b + a*x^3)) + (2*ArcTan[(a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/(3*a^(1/4)) + (2*ArcTanh[(a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/(3*a^(1/4))","A",1
1819,1,117,123,3.7631151,"\int \frac{\left(-1-x-x^2+x^4\right) \left(2+x+2 x^4\right)}{\sqrt{-1-x+x^4} \left(4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right)} \, dx","IntegrateAlgebraic[((-1 - x - x^2 + x^4)*(2 + x + 2*x^4))/(Sqrt[-1 - x + x^4]*(4 + 8*x + 3*x^2 - x^3 - 7*x^4 - 8*x^5 + x^6 + 4*x^8)),x]","-(-1)^{3/4} \sqrt{\frac{1}{10} \left(-7 i+\sqrt{15}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{8}-\frac{i \sqrt{15}}{8}} x}{\sqrt{-1-x+x^4}}\right)+\frac{\sqrt[4]{-1} \sqrt{7 i+\sqrt{15}} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{8}+\frac{i \sqrt{15}}{8}} x}{\sqrt{-1-x+x^4}}\right)}{\sqrt{10}}","-(-1)^{3/4} \sqrt{\frac{1}{10} \left(-7 i+\sqrt{15}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}-\frac{i \sqrt{15}}{2}} x}{2 \sqrt{-1-x+x^4}}\right)+\frac{\sqrt[4]{-1} \sqrt{7 i+\sqrt{15}} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{i \sqrt{15}}{2}} x}{2 \sqrt{-1-x+x^4}}\right)}{\sqrt{10}}",1,"-((-1)^(3/4)*Sqrt[(-7*I + Sqrt[15])/10]*ArcTan[(Sqrt[1/8 - (I/8)*Sqrt[15]]*x)/Sqrt[-1 - x + x^4]]) + ((-1)^(1/4)*Sqrt[7*I + Sqrt[15]]*ArcTan[(Sqrt[1/8 + (I/8)*Sqrt[15]]*x)/Sqrt[-1 - x + x^4]])/Sqrt[10]","A",1
1820,1,123,123,3.3412289,"\int \frac{\left(1+x^6\right)^2 \left(-1+2 x^6\right)}{\left(1-x^2+x^6\right)^{3/2} \left(1-x^2-x^4+2 x^6-x^8+x^{12}\right)} \, dx","IntegrateAlgebraic[((1 + x^6)^2*(-1 + 2*x^6))/((1 - x^2 + x^6)^(3/2)*(1 - x^2 - x^4 + 2*x^6 - x^8 + x^12)),x]","\frac{x}{\sqrt{1-x^2+x^6}}-\sqrt{\frac{2}{145+65 \sqrt{5}}} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt{1-x^2+x^6}}\right)-\sqrt{\frac{1}{10} \left(29+13 \sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt{1-x^2+x^6}}\right)","\frac{x}{\sqrt{1-x^2+x^6}}-\sqrt{\frac{2}{145+65 \sqrt{5}}} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt{1-x^2+x^6}}\right)-\sqrt{\frac{1}{10} \left(29+13 \sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt{1-x^2+x^6}}\right)",1,"x/Sqrt[1 - x^2 + x^6] - Sqrt[2/(145 + 65*Sqrt[5])]*ArcTan[(Sqrt[1/2 + Sqrt[5]/2]*x)/Sqrt[1 - x^2 + x^6]] - Sqrt[(29 + 13*Sqrt[5])/10]*ArcTanh[(Sqrt[-1/2 + Sqrt[5]/2]*x)/Sqrt[1 - x^2 + x^6]]","A",1
1821,1,78,123,0.0836877,"\int \frac{1}{\left(2 \sqrt{x}+\sqrt{1+x}\right)^2} \, dx","IntegrateAlgebraic[(2*Sqrt[x] + Sqrt[1 + x])^(-2),x]","-\frac{8}{9 (-1+3 x)}+\frac{4 \sqrt{x} \sqrt{1+x}}{3 (-1+3 x)}-\frac{2}{9} \log \left(-\sqrt{x}+\sqrt{1+x}\right)+\frac{10}{9} \log \left(1-x+\sqrt{x} \sqrt{1+x}\right)","\frac{\sqrt{x} \left(4-4 \sqrt{1+x}\right)}{3+3 x-3 \sqrt{1+x}+\sqrt{x} \left(-6+6 \sqrt{1+x}\right)}-\frac{2}{9} \log \left(-1-\sqrt{x}+\sqrt{1+x}\right)-2 \log \left(-1+\sqrt{x}+\sqrt{1+x}\right)+\frac{10}{9} \log \left(1+x-\sqrt{1+x}+\sqrt{x} \left(-2+2 \sqrt{1+x}\right)\right)",1,"-8/(9*(-1 + 3*x)) + (4*Sqrt[x]*Sqrt[1 + x])/(3*(-1 + 3*x)) - (2*Log[-Sqrt[x] + Sqrt[1 + x]])/9 + (10*Log[1 - x + Sqrt[x]*Sqrt[1 + x]])/9","A",1
1822,1,123,123,2.9777375,"\int \frac{1}{x^3 \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}} \, dx","IntegrateAlgebraic[1/(x^3*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]]),x]","\frac{4 \left(-2+a x^2\right) \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{15 x^2}-\frac{2 b \left(-3+2 a x^2\right) \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}} \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{15 a x^3}","\frac{4 \left(-2+a x^2\right) \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{15 x^2}-\frac{2 b \left(-3+2 a x^2\right) \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}} \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{15 a x^3}",1,"(4*(-2 + a*x^2)*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]])/(15*x^2) - (2*b*(-3 + 2*a*x^2)*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]])/(15*a*x^3)","A",1
1823,1,122,124,0.1886774,"\int \frac{1}{x \sqrt[4]{-b+a x^2}} \, dx","IntegrateAlgebraic[1/(x*(-b + a*x^2)^(1/4)),x]","\frac{\tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^2}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^2}}\right)}{\sqrt{2} \sqrt[4]{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^2}}{\sqrt{b}+\sqrt{-b+a x^2}}\right)}{\sqrt{2} \sqrt[4]{b}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^2}}{-\sqrt{b}+\sqrt{-b+a x^2}}\right)}{\sqrt{2} \sqrt[4]{b}}-\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^2}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^2}}\right)}{\sqrt{2} \sqrt[4]{b}}",1,"ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^2]/(Sqrt[2]*b^(1/4)))/(-b + a*x^2)^(1/4)]/(Sqrt[2]*b^(1/4)) - ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^2)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^2])]/(Sqrt[2]*b^(1/4))","A",1
1824,1,124,124,4.1318757,"\int \frac{1+k^{3/2} x^3}{\sqrt{\left(1-x^2\right) \left(1-k^2 x^2\right)} \left(-1+k^{3/2} x^3\right)} \, dx","IntegrateAlgebraic[(1 + k^(3/2)*x^3)/(Sqrt[(1 - x^2)*(1 - k^2*x^2)]*(-1 + k^(3/2)*x^3)),x]","-\frac{2 \tan ^{-1}\left(\frac{(-1+k) x}{1-2 \sqrt{k} x+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{3 (-1+k)}-\frac{4 \tan ^{-1}\left(\frac{\sqrt{1+k+k^2} x}{1+\sqrt{k} x+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{3 \sqrt{1+k+k^2}}","-\frac{2 \tan ^{-1}\left(\frac{(-1+k) x}{1-2 \sqrt{k} x+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{3 (-1+k)}-\frac{4 \tan ^{-1}\left(\frac{\sqrt{1+k+k^2} x}{1+\sqrt{k} x+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{3 \sqrt{1+k+k^2}}",1,"(-2*ArcTan[((-1 + k)*x)/(1 - 2*Sqrt[k]*x + k*x^2 + Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4])])/(3*(-1 + k)) - (4*ArcTan[(Sqrt[1 + k + k^2]*x)/(1 + Sqrt[k]*x + k*x^2 + Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4])])/(3*Sqrt[1 + k + k^2])","A",1
1825,1,124,124,0.6667384,"\int \frac{(-d+2 c x) \sqrt[4]{b x^3+a x^4}}{x} \, dx","IntegrateAlgebraic[((-d + 2*c*x)*(b*x^3 + a*x^4)^(1/4))/x,x]","\frac{(b c-4 a d+4 a c x) \sqrt[4]{b x^3+a x^4}}{4 a}+\frac{\left(3 b^2 c+4 a b d\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{8 a^{7/4}}+\frac{\left(-3 b^2 c-4 a b d\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{8 a^{7/4}}","\frac{(b c-4 a d+4 a c x) \sqrt[4]{b x^3+a x^4}}{4 a}+\frac{\left(3 b^2 c+4 a b d\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{8 a^{7/4}}+\frac{\left(-3 b^2 c-4 a b d\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{8 a^{7/4}}",1,"((b*c - 4*a*d + 4*a*c*x)*(b*x^3 + a*x^4)^(1/4))/(4*a) + ((3*b^2*c + 4*a*b*d)*ArcTan[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(8*a^(7/4)) + ((-3*b^2*c - 4*a*b*d)*ArcTanh[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(8*a^(7/4))","A",1
1826,1,128,124,1.0318128,"\int \frac{-1+x^6}{\left(1+x^6\right) \sqrt[3]{1+a^3 x^3+x^6}} \, dx","IntegrateAlgebraic[(-1 + x^6)/((1 + x^6)*(1 + a^3*x^3 + x^6)^(1/3)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{1+a^3 x^3+x^6}}\right)}{\sqrt{3} a}+\frac{\log \left(a^2 x-a \sqrt[3]{1+a^3 x^3+x^6}\right)}{3 a}-\frac{\log \left(a^2 x^2+a x \sqrt[3]{1+a^3 x^3+x^6}+\left(1+a^3 x^3+x^6\right)^{2/3}\right)}{6 a}","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{1+a^3 x^3+x^6}}\right)}{\sqrt{3} a}+\frac{\log \left(-a x+\sqrt[3]{1+a^3 x^3+x^6}\right)}{3 a}-\frac{\log \left(a^2 x^2+a x \sqrt[3]{1+a^3 x^3+x^6}+\left(1+a^3 x^3+x^6\right)^{2/3}\right)}{6 a}",1,"-(ArcTan[(Sqrt[3]*a*x)/(a*x + 2*(1 + a^3*x^3 + x^6)^(1/3))]/(Sqrt[3]*a)) + Log[a^2*x - a*(1 + a^3*x^3 + x^6)^(1/3)]/(3*a) - Log[a^2*x^2 + a*x*(1 + a^3*x^3 + x^6)^(1/3) + (1 + a^3*x^3 + x^6)^(2/3)]/(6*a)","A",1
1827,1,124,124,0.5105718,"\int \frac{\left(2+x^3+4 x^6\right) \sqrt[3]{x+2 x^3-x^4-x^7}}{\left(-1-2 x^2+x^3+x^6\right) \left(-1-x^2+x^3+x^6\right)} \, dx","IntegrateAlgebraic[((2 + x^3 + 4*x^6)*(x + 2*x^3 - x^4 - x^7)^(1/3))/((-1 - 2*x^2 + x^3 + x^6)*(-1 - x^2 + x^3 + x^6)),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+2 x^3-x^4-x^7}}\right)-\log \left(-x+\sqrt[3]{x+2 x^3-x^4-x^7}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{x+2 x^3-x^4-x^7}+\left(x+2 x^3-x^4-x^7\right)^{2/3}\right)","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+2 x^3-x^4-x^7}}\right)-\log \left(-x+\sqrt[3]{x+2 x^3-x^4-x^7}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{x+2 x^3-x^4-x^7}+\left(x+2 x^3-x^4-x^7\right)^{2/3}\right)",1,"-(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(x + 2*x^3 - x^4 - x^7)^(1/3))]) - Log[-x + (x + 2*x^3 - x^4 - x^7)^(1/3)] + Log[x^2 + x*(x + 2*x^3 - x^4 - x^7)^(1/3) + (x + 2*x^3 - x^4 - x^7)^(2/3)]/2","A",1
1828,1,124,124,0.2385317,"\int \frac{x^2}{\sqrt{a x^2+\sqrt{b+a^2 x^4}}} \, dx","IntegrateAlgebraic[x^2/Sqrt[a*x^2 + Sqrt[b + a^2*x^4]],x]","-\frac{b x}{8 a \left(a x^2+\sqrt{b+a^2 x^4}\right)^{3/2}}+\frac{x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{4 a}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{b}}\right)}{8 \sqrt{2} a^{3/2}}","-\frac{b x}{8 a \left(a x^2+\sqrt{b+a^2 x^4}\right)^{3/2}}+\frac{x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{4 a}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{b}}\right)}{8 \sqrt{2} a^{3/2}}",1,"-1/8*(b*x)/(a*(a*x^2 + Sqrt[b + a^2*x^4])^(3/2)) + (x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/(4*a) - (Sqrt[b]*ArcTan[(Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[b]])/(8*Sqrt[2]*a^(3/2))","A",1
1829,1,124,124,0.1917204,"\int \frac{\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1+x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/(1 + x^2)^(3/2),x]","-\frac{\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{1+x^2+x \sqrt{1+x^2}}+\frac{1}{4} \text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{-\text{$\#$1}+3 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","-\frac{\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{1+x^2+x \sqrt{1+x^2}}+\frac{1}{4} \text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{-\text{$\#$1}+3 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]",1,"-(Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/(1 + x^2 + x*Sqrt[1 + x^2])) + RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/(-#1 + 3*#1^3 - 3*#1^5 + #1^7) & ]/4","A",1
1830,1,124,124,0.00135,"\int \frac{\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1+x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/(1 + x^2)^(3/2),x]","-\frac{\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{1+x^2+x \sqrt{1+x^2}}+\frac{1}{4} \text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{-\text{$\#$1}+3 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","-\frac{\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{1+x^2+x \sqrt{1+x^2}}+\frac{1}{4} \text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{-\text{$\#$1}+3 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]",1,"-(Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/(1 + x^2 + x*Sqrt[1 + x^2])) + RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/(-#1 + 3*#1^3 - 3*#1^5 + #1^7) & ]/4","A",1
1831,1,124,124,0.0013208,"\int \frac{\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1+x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/(1 + x^2)^(3/2),x]","-\frac{\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{1+x^2+x \sqrt{1+x^2}}+\frac{1}{4} \text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{-\text{$\#$1}+3 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","-\frac{\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{1+x^2+x \sqrt{1+x^2}}+\frac{1}{4} \text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{-\text{$\#$1}+3 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]",1,"-(Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/(1 + x^2 + x*Sqrt[1 + x^2])) + RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/(-#1 + 3*#1^3 - 3*#1^5 + #1^7) & ]/4","A",1
1832,1,125,125,0.2261698,"\int \frac{1}{(-1+x) \sqrt[3]{-3-2 x+x^2}} \, dx","IntegrateAlgebraic[1/((-1 + x)*(-3 - 2*x + x^2)^(1/3)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{\sqrt[3]{2} \sqrt[3]{-3-2 x+x^2}}{\sqrt{3}}\right)}{2\ 2^{2/3}}-\frac{\log \left(2+\sqrt[3]{2} \sqrt[3]{-3-2 x+x^2}\right)}{2\ 2^{2/3}}+\frac{\log \left(4-2 \sqrt[3]{2} \sqrt[3]{-3-2 x+x^2}+2^{2/3} \left(-3-2 x+x^2\right)^{2/3}\right)}{4\ 2^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{\sqrt[3]{2} \sqrt[3]{-3-2 x+x^2}}{\sqrt{3}}\right)}{2\ 2^{2/3}}-\frac{\log \left(2+\sqrt[3]{2} \sqrt[3]{-3-2 x+x^2}\right)}{2\ 2^{2/3}}+\frac{\log \left(4-2 \sqrt[3]{2} \sqrt[3]{-3-2 x+x^2}+2^{2/3} \left(-3-2 x+x^2\right)^{2/3}\right)}{4\ 2^{2/3}}",1,"-1/2*(Sqrt[3]*ArcTan[1/Sqrt[3] - (2^(1/3)*(-3 - 2*x + x^2)^(1/3))/Sqrt[3]])/2^(2/3) - Log[2 + 2^(1/3)*(-3 - 2*x + x^2)^(1/3)]/(2*2^(2/3)) + Log[4 - 2*2^(1/3)*(-3 - 2*x + x^2)^(1/3) + 2^(2/3)*(-3 - 2*x + x^2)^(2/3)]/(4*2^(2/3))","A",1
1833,1,125,125,0.2134397,"\int \frac{1}{(-1+x) \sqrt[3]{-1-2 x+x^2}} \, dx","IntegrateAlgebraic[1/((-1 + x)*(-1 - 2*x + x^2)^(1/3)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2^{2/3} \sqrt[3]{-1-2 x+x^2}}{\sqrt{3}}\right)}{2 \sqrt[3]{2}}-\frac{\log \left(2+2^{2/3} \sqrt[3]{-1-2 x+x^2}\right)}{2 \sqrt[3]{2}}+\frac{\log \left(-2+2^{2/3} \sqrt[3]{-1-2 x+x^2}-\sqrt[3]{2} \left(-1-2 x+x^2\right)^{2/3}\right)}{4 \sqrt[3]{2}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2^{2/3} \sqrt[3]{-1-2 x+x^2}}{\sqrt{3}}\right)}{2 \sqrt[3]{2}}-\frac{\log \left(2+2^{2/3} \sqrt[3]{-1-2 x+x^2}\right)}{2 \sqrt[3]{2}}+\frac{\log \left(-2+2^{2/3} \sqrt[3]{-1-2 x+x^2}-\sqrt[3]{2} \left(-1-2 x+x^2\right)^{2/3}\right)}{4 \sqrt[3]{2}}",1,"-1/2*(Sqrt[3]*ArcTan[1/Sqrt[3] - (2^(2/3)*(-1 - 2*x + x^2)^(1/3))/Sqrt[3]])/2^(1/3) - Log[2 + 2^(2/3)*(-1 - 2*x + x^2)^(1/3)]/(2*2^(1/3)) + Log[-2 + 2^(2/3)*(-1 - 2*x + x^2)^(1/3) - 2^(1/3)*(-1 - 2*x + x^2)^(2/3)]/(4*2^(1/3))","A",1
1834,1,125,125,0.2124771,"\int \frac{1}{(1+x) \sqrt[3]{-1+2 x+x^2}} \, dx","IntegrateAlgebraic[1/((1 + x)*(-1 + 2*x + x^2)^(1/3)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2^{2/3} \sqrt[3]{-1+2 x+x^2}}{\sqrt{3}}\right)}{2 \sqrt[3]{2}}-\frac{\log \left(2+2^{2/3} \sqrt[3]{-1+2 x+x^2}\right)}{2 \sqrt[3]{2}}+\frac{\log \left(-2+2^{2/3} \sqrt[3]{-1+2 x+x^2}-\sqrt[3]{2} \left(-1+2 x+x^2\right)^{2/3}\right)}{4 \sqrt[3]{2}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2^{2/3} \sqrt[3]{-1+2 x+x^2}}{\sqrt{3}}\right)}{2 \sqrt[3]{2}}-\frac{\log \left(2+2^{2/3} \sqrt[3]{-1+2 x+x^2}\right)}{2 \sqrt[3]{2}}+\frac{\log \left(-2+2^{2/3} \sqrt[3]{-1+2 x+x^2}-\sqrt[3]{2} \left(-1+2 x+x^2\right)^{2/3}\right)}{4 \sqrt[3]{2}}",1,"-1/2*(Sqrt[3]*ArcTan[1/Sqrt[3] - (2^(2/3)*(-1 + 2*x + x^2)^(1/3))/Sqrt[3]])/2^(1/3) - Log[2 + 2^(2/3)*(-1 + 2*x + x^2)^(1/3)]/(2*2^(1/3)) + Log[-2 + 2^(2/3)*(-1 + 2*x + x^2)^(1/3) - 2^(1/3)*(-1 + 2*x + x^2)^(2/3)]/(4*2^(1/3))","A",1
1835,1,125,125,0.7238162,"\int \frac{-2 a b x^2+(a+b) x^3}{(-a+x) (-b+x) \sqrt[4]{x^2 (-a+x) (-b+x)} \left(-a b d+(a+b) d x+(1-d) x^2\right)} \, dx","IntegrateAlgebraic[(-2*a*b*x^2 + (a + b)*x^3)/((-a + x)*(-b + x)*(x^2*(-a + x)*(-b + x))^(1/4)*(-(a*b*d) + (a + b)*d*x + (1 - d)*x^2)),x]","-\frac{4 \left(a b x^2+(-a-b) x^3+x^4\right)^{3/4}}{x (-a+x) (-b+x)}+2 \sqrt[4]{d} \tan ^{-1}\left(\frac{x}{\sqrt[4]{d} \sqrt[4]{a b x^2+(-a-b) x^3+x^4}}\right)+2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{d} \sqrt[4]{a b x^2+(-a-b) x^3+x^4}}\right)","-\frac{4 \left(a b x^2+(-a-b) x^3+x^4\right)^{3/4}}{x (-a+x) (-b+x)}+2 \sqrt[4]{d} \tan ^{-1}\left(\frac{x}{\sqrt[4]{d} \sqrt[4]{a b x^2+(-a-b) x^3+x^4}}\right)+2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{d} \sqrt[4]{a b x^2+(-a-b) x^3+x^4}}\right)",1,"(-4*(a*b*x^2 + (-a - b)*x^3 + x^4)^(3/4))/(x*(-a + x)*(-b + x)) + 2*d^(1/4)*ArcTan[x/(d^(1/4)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/4))] + 2*d^(1/4)*ArcTanh[x/(d^(1/4)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/4))]","A",1
1836,1,125,125,3.669673,"\int \frac{\left(a^2-2 a x+x^2\right) \left(a b^2-2 (2 a-b) b x+(3 a-2 b) x^2\right)}{\left(x (-a+x) (-b+x)^2\right)^{3/4} \left(a^3 d+\left(b^2-3 a^2 d\right) x+(-2 b+3 a d) x^2+(1-d) x^3\right)} \, dx","IntegrateAlgebraic[((a^2 - 2*a*x + x^2)*(a*b^2 - 2*(2*a - b)*b*x + (3*a - 2*b)*x^2))/((x*(-a + x)*(-b + x)^2)^(3/4)*(a^3*d + (b^2 - 3*a^2*d)*x + (-2*b + 3*a*d)*x^2 + (1 - d)*x^3)),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{3/4}}{x (-b+x)^2}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{3/4}}{x (-b+x)^2}\right)}{d^{3/4}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{3/4}}{x (-b+x)^2}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{3/4}}{x (-b+x)^2}\right)}{d^{3/4}}",1,"(-2*ArcTan[(d^(1/4)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(3/4))/(x*(-b + x)^2)])/d^(3/4) + (2*ArcTanh[(d^(1/4)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(3/4))/(x*(-b + x)^2)])/d^(3/4)","A",1
1837,1,125,125,4.2598509,"\int \frac{-1+k^{3/2} x^3}{\sqrt{\left(1-x^2\right) \left(1-k^2 x^2\right)} \left(1+k^{3/2} x^3\right)} \, dx","IntegrateAlgebraic[(-1 + k^(3/2)*x^3)/(Sqrt[(1 - x^2)*(1 - k^2*x^2)]*(1 + k^(3/2)*x^3)),x]","-\frac{4 \tan ^{-1}\left(\frac{\sqrt{1+k+k^2} x}{1-\sqrt{k} x+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{3 \sqrt{1+k+k^2}}-\frac{2 \tan ^{-1}\left(\frac{(-1+k) x}{1+2 \sqrt{k} x+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{3 (-1+k)}","-\frac{4 \tan ^{-1}\left(\frac{\sqrt{1+k+k^2} x}{1-\sqrt{k} x+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{3 \sqrt{1+k+k^2}}-\frac{2 \tan ^{-1}\left(\frac{(-1+k) x}{1+2 \sqrt{k} x+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{3 (-1+k)}",1,"(-4*ArcTan[(Sqrt[1 + k + k^2]*x)/(1 - Sqrt[k]*x + k*x^2 + Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4])])/(3*Sqrt[1 + k + k^2]) - (2*ArcTan[((-1 + k)*x)/(1 + 2*Sqrt[k]*x + k*x^2 + Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4])])/(3*(-1 + k))","A",1
1838,1,125,125,0.2322381,"\int \frac{\sqrt{-x-x^2+x^3}}{-1+x^4} \, dx","IntegrateAlgebraic[Sqrt[-x - x^2 + x^3]/(-1 + x^4),x]","\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)-\frac{1}{4} \sqrt{1-2 i} \tan ^{-1}\left(\frac{\sqrt{1-2 i} \sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)-\frac{1}{4} \sqrt{1+2 i} \tan ^{-1}\left(\frac{\sqrt{1+2 i} \sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)","\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)-\frac{1}{4} \sqrt{1-2 i} \tan ^{-1}\left(\frac{\sqrt{1-2 i} \sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)-\frac{1}{4} \sqrt{1+2 i} \tan ^{-1}\left(\frac{\sqrt{1+2 i} \sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)",1,"ArcTan[Sqrt[-x - x^2 + x^3]/(-1 - x + x^2)]/2 - (Sqrt[1 - 2*I]*ArcTan[(Sqrt[1 - 2*I]*Sqrt[-x - x^2 + x^3])/(-1 - x + x^2)])/4 - (Sqrt[1 + 2*I]*ArcTan[(Sqrt[1 + 2*I]*Sqrt[-x - x^2 + x^3])/(-1 - x + x^2)])/4","A",1
1839,1,125,125,0.7470622,"\int \frac{\left(b+a x^3\right) \sqrt{x+x^4}}{-d+c x^3} \, dx","IntegrateAlgebraic[((b + a*x^3)*Sqrt[x + x^4])/(-d + c*x^3),x]","\frac{a x \sqrt{x+x^4}}{3 c}+\frac{2 \sqrt{-c-d} (b c+a d) \tan ^{-1}\left(\frac{\sqrt{-c-d} x \sqrt{x+x^4}}{\sqrt{d} (1+x) \left(1-x+x^2\right)}\right)}{3 c^2 \sqrt{d}}+\frac{(a c+2 b c+2 a d) \tanh ^{-1}\left(\frac{x^2}{\sqrt{x+x^4}}\right)}{3 c^2}","\frac{a x \sqrt{x+x^4}}{3 c}+\frac{2 \sqrt{-c-d} (b c+a d) \tan ^{-1}\left(\frac{\sqrt{-c-d} x \sqrt{x+x^4}}{\sqrt{d} (1+x) \left(1-x+x^2\right)}\right)}{3 c^2 \sqrt{d}}+\frac{(a c+2 b c+2 a d) \tanh ^{-1}\left(\frac{x^2}{\sqrt{x+x^4}}\right)}{3 c^2}",1,"(a*x*Sqrt[x + x^4])/(3*c) + (2*Sqrt[-c - d]*(b*c + a*d)*ArcTan[(Sqrt[-c - d]*x*Sqrt[x + x^4])/(Sqrt[d]*(1 + x)*(1 - x + x^2))])/(3*c^2*Sqrt[d]) + ((a*c + 2*b*c + 2*a*d)*ArcTanh[x^2/Sqrt[x + x^4]])/(3*c^2)","A",1
1840,1,120,125,4.463876,"\int \frac{1-2 x+2 x^4}{x \sqrt[4]{-1+x^4}} \, dx","IntegrateAlgebraic[(1 - 2*x + 2*x^4)/(x*(-1 + x^4)^(1/4)),x]","\frac{2}{3} \left(-1+x^4\right)^{3/4}+\tan ^{-1}\left(\frac{\sqrt[4]{-1+x^4}}{x}\right)-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^4}}{-1+\sqrt{-1+x^4}}\right)}{2 \sqrt{2}}-\tanh ^{-1}\left(\frac{\sqrt[4]{-1+x^4}}{x}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^4}}{1+\sqrt{-1+x^4}}\right)}{2 \sqrt{2}}","\frac{2}{3} \left(-1+x^4\right)^{3/4}+\tan ^{-1}\left(\frac{\sqrt[4]{-1+x^4}}{x}\right)+\frac{\tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{\sqrt{-1+x^4}}{\sqrt{2}}}{\sqrt[4]{-1+x^4}}\right)}{2 \sqrt{2}}-\tanh ^{-1}\left(\frac{\sqrt[4]{-1+x^4}}{x}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+x^4}}{1+\sqrt{-1+x^4}}\right)}{2 \sqrt{2}}",1,"(2*(-1 + x^4)^(3/4))/3 + ArcTan[(-1 + x^4)^(1/4)/x] - ArcTan[(Sqrt[2]*(-1 + x^4)^(1/4))/(-1 + Sqrt[-1 + x^4])]/(2*Sqrt[2]) - ArcTanh[(-1 + x^4)^(1/4)/x] - ArcTanh[(Sqrt[2]*(-1 + x^4)^(1/4))/(1 + Sqrt[-1 + x^4])]/(2*Sqrt[2])","A",1
1841,1,125,125,12.7673498,"\int \frac{x^2 \left(-4 b+a x^5\right)}{\left(b+a x^5\right)^{3/4} \left(b+c x^4+a x^5\right)} \, dx","IntegrateAlgebraic[(x^2*(-4*b + a*x^5))/((b + a*x^5)^(3/4)*(b + c*x^4 + a*x^5)),x]","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x \sqrt[4]{b+a x^5}}{-\sqrt{c} x^2+\sqrt{b+a x^5}}\right)}{c^{3/4}}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{c} x^2}{\sqrt{2}}+\frac{\sqrt{b+a x^5}}{\sqrt{2} \sqrt[4]{c}}}{x \sqrt[4]{b+a x^5}}\right)}{c^{3/4}}","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x \sqrt[4]{b+a x^5}}{-\sqrt{c} x^2+\sqrt{b+a x^5}}\right)}{c^{3/4}}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{c} x^2}{\sqrt{2}}+\frac{\sqrt{b+a x^5}}{\sqrt{2} \sqrt[4]{c}}}{x \sqrt[4]{b+a x^5}}\right)}{c^{3/4}}",1,"-((Sqrt[2]*ArcTan[(Sqrt[2]*c^(1/4)*x*(b + a*x^5)^(1/4))/(-(Sqrt[c]*x^2) + Sqrt[b + a*x^5])])/c^(3/4)) + (Sqrt[2]*ArcTanh[((c^(1/4)*x^2)/Sqrt[2] + Sqrt[b + a*x^5]/(Sqrt[2]*c^(1/4)))/(x*(b + a*x^5)^(1/4))])/c^(3/4)","A",1
1842,1,122,125,11.3407741,"\int \frac{1}{\sqrt[6]{1+2 x-x^2-4 x^3-x^4+2 x^5+x^6}} \, dx","IntegrateAlgebraic[(1 + 2*x - x^2 - 4*x^3 - x^4 + 2*x^5 + x^6)^(-1/6),x]","\frac{\sqrt[3]{-1+x} (1+x)^{2/3} \left(\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x}}{\sqrt{3} \sqrt[3]{-1+x}}\right)-\log \left(-1+\frac{\sqrt[3]{1+x}}{\sqrt[3]{-1+x}}\right)+\frac{1}{2} \log \left(1+\frac{\sqrt[3]{1+x}}{\sqrt[3]{-1+x}}+\frac{(1+x)^{2/3}}{(-1+x)^{2/3}}\right)\right)}{\sqrt[6]{(-1+x)^2 (1+x)^4}}","\frac{\sqrt[3]{-1+x} (1+x)^{2/3} \left(\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{1+x}}{2 \sqrt[3]{-1+x}+\sqrt[3]{1+x}}\right)-\log \left(\sqrt[3]{-1+x}-\sqrt[3]{1+x}\right)+\frac{1}{2} \log \left((-1+x)^{2/3}+\sqrt[3]{-1+x} \sqrt[3]{1+x}+(1+x)^{2/3}\right)\right)}{\sqrt[6]{(-1+x)^2 (1+x)^4}}",1,"((-1 + x)^(1/3)*(1 + x)^(2/3)*(Sqrt[3]*ArcTan[1/Sqrt[3] + (2*(1 + x)^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))] - Log[-1 + (1 + x)^(1/3)/(-1 + x)^(1/3)] + Log[1 + (1 + x)^(1/3)/(-1 + x)^(1/3) + (1 + x)^(2/3)/(-1 + x)^(2/3)]/2))/((-1 + x)^2*(1 + x)^4)^(1/6)","A",1
1843,1,125,125,15.6945238,"\int \frac{x^2 \left(-2 b+a x^6\right)}{\left(b+a x^6\right)^{3/4} \left(b+c x^4+a x^6\right)} \, dx","IntegrateAlgebraic[(x^2*(-2*b + a*x^6))/((b + a*x^6)^(3/4)*(b + c*x^4 + a*x^6)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x \sqrt[4]{b+a x^6}}{-\sqrt{c} x^2+\sqrt{b+a x^6}}\right)}{\sqrt{2} c^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[4]{c} x^2}{\sqrt{2}}+\frac{\sqrt{b+a x^6}}{\sqrt{2} \sqrt[4]{c}}}{x \sqrt[4]{b+a x^6}}\right)}{\sqrt{2} c^{3/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x \sqrt[4]{b+a x^6}}{-\sqrt{c} x^2+\sqrt{b+a x^6}}\right)}{\sqrt{2} c^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[4]{c} x^2}{\sqrt{2}}+\frac{\sqrt{b+a x^6}}{\sqrt{2} \sqrt[4]{c}}}{x \sqrt[4]{b+a x^6}}\right)}{\sqrt{2} c^{3/4}}",1,"-(ArcTan[(Sqrt[2]*c^(1/4)*x*(b + a*x^6)^(1/4))/(-(Sqrt[c]*x^2) + Sqrt[b + a*x^6])]/(Sqrt[2]*c^(3/4))) + ArcTanh[((c^(1/4)*x^2)/Sqrt[2] + Sqrt[b + a*x^6]/(Sqrt[2]*c^(1/4)))/(x*(b + a*x^6)^(1/4))]/(Sqrt[2]*c^(3/4))","A",1
1844,1,125,125,13.9745457,"\int \frac{\left(4+x^3\right) \left(1+2 x^3+x^6+x^8\right)}{x^4 \sqrt[4]{1+x^3} \left(-1-2 x^3-x^6+x^8\right)} \, dx","IntegrateAlgebraic[((4 + x^3)*(1 + 2*x^3 + x^6 + x^8))/(x^4*(1 + x^3)^(1/4)*(-1 - 2*x^3 - x^6 + x^8)),x]","\frac{4 \left(1+x^3\right)^{3/4}}{3 x^3}-2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^3}}\right)+\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{1+x^3}}{-x^2+\sqrt{1+x^3}}\right)-2 \tanh ^{-1}\left(\frac{\sqrt[4]{1+x^3}}{x}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{1+x^3}}{x^2+\sqrt{1+x^3}}\right)","\frac{4 \left(1+x^3\right)^{3/4}}{3 x^3}-2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^3}}\right)+\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{1+x^3}}{-x^2+\sqrt{1+x^3}}\right)-2 \tanh ^{-1}\left(\frac{\sqrt[4]{1+x^3}}{x}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{1+x^3}}{x^2+\sqrt{1+x^3}}\right)",1,"(4*(1 + x^3)^(3/4))/(3*x^3) - 2*ArcTan[x/(1 + x^3)^(1/4)] + Sqrt[2]*ArcTan[(Sqrt[2]*x*(1 + x^3)^(1/4))/(-x^2 + Sqrt[1 + x^3])] - 2*ArcTanh[(1 + x^3)^(1/4)/x] + Sqrt[2]*ArcTanh[(Sqrt[2]*x*(1 + x^3)^(1/4))/(x^2 + Sqrt[1 + x^3])]","A",1
1845,1,125,125,0.4439736,"\int \frac{2-x^4+2 x^8}{\sqrt[4]{1+x^4} \left(-2+x^4+x^8\right)} \, dx","IntegrateAlgebraic[(2 - x^4 + 2*x^8)/((1 + x^4)^(1/4)*(-2 + x^4 + x^8)),x]","\tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\sqrt[4]{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{2} \sqrt[4]{1+x^4}}\right)-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{1+x^4}}\right)}{2 \sqrt[4]{2}}+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\sqrt[4]{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{2} \sqrt[4]{1+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{1+x^4}}\right)}{2 \sqrt[4]{2}}","\tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\sqrt[4]{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{2} \sqrt[4]{1+x^4}}\right)-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{1+x^4}}\right)}{2 \sqrt[4]{2}}+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\sqrt[4]{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{2} \sqrt[4]{1+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{1+x^4}}\right)}{2 \sqrt[4]{2}}",1,"ArcTan[x/(1 + x^4)^(1/4)] - 2^(1/4)*ArcTan[x/(2^(1/4)*(1 + x^4)^(1/4))] - ArcTan[(2^(1/4)*x)/(1 + x^4)^(1/4)]/(2*2^(1/4)) + ArcTanh[x/(1 + x^4)^(1/4)] - 2^(1/4)*ArcTanh[x/(2^(1/4)*(1 + x^4)^(1/4))] - ArcTanh[(2^(1/4)*x)/(1 + x^4)^(1/4)]/(2*2^(1/4))","A",1
1846,1,125,125,0.2606356,"\int \frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\left(b^2+a x^2\right)^2} \, dx","IntegrateAlgebraic[Sqrt[b + Sqrt[b^2 + a*x^2]]/(b^2 + a*x^2)^2,x]","\frac{x}{2 b \left(b^2+a x^2\right) \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{3 x}{4 b^2 \sqrt{b^2+a x^2} \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{3 \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{4 \sqrt{a} b^{5/2}}","\frac{x}{2 b \left(b^2+a x^2\right) \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{3 x}{4 b^2 \sqrt{b^2+a x^2} \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{3 \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{4 \sqrt{a} b^{5/2}}",1,"x/(2*b*(b^2 + a*x^2)*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (3*x)/(4*b^2*Sqrt[b^2 + a*x^2]*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (3*ArcTan[(Sqrt[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/(4*Sqrt[a]*b^(5/2))","A",1
1847,1,126,126,0.6571892,"\int \frac{\left(b+a x^3\right) \sqrt{-x+x^4}}{-d+c x^3} \, dx","IntegrateAlgebraic[((b + a*x^3)*Sqrt[-x + x^4])/(-d + c*x^3),x]","\frac{a x \sqrt{-x+x^4}}{3 c}+\frac{2 \sqrt{c-d} (b c+a d) \tan ^{-1}\left(\frac{\sqrt{c-d} x \sqrt{-x+x^4}}{\sqrt{d} (-1+x) \left(1+x+x^2\right)}\right)}{3 c^2 \sqrt{d}}+\frac{(-a c+2 b c+2 a d) \tanh ^{-1}\left(\frac{x^2}{\sqrt{-x+x^4}}\right)}{3 c^2}","\frac{a x \sqrt{-x+x^4}}{3 c}+\frac{2 \sqrt{c-d} (b c+a d) \tan ^{-1}\left(\frac{\sqrt{c-d} x \sqrt{-x+x^4}}{\sqrt{d} (-1+x) \left(1+x+x^2\right)}\right)}{3 c^2 \sqrt{d}}+\frac{(-a c+2 b c+2 a d) \tanh ^{-1}\left(\frac{x^2}{\sqrt{-x+x^4}}\right)}{3 c^2}",1,"(a*x*Sqrt[-x + x^4])/(3*c) + (2*Sqrt[c - d]*(b*c + a*d)*ArcTan[(Sqrt[c - d]*x*Sqrt[-x + x^4])/(Sqrt[d]*(-1 + x)*(1 + x + x^2))])/(3*c^2*Sqrt[d]) + ((-(a*c) + 2*b*c + 2*a*d)*ArcTanh[x^2/Sqrt[-x + x^4]])/(3*c^2)","A",1
1848,1,126,126,0.4403865,"\int \frac{b+a x^3}{x^3 \left(-b+a x^3\right) \sqrt[4]{b x+a x^4}} \, dx","IntegrateAlgebraic[(b + a*x^3)/(x^3*(-b + a*x^3)*(b*x + a*x^4)^(1/4)),x]","\frac{4 \left(b x+a x^4\right)^{3/4}}{9 b x^3}-\frac{2\ 2^{3/4} a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 b}-\frac{2\ 2^{3/4} a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 b}","\frac{4 \left(b x+a x^4\right)^{3/4}}{9 b x^3}-\frac{2\ 2^{3/4} a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 b}-\frac{2\ 2^{3/4} a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 b}",1,"(4*(b*x + a*x^4)^(3/4))/(9*b*x^3) - (2*2^(3/4)*a^(3/4)*ArcTan[(2^(1/4)*a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/(3*b) - (2*2^(3/4)*a^(3/4)*ArcTanh[(2^(1/4)*a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/(3*b)","A",1
1849,1,126,126,0.2054686,"\int \frac{\left(1+2 x^6\right) \sqrt[3]{x+x^3-x^7}}{\left(-1+x^6\right)^2} \, dx","IntegrateAlgebraic[((1 + 2*x^6)*(x + x^3 - x^7)^(1/3))/(-1 + x^6)^2,x]","-\frac{x \sqrt[3]{x+x^3-x^7}}{2 \left(-1+x^6\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3-x^7}}\right)}{2 \sqrt{3}}-\frac{1}{6} \log \left(-x+\sqrt[3]{x+x^3-x^7}\right)+\frac{1}{12} \log \left(x^2+x \sqrt[3]{x+x^3-x^7}+\left(x+x^3-x^7\right)^{2/3}\right)","-\frac{x \sqrt[3]{x+x^3-x^7}}{2 \left(-1+x^6\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3-x^7}}\right)}{2 \sqrt{3}}-\frac{1}{6} \log \left(-x+\sqrt[3]{x+x^3-x^7}\right)+\frac{1}{12} \log \left(x^2+x \sqrt[3]{x+x^3-x^7}+\left(x+x^3-x^7\right)^{2/3}\right)",1,"-1/2*(x*(x + x^3 - x^7)^(1/3))/(-1 + x^6) - ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^3 - x^7)^(1/3))]/(2*Sqrt[3]) - Log[-x + (x + x^3 - x^7)^(1/3)]/6 + Log[x^2 + x*(x + x^3 - x^7)^(1/3) + (x + x^3 - x^7)^(2/3)]/12","A",1
1850,1,127,127,3.5145646,"\int \frac{-\left((2 a-b) b^2\right)+(4 a-b) b x-(2 a+b) x^2+x^3}{\left((-a+x) (-b+x)^2\right)^{3/4} \left(a+b^2 d-(1+2 b d) x+d x^2\right)} \, dx","IntegrateAlgebraic[(-((2*a - b)*b^2) + (4*a - b)*b*x - (2*a + b)*x^2 + x^3)/(((-a + x)*(-b + x)^2)^(3/4)*(a + b^2*d - (1 + 2*b*d)*x + d*x^2)),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{3/4}}{(b-x) (-a+x)}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{3/4}}{(b-x) (-a+x)}\right)}{d^{3/4}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{3/4}}{(b-x) (-a+x)}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{3/4}}{(b-x) (-a+x)}\right)}{d^{3/4}}",1,"(-2*ArcTan[(d^(1/4)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(3/4))/((b - x)*(-a + x))])/d^(3/4) + (2*ArcTanh[(d^(1/4)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(3/4))/((b - x)*(-a + x))])/d^(3/4)","A",1
1851,1,127,127,0.4151723,"\int \frac{2+x^4}{\sqrt[4]{x^2+x^4} \left(-1-x^4+2 x^8\right)} \, dx","IntegrateAlgebraic[(2 + x^4)/((x^2 + x^4)^(1/4)*(-1 - x^4 + 2*x^8)),x]","-\frac{\left(x^2+x^4\right)^{3/4}}{x \left(1+x^2\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2 \sqrt[4]{2}}+\frac{1}{4} \text{RootSum}\left[3-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{x^2+x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","-\frac{\left(x^2+x^4\right)^{3/4}}{x \left(1+x^2\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2 \sqrt[4]{2}}+\frac{1}{4} \text{RootSum}\left[3-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{x^2+x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"-((x^2 + x^4)^(3/4)/(x*(1 + x^2))) - ArcTan[(2^(1/4)*x)/(x^2 + x^4)^(1/4)]/(2*2^(1/4)) - ArcTanh[(2^(1/4)*x)/(x^2 + x^4)^(1/4)]/(2*2^(1/4)) + RootSum[3 - 2*#1^4 + #1^8 & , (-Log[x] + Log[(x^2 + x^4)^(1/4) - x*#1])/#1 & ]/4","A",1
1852,1,127,127,0.0012008,"\int \frac{2+x^4}{\sqrt[4]{x^2+x^4} \left(-1-x^4+2 x^8\right)} \, dx","IntegrateAlgebraic[(2 + x^4)/((x^2 + x^4)^(1/4)*(-1 - x^4 + 2*x^8)),x]","-\frac{\left(x^2+x^4\right)^{3/4}}{x \left(1+x^2\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2 \sqrt[4]{2}}+\frac{1}{4} \text{RootSum}\left[3-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{x^2+x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","-\frac{\left(x^2+x^4\right)^{3/4}}{x \left(1+x^2\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^4}}\right)}{2 \sqrt[4]{2}}+\frac{1}{4} \text{RootSum}\left[3-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{x^2+x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"-((x^2 + x^4)^(3/4)/(x*(1 + x^2))) - ArcTan[(2^(1/4)*x)/(x^2 + x^4)^(1/4)]/(2*2^(1/4)) - ArcTanh[(2^(1/4)*x)/(x^2 + x^4)^(1/4)]/(2*2^(1/4)) + RootSum[3 - 2*#1^4 + #1^8 & , (-Log[x] + Log[(x^2 + x^4)^(1/4) - x*#1])/#1 & ]/4","A",1
1853,1,77,127,0.2457611,"\int \frac{\sqrt{x+\sqrt{1+x}}}{x^2} \, dx","IntegrateAlgebraic[Sqrt[x + Sqrt[1 + x]]/x^2,x]","-\frac{\sqrt{x+\sqrt{1+x}}}{x}-\frac{1}{2} \tan ^{-1}\left(1+\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}\right)-\frac{3}{2} \tanh ^{-1}\left(1-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}\right)","-\frac{\sqrt{x+\sqrt{1+x}}}{x}-\frac{1}{4} \tan ^{-1}\left(\frac{1+2 x-2 \sqrt{x+\sqrt{1+x}}+\sqrt{1+x} \left(3-2 \sqrt{x+\sqrt{1+x}}\right)}{2+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}}\right)-\frac{3}{2} \tanh ^{-1}\left(1-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}\right)",1,"-(Sqrt[x + Sqrt[1 + x]]/x) - ArcTan[1 + Sqrt[1 + x] - Sqrt[x + Sqrt[1 + x]]]/2 - (3*ArcTanh[1 - Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]]])/2","A",1
1854,1,127,127,0.296523,"\int \frac{(-1+x) \sqrt{x+\sqrt{1+x^2}}}{1+x} \, dx","IntegrateAlgebraic[((-1 + x)*Sqrt[x + Sqrt[1 + x^2]])/(1 + x),x]","\frac{2 (-6+x) \sqrt{1+x^2}+2 \left(-1-6 x+x^2\right)}{3 \sqrt{x+\sqrt{1+x^2}}}+4 \sqrt{1+\sqrt{2}} \tan ^{-1}\left(\sqrt{-1+\sqrt{2}} \sqrt{x+\sqrt{1+x^2}}\right)+4 \sqrt{-1+\sqrt{2}} \tanh ^{-1}\left(\sqrt{1+\sqrt{2}} \sqrt{x+\sqrt{1+x^2}}\right)","\frac{2 (-6+x) \sqrt{1+x^2}+2 \left(-1-6 x+x^2\right)}{3 \sqrt{x+\sqrt{1+x^2}}}+4 \sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{x+\sqrt{1+x^2}}}{\sqrt{1+\sqrt{2}}}\right)+4 \sqrt{-1+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{x+\sqrt{1+x^2}}}{\sqrt{-1+\sqrt{2}}}\right)",1,"(2*(-6 + x)*Sqrt[1 + x^2] + 2*(-1 - 6*x + x^2))/(3*Sqrt[x + Sqrt[1 + x^2]]) + 4*Sqrt[1 + Sqrt[2]]*ArcTan[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] + 4*Sqrt[-1 + Sqrt[2]]*ArcTanh[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]]","A",1
1855,1,127,127,0.2601751,"\int \frac{(1+x) \sqrt{x+\sqrt{1+x^2}}}{-1+x} \, dx","IntegrateAlgebraic[((1 + x)*Sqrt[x + Sqrt[1 + x^2]])/(-1 + x),x]","\frac{2 (6+x) \sqrt{1+x^2}+2 \left(-1+6 x+x^2\right)}{3 \sqrt{x+\sqrt{1+x^2}}}-4 \sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\sqrt{1+\sqrt{2}} \sqrt{x+\sqrt{1+x^2}}\right)-4 \sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\sqrt{-1+\sqrt{2}} \sqrt{x+\sqrt{1+x^2}}\right)","\frac{2 (6+x) \sqrt{1+x^2}+2 \left(-1+6 x+x^2\right)}{3 \sqrt{x+\sqrt{1+x^2}}}-4 \sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{x+\sqrt{1+x^2}}}{\sqrt{-1+\sqrt{2}}}\right)-4 \sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{x+\sqrt{1+x^2}}}{\sqrt{1+\sqrt{2}}}\right)",1,"(2*(6 + x)*Sqrt[1 + x^2] + 2*(-1 + 6*x + x^2))/(3*Sqrt[x + Sqrt[1 + x^2]]) - 4*Sqrt[-1 + Sqrt[2]]*ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] - 4*Sqrt[1 + Sqrt[2]]*ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]]","A",1
1856,1,128,128,0.1808249,"\int \frac{6+2 x+x^2}{(1+x) \sqrt[3]{2+x+x^2} \left(2-x+2 x^2\right)} \, dx","IntegrateAlgebraic[(6 + 2*x + x^2)/((1 + x)*(2 + x + x^2)^(1/3)*(2 - x + 2*x^2)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{-\frac{2 \sqrt[3]{2} x}{\sqrt{3}}+\frac{\sqrt[3]{2+x+x^2}}{\sqrt{3}}}{\sqrt[3]{2+x+x^2}}\right)}{\sqrt[3]{2}}+\frac{\log \left(\sqrt[3]{2} x+\sqrt[3]{2+x+x^2}\right)}{\sqrt[3]{2}}-\frac{\log \left(2^{2/3} x^2-\sqrt[3]{2} x \sqrt[3]{2+x+x^2}+\left(2+x+x^2\right)^{2/3}\right)}{2 \sqrt[3]{2}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{-\frac{2 \sqrt[3]{2} x}{\sqrt{3}}+\frac{\sqrt[3]{2+x+x^2}}{\sqrt{3}}}{\sqrt[3]{2+x+x^2}}\right)}{\sqrt[3]{2}}+\frac{\log \left(\sqrt[3]{2} x+\sqrt[3]{2+x+x^2}\right)}{\sqrt[3]{2}}-\frac{\log \left(2^{2/3} x^2-\sqrt[3]{2} x \sqrt[3]{2+x+x^2}+\left(2+x+x^2\right)^{2/3}\right)}{2 \sqrt[3]{2}}",1,"-((Sqrt[3]*ArcTan[((-2*2^(1/3)*x)/Sqrt[3] + (2 + x + x^2)^(1/3)/Sqrt[3])/(2 + x + x^2)^(1/3)])/2^(1/3)) + Log[2^(1/3)*x + (2 + x + x^2)^(1/3)]/2^(1/3) - Log[2^(2/3)*x^2 - 2^(1/3)*x*(2 + x + x^2)^(1/3) + (2 + x + x^2)^(2/3)]/(2*2^(1/3))","A",1
1857,1,127,128,0.1807158,"\int \frac{1}{x \left(-b+a x^3\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x*(-b + a*x^3)^(3/4)),x]","\frac{\sqrt{2} \tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^3}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^3}}\right)}{3 b^{3/4}}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^3}}{\sqrt{b}+\sqrt{-b+a x^3}}\right)}{3 b^{3/4}}","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^3}}{-\sqrt{b}+\sqrt{-b+a x^3}}\right)}{3 b^{3/4}}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^3}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^3}}\right)}{3 b^{3/4}}",1,"(Sqrt[2]*ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^3]/(Sqrt[2]*b^(1/4)))/(-b + a*x^3)^(1/4)])/(3*b^(3/4)) + (Sqrt[2]*ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^3)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^3])])/(3*b^(3/4))","A",1
1858,1,127,128,0.1632863,"\int \frac{1}{x \sqrt[4]{-b+a x^3}} \, dx","IntegrateAlgebraic[1/(x*(-b + a*x^3)^(1/4)),x]","\frac{\sqrt{2} \tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^3}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^3}}\right)}{3 \sqrt[4]{b}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^3}}{\sqrt{b}+\sqrt{-b+a x^3}}\right)}{3 \sqrt[4]{b}}","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^3}}{-\sqrt{b}+\sqrt{-b+a x^3}}\right)}{3 \sqrt[4]{b}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^3}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^3}}\right)}{3 \sqrt[4]{b}}",1,"(Sqrt[2]*ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^3]/(Sqrt[2]*b^(1/4)))/(-b + a*x^3)^(1/4)])/(3*b^(1/4)) - (Sqrt[2]*ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^3)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^3])])/(3*b^(1/4))","A",1
1859,1,128,128,1.2347248,"\int \frac{\left(-3 b+a x^2\right) \left(b-a x^2+x^3\right)}{x^3 \left(-b+a x^2+x^3\right) \sqrt[4]{-b x+a x^3}} \, dx","IntegrateAlgebraic[((-3*b + a*x^2)*(b - a*x^2 + x^3))/(x^3*(-b + a*x^2 + x^3)*(-(b*x) + a*x^3)^(1/4)),x]","\frac{4 \left(-b x+a x^3\right)^{3/4}}{3 x^3}-2 \sqrt{2} \tan ^{-1}\left(\frac{-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{-b x+a x^3}}{\sqrt{2}}}{x \sqrt[4]{-b x+a x^3}}\right)+2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-b x+a x^3}}{x^2+\sqrt{-b x+a x^3}}\right)","\frac{4 \left(-b x+a x^3\right)^{3/4}}{3 x^3}-2 \sqrt{2} \tan ^{-1}\left(\frac{-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{-b x+a x^3}}{\sqrt{2}}}{x \sqrt[4]{-b x+a x^3}}\right)+2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-b x+a x^3}}{x^2+\sqrt{-b x+a x^3}}\right)",1,"(4*(-(b*x) + a*x^3)^(3/4))/(3*x^3) - 2*Sqrt[2]*ArcTan[(-(x^2/Sqrt[2]) + Sqrt[-(b*x) + a*x^3]/Sqrt[2])/(x*(-(b*x) + a*x^3)^(1/4))] + 2*Sqrt[2]*ArcTanh[(Sqrt[2]*x*(-(b*x) + a*x^3)^(1/4))/(x^2 + Sqrt[-(b*x) + a*x^3])]","A",1
1860,1,127,128,0.1596782,"\int \frac{1}{x \left(-b+a x^4\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x*(-b + a*x^4)^(3/4)),x]","\frac{\tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^4}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^4}}\right)}{2 \sqrt{2} b^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^4}}{\sqrt{b}+\sqrt{-b+a x^4}}\right)}{2 \sqrt{2} b^{3/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^4}}{-\sqrt{b}+\sqrt{-b+a x^4}}\right)}{2 \sqrt{2} b^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^4}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^4}}\right)}{2 \sqrt{2} b^{3/4}}",1,"ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^4]/(Sqrt[2]*b^(1/4)))/(-b + a*x^4)^(1/4)]/(2*Sqrt[2]*b^(3/4)) + ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^4)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^4])]/(2*Sqrt[2]*b^(3/4))","A",1
1861,1,127,128,0.1562522,"\int \frac{1}{x \sqrt[4]{-b+a x^4}} \, dx","IntegrateAlgebraic[1/(x*(-b + a*x^4)^(1/4)),x]","\frac{\tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^4}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^4}}\right)}{2 \sqrt{2} \sqrt[4]{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^4}}{\sqrt{b}+\sqrt{-b+a x^4}}\right)}{2 \sqrt{2} \sqrt[4]{b}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^4}}{-\sqrt{b}+\sqrt{-b+a x^4}}\right)}{2 \sqrt{2} \sqrt[4]{b}}-\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^4}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^4}}\right)}{2 \sqrt{2} \sqrt[4]{b}}",1,"ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^4]/(Sqrt[2]*b^(1/4)))/(-b + a*x^4)^(1/4)]/(2*Sqrt[2]*b^(1/4)) - ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^4)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^4])]/(2*Sqrt[2]*b^(1/4))","A",1
1862,1,464,128,0.3249208,"\int \frac{1-x^5}{\sqrt{a+b x} \left(1+x^5\right)} \, dx","IntegrateAlgebraic[(1 - x^5)/(Sqrt[a + b*x]*(1 + x^5)),x]","-\frac{2 \sqrt{a+b x}}{b}-\frac{4 \tan ^{-1}\left(\frac{\sqrt{-a+b} \sqrt{a+b x}}{a-b}\right)}{5 \sqrt{-a+b}}+\frac{2}{5} \text{RootSum}\left[a^4+a^3 b+a^2 b^2+a b^3+b^4-4 a^3 \text{$\#$1}^2-3 a^2 b \text{$\#$1}^2-2 a b^2 \text{$\#$1}^2-b^3 \text{$\#$1}^2+6 a^2 \text{$\#$1}^4+3 a b \text{$\#$1}^4+b^2 \text{$\#$1}^4-4 a \text{$\#$1}^6-b \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-a^3 \log \left(\sqrt{a+b x}-\text{$\#$1}\right)-2 a^2 b \log \left(\sqrt{a+b x}-\text{$\#$1}\right)-3 a b^2 \log \left(\sqrt{a+b x}-\text{$\#$1}\right)-4 b^3 \log \left(\sqrt{a+b x}-\text{$\#$1}\right)+3 a^2 \log \left(\sqrt{a+b x}-\text{$\#$1}\right) \text{$\#$1}^2+4 a b \log \left(\sqrt{a+b x}-\text{$\#$1}\right) \text{$\#$1}^2+3 b^2 \log \left(\sqrt{a+b x}-\text{$\#$1}\right) \text{$\#$1}^2-3 a \log \left(\sqrt{a+b x}-\text{$\#$1}\right) \text{$\#$1}^4-2 b \log \left(\sqrt{a+b x}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{a+b x}-\text{$\#$1}\right) \text{$\#$1}^6}{4 a^3 \text{$\#$1}+3 a^2 b \text{$\#$1}+2 a b^2 \text{$\#$1}+b^3 \text{$\#$1}-12 a^2 \text{$\#$1}^3-6 a b \text{$\#$1}^3-2 b^2 \text{$\#$1}^3+12 a \text{$\#$1}^5+3 b \text{$\#$1}^5-4 \text{$\#$1}^7}\&\right]","-\frac{2 \sqrt{a+b x}}{b}+\frac{2}{5} b^4 \text{RootSum}\left[a^5-b^5-5 a^4 \text{$\#$1}^2+10 a^3 \text{$\#$1}^4-10 a^2 \text{$\#$1}^6+5 a \text{$\#$1}^8-\text{$\#$1}^{10}\&,\frac{\log \left(\sqrt{a+b x}-\text{$\#$1}\right)}{a^4 \text{$\#$1}-4 a^3 \text{$\#$1}^3+6 a^2 \text{$\#$1}^5-4 a \text{$\#$1}^7+\text{$\#$1}^9}\&\right]",1,"(-2*Sqrt[a + b*x])/b - (4*ArcTan[(Sqrt[-a + b]*Sqrt[a + b*x])/(a - b)])/(5*Sqrt[-a + b]) + (2*RootSum[a^4 + a^3*b + a^2*b^2 + a*b^3 + b^4 - 4*a^3*#1^2 - 3*a^2*b*#1^2 - 2*a*b^2*#1^2 - b^3*#1^2 + 6*a^2*#1^4 + 3*a*b*#1^4 + b^2*#1^4 - 4*a*#1^6 - b*#1^6 + #1^8 & , (-(a^3*Log[Sqrt[a + b*x] - #1]) - 2*a^2*b*Log[Sqrt[a + b*x] - #1] - 3*a*b^2*Log[Sqrt[a + b*x] - #1] - 4*b^3*Log[Sqrt[a + b*x] - #1] + 3*a^2*Log[Sqrt[a + b*x] - #1]*#1^2 + 4*a*b*Log[Sqrt[a + b*x] - #1]*#1^2 + 3*b^2*Log[Sqrt[a + b*x] - #1]*#1^2 - 3*a*Log[Sqrt[a + b*x] - #1]*#1^4 - 2*b*Log[Sqrt[a + b*x] - #1]*#1^4 + Log[Sqrt[a + b*x] - #1]*#1^6)/(4*a^3*#1 + 3*a^2*b*#1 + 2*a*b^2*#1 + b^3*#1 - 12*a^2*#1^3 - 6*a*b*#1^3 - 2*b^2*#1^3 + 12*a*#1^5 + 3*b*#1^5 - 4*#1^7) & ])/5","B",1
1863,1,127,128,0.1641102,"\int \frac{1}{x \left(-b+a x^5\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x*(-b + a*x^5)^(3/4)),x]","\frac{\sqrt{2} \tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^5}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^5}}\right)}{5 b^{3/4}}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^5}}{\sqrt{b}+\sqrt{-b+a x^5}}\right)}{5 b^{3/4}}","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^5}}{-\sqrt{b}+\sqrt{-b+a x^5}}\right)}{5 b^{3/4}}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^5}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^5}}\right)}{5 b^{3/4}}",1,"(Sqrt[2]*ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^5]/(Sqrt[2]*b^(1/4)))/(-b + a*x^5)^(1/4)])/(5*b^(3/4)) + (Sqrt[2]*ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^5)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^5])])/(5*b^(3/4))","A",1
1864,1,127,128,0.1639713,"\int \frac{1}{x \left(-b+a x^6\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x*(-b + a*x^6)^(3/4)),x]","\frac{\tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^6}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^6}}\right)}{3 \sqrt{2} b^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^6}}{\sqrt{b}+\sqrt{-b+a x^6}}\right)}{3 \sqrt{2} b^{3/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^6}}{-\sqrt{b}+\sqrt{-b+a x^6}}\right)}{3 \sqrt{2} b^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^6}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^6}}\right)}{3 \sqrt{2} b^{3/4}}",1,"ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^6]/(Sqrt[2]*b^(1/4)))/(-b + a*x^6)^(1/4)]/(3*Sqrt[2]*b^(3/4)) + ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^6)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^6])]/(3*Sqrt[2]*b^(3/4))","A",1
1865,1,128,128,1.3162012,"\int \frac{\left(1+x^4\right) \left(-1+x^2+x^4\right)^{3/2}}{\left(-1+x^4\right) \left(1+x^2-x^4-x^6+x^8\right)} \, dx","IntegrateAlgebraic[((1 + x^4)*(-1 + x^2 + x^4)^(3/2))/((-1 + x^4)*(1 + x^2 - x^4 - x^6 + x^8)),x]","-\sqrt{\frac{1}{2} \left(3-i \sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{3}{2}-\frac{i \sqrt{3}}{2}} x}{\sqrt{-1+x^2+x^4}}\right)-\sqrt{\frac{1}{2} \left(3+i \sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{3}{2}+\frac{i \sqrt{3}}{2}} x}{\sqrt{-1+x^2+x^4}}\right)-\tanh ^{-1}\left(\frac{x}{\sqrt{-1+x^2+x^4}}\right)","-\sqrt{\frac{1}{2} \left(3-i \sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{3}{2}-\frac{i \sqrt{3}}{2}} x}{\sqrt{-1+x^2+x^4}}\right)-\sqrt{\frac{1}{2} \left(3+i \sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{3}{2}+\frac{i \sqrt{3}}{2}} x}{\sqrt{-1+x^2+x^4}}\right)-\tanh ^{-1}\left(\frac{x}{\sqrt{-1+x^2+x^4}}\right)",1,"-(Sqrt[(3 - I*Sqrt[3])/2]*ArcTan[(Sqrt[-3/2 - (I/2)*Sqrt[3]]*x)/Sqrt[-1 + x^2 + x^4]]) - Sqrt[(3 + I*Sqrt[3])/2]*ArcTan[(Sqrt[-3/2 + (I/2)*Sqrt[3]]*x)/Sqrt[-1 + x^2 + x^4]] - ArcTanh[x/Sqrt[-1 + x^2 + x^4]]","A",1
1866,1,128,128,0.4959021,"\int \frac{\left(-1+x^6\right) \left(1+x^3+x^6\right)^{2/3}}{1+x^6+x^{12}} \, dx","IntegrateAlgebraic[((-1 + x^6)*(1 + x^3 + x^6)^(2/3))/(1 + x^6 + x^12),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{1+x^3+x^6}}\right)}{\sqrt[3]{2} \sqrt{3}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{1+x^3+x^6}\right)}{3 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{1+x^3+x^6}+\sqrt[3]{2} \left(1+x^3+x^6\right)^{2/3}\right)}{6 \sqrt[3]{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{1+x^3+x^6}}\right)}{\sqrt[3]{2} \sqrt{3}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{1+x^3+x^6}\right)}{3 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{1+x^3+x^6}+\sqrt[3]{2} \left(1+x^3+x^6\right)^{2/3}\right)}{6 \sqrt[3]{2}}",1,"-(ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(1 + x^3 + x^6)^(1/3))]/(2^(1/3)*Sqrt[3])) + Log[-2*x + 2^(2/3)*(1 + x^3 + x^6)^(1/3)]/(3*2^(1/3)) - Log[2*x^2 + 2^(2/3)*x*(1 + x^3 + x^6)^(1/3) + 2^(1/3)*(1 + x^3 + x^6)^(2/3)]/(6*2^(1/3))","A",1
1867,1,128,128,3.7500767,"\int \frac{\left(-1+x^3\right)^3 \left(1+x^3\right) \sqrt{2+3 x^6+2 x^{12}}}{x^7 \left(1+x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^3*(1 + x^3)*Sqrt[2 + 3*x^6 + 2*x^12])/(x^7*(1 + x^6)),x]","\frac{\left(1-4 x^3+x^6\right) \sqrt{2+3 x^6+2 x^{12}}}{6 x^6}-\frac{4}{3} \tan ^{-1}\left(\frac{x^3}{\sqrt{2}+\sqrt{2} x^6+\sqrt{2+3 x^6+2 x^{12}}}\right)+\frac{\log (x)}{2 \sqrt{2}}-\frac{\log \left(\sqrt{2}+\sqrt{2} x^6+\sqrt{2+3 x^6+2 x^{12}}\right)}{6 \sqrt{2}}","\frac{\left(1-4 x^3+x^6\right) \sqrt{2+3 x^6+2 x^{12}}}{6 x^6}-\frac{4}{3} \tan ^{-1}\left(\frac{x^3}{\sqrt{2}+\sqrt{2} x^6+\sqrt{2+3 x^6+2 x^{12}}}\right)+\frac{\log (x)}{2 \sqrt{2}}-\frac{\log \left(\sqrt{2}+\sqrt{2} x^6+\sqrt{2+3 x^6+2 x^{12}}\right)}{6 \sqrt{2}}",1,"((1 - 4*x^3 + x^6)*Sqrt[2 + 3*x^6 + 2*x^12])/(6*x^6) - (4*ArcTan[x^3/(Sqrt[2] + Sqrt[2]*x^6 + Sqrt[2 + 3*x^6 + 2*x^12])])/3 + Log[x]/(2*Sqrt[2]) - Log[Sqrt[2] + Sqrt[2]*x^6 + Sqrt[2 + 3*x^6 + 2*x^12]]/(6*Sqrt[2])","A",1
1868,1,128,128,0.5283277,"\int \frac{\sqrt{a x-\sqrt{b+a^2 x^2}}}{a^2 x^2+\sqrt{b+a^2 x^2}} \, dx","IntegrateAlgebraic[Sqrt[a*x - Sqrt[b + a^2*x^2]]/(a^2*x^2 + Sqrt[b + a^2*x^2]),x]","-\frac{\text{RootSum}\left[b^2-2 b \text{$\#$1}^2-2 b \text{$\#$1}^4-2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{b \log \left(\sqrt{a x-\sqrt{b+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}+\log \left(\sqrt{a x-\sqrt{b+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^5}{b+2 b \text{$\#$1}^2+3 \text{$\#$1}^4-2 \text{$\#$1}^6}\&\right]}{a}","-\frac{\text{RootSum}\left[b^2-2 b \text{$\#$1}^2-2 b \text{$\#$1}^4-2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{b \log \left(\sqrt{a x-\sqrt{b+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}+\log \left(\sqrt{a x-\sqrt{b+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^5}{b+2 b \text{$\#$1}^2+3 \text{$\#$1}^4-2 \text{$\#$1}^6}\&\right]}{a}",1,"-(RootSum[b^2 - 2*b*#1^2 - 2*b*#1^4 - 2*#1^6 + #1^8 & , (b*Log[Sqrt[a*x - Sqrt[b + a^2*x^2]] - #1]*#1 + Log[Sqrt[a*x - Sqrt[b + a^2*x^2]] - #1]*#1^5)/(b + 2*b*#1^2 + 3*#1^4 - 2*#1^6) & ]/a)","A",1
1869,1,128,128,0.1985579,"\int \frac{d+c x}{\sqrt{a x+\sqrt{b^2+a^2 x^2}}} \, dx","IntegrateAlgebraic[(d + c*x)/Sqrt[a*x + Sqrt[b^2 + a^2*x^2]],x]","\frac{b^4 c}{10 a^2 \left(a x+\sqrt{b^2+a^2 x^2}\right)^{5/2}}-\frac{b^2 d}{3 a \left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}+\frac{d \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{a}+\frac{c \left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}{6 a^2}","\frac{b^4 c}{10 a^2 \left(a x+\sqrt{b^2+a^2 x^2}\right)^{5/2}}-\frac{b^2 d}{3 a \left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}+\frac{d \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{a}+\frac{c \left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}{6 a^2}",1,"(b^4*c)/(10*a^2*(a*x + Sqrt[b^2 + a^2*x^2])^(5/2)) - (b^2*d)/(3*a*(a*x + Sqrt[b^2 + a^2*x^2])^(3/2)) + (d*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/a + (c*(a*x + Sqrt[b^2 + a^2*x^2])^(3/2))/(6*a^2)","A",1
1870,1,82,128,0.0637353,"\int \frac{1}{\sqrt{1+\sqrt{1+\sqrt{1+x}}}} \, dx","IntegrateAlgebraic[1/Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]],x]","-\frac{16}{105} \left(-2+9 \sqrt{1+x}\right) \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\frac{8}{105} \sqrt{1+\sqrt{1+x}} \left(4+15 \sqrt{1+x}\right) \sqrt{1+\sqrt{1+\sqrt{1+x}}}","\frac{32}{105} \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\frac{32}{105} \sqrt{1+\sqrt{1+x}} \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\sqrt{1+x} \left(-\frac{48}{35} \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\frac{8}{7} \sqrt{1+\sqrt{1+x}} \sqrt{1+\sqrt{1+\sqrt{1+x}}}\right)",1,"(-16*(-2 + 9*Sqrt[1 + x])*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/105 + (8*Sqrt[1 + Sqrt[1 + x]]*(4 + 15*Sqrt[1 + x])*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/105","A",1
1871,1,129,129,0.3239924,"\int \frac{\left(-b+a x^2\right) \sqrt[3]{x+x^3}}{x^2} \, dx","IntegrateAlgebraic[((-b + a*x^2)*(x + x^3)^(1/3))/x^2,x]","\frac{\left(3 b+a x^2\right) \sqrt[3]{x+x^3}}{2 x}+\frac{1}{6} \left(-\sqrt{3} a+3 \sqrt{3} b\right) \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)+\frac{1}{6} (-a+3 b) \log \left(-x+\sqrt[3]{x+x^3}\right)+\frac{1}{12} (a-3 b) \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)","\frac{\left(3 b+a x^2\right) \sqrt[3]{x+x^3}}{2 x}+\frac{1}{6} \left(-\sqrt{3} a+3 \sqrt{3} b\right) \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)+\frac{1}{6} (-a+3 b) \log \left(-x+\sqrt[3]{x+x^3}\right)+\frac{1}{12} (a-3 b) \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)",1,"((3*b + a*x^2)*(x + x^3)^(1/3))/(2*x) + ((-(Sqrt[3]*a) + 3*Sqrt[3]*b)*ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^3)^(1/3))])/6 + ((-a + 3*b)*Log[-x + (x + x^3)^(1/3)])/6 + ((a - 3*b)*Log[x^2 + x*(x + x^3)^(1/3) + (x + x^3)^(2/3)])/12","A",1
1872,1,129,129,0.7017191,"\int \frac{1}{x^3 \sqrt[3]{-b x^2+a x^3} \left(d+c x^3\right)} \, dx","IntegrateAlgebraic[1/(x^3*(-(b*x^2) + a*x^3)^(1/3)*(d + c*x^3)),x]","\frac{3 \left(5 b^2+6 a b x+9 a^2 x^2\right) \left(-b x^2+a x^3\right)^{2/3}}{40 b^3 d x^4}+\frac{c \text{RootSum}\left[b^3 c+a^3 d-3 a^2 d \text{$\#$1}^3+3 a d \text{$\#$1}^6-d \text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{3 d^2}","\frac{3 \left(5 b^2+6 a b x+9 a^2 x^2\right) \left(-b x^2+a x^3\right)^{2/3}}{40 b^3 d x^4}+\frac{c \text{RootSum}\left[b^3 c+a^3 d-3 a^2 d \text{$\#$1}^3+3 a d \text{$\#$1}^6-d \text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{3 d^2}",1,"(3*(5*b^2 + 6*a*b*x + 9*a^2*x^2)*(-(b*x^2) + a*x^3)^(2/3))/(40*b^3*d*x^4) + (c*RootSum[b^3*c + a^3*d - 3*a^2*d*#1^3 + 3*a*d*#1^6 - d*#1^9 & , (-Log[x] + Log[(-(b*x^2) + a*x^3)^(1/3) - x*#1])/#1 & ])/(3*d^2)","A",1
1873,1,129,129,0.4076428,"\int \frac{-1+x^4}{\sqrt{-x-x^2+x^3} \left(1+x^4\right)} \, dx","IntegrateAlgebraic[(-1 + x^4)/(Sqrt[-x - x^2 + x^3]*(1 + x^4)),x]","-\sqrt{\frac{1}{3} \left(1+i \sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{1-i \sqrt{2}} \sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)-\sqrt{\frac{1}{3} \left(1-i \sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{1+i \sqrt{2}} \sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)","-\sqrt{\frac{1}{3} \left(1+i \sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{1-i \sqrt{2}} \sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)-\sqrt{\frac{1}{3} \left(1-i \sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{1+i \sqrt{2}} \sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)",1,"-(Sqrt[(1 + I*Sqrt[2])/3]*ArcTan[(Sqrt[1 - I*Sqrt[2]]*Sqrt[-x - x^2 + x^3])/(-1 - x + x^2)]) - Sqrt[(1 - I*Sqrt[2])/3]*ArcTan[(Sqrt[1 + I*Sqrt[2]]*Sqrt[-x - x^2 + x^3])/(-1 - x + x^2)]","A",1
1874,1,129,129,0.2412364,"\int \sqrt{-1-11 x-36 x^2-27 x^3+16 x^4+9 x^5+x^6} \, dx","IntegrateAlgebraic[Sqrt[-1 - 11*x - 36*x^2 - 27*x^3 + 16*x^4 + 9*x^5 + x^6],x]","\frac{\left(-185-6 x+104 x^2+16 x^3\right) \sqrt{-1-11 x-36 x^2-27 x^3+16 x^4+9 x^5+x^6}}{64 \left(1+5 x+x^2\right)}-\frac{325}{128} \log \left(1+5 x+x^2\right)+\frac{325}{128} \log \left(1+3 x-9 x^2-2 x^3+2 \sqrt{-1-11 x-36 x^2-27 x^3+16 x^4+9 x^5+x^6}\right)","\frac{\left(-185-6 x+104 x^2+16 x^3\right) \sqrt{-1-11 x-36 x^2-27 x^3+16 x^4+9 x^5+x^6}}{64 \left(1+5 x+x^2\right)}-\frac{325}{128} \log \left(1+5 x+x^2\right)+\frac{325}{128} \log \left(1+3 x-9 x^2-2 x^3+2 \sqrt{-1-11 x-36 x^2-27 x^3+16 x^4+9 x^5+x^6}\right)",1,"((-185 - 6*x + 104*x^2 + 16*x^3)*Sqrt[-1 - 11*x - 36*x^2 - 27*x^3 + 16*x^4 + 9*x^5 + x^6])/(64*(1 + 5*x + x^2)) - (325*Log[1 + 5*x + x^2])/128 + (325*Log[1 + 3*x - 9*x^2 - 2*x^3 + 2*Sqrt[-1 - 11*x - 36*x^2 - 27*x^3 + 16*x^4 + 9*x^5 + x^6]])/128","A",1
1875,1,129,129,0.439154,"\int \frac{\left(-q+2 p x^3\right) \left(a q+b x+a p x^3\right) \sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^4} \, dx","IntegrateAlgebraic[((-q + 2*p*x^3)*(a*q + b*x + a*p*x^3)*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6])/x^4,x]","\frac{\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left(2 a q^2+3 b q x-4 a p q x^2+4 a p q x^3+3 b p x^4+2 a p^2 x^6\right)}{6 x^3}+b p q \log (x)-b p q \log \left(q+p x^3+\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right)","\frac{\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left(2 a q^2+3 b q x-4 a p q x^2+4 a p q x^3+3 b p x^4+2 a p^2 x^6\right)}{6 x^3}+b p q \log (x)-b p q \log \left(q+p x^3+\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right)",1,"(Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]*(2*a*q^2 + 3*b*q*x - 4*a*p*q*x^2 + 4*a*p*q*x^3 + 3*b*p*x^4 + 2*a*p^2*x^6))/(6*x^3) + b*p*q*Log[x] - b*p*q*Log[q + p*x^3 + Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]]","A",1
1876,1,129,129,0.4528982,"\int \frac{-2-x^4+2 x^8}{\sqrt[4]{-1+x^4} \left(-2-x^4+x^8\right)} \, dx","IntegrateAlgebraic[(-2 - x^4 + 2*x^8)/((-1 + x^4)^(1/4)*(-2 - x^4 + x^8)),x]","\tan ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)-\frac{1}{3} \sqrt[4]{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{2} \sqrt[4]{-1+x^4}}\right)-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-1+x^4}}\right)}{6 \sqrt[4]{2}}+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)-\frac{1}{3} \sqrt[4]{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{2} \sqrt[4]{-1+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-1+x^4}}\right)}{6 \sqrt[4]{2}}","\tan ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)-\frac{1}{3} \sqrt[4]{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{2} \sqrt[4]{-1+x^4}}\right)-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-1+x^4}}\right)}{6 \sqrt[4]{2}}+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)-\frac{1}{3} \sqrt[4]{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{2} \sqrt[4]{-1+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-1+x^4}}\right)}{6 \sqrt[4]{2}}",1,"ArcTan[x/(-1 + x^4)^(1/4)] - (2^(1/4)*ArcTan[x/(2^(1/4)*(-1 + x^4)^(1/4))])/3 - ArcTan[(2^(1/4)*x)/(-1 + x^4)^(1/4)]/(6*2^(1/4)) + ArcTanh[x/(-1 + x^4)^(1/4)] - (2^(1/4)*ArcTanh[x/(2^(1/4)*(-1 + x^4)^(1/4))])/3 - ArcTanh[(2^(1/4)*x)/(-1 + x^4)^(1/4)]/(6*2^(1/4))","A",1
1877,1,114,129,0.1858143,"\int \frac{\left(-1+x^4\right) \sqrt{1+\sqrt{1+x}}}{1+x^4} \, dx","IntegrateAlgebraic[((-1 + x^4)*Sqrt[1 + Sqrt[1 + x]])/(1 + x^4),x]","\frac{4}{15} \sqrt{1+\sqrt{1+x}} \left(-2+\sqrt{1+x}+3 (1+x)\right)-\frac{1}{2} \text{RootSum}\left[1+16 \text{$\#$1}^8-32 \text{$\#$1}^{10}+24 \text{$\#$1}^{12}-8 \text{$\#$1}^{14}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)}{-8 \text{$\#$1}^5+12 \text{$\#$1}^7-6 \text{$\#$1}^9+\text{$\#$1}^{11}}\&\right]","\frac{4}{15} \sqrt{1+x} \sqrt{1+\sqrt{1+x}}+\frac{4}{15} (1+3 x) \sqrt{1+\sqrt{1+x}}-\frac{1}{2} \text{RootSum}\left[1+16 \text{$\#$1}^8-32 \text{$\#$1}^{10}+24 \text{$\#$1}^{12}-8 \text{$\#$1}^{14}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)}{-8 \text{$\#$1}^5+12 \text{$\#$1}^7-6 \text{$\#$1}^9+\text{$\#$1}^{11}}\&\right]",1,"(4*Sqrt[1 + Sqrt[1 + x]]*(-2 + Sqrt[1 + x] + 3*(1 + x)))/15 - RootSum[1 + 16*#1^8 - 32*#1^10 + 24*#1^12 - 8*#1^14 + #1^16 & , Log[Sqrt[1 + Sqrt[1 + x]] - #1]/(-8*#1^5 + 12*#1^7 - 6*#1^9 + #1^11) & ]/2","A",1
1878,1,114,129,0.0009784,"\int \frac{\left(-1+x^4\right) \sqrt{1+\sqrt{1+x}}}{1+x^4} \, dx","IntegrateAlgebraic[((-1 + x^4)*Sqrt[1 + Sqrt[1 + x]])/(1 + x^4),x]","\frac{4}{15} \sqrt{1+\sqrt{1+x}} \left(-2+\sqrt{1+x}+3 (1+x)\right)-\frac{1}{2} \text{RootSum}\left[1+16 \text{$\#$1}^8-32 \text{$\#$1}^{10}+24 \text{$\#$1}^{12}-8 \text{$\#$1}^{14}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)}{-8 \text{$\#$1}^5+12 \text{$\#$1}^7-6 \text{$\#$1}^9+\text{$\#$1}^{11}}\&\right]","\frac{4}{15} \sqrt{1+x} \sqrt{1+\sqrt{1+x}}+\frac{4}{15} (1+3 x) \sqrt{1+\sqrt{1+x}}-\frac{1}{2} \text{RootSum}\left[1+16 \text{$\#$1}^8-32 \text{$\#$1}^{10}+24 \text{$\#$1}^{12}-8 \text{$\#$1}^{14}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)}{-8 \text{$\#$1}^5+12 \text{$\#$1}^7-6 \text{$\#$1}^9+\text{$\#$1}^{11}}\&\right]",1,"(4*Sqrt[1 + Sqrt[1 + x]]*(-2 + Sqrt[1 + x] + 3*(1 + x)))/15 - RootSum[1 + 16*#1^8 - 32*#1^10 + 24*#1^12 - 8*#1^14 + #1^16 & , Log[Sqrt[1 + Sqrt[1 + x]] - #1]/(-8*#1^5 + 12*#1^7 - 6*#1^9 + #1^11) & ]/2","A",1
1879,1,130,130,0.3700053,"\int \frac{\left(b+a x^2\right) \sqrt{b x+a x^3}}{x^2 \left(-b+a x^2\right)} \, dx","IntegrateAlgebraic[((b + a*x^2)*Sqrt[b*x + a*x^3])/(x^2*(-b + a*x^2)),x]","\frac{2 \sqrt{b x+a x^3}}{x}+\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)","\frac{2 \sqrt{b x+a x^3}}{x}+\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)",1,"(2*Sqrt[b*x + a*x^3])/x + Sqrt[2]*a^(1/4)*b^(1/4)*ArcTan[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[b*x + a*x^3])/(b + a*x^2)] - Sqrt[2]*a^(1/4)*b^(1/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[b*x + a*x^3])/(b + a*x^2)]","A",1
1880,1,130,130,3.7211884,"\int \frac{(-2 a b+(3 a-b) x) \left(-a^3+3 a^2 x-3 a x^2+x^3\right)}{x (-b+x) \sqrt[4]{x^2 (-a+x) (-b+x)} \left(a^3-3 a^2 x+(3 a-b d) x^2+(-1+d) x^3\right)} \, dx","IntegrateAlgebraic[((-2*a*b + (3*a - b)*x)*(-a^3 + 3*a^2*x - 3*a*x^2 + x^3))/(x*(-b + x)*(x^2*(-a + x)*(-b + x))^(1/4)*(a^3 - 3*a^2*x + (3*a - b*d)*x^2 + (-1 + d)*x^3)),x]","-\frac{4 \left(a b x^2+(-a-b) x^3+x^4\right)^{3/4}}{x^2 (-b+x)}+2 \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b x^2+(-a-b) x^3+x^4}}{a-x}\right)-2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b x^2+(-a-b) x^3+x^4}}{a-x}\right)","-\frac{4 \left(a b x^2+(-a-b) x^3+x^4\right)^{3/4}}{x^2 (-b+x)}+2 \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b x^2+(-a-b) x^3+x^4}}{a-x}\right)-2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b x^2+(-a-b) x^3+x^4}}{a-x}\right)",1,"(-4*(a*b*x^2 + (-a - b)*x^3 + x^4)^(3/4))/(x^2*(-b + x)) + 2*d^(1/4)*ArcTan[(d^(1/4)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/4))/(a - x)] - 2*d^(1/4)*ArcTanh[(d^(1/4)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/4))/(a - x)]","A",1
1881,1,130,130,0.4155986,"\int \frac{2 b+a x^2}{\sqrt[4]{b+a x^2} \left(b n+a n x^2+2 x^4\right)} \, dx","IntegrateAlgebraic[(2*b + a*x^2)/((b + a*x^2)^(1/4)*(b*n + a*n*x^2 + 2*x^4)),x]","\frac{\tan ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2} \sqrt[4]{n}}-\frac{\sqrt[4]{n} \sqrt{b+a x^2}}{2^{3/4}}}{x \sqrt[4]{b+a x^2}}\right)}{2^{3/4} n^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2} \sqrt[4]{n}}+\frac{\sqrt[4]{n} \sqrt{b+a x^2}}{2^{3/4}}}{x \sqrt[4]{b+a x^2}}\right)}{2^{3/4} n^{3/4}}","\frac{\tan ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2} \sqrt[4]{n}}-\frac{\sqrt[4]{n} \sqrt{b+a x^2}}{2^{3/4}}}{x \sqrt[4]{b+a x^2}}\right)}{2^{3/4} n^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2} \sqrt[4]{n}}+\frac{\sqrt[4]{n} \sqrt{b+a x^2}}{2^{3/4}}}{x \sqrt[4]{b+a x^2}}\right)}{2^{3/4} n^{3/4}}",1,"ArcTan[(x^2/(2^(1/4)*n^(1/4)) - (n^(1/4)*Sqrt[b + a*x^2])/2^(3/4))/(x*(b + a*x^2)^(1/4))]/(2^(3/4)*n^(3/4)) + ArcTanh[(x^2/(2^(1/4)*n^(1/4)) + (n^(1/4)*Sqrt[b + a*x^2])/2^(3/4))/(x*(b + a*x^2)^(1/4))]/(2^(3/4)*n^(3/4))","A",1
1882,1,117,130,0.1154625,"\int \frac{\sqrt{1+x} \left(-1+x^2\right)}{\left(1+x^2\right) \sqrt{1+\sqrt{1+x}}} \, dx","IntegrateAlgebraic[(Sqrt[1 + x]*(-1 + x^2))/((1 + x^2)*Sqrt[1 + Sqrt[1 + x]]),x]","\frac{4}{15} \sqrt{1+\sqrt{1+x}} \left(8-4 \sqrt{1+x}+3 (1+x)\right)-\text{RootSum}\left[1+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)+\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2}{-2 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]","-\frac{16}{15} \sqrt{1+x} \sqrt{1+\sqrt{1+x}}+\frac{4}{15} (11+3 x) \sqrt{1+\sqrt{1+x}}-\text{RootSum}\left[1+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)+\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2}{-2 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]",1,"(4*Sqrt[1 + Sqrt[1 + x]]*(8 - 4*Sqrt[1 + x] + 3*(1 + x)))/15 - RootSum[1 + 4*#1^4 - 4*#1^6 + #1^8 & , (-Log[Sqrt[1 + Sqrt[1 + x]] - #1] + Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^2)/(-2*#1^3 + #1^5) & ]","A",1
1883,1,117,130,0.0010331,"\int \frac{\sqrt{1+x} \left(-1+x^2\right)}{\left(1+x^2\right) \sqrt{1+\sqrt{1+x}}} \, dx","IntegrateAlgebraic[(Sqrt[1 + x]*(-1 + x^2))/((1 + x^2)*Sqrt[1 + Sqrt[1 + x]]),x]","\frac{4}{15} \sqrt{1+\sqrt{1+x}} \left(8-4 \sqrt{1+x}+3 (1+x)\right)-\text{RootSum}\left[1+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)+\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2}{-2 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]","-\frac{16}{15} \sqrt{1+x} \sqrt{1+\sqrt{1+x}}+\frac{4}{15} (11+3 x) \sqrt{1+\sqrt{1+x}}-\text{RootSum}\left[1+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)+\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2}{-2 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]",1,"(4*Sqrt[1 + Sqrt[1 + x]]*(8 - 4*Sqrt[1 + x] + 3*(1 + x)))/15 - RootSum[1 + 4*#1^4 - 4*#1^6 + #1^8 & , (-Log[Sqrt[1 + Sqrt[1 + x]] - #1] + Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^2)/(-2*#1^3 + #1^5) & ]","A",1
1884,1,106,130,0.1803107,"\int \frac{\left(-1+x^4\right) \sqrt{1+\sqrt{1+x}}}{\sqrt{1+x} \left(1+x^4\right)} \, dx","IntegrateAlgebraic[((-1 + x^4)*Sqrt[1 + Sqrt[1 + x]])/(Sqrt[1 + x]*(1 + x^4)),x]","\frac{4}{3} \left(1+\sqrt{1+x}\right)^{3/2}-\frac{1}{2} \text{RootSum}\left[1+16 \text{$\#$1}^8-32 \text{$\#$1}^{10}+24 \text{$\#$1}^{12}-8 \text{$\#$1}^{14}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)}{8 \text{$\#$1}^5-20 \text{$\#$1}^7+18 \text{$\#$1}^9-7 \text{$\#$1}^{11}+\text{$\#$1}^{13}}\&\right]","\frac{4}{3} \sqrt{1+\sqrt{1+x}}+\frac{4}{3} \sqrt{1+x} \sqrt{1+\sqrt{1+x}}-\frac{1}{2} \text{RootSum}\left[1+16 \text{$\#$1}^8-32 \text{$\#$1}^{10}+24 \text{$\#$1}^{12}-8 \text{$\#$1}^{14}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)}{8 \text{$\#$1}^5-20 \text{$\#$1}^7+18 \text{$\#$1}^9-7 \text{$\#$1}^{11}+\text{$\#$1}^{13}}\&\right]",1,"(4*(1 + Sqrt[1 + x])^(3/2))/3 - RootSum[1 + 16*#1^8 - 32*#1^10 + 24*#1^12 - 8*#1^14 + #1^16 & , Log[Sqrt[1 + Sqrt[1 + x]] - #1]/(8*#1^5 - 20*#1^7 + 18*#1^9 - 7*#1^11 + #1^13) & ]/2","A",1
1885,1,106,130,0.0011086,"\int \frac{\left(-1+x^4\right) \sqrt{1+\sqrt{1+x}}}{\sqrt{1+x} \left(1+x^4\right)} \, dx","IntegrateAlgebraic[((-1 + x^4)*Sqrt[1 + Sqrt[1 + x]])/(Sqrt[1 + x]*(1 + x^4)),x]","\frac{4}{3} \left(1+\sqrt{1+x}\right)^{3/2}-\frac{1}{2} \text{RootSum}\left[1+16 \text{$\#$1}^8-32 \text{$\#$1}^{10}+24 \text{$\#$1}^{12}-8 \text{$\#$1}^{14}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)}{8 \text{$\#$1}^5-20 \text{$\#$1}^7+18 \text{$\#$1}^9-7 \text{$\#$1}^{11}+\text{$\#$1}^{13}}\&\right]","\frac{4}{3} \sqrt{1+\sqrt{1+x}}+\frac{4}{3} \sqrt{1+x} \sqrt{1+\sqrt{1+x}}-\frac{1}{2} \text{RootSum}\left[1+16 \text{$\#$1}^8-32 \text{$\#$1}^{10}+24 \text{$\#$1}^{12}-8 \text{$\#$1}^{14}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)}{8 \text{$\#$1}^5-20 \text{$\#$1}^7+18 \text{$\#$1}^9-7 \text{$\#$1}^{11}+\text{$\#$1}^{13}}\&\right]",1,"(4*(1 + Sqrt[1 + x])^(3/2))/3 - RootSum[1 + 16*#1^8 - 32*#1^10 + 24*#1^12 - 8*#1^14 + #1^16 & , Log[Sqrt[1 + Sqrt[1 + x]] - #1]/(8*#1^5 - 20*#1^7 + 18*#1^9 - 7*#1^11 + #1^13) & ]/2","A",1
1886,1,130,130,0.150086,"\int \frac{1+a x^2}{\left(-1+a x^2\right) \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[(1 + a*x^2)/((-1 + a*x^2)*Sqrt[x + Sqrt[1 + x^2]]),x]","-\frac{1}{3 \left(x+\sqrt{1+x^2}\right)^{3/2}}+\sqrt{x+\sqrt{1+x^2}}+\text{RootSum}\left[a-4 \text{$\#$1}^4-2 a \text{$\#$1}^4+a \text{$\#$1}^8\&,\frac{\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right)+\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{-2 \text{$\#$1}^3-a \text{$\#$1}^3+a \text{$\#$1}^7}\&\right]","-\frac{1}{3 \left(x+\sqrt{1+x^2}\right)^{3/2}}+\sqrt{x+\sqrt{1+x^2}}+\text{RootSum}\left[a-4 \text{$\#$1}^4-2 a \text{$\#$1}^4+a \text{$\#$1}^8\&,\frac{\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right)+\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{-2 \text{$\#$1}^3-a \text{$\#$1}^3+a \text{$\#$1}^7}\&\right]",1,"-1/3*1/(x + Sqrt[1 + x^2])^(3/2) + Sqrt[x + Sqrt[1 + x^2]] + RootSum[a - 4*#1^4 - 2*a*#1^4 + a*#1^8 & , (Log[Sqrt[x + Sqrt[1 + x^2]] - #1] + Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1^4)/(-2*#1^3 - a*#1^3 + a*#1^7) & ]","A",1
1887,1,130,130,4.5229661,"\int \frac{x \sqrt{x+x^2}}{\sqrt{x^2+x \sqrt{x+x^2}}} \, dx","IntegrateAlgebraic[(x*Sqrt[x + x^2])/Sqrt[x^2 + x*Sqrt[x + x^2]],x]","\frac{\left(255-136 x-384 x^2\right) \sqrt{x+x^2} \sqrt{x \left(x+\sqrt{x+x^2}\right)}}{960 x}+\sqrt{x \left(x+\sqrt{x+x^2}\right)} \left(\frac{1}{960} \left(-85+568 x+384 x^2\right)-\frac{17 \sqrt{-x+\sqrt{x+x^2}} \tanh ^{-1}\left(\sqrt{2} \sqrt{-x+\sqrt{x+x^2}}\right)}{64 \sqrt{2} x}\right)","\frac{\left(255-136 x-384 x^2\right) \sqrt{x+x^2} \sqrt{x \left(x+\sqrt{x+x^2}\right)}}{960 x}+\sqrt{x \left(x+\sqrt{x+x^2}\right)} \left(\frac{1}{960} \left(-85+568 x+384 x^2\right)-\frac{17 \sqrt{-x+\sqrt{x+x^2}} \tanh ^{-1}\left(\sqrt{2} \sqrt{-x+\sqrt{x+x^2}}\right)}{64 \sqrt{2} x}\right)",1,"((255 - 136*x - 384*x^2)*Sqrt[x + x^2]*Sqrt[x*(x + Sqrt[x + x^2])])/(960*x) + Sqrt[x*(x + Sqrt[x + x^2])]*((-85 + 568*x + 384*x^2)/960 - (17*Sqrt[-x + Sqrt[x + x^2]]*ArcTanh[Sqrt[2]*Sqrt[-x + Sqrt[x + x^2]]])/(64*Sqrt[2]*x))","A",1
1888,1,127,130,0.3449962,"\int \frac{\sqrt{1+x}}{1+\sqrt{x+\sqrt{1+x}}} \, dx","IntegrateAlgebraic[Sqrt[1 + x]/(1 + Sqrt[x + Sqrt[1 + x]]),x]","-2 \sqrt{1+x}+\frac{1}{2} \sqrt{x+\sqrt{1+x}} \left(-3+2 \sqrt{1+x}\right)+\frac{16}{3} \log \left(-1-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}\right)-\frac{4}{3} \log \left(2-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}\right)-\frac{23}{4} \log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}\right)","-\frac{3}{2} \sqrt{x+\sqrt{1+x}}+\sqrt{1+x} \left(-2+\sqrt{x+\sqrt{1+x}}\right)-\frac{23}{4} \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}\right)-\frac{4}{3} \log \left(-2+\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}\right)+\frac{16}{3} \log \left(1+\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}\right)",1,"-2*Sqrt[1 + x] + (Sqrt[x + Sqrt[1 + x]]*(-3 + 2*Sqrt[1 + x]))/2 + (16*Log[-1 - Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]]])/3 - (4*Log[2 - Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]]])/3 - (23*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]])/4","A",1
1889,1,131,131,0.1952972,"\int \frac{1+x}{(-1+x) (1+2 x) \sqrt[3]{-1+3 x^2}} \, dx","IntegrateAlgebraic[(1 + x)/((-1 + x)*(1 + 2*x)*(-1 + 3*x^2)^(1/3)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-1+3 x^2}}{2 \sqrt[3]{2} x+\sqrt[3]{-1+3 x^2}}\right)}{\sqrt[3]{2} \sqrt{3}}+\frac{\log \left(-\sqrt[3]{2} x+\sqrt[3]{-1+3 x^2}\right)}{3 \sqrt[3]{2}}-\frac{\log \left(2^{2/3} x^2+\sqrt[3]{2} x \sqrt[3]{-1+3 x^2}+\left(-1+3 x^2\right)^{2/3}\right)}{6 \sqrt[3]{2}}","\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-1+3 x^2}}{2 \sqrt[3]{2} x+\sqrt[3]{-1+3 x^2}}\right)}{\sqrt[3]{2} \sqrt{3}}+\frac{\log \left(-\sqrt[3]{2} x+\sqrt[3]{-1+3 x^2}\right)}{3 \sqrt[3]{2}}-\frac{\log \left(2^{2/3} x^2+\sqrt[3]{2} x \sqrt[3]{-1+3 x^2}+\left(-1+3 x^2\right)^{2/3}\right)}{6 \sqrt[3]{2}}",1,"ArcTan[(Sqrt[3]*(-1 + 3*x^2)^(1/3))/(2*2^(1/3)*x + (-1 + 3*x^2)^(1/3))]/(2^(1/3)*Sqrt[3]) + Log[-(2^(1/3)*x) + (-1 + 3*x^2)^(1/3)]/(3*2^(1/3)) - Log[2^(2/3)*x^2 + 2^(1/3)*x*(-1 + 3*x^2)^(1/3) + (-1 + 3*x^2)^(2/3)]/(6*2^(1/3))","A",1
1890,1,110,131,0.265693,"\int \frac{-b+a x^2}{x^2 \sqrt[3]{-x+x^3}} \, dx","IntegrateAlgebraic[(-b + a*x^2)/(x^2*(-x + x^3)^(1/3)),x]","-\frac{3 b \left(-x+x^3\right)^{2/3}}{4 x^2}+\frac{1}{2} \sqrt{3} a \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)-\frac{1}{2} a \log \left(-x+\sqrt[3]{-x+x^3}\right)+\frac{1}{4} a \log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)","-\frac{3 b \left(-x+x^3\right)^{2/3}}{4 x^2}+\frac{1}{4} \left(a-i \sqrt{3} a\right) \log \left(-i x+\sqrt{3} x-2 i \sqrt[3]{-x+x^3}\right)+\frac{1}{4} \left(a+i \sqrt{3} a\right) \log \left(i x+\sqrt{3} x+2 i \sqrt[3]{-x+x^3}\right)-\frac{1}{2} a \log \left(-x+\sqrt[3]{-x+x^3}\right)",1,"(-3*b*(-x + x^3)^(2/3))/(4*x^2) + (Sqrt[3]*a*ArcTan[(Sqrt[3]*x)/(x + 2*(-x + x^3)^(1/3))])/2 - (a*Log[-x + (-x + x^3)^(1/3)])/2 + (a*Log[x^2 + x*(-x + x^3)^(1/3) + (-x + x^3)^(2/3)])/4","A",1
1891,1,131,131,0.2955459,"\int \frac{1+x^4}{\sqrt{-x-x^2+x^3} \left(-1+x^4\right)} \, dx","IntegrateAlgebraic[(1 + x^4)/(Sqrt[-x - x^2 + x^3]*(-1 + x^4)),x]","-\tan ^{-1}\left(\frac{\sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)-\frac{1}{2} \sqrt{\frac{1}{5}+\frac{2 i}{5}} \tan ^{-1}\left(\frac{\sqrt{1-2 i} \sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)-\frac{1}{2} \sqrt{\frac{1}{5}-\frac{2 i}{5}} \tan ^{-1}\left(\frac{\sqrt{1+2 i} \sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)","-\tan ^{-1}\left(\frac{\sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)-\frac{1}{2} \sqrt{\frac{1}{5}+\frac{2 i}{5}} \tan ^{-1}\left(\frac{\sqrt{1-2 i} \sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)-\frac{1}{2} \sqrt{\frac{1}{5}-\frac{2 i}{5}} \tan ^{-1}\left(\frac{\sqrt{1+2 i} \sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)",1,"-ArcTan[Sqrt[-x - x^2 + x^3]/(-1 - x + x^2)] - (Sqrt[1/5 + (2*I)/5]*ArcTan[(Sqrt[1 - 2*I]*Sqrt[-x - x^2 + x^3])/(-1 - x + x^2)])/2 - (Sqrt[1/5 - (2*I)/5]*ArcTan[(Sqrt[1 + 2*I]*Sqrt[-x - x^2 + x^3])/(-1 - x + x^2)])/2","A",1
1892,1,131,131,0.662591,"\int \frac{\left(-1+x^3\right) \sqrt[3]{1+x^6}}{x^2 \left(1+x^3\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)*(1 + x^6)^(1/3))/(x^2*(1 + x^3)),x]","\frac{\sqrt[3]{1+x^6}}{x}+\frac{\sqrt[3]{2} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2^{2/3} \sqrt[3]{1+x^6}}\right)}{\sqrt{3}}-\frac{1}{3} \sqrt[3]{2} \log \left(2 x+2^{2/3} \sqrt[3]{1+x^6}\right)+\frac{\log \left(-2 x^2+2^{2/3} x \sqrt[3]{1+x^6}-\sqrt[3]{2} \left(1+x^6\right)^{2/3}\right)}{3\ 2^{2/3}}","\frac{\sqrt[3]{1+x^6}}{x}+\frac{\sqrt[3]{2} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2^{2/3} \sqrt[3]{1+x^6}}\right)}{\sqrt{3}}-\frac{1}{3} \sqrt[3]{2} \log \left(2 x+2^{2/3} \sqrt[3]{1+x^6}\right)+\frac{\log \left(-2 x^2+2^{2/3} x \sqrt[3]{1+x^6}-\sqrt[3]{2} \left(1+x^6\right)^{2/3}\right)}{3\ 2^{2/3}}",1,"(1 + x^6)^(1/3)/x + (2^(1/3)*ArcTan[(Sqrt[3]*x)/(-x + 2^(2/3)*(1 + x^6)^(1/3))])/Sqrt[3] - (2^(1/3)*Log[2*x + 2^(2/3)*(1 + x^6)^(1/3)])/3 + Log[-2*x^2 + 2^(2/3)*x*(1 + x^6)^(1/3) - 2^(1/3)*(1 + x^6)^(2/3)]/(3*2^(2/3))","A",1
1893,1,131,131,1.0922529,"\int \frac{\left(-2+x^6\right) \sqrt[3]{2+x^6}}{x^2 \left(2+2 x^3+x^6\right)} \, dx","IntegrateAlgebraic[((-2 + x^6)*(2 + x^6)^(1/3))/(x^2*(2 + 2*x^3 + x^6)),x]","\frac{\sqrt[3]{2+x^6}}{x}+\frac{\sqrt[3]{2} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2^{2/3} \sqrt[3]{2+x^6}}\right)}{\sqrt{3}}-\frac{1}{3} \sqrt[3]{2} \log \left(2 x+2^{2/3} \sqrt[3]{2+x^6}\right)+\frac{\log \left(-2 x^2+2^{2/3} x \sqrt[3]{2+x^6}-\sqrt[3]{2} \left(2+x^6\right)^{2/3}\right)}{3\ 2^{2/3}}","\frac{\sqrt[3]{2+x^6}}{x}+\frac{\sqrt[3]{2} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2^{2/3} \sqrt[3]{2+x^6}}\right)}{\sqrt{3}}-\frac{1}{3} \sqrt[3]{2} \log \left(2 x+2^{2/3} \sqrt[3]{2+x^6}\right)+\frac{\log \left(-2 x^2+2^{2/3} x \sqrt[3]{2+x^6}-\sqrt[3]{2} \left(2+x^6\right)^{2/3}\right)}{3\ 2^{2/3}}",1,"(2 + x^6)^(1/3)/x + (2^(1/3)*ArcTan[(Sqrt[3]*x)/(-x + 2^(2/3)*(2 + x^6)^(1/3))])/Sqrt[3] - (2^(1/3)*Log[2*x + 2^(2/3)*(2 + x^6)^(1/3)])/3 + Log[-2*x^2 + 2^(2/3)*x*(2 + x^6)^(1/3) - 2^(1/3)*(2 + x^6)^(2/3)]/(3*2^(2/3))","A",1
1894,1,131,131,0.3223056,"\int \frac{\left(1+x^3\right)^{2/3} \left(-1-2 x^3+2 x^6\right)}{x^6 \left(-1+x^3+2 x^6\right)} \, dx","IntegrateAlgebraic[((1 + x^3)^(2/3)*(-1 - 2*x^3 + 2*x^6))/(x^6*(-1 + x^3 + 2*x^6)),x]","\frac{\left(-2-17 x^3\right) \left(1+x^3\right)^{2/3}}{10 x^5}+2 \sqrt[6]{3} \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2 \sqrt[3]{1+x^3}}\right)-\frac{2 \log \left(-3 x+3^{2/3} \sqrt[3]{1+x^3}\right)}{\sqrt[3]{3}}+\frac{\log \left(3 x^2+3^{2/3} x \sqrt[3]{1+x^3}+\sqrt[3]{3} \left(1+x^3\right)^{2/3}\right)}{\sqrt[3]{3}}","\frac{\left(-2-17 x^3\right) \left(1+x^3\right)^{2/3}}{10 x^5}+2 \sqrt[6]{3} \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2 \sqrt[3]{1+x^3}}\right)-\frac{2 \log \left(-3 x+3^{2/3} \sqrt[3]{1+x^3}\right)}{\sqrt[3]{3}}+\frac{\log \left(3 x^2+3^{2/3} x \sqrt[3]{1+x^3}+\sqrt[3]{3} \left(1+x^3\right)^{2/3}\right)}{\sqrt[3]{3}}",1,"((-2 - 17*x^3)*(1 + x^3)^(2/3))/(10*x^5) + 2*3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*(1 + x^3)^(1/3))] - (2*Log[-3*x + 3^(2/3)*(1 + x^3)^(1/3)])/3^(1/3) + Log[3*x^2 + 3^(2/3)*x*(1 + x^3)^(1/3) + 3^(1/3)*(1 + x^3)^(2/3)]/3^(1/3)","A",1
1895,1,121,131,1.1618303,"\int \frac{\left(1+x^3\right) \sqrt{x-x^4}}{2+4 x^3+3 x^6} \, dx","IntegrateAlgebraic[((1 + x^3)*Sqrt[x - x^4])/(2 + 4*x^3 + 3*x^6),x]","-\frac{2}{9} \tan ^{-1}\left(\frac{x^2}{\sqrt{x-x^4}}\right)+\frac{1}{18} \sqrt{8+7 i \sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\frac{i}{\sqrt{2}}} x^2}{\sqrt{x-x^4}}\right)+\frac{1}{18} \sqrt{8-7 i \sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2+\frac{i}{\sqrt{2}}} x^2}{\sqrt{x-x^4}}\right)","-\frac{2}{9} \tan ^{-1}\left(\frac{x^2}{\sqrt{x-x^4}}\right)-\frac{1}{18} \sqrt{8+7 i \sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\frac{i}{\sqrt{2}}} x \sqrt{x-x^4}}{-1+x^3}\right)-\frac{1}{18} \sqrt{8-7 i \sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2+\frac{i}{\sqrt{2}}} x \sqrt{x-x^4}}{-1+x^3}\right)",1,"(-2*ArcTan[x^2/Sqrt[x - x^4]])/9 + (Sqrt[8 + (7*I)*Sqrt[2]]*ArcTan[(Sqrt[2 - I/Sqrt[2]]*x^2)/Sqrt[x - x^4]])/18 + (Sqrt[8 - (7*I)*Sqrt[2]]*ArcTan[(Sqrt[2 + I/Sqrt[2]]*x^2)/Sqrt[x - x^4]])/18","A",1
1896,1,131,131,1.5990381,"\int \frac{-2+a x^4}{\sqrt[4]{b+a x^4} \left(-b+a x^4+2 x^8\right)} \, dx","IntegrateAlgebraic[(-2 + a*x^4)/((b + a*x^4)^(1/4)*(-b + a*x^4 + 2*x^8)),x]","-\frac{\text{RootSum}\left[2 a^2-2 b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 a \log (x)+a b \log (x)-2 a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)-a b \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)-2 \log (x) \text{$\#$1}^4+2 \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]}{4 b}","-\frac{\text{RootSum}\left[2 a^2-2 b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 a \log (x)-a b \log (x)+2 a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)+a b \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)+2 \log (x) \text{$\#$1}^4-2 \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{3 a \text{$\#$1}-2 \text{$\#$1}^5}\&\right]}{4 b}",1,"-1/4*RootSum[2*a^2 - 2*b - 3*a*#1^4 + #1^8 & , (2*a*Log[x] + a*b*Log[x] - 2*a*Log[(b + a*x^4)^(1/4) - x*#1] - a*b*Log[(b + a*x^4)^(1/4) - x*#1] - 2*Log[x]*#1^4 + 2*Log[(b + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1 + 2*#1^5) & ]/b","A",1
1897,1,131,131,0.2141988,"\int \frac{-1+x}{(1+x) \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[(-1 + x)/((1 + x)*Sqrt[x + Sqrt[1 + x^2]]),x]","\frac{2 (-6+3 x) \sqrt{1+x^2}+2 \left(1-6 x+3 x^2\right)}{3 \left(x+\sqrt{1+x^2}\right)^{3/2}}-4 \sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\sqrt{-1+\sqrt{2}} \sqrt{x+\sqrt{1+x^2}}\right)+4 \sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\sqrt{1+\sqrt{2}} \sqrt{x+\sqrt{1+x^2}}\right)","\frac{2 (-6+3 x) \sqrt{1+x^2}+2 \left(1-6 x+3 x^2\right)}{3 \left(x+\sqrt{1+x^2}\right)^{3/2}}-4 \sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{x+\sqrt{1+x^2}}}{\sqrt{1+\sqrt{2}}}\right)+4 \sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{x+\sqrt{1+x^2}}}{\sqrt{-1+\sqrt{2}}}\right)",1,"(2*(-6 + 3*x)*Sqrt[1 + x^2] + 2*(1 - 6*x + 3*x^2))/(3*(x + Sqrt[1 + x^2])^(3/2)) - 4*Sqrt[-1 + Sqrt[2]]*ArcTan[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] + 4*Sqrt[1 + Sqrt[2]]*ArcTanh[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]]","A",1
1898,1,131,131,0.1916597,"\int \frac{1+x}{(-1+x) \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[(1 + x)/((-1 + x)*Sqrt[x + Sqrt[1 + x^2]]),x]","\frac{2 (6+3 x) \sqrt{1+x^2}+2 \left(1+6 x+3 x^2\right)}{3 \left(x+\sqrt{1+x^2}\right)^{3/2}}+4 \sqrt{1+\sqrt{2}} \tan ^{-1}\left(\sqrt{1+\sqrt{2}} \sqrt{x+\sqrt{1+x^2}}\right)-4 \sqrt{-1+\sqrt{2}} \tanh ^{-1}\left(\sqrt{-1+\sqrt{2}} \sqrt{x+\sqrt{1+x^2}}\right)","\frac{2 (6+3 x) \sqrt{1+x^2}+2 \left(1+6 x+3 x^2\right)}{3 \left(x+\sqrt{1+x^2}\right)^{3/2}}+4 \sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{x+\sqrt{1+x^2}}}{\sqrt{-1+\sqrt{2}}}\right)-4 \sqrt{-1+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{x+\sqrt{1+x^2}}}{\sqrt{1+\sqrt{2}}}\right)",1,"(2*(6 + 3*x)*Sqrt[1 + x^2] + 2*(1 + 6*x + 3*x^2))/(3*(x + Sqrt[1 + x^2])^(3/2)) + 4*Sqrt[1 + Sqrt[2]]*ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] - 4*Sqrt[-1 + Sqrt[2]]*ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]]","A",1
1899,1,131,131,4.2869402,"\int \frac{\sqrt{-x+x^2}}{\sqrt{x^2-x \sqrt{-x+x^2}}} \, dx","IntegrateAlgebraic[Sqrt[-x + x^2]/Sqrt[x^2 - x*Sqrt[-x + x^2]],x]","\frac{(-9+8 x) \sqrt{-x+x^2} \sqrt{-x \left(-x+\sqrt{-x+x^2}\right)}}{12 x}+\sqrt{x \left(x-\sqrt{-x+x^2}\right)} \left(\frac{1}{12} (-19+8 x)+\frac{3 \sqrt{x+\sqrt{-x+x^2}} \tanh ^{-1}\left(\sqrt{2} \sqrt{x+\sqrt{-x+x^2}}\right)}{4 \sqrt{2} x}\right)","\frac{(-9+8 x) \sqrt{-x+x^2} \sqrt{-x \left(-x+\sqrt{-x+x^2}\right)}}{12 x}+\sqrt{x \left(x-\sqrt{-x+x^2}\right)} \left(\frac{1}{12} (-19+8 x)+\frac{3 \sqrt{x+\sqrt{-x+x^2}} \tanh ^{-1}\left(\sqrt{2} \sqrt{x+\sqrt{-x+x^2}}\right)}{4 \sqrt{2} x}\right)",1,"((-9 + 8*x)*Sqrt[-x + x^2]*Sqrt[-(x*(-x + Sqrt[-x + x^2]))])/(12*x) + Sqrt[x*(x - Sqrt[-x + x^2])]*((-19 + 8*x)/12 + (3*Sqrt[x + Sqrt[-x + x^2]]*ArcTanh[Sqrt[2]*Sqrt[x + Sqrt[-x + x^2]]])/(4*Sqrt[2]*x))","A",1
1900,1,132,132,0.1845022,"\int \frac{6+2 x+x^2}{(2+x) \left(2+x^2\right) \sqrt[3]{2+x+x^2}} \, dx","IntegrateAlgebraic[(6 + 2*x + x^2)/((2 + x)*(2 + x^2)*(2 + x + x^2)^(1/3)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{-\frac{2^{2/3} x}{\sqrt{3}}+\frac{\sqrt[3]{2+x+x^2}}{\sqrt{3}}}{\sqrt[3]{2+x+x^2}}\right)}{2^{2/3}}+\frac{\log \left(2^{2/3} x+2 \sqrt[3]{2+x+x^2}\right)}{2^{2/3}}-\frac{\log \left(\sqrt[3]{2} x^2-2^{2/3} x \sqrt[3]{2+x+x^2}+2 \left(2+x+x^2\right)^{2/3}\right)}{2\ 2^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{-\frac{2^{2/3} x}{\sqrt{3}}+\frac{\sqrt[3]{2+x+x^2}}{\sqrt{3}}}{\sqrt[3]{2+x+x^2}}\right)}{2^{2/3}}+\frac{\log \left(2^{2/3} x+2 \sqrt[3]{2+x+x^2}\right)}{2^{2/3}}-\frac{\log \left(\sqrt[3]{2} x^2-2^{2/3} x \sqrt[3]{2+x+x^2}+2 \left(2+x+x^2\right)^{2/3}\right)}{2\ 2^{2/3}}",1,"-((Sqrt[3]*ArcTan[(-((2^(2/3)*x)/Sqrt[3]) + (2 + x + x^2)^(1/3)/Sqrt[3])/(2 + x + x^2)^(1/3)])/2^(2/3)) + Log[2^(2/3)*x + 2*(2 + x + x^2)^(1/3)]/2^(2/3) - Log[2^(1/3)*x^2 - 2^(2/3)*x*(2 + x + x^2)^(1/3) + 2*(2 + x + x^2)^(2/3)]/(2*2^(2/3))","A",1
1901,1,132,132,0.2617467,"\int \frac{-1+x}{x \sqrt[3]{1+2 x+2 x^2+x^3}} \, dx","IntegrateAlgebraic[(-1 + x)/(x*(1 + 2*x + 2*x^2 + x^3)^(1/3)),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{1+2 x+2 x^2+x^3}}{2+2 x+\sqrt[3]{1+2 x+2 x^2+x^3}}\right)-\log \left(-1-x+\sqrt[3]{1+2 x+2 x^2+x^3}\right)+\frac{1}{2} \log \left(1+2 x+x^2+(1+x) \sqrt[3]{1+2 x+2 x^2+x^3}+\left(1+2 x+2 x^2+x^3\right)^{2/3}\right)","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{1+2 x+2 x^2+x^3}}{2+2 x+\sqrt[3]{1+2 x+2 x^2+x^3}}\right)-\log \left(-1-x+\sqrt[3]{1+2 x+2 x^2+x^3}\right)+\frac{1}{2} \log \left(1+2 x+x^2+(1+x) \sqrt[3]{1+2 x+2 x^2+x^3}+\left(1+2 x+2 x^2+x^3\right)^{2/3}\right)",1,"-(Sqrt[3]*ArcTan[(Sqrt[3]*(1 + 2*x + 2*x^2 + x^3)^(1/3))/(2 + 2*x + (1 + 2*x + 2*x^2 + x^3)^(1/3))]) - Log[-1 - x + (1 + 2*x + 2*x^2 + x^3)^(1/3)] + Log[1 + 2*x + x^2 + (1 + x)*(1 + 2*x + 2*x^2 + x^3)^(1/3) + (1 + 2*x + 2*x^2 + x^3)^(2/3)]/2","A",1
1902,1,132,132,0.4933049,"\int \frac{\left(-2 q+p x^3\right) \left(a q+b x^2+a p x^3\right) \sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^7} \, dx","IntegrateAlgebraic[((-2*q + p*x^3)*(a*q + b*x^2 + a*p*x^3)*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6])/x^7,x]","\frac{\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left(2 a q^2+3 b q x^2+4 a p q x^3-4 a p q x^4+3 b p x^5+2 a p^2 x^6\right)}{6 x^6}+2 b p q \log (x)-b p q \log \left(q+p x^3+\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}\right)","\frac{\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left(2 a q^2+3 b q x^2+4 a p q x^3-4 a p q x^4+3 b p x^5+2 a p^2 x^6\right)}{6 x^6}+2 b p q \log (x)-b p q \log \left(q+p x^3+\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}\right)",1,"(Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]*(2*a*q^2 + 3*b*q*x^2 + 4*a*p*q*x^3 - 4*a*p*q*x^4 + 3*b*p*x^5 + 2*a*p^2*x^6))/(6*x^6) + 2*b*p*q*Log[x] - b*p*q*Log[q + p*x^3 + Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]]","A",1
1903,1,147,132,19.9704983,"\int \frac{x^2}{\left(1+x^2\right) \sqrt[10]{243-5265 x+47250 x^2-225810 x^3+615255 x^4-954733 x^5+820340 x^6-401440 x^7+112000 x^8-16640 x^9+1024 x^{10}}} \, dx","IntegrateAlgebraic[x^2/((1 + x^2)*(243 - 5265*x + 47250*x^2 - 225810*x^3 + 615255*x^4 - 954733*x^5 + 820340*x^6 - 401440*x^7 + 112000*x^8 - 16640*x^9 + 1024*x^10)^(1/10)),x]","\frac{\left(\left(3-13 x+4 x^2\right)^5\right)^{9/10} \left(-\frac{1}{2} \log \left(13-8 x+4 \sqrt{3-13 x+4 x^2}\right)+\frac{1}{2} \text{RootSum}\left[178+104 \text{$\#$1}+10 \text{$\#$1}^2+\text{$\#$1}^4\&,\frac{13 \log \left(-2 x+\sqrt{3-13 x+4 x^2}-\text{$\#$1}\right)+4 \log \left(-2 x+\sqrt{3-13 x+4 x^2}-\text{$\#$1}\right) \text{$\#$1}}{26+5 \text{$\#$1}+\text{$\#$1}^3}\&\right]\right)}{\left(3-13 x+4 x^2\right)^{9/2}}","\frac{\left(\left(3-13 x+4 x^2\right)^5\right)^{9/10} \left(-\frac{1}{2} \log \left(13-8 x+4 \sqrt{3-13 x+4 x^2}\right)+32 \text{RootSum}\left[14641-6292 \text{$\#$1}+1174 \text{$\#$1}^2-52 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{\log \left(13-8 x+4 \sqrt{3-13 x+4 x^2}-\text{$\#$1}\right) \text{$\#$1}}{-1573+587 \text{$\#$1}-39 \text{$\#$1}^2+\text{$\#$1}^3}\&\right]\right)}{\left(3-13 x+4 x^2\right)^{9/2}}",1,"(((3 - 13*x + 4*x^2)^5)^(9/10)*(-1/2*Log[13 - 8*x + 4*Sqrt[3 - 13*x + 4*x^2]] + RootSum[178 + 104*#1 + 10*#1^2 + #1^4 & , (13*Log[-2*x + Sqrt[3 - 13*x + 4*x^2] - #1] + 4*Log[-2*x + Sqrt[3 - 13*x + 4*x^2] - #1]*#1)/(26 + 5*#1 + #1^3) & ]/2))/(3 - 13*x + 4*x^2)^(9/2)","A",1
1904,1,132,132,5.2562483,"\int \frac{1}{x \sqrt{-b x+a^2 x^2} \left(a x^2+x \sqrt{-b x+a^2 x^2}\right)^{3/2}} \, dx","IntegrateAlgebraic[1/(x*Sqrt[-(b*x) + a^2*x^2]*(a*x^2 + x*Sqrt[-(b*x) + a^2*x^2])^(3/2)),x]","-\frac{4 \sqrt{-b x+a^2 x^2} \left(105 b^2-160 a^2 b x-256 a^4 x^2\right) \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{1155 b^4 x^4}+\frac{4 \left(245 a b^2+32 a^3 b x+256 a^5 x^2\right) \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{1155 b^4 x^3}","-\frac{4 \sqrt{-b x+a^2 x^2} \left(105 b^2-160 a^2 b x-256 a^4 x^2\right) \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{1155 b^4 x^4}+\frac{4 \left(245 a b^2+32 a^3 b x+256 a^5 x^2\right) \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{1155 b^4 x^3}",1,"(-4*Sqrt[-(b*x) + a^2*x^2]*(105*b^2 - 160*a^2*b*x - 256*a^4*x^2)*Sqrt[x*(a*x + Sqrt[-(b*x) + a^2*x^2])])/(1155*b^4*x^4) + (4*(245*a*b^2 + 32*a^3*b*x + 256*a^5*x^2)*Sqrt[x*(a*x + Sqrt[-(b*x) + a^2*x^2])])/(1155*b^4*x^3)","A",1
1905,1,190,132,0.5784514,"\int \frac{\sqrt{x^2+\sqrt{1+x^4}}}{\left(-1+x^2\right) \sqrt{1+x^4}} \, dx","IntegrateAlgebraic[Sqrt[x^2 + Sqrt[1 + x^4]]/((-1 + x^2)*Sqrt[1 + x^4]),x]","\sqrt{\frac{1}{2} \left(-1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{-\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)-\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{-\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)","\sqrt{\frac{1}{2} \left(-1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(-1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)",1,"Sqrt[(-1 + Sqrt[2])/2]*ArcTan[(-Sqrt[-1/2 + 1/Sqrt[2]] + Sqrt[-1/2 + 1/Sqrt[2]]*x^2 + Sqrt[-1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] - Sqrt[(1 + Sqrt[2])/2]*ArcTanh[(-Sqrt[1/2 + 1/Sqrt[2]] + Sqrt[1/2 + 1/Sqrt[2]]*x^2 + Sqrt[1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])]","A",1
1906,1,133,133,0.214804,"\int \frac{1}{(1+2 x) \sqrt[3]{-1+4 x+4 x^2}} \, dx","IntegrateAlgebraic[1/((1 + 2*x)*(-1 + 4*x + 4*x^2)^(1/3)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2^{2/3} \sqrt[3]{-1+4 x+4 x^2}}{\sqrt{3}}\right)}{4 \sqrt[3]{2}}-\frac{\log \left(2+2^{2/3} \sqrt[3]{-1+4 x+4 x^2}\right)}{4 \sqrt[3]{2}}+\frac{\log \left(-2+2^{2/3} \sqrt[3]{-1+4 x+4 x^2}-\sqrt[3]{2} \left(-1+4 x+4 x^2\right)^{2/3}\right)}{8 \sqrt[3]{2}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2^{2/3} \sqrt[3]{-1+4 x+4 x^2}}{\sqrt{3}}\right)}{4 \sqrt[3]{2}}-\frac{\log \left(2+2^{2/3} \sqrt[3]{-1+4 x+4 x^2}\right)}{4 \sqrt[3]{2}}+\frac{\log \left(-2+2^{2/3} \sqrt[3]{-1+4 x+4 x^2}-\sqrt[3]{2} \left(-1+4 x+4 x^2\right)^{2/3}\right)}{8 \sqrt[3]{2}}",1,"-1/4*(Sqrt[3]*ArcTan[1/Sqrt[3] - (2^(2/3)*(-1 + 4*x + 4*x^2)^(1/3))/Sqrt[3]])/2^(1/3) - Log[2 + 2^(2/3)*(-1 + 4*x + 4*x^2)^(1/3)]/(4*2^(1/3)) + Log[-2 + 2^(2/3)*(-1 + 4*x + 4*x^2)^(1/3) - 2^(1/3)*(-1 + 4*x + 4*x^2)^(2/3)]/(8*2^(1/3))","A",1
1907,1,133,133,0.4698094,"\int \frac{1}{x^3 \left(-b+a x^3\right) \sqrt[3]{b^2 x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[1/(x^3*(-b + a*x^3)*(b^2*x^2 + a^3*x^3)^(1/3)),x]","\frac{3 \left(5 b^4-6 a^3 b^2 x+9 a^6 x^2\right) \left(b^2 x^2+a^3 x^3\right)^{2/3}}{40 b^7 x^4}+\frac{a \text{RootSum}\left[a^9+a b^5-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{3 b^2}","\frac{3 \left(5 b^4-6 a^3 b^2 x+9 a^6 x^2\right) \left(b^2 x^2+a^3 x^3\right)^{2/3}}{40 b^7 x^4}+\frac{a \text{RootSum}\left[a^9+a b^5-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{3 b^2}",1,"(3*(5*b^4 - 6*a^3*b^2*x + 9*a^6*x^2)*(b^2*x^2 + a^3*x^3)^(2/3))/(40*b^7*x^4) + (a*RootSum[a^9 + a*b^5 - 3*a^6*#1^3 + 3*a^3*#1^6 - #1^9 & , (-Log[x] + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1])/#1 & ])/(3*b^2)","A",1
1908,1,133,133,0.0022671,"\int \frac{1}{x^3 \left(-b+a x^3\right) \sqrt[3]{b^2 x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[1/(x^3*(-b + a*x^3)*(b^2*x^2 + a^3*x^3)^(1/3)),x]","\frac{3 \left(5 b^4-6 a^3 b^2 x+9 a^6 x^2\right) \left(b^2 x^2+a^3 x^3\right)^{2/3}}{40 b^7 x^4}+\frac{a \text{RootSum}\left[a^9+a b^5-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{3 b^2}","\frac{3 \left(5 b^4-6 a^3 b^2 x+9 a^6 x^2\right) \left(b^2 x^2+a^3 x^3\right)^{2/3}}{40 b^7 x^4}+\frac{a \text{RootSum}\left[a^9+a b^5-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{3 b^2}",1,"(3*(5*b^4 - 6*a^3*b^2*x + 9*a^6*x^2)*(b^2*x^2 + a^3*x^3)^(2/3))/(40*b^7*x^4) + (a*RootSum[a^9 + a*b^5 - 3*a^6*#1^3 + 3*a^3*#1^6 - #1^9 & , (-Log[x] + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1])/#1 & ])/(3*b^2)","A",1
1909,1,133,133,6.6755922,"\int \frac{x^3 (-b+x) (-2 a b+(3 a-b) x)}{(-a+x) \left(x^2 (-a+x) (-b+x)\right)^{3/4} \left(-a^3 d+3 a^2 d x+(b-3 a d) x^2+(-1+d) x^3\right)} \, dx","IntegrateAlgebraic[(x^3*(-b + x)*(-2*a*b + (3*a - b)*x))/((-a + x)*(x^2*(-a + x)*(-b + x))^(3/4)*(-(a^3*d) + 3*a^2*d*x + (b - 3*a*d)*x^2 + (-1 + d)*x^3)),x]","-\frac{4 \sqrt[4]{a b x^2+(-a-b) x^3+x^4}}{a-x}+2 \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(a b x^2+(-a-b) x^3+x^4\right)^{3/4}}{x^2 (-b+x)}\right)-2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt[4]{d} \left(a b x^2+(-a-b) x^3+x^4\right)^{3/4}}{x^2 (-b+x)}\right)","-\frac{4 \sqrt[4]{a b x^2+(-a-b) x^3+x^4}}{a-x}+2 \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(a b x^2+(-a-b) x^3+x^4\right)^{3/4}}{x^2 (-b+x)}\right)-2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt[4]{d} \left(a b x^2+(-a-b) x^3+x^4\right)^{3/4}}{x^2 (-b+x)}\right)",1,"(-4*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/4))/(a - x) + 2*d^(1/4)*ArcTan[(d^(1/4)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(3/4))/(x^2*(-b + x))] - 2*d^(1/4)*ArcTanh[(d^(1/4)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(3/4))/(x^2*(-b + x))]","A",1
1910,1,133,133,0.2958732,"\int \frac{x^2}{\sqrt{-x-x^2+x^3} \left(-1+x^4\right)} \, dx","IntegrateAlgebraic[x^2/(Sqrt[-x - x^2 + x^3]*(-1 + x^4)),x]","-\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)+\frac{1}{4} \sqrt{\frac{1}{5}+\frac{2 i}{5}} \tan ^{-1}\left(\frac{\sqrt{1-2 i} \sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)+\frac{1}{4} \sqrt{\frac{1}{5}-\frac{2 i}{5}} \tan ^{-1}\left(\frac{\sqrt{1+2 i} \sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)","-\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)+\frac{1}{4} \sqrt{\frac{1}{5}+\frac{2 i}{5}} \tan ^{-1}\left(\frac{\sqrt{1-2 i} \sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)+\frac{1}{4} \sqrt{\frac{1}{5}-\frac{2 i}{5}} \tan ^{-1}\left(\frac{\sqrt{1+2 i} \sqrt{-x-x^2+x^3}}{-1-x+x^2}\right)",1,"-1/2*ArcTan[Sqrt[-x - x^2 + x^3]/(-1 - x + x^2)] + (Sqrt[1/5 + (2*I)/5]*ArcTan[(Sqrt[1 - 2*I]*Sqrt[-x - x^2 + x^3])/(-1 - x + x^2)])/4 + (Sqrt[1/5 - (2*I)/5]*ArcTan[(Sqrt[1 + 2*I]*Sqrt[-x - x^2 + x^3])/(-1 - x + x^2)])/4","A",1
1911,1,133,133,1.0617295,"\int \frac{\left(-1+x^2\right) \sqrt{1+x^2+x^4}}{\left(1+x^2\right) \left(1+x+x^2+x^3+x^4\right)} \, dx","IntegrateAlgebraic[((-1 + x^2)*Sqrt[1 + x^2 + x^4])/((1 + x^2)*(1 + x + x^2 + x^3 + x^4)),x]","-2 \tan ^{-1}\left(\frac{\sqrt{1+x^2+x^4}}{1-x+x^2}\right)+\sqrt{\frac{1}{5} \left(2+2 \sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-2+\sqrt{5}} \sqrt{1+x^2+x^4}}{1-x+x^2}\right)+\sqrt{\frac{1}{5} \left(-2+2 \sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{2+\sqrt{5}} \sqrt{1+x^2+x^4}}{1-x+x^2}\right)","-2 \tan ^{-1}\left(\frac{\sqrt{1+x^2+x^4}}{1-x+x^2}\right)+\sqrt{\frac{1}{5} \left(2+2 \sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{1+x^2+x^4}}{\sqrt{2+\sqrt{5}} \left(1-x+x^2\right)}\right)+\sqrt{\frac{1}{5} \left(-2+2 \sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{1+x^2+x^4}}{\sqrt{-2+\sqrt{5}} \left(1-x+x^2\right)}\right)",1,"-2*ArcTan[Sqrt[1 + x^2 + x^4]/(1 - x + x^2)] + Sqrt[(2 + 2*Sqrt[5])/5]*ArcTan[(Sqrt[-2 + Sqrt[5]]*Sqrt[1 + x^2 + x^4])/(1 - x + x^2)] + Sqrt[(-2 + 2*Sqrt[5])/5]*ArcTanh[(Sqrt[2 + Sqrt[5]]*Sqrt[1 + x^2 + x^4])/(1 - x + x^2)]","A",1
1912,1,133,133,0.5912507,"\int \frac{x^6}{\left(-b+a x^4\right) \left(b+a x^4\right)^{3/4}} \, dx","IntegrateAlgebraic[x^6/((-b + a*x^4)*(b + a*x^4)^(3/4)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2 a^{7/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2\ 2^{3/4} a^{7/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2 a^{7/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2\ 2^{3/4} a^{7/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2 a^{7/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2\ 2^{3/4} a^{7/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2 a^{7/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2\ 2^{3/4} a^{7/4}}",1,"-1/2*ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(7/4) + ArcTan[(2^(1/4)*a^(1/4)*x)/(b + a*x^4)^(1/4)]/(2*2^(3/4)*a^(7/4)) + ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/(2*a^(7/4)) - ArcTanh[(2^(1/4)*a^(1/4)*x)/(b + a*x^4)^(1/4)]/(2*2^(3/4)*a^(7/4))","A",1
1913,1,133,133,0.4417415,"\int \frac{-b+a x^3}{x^3 \left(b+a x^3\right) \sqrt[4]{-b x+a x^4}} \, dx","IntegrateAlgebraic[(-b + a*x^3)/(x^3*(b + a*x^3)*(-(b*x) + a*x^4)^(1/4)),x]","-\frac{4 \left(-b x+a x^4\right)^{3/4}}{9 b x^3}+\frac{2\ 2^{3/4} a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)}{3 b}+\frac{2\ 2^{3/4} a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)}{3 b}","-\frac{4 \left(-b x+a x^4\right)^{3/4}}{9 b x^3}+\frac{2\ 2^{3/4} a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)}{3 b}+\frac{2\ 2^{3/4} a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)}{3 b}",1,"(-4*(-(b*x) + a*x^4)^(3/4))/(9*b*x^3) + (2*2^(3/4)*a^(3/4)*ArcTan[(2^(1/4)*a^(1/4)*(-(b*x) + a*x^4)^(3/4))/(-b + a*x^3)])/(3*b) + (2*2^(3/4)*a^(3/4)*ArcTanh[(2^(1/4)*a^(1/4)*(-(b*x) + a*x^4)^(3/4))/(-b + a*x^3)])/(3*b)","A",1
1914,1,133,133,0.412918,"\int \frac{-b+a x^4}{\sqrt[4]{b+a x^4} \left(-b+3 a x^4\right)} \, dx","IntegrateAlgebraic[(-b + a*x^4)/((b + a*x^4)^(1/4)*(-b + 3*a*x^4)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{6 \sqrt[4]{a}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{3 \sqrt{2} \sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{6 \sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{3 \sqrt{2} \sqrt[4]{a}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{6 \sqrt[4]{a}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{3 \sqrt{2} \sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{6 \sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{3 \sqrt{2} \sqrt[4]{a}}",1,"ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/(6*a^(1/4)) + ArcTan[(Sqrt[2]*a^(1/4)*x)/(b + a*x^4)^(1/4)]/(3*Sqrt[2]*a^(1/4)) + ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/(6*a^(1/4)) + ArcTanh[(Sqrt[2]*a^(1/4)*x)/(b + a*x^4)^(1/4)]/(3*Sqrt[2]*a^(1/4))","A",1
1915,1,133,133,0.3501248,"\int \frac{-2 b+a x^2}{\sqrt[4]{-b+a x^2} \left(-b+a x^2+c x^4\right)} \, dx","IntegrateAlgebraic[(-2*b + a*x^2)/((-b + a*x^2)^(1/4)*(-b + a*x^2 + c*x^4)),x]","-\frac{\tan ^{-1}\left(\frac{-\frac{\sqrt[4]{c} x^2}{\sqrt{2}}+\frac{\sqrt{-b+a x^2}}{\sqrt{2} \sqrt[4]{c}}}{x \sqrt[4]{-b+a x^2}}\right)}{\sqrt{2} \sqrt[4]{c}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x \sqrt[4]{-b+a x^2}}{\sqrt{c} x^2+\sqrt{-b+a x^2}}\right)}{\sqrt{2} \sqrt[4]{c}}","-\frac{\tan ^{-1}\left(\frac{-\frac{\sqrt[4]{c} x^2}{\sqrt{2}}+\frac{\sqrt{-b+a x^2}}{\sqrt{2} \sqrt[4]{c}}}{x \sqrt[4]{-b+a x^2}}\right)}{\sqrt{2} \sqrt[4]{c}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x \sqrt[4]{-b+a x^2}}{\sqrt{c} x^2+\sqrt{-b+a x^2}}\right)}{\sqrt{2} \sqrt[4]{c}}",1,"-(ArcTan[(-((c^(1/4)*x^2)/Sqrt[2]) + Sqrt[-b + a*x^2]/(Sqrt[2]*c^(1/4)))/(x*(-b + a*x^2)^(1/4))]/(Sqrt[2]*c^(1/4))) + ArcTanh[(Sqrt[2]*c^(1/4)*x*(-b + a*x^2)^(1/4))/(Sqrt[c]*x^2 + Sqrt[-b + a*x^2])]/(Sqrt[2]*c^(1/4))","A",1
1916,1,133,133,1.5559927,"\int \frac{\left(-q+p x^2\right) \sqrt{q^2+p^2 x^4}}{x^2 \left(a q+b x+a p x^2\right)} \, dx","IntegrateAlgebraic[((-q + p*x^2)*Sqrt[q^2 + p^2*x^4])/(x^2*(a*q + b*x + a*p*x^2)),x]","\frac{\sqrt{q^2+p^2 x^4}}{a x}+\frac{2 \sqrt{-b^2+2 a^2 p q} \tan ^{-1}\left(\frac{\sqrt{-b^2+2 a^2 p q} x}{a q+b x+a p x^2+a \sqrt{q^2+p^2 x^4}}\right)}{a^2}+\frac{b \log (x)}{a^2}-\frac{b \log \left(q+p x^2+\sqrt{q^2+p^2 x^4}\right)}{a^2}","\frac{\sqrt{q^2+p^2 x^4}}{a x}+\frac{2 \sqrt{-b^2+2 a^2 p q} \tan ^{-1}\left(\frac{\sqrt{-b^2+2 a^2 p q} x}{a q+b x+a p x^2+a \sqrt{q^2+p^2 x^4}}\right)}{a^2}+\frac{b \log (x)}{a^2}-\frac{b \log \left(q+p x^2+\sqrt{q^2+p^2 x^4}\right)}{a^2}",1,"Sqrt[q^2 + p^2*x^4]/(a*x) + (2*Sqrt[-b^2 + 2*a^2*p*q]*ArcTan[(Sqrt[-b^2 + 2*a^2*p*q]*x)/(a*q + b*x + a*p*x^2 + a*Sqrt[q^2 + p^2*x^4])])/a^2 + (b*Log[x])/a^2 - (b*Log[q + p*x^2 + Sqrt[q^2 + p^2*x^4]])/a^2","A",1
1917,1,133,133,2.462898,"\int \frac{\sqrt[3]{1+x^5} \left(-3+2 x^5\right)}{x^2 \left(2-x^3+2 x^5\right)} \, dx","IntegrateAlgebraic[((1 + x^5)^(1/3)*(-3 + 2*x^5))/(x^2*(2 - x^3 + 2*x^5)),x]","\frac{3 \sqrt[3]{1+x^5}}{2 x}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2} \sqrt[3]{1+x^5}}\right)}{2 \sqrt[3]{2}}+\frac{\log \left(-x+\sqrt[3]{2} \sqrt[3]{1+x^5}\right)}{2 \sqrt[3]{2}}-\frac{\log \left(x^2+\sqrt[3]{2} x \sqrt[3]{1+x^5}+2^{2/3} \left(1+x^5\right)^{2/3}\right)}{4 \sqrt[3]{2}}","\frac{3 \sqrt[3]{1+x^5}}{2 x}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2} \sqrt[3]{1+x^5}}\right)}{2 \sqrt[3]{2}}+\frac{\log \left(-x+\sqrt[3]{2} \sqrt[3]{1+x^5}\right)}{2 \sqrt[3]{2}}-\frac{\log \left(x^2+\sqrt[3]{2} x \sqrt[3]{1+x^5}+2^{2/3} \left(1+x^5\right)^{2/3}\right)}{4 \sqrt[3]{2}}",1,"(3*(1 + x^5)^(1/3))/(2*x) + (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*2^(1/3)*(1 + x^5)^(1/3))])/(2*2^(1/3)) + Log[-x + 2^(1/3)*(1 + x^5)^(1/3)]/(2*2^(1/3)) - Log[x^2 + 2^(1/3)*x*(1 + x^5)^(1/3) + 2^(2/3)*(1 + x^5)^(2/3)]/(4*2^(1/3))","A",1
1918,1,133,133,13.0452766,"\int \frac{x^2 \left(4 b+a x^5\right)}{\left(-b+a x^5\right)^{3/4} \left(-b+c x^4+a x^5\right)} \, dx","IntegrateAlgebraic[(x^2*(4*b + a*x^5))/((-b + a*x^5)^(3/4)*(-b + c*x^4 + a*x^5)),x]","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x \sqrt[4]{-b+a x^5}}{-\sqrt{c} x^2+\sqrt{-b+a x^5}}\right)}{c^{3/4}}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{c} x^2}{\sqrt{2}}+\frac{\sqrt{-b+a x^5}}{\sqrt{2} \sqrt[4]{c}}}{x \sqrt[4]{-b+a x^5}}\right)}{c^{3/4}}","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x \sqrt[4]{-b+a x^5}}{-\sqrt{c} x^2+\sqrt{-b+a x^5}}\right)}{c^{3/4}}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{c} x^2}{\sqrt{2}}+\frac{\sqrt{-b+a x^5}}{\sqrt{2} \sqrt[4]{c}}}{x \sqrt[4]{-b+a x^5}}\right)}{c^{3/4}}",1,"-((Sqrt[2]*ArcTan[(Sqrt[2]*c^(1/4)*x*(-b + a*x^5)^(1/4))/(-(Sqrt[c]*x^2) + Sqrt[-b + a*x^5])])/c^(3/4)) + (Sqrt[2]*ArcTanh[((c^(1/4)*x^2)/Sqrt[2] + Sqrt[-b + a*x^5]/(Sqrt[2]*c^(1/4)))/(x*(-b + a*x^5)^(1/4))])/c^(3/4)","A",1
1919,1,133,133,1.0977168,"\int \frac{\left(-1+x^6\right) \left(1+x^6\right)^{2/3}}{x^3 \left(2-x^3+2 x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^6)*(1 + x^6)^(2/3))/(x^3*(2 - x^3 + 2*x^6)),x]","\frac{\left(1+x^6\right)^{2/3}}{4 x^2}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2} \sqrt[3]{1+x^6}}\right)}{2\ 2^{2/3} \sqrt{3}}+\frac{\log \left(-x+\sqrt[3]{2} \sqrt[3]{1+x^6}\right)}{6\ 2^{2/3}}-\frac{\log \left(x^2+\sqrt[3]{2} x \sqrt[3]{1+x^6}+2^{2/3} \left(1+x^6\right)^{2/3}\right)}{12\ 2^{2/3}}","\frac{\left(1+x^6\right)^{2/3}}{4 x^2}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2} \sqrt[3]{1+x^6}}\right)}{2\ 2^{2/3} \sqrt{3}}+\frac{\log \left(-x+\sqrt[3]{2} \sqrt[3]{1+x^6}\right)}{6\ 2^{2/3}}-\frac{\log \left(x^2+\sqrt[3]{2} x \sqrt[3]{1+x^6}+2^{2/3} \left(1+x^6\right)^{2/3}\right)}{12\ 2^{2/3}}",1,"(1 + x^6)^(2/3)/(4*x^2) - ArcTan[(Sqrt[3]*x)/(x + 2*2^(1/3)*(1 + x^6)^(1/3))]/(2*2^(2/3)*Sqrt[3]) + Log[-x + 2^(1/3)*(1 + x^6)^(1/3)]/(6*2^(2/3)) - Log[x^2 + 2^(1/3)*x*(1 + x^6)^(1/3) + 2^(2/3)*(1 + x^6)^(2/3)]/(12*2^(2/3))","A",1
1920,1,133,133,15.7427752,"\int \frac{x^2 \left(2 b+a x^6\right)}{\left(-b+a x^6\right)^{3/4} \left(-b+c x^4+a x^6\right)} \, dx","IntegrateAlgebraic[(x^2*(2*b + a*x^6))/((-b + a*x^6)^(3/4)*(-b + c*x^4 + a*x^6)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x \sqrt[4]{-b+a x^6}}{-\sqrt{c} x^2+\sqrt{-b+a x^6}}\right)}{\sqrt{2} c^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[4]{c} x^2}{\sqrt{2}}+\frac{\sqrt{-b+a x^6}}{\sqrt{2} \sqrt[4]{c}}}{x \sqrt[4]{-b+a x^6}}\right)}{\sqrt{2} c^{3/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x \sqrt[4]{-b+a x^6}}{-\sqrt{c} x^2+\sqrt{-b+a x^6}}\right)}{\sqrt{2} c^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[4]{c} x^2}{\sqrt{2}}+\frac{\sqrt{-b+a x^6}}{\sqrt{2} \sqrt[4]{c}}}{x \sqrt[4]{-b+a x^6}}\right)}{\sqrt{2} c^{3/4}}",1,"-(ArcTan[(Sqrt[2]*c^(1/4)*x*(-b + a*x^6)^(1/4))/(-(Sqrt[c]*x^2) + Sqrt[-b + a*x^6])]/(Sqrt[2]*c^(3/4))) + ArcTanh[((c^(1/4)*x^2)/Sqrt[2] + Sqrt[-b + a*x^6]/(Sqrt[2]*c^(1/4)))/(x*(-b + a*x^6)^(1/4))]/(Sqrt[2]*c^(3/4))","A",1
1921,1,133,133,0.1616853,"\int x^2 \sqrt{1+x^4} \sqrt{x^2+\sqrt{1+x^4}} \, dx","IntegrateAlgebraic[x^2*Sqrt[1 + x^4]*Sqrt[x^2 + Sqrt[1 + x^4]],x]","\frac{x \sqrt{1+x^4} \left(28 x^2+16 x^6\right) \sqrt{x^2+\sqrt{1+x^4}}+x \left(9+36 x^4+16 x^8\right) \sqrt{x^2+\sqrt{1+x^4}}}{96 x^2 \sqrt{1+x^4}+48 \left(1+2 x^4\right)}-\frac{3 \tan ^{-1}\left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}\right)}{16 \sqrt{2}}","\frac{x \sqrt{1+x^4} \left(28 x^2+16 x^6\right) \sqrt{x^2+\sqrt{1+x^4}}+x \left(9+36 x^4+16 x^8\right) \sqrt{x^2+\sqrt{1+x^4}}}{96 x^2 \sqrt{1+x^4}+48 \left(1+2 x^4\right)}-\frac{3 \tan ^{-1}\left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}\right)}{16 \sqrt{2}}",1,"(x*Sqrt[1 + x^4]*(28*x^2 + 16*x^6)*Sqrt[x^2 + Sqrt[1 + x^4]] + x*(9 + 36*x^4 + 16*x^8)*Sqrt[x^2 + Sqrt[1 + x^4]])/(96*x^2*Sqrt[1 + x^4] + 48*(1 + 2*x^4)) - (3*ArcTan[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]]])/(16*Sqrt[2])","A",1
1922,1,85,133,0.0631987,"\int \sqrt{1+\sqrt{1+\sqrt{1+x}}} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]],x]","\frac{8}{315} \sqrt{1+\sqrt{1+x}} \left(-8+5 \sqrt{1+x}\right) \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\frac{8}{315} \left(-8+\sqrt{1+x}+35 (1+x)\right) \sqrt{1+\sqrt{1+\sqrt{1+x}}}","\frac{8}{315} (27+35 x) \sqrt{1+\sqrt{1+\sqrt{1+x}}}-\frac{64}{315} \sqrt{1+\sqrt{1+x}} \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\sqrt{1+x} \left(\frac{8}{315} \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\frac{8}{63} \sqrt{1+\sqrt{1+x}} \sqrt{1+\sqrt{1+\sqrt{1+x}}}\right)",1,"(8*Sqrt[1 + Sqrt[1 + x]]*(-8 + 5*Sqrt[1 + x])*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/315 + (8*(-8 + Sqrt[1 + x] + 35*(1 + x))*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/315","A",1
1923,1,133,133,0.2260844,"\int \frac{\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1-x^2\right) \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((1 - x^2)*Sqrt[x + Sqrt[1 + x^2]]),x]","-\frac{1}{2} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^2}\&\right]+\frac{1}{2} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^2}\&\right]","-\frac{1}{2} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^2}\&\right]+\frac{1}{2} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^2}\&\right]",1,"-1/2*RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(-1 + #1^2) & ] + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(-1 + #1^2) & ]/2","A",1
1924,1,133,133,0.0020443,"\int \frac{\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1-x^2\right) \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((1 - x^2)*Sqrt[x + Sqrt[1 + x^2]]),x]","-\frac{1}{2} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^2}\&\right]+\frac{1}{2} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^2}\&\right]","-\frac{1}{2} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^2}\&\right]+\frac{1}{2} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^2}\&\right]",1,"-1/2*RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(-1 + #1^2) & ] + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(-1 + #1^2) & ]/2","A",1
1925,1,134,134,0.2838559,"\int \frac{\left(-2+x^3\right) \left(1+x^3\right)^{2/3}}{x^6 \left(-1+2 x^3\right)} \, dx","IntegrateAlgebraic[((-2 + x^3)*(1 + x^3)^(2/3))/(x^6*(-1 + 2*x^3)),x]","\frac{\left(-4-19 x^3\right) \left(1+x^3\right)^{2/3}}{10 x^5}+3 \sqrt[6]{3} \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2 \sqrt[3]{1+x^3}}\right)-3^{2/3} \log \left(-3 x+3^{2/3} \sqrt[3]{1+x^3}\right)+\frac{1}{2} 3^{2/3} \log \left(3 x^2+3^{2/3} x \sqrt[3]{1+x^3}+\sqrt[3]{3} \left(1+x^3\right)^{2/3}\right)","\frac{\left(-4-19 x^3\right) \left(1+x^3\right)^{2/3}}{10 x^5}+3 \sqrt[6]{3} \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2 \sqrt[3]{1+x^3}}\right)-3^{2/3} \log \left(-3 x+3^{2/3} \sqrt[3]{1+x^3}\right)+\frac{1}{2} 3^{2/3} \log \left(3 x^2+3^{2/3} x \sqrt[3]{1+x^3}+\sqrt[3]{3} \left(1+x^3\right)^{2/3}\right)",1,"((-4 - 19*x^3)*(1 + x^3)^(2/3))/(10*x^5) + 3*3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*(1 + x^3)^(1/3))] - 3^(2/3)*Log[-3*x + 3^(2/3)*(1 + x^3)^(1/3)] + (3^(2/3)*Log[3*x^2 + 3^(2/3)*x*(1 + x^3)^(1/3) + 3^(1/3)*(1 + x^3)^(2/3)])/2","A",1
1926,1,134,134,0.4574983,"\int \frac{1}{x^3 \left(b+a x^3\right) \sqrt[3]{b^2 x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[1/(x^3*(b + a*x^3)*(b^2*x^2 + a^3*x^3)^(1/3)),x]","-\frac{3 \left(5 b^4-6 a^3 b^2 x+9 a^6 x^2\right) \left(b^2 x^2+a^3 x^3\right)^{2/3}}{40 b^7 x^4}+\frac{a \text{RootSum}\left[a^9-a b^5-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{3 b^2}","-\frac{3 \left(5 b^4-6 a^3 b^2 x+9 a^6 x^2\right) \left(b^2 x^2+a^3 x^3\right)^{2/3}}{40 b^7 x^4}+\frac{a \text{RootSum}\left[a^9-a b^5-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{3 b^2}",1,"(-3*(5*b^4 - 6*a^3*b^2*x + 9*a^6*x^2)*(b^2*x^2 + a^3*x^3)^(2/3))/(40*b^7*x^4) + (a*RootSum[a^9 - a*b^5 - 3*a^6*#1^3 + 3*a^3*#1^6 - #1^9 & , (-Log[x] + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1])/#1 & ])/(3*b^2)","A",1
1927,1,134,134,0.0021458,"\int \frac{1}{x^3 \left(b+a x^3\right) \sqrt[3]{b^2 x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[1/(x^3*(b + a*x^3)*(b^2*x^2 + a^3*x^3)^(1/3)),x]","-\frac{3 \left(5 b^4-6 a^3 b^2 x+9 a^6 x^2\right) \left(b^2 x^2+a^3 x^3\right)^{2/3}}{40 b^7 x^4}+\frac{a \text{RootSum}\left[a^9-a b^5-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{3 b^2}","-\frac{3 \left(5 b^4-6 a^3 b^2 x+9 a^6 x^2\right) \left(b^2 x^2+a^3 x^3\right)^{2/3}}{40 b^7 x^4}+\frac{a \text{RootSum}\left[a^9-a b^5-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{3 b^2}",1,"(-3*(5*b^4 - 6*a^3*b^2*x + 9*a^6*x^2)*(b^2*x^2 + a^3*x^3)^(2/3))/(40*b^7*x^4) + (a*RootSum[a^9 - a*b^5 - 3*a^6*#1^3 + 3*a^3*#1^6 - #1^9 & , (-Log[x] + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1])/#1 & ])/(3*b^2)","A",1
1928,1,134,134,0.7599116,"\int \frac{\left(-1+x+x^4\right) \sqrt[4]{-x^3+x^4}}{1+x} \, dx","IntegrateAlgebraic[((-1 + x + x^4)*(-x^3 + x^4)^(1/4))/(1 + x),x]","\frac{\sqrt[4]{-x^3+x^4} \left(-32575-1060 x+10400 x^2-8064 x^3+6144 x^4\right)}{30720}-\frac{9869 \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)}{4096}+2 \sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)+\frac{9869 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)}{4096}-2 \sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)","\frac{\sqrt[4]{-x^3+x^4} \left(-32575-1060 x+10400 x^2-8064 x^3+6144 x^4\right)}{30720}-\frac{9869 \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)}{4096}+2 \sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)+\frac{9869 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)}{4096}-2 \sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)",1,"((-x^3 + x^4)^(1/4)*(-32575 - 1060*x + 10400*x^2 - 8064*x^3 + 6144*x^4))/30720 - (9869*ArcTan[x/(-x^3 + x^4)^(1/4)])/4096 + 2*2^(1/4)*ArcTan[(2^(1/4)*x)/(-x^3 + x^4)^(1/4)] + (9869*ArcTanh[x/(-x^3 + x^4)^(1/4)])/4096 - 2*2^(1/4)*ArcTanh[(2^(1/4)*x)/(-x^3 + x^4)^(1/4)]","A",1
1929,1,134,134,1.1257015,"\int \frac{\left(-3+x^4\right) \left(1+x^4\right)^{2/3}}{x^3 \left(2+x^3+2 x^4\right)} \, dx","IntegrateAlgebraic[((-3 + x^4)*(1 + x^4)^(2/3))/(x^3*(2 + x^3 + 2*x^4)),x]","\frac{3 \left(1+x^4\right)^{2/3}}{4 x^2}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{2} \sqrt[3]{1+x^4}}\right)}{2\ 2^{2/3}}+\frac{\log \left(x+\sqrt[3]{2} \sqrt[3]{1+x^4}\right)}{2\ 2^{2/3}}-\frac{\log \left(x^2-\sqrt[3]{2} x \sqrt[3]{1+x^4}+2^{2/3} \left(1+x^4\right)^{2/3}\right)}{4\ 2^{2/3}}","\frac{3 \left(1+x^4\right)^{2/3}}{4 x^2}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{2} \sqrt[3]{1+x^4}}\right)}{2\ 2^{2/3}}+\frac{\log \left(x+\sqrt[3]{2} \sqrt[3]{1+x^4}\right)}{2\ 2^{2/3}}-\frac{\log \left(x^2-\sqrt[3]{2} x \sqrt[3]{1+x^4}+2^{2/3} \left(1+x^4\right)^{2/3}\right)}{4\ 2^{2/3}}",1,"(3*(1 + x^4)^(2/3))/(4*x^2) + (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*2^(1/3)*(1 + x^4)^(1/3))])/(2*2^(2/3)) + Log[x + 2^(1/3)*(1 + x^4)^(1/3)]/(2*2^(2/3)) - Log[x^2 - 2^(1/3)*x*(1 + x^4)^(1/3) + 2^(2/3)*(1 + x^4)^(2/3)]/(4*2^(2/3))","A",1
1930,1,134,134,0.5610009,"\int \frac{\left(-4 b+a x^4\right) \left(b+a x^4\right)^{3/4}}{x^8 \left(4 b+a x^4\right)} \, dx","IntegrateAlgebraic[((-4*b + a*x^4)*(b + a*x^4)^(3/4))/(x^8*(4*b + a*x^4)),x]","\frac{\left(6 b-a x^4\right) \left(b+a x^4\right)^{3/4}}{42 b x^7}+\frac{3^{3/4} a^{7/4} \tan ^{-1}\left(\frac{\sqrt[4]{3} \sqrt[4]{a} x}{\sqrt{2} \sqrt[4]{b+a x^4}}\right)}{8 \sqrt{2} b}+\frac{3^{3/4} a^{7/4} \tanh ^{-1}\left(\frac{\sqrt[4]{3} \sqrt[4]{a} x}{\sqrt{2} \sqrt[4]{b+a x^4}}\right)}{8 \sqrt{2} b}","\frac{\left(6 b-a x^4\right) \left(b+a x^4\right)^{3/4}}{42 b x^7}+\frac{3^{3/4} a^{7/4} \tan ^{-1}\left(\frac{\sqrt[4]{3} \sqrt[4]{a} x}{\sqrt{2} \sqrt[4]{b+a x^4}}\right)}{8 \sqrt{2} b}+\frac{3^{3/4} a^{7/4} \tanh ^{-1}\left(\frac{\sqrt[4]{3} \sqrt[4]{a} x}{\sqrt{2} \sqrt[4]{b+a x^4}}\right)}{8 \sqrt{2} b}",1,"((6*b - a*x^4)*(b + a*x^4)^(3/4))/(42*b*x^7) + (3^(3/4)*a^(7/4)*ArcTan[(3^(1/4)*a^(1/4)*x)/(Sqrt[2]*(b + a*x^4)^(1/4))])/(8*Sqrt[2]*b) + (3^(3/4)*a^(7/4)*ArcTanh[(3^(1/4)*a^(1/4)*x)/(Sqrt[2]*(b + a*x^4)^(1/4))])/(8*Sqrt[2]*b)","A",1
1931,1,134,134,0.545958,"\int \frac{\left(b+a x^4\right)^{3/4} \left(2 b+a x^4\right)}{x^8 \left(4 b+a x^4\right)} \, dx","IntegrateAlgebraic[((b + a*x^4)^(3/4)*(2*b + a*x^4))/(x^8*(4*b + a*x^4)),x]","\frac{\left(-12 b-19 a x^4\right) \left(b+a x^4\right)^{3/4}}{168 b x^7}+\frac{3^{3/4} a^{7/4} \tan ^{-1}\left(\frac{\sqrt[4]{3} \sqrt[4]{a} x}{\sqrt{2} \sqrt[4]{b+a x^4}}\right)}{32 \sqrt{2} b}+\frac{3^{3/4} a^{7/4} \tanh ^{-1}\left(\frac{\sqrt[4]{3} \sqrt[4]{a} x}{\sqrt{2} \sqrt[4]{b+a x^4}}\right)}{32 \sqrt{2} b}","\frac{\left(-12 b-19 a x^4\right) \left(b+a x^4\right)^{3/4}}{168 b x^7}+\frac{3^{3/4} a^{7/4} \tan ^{-1}\left(\frac{\sqrt[4]{3} \sqrt[4]{a} x}{\sqrt{2} \sqrt[4]{b+a x^4}}\right)}{32 \sqrt{2} b}+\frac{3^{3/4} a^{7/4} \tanh ^{-1}\left(\frac{\sqrt[4]{3} \sqrt[4]{a} x}{\sqrt{2} \sqrt[4]{b+a x^4}}\right)}{32 \sqrt{2} b}",1,"((-12*b - 19*a*x^4)*(b + a*x^4)^(3/4))/(168*b*x^7) + (3^(3/4)*a^(7/4)*ArcTan[(3^(1/4)*a^(1/4)*x)/(Sqrt[2]*(b + a*x^4)^(1/4))])/(32*Sqrt[2]*b) + (3^(3/4)*a^(7/4)*ArcTanh[(3^(1/4)*a^(1/4)*x)/(Sqrt[2]*(b + a*x^4)^(1/4))])/(32*Sqrt[2]*b)","A",1
1932,1,134,134,0.5184974,"\int \frac{x^4 \sqrt[4]{b x^3+a x^4}}{b+a x} \, dx","IntegrateAlgebraic[(x^4*(b*x^3 + a*x^4)^(1/4))/(b + a*x),x]","\frac{\sqrt[4]{b x^3+a x^4} \left(7315 b^4-4180 a b^3 x+3040 a^2 b^2 x^2-2432 a^3 b x^3+2048 a^4 x^4\right)}{10240 a^5}+\frac{4389 b^5 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{4096 a^{23/4}}-\frac{4389 b^5 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{4096 a^{23/4}}","\frac{\sqrt[4]{b x^3+a x^4} \left(7315 b^4-4180 a b^3 x+3040 a^2 b^2 x^2-2432 a^3 b x^3+2048 a^4 x^4\right)}{10240 a^5}+\frac{4389 b^5 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{4096 a^{23/4}}-\frac{4389 b^5 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{4096 a^{23/4}}",1,"((b*x^3 + a*x^4)^(1/4)*(7315*b^4 - 4180*a*b^3*x + 3040*a^2*b^2*x^2 - 2432*a^3*b*x^3 + 2048*a^4*x^4))/(10240*a^5) + (4389*b^5*ArcTan[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(4096*a^(23/4)) - (4389*b^5*ArcTanh[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(4096*a^(23/4))","A",1
1933,1,50,134,0.4811256,"\int \frac{b+a x^2}{\left(-b+a x^2\right) \sqrt{b^2+a^2 x^4}} \, dx","IntegrateAlgebraic[(b + a*x^2)/((-b + a*x^2)*Sqrt[b^2 + a^2*x^4]),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{b} x}{\sqrt{b^2+a^2 x^4}}\right)}{\sqrt{2} \sqrt{a} \sqrt{b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{6-4 \sqrt{2}} \sqrt{a} \sqrt{b} x}{b+a x^2+\sqrt{b^2+a^2 x^4}}\right)}{\sqrt{2} \sqrt{a} \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{6+4 \sqrt{2}} \sqrt{a} \sqrt{b} x}{b+a x^2+\sqrt{b^2+a^2 x^4}}\right)}{\sqrt{2} \sqrt{a} \sqrt{b}}",1,"-(ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[b]*x)/Sqrt[b^2 + a^2*x^4]]/(Sqrt[2]*Sqrt[a]*Sqrt[b]))","A",1
1934,1,134,134,0.3432821,"\int \frac{\left(1+x^2\right) \sqrt{1+\sqrt{1+x^2}}}{-1+x^2} \, dx","IntegrateAlgebraic[((1 + x^2)*Sqrt[1 + Sqrt[1 + x^2]])/(-1 + x^2),x]","\frac{4 x}{3 \sqrt{1+\sqrt{1+x^2}}}+\frac{2 x \sqrt{1+x^2}}{3 \sqrt{1+\sqrt{1+x^2}}}+2 \sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{-1+\sqrt{2}} x}{\sqrt{1+\sqrt{1+x^2}}}\right)-2 \sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{1+\sqrt{2}} x}{\sqrt{1+\sqrt{1+x^2}}}\right)","\frac{4 x}{3 \sqrt{1+\sqrt{1+x^2}}}+\frac{2 x \sqrt{1+x^2}}{3 \sqrt{1+\sqrt{1+x^2}}}+2 \sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\frac{x}{\sqrt{1+\sqrt{2}} \sqrt{1+\sqrt{1+x^2}}}\right)-2 \sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{x}{\sqrt{-1+\sqrt{2}} \sqrt{1+\sqrt{1+x^2}}}\right)",1,"(4*x)/(3*Sqrt[1 + Sqrt[1 + x^2]]) + (2*x*Sqrt[1 + x^2])/(3*Sqrt[1 + Sqrt[1 + x^2]]) + 2*Sqrt[-1 + Sqrt[2]]*ArcTan[(Sqrt[-1 + Sqrt[2]]*x)/Sqrt[1 + Sqrt[1 + x^2]]] - 2*Sqrt[1 + Sqrt[2]]*ArcTanh[(Sqrt[1 + Sqrt[2]]*x)/Sqrt[1 + Sqrt[1 + x^2]]]","A",1
1935,1,135,135,0.2306187,"\int \frac{1}{\sqrt[3]{-b+a x^3}} \, dx","IntegrateAlgebraic[(-b + a*x^3)^(-1/3),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a} x}{\sqrt[3]{a} x+2 \sqrt[3]{-b+a x^3}}\right)}{\sqrt{3} \sqrt[3]{a}}-\frac{\log \left(-\sqrt[3]{a} x+\sqrt[3]{-b+a x^3}\right)}{3 \sqrt[3]{a}}+\frac{\log \left(a^{2/3} x^2+\sqrt[3]{a} x \sqrt[3]{-b+a x^3}+\left(-b+a x^3\right)^{2/3}\right)}{6 \sqrt[3]{a}}","\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a} x}{\sqrt[3]{a} x+2 \sqrt[3]{-b+a x^3}}\right)}{\sqrt{3} \sqrt[3]{a}}-\frac{\log \left(-\sqrt[3]{a} x+\sqrt[3]{-b+a x^3}\right)}{3 \sqrt[3]{a}}+\frac{\log \left(a^{2/3} x^2+\sqrt[3]{a} x \sqrt[3]{-b+a x^3}+\left(-b+a x^3\right)^{2/3}\right)}{6 \sqrt[3]{a}}",1,"ArcTan[(Sqrt[3]*a^(1/3)*x)/(a^(1/3)*x + 2*(-b + a*x^3)^(1/3))]/(Sqrt[3]*a^(1/3)) - Log[-(a^(1/3)*x) + (-b + a*x^3)^(1/3)]/(3*a^(1/3)) + Log[a^(2/3)*x^2 + a^(1/3)*x*(-b + a*x^3)^(1/3) + (-b + a*x^3)^(2/3)]/(6*a^(1/3))","A",1
1936,1,135,135,4.0496679,"\int \frac{\left(a^2-2 a x+x^2\right) \left(-a b-a c+3 b c+2 (a-b-c) x+x^2\right)}{((-a+x) (-b+x) (-c+x))^{3/4} \left(-b c-a^3 d+\left(b+c+3 a^2 d\right) x-(1+3 a d) x^2+d x^3\right)} \, dx","IntegrateAlgebraic[((a^2 - 2*a*x + x^2)*(-(a*b) - a*c + 3*b*c + 2*(a - b - c)*x + x^2))/(((-a + x)*(-b + x)*(-c + x))^(3/4)*(-(b*c) - a^3*d + (b + c + 3*a^2*d)*x - (1 + 3*a*d)*x^2 + d*x^3)),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3\right)^{3/4}}{(b-x) (-c+x)}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \left(-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3\right)^{3/4}}{(b-x) (-c+x)}\right)}{d^{3/4}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3\right)^{3/4}}{(b-x) (-c+x)}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \left(-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3\right)^{3/4}}{(b-x) (-c+x)}\right)}{d^{3/4}}",1,"(-2*ArcTan[(d^(1/4)*(-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3)^(3/4))/((b - x)*(-c + x))])/d^(3/4) + (2*ArcTanh[(d^(1/4)*(-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3)^(3/4))/((b - x)*(-c + x))])/d^(3/4)","A",1
1937,1,135,135,1.953726,"\int \frac{\left(-4+x^5\right) \sqrt[4]{1-2 x^4+x^5}}{x^2 \left(1+x^5\right)} \, dx","IntegrateAlgebraic[((-4 + x^5)*(1 - 2*x^4 + x^5)^(1/4))/(x^2*(1 + x^5)),x]","\frac{4 \sqrt[4]{1-2 x^4+x^5}}{x}-2^{3/4} \tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{1-2 x^4+x^5}}{\sqrt{2} x^2-\sqrt{1-2 x^4+x^5}}\right)-2^{3/4} \tanh ^{-1}\left(\frac{2 \sqrt[4]{2} x \sqrt[4]{1-2 x^4+x^5}}{2 x^2+\sqrt{2} \sqrt{1-2 x^4+x^5}}\right)","\frac{4 \sqrt[4]{1-2 x^4+x^5}}{x}-2^{3/4} \tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{1-2 x^4+x^5}}{\sqrt{2} x^2-\sqrt{1-2 x^4+x^5}}\right)-2^{3/4} \tanh ^{-1}\left(\frac{2 \sqrt[4]{2} x \sqrt[4]{1-2 x^4+x^5}}{2 x^2+\sqrt{2} \sqrt{1-2 x^4+x^5}}\right)",1,"(4*(1 - 2*x^4 + x^5)^(1/4))/x - 2^(3/4)*ArcTan[(2^(3/4)*x*(1 - 2*x^4 + x^5)^(1/4))/(Sqrt[2]*x^2 - Sqrt[1 - 2*x^4 + x^5])] - 2^(3/4)*ArcTanh[(2*2^(1/4)*x*(1 - 2*x^4 + x^5)^(1/4))/(2*x^2 + Sqrt[2]*Sqrt[1 - 2*x^4 + x^5])]","A",1
1938,1,112,135,0.3040778,"\int \frac{b+a x^6}{x^6 \sqrt[3]{x+x^3}} \, dx","IntegrateAlgebraic[(b + a*x^6)/(x^6*(x + x^3)^(1/3)),x]","-\frac{3 b \left(x+x^3\right)^{2/3} \left(5-6 x^2+9 x^4\right)}{80 x^6}+\frac{1}{2} \sqrt{3} a \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)-\frac{1}{2} a \log \left(-x+\sqrt[3]{x+x^3}\right)+\frac{1}{4} a \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)","-\frac{3 b \left(x+x^3\right)^{2/3} \left(5-6 x^2+9 x^4\right)}{80 x^6}+\frac{1}{4} \left(a-i \sqrt{3} a\right) \log \left(-i x+\sqrt{3} x-2 i \sqrt[3]{x+x^3}\right)+\frac{1}{4} \left(a+i \sqrt{3} a\right) \log \left(i x+\sqrt{3} x+2 i \sqrt[3]{x+x^3}\right)-\frac{1}{2} a \log \left(-x+\sqrt[3]{x+x^3}\right)",1,"(-3*b*(x + x^3)^(2/3)*(5 - 6*x^2 + 9*x^4))/(80*x^6) + (Sqrt[3]*a*ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^3)^(1/3))])/2 - (a*Log[-x + (x + x^3)^(1/3)])/2 + (a*Log[x^2 + x*(x + x^3)^(1/3) + (x + x^3)^(2/3)])/4","A",1
1939,1,135,135,0.1767635,"\int \frac{b^2+a^2 x^2}{\sqrt{a x+\sqrt{b^2+a^2 x^2}}} \, dx","IntegrateAlgebraic[(b^2 + a^2*x^2)/Sqrt[a*x + Sqrt[b^2 + a^2*x^2]],x]","\frac{2 \sqrt{b^2+a^2 x^2} \left(7 b^4 x+19 a^2 b^2 x^3+4 a^4 x^5\right)}{5 \left(a x+\sqrt{b^2+a^2 x^2}\right)^{7/2}}+\frac{2 \left(9 b^6+112 a^2 b^4 x^2+147 a^4 b^2 x^4+28 a^6 x^6\right)}{35 a \left(a x+\sqrt{b^2+a^2 x^2}\right)^{7/2}}","\frac{2 \sqrt{b^2+a^2 x^2} \left(7 b^4 x+19 a^2 b^2 x^3+4 a^4 x^5\right)}{5 \left(a x+\sqrt{b^2+a^2 x^2}\right)^{7/2}}+\frac{2 \left(9 b^6+112 a^2 b^4 x^2+147 a^4 b^2 x^4+28 a^6 x^6\right)}{35 a \left(a x+\sqrt{b^2+a^2 x^2}\right)^{7/2}}",1,"(2*Sqrt[b^2 + a^2*x^2]*(7*b^4*x + 19*a^2*b^2*x^3 + 4*a^4*x^5))/(5*(a*x + Sqrt[b^2 + a^2*x^2])^(7/2)) + (2*(9*b^6 + 112*a^2*b^4*x^2 + 147*a^4*b^2*x^4 + 28*a^6*x^6))/(35*a*(a*x + Sqrt[b^2 + a^2*x^2])^(7/2))","A",1
1940,1,136,136,2.3920953,"\int \frac{x^2}{\left(b+a x^2\right)^{3/4} \left(2 b+a x^2\right)} \, dx","IntegrateAlgebraic[x^2/((b + a*x^2)^(3/4)*(2*b + a*x^2)),x]","-\frac{\tan ^{-1}\left(\frac{-\frac{\sqrt{a} x^2}{2 \sqrt[4]{b}}+\frac{\sqrt[4]{b} \sqrt{b+a x^2}}{\sqrt{a}}}{x \sqrt[4]{b+a x^2}}\right)}{2 a^{3/2} \sqrt[4]{b}}-\frac{\tanh ^{-1}\left(\frac{2 \sqrt{a} \sqrt[4]{b} x \sqrt[4]{b+a x^2}}{a x^2+2 \sqrt{b} \sqrt{b+a x^2}}\right)}{2 a^{3/2} \sqrt[4]{b}}","-\frac{\tan ^{-1}\left(\frac{-\frac{\sqrt{a} x^2}{2 \sqrt[4]{b}}+\frac{\sqrt[4]{b} \sqrt{b+a x^2}}{\sqrt{a}}}{x \sqrt[4]{b+a x^2}}\right)}{2 a^{3/2} \sqrt[4]{b}}-\frac{\tanh ^{-1}\left(\frac{2 \sqrt{a} \sqrt[4]{b} x \sqrt[4]{b+a x^2}}{a x^2+2 \sqrt{b} \sqrt{b+a x^2}}\right)}{2 a^{3/2} \sqrt[4]{b}}",1,"-1/2*ArcTan[(-1/2*(Sqrt[a]*x^2)/b^(1/4) + (b^(1/4)*Sqrt[b + a*x^2])/Sqrt[a])/(x*(b + a*x^2)^(1/4))]/(a^(3/2)*b^(1/4)) - ArcTanh[(2*Sqrt[a]*b^(1/4)*x*(b + a*x^2)^(1/4))/(a*x^2 + 2*Sqrt[b]*Sqrt[b + a*x^2])]/(2*a^(3/2)*b^(1/4))","A",1
1941,1,136,136,0.231749,"\int \frac{x}{\left(-b+a x^3\right)^{2/3}} \, dx","IntegrateAlgebraic[x/(-b + a*x^3)^(2/3),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a} x}{\sqrt[3]{a} x+2 \sqrt[3]{-b+a x^3}}\right)}{\sqrt{3} a^{2/3}}-\frac{\log \left(-\sqrt[3]{a} x+\sqrt[3]{-b+a x^3}\right)}{3 a^{2/3}}+\frac{\log \left(a^{2/3} x^2+\sqrt[3]{a} x \sqrt[3]{-b+a x^3}+\left(-b+a x^3\right)^{2/3}\right)}{6 a^{2/3}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a} x}{\sqrt[3]{a} x+2 \sqrt[3]{-b+a x^3}}\right)}{\sqrt{3} a^{2/3}}-\frac{\log \left(-\sqrt[3]{a} x+\sqrt[3]{-b+a x^3}\right)}{3 a^{2/3}}+\frac{\log \left(a^{2/3} x^2+\sqrt[3]{a} x \sqrt[3]{-b+a x^3}+\left(-b+a x^3\right)^{2/3}\right)}{6 a^{2/3}}",1,"-(ArcTan[(Sqrt[3]*a^(1/3)*x)/(a^(1/3)*x + 2*(-b + a*x^3)^(1/3))]/(Sqrt[3]*a^(2/3))) - Log[-(a^(1/3)*x) + (-b + a*x^3)^(1/3)]/(3*a^(2/3)) + Log[a^(2/3)*x^2 + a^(1/3)*x*(-b + a*x^3)^(1/3) + (-b + a*x^3)^(2/3)]/(6*a^(2/3))","A",1
1942,1,136,136,0.254377,"\int \frac{x \sqrt[3]{x^2+x^4}}{1+2 x^2} \, dx","IntegrateAlgebraic[(x*(x^2 + x^4)^(1/3))/(1 + 2*x^2),x]","\frac{3}{8} \sqrt[3]{x^2+x^4}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2\ 2^{2/3} \sqrt[3]{x^2+x^4}}{\sqrt{3}}\right)}{8\ 2^{2/3}}-\frac{\log \left(1+2^{2/3} \sqrt[3]{x^2+x^4}\right)}{8\ 2^{2/3}}+\frac{\log \left(-1+2^{2/3} \sqrt[3]{x^2+x^4}-2 \sqrt[3]{2} \left(x^2+x^4\right)^{2/3}\right)}{16\ 2^{2/3}}","\frac{3}{8} \sqrt[3]{x^2+x^4}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2\ 2^{2/3} \sqrt[3]{x^2+x^4}}{\sqrt{3}}\right)}{8\ 2^{2/3}}-\frac{\log \left(1+2^{2/3} \sqrt[3]{x^2+x^4}\right)}{8\ 2^{2/3}}+\frac{\log \left(-1+2^{2/3} \sqrt[3]{x^2+x^4}-2 \sqrt[3]{2} \left(x^2+x^4\right)^{2/3}\right)}{16\ 2^{2/3}}",1,"(3*(x^2 + x^4)^(1/3))/8 + (Sqrt[3]*ArcTan[1/Sqrt[3] - (2*2^(2/3)*(x^2 + x^4)^(1/3))/Sqrt[3]])/(8*2^(2/3)) - Log[1 + 2^(2/3)*(x^2 + x^4)^(1/3)]/(8*2^(2/3)) + Log[-1 + 2^(2/3)*(x^2 + x^4)^(1/3) - 2*2^(1/3)*(x^2 + x^4)^(2/3)]/(16*2^(2/3))","A",1
1943,1,136,136,0.4892474,"\int \frac{b+a x^3}{x^6 \left(-b+a x^3\right) \sqrt[4]{b x+a x^4}} \, dx","IntegrateAlgebraic[(b + a*x^3)/(x^6*(-b + a*x^3)*(b*x + a*x^4)^(1/4)),x]","\frac{4 \left(3 b+10 a x^3\right) \left(b x+a x^4\right)^{3/4}}{63 b^2 x^6}-\frac{2\ 2^{3/4} a^{7/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 b^2}-\frac{2\ 2^{3/4} a^{7/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 b^2}","\frac{4 \left(3 b+10 a x^3\right) \left(b x+a x^4\right)^{3/4}}{63 b^2 x^6}-\frac{2\ 2^{3/4} a^{7/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 b^2}-\frac{2\ 2^{3/4} a^{7/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 b^2}",1,"(4*(3*b + 10*a*x^3)*(b*x + a*x^4)^(3/4))/(63*b^2*x^6) - (2*2^(3/4)*a^(7/4)*ArcTan[(2^(1/4)*a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/(3*b^2) - (2*2^(3/4)*a^(7/4)*ArcTanh[(2^(1/4)*a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/(3*b^2)","A",1
1944,1,136,136,0.475028,"\int \frac{\left(-1+x^4\right) \sqrt[4]{x^2+x^6}}{1+x^4+x^8} \, dx","IntegrateAlgebraic[((-1 + x^4)*(x^2 + x^6)^(1/4))/(1 + x^4 + x^8),x]","\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^2+x^6}}\right)-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{x^2+x^6}}{-x^2+\sqrt{x^2+x^6}}\right)}{2 \sqrt{2}}-\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^2+x^6}}\right)+\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{x^2+x^6}}{\sqrt{2}}}{x \sqrt[4]{x^2+x^6}}\right)}{2 \sqrt{2}}","\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^2+x^6}}\right)-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{x^2+x^6}}{-x^2+\sqrt{x^2+x^6}}\right)}{2 \sqrt{2}}-\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^2+x^6}}\right)+\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{x^2+x^6}}{\sqrt{2}}}{x \sqrt[4]{x^2+x^6}}\right)}{2 \sqrt{2}}",1,"ArcTan[x/(x^2 + x^6)^(1/4)]/2 - ArcTan[(Sqrt[2]*x*(x^2 + x^6)^(1/4))/(-x^2 + Sqrt[x^2 + x^6])]/(2*Sqrt[2]) - ArcTanh[x/(x^2 + x^6)^(1/4)]/2 + ArcTanh[(x^2/Sqrt[2] + Sqrt[x^2 + x^6]/Sqrt[2])/(x*(x^2 + x^6)^(1/4))]/(2*Sqrt[2])","A",1
1945,1,136,136,0.0018281,"\int \frac{\left(-1+x^4\right) \sqrt[4]{x^2+x^6}}{1+x^4+x^8} \, dx","IntegrateAlgebraic[((-1 + x^4)*(x^2 + x^6)^(1/4))/(1 + x^4 + x^8),x]","\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^2+x^6}}\right)-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{x^2+x^6}}{-x^2+\sqrt{x^2+x^6}}\right)}{2 \sqrt{2}}-\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^2+x^6}}\right)+\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{x^2+x^6}}{\sqrt{2}}}{x \sqrt[4]{x^2+x^6}}\right)}{2 \sqrt{2}}","\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^2+x^6}}\right)-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{x^2+x^6}}{-x^2+\sqrt{x^2+x^6}}\right)}{2 \sqrt{2}}-\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^2+x^6}}\right)+\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{x^2+x^6}}{\sqrt{2}}}{x \sqrt[4]{x^2+x^6}}\right)}{2 \sqrt{2}}",1,"ArcTan[x/(x^2 + x^6)^(1/4)]/2 - ArcTan[(Sqrt[2]*x*(x^2 + x^6)^(1/4))/(-x^2 + Sqrt[x^2 + x^6])]/(2*Sqrt[2]) - ArcTanh[x/(x^2 + x^6)^(1/4)]/2 + ArcTanh[(x^2/Sqrt[2] + Sqrt[x^2 + x^6]/Sqrt[2])/(x*(x^2 + x^6)^(1/4))]/(2*Sqrt[2])","A",1
1946,1,135,136,0.3038593,"\int \frac{x \sqrt{a x+\sqrt{-b+a x}}}{\sqrt{-b+a x}} \, dx","IntegrateAlgebraic[(x*Sqrt[a*x + Sqrt[-b + a*x]])/Sqrt[-b + a*x],x]","\frac{\sqrt{a x+\sqrt{-b+a x}} \left(15-4 b-10 \sqrt{-b+a x}+120 b \sqrt{-b+a x}+8 (-b+a x)+48 (-b+a x)^{3/2}\right)}{96 a^2}+\frac{\left(5-8 b-48 b^2\right) \log \left(-1-2 \sqrt{-b+a x}+2 \sqrt{a x+\sqrt{-b+a x}}\right)}{64 a^2}","\frac{(15-12 b+8 a x) \sqrt{a x+\sqrt{-b+a x}}}{96 a^2}+\frac{\sqrt{-b+a x} (-5+36 b+24 a x) \sqrt{a x+\sqrt{-b+a x}}}{48 a^2}+\frac{\left(5-8 b-48 b^2\right) \log \left(1+2 \sqrt{-b+a x}-2 \sqrt{a x+\sqrt{-b+a x}}\right)}{64 a^2}",1,"(Sqrt[a*x + Sqrt[-b + a*x]]*(15 - 4*b - 10*Sqrt[-b + a*x] + 120*b*Sqrt[-b + a*x] + 8*(-b + a*x) + 48*(-b + a*x)^(3/2)))/(96*a^2) + ((5 - 8*b - 48*b^2)*Log[-1 - 2*Sqrt[-b + a*x] + 2*Sqrt[a*x + Sqrt[-b + a*x]]])/(64*a^2)","A",1
1947,1,139,136,0.3944081,"\int \frac{(-1+a x) \sqrt{a x+\sqrt{-b+a x}}}{\sqrt{-b+a x}} \, dx","IntegrateAlgebraic[((-1 + a*x)*Sqrt[a*x + Sqrt[-b + a*x]])/Sqrt[-b + a*x],x]","\frac{\sqrt{a x+\sqrt{-b+a x}} \left(-33-4 b-106 \sqrt{-b+a x}+120 b \sqrt{-b+a x}+8 (-b+a x)+48 (-b+a x)^{3/2}\right)}{96 a}+\frac{\left(-11+56 b-48 b^2\right) \log \left(a \left(-1-2 \sqrt{-b+a x}\right)+2 a \sqrt{a x+\sqrt{-b+a x}}\right)}{64 a}","\frac{(-33-12 b+8 a x) \sqrt{a x+\sqrt{-b+a x}}}{96 a}+\frac{\sqrt{-b+a x} (-53+36 b+24 a x) \sqrt{a x+\sqrt{-b+a x}}}{48 a}+\frac{\left(-11+56 b-48 b^2\right) \log \left(1+2 \sqrt{-b+a x}-2 \sqrt{a x+\sqrt{-b+a x}}\right)}{64 a}",1,"(Sqrt[a*x + Sqrt[-b + a*x]]*(-33 - 4*b - 106*Sqrt[-b + a*x] + 120*b*Sqrt[-b + a*x] + 8*(-b + a*x) + 48*(-b + a*x)^(3/2)))/(96*a) + ((-11 + 56*b - 48*b^2)*Log[a*(-1 - 2*Sqrt[-b + a*x]) + 2*a*Sqrt[a*x + Sqrt[-b + a*x]]])/(64*a)","A",1
1948,1,136,136,0.1809224,"\int \frac{-1+x^4}{\left(1+x^4\right) \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[(-1 + x^4)/((1 + x^4)*Sqrt[x + Sqrt[1 + x^2]]),x]","-\frac{1}{3 \left(x+\sqrt{1+x^2}\right)^{3/2}}+\sqrt{x+\sqrt{1+x^2}}-2 \text{RootSum}\left[1-4 \text{$\#$1}^4+22 \text{$\#$1}^8-4 \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}+\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^5}{-1+11 \text{$\#$1}^4-3 \text{$\#$1}^8+\text{$\#$1}^{12}}\&\right]","-\frac{1}{3 \left(x+\sqrt{1+x^2}\right)^{3/2}}+\sqrt{x+\sqrt{1+x^2}}-2 \text{RootSum}\left[1-4 \text{$\#$1}^4+22 \text{$\#$1}^8-4 \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}+\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^5}{-1+11 \text{$\#$1}^4-3 \text{$\#$1}^8+\text{$\#$1}^{12}}\&\right]",1,"-1/3*1/(x + Sqrt[1 + x^2])^(3/2) + Sqrt[x + Sqrt[1 + x^2]] - 2*RootSum[1 - 4*#1^4 + 22*#1^8 - 4*#1^12 + #1^16 & , (Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1 + Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1^5)/(-1 + 11*#1^4 - 3*#1^8 + #1^12) & ]","A",1
1949,1,136,136,0.0012498,"\int \frac{-1+x^4}{\left(1+x^4\right) \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[(-1 + x^4)/((1 + x^4)*Sqrt[x + Sqrt[1 + x^2]]),x]","-\frac{1}{3 \left(x+\sqrt{1+x^2}\right)^{3/2}}+\sqrt{x+\sqrt{1+x^2}}-2 \text{RootSum}\left[1-4 \text{$\#$1}^4+22 \text{$\#$1}^8-4 \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}+\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^5}{-1+11 \text{$\#$1}^4-3 \text{$\#$1}^8+\text{$\#$1}^{12}}\&\right]","-\frac{1}{3 \left(x+\sqrt{1+x^2}\right)^{3/2}}+\sqrt{x+\sqrt{1+x^2}}-2 \text{RootSum}\left[1-4 \text{$\#$1}^4+22 \text{$\#$1}^8-4 \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}+\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^5}{-1+11 \text{$\#$1}^4-3 \text{$\#$1}^8+\text{$\#$1}^{12}}\&\right]",1,"-1/3*1/(x + Sqrt[1 + x^2])^(3/2) + Sqrt[x + Sqrt[1 + x^2]] - 2*RootSum[1 - 4*#1^4 + 22*#1^8 - 4*#1^12 + #1^16 & , (Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1 + Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1^5)/(-1 + 11*#1^4 - 3*#1^8 + #1^12) & ]","A",1
1950,1,137,137,0.3278308,"\int \frac{1}{\sqrt[4]{-x^2+x^4} \left(-1+x^8\right)} \, dx","IntegrateAlgebraic[1/((-x^2 + x^4)^(1/4)*(-1 + x^8)),x]","-\frac{\left(-x^2+x^4\right)^{3/4}}{2 x \left(-1+x^2\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{4 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{4 \sqrt[4]{2}}+\frac{1}{8} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","-\frac{\left(-x^2+x^4\right)^{3/4}}{2 x \left(-1+x^2\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{4 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{4 \sqrt[4]{2}}+\frac{1}{8} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"-1/2*(-x^2 + x^4)^(3/4)/(x*(-1 + x^2)) - ArcTan[(2^(1/4)*x)/(-x^2 + x^4)^(1/4)]/(4*2^(1/4)) - ArcTanh[(2^(1/4)*x)/(-x^2 + x^4)^(1/4)]/(4*2^(1/4)) + RootSum[2 - 2*#1^4 + #1^8 & , (-Log[x] + Log[(-x^2 + x^4)^(1/4) - x*#1])/#1 & ]/8","A",1
1951,1,137,137,0.0009497,"\int \frac{1}{\sqrt[4]{-x^2+x^4} \left(-1+x^8\right)} \, dx","IntegrateAlgebraic[1/((-x^2 + x^4)^(1/4)*(-1 + x^8)),x]","-\frac{\left(-x^2+x^4\right)^{3/4}}{2 x \left(-1+x^2\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{4 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{4 \sqrt[4]{2}}+\frac{1}{8} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","-\frac{\left(-x^2+x^4\right)^{3/4}}{2 x \left(-1+x^2\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{4 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{4 \sqrt[4]{2}}+\frac{1}{8} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"-1/2*(-x^2 + x^4)^(3/4)/(x*(-1 + x^2)) - ArcTan[(2^(1/4)*x)/(-x^2 + x^4)^(1/4)]/(4*2^(1/4)) - ArcTanh[(2^(1/4)*x)/(-x^2 + x^4)^(1/4)]/(4*2^(1/4)) + RootSum[2 - 2*#1^4 + #1^8 & , (-Log[x] + Log[(-x^2 + x^4)^(1/4) - x*#1])/#1 & ]/8","A",1
1952,1,137,137,0.4000266,"\int \frac{-2+x^4}{\sqrt[4]{1+x^4} \left(-1-x^4+2 x^8\right)} \, dx","IntegrateAlgebraic[(-2 + x^4)/((1 + x^4)^(1/4)*(-1 - x^4 + 2*x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{1+x^4}}\right)}{6 \sqrt[4]{2}}+\frac{5 \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{1+x^4}}{-x^2+\sqrt{1+x^4}}\right)}{6 \sqrt{2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{1+x^4}}\right)}{6 \sqrt[4]{2}}+\frac{5 \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{1+x^4}}{x^2+\sqrt{1+x^4}}\right)}{6 \sqrt{2}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{1+x^4}}\right)}{6 \sqrt[4]{2}}+\frac{5 \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{1+x^4}}{-x^2+\sqrt{1+x^4}}\right)}{6 \sqrt{2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{1+x^4}}\right)}{6 \sqrt[4]{2}}+\frac{5 \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{1+x^4}}{x^2+\sqrt{1+x^4}}\right)}{6 \sqrt{2}}",1,"ArcTan[(2^(1/4)*x)/(1 + x^4)^(1/4)]/(6*2^(1/4)) + (5*ArcTan[(Sqrt[2]*x*(1 + x^4)^(1/4))/(-x^2 + Sqrt[1 + x^4])])/(6*Sqrt[2]) + ArcTanh[(2^(1/4)*x)/(1 + x^4)^(1/4)]/(6*2^(1/4)) + (5*ArcTanh[(Sqrt[2]*x*(1 + x^4)^(1/4))/(x^2 + Sqrt[1 + x^4])])/(6*Sqrt[2])","A",1
1953,1,137,137,0.2835593,"\int \frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\left(b^2+a x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[Sqrt[b + Sqrt[b^2 + a*x^2]]/(b^2 + a*x^2)^(5/2),x]","\frac{5 x}{12 b^2 \left(b^2+a x^2\right) \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{x \left(23 b^2+15 a x^2\right)}{24 b^3 \left(b^2+a x^2\right)^{3/2} \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{5 \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{8 \sqrt{a} b^{7/2}}","\frac{5 x}{12 b^2 \left(b^2+a x^2\right) \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{x \left(23 b^2+15 a x^2\right)}{24 b^3 \left(b^2+a x^2\right)^{3/2} \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{5 \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{8 \sqrt{a} b^{7/2}}",1,"(5*x)/(12*b^2*(b^2 + a*x^2)*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (x*(23*b^2 + 15*a*x^2))/(24*b^3*(b^2 + a*x^2)^(3/2)*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (5*ArcTan[(Sqrt[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/(8*Sqrt[a]*b^(7/2))","A",1
1954,1,138,138,0.1900046,"\int \frac{\sqrt[4]{-b+a x^2}}{x} \, dx","IntegrateAlgebraic[(-b + a*x^2)^(1/4)/x,x]","2 \sqrt[4]{-b+a x^2}-\frac{\sqrt[4]{b} \tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^2}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^2}}\right)}{\sqrt{2}}-\frac{\sqrt[4]{b} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^2}}{\sqrt{b}+\sqrt{-b+a x^2}}\right)}{\sqrt{2}}","2 \sqrt[4]{-b+a x^2}+\frac{\sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^2}}{-\sqrt{b}+\sqrt{-b+a x^2}}\right)}{\sqrt{2}}-\frac{\sqrt[4]{b} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^2}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^2}}\right)}{\sqrt{2}}",1,"2*(-b + a*x^2)^(1/4) - (b^(1/4)*ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^2]/(Sqrt[2]*b^(1/4)))/(-b + a*x^2)^(1/4)])/Sqrt[2] - (b^(1/4)*ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^2)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^2])])/Sqrt[2]","A",1
1955,1,138,138,0.3434195,"\int \frac{\left(b+a x^2\right) \sqrt[3]{-x+x^3}}{x^2} \, dx","IntegrateAlgebraic[((b + a*x^2)*(-x + x^3)^(1/3))/x^2,x]","\frac{\left(-3 b+a x^2\right) \sqrt[3]{-x+x^3}}{2 x}+\frac{1}{6} \left(\sqrt{3} a-3 \sqrt{3} b\right) \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)+\frac{1}{6} (a-3 b) \log \left(-x+\sqrt[3]{-x+x^3}\right)+\frac{1}{12} (-a+3 b) \log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)","\frac{\left(-3 b+a x^2\right) \sqrt[3]{-x+x^3}}{2 x}+\frac{1}{6} \left(\sqrt{3} a-3 \sqrt{3} b\right) \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)+\frac{1}{6} (a-3 b) \log \left(-x+\sqrt[3]{-x+x^3}\right)+\frac{1}{12} (-a+3 b) \log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)",1,"((-3*b + a*x^2)*(-x + x^3)^(1/3))/(2*x) + ((Sqrt[3]*a - 3*Sqrt[3]*b)*ArcTan[(Sqrt[3]*x)/(x + 2*(-x + x^3)^(1/3))])/6 + ((a - 3*b)*Log[-x + (-x + x^3)^(1/3)])/6 + ((-a + 3*b)*Log[x^2 + x*(-x + x^3)^(1/3) + (-x + x^3)^(2/3)])/12","A",1
1956,1,138,138,0.6087865,"\int \frac{-b+a x}{x \sqrt[3]{b^3+a^3 x^3}} \, dx","IntegrateAlgebraic[(-b + a*x)/(x*(b^3 + a^3*x^3)^(1/3)),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b^3+a^3 x^3}}{2 b+2 a x+\sqrt[3]{b^3+a^3 x^3}}\right)-\log \left(-b-a x+\sqrt[3]{b^3+a^3 x^3}\right)+\frac{1}{2} \log \left(b^2+2 a b x+a^2 x^2+(b+a x) \sqrt[3]{b^3+a^3 x^3}+\left(b^3+a^3 x^3\right)^{2/3}\right)","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b^3+a^3 x^3}}{2 b+2 a x+\sqrt[3]{b^3+a^3 x^3}}\right)-\log \left(-b-a x+\sqrt[3]{b^3+a^3 x^3}\right)+\frac{1}{2} \log \left(b^2+2 a b x+a^2 x^2+(b+a x) \sqrt[3]{b^3+a^3 x^3}+\left(b^3+a^3 x^3\right)^{2/3}\right)",1,"-(Sqrt[3]*ArcTan[(Sqrt[3]*(b^3 + a^3*x^3)^(1/3))/(2*b + 2*a*x + (b^3 + a^3*x^3)^(1/3))]) - Log[-b - a*x + (b^3 + a^3*x^3)^(1/3)] + Log[b^2 + 2*a*b*x + a^2*x^2 + (b + a*x)*(b^3 + a^3*x^3)^(1/3) + (b^3 + a^3*x^3)^(2/3)]/2","A",1
1957,1,138,138,0.6165874,"\int \frac{x^2 \sqrt[4]{-x^3+x^4}}{2+x} \, dx","IntegrateAlgebraic[(x^2*(-x^3 + x^4)^(1/4))/(2 + x),x]","\frac{1}{96} \left(401-100 x+32 x^2\right) \sqrt[4]{-x^3+x^4}+\frac{1135}{64} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)-8\ 2^{3/4} \sqrt[4]{3} \tan ^{-1}\left(\frac{\sqrt[4]{\frac{3}{2}} x}{\sqrt[4]{-x^3+x^4}}\right)-\frac{1135}{64} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)+8\ 2^{3/4} \sqrt[4]{3} \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{3}{2}} x}{\sqrt[4]{-x^3+x^4}}\right)","\frac{1}{96} \left(401-100 x+32 x^2\right) \sqrt[4]{-x^3+x^4}+\frac{1135}{64} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)-8\ 2^{3/4} \sqrt[4]{3} \tan ^{-1}\left(\frac{\sqrt[4]{\frac{3}{2}} x}{\sqrt[4]{-x^3+x^4}}\right)-\frac{1135}{64} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)+8\ 2^{3/4} \sqrt[4]{3} \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{3}{2}} x}{\sqrt[4]{-x^3+x^4}}\right)",1,"((401 - 100*x + 32*x^2)*(-x^3 + x^4)^(1/4))/96 + (1135*ArcTan[x/(-x^3 + x^4)^(1/4)])/64 - 8*2^(3/4)*3^(1/4)*ArcTan[((3/2)^(1/4)*x)/(-x^3 + x^4)^(1/4)] - (1135*ArcTanh[x/(-x^3 + x^4)^(1/4)])/64 + 8*2^(3/4)*3^(1/4)*ArcTanh[((3/2)^(1/4)*x)/(-x^3 + x^4)^(1/4)]","A",1
1958,1,138,138,3.0855888,"\int \frac{(-1+x) \left(-3+8 x-8 x^2+12 x^4\right)}{x \left(\frac{1-2 x^2}{1+2 x^2}\right)^{2/3} \left(1+2 x^2\right) \left(3-7 x+7 x^2-6 x^3+2 x^4\right)} \, dx","IntegrateAlgebraic[((-1 + x)*(-3 + 8*x - 8*x^2 + 12*x^4))/(x*((1 - 2*x^2)/(1 + 2*x^2))^(2/3)*(1 + 2*x^2)*(3 - 7*x + 7*x^2 - 6*x^3 + 2*x^4)),x]","\sqrt{3} \tan ^{-1}\left(\frac{-\sqrt{3}+\sqrt{3} x}{-1+x-2 \sqrt[3]{\frac{1-2 x^2}{1+2 x^2}}}\right)+\log \left(-1+x+\sqrt[3]{\frac{1-2 x^2}{1+2 x^2}}\right)-\frac{1}{2} \log \left(1-2 x+x^2+(1-x) \sqrt[3]{\frac{1-2 x^2}{1+2 x^2}}+\left(\frac{1-2 x^2}{1+2 x^2}\right)^{2/3}\right)","\sqrt{3} \tan ^{-1}\left(\frac{-\sqrt{3}+\sqrt{3} x}{-1+x-2 \sqrt[3]{\frac{1-2 x^2}{1+2 x^2}}}\right)+\log \left(-1+x+\sqrt[3]{\frac{1-2 x^2}{1+2 x^2}}\right)-\frac{1}{2} \log \left(1-2 x+x^2+(1-x) \sqrt[3]{\frac{1-2 x^2}{1+2 x^2}}+\left(\frac{1-2 x^2}{1+2 x^2}\right)^{2/3}\right)",1,"Sqrt[3]*ArcTan[(-Sqrt[3] + Sqrt[3]*x)/(-1 + x - 2*((1 - 2*x^2)/(1 + 2*x^2))^(1/3))] + Log[-1 + x + ((1 - 2*x^2)/(1 + 2*x^2))^(1/3)] - Log[1 - 2*x + x^2 + (1 - x)*((1 - 2*x^2)/(1 + 2*x^2))^(1/3) + ((1 - 2*x^2)/(1 + 2*x^2))^(2/3)]/2","A",1
1959,1,138,138,0.1110223,"\int \frac{b-a x^5}{\sqrt{a+b x} \left(a b+x^5\right)} \, dx","IntegrateAlgebraic[(b - a*x^5)/(Sqrt[a + b*x]*(a*b + x^5)),x]","-\frac{2 a \sqrt{a+b x}}{b}+\frac{1}{5} \left(b^5+a^2 b^5\right) \text{RootSum}\left[a^5-a b^6-5 a^4 \text{$\#$1}^2+10 a^3 \text{$\#$1}^4-10 a^2 \text{$\#$1}^6+5 a \text{$\#$1}^8-\text{$\#$1}^{10}\&,\frac{\log \left(\sqrt{a+b x}-\text{$\#$1}\right)}{a^4 \text{$\#$1}-4 a^3 \text{$\#$1}^3+6 a^2 \text{$\#$1}^5-4 a \text{$\#$1}^7+\text{$\#$1}^9}\&\right]","-\frac{2 a \sqrt{a+b x}}{b}+\frac{1}{5} \left(b^5+a^2 b^5\right) \text{RootSum}\left[a^5-a b^6-5 a^4 \text{$\#$1}^2+10 a^3 \text{$\#$1}^4-10 a^2 \text{$\#$1}^6+5 a \text{$\#$1}^8-\text{$\#$1}^{10}\&,\frac{\log \left(\sqrt{a+b x}-\text{$\#$1}\right)}{a^4 \text{$\#$1}-4 a^3 \text{$\#$1}^3+6 a^2 \text{$\#$1}^5-4 a \text{$\#$1}^7+\text{$\#$1}^9}\&\right]",1,"(-2*a*Sqrt[a + b*x])/b + ((b^5 + a^2*b^5)*RootSum[a^5 - a*b^6 - 5*a^4*#1^2 + 10*a^3*#1^4 - 10*a^2*#1^6 + 5*a*#1^8 - #1^10 & , Log[Sqrt[a + b*x] - #1]/(a^4*#1 - 4*a^3*#1^3 + 6*a^2*#1^5 - 4*a*#1^7 + #1^9) & ])/5","A",1
1960,1,138,138,0.3653199,"\int \frac{\left(-1+x^3\right)^{2/3} \left(1+x^3+x^6\right)}{x^6 \left(-1+x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(1 + x^3 + x^6))/(x^6*(-1 + x^6)),x]","\frac{\left(-1+x^3\right)^{2/3} \left(2+3 x^3\right)}{10 x^5}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-1+x^3}}\right)}{\sqrt[3]{2} \sqrt{3}}-\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{-1+x^3}\right)}{3 \sqrt[3]{2}}+\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-1+x^3}+\sqrt[3]{2} \left(-1+x^3\right)^{2/3}\right)}{6 \sqrt[3]{2}}","\frac{\left(-1+x^3\right)^{2/3} \left(2+3 x^3\right)}{10 x^5}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-1+x^3}}\right)}{\sqrt[3]{2} \sqrt{3}}-\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{-1+x^3}\right)}{3 \sqrt[3]{2}}+\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-1+x^3}+\sqrt[3]{2} \left(-1+x^3\right)^{2/3}\right)}{6 \sqrt[3]{2}}",1,"((-1 + x^3)^(2/3)*(2 + 3*x^3))/(10*x^5) + ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-1 + x^3)^(1/3))]/(2^(1/3)*Sqrt[3]) - Log[-2*x + 2^(2/3)*(-1 + x^3)^(1/3)]/(3*2^(1/3)) + Log[2*x^2 + 2^(2/3)*x*(-1 + x^3)^(1/3) + 2^(1/3)*(-1 + x^3)^(2/3)]/(6*2^(1/3))","A",1
1961,1,138,138,0.2312739,"\int \frac{-1+x^2}{\sqrt{1+x} \left(1+x^2\right) \sqrt{1+\sqrt{1+x}} \sqrt{1+\sqrt{1+\sqrt{1+x}}}} \, dx","IntegrateAlgebraic[(-1 + x^2)/(Sqrt[1 + x]*(1 + x^2)*Sqrt[1 + Sqrt[1 + x]]*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]),x]","8 \sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{RootSum}\left[2-8 \text{$\#$1}^4+8 \text{$\#$1}^6+14 \text{$\#$1}^8-32 \text{$\#$1}^{10}+24 \text{$\#$1}^{12}-8 \text{$\#$1}^{14}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{$\#$1}\right)}{-2 \text{$\#$1}^3+3 \text{$\#$1}^5+7 \text{$\#$1}^7-20 \text{$\#$1}^9+18 \text{$\#$1}^{11}-7 \text{$\#$1}^{13}+\text{$\#$1}^{15}}\&\right]","8 \sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{RootSum}\left[2-8 \text{$\#$1}^4+8 \text{$\#$1}^6+14 \text{$\#$1}^8-32 \text{$\#$1}^{10}+24 \text{$\#$1}^{12}-8 \text{$\#$1}^{14}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{$\#$1}\right)}{-2 \text{$\#$1}^3+3 \text{$\#$1}^5+7 \text{$\#$1}^7-20 \text{$\#$1}^9+18 \text{$\#$1}^{11}-7 \text{$\#$1}^{13}+\text{$\#$1}^{15}}\&\right]",1,"8*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]] - RootSum[2 - 8*#1^4 + 8*#1^6 + 14*#1^8 - 32*#1^10 + 24*#1^12 - 8*#1^14 + #1^16 & , Log[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]] - #1]/(-2*#1^3 + 3*#1^5 + 7*#1^7 - 20*#1^9 + 18*#1^11 - 7*#1^13 + #1^15) & ]","A",1
1962,1,138,138,0.0027916,"\int \frac{-1+x^2}{\sqrt{1+x} \left(1+x^2\right) \sqrt{1+\sqrt{1+x}} \sqrt{1+\sqrt{1+\sqrt{1+x}}}} \, dx","IntegrateAlgebraic[(-1 + x^2)/(Sqrt[1 + x]*(1 + x^2)*Sqrt[1 + Sqrt[1 + x]]*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]),x]","8 \sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{RootSum}\left[2-8 \text{$\#$1}^4+8 \text{$\#$1}^6+14 \text{$\#$1}^8-32 \text{$\#$1}^{10}+24 \text{$\#$1}^{12}-8 \text{$\#$1}^{14}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{$\#$1}\right)}{-2 \text{$\#$1}^3+3 \text{$\#$1}^5+7 \text{$\#$1}^7-20 \text{$\#$1}^9+18 \text{$\#$1}^{11}-7 \text{$\#$1}^{13}+\text{$\#$1}^{15}}\&\right]","8 \sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{RootSum}\left[2-8 \text{$\#$1}^4+8 \text{$\#$1}^6+14 \text{$\#$1}^8-32 \text{$\#$1}^{10}+24 \text{$\#$1}^{12}-8 \text{$\#$1}^{14}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{$\#$1}\right)}{-2 \text{$\#$1}^3+3 \text{$\#$1}^5+7 \text{$\#$1}^7-20 \text{$\#$1}^9+18 \text{$\#$1}^{11}-7 \text{$\#$1}^{13}+\text{$\#$1}^{15}}\&\right]",1,"8*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]] - RootSum[2 - 8*#1^4 + 8*#1^6 + 14*#1^8 - 32*#1^10 + 24*#1^12 - 8*#1^14 + #1^16 & , Log[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]] - #1]/(-2*#1^3 + 3*#1^5 + 7*#1^7 - 20*#1^9 + 18*#1^11 - 7*#1^13 + #1^15) & ]","A",1
1963,1,139,139,0.4016187,"\int \frac{x}{\left(x^2 (-a+x)\right)^{2/3} (-a d+(-1+d) x)} \, dx","IntegrateAlgebraic[x/((x^2*(-a + x))^(2/3)*(-(a*d) + (-1 + d)*x)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{d} \sqrt[3]{-a x^2+x^3}}\right)}{a \sqrt[3]{d}}+\frac{\log \left(x-\sqrt[3]{d} \sqrt[3]{-a x^2+x^3}\right)}{a \sqrt[3]{d}}-\frac{\log \left(x^2+\sqrt[3]{d} x \sqrt[3]{-a x^2+x^3}+d^{2/3} \left(-a x^2+x^3\right)^{2/3}\right)}{2 a \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{d} \sqrt[3]{-a x^2+x^3}}\right)}{a \sqrt[3]{d}}+\frac{\log \left(x-\sqrt[3]{d} \sqrt[3]{-a x^2+x^3}\right)}{a \sqrt[3]{d}}-\frac{\log \left(x^2+\sqrt[3]{d} x \sqrt[3]{-a x^2+x^3}+d^{2/3} \left(-a x^2+x^3\right)^{2/3}\right)}{2 a \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*d^(1/3)*(-(a*x^2) + x^3)^(1/3))])/(a*d^(1/3)) + Log[x - d^(1/3)*(-(a*x^2) + x^3)^(1/3)]/(a*d^(1/3)) - Log[x^2 + d^(1/3)*x*(-(a*x^2) + x^3)^(1/3) + d^(2/3)*(-(a*x^2) + x^3)^(2/3)]/(2*a*d^(1/3))","A",1
1964,1,139,139,0.1877565,"\int \frac{\left(-b+a x^2\right)^{3/4}}{x} \, dx","IntegrateAlgebraic[(-b + a*x^2)^(3/4)/x,x]","\frac{2}{3} \left(-b+a x^2\right)^{3/4}-\frac{b^{3/4} \tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^2}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^2}}\right)}{\sqrt{2}}+\frac{b^{3/4} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^2}}{\sqrt{b}+\sqrt{-b+a x^2}}\right)}{\sqrt{2}}","\frac{2}{3} \left(-b+a x^2\right)^{3/4}+\frac{b^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^2}}{-\sqrt{b}+\sqrt{-b+a x^2}}\right)}{\sqrt{2}}+\frac{b^{3/4} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^2}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^2}}\right)}{\sqrt{2}}",1,"(2*(-b + a*x^2)^(3/4))/3 - (b^(3/4)*ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^2]/(Sqrt[2]*b^(1/4)))/(-b + a*x^2)^(1/4)])/Sqrt[2] + (b^(3/4)*ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^2)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^2])])/Sqrt[2]","A",1
1965,1,139,139,12.0076802,"\int \frac{-2 x+x^2}{\left(1-x+x^2\right) \sqrt[4]{1+x^4}} \, dx","IntegrateAlgebraic[(-2*x + x^2)/((1 - x + x^2)*(1 + x^4)^(1/4)),x]","\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\frac{\tan ^{-1}\left(\frac{\left(-\sqrt{2}+\sqrt{2} x\right) \sqrt[4]{1+x^4}}{-1+2 x-x^2+\sqrt{1+x^4}}\right)}{\sqrt{2}}+\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\left(-\sqrt{2}+\sqrt{2} x\right) \sqrt[4]{1+x^4}}{1-2 x+x^2+\sqrt{1+x^4}}\right)}{\sqrt{2}}","\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\frac{\tan ^{-1}\left(\frac{\left(-\sqrt{2}+\sqrt{2} x\right) \sqrt[4]{1+x^4}}{-1+2 x-x^2+\sqrt{1+x^4}}\right)}{\sqrt{2}}+\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\left(-\sqrt{2}+\sqrt{2} x\right) \sqrt[4]{1+x^4}}{1-2 x+x^2+\sqrt{1+x^4}}\right)}{\sqrt{2}}",1,"ArcTan[x/(1 + x^4)^(1/4)]/2 - ArcTan[((-Sqrt[2] + Sqrt[2]*x)*(1 + x^4)^(1/4))/(-1 + 2*x - x^2 + Sqrt[1 + x^4])]/Sqrt[2] + ArcTanh[x/(1 + x^4)^(1/4)]/2 - ArcTanh[((-Sqrt[2] + Sqrt[2]*x)*(1 + x^4)^(1/4))/(1 - 2*x + x^2 + Sqrt[1 + x^4])]/Sqrt[2]","A",1
1966,1,139,139,0.2977484,"\int \frac{1+k^4 x^4}{\sqrt{(1-x) x \left(1-k^2 x\right)} \left(-1+k^4 x^4\right)} \, dx","IntegrateAlgebraic[(1 + k^4*x^4)/(Sqrt[(1 - x)*x*(1 - k^2*x)]*(-1 + k^4*x^4)),x]","-\frac{\tan ^{-1}\left(\frac{(-1+k) x}{\sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}\right)}{2 (-1+k)}-\frac{\tan ^{-1}\left(\frac{(1+k) x}{\sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}\right)}{2 (1+k)}-\frac{\tan ^{-1}\left(\frac{\sqrt{1+k^2} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{(-1+x) \left(-1+k^2 x\right)}\right)}{\sqrt{1+k^2}}","-\frac{\tan ^{-1}\left(\frac{(-1+k) x}{\sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}\right)}{2 (-1+k)}-\frac{\tan ^{-1}\left(\frac{(1+k) x}{\sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}\right)}{2 (1+k)}-\frac{\tan ^{-1}\left(\frac{\sqrt{1+k^2} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{(-1+x) \left(-1+k^2 x\right)}\right)}{\sqrt{1+k^2}}",1,"-1/2*ArcTan[((-1 + k)*x)/Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3]]/(-1 + k) - ArcTan[((1 + k)*x)/Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3]]/(2*(1 + k)) - ArcTan[(Sqrt[1 + k^2]*Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3])/((-1 + x)*(-1 + k^2*x))]/Sqrt[1 + k^2]","A",1
1967,1,139,139,2.6562694,"\int \frac{-b+a x^8}{\sqrt[4]{-b x^2+a x^4} \left(b+a x^8\right)} \, dx","IntegrateAlgebraic[(-b + a*x^8)/((-(b*x^2) + a*x^4)^(1/4)*(b + a*x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{1}{4} \text{RootSum}\left[a^4+a b^3-4 a^3 \text{$\#$1}^4+6 a^2 \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{1}{4} \text{RootSum}\left[a^4+a b^3-4 a^3 \text{$\#$1}^4+6 a^2 \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"ArcTan[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)]/a^(1/4) + ArcTanh[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)]/a^(1/4) + RootSum[a^4 + a*b^3 - 4*a^3*#1^4 + 6*a^2*#1^8 - 4*a*#1^12 + #1^16 & , (-Log[x] + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1])/#1 & ]/4","A",1
1968,1,139,139,0.0020723,"\int \frac{-b+a x^8}{\sqrt[4]{-b x^2+a x^4} \left(b+a x^8\right)} \, dx","IntegrateAlgebraic[(-b + a*x^8)/((-(b*x^2) + a*x^4)^(1/4)*(b + a*x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{1}{4} \text{RootSum}\left[a^4+a b^3-4 a^3 \text{$\#$1}^4+6 a^2 \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{1}{4} \text{RootSum}\left[a^4+a b^3-4 a^3 \text{$\#$1}^4+6 a^2 \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"ArcTan[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)]/a^(1/4) + ArcTanh[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)]/a^(1/4) + RootSum[a^4 + a*b^3 - 4*a^3*#1^4 + 6*a^2*#1^8 - 4*a*#1^12 + #1^16 & , (-Log[x] + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1])/#1 & ]/4","A",1
1969,1,139,139,2.3931442,"\int \frac{\left(-1+x^8\right) \left(1+x^8\right)}{\sqrt[4]{-1-x^4+x^8} \left(1-3 x^8+x^{16}\right)} \, dx","IntegrateAlgebraic[((-1 + x^8)*(1 + x^8))/((-1 - x^4 + x^8)^(1/4)*(1 - 3*x^8 + x^16)),x]","-\frac{x}{2 \sqrt[4]{-1-x^4+x^8}}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{-1-x^4+x^8}}{\sqrt{2} x^2-\sqrt{-1-x^4+x^8}}\right)}{4\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{2 \sqrt[4]{2} x \sqrt[4]{-1-x^4+x^8}}{2 x^2+\sqrt{2} \sqrt{-1-x^4+x^8}}\right)}{4\ 2^{3/4}}","-\frac{x}{2 \sqrt[4]{-1-x^4+x^8}}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{-1-x^4+x^8}}{\sqrt{2} x^2-\sqrt{-1-x^4+x^8}}\right)}{4\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{2 \sqrt[4]{2} x \sqrt[4]{-1-x^4+x^8}}{2 x^2+\sqrt{2} \sqrt{-1-x^4+x^8}}\right)}{4\ 2^{3/4}}",1,"-1/2*x/(-1 - x^4 + x^8)^(1/4) + ArcTan[(2^(3/4)*x*(-1 - x^4 + x^8)^(1/4))/(Sqrt[2]*x^2 - Sqrt[-1 - x^4 + x^8])]/(4*2^(3/4)) - ArcTanh[(2*2^(1/4)*x*(-1 - x^4 + x^8)^(1/4))/(2*x^2 + Sqrt[2]*Sqrt[-1 - x^4 + x^8])]/(4*2^(3/4))","A",1
1970,1,140,140,0.3514446,"\int \frac{1}{\sqrt[3]{x^2 (-a+x)} (-a d+(-1+d) x)} \, dx","IntegrateAlgebraic[1/((x^2*(-a + x))^(1/3)*(-(a*d) + (-1 + d)*x)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{d} \sqrt[3]{-a x^2+x^3}}\right)}{a d^{2/3}}+\frac{\log \left(x-\sqrt[3]{d} \sqrt[3]{-a x^2+x^3}\right)}{a d^{2/3}}-\frac{\log \left(x^2+\sqrt[3]{d} x \sqrt[3]{-a x^2+x^3}+d^{2/3} \left(-a x^2+x^3\right)^{2/3}\right)}{2 a d^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{d} \sqrt[3]{-a x^2+x^3}}\right)}{a d^{2/3}}+\frac{\log \left(x-\sqrt[3]{d} \sqrt[3]{-a x^2+x^3}\right)}{a d^{2/3}}-\frac{\log \left(x^2+\sqrt[3]{d} x \sqrt[3]{-a x^2+x^3}+d^{2/3} \left(-a x^2+x^3\right)^{2/3}\right)}{2 a d^{2/3}}",1,"-((Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*d^(1/3)*(-(a*x^2) + x^3)^(1/3))])/(a*d^(2/3))) + Log[x - d^(1/3)*(-(a*x^2) + x^3)^(1/3)]/(a*d^(2/3)) - Log[x^2 + d^(1/3)*x*(-(a*x^2) + x^3)^(1/3) + d^(2/3)*(-(a*x^2) + x^3)^(2/3)]/(2*a*d^(2/3))","A",1
1971,1,140,140,0.2300769,"\int \frac{\sqrt[3]{6+2 x+x^2}}{1+x} \, dx","IntegrateAlgebraic[(6 + 2*x + x^2)^(1/3)/(1 + x),x]","\frac{3}{2} \sqrt[3]{6+2 x+x^2}-\frac{1}{2} \sqrt{3} \sqrt[3]{5} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{6+2 x+x^2}}{\sqrt{3} \sqrt[3]{5}}\right)+\frac{1}{2} \sqrt[3]{5} \log \left(-5+5^{2/3} \sqrt[3]{6+2 x+x^2}\right)-\frac{1}{4} \sqrt[3]{5} \log \left(5+5^{2/3} \sqrt[3]{6+2 x+x^2}+\sqrt[3]{5} \left(6+2 x+x^2\right)^{2/3}\right)","\frac{3}{2} \sqrt[3]{6+2 x+x^2}-\frac{1}{2} \sqrt{3} \sqrt[3]{5} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{6+2 x+x^2}}{\sqrt{3} \sqrt[3]{5}}\right)+\frac{1}{2} \sqrt[3]{5} \log \left(-5+5^{2/3} \sqrt[3]{6+2 x+x^2}\right)-\frac{1}{4} \sqrt[3]{5} \log \left(5+5^{2/3} \sqrt[3]{6+2 x+x^2}+\sqrt[3]{5} \left(6+2 x+x^2\right)^{2/3}\right)",1,"(3*(6 + 2*x + x^2)^(1/3))/2 - (Sqrt[3]*5^(1/3)*ArcTan[1/Sqrt[3] + (2*(6 + 2*x + x^2)^(1/3))/(Sqrt[3]*5^(1/3))])/2 + (5^(1/3)*Log[-5 + 5^(2/3)*(6 + 2*x + x^2)^(1/3)])/2 - (5^(1/3)*Log[5 + 5^(2/3)*(6 + 2*x + x^2)^(1/3) + 5^(1/3)*(6 + 2*x + x^2)^(2/3)])/4","A",1
1972,1,140,140,3.528086,"\int \frac{-1+x}{x \sqrt[3]{-1+x^3}} \, dx","IntegrateAlgebraic[(-1 + x)/(x*(-1 + x^3)^(1/3)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{\sqrt{3}}+\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^3}}{\sqrt{3}}\right)}{\sqrt{3}}-\frac{2}{3} \tanh ^{-1}\left(\frac{1+x}{-1+x-2 \sqrt[3]{-1+x^3}}\right)-\frac{1}{6} \log \left(1-\sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)+\frac{1}{6} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)","\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{\sqrt{3}}+\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x^3}}{\sqrt{3}}\right)}{\sqrt{3}}-\frac{2}{3} \tanh ^{-1}\left(\frac{1+x}{-1+x-2 \sqrt[3]{-1+x^3}}\right)-\frac{1}{6} \log \left(1-\sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)+\frac{1}{6} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)",1,"ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^3)^(1/3))]/Sqrt[3] + ArcTan[1/Sqrt[3] - (2*(-1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3] - (2*ArcTanh[(1 + x)/(-1 + x - 2*(-1 + x^3)^(1/3))])/3 - Log[1 - (-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)]/6 + Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)]/6","A",1
1973,1,140,140,0.3120497,"\int \frac{\left(-1+x^3\right)^{2/3} \left(1+x^3\right)}{x^6 \left(-2+x^3\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(1 + x^3))/(x^6*(-2 + x^3)),x]","\frac{\left(-1+x^3\right)^{2/3} \left(4+11 x^3\right)}{40 x^5}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2} \sqrt[3]{-1+x^3}}\right)}{4\ 2^{2/3}}+\frac{\log \left(-x+\sqrt[3]{2} \sqrt[3]{-1+x^3}\right)}{4\ 2^{2/3}}-\frac{\log \left(x^2+\sqrt[3]{2} x \sqrt[3]{-1+x^3}+2^{2/3} \left(-1+x^3\right)^{2/3}\right)}{8\ 2^{2/3}}","\frac{\left(-1+x^3\right)^{2/3} \left(4+11 x^3\right)}{40 x^5}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2} \sqrt[3]{-1+x^3}}\right)}{4\ 2^{2/3}}+\frac{\log \left(-x+\sqrt[3]{2} \sqrt[3]{-1+x^3}\right)}{4\ 2^{2/3}}-\frac{\log \left(x^2+\sqrt[3]{2} x \sqrt[3]{-1+x^3}+2^{2/3} \left(-1+x^3\right)^{2/3}\right)}{8\ 2^{2/3}}",1,"((-1 + x^3)^(2/3)*(4 + 11*x^3))/(40*x^5) - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*2^(1/3)*(-1 + x^3)^(1/3))])/(4*2^(2/3)) + Log[-x + 2^(1/3)*(-1 + x^3)^(1/3)]/(4*2^(2/3)) - Log[x^2 + 2^(1/3)*x*(-1 + x^3)^(1/3) + 2^(2/3)*(-1 + x^3)^(2/3)]/(8*2^(2/3))","A",1
1974,1,140,140,0.4105287,"\int \frac{\left(-4+x^2\right) \sqrt[3]{x+x^3}}{x^4 \left(2+x^2\right)} \, dx","IntegrateAlgebraic[((-4 + x^2)*(x + x^3)^(1/3))/(x^4*(2 + x^2)),x]","-\frac{3 \left(-1+2 x^2\right) \sqrt[3]{x+x^3}}{4 x^3}-\frac{3 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2} \sqrt[3]{x+x^3}}\right)}{4 \sqrt[3]{2}}-\frac{3 \log \left(-x+\sqrt[3]{2} \sqrt[3]{x+x^3}\right)}{4 \sqrt[3]{2}}+\frac{3 \log \left(x^2+\sqrt[3]{2} x \sqrt[3]{x+x^3}+2^{2/3} \left(x+x^3\right)^{2/3}\right)}{8 \sqrt[3]{2}}","-\frac{3 \left(-1+2 x^2\right) \sqrt[3]{x+x^3}}{4 x^3}-\frac{3 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2} \sqrt[3]{x+x^3}}\right)}{4 \sqrt[3]{2}}-\frac{3 \log \left(-x+\sqrt[3]{2} \sqrt[3]{x+x^3}\right)}{4 \sqrt[3]{2}}+\frac{3 \log \left(x^2+\sqrt[3]{2} x \sqrt[3]{x+x^3}+2^{2/3} \left(x+x^3\right)^{2/3}\right)}{8 \sqrt[3]{2}}",1,"(-3*(-1 + 2*x^2)*(x + x^3)^(1/3))/(4*x^3) - (3*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*2^(1/3)*(x + x^3)^(1/3))])/(4*2^(1/3)) - (3*Log[-x + 2^(1/3)*(x + x^3)^(1/3)])/(4*2^(1/3)) + (3*Log[x^2 + 2^(1/3)*x*(x + x^3)^(1/3) + 2^(2/3)*(x + x^3)^(2/3)])/(8*2^(1/3))","A",1
1975,1,140,140,0.3492558,"\int \frac{2+x^2}{\left(1+x^2\right) \sqrt[3]{-x^2+x^3}} \, dx","IntegrateAlgebraic[(2 + x^2)/((1 + x^2)*(-x^2 + x^3)^(1/3)),x]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{-x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)-\frac{1}{2} \text{RootSum}\left[2-2 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{-x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)-\frac{1}{2} \text{RootSum}\left[2-2 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-x^2 + x^3)^(1/3))] - Log[-x + (-x^2 + x^3)^(1/3)] + Log[x^2 + x*(-x^2 + x^3)^(1/3) + (-x^2 + x^3)^(2/3)]/2 - RootSum[2 - 2*#1^3 + #1^6 & , (-Log[x] + Log[(-x^2 + x^3)^(1/3) - x*#1])/#1 & ]/2","A",1
1976,1,150,140,0.4319133,"\int \frac{x}{\left(b+a x^2\right) \sqrt{-b x+a x^3}} \, dx","IntegrateAlgebraic[x/((b + a*x^2)*Sqrt[-(b*x) + a*x^3]),x]","\frac{\tan ^{-1}\left(\frac{-\frac{b^{3/4}}{2 \sqrt[4]{a}}-\sqrt[4]{a} \sqrt[4]{b} x+\frac{a^{3/4} x^2}{2 \sqrt[4]{b}}}{\sqrt{-b x+a x^3}}\right)}{4 a^{3/4} b^{3/4}}+\frac{\tanh ^{-1}\left(\frac{-\frac{b^{3/4}}{2 \sqrt[4]{a}}+\sqrt[4]{a} \sqrt[4]{b} x+\frac{a^{3/4} x^2}{2 \sqrt[4]{b}}}{\sqrt{-b x+a x^3}}\right)}{4 a^{3/4} b^{3/4}}","-\frac{\tan ^{-1}\left(\frac{2 \sqrt[4]{a} \sqrt[4]{b} \sqrt{-b x+a x^3}}{-b-2 \sqrt{a} \sqrt{b} x+a x^2}\right)}{4 a^{3/4} b^{3/4}}+\frac{\tanh ^{-1}\left(\frac{-\frac{b^{3/4}}{2 \sqrt[4]{a}}+\sqrt[4]{a} \sqrt[4]{b} x+\frac{a^{3/4} x^2}{2 \sqrt[4]{b}}}{\sqrt{-b x+a x^3}}\right)}{4 a^{3/4} b^{3/4}}",1,"ArcTan[(-1/2*b^(3/4)/a^(1/4) - a^(1/4)*b^(1/4)*x + (a^(3/4)*x^2)/(2*b^(1/4)))/Sqrt[-(b*x) + a*x^3]]/(4*a^(3/4)*b^(3/4)) + ArcTanh[(-1/2*b^(3/4)/a^(1/4) + a^(1/4)*b^(1/4)*x + (a^(3/4)*x^2)/(2*b^(1/4)))/Sqrt[-(b*x) + a*x^3]]/(4*a^(3/4)*b^(3/4))","A",1
1977,1,150,140,0.4208842,"\int \frac{-b+a x^2}{\left(b+a x^2\right) \sqrt{-b x+a x^3}} \, dx","IntegrateAlgebraic[(-b + a*x^2)/((b + a*x^2)*Sqrt[-(b*x) + a*x^3]),x]","\frac{\tan ^{-1}\left(\frac{-\frac{b^{3/4}}{2 \sqrt[4]{a}}-\sqrt[4]{a} \sqrt[4]{b} x+\frac{a^{3/4} x^2}{2 \sqrt[4]{b}}}{\sqrt{-b x+a x^3}}\right)}{2 \sqrt[4]{a} \sqrt[4]{b}}-\frac{\tanh ^{-1}\left(\frac{-\frac{b^{3/4}}{2 \sqrt[4]{a}}+\sqrt[4]{a} \sqrt[4]{b} x+\frac{a^{3/4} x^2}{2 \sqrt[4]{b}}}{\sqrt{-b x+a x^3}}\right)}{2 \sqrt[4]{a} \sqrt[4]{b}}","-\frac{\tan ^{-1}\left(\frac{2 \sqrt[4]{a} \sqrt[4]{b} \sqrt{-b x+a x^3}}{-b-2 \sqrt{a} \sqrt{b} x+a x^2}\right)}{2 \sqrt[4]{a} \sqrt[4]{b}}-\frac{\tanh ^{-1}\left(\frac{-\frac{b^{3/4}}{2 \sqrt[4]{a}}+\sqrt[4]{a} \sqrt[4]{b} x+\frac{a^{3/4} x^2}{2 \sqrt[4]{b}}}{\sqrt{-b x+a x^3}}\right)}{2 \sqrt[4]{a} \sqrt[4]{b}}",1,"ArcTan[(-1/2*b^(3/4)/a^(1/4) - a^(1/4)*b^(1/4)*x + (a^(3/4)*x^2)/(2*b^(1/4)))/Sqrt[-(b*x) + a*x^3]]/(2*a^(1/4)*b^(1/4)) - ArcTanh[(-1/2*b^(3/4)/a^(1/4) + a^(1/4)*b^(1/4)*x + (a^(3/4)*x^2)/(2*b^(1/4)))/Sqrt[-(b*x) + a*x^3]]/(2*a^(1/4)*b^(1/4))","A",1
1978,1,150,140,0.4194724,"\int \frac{\sqrt{-b x+a x^3}}{-b^2+a^2 x^4} \, dx","IntegrateAlgebraic[Sqrt[-(b*x) + a*x^3]/(-b^2 + a^2*x^4),x]","\frac{\tan ^{-1}\left(\frac{-\frac{b^{3/4}}{2 \sqrt[4]{a}}-\sqrt[4]{a} \sqrt[4]{b} x+\frac{a^{3/4} x^2}{2 \sqrt[4]{b}}}{\sqrt{-b x+a x^3}}\right)}{4 a^{3/4} b^{3/4}}+\frac{\tanh ^{-1}\left(\frac{-\frac{b^{3/4}}{2 \sqrt[4]{a}}+\sqrt[4]{a} \sqrt[4]{b} x+\frac{a^{3/4} x^2}{2 \sqrt[4]{b}}}{\sqrt{-b x+a x^3}}\right)}{4 a^{3/4} b^{3/4}}","-\frac{\tan ^{-1}\left(\frac{2 \sqrt[4]{a} \sqrt[4]{b} \sqrt{-b x+a x^3}}{-b-2 \sqrt{a} \sqrt{b} x+a x^2}\right)}{4 a^{3/4} b^{3/4}}+\frac{\tanh ^{-1}\left(\frac{-\frac{b^{3/4}}{2 \sqrt[4]{a}}+\sqrt[4]{a} \sqrt[4]{b} x+\frac{a^{3/4} x^2}{2 \sqrt[4]{b}}}{\sqrt{-b x+a x^3}}\right)}{4 a^{3/4} b^{3/4}}",1,"ArcTan[(-1/2*b^(3/4)/a^(1/4) - a^(1/4)*b^(1/4)*x + (a^(3/4)*x^2)/(2*b^(1/4)))/Sqrt[-(b*x) + a*x^3]]/(4*a^(3/4)*b^(3/4)) + ArcTanh[(-1/2*b^(3/4)/a^(1/4) + a^(1/4)*b^(1/4)*x + (a^(3/4)*x^2)/(2*b^(1/4)))/Sqrt[-(b*x) + a*x^3]]/(4*a^(3/4)*b^(3/4))","A",1
1979,1,140,140,3.3611748,"\int \frac{\left(1+x^5\right)^{2/3} \left(-3+2 x^5\right) \left(2+x^3+2 x^5\right)}{x^6 \left(2-x^3+2 x^5\right)} \, dx","IntegrateAlgebraic[((1 + x^5)^(2/3)*(-3 + 2*x^5)*(2 + x^3 + 2*x^5))/(x^6*(2 - x^3 + 2*x^5)),x]","\frac{3 \left(1+x^5\right)^{2/3} \left(2+5 x^3+2 x^5\right)}{10 x^5}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2} \sqrt[3]{1+x^5}}\right)}{2^{2/3}}+\frac{\log \left(-x+\sqrt[3]{2} \sqrt[3]{1+x^5}\right)}{2^{2/3}}-\frac{\log \left(x^2+\sqrt[3]{2} x \sqrt[3]{1+x^5}+2^{2/3} \left(1+x^5\right)^{2/3}\right)}{2\ 2^{2/3}}","\frac{3 \left(1+x^5\right)^{2/3} \left(2+5 x^3+2 x^5\right)}{10 x^5}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2} \sqrt[3]{1+x^5}}\right)}{2^{2/3}}+\frac{\log \left(-x+\sqrt[3]{2} \sqrt[3]{1+x^5}\right)}{2^{2/3}}-\frac{\log \left(x^2+\sqrt[3]{2} x \sqrt[3]{1+x^5}+2^{2/3} \left(1+x^5\right)^{2/3}\right)}{2\ 2^{2/3}}",1,"(3*(1 + x^5)^(2/3)*(2 + 5*x^3 + 2*x^5))/(10*x^5) - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*2^(1/3)*(1 + x^5)^(1/3))])/2^(2/3) + Log[-x + 2^(1/3)*(1 + x^5)^(1/3)]/2^(2/3) - Log[x^2 + 2^(1/3)*x*(1 + x^5)^(1/3) + 2^(2/3)*(1 + x^5)^(2/3)]/(2*2^(2/3))","A",1
1980,1,144,140,2.7834593,"\int \frac{b+a x^6}{\left(-b+a x^6\right) \sqrt[3]{-b+a^3 x^3+a x^6}} \, dx","IntegrateAlgebraic[(b + a*x^6)/((-b + a*x^6)*(-b + a^3*x^3 + a*x^6)^(1/3)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{-b+a^3 x^3+a x^6}}\right)}{\sqrt{3} a}+\frac{\log \left(a^2 x-a \sqrt[3]{-b+a^3 x^3+a x^6}\right)}{3 a}-\frac{\log \left(a^2 x^2+a x \sqrt[3]{-b+a^3 x^3+a x^6}+\left(-b+a^3 x^3+a x^6\right)^{2/3}\right)}{6 a}","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{-b+a^3 x^3+a x^6}}\right)}{\sqrt{3} a}+\frac{\log \left(-a x+\sqrt[3]{-b+a^3 x^3+a x^6}\right)}{3 a}-\frac{\log \left(a^2 x^2+a x \sqrt[3]{-b+a^3 x^3+a x^6}+\left(-b+a^3 x^3+a x^6\right)^{2/3}\right)}{6 a}",1,"-(ArcTan[(Sqrt[3]*a*x)/(a*x + 2*(-b + a^3*x^3 + a*x^6)^(1/3))]/(Sqrt[3]*a)) + Log[a^2*x - a*(-b + a^3*x^3 + a*x^6)^(1/3)]/(3*a) - Log[a^2*x^2 + a*x*(-b + a^3*x^3 + a*x^6)^(1/3) + (-b + a^3*x^3 + a*x^6)^(2/3)]/(6*a)","A",1
1981,1,140,140,0.4905736,"\int \frac{\left(1+x^2\right) \left(1+x^8\right) \sqrt{1+x^2+x^4+x^6+x^8}}{x^7 \left(-1+x^2\right)} \, dx","IntegrateAlgebraic[((1 + x^2)*(1 + x^8)*Sqrt[1 + x^2 + x^4 + x^6 + x^8])/(x^7*(-1 + x^2)),x]","\frac{\sqrt{1+x^2+x^4+x^6+x^8} \left(8+26 x^2+65 x^4+26 x^6+8 x^8\right)}{48 x^6}+2 \sqrt{5} \tanh ^{-1}\left(\frac{\sqrt{5} x^2}{1-2 x^2+x^4-\sqrt{1+x^2+x^4+x^6+x^8}}\right)+\frac{65 \log (x)}{16}-\frac{65}{32} \log \left(-2-x^2-2 x^4+2 \sqrt{1+x^2+x^4+x^6+x^8}\right)","\frac{\sqrt{1+x^2+x^4+x^6+x^8} \left(8+26 x^2+65 x^4+26 x^6+8 x^8\right)}{48 x^6}+2 \sqrt{5} \tanh ^{-1}\left(\frac{\sqrt{5} x^2}{1-2 x^2+x^4-\sqrt{1+x^2+x^4+x^6+x^8}}\right)+\frac{65 \log (x)}{16}-\frac{65}{32} \log \left(-2-x^2-2 x^4+2 \sqrt{1+x^2+x^4+x^6+x^8}\right)",1,"(Sqrt[1 + x^2 + x^4 + x^6 + x^8]*(8 + 26*x^2 + 65*x^4 + 26*x^6 + 8*x^8))/(48*x^6) + 2*Sqrt[5]*ArcTanh[(Sqrt[5]*x^2)/(1 - 2*x^2 + x^4 - Sqrt[1 + x^2 + x^4 + x^6 + x^8])] + (65*Log[x])/16 - (65*Log[-2 - x^2 - 2*x^4 + 2*Sqrt[1 + x^2 + x^4 + x^6 + x^8]])/32","A",1
1982,1,140,140,2.6478011,"\int \frac{b+a x^8}{\sqrt[4]{-b x^2+a x^4} \left(-b+a x^8\right)} \, dx","IntegrateAlgebraic[(b + a*x^8)/((-(b*x^2) + a*x^4)^(1/4)*(-b + a*x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{1}{4} \text{RootSum}\left[a^4-a b^3-4 a^3 \text{$\#$1}^4+6 a^2 \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{1}{4} \text{RootSum}\left[a^4-a b^3-4 a^3 \text{$\#$1}^4+6 a^2 \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"ArcTan[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)]/a^(1/4) + ArcTanh[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)]/a^(1/4) + RootSum[a^4 - a*b^3 - 4*a^3*#1^4 + 6*a^2*#1^8 - 4*a*#1^12 + #1^16 & , (-Log[x] + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1])/#1 & ]/4","A",1
1983,1,140,140,0.0018301,"\int \frac{b+a x^8}{\sqrt[4]{-b x^2+a x^4} \left(-b+a x^8\right)} \, dx","IntegrateAlgebraic[(b + a*x^8)/((-(b*x^2) + a*x^4)^(1/4)*(-b + a*x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{1}{4} \text{RootSum}\left[a^4-a b^3-4 a^3 \text{$\#$1}^4+6 a^2 \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{1}{4} \text{RootSum}\left[a^4-a b^3-4 a^3 \text{$\#$1}^4+6 a^2 \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"ArcTan[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)]/a^(1/4) + ArcTanh[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)]/a^(1/4) + RootSum[a^4 - a*b^3 - 4*a^3*#1^4 + 6*a^2*#1^8 - 4*a*#1^12 + #1^16 & , (-Log[x] + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1])/#1 & ]/4","A",1
1984,1,140,140,0.1798601,"\int \frac{\left(-1+x^4\right) \sqrt{x+\sqrt{1+x^2}}}{1+x^4} \, dx","IntegrateAlgebraic[((-1 + x^4)*Sqrt[x + Sqrt[1 + x^2]])/(1 + x^4),x]","-\frac{1}{\sqrt{x+\sqrt{1+x^2}}}+\frac{1}{3} \left(x+\sqrt{1+x^2}\right)^{3/2}-2 \text{RootSum}\left[1-4 \text{$\#$1}^4+22 \text{$\#$1}^8-4 \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^3+\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^7}{-1+11 \text{$\#$1}^4-3 \text{$\#$1}^8+\text{$\#$1}^{12}}\&\right]","-\frac{1}{\sqrt{x+\sqrt{1+x^2}}}+\frac{1}{3} \left(x+\sqrt{1+x^2}\right)^{3/2}-2 \text{RootSum}\left[1-4 \text{$\#$1}^4+22 \text{$\#$1}^8-4 \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^3+\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^7}{-1+11 \text{$\#$1}^4-3 \text{$\#$1}^8+\text{$\#$1}^{12}}\&\right]",1,"-(1/Sqrt[x + Sqrt[1 + x^2]]) + (x + Sqrt[1 + x^2])^(3/2)/3 - 2*RootSum[1 - 4*#1^4 + 22*#1^8 - 4*#1^12 + #1^16 & , (Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1^3 + Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1^7)/(-1 + 11*#1^4 - 3*#1^8 + #1^12) & ]","A",1
1985,1,140,140,0.001026,"\int \frac{\left(-1+x^4\right) \sqrt{x+\sqrt{1+x^2}}}{1+x^4} \, dx","IntegrateAlgebraic[((-1 + x^4)*Sqrt[x + Sqrt[1 + x^2]])/(1 + x^4),x]","-\frac{1}{\sqrt{x+\sqrt{1+x^2}}}+\frac{1}{3} \left(x+\sqrt{1+x^2}\right)^{3/2}-2 \text{RootSum}\left[1-4 \text{$\#$1}^4+22 \text{$\#$1}^8-4 \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^3+\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^7}{-1+11 \text{$\#$1}^4-3 \text{$\#$1}^8+\text{$\#$1}^{12}}\&\right]","-\frac{1}{\sqrt{x+\sqrt{1+x^2}}}+\frac{1}{3} \left(x+\sqrt{1+x^2}\right)^{3/2}-2 \text{RootSum}\left[1-4 \text{$\#$1}^4+22 \text{$\#$1}^8-4 \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^3+\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^7}{-1+11 \text{$\#$1}^4-3 \text{$\#$1}^8+\text{$\#$1}^{12}}\&\right]",1,"-(1/Sqrt[x + Sqrt[1 + x^2]]) + (x + Sqrt[1 + x^2])^(3/2)/3 - 2*RootSum[1 - 4*#1^4 + 22*#1^8 - 4*#1^12 + #1^16 & , (Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1^3 + Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1^7)/(-1 + 11*#1^4 - 3*#1^8 + #1^12) & ]","A",1
1986,1,140,140,4.8016327,"\int \frac{x^2 \sqrt{x+x^2}}{\sqrt{x^2+x \sqrt{x+x^2}}} \, dx","IntegrateAlgebraic[(x^2*Sqrt[x + x^2])/Sqrt[x^2 + x*Sqrt[x + x^2]],x]","\frac{\sqrt{x+x^2} \left(-1575+840 x-640 x^2-3072 x^3\right) \sqrt{x \left(x+\sqrt{x+x^2}\right)}}{10752 x}+\sqrt{x \left(x+\sqrt{x+x^2}\right)} \left(\frac{525-120 x+3968 x^2+3072 x^3}{10752}+\frac{75 \sqrt{-x+\sqrt{x+x^2}} \tanh ^{-1}\left(\sqrt{2} \sqrt{-x+\sqrt{x+x^2}}\right)}{512 \sqrt{2} x}\right)","\frac{\sqrt{x+x^2} \left(-1575+840 x-640 x^2-3072 x^3\right) \sqrt{x \left(x+\sqrt{x+x^2}\right)}}{10752 x}+\sqrt{x \left(x+\sqrt{x+x^2}\right)} \left(\frac{525-120 x+3968 x^2+3072 x^3}{10752}+\frac{75 \sqrt{-x+\sqrt{x+x^2}} \tanh ^{-1}\left(\sqrt{2} \sqrt{-x+\sqrt{x+x^2}}\right)}{512 \sqrt{2} x}\right)",1,"(Sqrt[x + x^2]*(-1575 + 840*x - 640*x^2 - 3072*x^3)*Sqrt[x*(x + Sqrt[x + x^2])])/(10752*x) + Sqrt[x*(x + Sqrt[x + x^2])]*((525 - 120*x + 3968*x^2 + 3072*x^3)/10752 + (75*Sqrt[-x + Sqrt[x + x^2]]*ArcTanh[Sqrt[2]*Sqrt[-x + Sqrt[x + x^2]]])/(512*Sqrt[2]*x))","A",1
1987,1,140,140,0.2080467,"\int \frac{1}{\left(d+c x^2\right) \sqrt{a x+\sqrt{b^2+a^2 x^2}}} \, dx","IntegrateAlgebraic[1/((d + c*x^2)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]),x]","\frac{1}{2} a \text{RootSum}\left[b^4 c-2 b^2 c \text{$\#$1}^4+4 a^2 d \text{$\#$1}^4+c \text{$\#$1}^8\&,\frac{-b^2 \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-\log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{b^2 c \text{$\#$1}^3-2 a^2 d \text{$\#$1}^3-c \text{$\#$1}^7}\&\right]","\frac{1}{2} a \text{RootSum}\left[b^4 c-2 b^2 c \text{$\#$1}^4+4 a^2 d \text{$\#$1}^4+c \text{$\#$1}^8\&,\frac{-b^2 \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-\log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{b^2 c \text{$\#$1}^3-2 a^2 d \text{$\#$1}^3-c \text{$\#$1}^7}\&\right]",1,"(a*RootSum[b^4*c - 2*b^2*c*#1^4 + 4*a^2*d*#1^4 + c*#1^8 & , (-(b^2*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]) - Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]*#1^4)/(b^2*c*#1^3 - 2*a^2*d*#1^3 - c*#1^7) & ])/2","A",1
1988,1,140,140,0.0021391,"\int \frac{1}{\left(d+c x^2\right) \sqrt{a x+\sqrt{b^2+a^2 x^2}}} \, dx","IntegrateAlgebraic[1/((d + c*x^2)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]),x]","\frac{1}{2} a \text{RootSum}\left[b^4 c-2 b^2 c \text{$\#$1}^4+4 a^2 d \text{$\#$1}^4+c \text{$\#$1}^8\&,\frac{-b^2 \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-\log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{b^2 c \text{$\#$1}^3-2 a^2 d \text{$\#$1}^3-c \text{$\#$1}^7}\&\right]","\frac{1}{2} a \text{RootSum}\left[b^4 c-2 b^2 c \text{$\#$1}^4+4 a^2 d \text{$\#$1}^4+c \text{$\#$1}^8\&,\frac{-b^2 \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-\log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{b^2 c \text{$\#$1}^3-2 a^2 d \text{$\#$1}^3-c \text{$\#$1}^7}\&\right]",1,"(a*RootSum[b^4*c - 2*b^2*c*#1^4 + 4*a^2*d*#1^4 + c*#1^8 & , (-(b^2*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]) - Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]*#1^4)/(b^2*c*#1^3 - 2*a^2*d*#1^3 - c*#1^7) & ])/2","A",1
1989,1,140,140,0.2034429,"\int \frac{\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1-x^2\right) \sqrt{1+x^2}} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((1 - x^2)*Sqrt[1 + x^2]),x]","-\frac{1}{2} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{-2 \text{$\#$1}+\text{$\#$1}^3}\&\right]+\frac{1}{2} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{2-2 \text{$\#$1}^2+\text{$\#$1}^4}\&\right]","-\frac{1}{2} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{-2 \text{$\#$1}+\text{$\#$1}^3}\&\right]+\frac{1}{2} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{2-2 \text{$\#$1}^2+\text{$\#$1}^4}\&\right]",1,"-1/2*RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/(-2*#1 + #1^3) & ] + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(2 - 2*#1^2 + #1^4) & ]/2","A",1
1990,1,140,140,0.0015642,"\int \frac{\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1-x^2\right) \sqrt{1+x^2}} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((1 - x^2)*Sqrt[1 + x^2]),x]","-\frac{1}{2} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{-2 \text{$\#$1}+\text{$\#$1}^3}\&\right]+\frac{1}{2} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{2-2 \text{$\#$1}^2+\text{$\#$1}^4}\&\right]","-\frac{1}{2} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{-2 \text{$\#$1}+\text{$\#$1}^3}\&\right]+\frac{1}{2} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{2-2 \text{$\#$1}^2+\text{$\#$1}^4}\&\right]",1,"-1/2*RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/(-2*#1 + #1^3) & ] + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(2 - 2*#1^2 + #1^4) & ]/2","A",1
1991,1,140,140,3.9214412,"\int \frac{\sqrt{b+a^2 x^2}}{x^2-\sqrt[3]{a x-\sqrt{b+a^2 x^2}}} \, dx","IntegrateAlgebraic[Sqrt[b + a^2*x^2]/(x^2 - (a*x - Sqrt[b + a^2*x^2])^(1/3)),x]","-a \log \left(-a x+\sqrt{b+a^2 x^2}\right)+3 a \text{RootSum}\left[b^2-2 b \text{$\#$1}^6-4 a^2 \text{$\#$1}^7+\text{$\#$1}^{12}\&,\frac{b \log \left(\sqrt[3]{a x-\sqrt{b+a^2 x^2}}-\text{$\#$1}\right)+a^2 \log \left(\sqrt[3]{a x-\sqrt{b+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}}{3 b+7 a^2 \text{$\#$1}-3 \text{$\#$1}^6}\&\right]","-a \log \left(-a x+\sqrt{b+a^2 x^2}\right)+3 a \text{RootSum}\left[b^2-2 b \text{$\#$1}^6-4 a^2 \text{$\#$1}^7+\text{$\#$1}^{12}\&,\frac{b \log \left(\sqrt[3]{a x-\sqrt{b+a^2 x^2}}-\text{$\#$1}\right)+a^2 \log \left(\sqrt[3]{a x-\sqrt{b+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}}{3 b+7 a^2 \text{$\#$1}-3 \text{$\#$1}^6}\&\right]",1,"-(a*Log[-(a*x) + Sqrt[b + a^2*x^2]]) + 3*a*RootSum[b^2 - 2*b*#1^6 - 4*a^2*#1^7 + #1^12 & , (b*Log[(a*x - Sqrt[b + a^2*x^2])^(1/3) - #1] + a^2*Log[(a*x - Sqrt[b + a^2*x^2])^(1/3) - #1]*#1)/(3*b + 7*a^2*#1 - 3*#1^6) & ]","A",1
1992,1,140,140,1.6145829,"\int \frac{\sqrt{b+a^2 x^2}}{x^2-\sqrt{a x-\sqrt{b+a^2 x^2}}} \, dx","IntegrateAlgebraic[Sqrt[b + a^2*x^2]/(x^2 - Sqrt[a*x - Sqrt[b + a^2*x^2]]),x]","-a \log \left(-a x+\sqrt{b+a^2 x^2}\right)+2 a \text{RootSum}\left[b^2-2 b \text{$\#$1}^4-4 a^2 \text{$\#$1}^5+\text{$\#$1}^8\&,\frac{b \log \left(\sqrt{a x-\sqrt{b+a^2 x^2}}-\text{$\#$1}\right)+a^2 \log \left(\sqrt{a x-\sqrt{b+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}}{2 b+5 a^2 \text{$\#$1}-2 \text{$\#$1}^4}\&\right]","-a \log \left(-a x+\sqrt{b+a^2 x^2}\right)+2 a \text{RootSum}\left[b^2-2 b \text{$\#$1}^4-4 a^2 \text{$\#$1}^5+\text{$\#$1}^8\&,\frac{b \log \left(\sqrt{a x-\sqrt{b+a^2 x^2}}-\text{$\#$1}\right)+a^2 \log \left(\sqrt{a x-\sqrt{b+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}}{2 b+5 a^2 \text{$\#$1}-2 \text{$\#$1}^4}\&\right]",1,"-(a*Log[-(a*x) + Sqrt[b + a^2*x^2]]) + 2*a*RootSum[b^2 - 2*b*#1^4 - 4*a^2*#1^5 + #1^8 & , (b*Log[Sqrt[a*x - Sqrt[b + a^2*x^2]] - #1] + a^2*Log[Sqrt[a*x - Sqrt[b + a^2*x^2]] - #1]*#1)/(2*b + 5*a^2*#1 - 2*#1^4) & ]","A",1
1993,1,141,141,0.2450613,"\int \frac{\sqrt[3]{-6-2 x+x^2}}{-1+x} \, dx","IntegrateAlgebraic[(-6 - 2*x + x^2)^(1/3)/(-1 + x),x]","\frac{3}{2} \sqrt[3]{-6-2 x+x^2}+\frac{1}{2} \sqrt{3} \sqrt[3]{7} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-6-2 x+x^2}}{\sqrt{3} \sqrt[3]{7}}\right)-\frac{1}{2} \sqrt[3]{7} \log \left(7+7^{2/3} \sqrt[3]{-6-2 x+x^2}\right)+\frac{1}{4} \sqrt[3]{7} \log \left(-7+7^{2/3} \sqrt[3]{-6-2 x+x^2}-\sqrt[3]{7} \left(-6-2 x+x^2\right)^{2/3}\right)","\frac{3}{2} \sqrt[3]{-6-2 x+x^2}+\frac{1}{2} \sqrt{3} \sqrt[3]{7} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-6-2 x+x^2}}{\sqrt{3} \sqrt[3]{7}}\right)-\frac{1}{2} \sqrt[3]{7} \log \left(7+7^{2/3} \sqrt[3]{-6-2 x+x^2}\right)+\frac{1}{4} \sqrt[3]{7} \log \left(-7+7^{2/3} \sqrt[3]{-6-2 x+x^2}-\sqrt[3]{7} \left(-6-2 x+x^2\right)^{2/3}\right)",1,"(3*(-6 - 2*x + x^2)^(1/3))/2 + (Sqrt[3]*7^(1/3)*ArcTan[1/Sqrt[3] - (2*(-6 - 2*x + x^2)^(1/3))/(Sqrt[3]*7^(1/3))])/2 - (7^(1/3)*Log[7 + 7^(2/3)*(-6 - 2*x + x^2)^(1/3)])/2 + (7^(1/3)*Log[-7 + 7^(2/3)*(-6 - 2*x + x^2)^(1/3) - 7^(1/3)*(-6 - 2*x + x^2)^(2/3)])/4","A",1
1994,1,141,141,5.0444707,"\int \frac{-1+x}{\sqrt[3]{1-x-x^2+x^3}} \, dx","IntegrateAlgebraic[(-1 + x)/(1 - x - x^2 + x^3)^(1/3),x]","\frac{(-1+x)^{2/3} \sqrt[3]{1+x} \left(\sqrt[3]{-1+x} (1+x)^{2/3}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{1+x}}{2 \sqrt[3]{-1+x}+\sqrt[3]{1+x}}\right)}{\sqrt{3}}+\frac{2}{3} \log \left(\sqrt[3]{-1+x}-\sqrt[3]{1+x}\right)-\frac{1}{3} \log \left((-1+x)^{2/3}+\sqrt[3]{-1+x} \sqrt[3]{1+x}+(1+x)^{2/3}\right)\right)}{\sqrt[3]{(-1+x)^2 (1+x)}}","\frac{(-1+x)^{2/3} \sqrt[3]{1+x} \left(\sqrt[3]{-1+x} (1+x)^{2/3}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{1+x}}{2 \sqrt[3]{-1+x}+\sqrt[3]{1+x}}\right)}{\sqrt{3}}+\frac{2}{3} \log \left(\sqrt[3]{-1+x}-\sqrt[3]{1+x}\right)-\frac{1}{3} \log \left((-1+x)^{2/3}+\sqrt[3]{-1+x} \sqrt[3]{1+x}+(1+x)^{2/3}\right)\right)}{\sqrt[3]{(-1+x)^2 (1+x)}}",1,"((-1 + x)^(2/3)*(1 + x)^(1/3)*((-1 + x)^(1/3)*(1 + x)^(2/3) + (2*ArcTan[(Sqrt[3]*(1 + x)^(1/3))/(2*(-1 + x)^(1/3) + (1 + x)^(1/3))])/Sqrt[3] + (2*Log[(-1 + x)^(1/3) - (1 + x)^(1/3)])/3 - Log[(-1 + x)^(2/3) + (-1 + x)^(1/3)*(1 + x)^(1/3) + (1 + x)^(2/3)]/3))/((-1 + x)^2*(1 + x))^(1/3)","A",1
1995,1,152,141,5.0680326,"\int \frac{x}{\sqrt[3]{-1-x+x^2+x^3}} \, dx","IntegrateAlgebraic[x/(-1 - x + x^2 + x^3)^(1/3),x]","\frac{\sqrt[3]{-1+x} (1+x)^{2/3} \left(\frac{2 \sqrt[3]{1+x}}{\sqrt[3]{-1+x} \left(-1+\frac{1+x}{-1+x}\right)}-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1+x}}{\sqrt{3} \sqrt[3]{-1+x}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(-1+\frac{\sqrt[3]{1+x}}{\sqrt[3]{-1+x}}\right)-\frac{1}{6} \log \left(1+\frac{\sqrt[3]{1+x}}{\sqrt[3]{-1+x}}+\frac{(1+x)^{2/3}}{(-1+x)^{2/3}}\right)\right)}{\sqrt[3]{(-1+x) (1+x)^2}}","\frac{\sqrt[3]{-1+x} (1+x)^{2/3} \left((-1+x)^{2/3} \sqrt[3]{1+x}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{1+x}}{2 \sqrt[3]{-1+x}+\sqrt[3]{1+x}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(\sqrt[3]{-1+x}-\sqrt[3]{1+x}\right)-\frac{1}{6} \log \left((-1+x)^{2/3}+\sqrt[3]{-1+x} \sqrt[3]{1+x}+(1+x)^{2/3}\right)\right)}{\sqrt[3]{(-1+x) (1+x)^2}}",1,"((-1 + x)^(1/3)*(1 + x)^(2/3)*((2*(1 + x)^(1/3))/((-1 + x)^(1/3)*(-1 + (1 + x)/(-1 + x))) - ArcTan[1/Sqrt[3] + (2*(1 + x)^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))]/Sqrt[3] + Log[-1 + (1 + x)^(1/3)/(-1 + x)^(1/3)]/3 - Log[1 + (1 + x)^(1/3)/(-1 + x)^(1/3) + (1 + x)^(2/3)/(-1 + x)^(2/3)]/6))/((-1 + x)*(1 + x)^2)^(1/3)","A",1
1996,1,141,141,0.461585,"\int \frac{(3+2 x) \left(1+x+3 x^3\right)^{2/3}}{x^3 \left(1+x+x^3\right)} \, dx","IntegrateAlgebraic[((3 + 2*x)*(1 + x + 3*x^3)^(2/3))/(x^3*(1 + x + x^3)),x]","-\frac{3 \left(1+x+3 x^3\right)^{2/3}}{2 x^2}+2^{2/3} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{1+x+3 x^3}}\right)-2^{2/3} \log \left(-2 x+2^{2/3} \sqrt[3]{1+x+3 x^3}\right)+\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{1+x+3 x^3}+\sqrt[3]{2} \left(1+x+3 x^3\right)^{2/3}\right)}{\sqrt[3]{2}}","-\frac{3 \left(1+x+3 x^3\right)^{2/3}}{2 x^2}+2^{2/3} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{1+x+3 x^3}}\right)-2^{2/3} \log \left(-2 x+2^{2/3} \sqrt[3]{1+x+3 x^3}\right)+\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{1+x+3 x^3}+\sqrt[3]{2} \left(1+x+3 x^3\right)^{2/3}\right)}{\sqrt[3]{2}}",1,"(-3*(1 + x + 3*x^3)^(2/3))/(2*x^2) + 2^(2/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(1 + x + 3*x^3)^(1/3))] - 2^(2/3)*Log[-2*x + 2^(2/3)*(1 + x + 3*x^3)^(1/3)] + Log[2*x^2 + 2^(2/3)*x*(1 + x + 3*x^3)^(1/3) + 2^(1/3)*(1 + x + 3*x^3)^(2/3)]/2^(1/3)","A",1
1997,1,141,141,3.9317354,"\int \frac{x^2 \left(-3 a b^3+2 b^2 (3 a+b) x-3 b (a+b) x^2+x^4\right)}{\left(x (-a+x) (-b+x)^3\right)^{3/4} \left(a b^3-b^2 (3 a+b) x+3 b (a+b) x^2-(a+3 b+d) x^3+x^4\right)} \, dx","IntegrateAlgebraic[(x^2*(-3*a*b^3 + 2*b^2*(3*a + b)*x - 3*b*(a + b)*x^2 + x^4))/((x*(-a + x)*(-b + x)^3)^(3/4)*(a*b^3 - b^2*(3*a + b)*x + 3*b*(a + b)*x^2 - (a + 3*b + d)*x^3 + x^4)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{a b^3 x+\left(-3 a b^2-b^3\right) x^2+\left(3 a b+3 b^2\right) x^3+(-a-3 b) x^4+x^5}}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{a b^3 x+\left(-3 a b^2-b^3\right) x^2+\left(3 a b+3 b^2\right) x^3+(-a-3 b) x^4+x^5}}\right)}{d^{3/4}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{a b^3 x+\left(-3 a b^2-b^3\right) x^2+\left(3 a b+3 b^2\right) x^3+(-a-3 b) x^4+x^5}}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{a b^3 x+\left(-3 a b^2-b^3\right) x^2+\left(3 a b+3 b^2\right) x^3+(-a-3 b) x^4+x^5}}\right)}{d^{3/4}}",1,"(2*ArcTan[(d^(1/4)*x)/(a*b^3*x + (-3*a*b^2 - b^3)*x^2 + (3*a*b + 3*b^2)*x^3 + (-a - 3*b)*x^4 + x^5)^(1/4)])/d^(3/4) - (2*ArcTanh[(d^(1/4)*x)/(a*b^3*x + (-3*a*b^2 - b^3)*x^2 + (3*a*b + 3*b^2)*x^3 + (-a - 3*b)*x^4 + x^5)^(1/4)])/d^(3/4)","A",1
1998,1,141,141,0.7338876,"\int \frac{x^6}{\left(-b+a x^4\right)^{3/4} \left(b+a x^4\right)} \, dx","IntegrateAlgebraic[x^6/((-b + a*x^4)^(3/4)*(b + a*x^4)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2 a^{7/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2\ 2^{3/4} a^{7/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2 a^{7/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2\ 2^{3/4} a^{7/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2 a^{7/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2\ 2^{3/4} a^{7/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2 a^{7/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2\ 2^{3/4} a^{7/4}}",1,"-1/2*ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/a^(7/4) + ArcTan[(2^(1/4)*a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(2*2^(3/4)*a^(7/4)) + ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(2*a^(7/4)) - ArcTanh[(2^(1/4)*a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(2*2^(3/4)*a^(7/4))","A",1
1999,1,141,141,0.3957719,"\int \frac{-b+2 a x^2}{\left(-b+a x^2\right) \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[(-b + 2*a*x^2)/((-b + a*x^2)*(b*x^2 + a*x^4)^(1/4)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{\sqrt[4]{a}}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{\sqrt[4]{2} \sqrt[4]{a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{\sqrt[4]{a}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{\sqrt[4]{2} \sqrt[4]{a}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{\sqrt[4]{a}}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{\sqrt[4]{2} \sqrt[4]{a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{\sqrt[4]{a}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{\sqrt[4]{2} \sqrt[4]{a}}",1,"(2*ArcTan[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/a^(1/4) - ArcTan[(2^(1/4)*a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)]/(2^(1/4)*a^(1/4)) + (2*ArcTanh[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/a^(1/4) - ArcTanh[(2^(1/4)*a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)]/(2^(1/4)*a^(1/4))","A",1
2000,1,141,141,0.5509337,"\int \frac{-3 b+2 a x^4}{\left(-2 b+a x^4\right) \sqrt[4]{b+a x^4}} \, dx","IntegrateAlgebraic[(-3*b + 2*a*x^4)/((-2*b + a*x^4)*(b + a*x^4)^(1/4)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{\frac{3}{2}} \sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2\ 2^{3/4} \sqrt[4]{3} \sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{\frac{3}{2}} \sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2\ 2^{3/4} \sqrt[4]{3} \sqrt[4]{a}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{\frac{3}{2}} \sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2\ 2^{3/4} \sqrt[4]{3} \sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{\frac{3}{2}} \sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{2\ 2^{3/4} \sqrt[4]{3} \sqrt[4]{a}}",1,"ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(1/4) - ArcTan[((3/2)^(1/4)*a^(1/4)*x)/(b + a*x^4)^(1/4)]/(2*2^(3/4)*3^(1/4)*a^(1/4)) + ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(1/4) - ArcTanh[((3/2)^(1/4)*a^(1/4)*x)/(b + a*x^4)^(1/4)]/(2*2^(3/4)*3^(1/4)*a^(1/4))","A",1
2001,1,141,141,0.426159,"\int \frac{b+a x^4}{\sqrt[4]{-b+a x^4} \left(b+3 a x^4\right)} \, dx","IntegrateAlgebraic[(b + a*x^4)/((-b + a*x^4)^(1/4)*(b + 3*a*x^4)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{6 \sqrt[4]{a}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{3 \sqrt{2} \sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{6 \sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{3 \sqrt{2} \sqrt[4]{a}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{6 \sqrt[4]{a}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{3 \sqrt{2} \sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{6 \sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{3 \sqrt{2} \sqrt[4]{a}}",1,"ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(6*a^(1/4)) + ArcTan[(Sqrt[2]*a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(3*Sqrt[2]*a^(1/4)) + ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(6*a^(1/4)) + ArcTanh[(Sqrt[2]*a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(3*Sqrt[2]*a^(1/4))","A",1
2002,1,141,141,2.8519584,"\int \frac{\left(2+x^6\right) \left(-1-x^4+x^6\right)}{\sqrt[4]{1-x^4-x^6} \left(-1+x^6\right)^2} \, dx","IntegrateAlgebraic[((2 + x^6)*(-1 - x^4 + x^6))/((1 - x^4 - x^6)^(1/4)*(-1 + x^6)^2),x]","-\frac{x \left(1-x^4-x^6\right)^{3/4}}{2 \left(-1+x^6\right)}-\frac{5 \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{1-x^4-x^6}}{-x^2+\sqrt{1-x^4-x^6}}\right)}{4 \sqrt{2}}-\frac{5 \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{1-x^4-x^6}}{x^2+\sqrt{1-x^4-x^6}}\right)}{4 \sqrt{2}}","-\frac{x \left(1-x^4-x^6\right)^{3/4}}{2 \left(-1+x^6\right)}-\frac{5 \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{1-x^4-x^6}}{-x^2+\sqrt{1-x^4-x^6}}\right)}{4 \sqrt{2}}-\frac{5 \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{1-x^4-x^6}}{x^2+\sqrt{1-x^4-x^6}}\right)}{4 \sqrt{2}}",1,"-1/2*(x*(1 - x^4 - x^6)^(3/4))/(-1 + x^6) - (5*ArcTan[(Sqrt[2]*x*(1 - x^4 - x^6)^(1/4))/(-x^2 + Sqrt[1 - x^4 - x^6])])/(4*Sqrt[2]) - (5*ArcTanh[(Sqrt[2]*x*(1 - x^4 - x^6)^(1/4))/(x^2 + Sqrt[1 - x^4 - x^6])])/(4*Sqrt[2])","A",1
2003,1,141,141,15.6182647,"\int \frac{\left(-2 b+a x^6\right) \left(b-c x^4+a x^6\right)}{x^2 \left(b+a x^6\right)^{3/4} \left(b+c x^4+a x^6\right)} \, dx","IntegrateAlgebraic[((-2*b + a*x^6)*(b - c*x^4 + a*x^6))/(x^2*(b + a*x^6)^(3/4)*(b + c*x^4 + a*x^6)),x]","\frac{2 \sqrt[4]{b+a x^6}}{x}+\sqrt{2} \sqrt[4]{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x \sqrt[4]{b+a x^6}}{-\sqrt{c} x^2+\sqrt{b+a x^6}}\right)-\sqrt{2} \sqrt[4]{c} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{c} x^2}{\sqrt{2}}+\frac{\sqrt{b+a x^6}}{\sqrt{2} \sqrt[4]{c}}}{x \sqrt[4]{b+a x^6}}\right)","\frac{2 \sqrt[4]{b+a x^6}}{x}+\sqrt{2} \sqrt[4]{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x \sqrt[4]{b+a x^6}}{-\sqrt{c} x^2+\sqrt{b+a x^6}}\right)-\sqrt{2} \sqrt[4]{c} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{c} x^2}{\sqrt{2}}+\frac{\sqrt{b+a x^6}}{\sqrt{2} \sqrt[4]{c}}}{x \sqrt[4]{b+a x^6}}\right)",1,"(2*(b + a*x^6)^(1/4))/x + Sqrt[2]*c^(1/4)*ArcTan[(Sqrt[2]*c^(1/4)*x*(b + a*x^6)^(1/4))/(-(Sqrt[c]*x^2) + Sqrt[b + a*x^6])] - Sqrt[2]*c^(1/4)*ArcTanh[((c^(1/4)*x^2)/Sqrt[2] + Sqrt[b + a*x^6]/(Sqrt[2]*c^(1/4)))/(x*(b + a*x^6)^(1/4))]","A",1
2004,1,141,141,0.4402615,"\int \frac{\sqrt[4]{1+2 x^4} \left(-1-x^4+2 x^8\right)}{x^6 \left(2+x^4\right)} \, dx","IntegrateAlgebraic[((1 + 2*x^4)^(1/4)*(-1 - x^4 + 2*x^8))/(x^6*(2 + x^4)),x]","\frac{\sqrt[4]{1+2 x^4} \left(2+9 x^4\right)}{20 x^5}+\frac{9}{8} \sqrt[4]{\frac{3}{2}} \tan ^{-1}\left(\frac{\sqrt[4]{\frac{3}{2}} x}{\sqrt[4]{1+2 x^4}}\right)-\sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{1+2 x^4}}\right)-\frac{9}{8} \sqrt[4]{\frac{3}{2}} \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{3}{2}} x}{\sqrt[4]{1+2 x^4}}\right)+\sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{1+2 x^4}}\right)","\frac{\sqrt[4]{1+2 x^4} \left(2+9 x^4\right)}{20 x^5}+\frac{9}{8} \sqrt[4]{\frac{3}{2}} \tan ^{-1}\left(\frac{\sqrt[4]{\frac{3}{2}} x}{\sqrt[4]{1+2 x^4}}\right)-\sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{1+2 x^4}}\right)-\frac{9}{8} \sqrt[4]{\frac{3}{2}} \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{3}{2}} x}{\sqrt[4]{1+2 x^4}}\right)+\sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{1+2 x^4}}\right)",1,"((1 + 2*x^4)^(1/4)*(2 + 9*x^4))/(20*x^5) + (9*(3/2)^(1/4)*ArcTan[((3/2)^(1/4)*x)/(1 + 2*x^4)^(1/4)])/8 - 2^(1/4)*ArcTan[(2^(1/4)*x)/(1 + 2*x^4)^(1/4)] - (9*(3/2)^(1/4)*ArcTanh[((3/2)^(1/4)*x)/(1 + 2*x^4)^(1/4)])/8 + 2^(1/4)*ArcTanh[(2^(1/4)*x)/(1 + 2*x^4)^(1/4)]","A",1
2005,1,142,141,1.5559851,"\int \frac{-b^8+2 a^4 x^4}{\sqrt[4]{-b^8+a^4 x^4} \left(-b^8-c x^4+a^8 x^8\right)} \, dx","IntegrateAlgebraic[(-b^8 + 2*a^4*x^4)/((-b^8 + a^4*x^4)^(1/4)*(-b^8 - c*x^4 + a^8*x^8)),x]","\frac{1}{4} \text{RootSum}\left[-a^8+a^8 b^8-a^4 c+2 a^4 \text{$\#$1}^4+c \text{$\#$1}^4-\text{$\#$1}^8\&,\frac{a^4 \log (x)-a^4 \log \left(\sqrt[4]{-b^8+a^4 x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b^8+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-2 a^4 \text{$\#$1}-c \text{$\#$1}+2 \text{$\#$1}^5}\&\right]","\frac{1}{4} \text{RootSum}\left[-a^8+a^8 b^8-a^4 c+2 a^4 \text{$\#$1}^4+c \text{$\#$1}^4-\text{$\#$1}^8\&,\frac{-a^4 \log (x)+a^4 \log \left(\sqrt[4]{-b^8+a^4 x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{-b^8+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{2 a^4 \text{$\#$1}+c \text{$\#$1}-2 \text{$\#$1}^5}\&\right]",1,"RootSum[-a^8 + a^8*b^8 - a^4*c + 2*a^4*#1^4 + c*#1^4 - #1^8 & , (a^4*Log[x] - a^4*Log[(-b^8 + a^4*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-b^8 + a^4*x^4)^(1/4) - x*#1]*#1^4)/(-2*a^4*#1 - c*#1 + 2*#1^5) & ]/4","A",1
2006,1,141,141,0.179036,"\int \frac{-1+x^2}{\left(1+x^2\right) \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[(-1 + x^2)/((1 + x^2)*Sqrt[x + Sqrt[1 + x^2]]),x]","-\frac{1}{3 \left(x+\sqrt{1+x^2}\right)^{3/2}}+\sqrt{x+\sqrt{1+x^2}}-2 \sqrt{2} \tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x}{\sqrt{2}}+\frac{\sqrt{1+x^2}}{\sqrt{2}}}{\sqrt{x+\sqrt{1+x^2}}}\right)-2 \sqrt{2} \tanh ^{-1}\left(\frac{\frac{1}{\sqrt{2}}+\frac{x}{\sqrt{2}}+\frac{\sqrt{1+x^2}}{\sqrt{2}}}{\sqrt{x+\sqrt{1+x^2}}}\right)","-\frac{1}{3 \left(x+\sqrt{1+x^2}\right)^{3/2}}+\sqrt{x+\sqrt{1+x^2}}-2 \sqrt{2} \tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x}{\sqrt{2}}+\frac{\sqrt{1+x^2}}{\sqrt{2}}}{\sqrt{x+\sqrt{1+x^2}}}\right)-2 \sqrt{2} \tanh ^{-1}\left(\frac{\frac{1}{\sqrt{2}}+\frac{x}{\sqrt{2}}+\frac{\sqrt{1+x^2}}{\sqrt{2}}}{\sqrt{x+\sqrt{1+x^2}}}\right)",1,"-1/3*1/(x + Sqrt[1 + x^2])^(3/2) + Sqrt[x + Sqrt[1 + x^2]] - 2*Sqrt[2]*ArcTan[(-(1/Sqrt[2]) + x/Sqrt[2] + Sqrt[1 + x^2]/Sqrt[2])/Sqrt[x + Sqrt[1 + x^2]]] - 2*Sqrt[2]*ArcTanh[(1/Sqrt[2] + x/Sqrt[2] + Sqrt[1 + x^2]/Sqrt[2])/Sqrt[x + Sqrt[1 + x^2]]]","A",1
2007,1,142,142,0.3877012,"\int \frac{\left(1+x^2\right) \sqrt[3]{x+2 x^3}}{x^4 \left(-1+x^2\right)} \, dx","IntegrateAlgebraic[((1 + x^2)*(x + 2*x^3)^(1/3))/(x^4*(-1 + x^2)),x]","\frac{3 \left(1+10 x^2\right) \sqrt[3]{x+2 x^3}}{8 x^3}+3^{5/6} \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2 \sqrt[3]{x+2 x^3}}\right)+\sqrt[3]{3} \log \left(-3 x+3^{2/3} \sqrt[3]{x+2 x^3}\right)-\frac{1}{2} \sqrt[3]{3} \log \left(3 x^2+3^{2/3} x \sqrt[3]{x+2 x^3}+\sqrt[3]{3} \left(x+2 x^3\right)^{2/3}\right)","\frac{3 \left(1+10 x^2\right) \sqrt[3]{x+2 x^3}}{8 x^3}+3^{5/6} \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2 \sqrt[3]{x+2 x^3}}\right)+\sqrt[3]{3} \log \left(-3 x+3^{2/3} \sqrt[3]{x+2 x^3}\right)-\frac{1}{2} \sqrt[3]{3} \log \left(3 x^2+3^{2/3} x \sqrt[3]{x+2 x^3}+\sqrt[3]{3} \left(x+2 x^3\right)^{2/3}\right)",1,"(3*(1 + 10*x^2)*(x + 2*x^3)^(1/3))/(8*x^3) + 3^(5/6)*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*(x + 2*x^3)^(1/3))] + 3^(1/3)*Log[-3*x + 3^(2/3)*(x + 2*x^3)^(1/3)] - (3^(1/3)*Log[3*x^2 + 3^(2/3)*x*(x + 2*x^3)^(1/3) + 3^(1/3)*(x + 2*x^3)^(2/3)])/2","A",1
2008,1,142,142,0.2577162,"\int \frac{\left(-b+a^3 x^3\right) \sqrt[3]{b+a^3 x^3}}{x^5} \, dx","IntegrateAlgebraic[((-b + a^3*x^3)*(b + a^3*x^3)^(1/3))/x^5,x]","\frac{\left(b-3 a^3 x^3\right) \sqrt[3]{b+a^3 x^3}}{4 x^4}-\frac{a^4 \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{b+a^3 x^3}}\right)}{\sqrt{3}}-\frac{1}{3} a^4 \log \left(-a x+\sqrt[3]{b+a^3 x^3}\right)+\frac{1}{6} a^4 \log \left(a^2 x^2+a x \sqrt[3]{b+a^3 x^3}+\left(b+a^3 x^3\right)^{2/3}\right)","\frac{\left(b-3 a^3 x^3\right) \sqrt[3]{b+a^3 x^3}}{4 x^4}-\frac{a^4 \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{b+a^3 x^3}}\right)}{\sqrt{3}}-\frac{1}{3} a^4 \log \left(-a x+\sqrt[3]{b+a^3 x^3}\right)+\frac{1}{6} a^4 \log \left(a^2 x^2+a x \sqrt[3]{b+a^3 x^3}+\left(b+a^3 x^3\right)^{2/3}\right)",1,"((b - 3*a^3*x^3)*(b + a^3*x^3)^(1/3))/(4*x^4) - (a^4*ArcTan[(Sqrt[3]*a*x)/(a*x + 2*(b + a^3*x^3)^(1/3))])/Sqrt[3] - (a^4*Log[-(a*x) + (b + a^3*x^3)^(1/3)])/3 + (a^4*Log[a^2*x^2 + a*x*(b + a^3*x^3)^(1/3) + (b + a^3*x^3)^(2/3)])/6","A",1
2009,1,142,142,0.436885,"\int \frac{-3 b+a x^2}{\sqrt[4]{3 b-2 a x^2} \left(3 b-2 a x^2+3 x^4\right)} \, dx","IntegrateAlgebraic[(-3*b + a*x^2)/((3*b - 2*a*x^2)^(1/4)*(3*b - 2*a*x^2 + 3*x^4)),x]","-\frac{\tan ^{-1}\left(\frac{\frac{\sqrt[4]{3} x^2}{\sqrt{2}}-\frac{\sqrt{3 b-2 a x^2}}{\sqrt{2} \sqrt[4]{3}}}{x \sqrt[4]{3 b-2 a x^2}}\right)}{2 \sqrt{2} \sqrt[4]{3}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{3} x \sqrt[4]{3 b-2 a x^2}}{\sqrt{3} x^2+\sqrt{3 b-2 a x^2}}\right)}{2 \sqrt{2} \sqrt[4]{3}}","-\frac{\tan ^{-1}\left(\frac{\frac{\sqrt[4]{3} x^2}{\sqrt{2}}-\frac{\sqrt{3 b-2 a x^2}}{\sqrt{2} \sqrt[4]{3}}}{x \sqrt[4]{3 b-2 a x^2}}\right)}{2 \sqrt{2} \sqrt[4]{3}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{3} x \sqrt[4]{3 b-2 a x^2}}{\sqrt{3} x^2+\sqrt{3 b-2 a x^2}}\right)}{2 \sqrt{2} \sqrt[4]{3}}",1,"-1/2*ArcTan[((3^(1/4)*x^2)/Sqrt[2] - Sqrt[3*b - 2*a*x^2]/(Sqrt[2]*3^(1/4)))/(x*(3*b - 2*a*x^2)^(1/4))]/(Sqrt[2]*3^(1/4)) - ArcTanh[(Sqrt[2]*3^(1/4)*x*(3*b - 2*a*x^2)^(1/4))/(Sqrt[3]*x^2 + Sqrt[3*b - 2*a*x^2])]/(2*Sqrt[2]*3^(1/4))","A",1
2010,1,142,142,13.1419988,"\int \frac{\left(-4 b+a x^5\right) \left(b-c x^4+a x^5\right)}{x^2 \left(b+a x^5\right)^{3/4} \left(b+c x^4+a x^5\right)} \, dx","IntegrateAlgebraic[((-4*b + a*x^5)*(b - c*x^4 + a*x^5))/(x^2*(b + a*x^5)^(3/4)*(b + c*x^4 + a*x^5)),x]","\frac{4 \sqrt[4]{b+a x^5}}{x}+2 \sqrt{2} \sqrt[4]{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x \sqrt[4]{b+a x^5}}{-\sqrt{c} x^2+\sqrt{b+a x^5}}\right)-2 \sqrt{2} \sqrt[4]{c} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{c} x^2}{\sqrt{2}}+\frac{\sqrt{b+a x^5}}{\sqrt{2} \sqrt[4]{c}}}{x \sqrt[4]{b+a x^5}}\right)","\frac{4 \sqrt[4]{b+a x^5}}{x}+2 \sqrt{2} \sqrt[4]{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x \sqrt[4]{b+a x^5}}{-\sqrt{c} x^2+\sqrt{b+a x^5}}\right)-2 \sqrt{2} \sqrt[4]{c} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{c} x^2}{\sqrt{2}}+\frac{\sqrt{b+a x^5}}{\sqrt{2} \sqrt[4]{c}}}{x \sqrt[4]{b+a x^5}}\right)",1,"(4*(b + a*x^5)^(1/4))/x + 2*Sqrt[2]*c^(1/4)*ArcTan[(Sqrt[2]*c^(1/4)*x*(b + a*x^5)^(1/4))/(-(Sqrt[c]*x^2) + Sqrt[b + a*x^5])] - 2*Sqrt[2]*c^(1/4)*ArcTanh[((c^(1/4)*x^2)/Sqrt[2] + Sqrt[b + a*x^5]/(Sqrt[2]*c^(1/4)))/(x*(b + a*x^5)^(1/4))]","A",1
2011,1,142,142,1.4067224,"\int \frac{\left(-1+x^3\right) \left(1+x^6\right)^{2/3} \left(1-x^3+x^6\right)}{x^6 \left(1+x^3\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)*(1 + x^6)^(2/3)*(1 - x^3 + x^6))/(x^6*(1 + x^3)),x]","\frac{\left(1+x^6\right)^{2/3} \left(2-15 x^3+2 x^6\right)}{10 x^5}-2^{2/3} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2^{2/3} \sqrt[3]{1+x^6}}\right)-2^{2/3} \log \left(2 x+2^{2/3} \sqrt[3]{1+x^6}\right)+\frac{\log \left(-2 x^2+2^{2/3} x \sqrt[3]{1+x^6}-\sqrt[3]{2} \left(1+x^6\right)^{2/3}\right)}{\sqrt[3]{2}}","\frac{\left(1+x^6\right)^{2/3} \left(2-15 x^3+2 x^6\right)}{10 x^5}-2^{2/3} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2^{2/3} \sqrt[3]{1+x^6}}\right)-2^{2/3} \log \left(2 x+2^{2/3} \sqrt[3]{1+x^6}\right)+\frac{\log \left(-2 x^2+2^{2/3} x \sqrt[3]{1+x^6}-\sqrt[3]{2} \left(1+x^6\right)^{2/3}\right)}{\sqrt[3]{2}}",1,"((1 + x^6)^(2/3)*(2 - 15*x^3 + 2*x^6))/(10*x^5) - 2^(2/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2^(2/3)*(1 + x^6)^(1/3))] - 2^(2/3)*Log[2*x + 2^(2/3)*(1 + x^6)^(1/3)] + Log[-2*x^2 + 2^(2/3)*x*(1 + x^6)^(1/3) - 2^(1/3)*(1 + x^6)^(2/3)]/2^(1/3)","A",1
2012,1,142,142,0.3336476,"\int \frac{\left(-b^2+a x^2\right)^2 \sqrt{b+\sqrt{b^2+a x^2}}}{\left(b^2+a x^2\right)^2} \, dx","IntegrateAlgebraic[((-b^2 + a*x^2)^2*Sqrt[b + Sqrt[b^2 + a*x^2]])/(b^2 + a*x^2)^2,x]","\frac{x \left(11 b^2+2 a x^2\right)}{3 \sqrt{b^2+a x^2} \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{2 x \left(5 b^3+2 a b x^2\right)}{3 \left(b^2+a x^2\right) \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{5 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{\sqrt{a}}","\frac{x \left(11 b^2+2 a x^2\right)}{3 \sqrt{b^2+a x^2} \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{2 x \left(5 b^3+2 a b x^2\right)}{3 \left(b^2+a x^2\right) \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{5 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{\sqrt{a}}",1,"(x*(11*b^2 + 2*a*x^2))/(3*Sqrt[b^2 + a*x^2]*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (2*x*(5*b^3 + 2*a*b*x^2))/(3*(b^2 + a*x^2)*Sqrt[b + Sqrt[b^2 + a*x^2]]) - (5*b^(3/2)*ArcTan[(Sqrt[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/Sqrt[a]","A",1
2013,1,143,143,0.312044,"\int \frac{\left(2+x^3\right) \left(1+2 x^3\right)^{2/3}}{x^6 \left(-1+x^3\right)} \, dx","IntegrateAlgebraic[((2 + x^3)*(1 + 2*x^3)^(2/3))/(x^6*(-1 + x^3)),x]","\frac{\left(1+2 x^3\right)^{2/3} \left(4+23 x^3\right)}{10 x^5}-3 \sqrt[6]{3} \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2 \sqrt[3]{1+2 x^3}}\right)+3^{2/3} \log \left(-3 x+3^{2/3} \sqrt[3]{1+2 x^3}\right)-\frac{1}{2} 3^{2/3} \log \left(3 x^2+3^{2/3} x \sqrt[3]{1+2 x^3}+\sqrt[3]{3} \left(1+2 x^3\right)^{2/3}\right)","\frac{\left(1+2 x^3\right)^{2/3} \left(4+23 x^3\right)}{10 x^5}-3 \sqrt[6]{3} \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2 \sqrt[3]{1+2 x^3}}\right)+3^{2/3} \log \left(-3 x+3^{2/3} \sqrt[3]{1+2 x^3}\right)-\frac{1}{2} 3^{2/3} \log \left(3 x^2+3^{2/3} x \sqrt[3]{1+2 x^3}+\sqrt[3]{3} \left(1+2 x^3\right)^{2/3}\right)",1,"((1 + 2*x^3)^(2/3)*(4 + 23*x^3))/(10*x^5) - 3*3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*(1 + 2*x^3)^(1/3))] + 3^(2/3)*Log[-3*x + 3^(2/3)*(1 + 2*x^3)^(1/3)] - (3^(2/3)*Log[3*x^2 + 3^(2/3)*x*(1 + 2*x^3)^(1/3) + 3^(1/3)*(1 + 2*x^3)^(2/3)])/2","A",1
2014,1,143,143,0.2957802,"\int \frac{\left(-4+x^2\right) \sqrt[4]{2-x^2-2 x^4}}{x^2 \left(-2+x^2\right)} \, dx","IntegrateAlgebraic[((-4 + x^2)*(2 - x^2 - 2*x^4)^(1/4))/(x^2*(-2 + x^2)),x]","-\frac{2 \sqrt[4]{2-x^2-2 x^4}}{x}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{2-x^2-2 x^4}}{\sqrt{2} x^2-\sqrt{2-x^2-2 x^4}}\right)}{\sqrt[4]{2}}+\frac{\tanh ^{-1}\left(\frac{2 \sqrt[4]{2} x \sqrt[4]{2-x^2-2 x^4}}{2 x^2+\sqrt{2} \sqrt{2-x^2-2 x^4}}\right)}{\sqrt[4]{2}}","-\frac{2 \sqrt[4]{2-x^2-2 x^4}}{x}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{2-x^2-2 x^4}}{\sqrt{2} x^2-\sqrt{2-x^2-2 x^4}}\right)}{\sqrt[4]{2}}+\frac{\tanh ^{-1}\left(\frac{2 \sqrt[4]{2} x \sqrt[4]{2-x^2-2 x^4}}{2 x^2+\sqrt{2} \sqrt{2-x^2-2 x^4}}\right)}{\sqrt[4]{2}}",1,"(-2*(2 - x^2 - 2*x^4)^(1/4))/x + ArcTan[(2^(3/4)*x*(2 - x^2 - 2*x^4)^(1/4))/(Sqrt[2]*x^2 - Sqrt[2 - x^2 - 2*x^4])]/2^(1/4) + ArcTanh[(2*2^(1/4)*x*(2 - x^2 - 2*x^4)^(1/4))/(2*x^2 + Sqrt[2]*Sqrt[2 - x^2 - 2*x^4])]/2^(1/4)","A",1
2015,1,143,143,8.7431663,"\int \frac{1}{(1+x) \left(-2+2 x+x^2-x^4\right)^{3/2}} \, dx","IntegrateAlgebraic[1/((1 + x)*(-2 + 2*x + x^2 - x^4)^(3/2)),x]","-\frac{i \sqrt{-2+2 x+x^2-x^4} \left(-\frac{i \left(4-2 x-2 x^2+x^3\right)}{20 (-1+x)^2 \sqrt{2+2 x+x^2}}-\frac{1}{4} i \tanh ^{-1}\left(1+x-\sqrt{2+2 x+x^2}\right)-\frac{7 i \tanh ^{-1}\left(\frac{1}{\sqrt{5}}-\frac{x}{\sqrt{5}}+\frac{\sqrt{2+2 x+x^2}}{\sqrt{5}}\right)}{20 \sqrt{5}}\right)}{(-1+x) \sqrt{2+2 x+x^2}}","-\frac{i \sqrt{-2+2 x+x^2-x^4} \left(-\frac{i \left(4-2 x-2 x^2+x^3\right)}{20 (-1+x)^2 \sqrt{2+2 x+x^2}}-\frac{1}{4} i \tanh ^{-1}\left(1+x-\sqrt{2+2 x+x^2}\right)-\frac{7 i \tanh ^{-1}\left(\frac{1}{\sqrt{5}}-\frac{x}{\sqrt{5}}+\frac{\sqrt{2+2 x+x^2}}{\sqrt{5}}\right)}{20 \sqrt{5}}\right)}{(-1+x) \sqrt{2+2 x+x^2}}",1,"((-I)*Sqrt[-2 + 2*x + x^2 - x^4]*(((-1/20*I)*(4 - 2*x - 2*x^2 + x^3))/((-1 + x)^2*Sqrt[2 + 2*x + x^2]) - (I/4)*ArcTanh[1 + x - Sqrt[2 + 2*x + x^2]] - (((7*I)/20)*ArcTanh[1/Sqrt[5] - x/Sqrt[5] + Sqrt[2 + 2*x + x^2]/Sqrt[5]])/Sqrt[5]))/((-1 + x)*Sqrt[2 + 2*x + x^2])","A",1
2016,1,143,143,1.1100098,"\int \frac{\left(-3+2 x^4\right) \left(1+2 x^4\right)^{2/3}}{x^3 \left(2-x^3+4 x^4\right)} \, dx","IntegrateAlgebraic[((-3 + 2*x^4)*(1 + 2*x^4)^(2/3))/(x^3*(2 - x^3 + 4*x^4)),x]","\frac{3 \left(1+2 x^4\right)^{2/3}}{4 x^2}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2} \sqrt[3]{1+2 x^4}}\right)}{2\ 2^{2/3}}+\frac{\log \left(-x+\sqrt[3]{2} \sqrt[3]{1+2 x^4}\right)}{2\ 2^{2/3}}-\frac{\log \left(x^2+\sqrt[3]{2} x \sqrt[3]{1+2 x^4}+2^{2/3} \left(1+2 x^4\right)^{2/3}\right)}{4\ 2^{2/3}}","\frac{3 \left(1+2 x^4\right)^{2/3}}{4 x^2}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2} \sqrt[3]{1+2 x^4}}\right)}{2\ 2^{2/3}}+\frac{\log \left(-x+\sqrt[3]{2} \sqrt[3]{1+2 x^4}\right)}{2\ 2^{2/3}}-\frac{\log \left(x^2+\sqrt[3]{2} x \sqrt[3]{1+2 x^4}+2^{2/3} \left(1+2 x^4\right)^{2/3}\right)}{4\ 2^{2/3}}",1,"(3*(1 + 2*x^4)^(2/3))/(4*x^2) - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*2^(1/3)*(1 + 2*x^4)^(1/3))])/(2*2^(2/3)) + Log[-x + 2^(1/3)*(1 + 2*x^4)^(1/3)]/(2*2^(2/3)) - Log[x^2 + 2^(1/3)*x*(1 + 2*x^4)^(1/3) + 2^(2/3)*(1 + 2*x^4)^(2/3)]/(4*2^(2/3))","A",1
2017,1,143,143,0.4857756,"\int \frac{-b+a x^3}{x^6 \left(b+a x^3\right) \sqrt[4]{-b x+a x^4}} \, dx","IntegrateAlgebraic[(-b + a*x^3)/(x^6*(b + a*x^3)*(-(b*x) + a*x^4)^(1/4)),x]","-\frac{4 \left(3 b-10 a x^3\right) \left(-b x+a x^4\right)^{3/4}}{63 b^2 x^6}-\frac{2\ 2^{3/4} a^{7/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)}{3 b^2}-\frac{2\ 2^{3/4} a^{7/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)}{3 b^2}","-\frac{4 \left(3 b-10 a x^3\right) \left(-b x+a x^4\right)^{3/4}}{63 b^2 x^6}-\frac{2\ 2^{3/4} a^{7/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)}{3 b^2}-\frac{2\ 2^{3/4} a^{7/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)}{3 b^2}",1,"(-4*(3*b - 10*a*x^3)*(-(b*x) + a*x^4)^(3/4))/(63*b^2*x^6) - (2*2^(3/4)*a^(7/4)*ArcTan[(2^(1/4)*a^(1/4)*(-(b*x) + a*x^4)^(3/4))/(-b + a*x^3)])/(3*b^2) - (2*2^(3/4)*a^(7/4)*ArcTanh[(2^(1/4)*a^(1/4)*(-(b*x) + a*x^4)^(3/4))/(-b + a*x^3)])/(3*b^2)","A",1
2018,1,143,143,0.3434141,"\int \frac{\left(-1+x^3\right)^{2/3} \left(4+4 x^3+x^6\right)}{x^9 \left(1+x^3\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(4 + 4*x^3 + x^6))/(x^9*(1 + x^3)),x]","\frac{\left(-1+x^3\right)^{2/3} \left(-5+2 x^3-2 x^6\right)}{10 x^8}+\frac{2^{2/3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-1+x^3}}\right)}{\sqrt{3}}-\frac{1}{3} 2^{2/3} \log \left(-2 x+2^{2/3} \sqrt[3]{-1+x^3}\right)+\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-1+x^3}+\sqrt[3]{2} \left(-1+x^3\right)^{2/3}\right)}{3 \sqrt[3]{2}}","\frac{\left(-1+x^3\right)^{2/3} \left(-5+2 x^3-2 x^6\right)}{10 x^8}+\frac{2^{2/3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-1+x^3}}\right)}{\sqrt{3}}-\frac{1}{3} 2^{2/3} \log \left(-2 x+2^{2/3} \sqrt[3]{-1+x^3}\right)+\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-1+x^3}+\sqrt[3]{2} \left(-1+x^3\right)^{2/3}\right)}{3 \sqrt[3]{2}}",1,"((-1 + x^3)^(2/3)*(-5 + 2*x^3 - 2*x^6))/(10*x^8) + (2^(2/3)*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-1 + x^3)^(1/3))])/Sqrt[3] - (2^(2/3)*Log[-2*x + 2^(2/3)*(-1 + x^3)^(1/3)])/3 + Log[2*x^2 + 2^(2/3)*x*(-1 + x^3)^(1/3) + 2^(1/3)*(-1 + x^3)^(2/3)]/(3*2^(1/3))","A",1
2019,1,143,143,0.3529456,"\int \frac{\left(1+x^3\right)^{2/3} \left(-1-2 x^3+2 x^6\right)}{x^9 \left(-1+x^3\right)} \, dx","IntegrateAlgebraic[((1 + x^3)^(2/3)*(-1 - 2*x^3 + 2*x^6))/(x^9*(-1 + x^3)),x]","\frac{\left(1+x^3\right)^{2/3} \left(-5-26 x^3-41 x^6\right)}{40 x^8}+\frac{2^{2/3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{1+x^3}}\right)}{\sqrt{3}}-\frac{1}{3} 2^{2/3} \log \left(-2 x+2^{2/3} \sqrt[3]{1+x^3}\right)+\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{1+x^3}+\sqrt[3]{2} \left(1+x^3\right)^{2/3}\right)}{3 \sqrt[3]{2}}","\frac{\left(1+x^3\right)^{2/3} \left(-5-26 x^3-41 x^6\right)}{40 x^8}+\frac{2^{2/3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{1+x^3}}\right)}{\sqrt{3}}-\frac{1}{3} 2^{2/3} \log \left(-2 x+2^{2/3} \sqrt[3]{1+x^3}\right)+\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{1+x^3}+\sqrt[3]{2} \left(1+x^3\right)^{2/3}\right)}{3 \sqrt[3]{2}}",1,"((1 + x^3)^(2/3)*(-5 - 26*x^3 - 41*x^6))/(40*x^8) + (2^(2/3)*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(1 + x^3)^(1/3))])/Sqrt[3] - (2^(2/3)*Log[-2*x + 2^(2/3)*(1 + x^3)^(1/3)])/3 + Log[2*x^2 + 2^(2/3)*x*(1 + x^3)^(1/3) + 2^(1/3)*(1 + x^3)^(2/3)]/(3*2^(1/3))","A",1
2020,1,144,143,0.7795136,"\int \frac{1}{\sqrt[4]{-b^4+a^4 x^4} \left(-b^8-c x^4+a^8 x^8\right)} \, dx","IntegrateAlgebraic[1/((-b^4 + a^4*x^4)^(1/4)*(-b^8 - c*x^4 + a^8*x^8)),x]","-\frac{\text{RootSum}\left[a^4 c-2 a^4 b^4 \text{$\#$1}^4-c \text{$\#$1}^4+b^4 \text{$\#$1}^8\&,\frac{-a^4 \log (x)+a^4 \log \left(\sqrt[4]{-b^4+a^4 x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-2 a^4 b^4 \text{$\#$1}-c \text{$\#$1}+2 b^4 \text{$\#$1}^5}\&\right]}{4 b^4}","-\frac{\text{RootSum}\left[a^4 c-2 a^4 b^4 \text{$\#$1}^4-c \text{$\#$1}^4+b^4 \text{$\#$1}^8\&,\frac{a^4 \log (x)-a^4 \log \left(\sqrt[4]{-b^4+a^4 x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{-b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{2 a^4 b^4 \text{$\#$1}+c \text{$\#$1}-2 b^4 \text{$\#$1}^5}\&\right]}{4 b^4}",1,"-1/4*RootSum[a^4*c - 2*a^4*b^4*#1^4 - c*#1^4 + b^4*#1^8 & , (-(a^4*Log[x]) + a^4*Log[(-b^4 + a^4*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-b^4 + a^4*x^4)^(1/4) - x*#1]*#1^4)/(-2*a^4*b^4*#1 - c*#1 + 2*b^4*#1^5) & ]/b^4","A",1
2021,1,144,143,0.0017155,"\int \frac{1}{\sqrt[4]{-b^4+a^4 x^4} \left(-b^8-c x^4+a^8 x^8\right)} \, dx","IntegrateAlgebraic[1/((-b^4 + a^4*x^4)^(1/4)*(-b^8 - c*x^4 + a^8*x^8)),x]","-\frac{\text{RootSum}\left[a^4 c-2 a^4 b^4 \text{$\#$1}^4-c \text{$\#$1}^4+b^4 \text{$\#$1}^8\&,\frac{-a^4 \log (x)+a^4 \log \left(\sqrt[4]{-b^4+a^4 x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-2 a^4 b^4 \text{$\#$1}-c \text{$\#$1}+2 b^4 \text{$\#$1}^5}\&\right]}{4 b^4}","-\frac{\text{RootSum}\left[a^4 c-2 a^4 b^4 \text{$\#$1}^4-c \text{$\#$1}^4+b^4 \text{$\#$1}^8\&,\frac{a^4 \log (x)-a^4 \log \left(\sqrt[4]{-b^4+a^4 x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{-b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{2 a^4 b^4 \text{$\#$1}+c \text{$\#$1}-2 b^4 \text{$\#$1}^5}\&\right]}{4 b^4}",1,"-1/4*RootSum[a^4*c - 2*a^4*b^4*#1^4 - c*#1^4 + b^4*#1^8 & , (-(a^4*Log[x]) + a^4*Log[(-b^4 + a^4*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-b^4 + a^4*x^4)^(1/4) - x*#1]*#1^4)/(-2*a^4*b^4*#1 - c*#1 + 2*b^4*#1^5) & ]/b^4","A",1
2022,1,143,143,0.3938182,"\int \frac{x-\sqrt{1+x^2}}{\sqrt{1+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[(x - Sqrt[1 + x^2])/Sqrt[1 + Sqrt[1 + x^2]],x]","\frac{2 x}{3 \sqrt{1+\sqrt{1+x^2}}}-\frac{4}{3} \sqrt{1+\sqrt{1+x^2}}+\sqrt{1+x^2} \left(-\frac{2 x}{3 \sqrt{1+\sqrt{1+x^2}}}+\frac{2}{3} \sqrt{1+\sqrt{1+x^2}}\right)-2 \sqrt{2} \tan ^{-1}\left(\frac{x}{\sqrt{2} \sqrt{1+\sqrt{1+x^2}}}-\frac{\sqrt{1+\sqrt{1+x^2}}}{\sqrt{2}}\right)","\frac{2 x}{3 \sqrt{1+\sqrt{1+x^2}}}-\frac{4}{3} \sqrt{1+\sqrt{1+x^2}}+\sqrt{1+x^2} \left(-\frac{2 x}{3 \sqrt{1+\sqrt{1+x^2}}}+\frac{2}{3} \sqrt{1+\sqrt{1+x^2}}\right)-2 \sqrt{2} \tan ^{-1}\left(\frac{x}{\sqrt{2} \sqrt{1+\sqrt{1+x^2}}}-\frac{\sqrt{1+\sqrt{1+x^2}}}{\sqrt{2}}\right)",1,"(2*x)/(3*Sqrt[1 + Sqrt[1 + x^2]]) - (4*Sqrt[1 + Sqrt[1 + x^2]])/3 + Sqrt[1 + x^2]*((-2*x)/(3*Sqrt[1 + Sqrt[1 + x^2]]) + (2*Sqrt[1 + Sqrt[1 + x^2]])/3) - 2*Sqrt[2]*ArcTan[x/(Sqrt[2]*Sqrt[1 + Sqrt[1 + x^2]]) - Sqrt[1 + Sqrt[1 + x^2]]/Sqrt[2]]","A",1
2023,1,144,144,0.3349427,"\int \frac{\left(-4+x^3\right) \left(-2+x^3\right) \left(-1+x^3\right)^{2/3}}{x^9 \left(-2+3 x^3\right)} \, dx","IntegrateAlgebraic[((-4 + x^3)*(-2 + x^3)*(-1 + x^3)^(2/3))/(x^9*(-2 + 3*x^3)),x]","\frac{\left(-1+x^3\right)^{2/3} \left(5+4 x^3+16 x^6\right)}{10 x^8}+\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{2} \sqrt[3]{-1+x^3}}\right)}{2^{2/3} \sqrt{3}}+\frac{5 \log \left(x+\sqrt[3]{2} \sqrt[3]{-1+x^3}\right)}{3\ 2^{2/3}}-\frac{5 \log \left(x^2-\sqrt[3]{2} x \sqrt[3]{-1+x^3}+2^{2/3} \left(-1+x^3\right)^{2/3}\right)}{6\ 2^{2/3}}","\frac{\left(-1+x^3\right)^{2/3} \left(5+4 x^3+16 x^6\right)}{10 x^8}+\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{2} \sqrt[3]{-1+x^3}}\right)}{2^{2/3} \sqrt{3}}+\frac{5 \log \left(x+\sqrt[3]{2} \sqrt[3]{-1+x^3}\right)}{3\ 2^{2/3}}-\frac{5 \log \left(x^2-\sqrt[3]{2} x \sqrt[3]{-1+x^3}+2^{2/3} \left(-1+x^3\right)^{2/3}\right)}{6\ 2^{2/3}}",1,"((-1 + x^3)^(2/3)*(5 + 4*x^3 + 16*x^6))/(10*x^8) + (5*ArcTan[(Sqrt[3]*x)/(-x + 2*2^(1/3)*(-1 + x^3)^(1/3))])/(2^(2/3)*Sqrt[3]) + (5*Log[x + 2^(1/3)*(-1 + x^3)^(1/3)])/(3*2^(2/3)) - (5*Log[x^2 - 2^(1/3)*x*(-1 + x^3)^(1/3) + 2^(2/3)*(-1 + x^3)^(2/3)])/(6*2^(2/3))","A",1
2024,1,144,144,1.3015828,"\int \frac{1}{(-1+x) \left(-2 x^2-3 x^3+x^4\right)^{3/2}} \, dx","IntegrateAlgebraic[1/((-1 + x)*(-2*x^2 - 3*x^3 + x^4)^(3/2)),x]","\frac{\left(-136+238 x+1555 x^2-453 x^3\right) \sqrt{-2 x^2-3 x^3+x^4}}{544 x^3 \left(-2-3 x+x^2\right)}+\frac{1}{4} \tan ^{-1}\left(\frac{-\frac{x}{2}+\frac{x^2}{2}-\frac{1}{2} \sqrt{-2 x^2-3 x^3+x^4}}{x}\right)-\frac{119 \tan ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}-\frac{\sqrt{-2 x^2-3 x^3+x^4}}{\sqrt{2}}}{x}\right)}{32 \sqrt{2}}","\frac{\left(-136+238 x+1555 x^2-453 x^3\right) \sqrt{-2 x^2-3 x^3+x^4}}{544 x^3 \left(-2-3 x+x^2\right)}+\frac{1}{4} \tan ^{-1}\left(\frac{-\frac{x}{2}+\frac{x^2}{2}-\frac{1}{2} \sqrt{-2 x^2-3 x^3+x^4}}{x}\right)-\frac{119 \tan ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}-\frac{\sqrt{-2 x^2-3 x^3+x^4}}{\sqrt{2}}}{x}\right)}{32 \sqrt{2}}",1,"((-136 + 238*x + 1555*x^2 - 453*x^3)*Sqrt[-2*x^2 - 3*x^3 + x^4])/(544*x^3*(-2 - 3*x + x^2)) + ArcTan[(-1/2*x + x^2/2 - Sqrt[-2*x^2 - 3*x^3 + x^4]/2)/x]/4 - (119*ArcTan[(x^2/Sqrt[2] - Sqrt[-2*x^2 - 3*x^3 + x^4]/Sqrt[2])/x])/(32*Sqrt[2])","A",1
2025,1,144,144,0.459915,"\int \frac{1}{\left(b+2 a x^3\right) \sqrt[4]{b x+a x^4}} \, dx","IntegrateAlgebraic[1/((b + 2*a*x^3)*(b*x + a*x^4)^(1/4)),x]","\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} x \sqrt[4]{b x+a x^4}}{-\sqrt{a} x^2+\sqrt{b x+a x^4}}\right)}{3 \sqrt[4]{a} b}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{a} x^2}{\sqrt{2}}+\frac{\sqrt{b x+a x^4}}{\sqrt{2} \sqrt[4]{a}}}{x \sqrt[4]{b x+a x^4}}\right)}{3 \sqrt[4]{a} b}","\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} x \sqrt[4]{b x+a x^4}}{-\sqrt{a} x^2+\sqrt{b x+a x^4}}\right)}{3 \sqrt[4]{a} b}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{a} x^2}{\sqrt{2}}+\frac{\sqrt{b x+a x^4}}{\sqrt{2} \sqrt[4]{a}}}{x \sqrt[4]{b x+a x^4}}\right)}{3 \sqrt[4]{a} b}",1,"(Sqrt[2]*ArcTan[(Sqrt[2]*a^(1/4)*x*(b*x + a*x^4)^(1/4))/(-(Sqrt[a]*x^2) + Sqrt[b*x + a*x^4])])/(3*a^(1/4)*b) + (Sqrt[2]*ArcTanh[((a^(1/4)*x^2)/Sqrt[2] + Sqrt[b*x + a*x^4]/(Sqrt[2]*a^(1/4)))/(x*(b*x + a*x^4)^(1/4))])/(3*a^(1/4)*b)","A",1
2026,1,144,144,18.0945534,"\int \frac{\sqrt[3]{1-x^7} \left(-2+x^3+2 x^7\right) \left(3+4 x^7\right)}{x^2 \left(-1+x^7\right) \left(-4+x^3+4 x^7\right)} \, dx","IntegrateAlgebraic[((1 - x^7)^(1/3)*(-2 + x^3 + 2*x^7)*(3 + 4*x^7))/(x^2*(-1 + x^7)*(-4 + x^3 + 4*x^7)),x]","\frac{3 \sqrt[3]{1-x^7}}{2 x}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2\ 2^{2/3} \sqrt[3]{1-x^7}}\right)}{2\ 2^{2/3}}-\frac{\log \left(-x+2^{2/3} \sqrt[3]{1-x^7}\right)}{2\ 2^{2/3}}+\frac{\log \left(x^2+2^{2/3} x \sqrt[3]{1-x^7}+2 \sqrt[3]{2} \left(1-x^7\right)^{2/3}\right)}{4\ 2^{2/3}}","\frac{3 \sqrt[3]{1-x^7}}{2 x}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2\ 2^{2/3} \sqrt[3]{1-x^7}}\right)}{2\ 2^{2/3}}-\frac{\log \left(-x+2^{2/3} \sqrt[3]{1-x^7}\right)}{2\ 2^{2/3}}+\frac{\log \left(x^2+2^{2/3} x \sqrt[3]{1-x^7}+2 \sqrt[3]{2} \left(1-x^7\right)^{2/3}\right)}{4\ 2^{2/3}}",1,"(3*(1 - x^7)^(1/3))/(2*x) - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*2^(2/3)*(1 - x^7)^(1/3))])/(2*2^(2/3)) - Log[-x + 2^(2/3)*(1 - x^7)^(1/3)]/(2*2^(2/3)) + Log[x^2 + 2^(2/3)*x*(1 - x^7)^(1/3) + 2*2^(1/3)*(1 - x^7)^(2/3)]/(4*2^(2/3))","A",1
2027,1,134,144,0.96564,"\int \frac{\left(-1+x^2\right) \left(1+x^2\right)^3 \sqrt{1+2 x^2+x^4}}{\left(1+x^4\right) \left(1-x^2+x^4-x^6+x^8\right)} \, dx","IntegrateAlgebraic[((-1 + x^2)*(1 + x^2)^3*Sqrt[1 + 2*x^2 + x^4])/((1 + x^4)*(1 - x^2 + x^4 - x^6 + x^8)),x]","2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{1+2 x^2+x^4}}\right)-\sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{5}{2}-\frac{\sqrt{5}}{2}} x}{\sqrt{1+2 x^2+x^4}}\right)-\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{5}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt{1+2 x^2+x^4}}\right)","\frac{\sqrt{\left(1+x^2\right)^2} \left(2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{1+x^2}\right)+\frac{5}{2} \text{RootSum}\left[1-\text{$\#$1}^2+\text{$\#$1}^4-\text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log (x-\text{$\#$1})+\log (x-\text{$\#$1}) \text{$\#$1}^2-\log (x-\text{$\#$1}) \text{$\#$1}^4+\log (x-\text{$\#$1}) \text{$\#$1}^6}{-\text{$\#$1}+2 \text{$\#$1}^3-3 \text{$\#$1}^5+4 \text{$\#$1}^7}\&\right]\right)}{1+x^2}",1,"2*Sqrt[2]*ArcTanh[(Sqrt[2]*x)/Sqrt[1 + 2*x^2 + x^4]] - Sqrt[(5 - Sqrt[5])/2]*ArcTanh[(Sqrt[5/2 - Sqrt[5]/2]*x)/Sqrt[1 + 2*x^2 + x^4]] - Sqrt[(5 + Sqrt[5])/2]*ArcTanh[(Sqrt[5/2 + Sqrt[5]/2]*x)/Sqrt[1 + 2*x^2 + x^4]]","A",1
2028,1,112,144,0.2342282,"\int \frac{1}{x^2 \sqrt{b+\sqrt{b^2+a x^2}}} \, dx","IntegrateAlgebraic[1/(x^2*Sqrt[b + Sqrt[b^2 + a*x^2]]),x]","-\frac{1}{2 x \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{4 b x}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{4 \sqrt{2} b^{3/2}}","-\frac{1}{2 x \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{4 b x}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2} \sqrt{b}}\right)}{2 \sqrt{2} b^{3/2}}",1,"-1/2*1/(x*Sqrt[b + Sqrt[b^2 + a*x^2]]) - Sqrt[b + Sqrt[b^2 + a*x^2]]/(4*b*x) - (Sqrt[a]*ArcTan[(Sqrt[a]*x)/(Sqrt[2]*Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/(4*Sqrt[2]*b^(3/2))","A",1
2029,1,144,145,0.1913943,"\int \frac{\sqrt[4]{-b+a x^3}}{x} \, dx","IntegrateAlgebraic[(-b + a*x^3)^(1/4)/x,x]","\frac{4}{3} \sqrt[4]{-b+a x^3}-\frac{1}{3} \sqrt{2} \sqrt[4]{b} \tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^3}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^3}}\right)-\frac{1}{3} \sqrt{2} \sqrt[4]{b} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^3}}{\sqrt{b}+\sqrt{-b+a x^3}}\right)","\frac{4}{3} \sqrt[4]{-b+a x^3}+\frac{1}{3} \sqrt{2} \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^3}}{-\sqrt{b}+\sqrt{-b+a x^3}}\right)-\frac{1}{3} \sqrt{2} \sqrt[4]{b} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^3}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^3}}\right)",1,"(4*(-b + a*x^3)^(1/4))/3 - (Sqrt[2]*b^(1/4)*ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^3]/(Sqrt[2]*b^(1/4)))/(-b + a*x^3)^(1/4)])/3 - (Sqrt[2]*b^(1/4)*ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^3)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^3])])/3","A",1
2030,1,144,145,0.1782259,"\int \frac{\left(-b+a x^3\right)^{3/4}}{x} \, dx","IntegrateAlgebraic[(-b + a*x^3)^(3/4)/x,x]","\frac{4}{9} \left(-b+a x^3\right)^{3/4}-\frac{1}{3} \sqrt{2} b^{3/4} \tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^3}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^3}}\right)+\frac{1}{3} \sqrt{2} b^{3/4} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^3}}{\sqrt{b}+\sqrt{-b+a x^3}}\right)","\frac{4}{9} \left(-b+a x^3\right)^{3/4}+\frac{1}{3} \sqrt{2} b^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^3}}{-\sqrt{b}+\sqrt{-b+a x^3}}\right)+\frac{1}{3} \sqrt{2} b^{3/4} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^3}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^3}}\right)",1,"(4*(-b + a*x^3)^(3/4))/9 - (Sqrt[2]*b^(3/4)*ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^3]/(Sqrt[2]*b^(1/4)))/(-b + a*x^3)^(1/4)])/3 + (Sqrt[2]*b^(3/4)*ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^3)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^3])])/3","A",1
2031,1,145,145,3.295916,"\int \frac{\left(-1+x^4\right)^{2/3} \left(3+x^4\right) \left(-1-x^3+x^4\right)}{x^6 \left(-2-x^3+2 x^4\right)} \, dx","IntegrateAlgebraic[((-1 + x^4)^(2/3)*(3 + x^4)*(-1 - x^3 + x^4))/(x^6*(-2 - x^3 + 2*x^4)),x]","\frac{3 \left(-1+x^4\right)^{2/3} \left(-4-5 x^3+4 x^4\right)}{40 x^5}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2} \sqrt[3]{-1+x^4}}\right)}{4\ 2^{2/3}}-\frac{\log \left(-x+\sqrt[3]{2} \sqrt[3]{-1+x^4}\right)}{4\ 2^{2/3}}+\frac{\log \left(x^2+\sqrt[3]{2} x \sqrt[3]{-1+x^4}+2^{2/3} \left(-1+x^4\right)^{2/3}\right)}{8\ 2^{2/3}}","\frac{3 \left(-1+x^4\right)^{2/3} \left(-4-5 x^3+4 x^4\right)}{40 x^5}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2} \sqrt[3]{-1+x^4}}\right)}{4\ 2^{2/3}}-\frac{\log \left(-x+\sqrt[3]{2} \sqrt[3]{-1+x^4}\right)}{4\ 2^{2/3}}+\frac{\log \left(x^2+\sqrt[3]{2} x \sqrt[3]{-1+x^4}+2^{2/3} \left(-1+x^4\right)^{2/3}\right)}{8\ 2^{2/3}}",1,"(3*(-1 + x^4)^(2/3)*(-4 - 5*x^3 + 4*x^4))/(40*x^5) + (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*2^(1/3)*(-1 + x^4)^(1/3))])/(4*2^(2/3)) - Log[-x + 2^(1/3)*(-1 + x^4)^(1/3)]/(4*2^(2/3)) + Log[x^2 + 2^(1/3)*x*(-1 + x^4)^(1/3) + 2^(2/3)*(-1 + x^4)^(2/3)]/(8*2^(2/3))","A",1
2032,1,144,145,0.1698123,"\int \frac{\left(-b+a x^4\right)^{3/4}}{x} \, dx","IntegrateAlgebraic[(-b + a*x^4)^(3/4)/x,x]","\frac{1}{3} \left(-b+a x^4\right)^{3/4}-\frac{b^{3/4} \tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^4}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^4}}\right)}{2 \sqrt{2}}+\frac{b^{3/4} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^4}}{\sqrt{b}+\sqrt{-b+a x^4}}\right)}{2 \sqrt{2}}","\frac{1}{3} \left(-b+a x^4\right)^{3/4}+\frac{b^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^4}}{-\sqrt{b}+\sqrt{-b+a x^4}}\right)}{2 \sqrt{2}}+\frac{b^{3/4} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^4}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^4}}\right)}{2 \sqrt{2}}",1,"(-b + a*x^4)^(3/4)/3 - (b^(3/4)*ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^4]/(Sqrt[2]*b^(1/4)))/(-b + a*x^4)^(1/4)])/(2*Sqrt[2]) + (b^(3/4)*ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^4)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^4])])/(2*Sqrt[2])","A",1
2033,1,145,145,0.4858603,"\int \frac{-3 a b^3+2 b^2 (3 a+b) x-3 b (a+b) x^2+x^4}{\sqrt[4]{x (-a+x) (-b+x)^3} \left(a b^3 d-b^2 (3 a+b) d x+3 b (a+b) d x^2-(1+a d+3 b d) x^3+d x^4\right)} \, dx","IntegrateAlgebraic[(-3*a*b^3 + 2*b^2*(3*a + b)*x - 3*b*(a + b)*x^2 + x^4)/((x*(-a + x)*(-b + x)^3)^(1/4)*(a*b^3*d - b^2*(3*a + b)*d*x + 3*b*(a + b)*d*x^2 - (1 + a*d + 3*b*d)*x^3 + d*x^4)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b^3 x+\left(-3 a b^2-b^3\right) x^2+\left(3 a b+3 b^2\right) x^3+(-a-3 b) x^4+x^5}}{x}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b^3 x+\left(-3 a b^2-b^3\right) x^2+\left(3 a b+3 b^2\right) x^3+(-a-3 b) x^4+x^5}}{x}\right)}{d^{3/4}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b^3 x+\left(-3 a b^2-b^3\right) x^2+\left(3 a b+3 b^2\right) x^3+(-a-3 b) x^4+x^5}}{x}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b^3 x+\left(-3 a b^2-b^3\right) x^2+\left(3 a b+3 b^2\right) x^3+(-a-3 b) x^4+x^5}}{x}\right)}{d^{3/4}}",1,"(2*ArcTan[(d^(1/4)*(a*b^3*x + (-3*a*b^2 - b^3)*x^2 + (3*a*b + 3*b^2)*x^3 + (-a - 3*b)*x^4 + x^5)^(1/4))/x])/d^(3/4) - (2*ArcTanh[(d^(1/4)*(a*b^3*x + (-3*a*b^2 - b^3)*x^2 + (3*a*b + 3*b^2)*x^3 + (-a - 3*b)*x^4 + x^5)^(1/4))/x])/d^(3/4)","A",1
2034,1,144,145,0.1756865,"\int \frac{\left(-b+a x^5\right)^{3/4}}{x} \, dx","IntegrateAlgebraic[(-b + a*x^5)^(3/4)/x,x]","\frac{4}{15} \left(-b+a x^5\right)^{3/4}-\frac{1}{5} \sqrt{2} b^{3/4} \tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^5}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^5}}\right)+\frac{1}{5} \sqrt{2} b^{3/4} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^5}}{\sqrt{b}+\sqrt{-b+a x^5}}\right)","\frac{4}{15} \left(-b+a x^5\right)^{3/4}+\frac{1}{5} \sqrt{2} b^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^5}}{-\sqrt{b}+\sqrt{-b+a x^5}}\right)+\frac{1}{5} \sqrt{2} b^{3/4} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^5}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^5}}\right)",1,"(4*(-b + a*x^5)^(3/4))/15 - (Sqrt[2]*b^(3/4)*ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^5]/(Sqrt[2]*b^(1/4)))/(-b + a*x^5)^(1/4)])/5 + (Sqrt[2]*b^(3/4)*ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^5)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^5])])/5","A",1
2035,1,145,145,0.3360825,"\int \frac{\left(-1+x^3\right)^{2/3} \left(4+x^3+x^6\right)}{x^9 \left(-2+x^3\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(4 + x^3 + x^6))/(x^9*(-2 + x^3)),x]","\frac{\left(-1+x^3\right)^{2/3} \left(10+8 x^3+7 x^6\right)}{40 x^8}-\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2} \sqrt[3]{-1+x^3}}\right)}{4\ 2^{2/3} \sqrt{3}}+\frac{5 \log \left(-x+\sqrt[3]{2} \sqrt[3]{-1+x^3}\right)}{12\ 2^{2/3}}-\frac{5 \log \left(x^2+\sqrt[3]{2} x \sqrt[3]{-1+x^3}+2^{2/3} \left(-1+x^3\right)^{2/3}\right)}{24\ 2^{2/3}}","\frac{\left(-1+x^3\right)^{2/3} \left(10+8 x^3+7 x^6\right)}{40 x^8}-\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2} \sqrt[3]{-1+x^3}}\right)}{4\ 2^{2/3} \sqrt{3}}+\frac{5 \log \left(-x+\sqrt[3]{2} \sqrt[3]{-1+x^3}\right)}{12\ 2^{2/3}}-\frac{5 \log \left(x^2+\sqrt[3]{2} x \sqrt[3]{-1+x^3}+2^{2/3} \left(-1+x^3\right)^{2/3}\right)}{24\ 2^{2/3}}",1,"((-1 + x^3)^(2/3)*(10 + 8*x^3 + 7*x^6))/(40*x^8) - (5*ArcTan[(Sqrt[3]*x)/(x + 2*2^(1/3)*(-1 + x^3)^(1/3))])/(4*2^(2/3)*Sqrt[3]) + (5*Log[-x + 2^(1/3)*(-1 + x^3)^(1/3)])/(12*2^(2/3)) - (5*Log[x^2 + 2^(1/3)*x*(-1 + x^3)^(1/3) + 2^(2/3)*(-1 + x^3)^(2/3)])/(24*2^(2/3))","A",1
2036,1,145,145,0.5259154,"\int \frac{-3-4 x+3 x^6}{\left(1+2 x+x^6\right) \sqrt[3]{1+2 x+2 x^3+x^6}} \, dx","IntegrateAlgebraic[(-3 - 4*x + 3*x^6)/((1 + 2*x + x^6)*(1 + 2*x + 2*x^3 + x^6)^(1/3)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{1+2 x+2 x^3+x^6}}\right)}{\sqrt[3]{2}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{1+2 x+2 x^3+x^6}\right)}{\sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{1+2 x+2 x^3+x^6}+\sqrt[3]{2} \left(1+2 x+2 x^3+x^6\right)^{2/3}\right)}{2 \sqrt[3]{2}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{1+2 x+2 x^3+x^6}}\right)}{\sqrt[3]{2}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{1+2 x+2 x^3+x^6}\right)}{\sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{1+2 x+2 x^3+x^6}+\sqrt[3]{2} \left(1+2 x+2 x^3+x^6\right)^{2/3}\right)}{2 \sqrt[3]{2}}",1,"-((Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(1 + 2*x + 2*x^3 + x^6)^(1/3))])/2^(1/3)) + Log[-2*x + 2^(2/3)*(1 + 2*x + 2*x^3 + x^6)^(1/3)]/2^(1/3) - Log[2*x^2 + 2^(2/3)*x*(1 + 2*x + 2*x^3 + x^6)^(1/3) + 2^(1/3)*(1 + 2*x + 2*x^3 + x^6)^(2/3)]/(2*2^(1/3))","A",1
2037,1,144,145,0.1720932,"\int \frac{\left(-b+a x^6\right)^{3/4}}{x} \, dx","IntegrateAlgebraic[(-b + a*x^6)^(3/4)/x,x]","\frac{2}{9} \left(-b+a x^6\right)^{3/4}-\frac{b^{3/4} \tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^6}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^6}}\right)}{3 \sqrt{2}}+\frac{b^{3/4} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^6}}{\sqrt{b}+\sqrt{-b+a x^6}}\right)}{3 \sqrt{2}}","\frac{2}{9} \left(-b+a x^6\right)^{3/4}+\frac{b^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^6}}{-\sqrt{b}+\sqrt{-b+a x^6}}\right)}{3 \sqrt{2}}+\frac{b^{3/4} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^6}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^6}}\right)}{3 \sqrt{2}}",1,"(2*(-b + a*x^6)^(3/4))/9 - (b^(3/4)*ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^6]/(Sqrt[2]*b^(1/4)))/(-b + a*x^6)^(1/4)])/(3*Sqrt[2]) + (b^(3/4)*ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^6)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^6])])/(3*Sqrt[2])","A",1
2038,1,145,145,0.4705507,"\int \frac{b+a x^6}{\sqrt[3]{-x+x^3}} \, dx","IntegrateAlgebraic[(b + a*x^6)/(-x + x^3)^(1/3),x]","\frac{1}{108} \left(-x+x^3\right)^{2/3} \left(28 a+21 a x^2+18 a x^4\right)+\frac{1}{162} \left(14 \sqrt{3} a+81 \sqrt{3} b\right) \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)+\frac{1}{162} (-14 a-81 b) \log \left(-x+\sqrt[3]{-x+x^3}\right)+\frac{1}{324} (14 a+81 b) \log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)","\frac{1}{108} \left(-x+x^3\right)^{2/3} \left(28 a+21 a x^2+18 a x^4\right)+\frac{1}{162} \left(14 \sqrt{3} a+81 \sqrt{3} b\right) \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)+\frac{1}{162} (-14 a-81 b) \log \left(-x+\sqrt[3]{-x+x^3}\right)+\frac{1}{324} (14 a+81 b) \log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)",1,"((-x + x^3)^(2/3)*(28*a + 21*a*x^2 + 18*a*x^4))/108 + ((14*Sqrt[3]*a + 81*Sqrt[3]*b)*ArcTan[(Sqrt[3]*x)/(x + 2*(-x + x^3)^(1/3))])/162 + ((-14*a - 81*b)*Log[-x + (-x + x^3)^(1/3)])/162 + ((14*a + 81*b)*Log[x^2 + x*(-x + x^3)^(1/3) + (-x + x^3)^(2/3)])/324","A",1
2039,1,206,145,3.7079754,"\int \frac{\sqrt[4]{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{\sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}} \, dx","IntegrateAlgebraic[(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])^(1/4)/Sqrt[-(a/b^2) + (a^2*x^2)/b^2],x]","\frac{2 b \sqrt[4]{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{a}-\frac{b \tan ^{-1}\left(\sqrt[4]{2} \sqrt[4]{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}\right)}{\sqrt[4]{2} a}+\frac{b \log \left(-1+\sqrt[4]{2} \sqrt[4]{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}\right)}{2 \sqrt[4]{2} a}-\frac{b \log \left(a+\sqrt[4]{2} a \sqrt[4]{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}\right)}{2 \sqrt[4]{2} a}","\frac{2 b \sqrt[4]{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{a}-\frac{b \tan ^{-1}\left(\sqrt[4]{2} \sqrt[4]{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}\right)}{\sqrt[4]{2} a}-\frac{b \tanh ^{-1}\left(\sqrt[4]{2} \sqrt[4]{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}\right)}{\sqrt[4]{2} a}",1,"(2*b*(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])^(1/4))/a - (b*ArcTan[2^(1/4)*(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])^(1/4)])/(2^(1/4)*a) + (b*Log[-1 + 2^(1/4)*(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])^(1/4)])/(2*2^(1/4)*a) - (b*Log[a + 2^(1/4)*a*(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])^(1/4)])/(2*2^(1/4)*a)","A",1
2040,1,146,146,0.3762315,"\int \frac{1}{\sqrt[3]{x^2 (-a+x)} (a+(-1+d) x)} \, dx","IntegrateAlgebraic[1/((x^2*(-a + x))^(1/3)*(a + (-1 + d)*x)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} x}{\sqrt[3]{d} x+2 \sqrt[3]{-a x^2+x^3}}\right)}{a \sqrt[3]{d}}-\frac{\log \left(-\sqrt[3]{d} x+\sqrt[3]{-a x^2+x^3}\right)}{a \sqrt[3]{d}}+\frac{\log \left(d^{2/3} x^2+\sqrt[3]{d} x \sqrt[3]{-a x^2+x^3}+\left(-a x^2+x^3\right)^{2/3}\right)}{2 a \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} x}{\sqrt[3]{d} x+2 \sqrt[3]{-a x^2+x^3}}\right)}{a \sqrt[3]{d}}-\frac{\log \left(-\sqrt[3]{d} x+\sqrt[3]{-a x^2+x^3}\right)}{a \sqrt[3]{d}}+\frac{\log \left(d^{2/3} x^2+\sqrt[3]{d} x \sqrt[3]{-a x^2+x^3}+\left(-a x^2+x^3\right)^{2/3}\right)}{2 a \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*x)/(d^(1/3)*x + 2*(-(a*x^2) + x^3)^(1/3))])/(a*d^(1/3)) - Log[-(d^(1/3)*x) + (-(a*x^2) + x^3)^(1/3)]/(a*d^(1/3)) + Log[d^(2/3)*x^2 + d^(1/3)*x*(-(a*x^2) + x^3)^(1/3) + (-(a*x^2) + x^3)^(2/3)]/(2*a*d^(1/3))","A",1
2041,1,146,146,0.2127307,"\int \frac{\left(2+x-x^3-x^4\right)^{2/3} \left(6+2 x+x^4\right) \left(-2-x+x^3+x^4\right)}{x^6 \left(-2-x+2 x^3+x^4\right)} \, dx","IntegrateAlgebraic[((2 + x - x^3 - x^4)^(2/3)*(6 + 2*x + x^4)*(-2 - x + x^3 + x^4))/(x^6*(-2 - x + 2*x^3 + x^4)),x]","\frac{3 \left(2+x-x^3-x^4\right)^{2/3} \left(-4-2 x-3 x^3+2 x^4\right)}{10 x^5}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2+x-x^3-x^4}}\right)-\log \left(-x+\sqrt[3]{2+x-x^3-x^4}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{2+x-x^3-x^4}+\left(2+x-x^3-x^4\right)^{2/3}\right)","\frac{3 \left(2+x-x^3-x^4\right)^{2/3} \left(-4-2 x-3 x^3+2 x^4\right)}{10 x^5}+\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2+x-x^3-x^4}}\right)-\log \left(-x+\sqrt[3]{2+x-x^3-x^4}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{2+x-x^3-x^4}+\left(2+x-x^3-x^4\right)^{2/3}\right)",1,"(3*(2 + x - x^3 - x^4)^(2/3)*(-4 - 2*x - 3*x^3 + 2*x^4))/(10*x^5) + Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(2 + x - x^3 - x^4)^(1/3))] - Log[-x + (2 + x - x^3 - x^4)^(1/3)] + Log[x^2 + x*(2 + x - x^3 - x^4)^(1/3) + (2 + x - x^3 - x^4)^(2/3)]/2","A",1
2042,1,146,146,3.3620878,"\int \frac{\left(-3+x^4\right) \left(1+x^4\right)^{2/3} \left(2+x^3+2 x^4\right)}{x^6 \left(4-x^3+4 x^4\right)} \, dx","IntegrateAlgebraic[((-3 + x^4)*(1 + x^4)^(2/3)*(2 + x^3 + 2*x^4))/(x^6*(4 - x^3 + 4*x^4)),x]","\frac{3 \left(1+x^4\right)^{2/3} \left(8+15 x^3+8 x^4\right)}{80 x^5}-\frac{3 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2\ 2^{2/3} \sqrt[3]{1+x^4}}\right)}{16 \sqrt[3]{2}}+\frac{3 \log \left(-x+2^{2/3} \sqrt[3]{1+x^4}\right)}{16 \sqrt[3]{2}}-\frac{3 \log \left(x^2+2^{2/3} x \sqrt[3]{1+x^4}+2 \sqrt[3]{2} \left(1+x^4\right)^{2/3}\right)}{32 \sqrt[3]{2}}","\frac{3 \left(1+x^4\right)^{2/3} \left(8+15 x^3+8 x^4\right)}{80 x^5}-\frac{3 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2\ 2^{2/3} \sqrt[3]{1+x^4}}\right)}{16 \sqrt[3]{2}}+\frac{3 \log \left(-x+2^{2/3} \sqrt[3]{1+x^4}\right)}{16 \sqrt[3]{2}}-\frac{3 \log \left(x^2+2^{2/3} x \sqrt[3]{1+x^4}+2 \sqrt[3]{2} \left(1+x^4\right)^{2/3}\right)}{32 \sqrt[3]{2}}",1,"(3*(1 + x^4)^(2/3)*(8 + 15*x^3 + 8*x^4))/(80*x^5) - (3*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*2^(2/3)*(1 + x^4)^(1/3))])/(16*2^(1/3)) + (3*Log[-x + 2^(2/3)*(1 + x^4)^(1/3)])/(16*2^(1/3)) - (3*Log[x^2 + 2^(2/3)*x*(1 + x^4)^(1/3) + 2*2^(1/3)*(1 + x^4)^(2/3)])/(32*2^(1/3))","A",1
2043,1,146,146,4.9220645,"\int \frac{\left(-1-x^4+2 x^6\right) \sqrt[3]{x-x^5+x^7}}{\left(1+x^2-x^4+x^6\right)^2} \, dx","IntegrateAlgebraic[((-1 - x^4 + 2*x^6)*(x - x^5 + x^7)^(1/3))/(1 + x^2 - x^4 + x^6)^2,x]","-\frac{x \sqrt[3]{x-x^5+x^7}}{2 \left(1+x^2-x^4+x^6\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{x-x^5+x^7}}{-2 x+\sqrt[3]{x-x^5+x^7}}\right)}{2 \sqrt{3}}+\frac{1}{6} \log \left(x+\sqrt[3]{x-x^5+x^7}\right)-\frac{1}{12} \log \left(x^2-x \sqrt[3]{x-x^5+x^7}+\left(x-x^5+x^7\right)^{2/3}\right)","-\frac{x \sqrt[3]{x-x^5+x^7}}{2 \left(1+x^2-x^4+x^6\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{x-x^5+x^7}}{-2 x+\sqrt[3]{x-x^5+x^7}}\right)}{2 \sqrt{3}}+\frac{1}{6} \log \left(x+\sqrt[3]{x-x^5+x^7}\right)-\frac{1}{12} \log \left(x^2-x \sqrt[3]{x-x^5+x^7}+\left(x-x^5+x^7\right)^{2/3}\right)",1,"-1/2*(x*(x - x^5 + x^7)^(1/3))/(1 + x^2 - x^4 + x^6) - ArcTan[(Sqrt[3]*(x - x^5 + x^7)^(1/3))/(-2*x + (x - x^5 + x^7)^(1/3))]/(2*Sqrt[3]) + Log[x + (x - x^5 + x^7)^(1/3)]/6 - Log[x^2 - x*(x - x^5 + x^7)^(1/3) + (x - x^5 + x^7)^(2/3)]/12","A",1
2044,1,146,146,1.772136,"\int \frac{x^4 \left(-4 b+a x^3\right)}{\sqrt[4]{-b+a x^3} \left(-b^2+2 a b x^3-a^2 x^6+x^8\right)} \, dx","IntegrateAlgebraic[(x^4*(-4*b + a*x^3))/((-b + a*x^3)^(1/4)*(-b^2 + 2*a*b*x^3 - a^2*x^6 + x^8)),x]","\tan ^{-1}\left(\frac{\sqrt[4]{-b+a x^3}}{x}\right)-\frac{\tan ^{-1}\left(\frac{-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{-b+a x^3}}{\sqrt{2}}}{x \sqrt[4]{-b+a x^3}}\right)}{\sqrt{2}}+\tanh ^{-1}\left(\frac{x \left(-b+a x^3\right)^{3/4}}{b-a x^3}\right)+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-b+a x^3}}{x^2+\sqrt{-b+a x^3}}\right)}{\sqrt{2}}","\tan ^{-1}\left(\frac{\sqrt[4]{-b+a x^3}}{x}\right)-\frac{\tan ^{-1}\left(\frac{-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{-b+a x^3}}{\sqrt{2}}}{x \sqrt[4]{-b+a x^3}}\right)}{\sqrt{2}}+\tanh ^{-1}\left(\frac{x \left(-b+a x^3\right)^{3/4}}{b-a x^3}\right)+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-b+a x^3}}{x^2+\sqrt{-b+a x^3}}\right)}{\sqrt{2}}",1,"ArcTan[(-b + a*x^3)^(1/4)/x] - ArcTan[(-(x^2/Sqrt[2]) + Sqrt[-b + a*x^3]/Sqrt[2])/(x*(-b + a*x^3)^(1/4))]/Sqrt[2] + ArcTanh[(x*(-b + a*x^3)^(3/4))/(b - a*x^3)] + ArcTanh[(Sqrt[2]*x*(-b + a*x^3)^(1/4))/(x^2 + Sqrt[-b + a*x^3])]/Sqrt[2]","A",1
2045,1,146,146,2.0728605,"\int \frac{\sqrt{-1+x^2} \sqrt{x^2+x \sqrt{-1+x^2}}}{1+x^2} \, dx","IntegrateAlgebraic[(Sqrt[-1 + x^2]*Sqrt[x^2 + x*Sqrt[-1 + x^2]])/(1 + x^2),x]","\sqrt{x^2+x \sqrt{-1+x^2}}-\sqrt{2 \left(1+\sqrt{2}\right)} \tan ^{-1}\left(\sqrt{-1+\sqrt{2}} \sqrt{x^2+x \sqrt{-1+x^2}}\right)-\frac{\tanh ^{-1}\left(\sqrt{2} \sqrt{x^2+x \sqrt{-1+x^2}}\right)}{\sqrt{2}}+\sqrt{2 \left(-1+\sqrt{2}\right)} \tanh ^{-1}\left(\sqrt{1+\sqrt{2}} \sqrt{x^2+x \sqrt{-1+x^2}}\right)","\sqrt{x^2+x \sqrt{-1+x^2}}-\sqrt{2 \left(1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{x^2+x \sqrt{-1+x^2}}}{\sqrt{1+\sqrt{2}}}\right)-\frac{\tanh ^{-1}\left(\sqrt{2} \sqrt{x^2+x \sqrt{-1+x^2}}\right)}{\sqrt{2}}+\sqrt{2 \left(-1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{x^2+x \sqrt{-1+x^2}}}{\sqrt{-1+\sqrt{2}}}\right)",1,"Sqrt[x^2 + x*Sqrt[-1 + x^2]] - Sqrt[2*(1 + Sqrt[2])]*ArcTan[Sqrt[-1 + Sqrt[2]]*Sqrt[x^2 + x*Sqrt[-1 + x^2]]] - ArcTanh[Sqrt[2]*Sqrt[x^2 + x*Sqrt[-1 + x^2]]]/Sqrt[2] + Sqrt[2*(-1 + Sqrt[2])]*ArcTanh[Sqrt[1 + Sqrt[2]]*Sqrt[x^2 + x*Sqrt[-1 + x^2]]]","A",1
2046,1,147,147,0.3812952,"\int \frac{x}{\left(x^2 (-a+x)\right)^{2/3} (a+(-1+d) x)} \, dx","IntegrateAlgebraic[x/((x^2*(-a + x))^(2/3)*(a + (-1 + d)*x)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} x}{\sqrt[3]{d} x+2 \sqrt[3]{-a x^2+x^3}}\right)}{a d^{2/3}}-\frac{\log \left(-\sqrt[3]{d} x+\sqrt[3]{-a x^2+x^3}\right)}{a d^{2/3}}+\frac{\log \left(d^{2/3} x^2+\sqrt[3]{d} x \sqrt[3]{-a x^2+x^3}+\left(-a x^2+x^3\right)^{2/3}\right)}{2 a d^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} x}{\sqrt[3]{d} x+2 \sqrt[3]{-a x^2+x^3}}\right)}{a d^{2/3}}-\frac{\log \left(-\sqrt[3]{d} x+\sqrt[3]{-a x^2+x^3}\right)}{a d^{2/3}}+\frac{\log \left(d^{2/3} x^2+\sqrt[3]{d} x \sqrt[3]{-a x^2+x^3}+\left(-a x^2+x^3\right)^{2/3}\right)}{2 a d^{2/3}}",1,"-((Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*x)/(d^(1/3)*x + 2*(-(a*x^2) + x^3)^(1/3))])/(a*d^(2/3))) - Log[-(d^(1/3)*x) + (-(a*x^2) + x^3)^(1/3)]/(a*d^(2/3)) + Log[d^(2/3)*x^2 + d^(1/3)*x*(-(a*x^2) + x^3)^(1/3) + (-(a*x^2) + x^3)^(2/3)]/(2*a*d^(2/3))","A",1
2047,1,147,147,0.3505322,"\int \frac{2+x+x^2}{\left(3+2 x+x^2\right) \sqrt[3]{x^2+x^3}} \, dx","IntegrateAlgebraic[(2 + x + x^2)/((3 + 2*x + x^2)*(x^2 + x^3)^(1/3)),x]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{x^2+x^3}+\left(x^2+x^3\right)^{2/3}\right)+\frac{1}{2} \text{RootSum}\left[2-4 \text{$\#$1}^3+3 \text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-2+3 \text{$\#$1}^3}\&\right]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{x^2+x^3}+\left(x^2+x^3\right)^{2/3}\right)+\frac{1}{2} \text{RootSum}\left[2-4 \text{$\#$1}^3+3 \text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-2+3 \text{$\#$1}^3}\&\right]",1,"Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(x^2 + x^3)^(1/3))] - Log[-x + (x^2 + x^3)^(1/3)] + Log[x^2 + x*(x^2 + x^3)^(1/3) + (x^2 + x^3)^(2/3)]/2 + RootSum[2 - 4*#1^3 + 3*#1^6 & , (-(Log[x]*#1^2) + Log[(x^2 + x^3)^(1/3) - x*#1]*#1^2)/(-2 + 3*#1^3) & ]/2","A",1
2048,1,147,147,0.0011272,"\int \frac{2+x+x^2}{\left(3+2 x+x^2\right) \sqrt[3]{x^2+x^3}} \, dx","IntegrateAlgebraic[(2 + x + x^2)/((3 + 2*x + x^2)*(x^2 + x^3)^(1/3)),x]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{x^2+x^3}+\left(x^2+x^3\right)^{2/3}\right)+\frac{1}{2} \text{RootSum}\left[2-4 \text{$\#$1}^3+3 \text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-2+3 \text{$\#$1}^3}\&\right]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{x^2+x^3}+\left(x^2+x^3\right)^{2/3}\right)+\frac{1}{2} \text{RootSum}\left[2-4 \text{$\#$1}^3+3 \text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-2+3 \text{$\#$1}^3}\&\right]",1,"Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(x^2 + x^3)^(1/3))] - Log[-x + (x^2 + x^3)^(1/3)] + Log[x^2 + x*(x^2 + x^3)^(1/3) + (x^2 + x^3)^(2/3)]/2 + RootSum[2 - 4*#1^3 + 3*#1^6 & , (-(Log[x]*#1^2) + Log[(x^2 + x^3)^(1/3) - x*#1]*#1^2)/(-2 + 3*#1^3) & ]/2","A",1
2049,1,147,147,0.4257336,"\int \frac{b-c x+a x^2}{\left(-b+a x^2\right) \sqrt{b x+a x^3}} \, dx","IntegrateAlgebraic[(b - c*x + a*x^2)/((-b + a*x^2)*Sqrt[b*x + a*x^3]),x]","-\frac{\left(2 \sqrt{a} \sqrt{b}+c\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)}{2 \sqrt{2} a^{3/4} b^{3/4}}-\frac{\left(2 \sqrt{a} \sqrt{b}-c\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)}{2 \sqrt{2} a^{3/4} b^{3/4}}","-\frac{\left(2 \sqrt{a} \sqrt{b}+c\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)}{2 \sqrt{2} a^{3/4} b^{3/4}}-\frac{\left(2 \sqrt{a} \sqrt{b}-c\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)}{2 \sqrt{2} a^{3/4} b^{3/4}}",1,"-1/2*((2*Sqrt[a]*Sqrt[b] + c)*ArcTan[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[b*x + a*x^3])/(b + a*x^2)])/(Sqrt[2]*a^(3/4)*b^(3/4)) - ((2*Sqrt[a]*Sqrt[b] - c)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[b*x + a*x^3])/(b + a*x^2)])/(2*Sqrt[2]*a^(3/4)*b^(3/4))","A",1
2050,1,147,147,0.4128187,"\int \frac{b+c x+a x^2}{\left(-b+a x^2\right) \sqrt{b x+a x^3}} \, dx","IntegrateAlgebraic[(b + c*x + a*x^2)/((-b + a*x^2)*Sqrt[b*x + a*x^3]),x]","-\frac{\left(2 \sqrt{a} \sqrt{b}-c\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)}{2 \sqrt{2} a^{3/4} b^{3/4}}-\frac{\left(2 \sqrt{a} \sqrt{b}+c\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)}{2 \sqrt{2} a^{3/4} b^{3/4}}","-\frac{\left(2 \sqrt{a} \sqrt{b}-c\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)}{2 \sqrt{2} a^{3/4} b^{3/4}}-\frac{\left(2 \sqrt{a} \sqrt{b}+c\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)}{2 \sqrt{2} a^{3/4} b^{3/4}}",1,"-1/2*((2*Sqrt[a]*Sqrt[b] - c)*ArcTan[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[b*x + a*x^3])/(b + a*x^2)])/(Sqrt[2]*a^(3/4)*b^(3/4)) - ((2*Sqrt[a]*Sqrt[b] + c)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[b*x + a*x^3])/(b + a*x^2)])/(2*Sqrt[2]*a^(3/4)*b^(3/4))","A",1
2051,1,147,147,0.3202218,"\int \frac{(1+x) \sqrt[4]{-x^3+x^4}}{1-x+x^2} \, dx","IntegrateAlgebraic[((1 + x)*(-x^3 + x^4)^(1/4))/(1 - x + x^2),x]","\sqrt[4]{-x^3+x^4}-\frac{7}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)+\frac{7}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)-\text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 \log (x)+2 \log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]","\sqrt[4]{-x^3+x^4}-\frac{7}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)+\frac{7}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)-\text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 \log (x)+2 \log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]",1,"(-x^3 + x^4)^(1/4) - (7*ArcTan[x/(-x^3 + x^4)^(1/4)])/2 + (7*ArcTanh[x/(-x^3 + x^4)^(1/4)])/2 - RootSum[1 - #1^4 + #1^8 & , (-2*Log[x] + 2*Log[(-x^3 + x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-x^3 + x^4)^(1/4) - x*#1]*#1^4)/(-#1^3 + 2*#1^7) & ]","A",1
2052,1,147,147,0.0008956,"\int \frac{(1+x) \sqrt[4]{-x^3+x^4}}{1-x+x^2} \, dx","IntegrateAlgebraic[((1 + x)*(-x^3 + x^4)^(1/4))/(1 - x + x^2),x]","\sqrt[4]{-x^3+x^4}-\frac{7}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)+\frac{7}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)-\text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 \log (x)+2 \log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]","\sqrt[4]{-x^3+x^4}-\frac{7}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)+\frac{7}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)-\text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 \log (x)+2 \log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]",1,"(-x^3 + x^4)^(1/4) - (7*ArcTan[x/(-x^3 + x^4)^(1/4)])/2 + (7*ArcTanh[x/(-x^3 + x^4)^(1/4)])/2 - RootSum[1 - #1^4 + #1^8 & , (-2*Log[x] + 2*Log[(-x^3 + x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-x^3 + x^4)^(1/4) - x*#1]*#1^4)/(-#1^3 + 2*#1^7) & ]","A",1
2053,1,147,147,2.1018061,"\int \frac{\sqrt[3]{b-a x^6} \left(b+a x^6\right)}{x^2 \left(-b+c x^3+a x^6\right)} \, dx","IntegrateAlgebraic[((b - a*x^6)^(1/3)*(b + a*x^6))/(x^2*(-b + c*x^3 + a*x^6)),x]","\frac{\sqrt[3]{b-a x^6}}{x}+\frac{\sqrt[3]{c} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{c} x}{\sqrt[3]{c} x+2 \sqrt[3]{b-a x^6}}\right)}{\sqrt{3}}+\frac{1}{3} \sqrt[3]{c} \log \left(-\sqrt[3]{c} x+\sqrt[3]{b-a x^6}\right)-\frac{1}{6} \sqrt[3]{c} \log \left(c^{2/3} x^2+\sqrt[3]{c} x \sqrt[3]{b-a x^6}+\left(b-a x^6\right)^{2/3}\right)","\frac{\sqrt[3]{b-a x^6}}{x}+\frac{\sqrt[3]{c} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{c} x}{\sqrt[3]{c} x+2 \sqrt[3]{b-a x^6}}\right)}{\sqrt{3}}+\frac{1}{3} \sqrt[3]{c} \log \left(-\sqrt[3]{c} x+\sqrt[3]{b-a x^6}\right)-\frac{1}{6} \sqrt[3]{c} \log \left(c^{2/3} x^2+\sqrt[3]{c} x \sqrt[3]{b-a x^6}+\left(b-a x^6\right)^{2/3}\right)",1,"(b - a*x^6)^(1/3)/x + (c^(1/3)*ArcTan[(Sqrt[3]*c^(1/3)*x)/(c^(1/3)*x + 2*(b - a*x^6)^(1/3))])/Sqrt[3] + (c^(1/3)*Log[-(c^(1/3)*x) + (b - a*x^6)^(1/3)])/3 - (c^(1/3)*Log[c^(2/3)*x^2 + c^(1/3)*x*(b - a*x^6)^(1/3) + (b - a*x^6)^(2/3)])/6","A",1
2054,1,147,147,9.9059951,"\int \frac{x^2 \left(4+7 x^3\right)}{\sqrt[3]{x+x^4} \left(-1+x^4+x^7\right)} \, dx","IntegrateAlgebraic[(x^2*(4 + 7*x^3))/((x + x^4)^(1/3)*(-1 + x^4 + x^7)),x]","-\sqrt{3} \tan ^{-1}\left(\frac{3 \sqrt{3} x \sqrt[3]{x+x^4}-3 x^2 \sqrt[3]{x+x^4}}{-6+2 \sqrt{3} x-3 x \sqrt[3]{x+x^4}+\sqrt{3} x^2 \sqrt[3]{x+x^4}}\right)+2 \tanh ^{-1}\left(1-2 x \sqrt[3]{x+x^4}\right)-\tanh ^{-1}\left(\frac{1+x \sqrt[3]{x+x^4}}{1+x \sqrt[3]{x+x^4}+2 x^2 \left(x+x^4\right)^{2/3}}\right)","-\sqrt{3} \tan ^{-1}\left(\frac{3 \sqrt{3} x \sqrt[3]{x+x^4}-3 x^2 \sqrt[3]{x+x^4}}{-6+2 \sqrt{3} x-3 x \sqrt[3]{x+x^4}+\sqrt{3} x^2 \sqrt[3]{x+x^4}}\right)+2 \tanh ^{-1}\left(1-2 x \sqrt[3]{x+x^4}\right)-\tanh ^{-1}\left(\frac{1+x \sqrt[3]{x+x^4}}{1+x \sqrt[3]{x+x^4}+2 x^2 \left(x+x^4\right)^{2/3}}\right)",1,"-(Sqrt[3]*ArcTan[(3*Sqrt[3]*x*(x + x^4)^(1/3) - 3*x^2*(x + x^4)^(1/3))/(-6 + 2*Sqrt[3]*x - 3*x*(x + x^4)^(1/3) + Sqrt[3]*x^2*(x + x^4)^(1/3))]) + 2*ArcTanh[1 - 2*x*(x + x^4)^(1/3)] - ArcTanh[(1 + x*(x + x^4)^(1/3))/(1 + x*(x + x^4)^(1/3) + 2*x^2*(x + x^4)^(2/3))]","A",1
2055,1,146,147,2.2220552,"\int \frac{2 d+c x^4}{\sqrt[4]{-b+a x^4} \left(-2 f+e x^8\right)} \, dx","IntegrateAlgebraic[(2*d + c*x^4)/((-b + a*x^4)^(1/4)*(-2*f + e*x^8)),x]","\frac{\text{RootSum}\left[b^2 e-2 a^2 f+4 a f \text{$\#$1}^4-2 f \text{$\#$1}^8\&,\frac{b c \log (x)+2 a d \log (x)-b c \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-2 a d \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-2 d \log (x) \text{$\#$1}^4+2 d \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}+\text{$\#$1}^5}\&\right]}{16 f}","\frac{\text{RootSum}\left[b^2 e-2 a^2 f+4 a f \text{$\#$1}^4-2 f \text{$\#$1}^8\&,\frac{-b c \log (x)-2 a d \log (x)+b c \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+2 a d \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+2 d \log (x) \text{$\#$1}^4-2 d \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}-\text{$\#$1}^5}\&\right]}{16 f}",1,"RootSum[b^2*e - 2*a^2*f + 4*a*f*#1^4 - 2*f*#1^8 & , (b*c*Log[x] + 2*a*d*Log[x] - b*c*Log[(-b + a*x^4)^(1/4) - x*#1] - 2*a*d*Log[(-b + a*x^4)^(1/4) - x*#1] - 2*d*Log[x]*#1^4 + 2*d*Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1) + #1^5) & ]/(16*f)","A",1
2056,1,157,147,0.380913,"\int \frac{\sqrt{1+x} \left(1+x^2\right)}{\left(-1+x^2\right) \sqrt{x+\sqrt{1+x}}} \, dx","IntegrateAlgebraic[(Sqrt[1 + x]*(1 + x^2))/((-1 + x^2)*Sqrt[x + Sqrt[1 + x]]),x]","\frac{1}{2} \sqrt{x+\sqrt{1+x}} \left(-3+2 \sqrt{1+x}\right)-\frac{7}{4} \log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}\right)-2 \text{RootSum}\left[-1+8 \text{$\#$1}-6 \text{$\#$1}^2+\text{$\#$1}^4\&,\frac{-\log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right)+2 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}}{2-3 \text{$\#$1}+\text{$\#$1}^3}\&\right]","-\frac{3}{2} \sqrt{x+\sqrt{1+x}}+\sqrt{1+x} \sqrt{x+\sqrt{1+x}}-\frac{7}{4} \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}\right)-16 \text{RootSum}\left[25+20 \text{$\#$1}-18 \text{$\#$1}^2+4 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{\log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}}{5-9 \text{$\#$1}+3 \text{$\#$1}^2+\text{$\#$1}^3}\&\right]",1,"(Sqrt[x + Sqrt[1 + x]]*(-3 + 2*Sqrt[1 + x]))/2 - (7*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]])/4 - 2*RootSum[-1 + 8*#1 - 6*#1^2 + #1^4 & , (-Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1] + 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1)/(2 - 3*#1 + #1^3) & ]","A",1
2057,1,157,147,0.0011897,"\int \frac{\sqrt{1+x} \left(1+x^2\right)}{\left(-1+x^2\right) \sqrt{x+\sqrt{1+x}}} \, dx","IntegrateAlgebraic[(Sqrt[1 + x]*(1 + x^2))/((-1 + x^2)*Sqrt[x + Sqrt[1 + x]]),x]","\frac{1}{2} \sqrt{x+\sqrt{1+x}} \left(-3+2 \sqrt{1+x}\right)-\frac{7}{4} \log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}\right)-2 \text{RootSum}\left[-1+8 \text{$\#$1}-6 \text{$\#$1}^2+\text{$\#$1}^4\&,\frac{-\log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right)+2 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}}{2-3 \text{$\#$1}+\text{$\#$1}^3}\&\right]","-\frac{3}{2} \sqrt{x+\sqrt{1+x}}+\sqrt{1+x} \sqrt{x+\sqrt{1+x}}-\frac{7}{4} \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}\right)-16 \text{RootSum}\left[25+20 \text{$\#$1}-18 \text{$\#$1}^2+4 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{\log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}}{5-9 \text{$\#$1}+3 \text{$\#$1}^2+\text{$\#$1}^3}\&\right]",1,"(Sqrt[x + Sqrt[1 + x]]*(-3 + 2*Sqrt[1 + x]))/2 - (7*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]])/4 - 2*RootSum[-1 + 8*#1 - 6*#1^2 + #1^4 & , (-Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1] + 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1)/(2 - 3*#1 + #1^3) & ]","A",1
2058,1,135,147,0.2444805,"\int \frac{\sqrt{x-\sqrt{-1+x^2}}}{x^2} \, dx","IntegrateAlgebraic[Sqrt[x - Sqrt[-1 + x^2]]/x^2,x]","-\frac{\sqrt{x-\sqrt{-1+x^2}}}{x}+\frac{\tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x}{\sqrt{2}}-\frac{\sqrt{-1+x^2}}{\sqrt{2}}}{\sqrt{x-\sqrt{-1+x^2}}}\right)}{\sqrt{2}}+\frac{\tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}-\frac{x}{\sqrt{2}}+\frac{\sqrt{-1+x^2}}{\sqrt{2}}}{\sqrt{x-\sqrt{-1+x^2}}}\right)}{\sqrt{2}}","\sqrt{x-\sqrt{-1+x^2}} \left(-\frac{1}{x}+\sqrt{x+\sqrt{-1+x^2}} \left(-\frac{\tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x}{\sqrt{2}}+\frac{\sqrt{-1+x^2}}{\sqrt{2}}}{\sqrt{x+\sqrt{-1+x^2}}}\right)}{\sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{1}{\sqrt{2}}+\frac{x}{\sqrt{2}}+\frac{\sqrt{-1+x^2}}{\sqrt{2}}}{\sqrt{x+\sqrt{-1+x^2}}}\right)}{\sqrt{2}}\right)\right)",1,"-(Sqrt[x - Sqrt[-1 + x^2]]/x) + ArcTan[(-(1/Sqrt[2]) + x/Sqrt[2] - Sqrt[-1 + x^2]/Sqrt[2])/Sqrt[x - Sqrt[-1 + x^2]]]/Sqrt[2] + ArcTanh[(-(1/Sqrt[2]) - x/Sqrt[2] + Sqrt[-1 + x^2]/Sqrt[2])/Sqrt[x - Sqrt[-1 + x^2]]]/Sqrt[2]","A",1
2059,0,0,147,16.3417689,"\int \frac{\left(a x+\sqrt{-b x+a^2 x^2}\right)^{3/4}}{\sqrt{-b x+a^2 x^2}} \, dx","IntegrateAlgebraic[(a*x + Sqrt[-(b*x) + a^2*x^2])^(3/4)/Sqrt[-(b*x) + a^2*x^2],x]","\int \frac{\left(a x+\sqrt{-b x+a^2 x^2}\right)^{3/4}}{\sqrt{-b x+a^2 x^2}} \, dx","\frac{4 \left(a x+\sqrt{-b x+a^2 x^2}\right)^{3/4}}{3 a}+\frac{\sqrt[4]{2} b^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \sqrt[4]{a x+\sqrt{-b x+a^2 x^2}}}{\sqrt[4]{b}}\right)}{a^{7/4}}-\frac{\sqrt[4]{2} b^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \sqrt[4]{a x+\sqrt{-b x+a^2 x^2}}}{\sqrt[4]{b}}\right)}{a^{7/4}}",1,"Could not integrate","F",-1
2060,1,148,148,0.4077126,"\int \frac{(1+x) \sqrt[4]{x^3+x^4}}{x \left(-1+x^3\right)} \, dx","IntegrateAlgebraic[((1 + x)*(x^3 + x^4)^(1/4))/(x*(-1 + x^3)),x]","\frac{4}{3} \sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^4}}\right)-\frac{4}{3} \sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^4}}\right)+\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 \log (x)+2 \log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]","\frac{4}{3} \sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^4}}\right)-\frac{4}{3} \sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^4}}\right)+\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 \log (x)+2 \log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]",1,"(4*2^(1/4)*ArcTan[(2^(1/4)*x)/(x^3 + x^4)^(1/4)])/3 - (4*2^(1/4)*ArcTanh[(2^(1/4)*x)/(x^3 + x^4)^(1/4)])/3 + RootSum[1 - #1^4 + #1^8 & , (-2*Log[x] + 2*Log[(x^3 + x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(x^3 + x^4)^(1/4) - x*#1]*#1^4)/(-#1^3 + 2*#1^7) & ]/3","A",1
2061,1,148,148,0.8341231,"\int \frac{\sqrt[4]{-b x^3+a x^4} \left(-d+c x^4\right)}{x^4} \, dx","IntegrateAlgebraic[((-(b*x^3) + a*x^4)^(1/4)*(-d + c*x^4))/x^4,x]","\frac{\sqrt[4]{-b x^3+a x^4} \left(160 a b^2 d-32 a^2 b d x-128 a^3 d x^2-45 b^3 c x^3+180 a b^2 c x^4\right)}{360 a b^2 x^3}+\frac{3 b^2 c \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)}{16 a^{7/4}}-\frac{3 b^2 c \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)}{16 a^{7/4}}","\frac{\sqrt[4]{-b x^3+a x^4} \left(160 a b^2 d-32 a^2 b d x-128 a^3 d x^2-45 b^3 c x^3+180 a b^2 c x^4\right)}{360 a b^2 x^3}+\frac{3 b^2 c \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)}{16 a^{7/4}}-\frac{3 b^2 c \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)}{16 a^{7/4}}",1,"((-(b*x^3) + a*x^4)^(1/4)*(160*a*b^2*d - 32*a^2*b*d*x - 128*a^3*d*x^2 - 45*b^3*c*x^3 + 180*a*b^2*c*x^4))/(360*a*b^2*x^3) + (3*b^2*c*ArcTan[(a^(1/4)*x)/(-(b*x^3) + a*x^4)^(1/4)])/(16*a^(7/4)) - (3*b^2*c*ArcTanh[(a^(1/4)*x)/(-(b*x^3) + a*x^4)^(1/4)])/(16*a^(7/4))","A",1
2062,1,149,149,0.3914909,"\int \frac{\left(1+x^3\right)^{2/3} \left(2+x^3\right)}{x^6 \left(4+x^3\right)} \, dx","IntegrateAlgebraic[((1 + x^3)^(2/3)*(2 + x^3))/(x^6*(4 + x^3)),x]","\frac{\left(-8-13 x^3\right) \left(1+x^3\right)^{2/3}}{80 x^5}+\frac{\sqrt[6]{3} \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2\ 2^{2/3} \sqrt[3]{1+x^3}}\right)}{16 \sqrt[3]{2}}-\frac{\log \left(-3 x+6^{2/3} \sqrt[3]{1+x^3}\right)}{16 \sqrt[3]{6}}+\frac{\log \left(3 x^2+6^{2/3} x \sqrt[3]{1+x^3}+2 \sqrt[3]{6} \left(1+x^3\right)^{2/3}\right)}{32 \sqrt[3]{6}}","\frac{\left(-8-13 x^3\right) \left(1+x^3\right)^{2/3}}{80 x^5}+\frac{\sqrt[6]{3} \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2\ 2^{2/3} \sqrt[3]{1+x^3}}\right)}{16 \sqrt[3]{2}}-\frac{\log \left(-3 x+6^{2/3} \sqrt[3]{1+x^3}\right)}{16 \sqrt[3]{6}}+\frac{\log \left(3 x^2+6^{2/3} x \sqrt[3]{1+x^3}+2 \sqrt[3]{6} \left(1+x^3\right)^{2/3}\right)}{32 \sqrt[3]{6}}",1,"((-8 - 13*x^3)*(1 + x^3)^(2/3))/(80*x^5) + (3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*2^(2/3)*(1 + x^3)^(1/3))])/(16*2^(1/3)) - Log[-3*x + 6^(2/3)*(1 + x^3)^(1/3)]/(16*6^(1/3)) + Log[3*x^2 + 6^(2/3)*x*(1 + x^3)^(1/3) + 2*6^(1/3)*(1 + x^3)^(2/3)]/(32*6^(1/3))","A",1
2063,1,149,149,0.3762529,"\int \frac{\left(-1+x^3\right) \left(1+3 x^3\right)^{2/3}}{x^6 \left(1+x^3\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)*(1 + 3*x^3)^(2/3))/(x^6*(1 + x^3)),x]","\frac{\left(1-2 x^3\right) \left(1+3 x^3\right)^{2/3}}{5 x^5}+\frac{2\ 2^{2/3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{1+3 x^3}}\right)}{\sqrt{3}}-\frac{2}{3} 2^{2/3} \log \left(-2 x+2^{2/3} \sqrt[3]{1+3 x^3}\right)+\frac{1}{3} 2^{2/3} \log \left(2 x^2+2^{2/3} x \sqrt[3]{1+3 x^3}+\sqrt[3]{2} \left(1+3 x^3\right)^{2/3}\right)","\frac{\left(1-2 x^3\right) \left(1+3 x^3\right)^{2/3}}{5 x^5}+\frac{2\ 2^{2/3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{1+3 x^3}}\right)}{\sqrt{3}}-\frac{2}{3} 2^{2/3} \log \left(-2 x+2^{2/3} \sqrt[3]{1+3 x^3}\right)+\frac{1}{3} 2^{2/3} \log \left(2 x^2+2^{2/3} x \sqrt[3]{1+3 x^3}+\sqrt[3]{2} \left(1+3 x^3\right)^{2/3}\right)",1,"((1 - 2*x^3)*(1 + 3*x^3)^(2/3))/(5*x^5) + (2*2^(2/3)*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(1 + 3*x^3)^(1/3))])/Sqrt[3] - (2*2^(2/3)*Log[-2*x + 2^(2/3)*(1 + 3*x^3)^(1/3)])/3 + (2^(2/3)*Log[2*x^2 + 2^(2/3)*x*(1 + 3*x^3)^(1/3) + 2^(1/3)*(1 + 3*x^3)^(2/3)])/3","A",1
2064,1,149,149,0.3971034,"\int \frac{3-8 x+8 x^2-12 x^4}{x \sqrt[3]{\frac{1-2 x^2}{1+2 x^2}} \left(1+2 x^2\right) \left(3-7 x+7 x^2-6 x^3+2 x^4\right)} \, dx","IntegrateAlgebraic[(3 - 8*x + 8*x^2 - 12*x^4)/(x*((1 - 2*x^2)/(1 + 2*x^2))^(1/3)*(1 + 2*x^2)*(3 - 7*x + 7*x^2 - 6*x^3 + 2*x^4)),x]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{\frac{1-2 x^2}{1+2 x^2}}}{2-2 x+\sqrt[3]{\frac{1-2 x^2}{1+2 x^2}}}\right)+\log \left(-1+x+\sqrt[3]{\frac{1-2 x^2}{1+2 x^2}}\right)-\frac{1}{2} \log \left(1-2 x+x^2+(1-x) \sqrt[3]{\frac{1-2 x^2}{1+2 x^2}}+\left(\frac{1-2 x^2}{1+2 x^2}\right)^{2/3}\right)","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{\frac{1-2 x^2}{1+2 x^2}}}{2-2 x+\sqrt[3]{\frac{1-2 x^2}{1+2 x^2}}}\right)+\log \left(-1+x+\sqrt[3]{\frac{1-2 x^2}{1+2 x^2}}\right)-\frac{1}{2} \log \left(1-2 x+x^2+(1-x) \sqrt[3]{\frac{1-2 x^2}{1+2 x^2}}+\left(\frac{1-2 x^2}{1+2 x^2}\right)^{2/3}\right)",1,"Sqrt[3]*ArcTan[(Sqrt[3]*((1 - 2*x^2)/(1 + 2*x^2))^(1/3))/(2 - 2*x + ((1 - 2*x^2)/(1 + 2*x^2))^(1/3))] + Log[-1 + x + ((1 - 2*x^2)/(1 + 2*x^2))^(1/3)] - Log[1 - 2*x + x^2 + (1 - x)*((1 - 2*x^2)/(1 + 2*x^2))^(1/3) + ((1 - 2*x^2)/(1 + 2*x^2))^(2/3)]/2","A",1
2065,1,149,149,0.430477,"\int \frac{x^2}{\sqrt{b x+a x^3} \left(-b^2+a^2 x^4\right)} \, dx","IntegrateAlgebraic[x^2/(Sqrt[b*x + a*x^3]*(-b^2 + a^2*x^4)),x]","\frac{\sqrt{b x+a x^3}}{2 a b \left(b+a x^2\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)}{4 \sqrt{2} a^{5/4} b^{5/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)}{4 \sqrt{2} a^{5/4} b^{5/4}}","\frac{\sqrt{b x+a x^3}}{2 a b \left(b+a x^2\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)}{4 \sqrt{2} a^{5/4} b^{5/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)}{4 \sqrt{2} a^{5/4} b^{5/4}}",1,"Sqrt[b*x + a*x^3]/(2*a*b*(b + a*x^2)) - ArcTan[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[b*x + a*x^3])/(b + a*x^2)]/(4*Sqrt[2]*a^(5/4)*b^(5/4)) - ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[b*x + a*x^3])/(b + a*x^2)]/(4*Sqrt[2]*a^(5/4)*b^(5/4))","A",1
2066,1,149,149,0.9295459,"\int \frac{b+a x^6}{x^6 \left(-b+a x^3\right) \sqrt[4]{b x+a x^4}} \, dx","IntegrateAlgebraic[(b + a*x^6)/(x^6*(-b + a*x^3)*(b*x + a*x^4)^(1/4)),x]","\frac{4 \left(b+a x^3\right) \left(b x+a x^4\right)^{3/4}}{21 b^2 x^6}-\frac{2^{3/4} \left(a^{7/4}+a^{3/4} b\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 b^2}-\frac{2^{3/4} \left(a^{7/4}+a^{3/4} b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 b^2}","\frac{4 \left(b+a x^3\right) \left(b x+a x^4\right)^{3/4}}{21 b^2 x^6}-\frac{2^{3/4} \left(a^{7/4}+a^{3/4} b\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 b^2}-\frac{2^{3/4} \left(a^{7/4}+a^{3/4} b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 b^2}",1,"(4*(b + a*x^3)*(b*x + a*x^4)^(3/4))/(21*b^2*x^6) - (2^(3/4)*(a^(7/4) + a^(3/4)*b)*ArcTan[(2^(1/4)*a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/(3*b^2) - (2^(3/4)*(a^(7/4) + a^(3/4)*b)*ArcTanh[(2^(1/4)*a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/(3*b^2)","A",1
2067,1,149,149,2.8373396,"\int \frac{\sqrt[3]{-1+2 x^3+x^8} \left(3+5 x^8\right)}{x^2 \left(-1+x^8\right)} \, dx","IntegrateAlgebraic[((-1 + 2*x^3 + x^8)^(1/3)*(3 + 5*x^8))/(x^2*(-1 + x^8)),x]","\frac{3 \sqrt[3]{-1+2 x^3+x^8}}{x}+\sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-1+2 x^3+x^8}}\right)+\sqrt[3]{2} \log \left(-2 x+2^{2/3} \sqrt[3]{-1+2 x^3+x^8}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-1+2 x^3+x^8}+\sqrt[3]{2} \left(-1+2 x^3+x^8\right)^{2/3}\right)}{2^{2/3}}","\frac{3 \sqrt[3]{-1+2 x^3+x^8}}{x}+\sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-1+2 x^3+x^8}}\right)+\sqrt[3]{2} \log \left(-2 x+2^{2/3} \sqrt[3]{-1+2 x^3+x^8}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-1+2 x^3+x^8}+\sqrt[3]{2} \left(-1+2 x^3+x^8\right)^{2/3}\right)}{2^{2/3}}",1,"(3*(-1 + 2*x^3 + x^8)^(1/3))/x + 2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-1 + 2*x^3 + x^8)^(1/3))] + 2^(1/3)*Log[-2*x + 2^(2/3)*(-1 + 2*x^3 + x^8)^(1/3)] - Log[2*x^2 + 2^(2/3)*x*(-1 + 2*x^3 + x^8)^(1/3) + 2^(1/3)*(-1 + 2*x^3 + x^8)^(2/3)]/2^(2/3)","A",1
2068,1,81,149,0.2158213,"\int \frac{\sqrt{b^2+a x^2}}{\sqrt{b+\sqrt{b^2+a x^2}}} \, dx","IntegrateAlgebraic[Sqrt[b^2 + a*x^2]/Sqrt[b + Sqrt[b^2 + a*x^2]],x]","\frac{2 a x^3}{3 \left(b+\sqrt{b^2+a x^2}\right)^{3/2}}+\frac{\sqrt{2} b^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{\sqrt{a}}","-\frac{2 b x}{3 \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{2 x \sqrt{b^2+a x^2}}{3 \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{2 \sqrt{2} b^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2} \sqrt{b}}\right)}{\sqrt{a}}",1,"(2*a*x^3)/(3*(b + Sqrt[b^2 + a*x^2])^(3/2)) + (Sqrt[2]*b^(3/2)*ArcTan[(Sqrt[a]*x)/(Sqrt[2]*Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/Sqrt[a]","A",1
2069,1,149,149,0.3412726,"\int \frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\left(b^2+a x^2\right)^3} \, dx","IntegrateAlgebraic[Sqrt[b + Sqrt[b^2 + a*x^2]]/(b^2 + a*x^2)^3,x]","\frac{7 x \left(23 b^2+15 a x^2\right)}{192 b^4 \left(b^2+a x^2\right)^{3/2} \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{x \left(59 b^2+35 a x^2\right)}{96 b^3 \left(b^2+a x^2\right)^2 \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{35 \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{64 \sqrt{a} b^{9/2}}","\frac{7 x \left(23 b^2+15 a x^2\right)}{192 b^4 \left(b^2+a x^2\right)^{3/2} \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{x \left(59 b^2+35 a x^2\right)}{96 b^3 \left(b^2+a x^2\right)^2 \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{35 \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{64 \sqrt{a} b^{9/2}}",1,"(7*x*(23*b^2 + 15*a*x^2))/(192*b^4*(b^2 + a*x^2)^(3/2)*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (x*(59*b^2 + 35*a*x^2))/(96*b^3*(b^2 + a*x^2)^2*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (35*ArcTan[(Sqrt[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/(64*Sqrt[a]*b^(9/2))","A",1
2070,1,149,150,0.3522886,"\int \frac{\left(-b+a x^2\right)^{3/4}}{x^3} \, dx","IntegrateAlgebraic[(-b + a*x^2)^(3/4)/x^3,x]","-\frac{\left(-b+a x^2\right)^{3/4}}{2 x^2}+\frac{3 a \tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^2}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^2}}\right)}{4 \sqrt{2} \sqrt[4]{b}}-\frac{3 a \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^2}}{\sqrt{b}+\sqrt{-b+a x^2}}\right)}{4 \sqrt{2} \sqrt[4]{b}}","-\frac{\left(-b+a x^2\right)^{3/4}}{2 x^2}-\frac{3 a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^2}}{-\sqrt{b}+\sqrt{-b+a x^2}}\right)}{4 \sqrt{2} \sqrt[4]{b}}-\frac{3 a \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^2}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^2}}\right)}{4 \sqrt{2} \sqrt[4]{b}}",1,"-1/2*(-b + a*x^2)^(3/4)/x^2 + (3*a*ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^2]/(Sqrt[2]*b^(1/4)))/(-b + a*x^2)^(1/4)])/(4*Sqrt[2]*b^(1/4)) - (3*a*ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^2)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^2])])/(4*Sqrt[2]*b^(1/4))","A",1
2071,1,150,150,0.3728068,"\int \frac{\left(1+x^3\right) \left(-1+2 x^3\right)^{2/3}}{x^6 \left(1+2 x^3\right)} \, dx","IntegrateAlgebraic[((1 + x^3)*(-1 + 2*x^3)^(2/3))/(x^6*(1 + 2*x^3)),x]","\frac{\left(-1+2 x^3\right)^{2/3} \left(-2+9 x^3\right)}{10 x^5}-\frac{2 \sqrt[3]{2} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+\sqrt[3]{2} \sqrt[3]{-1+2 x^3}}\right)}{\sqrt{3}}+\frac{2}{3} \sqrt[3]{2} \log \left(-2 x+\sqrt[3]{2} \sqrt[3]{-1+2 x^3}\right)-\frac{1}{3} \sqrt[3]{2} \log \left(4 x^2+2 \sqrt[3]{2} x \sqrt[3]{-1+2 x^3}+2^{2/3} \left(-1+2 x^3\right)^{2/3}\right)","\frac{\left(-1+2 x^3\right)^{2/3} \left(-2+9 x^3\right)}{10 x^5}-\frac{2 \sqrt[3]{2} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+\sqrt[3]{2} \sqrt[3]{-1+2 x^3}}\right)}{\sqrt{3}}+\frac{2}{3} \sqrt[3]{2} \log \left(-2 x+\sqrt[3]{2} \sqrt[3]{-1+2 x^3}\right)-\frac{1}{3} \sqrt[3]{2} \log \left(4 x^2+2 \sqrt[3]{2} x \sqrt[3]{-1+2 x^3}+2^{2/3} \left(-1+2 x^3\right)^{2/3}\right)",1,"((-1 + 2*x^3)^(2/3)*(-2 + 9*x^3))/(10*x^5) - (2*2^(1/3)*ArcTan[(Sqrt[3]*x)/(x + 2^(1/3)*(-1 + 2*x^3)^(1/3))])/Sqrt[3] + (2*2^(1/3)*Log[-2*x + 2^(1/3)*(-1 + 2*x^3)^(1/3)])/3 - (2^(1/3)*Log[4*x^2 + 2*2^(1/3)*x*(-1 + 2*x^3)^(1/3) + 2^(2/3)*(-1 + 2*x^3)^(2/3)])/3","A",1
2072,1,150,150,0.35461,"\int \frac{\left(1-x^3\right)^{2/3} \left(-1+4 x^3\right)}{x^6 \left(-2+3 x^3\right)} \, dx","IntegrateAlgebraic[((1 - x^3)^(2/3)*(-1 + 4*x^3))/(x^6*(-2 + 3*x^3)),x]","\frac{\left(1-x^3\right)^{2/3} \left(-4+29 x^3\right)}{40 x^5}-\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2} \sqrt[3]{1-x^3}}\right)}{4\ 2^{2/3} \sqrt{3}}+\frac{5 \log \left(-x+\sqrt[3]{2} \sqrt[3]{1-x^3}\right)}{12\ 2^{2/3}}-\frac{5 \log \left(x^2+\sqrt[3]{2} x \sqrt[3]{1-x^3}+2^{2/3} \left(1-x^3\right)^{2/3}\right)}{24\ 2^{2/3}}","\frac{\left(1-x^3\right)^{2/3} \left(-4+29 x^3\right)}{40 x^5}-\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2} \sqrt[3]{1-x^3}}\right)}{4\ 2^{2/3} \sqrt{3}}+\frac{5 \log \left(-x+\sqrt[3]{2} \sqrt[3]{1-x^3}\right)}{12\ 2^{2/3}}-\frac{5 \log \left(x^2+\sqrt[3]{2} x \sqrt[3]{1-x^3}+2^{2/3} \left(1-x^3\right)^{2/3}\right)}{24\ 2^{2/3}}",1,"((1 - x^3)^(2/3)*(-4 + 29*x^3))/(40*x^5) - (5*ArcTan[(Sqrt[3]*x)/(x + 2*2^(1/3)*(1 - x^3)^(1/3))])/(4*2^(2/3)*Sqrt[3]) + (5*Log[-x + 2^(1/3)*(1 - x^3)^(1/3)])/(12*2^(2/3)) - (5*Log[x^2 + 2^(1/3)*x*(1 - x^3)^(1/3) + 2^(2/3)*(1 - x^3)^(2/3)])/(24*2^(2/3))","A",1
2073,1,149,150,0.3315566,"\int \frac{\sqrt[4]{-b+a x^3}}{x^4} \, dx","IntegrateAlgebraic[(-b + a*x^3)^(1/4)/x^4,x]","-\frac{\sqrt[4]{-b+a x^3}}{3 x^3}+\frac{a \tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^3}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^3}}\right)}{6 \sqrt{2} b^{3/4}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^3}}{\sqrt{b}+\sqrt{-b+a x^3}}\right)}{6 \sqrt{2} b^{3/4}}","-\frac{\sqrt[4]{-b+a x^3}}{3 x^3}-\frac{a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^3}}{-\sqrt{b}+\sqrt{-b+a x^3}}\right)}{6 \sqrt{2} b^{3/4}}+\frac{a \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^3}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^3}}\right)}{6 \sqrt{2} b^{3/4}}",1,"-1/3*(-b + a*x^3)^(1/4)/x^3 + (a*ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^3]/(Sqrt[2]*b^(1/4)))/(-b + a*x^3)^(1/4)])/(6*Sqrt[2]*b^(3/4)) + (a*ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^3)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^3])])/(6*Sqrt[2]*b^(3/4))","A",1
2074,1,149,150,0.2114574,"\int \frac{\left(-b+a x^3\right)^{3/4}}{x^4} \, dx","IntegrateAlgebraic[(-b + a*x^3)^(3/4)/x^4,x]","-\frac{\left(-b+a x^3\right)^{3/4}}{3 x^3}+\frac{a \tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^3}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^3}}\right)}{2 \sqrt{2} \sqrt[4]{b}}-\frac{a \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^3}}{\sqrt{b}+\sqrt{-b+a x^3}}\right)}{2 \sqrt{2} \sqrt[4]{b}}","-\frac{\left(-b+a x^3\right)^{3/4}}{3 x^3}-\frac{a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^3}}{-\sqrt{b}+\sqrt{-b+a x^3}}\right)}{2 \sqrt{2} \sqrt[4]{b}}-\frac{a \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^3}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^3}}\right)}{2 \sqrt{2} \sqrt[4]{b}}",1,"-1/3*(-b + a*x^3)^(3/4)/x^3 + (a*ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^3]/(Sqrt[2]*b^(1/4)))/(-b + a*x^3)^(1/4)])/(2*Sqrt[2]*b^(1/4)) - (a*ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^3)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^3])])/(2*Sqrt[2]*b^(1/4))","A",1
2075,1,150,150,4.8248937,"\int \frac{(-a+x) (-3 a b+(a+2 b) x) \left(-b^3+3 b^2 x-3 b x^2+x^3\right)}{x \left(x (-a+x) (-b+x)^2\right)^{3/4} \left(a b^2-b (2 a+b) x+(a+2 b) x^2+(-1+d) x^3\right)} \, dx","IntegrateAlgebraic[((-a + x)*(-3*a*b + (a + 2*b)*x)*(-b^3 + 3*b^2*x - 3*b*x^2 + x^3))/(x*(x*(-a + x)*(-b + x)^2)^(3/4)*(a*b^2 - b*(2*a + b)*x + (a + 2*b)*x^2 + (-1 + d)*x^3)),x]","-\frac{4 \sqrt[4]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{x}-2 \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)+2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)","-\frac{4 \sqrt[4]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{x}-2 \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)+2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt[4]{d} x}{\sqrt[4]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)",1,"(-4*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/4))/x - 2*d^(1/4)*ArcTan[(d^(1/4)*x)/(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/4)] + 2*d^(1/4)*ArcTanh[(d^(1/4)*x)/(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/4)]","A",1
2076,1,149,150,0.2132385,"\int \frac{\left(-b+a x^5\right)^{3/4}}{x^6} \, dx","IntegrateAlgebraic[(-b + a*x^5)^(3/4)/x^6,x]","-\frac{\left(-b+a x^5\right)^{3/4}}{5 x^5}+\frac{3 a \tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^5}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^5}}\right)}{10 \sqrt{2} \sqrt[4]{b}}-\frac{3 a \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^5}}{\sqrt{b}+\sqrt{-b+a x^5}}\right)}{10 \sqrt{2} \sqrt[4]{b}}","-\frac{\left(-b+a x^5\right)^{3/4}}{5 x^5}-\frac{3 a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^5}}{-\sqrt{b}+\sqrt{-b+a x^5}}\right)}{10 \sqrt{2} \sqrt[4]{b}}-\frac{3 a \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^5}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^5}}\right)}{10 \sqrt{2} \sqrt[4]{b}}",1,"-1/5*(-b + a*x^5)^(3/4)/x^5 + (3*a*ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^5]/(Sqrt[2]*b^(1/4)))/(-b + a*x^5)^(1/4)])/(10*Sqrt[2]*b^(1/4)) - (3*a*ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^5)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^5])])/(10*Sqrt[2]*b^(1/4))","A",1
2077,1,149,150,0.2143293,"\int \frac{\left(-b+a x^6\right)^{3/4}}{x^7} \, dx","IntegrateAlgebraic[(-b + a*x^6)^(3/4)/x^7,x]","-\frac{\left(-b+a x^6\right)^{3/4}}{6 x^6}+\frac{a \tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^6}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^6}}\right)}{4 \sqrt{2} \sqrt[4]{b}}-\frac{a \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^6}}{\sqrt{b}+\sqrt{-b+a x^6}}\right)}{4 \sqrt{2} \sqrt[4]{b}}","-\frac{\left(-b+a x^6\right)^{3/4}}{6 x^6}-\frac{a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^6}}{-\sqrt{b}+\sqrt{-b+a x^6}}\right)}{4 \sqrt{2} \sqrt[4]{b}}-\frac{a \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^6}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^6}}\right)}{4 \sqrt{2} \sqrt[4]{b}}",1,"-1/6*(-b + a*x^6)^(3/4)/x^6 + (a*ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^6]/(Sqrt[2]*b^(1/4)))/(-b + a*x^6)^(1/4)])/(4*Sqrt[2]*b^(1/4)) - (a*ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^6)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^6])])/(4*Sqrt[2]*b^(1/4))","A",1
2078,1,150,150,0.4404088,"\int \frac{\left(-4+5 x^7\right) \sqrt[3]{-2 x+2 x^3-x^8}}{\left(2+x^7\right) \left(2-2 x^2+x^7\right)} \, dx","IntegrateAlgebraic[((-4 + 5*x^7)*(-2*x + 2*x^3 - x^8)^(1/3))/((2 + x^7)*(2 - 2*x^2 + x^7)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-2 x+2 x^3-x^8}}\right)}{2^{2/3}}-\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{-2 x+2 x^3-x^8}\right)}{2^{2/3}}+\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-2 x+2 x^3-x^8}+\sqrt[3]{2} \left(-2 x+2 x^3-x^8\right)^{2/3}\right)}{2\ 2^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-2 x+2 x^3-x^8}}\right)}{2^{2/3}}-\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{-2 x+2 x^3-x^8}\right)}{2^{2/3}}+\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-2 x+2 x^3-x^8}+\sqrt[3]{2} \left(-2 x+2 x^3-x^8\right)^{2/3}\right)}{2\ 2^{2/3}}",1,"-((Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-2*x + 2*x^3 - x^8)^(1/3))])/2^(2/3)) - Log[-2*x + 2^(2/3)*(-2*x + 2*x^3 - x^8)^(1/3)]/2^(2/3) + Log[2*x^2 + 2^(2/3)*x*(-2*x + 2*x^3 - x^8)^(1/3) + 2^(1/3)*(-2*x + 2*x^3 - x^8)^(2/3)]/(2*2^(2/3))","A",1
2079,1,150,150,1.2647146,"\int \frac{b-2 a x^4+2 x^8}{\sqrt[4]{b+a x^4} \left(-2 b-a x^4+x^8\right)} \, dx","IntegrateAlgebraic[(b - 2*a*x^4 + 2*x^8)/((b + a*x^4)^(1/4)*(-2*b - a*x^4 + x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}-\frac{5}{4} \text{RootSum}\left[a^2-b-3 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-a \log (x)+a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 a \text{$\#$1}+4 \text{$\#$1}^5}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}-\frac{5}{4} \text{RootSum}\left[a^2-b-3 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{a \log (x)-a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{3 a \text{$\#$1}-4 \text{$\#$1}^5}\&\right]",1,"ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(1/4) + ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(1/4) - (5*RootSum[a^2 - b - 3*a*#1^4 + 2*#1^8 & , (-(a*Log[x]) + a*Log[(b + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(b + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1 + 4*#1^5) & ])/4","A",1
2080,1,150,150,1.2728443,"\int \frac{-b+2 a x^4+2 x^8}{\sqrt[4]{b+a x^4} \left(-b+a x^4+x^8\right)} \, dx","IntegrateAlgebraic[(-b + 2*a*x^4 + 2*x^8)/((b + a*x^4)^(1/4)*(-b + a*x^4 + x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}-\frac{1}{4} \text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a \log (x)+a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}-\frac{1}{4} \text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a \log (x)-a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{3 a \text{$\#$1}-2 \text{$\#$1}^5}\&\right]",1,"ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(1/4) + ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(1/4) - RootSum[2*a^2 - b - 3*a*#1^4 + #1^8 & , (-(a*Log[x]) + a*Log[(b + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(b + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1 + 2*#1^5) & ]/4","A",1
2081,1,150,150,0.0014528,"\int \frac{-b+2 a x^4+2 x^8}{\sqrt[4]{b+a x^4} \left(-b+a x^4+x^8\right)} \, dx","IntegrateAlgebraic[(-b + 2*a*x^4 + 2*x^8)/((b + a*x^4)^(1/4)*(-b + a*x^4 + x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}-\frac{1}{4} \text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a \log (x)+a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}-\frac{1}{4} \text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a \log (x)-a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{3 a \text{$\#$1}-2 \text{$\#$1}^5}\&\right]",1,"ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(1/4) + ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(1/4) - RootSum[2*a^2 - b - 3*a*#1^4 + #1^8 & , (-(a*Log[x]) + a*Log[(b + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(b + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1 + 2*#1^5) & ]/4","A",1
2082,1,156,150,0.658943,"\int \frac{\sqrt{-b^4+a^4 x^4} \left(b^4+a^4 x^4\right)}{b^8+a^8 x^8} \, dx","IntegrateAlgebraic[(Sqrt[-b^4 + a^4*x^4]*(b^4 + a^4*x^4))/(b^8 + a^8*x^8),x]","-\frac{\tan ^{-1}\left(\frac{\frac{b^3}{2^{3/4} a}+\frac{a b x^2}{\sqrt[4]{2}}-\frac{a^3 x^4}{2^{3/4} b}}{x \sqrt{-b^4+a^4 x^4}}\right)}{2\ 2^{3/4} a b}+\frac{\tanh ^{-1}\left(\frac{\frac{b^3}{2^{3/4} a}-\frac{a b x^2}{\sqrt[4]{2}}-\frac{a^3 x^4}{2^{3/4} b}}{x \sqrt{-b^4+a^4 x^4}}\right)}{2\ 2^{3/4} a b}","-\frac{\tan ^{-1}\left(\frac{\frac{b^3}{2^{3/4} a}+\frac{a b x^2}{\sqrt[4]{2}}-\frac{a^3 x^4}{2^{3/4} b}}{x \sqrt{-b^4+a^4 x^4}}\right)}{2\ 2^{3/4} a b}-\frac{\tanh ^{-1}\left(\frac{2^{3/4} a b x \sqrt{-b^4+a^4 x^4}}{-b^4+\sqrt{2} a^2 b^2 x^2+a^4 x^4}\right)}{2\ 2^{3/4} a b}",1,"-1/2*ArcTan[(b^3/(2^(3/4)*a) + (a*b*x^2)/2^(1/4) - (a^3*x^4)/(2^(3/4)*b))/(x*Sqrt[-b^4 + a^4*x^4])]/(2^(3/4)*a*b) + ArcTanh[(b^3/(2^(3/4)*a) - (a*b*x^2)/2^(1/4) - (a^3*x^4)/(2^(3/4)*b))/(x*Sqrt[-b^4 + a^4*x^4])]/(2*2^(3/4)*a*b)","A",1
2083,1,156,150,0.7209479,"\int \frac{-b^8+a^8 x^8}{\sqrt{-b^4+a^4 x^4} \left(b^8+a^8 x^8\right)} \, dx","IntegrateAlgebraic[(-b^8 + a^8*x^8)/(Sqrt[-b^4 + a^4*x^4]*(b^8 + a^8*x^8)),x]","-\frac{\tan ^{-1}\left(\frac{\frac{b^3}{2^{3/4} a}+\frac{a b x^2}{\sqrt[4]{2}}-\frac{a^3 x^4}{2^{3/4} b}}{x \sqrt{-b^4+a^4 x^4}}\right)}{2\ 2^{3/4} a b}+\frac{\tanh ^{-1}\left(\frac{\frac{b^3}{2^{3/4} a}-\frac{a b x^2}{\sqrt[4]{2}}-\frac{a^3 x^4}{2^{3/4} b}}{x \sqrt{-b^4+a^4 x^4}}\right)}{2\ 2^{3/4} a b}","-\frac{\tan ^{-1}\left(\frac{\frac{b^3}{2^{3/4} a}+\frac{a b x^2}{\sqrt[4]{2}}-\frac{a^3 x^4}{2^{3/4} b}}{x \sqrt{-b^4+a^4 x^4}}\right)}{2\ 2^{3/4} a b}-\frac{\tanh ^{-1}\left(\frac{2^{3/4} a b x \sqrt{-b^4+a^4 x^4}}{-b^4+\sqrt{2} a^2 b^2 x^2+a^4 x^4}\right)}{2\ 2^{3/4} a b}",1,"-1/2*ArcTan[(b^3/(2^(3/4)*a) + (a*b*x^2)/2^(1/4) - (a^3*x^4)/(2^(3/4)*b))/(x*Sqrt[-b^4 + a^4*x^4])]/(2^(3/4)*a*b) + ArcTanh[(b^3/(2^(3/4)*a) - (a*b*x^2)/2^(1/4) - (a^3*x^4)/(2^(3/4)*b))/(x*Sqrt[-b^4 + a^4*x^4])]/(2*2^(3/4)*a*b)","A",1
2084,1,150,150,5.2348963,"\int \frac{x^3 \sqrt{x+x^2}}{\sqrt{x^2+x \sqrt{x+x^2}}} \, dx","IntegrateAlgebraic[(x^3*Sqrt[x + x^2])/Sqrt[x^2 + x*Sqrt[x + x^2]],x]","\frac{\sqrt{x+x^2} \left(495495-264264 x+201344 x^2-168960 x^3-1146880 x^4\right) \sqrt{x \left(x+\sqrt{x+x^2}\right)}}{5160960 x}+\sqrt{x \left(x+\sqrt{x+x^2}\right)} \left(\frac{-165165+37752 x-18304 x^2+1387520 x^3+1146880 x^4}{5160960}-\frac{1573 \sqrt{-x+\sqrt{x+x^2}} \tanh ^{-1}\left(\sqrt{2} \sqrt{-x+\sqrt{x+x^2}}\right)}{16384 \sqrt{2} x}\right)","\frac{\sqrt{x+x^2} \left(495495-264264 x+201344 x^2-168960 x^3-1146880 x^4\right) \sqrt{x \left(x+\sqrt{x+x^2}\right)}}{5160960 x}+\sqrt{x \left(x+\sqrt{x+x^2}\right)} \left(\frac{-165165+37752 x-18304 x^2+1387520 x^3+1146880 x^4}{5160960}-\frac{1573 \sqrt{-x+\sqrt{x+x^2}} \tanh ^{-1}\left(\sqrt{2} \sqrt{-x+\sqrt{x+x^2}}\right)}{16384 \sqrt{2} x}\right)",1,"(Sqrt[x + x^2]*(495495 - 264264*x + 201344*x^2 - 168960*x^3 - 1146880*x^4)*Sqrt[x*(x + Sqrt[x + x^2])])/(5160960*x) + Sqrt[x*(x + Sqrt[x + x^2])]*((-165165 + 37752*x - 18304*x^2 + 1387520*x^3 + 1146880*x^4)/5160960 - (1573*Sqrt[-x + Sqrt[x + x^2]]*ArcTanh[Sqrt[2]*Sqrt[-x + Sqrt[x + x^2]]])/(16384*Sqrt[2]*x))","A",1
2085,1,150,151,0.2163234,"\int \frac{\left(-b+a x^2\right)^{3/4} \left(3 b+2 a x^2\right)}{x} \, dx","IntegrateAlgebraic[((-b + a*x^2)^(3/4)*(3*b + 2*a*x^2))/x,x]","\frac{2}{7} \left(-b+a x^2\right)^{3/4} \left(5 b+2 a x^2\right)-\frac{3 b^{7/4} \tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^2}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^2}}\right)}{\sqrt{2}}+\frac{3 b^{7/4} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^2}}{\sqrt{b}+\sqrt{-b+a x^2}}\right)}{\sqrt{2}}","\frac{2}{7} \left(-b+a x^2\right)^{3/4} \left(5 b+2 a x^2\right)+\frac{3 b^{7/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^2}}{-\sqrt{b}+\sqrt{-b+a x^2}}\right)}{\sqrt{2}}+\frac{3 b^{7/4} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^2}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^2}}\right)}{\sqrt{2}}",1,"(2*(-b + a*x^2)^(3/4)*(5*b + 2*a*x^2))/7 - (3*b^(7/4)*ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^2]/(Sqrt[2]*b^(1/4)))/(-b + a*x^2)^(1/4)])/Sqrt[2] + (3*b^(7/4)*ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^2)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^2])])/Sqrt[2]","A",1
2086,1,151,151,0.4796477,"\int \frac{\left(-6+x^2\right) \left(2-x^2+x^3\right)^{2/3}}{x^3 \left(-2+x^2+x^3\right)} \, dx","IntegrateAlgebraic[((-6 + x^2)*(2 - x^2 + x^3)^(2/3))/(x^3*(-2 + x^2 + x^3)),x]","-\frac{3 \left(2-x^2+x^3\right)^{2/3}}{2 x^2}+2^{2/3} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{2-x^2+x^3}}\right)-2^{2/3} \log \left(-2 x+2^{2/3} \sqrt[3]{2-x^2+x^3}\right)+\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{2-x^2+x^3}+\sqrt[3]{2} \left(2-x^2+x^3\right)^{2/3}\right)}{\sqrt[3]{2}}","-\frac{3 \left(2-x^2+x^3\right)^{2/3}}{2 x^2}+2^{2/3} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{2-x^2+x^3}}\right)-2^{2/3} \log \left(-2 x+2^{2/3} \sqrt[3]{2-x^2+x^3}\right)+\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{2-x^2+x^3}+\sqrt[3]{2} \left(2-x^2+x^3\right)^{2/3}\right)}{\sqrt[3]{2}}",1,"(-3*(2 - x^2 + x^3)^(2/3))/(2*x^2) + 2^(2/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(2 - x^2 + x^3)^(1/3))] - 2^(2/3)*Log[-2*x + 2^(2/3)*(2 - x^2 + x^3)^(1/3)] + Log[2*x^2 + 2^(2/3)*x*(2 - x^2 + x^3)^(1/3) + 2^(1/3)*(2 - x^2 + x^3)^(2/3)]/2^(1/3)","A",1
2087,1,151,151,0.5311807,"\int \frac{x^3 (-4 a+3 x)}{\left(x^2 (-a+x)\right)^{2/3} \left(a d-d x+x^4\right)} \, dx","IntegrateAlgebraic[(x^3*(-4*a + 3*x))/((x^2*(-a + x))^(2/3)*(a*d - d*x + x^4)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{d} \sqrt[3]{-a x^2+x^3}}\right)}{\sqrt[3]{d}}+\frac{\log \left(a x^2-a \sqrt[3]{d} \sqrt[3]{-a x^2+x^3}\right)}{\sqrt[3]{d}}-\frac{\log \left(a^2 x^4+a^2 \sqrt[3]{d} x^2 \sqrt[3]{-a x^2+x^3}+a^2 d^{2/3} \left(-a x^2+x^3\right)^{2/3}\right)}{2 \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{d} \sqrt[3]{-a x^2+x^3}}\right)}{\sqrt[3]{d}}+\frac{\log \left(a x^2-a \sqrt[3]{d} \sqrt[3]{-a x^2+x^3}\right)}{\sqrt[3]{d}}-\frac{\log \left(a^2 x^4+a^2 \sqrt[3]{d} x^2 \sqrt[3]{-a x^2+x^3}+a^2 d^{2/3} \left(-a x^2+x^3\right)^{2/3}\right)}{2 \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*d^(1/3)*(-(a*x^2) + x^3)^(1/3))])/d^(1/3) + Log[a*x^2 - a*d^(1/3)*(-(a*x^2) + x^3)^(1/3)]/d^(1/3) - Log[a^2*x^4 + a^2*d^(1/3)*x^2*(-(a*x^2) + x^3)^(1/3) + a^2*d^(2/3)*(-(a*x^2) + x^3)^(2/3)]/(2*d^(1/3))","A",1
2088,1,151,151,0.3535979,"\int \frac{-1+k^4 x^4}{\sqrt{(1-x) x \left(1-k^2 x\right)} \left(1+k^4 x^4\right)} \, dx","IntegrateAlgebraic[(-1 + k^4*x^4)/(Sqrt[(1 - x)*x*(1 - k^2*x)]*(1 + k^4*x^4)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{1-\sqrt{2} k+k^2} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{(-1+x) \left(-1+k^2 x\right)}\right)}{\sqrt{1-\sqrt{2} k+k^2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{1+\sqrt{2} k+k^2} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{(-1+x) \left(-1+k^2 x\right)}\right)}{\sqrt{1+\sqrt{2} k+k^2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{1-\sqrt{2} k+k^2} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{(-1+x) \left(-1+k^2 x\right)}\right)}{\sqrt{1-\sqrt{2} k+k^2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{1+\sqrt{2} k+k^2} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{(-1+x) \left(-1+k^2 x\right)}\right)}{\sqrt{1+\sqrt{2} k+k^2}}",1,"-(ArcTan[(Sqrt[1 - Sqrt[2]*k + k^2]*Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3])/((-1 + x)*(-1 + k^2*x))]/Sqrt[1 - Sqrt[2]*k + k^2]) - ArcTan[(Sqrt[1 + Sqrt[2]*k + k^2]*Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3])/((-1 + x)*(-1 + k^2*x))]/Sqrt[1 + Sqrt[2]*k + k^2]","A",1
2089,1,151,151,0.5738854,"\int \frac{1-x^4}{\left(1+x^4\right) \sqrt[4]{x^3+x^5}} \, dx","IntegrateAlgebraic[(1 - x^4)/((1 + x^4)*(x^3 + x^5)^(1/4)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{x^3+x^5}}\right)}{\sqrt[8]{2}}-\frac{\tan ^{-1}\left(\frac{2^{5/8} x \sqrt[4]{x^3+x^5}}{\sqrt[4]{2} x^2-\sqrt{x^3+x^5}}\right)}{2^{5/8}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{x^3+x^5}}\right)}{\sqrt[8]{2}}+\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{2^{3/8}}+\frac{\sqrt{x^3+x^5}}{2^{5/8}}}{x \sqrt[4]{x^3+x^5}}\right)}{2^{5/8}}","\frac{\tan ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{x^3+x^5}}\right)}{\sqrt[8]{2}}-\frac{\tan ^{-1}\left(\frac{2^{5/8} x \sqrt[4]{x^3+x^5}}{\sqrt[4]{2} x^2-\sqrt{x^3+x^5}}\right)}{2^{5/8}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{x^3+x^5}}\right)}{\sqrt[8]{2}}+\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{2^{3/8}}+\frac{\sqrt{x^3+x^5}}{2^{5/8}}}{x \sqrt[4]{x^3+x^5}}\right)}{2^{5/8}}",1,"ArcTan[(2^(1/8)*x)/(x^3 + x^5)^(1/4)]/2^(1/8) - ArcTan[(2^(5/8)*x*(x^3 + x^5)^(1/4))/(2^(1/4)*x^2 - Sqrt[x^3 + x^5])]/2^(5/8) + ArcTanh[(2^(1/8)*x)/(x^3 + x^5)^(1/4)]/2^(1/8) + ArcTanh[(x^2/2^(3/8) + Sqrt[x^3 + x^5]/2^(5/8))/(x*(x^3 + x^5)^(1/4))]/2^(5/8)","A",1
2090,1,151,151,0.4757519,"\int \frac{\left(1+x^3\right)^{2/3} \left(2+x^6\right)}{x^6 \left(-1+x^3\right)^2} \, dx","IntegrateAlgebraic[((1 + x^3)^(2/3)*(2 + x^6))/(x^6*(-1 + x^3)^2),x]","\frac{\left(1+x^3\right)^{2/3} \left(2+10 x^3-17 x^6\right)}{5 x^5 \left(-1+x^3\right)}+\frac{5\ 2^{2/3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{1+x^3}}\right)}{\sqrt{3}}-\frac{5}{3} 2^{2/3} \log \left(-2 x+2^{2/3} \sqrt[3]{1+x^3}\right)+\frac{5 \log \left(2 x^2+2^{2/3} x \sqrt[3]{1+x^3}+\sqrt[3]{2} \left(1+x^3\right)^{2/3}\right)}{3 \sqrt[3]{2}}","\frac{\left(1+x^3\right)^{2/3} \left(2+10 x^3-17 x^6\right)}{5 x^5 \left(-1+x^3\right)}+\frac{5\ 2^{2/3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{1+x^3}}\right)}{\sqrt{3}}-\frac{5}{3} 2^{2/3} \log \left(-2 x+2^{2/3} \sqrt[3]{1+x^3}\right)+\frac{5 \log \left(2 x^2+2^{2/3} x \sqrt[3]{1+x^3}+\sqrt[3]{2} \left(1+x^3\right)^{2/3}\right)}{3 \sqrt[3]{2}}",1,"((1 + x^3)^(2/3)*(2 + 10*x^3 - 17*x^6))/(5*x^5*(-1 + x^3)) + (5*2^(2/3)*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(1 + x^3)^(1/3))])/Sqrt[3] - (5*2^(2/3)*Log[-2*x + 2^(2/3)*(1 + x^3)^(1/3)])/3 + (5*Log[2*x^2 + 2^(2/3)*x*(1 + x^3)^(1/3) + 2^(1/3)*(1 + x^3)^(2/3)])/(3*2^(1/3))","A",1
2091,1,151,151,0.4789519,"\int \frac{\left(-1+x^3\right)^{2/3} \left(-2+2 x^3+x^6\right)}{x^6 \left(1+x^3\right)^2} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(-2 + 2*x^3 + x^6))/(x^6*(1 + x^3)^2),x]","\frac{\left(-1+x^3\right)^{2/3} \left(2-15 x^3-22 x^6\right)}{5 x^5 \left(1+x^3\right)}+\frac{7\ 2^{2/3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-1+x^3}}\right)}{\sqrt{3}}-\frac{7}{3} 2^{2/3} \log \left(-2 x+2^{2/3} \sqrt[3]{-1+x^3}\right)+\frac{7 \log \left(2 x^2+2^{2/3} x \sqrt[3]{-1+x^3}+\sqrt[3]{2} \left(-1+x^3\right)^{2/3}\right)}{3 \sqrt[3]{2}}","\frac{\left(-1+x^3\right)^{2/3} \left(2-15 x^3-22 x^6\right)}{5 x^5 \left(1+x^3\right)}+\frac{7\ 2^{2/3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-1+x^3}}\right)}{\sqrt{3}}-\frac{7}{3} 2^{2/3} \log \left(-2 x+2^{2/3} \sqrt[3]{-1+x^3}\right)+\frac{7 \log \left(2 x^2+2^{2/3} x \sqrt[3]{-1+x^3}+\sqrt[3]{2} \left(-1+x^3\right)^{2/3}\right)}{3 \sqrt[3]{2}}",1,"((-1 + x^3)^(2/3)*(2 - 15*x^3 - 22*x^6))/(5*x^5*(1 + x^3)) + (7*2^(2/3)*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-1 + x^3)^(1/3))])/Sqrt[3] - (7*2^(2/3)*Log[-2*x + 2^(2/3)*(-1 + x^3)^(1/3)])/3 + (7*Log[2*x^2 + 2^(2/3)*x*(-1 + x^3)^(1/3) + 2^(1/3)*(-1 + x^3)^(2/3)])/(3*2^(1/3))","A",1
2092,1,151,151,15.6630645,"\int \frac{\left(2 b+a x^6\right) \left(-b-c x^4+a x^6\right)}{x^2 \left(-b+a x^6\right)^{3/4} \left(-b+c x^4+a x^6\right)} \, dx","IntegrateAlgebraic[((2*b + a*x^6)*(-b - c*x^4 + a*x^6))/(x^2*(-b + a*x^6)^(3/4)*(-b + c*x^4 + a*x^6)),x]","\frac{2 \sqrt[4]{-b+a x^6}}{x}+\sqrt{2} \sqrt[4]{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x \sqrt[4]{-b+a x^6}}{-\sqrt{c} x^2+\sqrt{-b+a x^6}}\right)-\sqrt{2} \sqrt[4]{c} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{c} x^2}{\sqrt{2}}+\frac{\sqrt{-b+a x^6}}{\sqrt{2} \sqrt[4]{c}}}{x \sqrt[4]{-b+a x^6}}\right)","\frac{2 \sqrt[4]{-b+a x^6}}{x}+\sqrt{2} \sqrt[4]{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x \sqrt[4]{-b+a x^6}}{-\sqrt{c} x^2+\sqrt{-b+a x^6}}\right)-\sqrt{2} \sqrt[4]{c} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{c} x^2}{\sqrt{2}}+\frac{\sqrt{-b+a x^6}}{\sqrt{2} \sqrt[4]{c}}}{x \sqrt[4]{-b+a x^6}}\right)",1,"(2*(-b + a*x^6)^(1/4))/x + Sqrt[2]*c^(1/4)*ArcTan[(Sqrt[2]*c^(1/4)*x*(-b + a*x^6)^(1/4))/(-(Sqrt[c]*x^2) + Sqrt[-b + a*x^6])] - Sqrt[2]*c^(1/4)*ArcTanh[((c^(1/4)*x^2)/Sqrt[2] + Sqrt[-b + a*x^6]/(Sqrt[2]*c^(1/4)))/(x*(-b + a*x^6)^(1/4))]","A",1
2093,1,151,151,0.3916378,"\int \frac{\sqrt{c+d \sqrt{b+a x^2}}}{x} \, dx","IntegrateAlgebraic[Sqrt[c + d*Sqrt[b + a*x^2]]/x,x]","2 \sqrt{c+d \sqrt{b+a x^2}}+\sqrt{-c+\sqrt{b} d} \tan ^{-1}\left(\frac{\sqrt{-c+\sqrt{b} d} \sqrt{c+d \sqrt{b+a x^2}}}{c-\sqrt{b} d}\right)+\sqrt{-c-\sqrt{b} d} \tan ^{-1}\left(\frac{\sqrt{-c-\sqrt{b} d} \sqrt{c+d \sqrt{b+a x^2}}}{c+\sqrt{b} d}\right)","2 \sqrt{c+d \sqrt{b+a x^2}}+\sqrt{-c+\sqrt{b} d} \tan ^{-1}\left(\frac{\sqrt{-c+\sqrt{b} d} \sqrt{c+d \sqrt{b+a x^2}}}{c-\sqrt{b} d}\right)+\sqrt{-c-\sqrt{b} d} \tan ^{-1}\left(\frac{\sqrt{-c-\sqrt{b} d} \sqrt{c+d \sqrt{b+a x^2}}}{c+\sqrt{b} d}\right)",1,"2*Sqrt[c + d*Sqrt[b + a*x^2]] + Sqrt[-c + Sqrt[b]*d]*ArcTan[(Sqrt[-c + Sqrt[b]*d]*Sqrt[c + d*Sqrt[b + a*x^2]])/(c - Sqrt[b]*d)] + Sqrt[-c - Sqrt[b]*d]*ArcTan[(Sqrt[-c - Sqrt[b]*d]*Sqrt[c + d*Sqrt[b + a*x^2]])/(c + Sqrt[b]*d)]","A",1
2094,1,151,151,0.8372548,"\int \frac{\left(-b+a^2 x^4\right) \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{b+a^2 x^4}} \, dx","IntegrateAlgebraic[((-b + a^2*x^4)*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[b + a^2*x^4],x]","\frac{3 \sqrt{a} b x^2+2 a^{5/2} x^6+2 a^{3/2} x^4 \sqrt{b+a^2 x^4}}{8 \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}-\frac{11 b \log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{8 \sqrt{2} \sqrt{a}}","\frac{3 \sqrt{a} b x^2+2 a^{5/2} x^6+2 a^{3/2} x^4 \sqrt{b+a^2 x^4}}{8 \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}-\frac{11 b \log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{8 \sqrt{2} \sqrt{a}}",1,"(3*Sqrt[a]*b*x^2 + 2*a^(5/2)*x^6 + 2*a^(3/2)*x^4*Sqrt[b + a^2*x^4])/(8*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]) - (11*b*Log[a*x^2 + Sqrt[b + a^2*x^4] + Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]])/(8*Sqrt[2]*Sqrt[a])","A",1
2095,1,151,151,0.778312,"\int \sqrt{b+a^2 x^4} \sqrt{a x^2+\sqrt{b+a^2 x^4}} \, dx","IntegrateAlgebraic[Sqrt[b + a^2*x^4]*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]],x]","\frac{3 \sqrt{a} b x^2+2 a^{5/2} x^6+2 a^{3/2} x^4 \sqrt{b+a^2 x^4}}{8 \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}+\frac{5 b \log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{8 \sqrt{2} \sqrt{a}}","\frac{3 \sqrt{a} b x^2+2 a^{5/2} x^6+2 a^{3/2} x^4 \sqrt{b+a^2 x^4}}{8 \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}+\frac{5 b \log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{8 \sqrt{2} \sqrt{a}}",1,"(3*Sqrt[a]*b*x^2 + 2*a^(5/2)*x^6 + 2*a^(3/2)*x^4*Sqrt[b + a^2*x^4])/(8*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]) + (5*b*Log[a*x^2 + Sqrt[b + a^2*x^4] + Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]])/(8*Sqrt[2]*Sqrt[a])","A",1
2096,1,152,152,0.4576739,"\int \frac{1}{(2 b+a x) \sqrt[4]{b x^2+a x^3}} \, dx","IntegrateAlgebraic[1/((2*b + a*x)*(b*x^2 + a*x^3)^(1/4)),x]","-\frac{\tan ^{-1}\left(\frac{-\frac{\sqrt{a} x^2}{2 \sqrt[4]{b}}+\frac{\sqrt[4]{b} \sqrt{b x^2+a x^3}}{\sqrt{a}}}{x \sqrt[4]{b x^2+a x^3}}\right)}{2 \sqrt{a} b^{3/4}}+\frac{\tanh ^{-1}\left(\frac{2 \sqrt{a} \sqrt[4]{b} x \sqrt[4]{b x^2+a x^3}}{a x^2+2 \sqrt{b} \sqrt{b x^2+a x^3}}\right)}{2 \sqrt{a} b^{3/4}}","-\frac{\tan ^{-1}\left(\frac{-\frac{\sqrt{a} x^2}{2 \sqrt[4]{b}}+\frac{\sqrt[4]{b} \sqrt{b x^2+a x^3}}{\sqrt{a}}}{x \sqrt[4]{b x^2+a x^3}}\right)}{2 \sqrt{a} b^{3/4}}+\frac{\tanh ^{-1}\left(\frac{2 \sqrt{a} \sqrt[4]{b} x \sqrt[4]{b x^2+a x^3}}{a x^2+2 \sqrt{b} \sqrt{b x^2+a x^3}}\right)}{2 \sqrt{a} b^{3/4}}",1,"-1/2*ArcTan[(-1/2*(Sqrt[a]*x^2)/b^(1/4) + (b^(1/4)*Sqrt[b*x^2 + a*x^3])/Sqrt[a])/(x*(b*x^2 + a*x^3)^(1/4))]/(Sqrt[a]*b^(3/4)) + ArcTanh[(2*Sqrt[a]*b^(1/4)*x*(b*x^2 + a*x^3)^(1/4))/(a*x^2 + 2*Sqrt[b]*Sqrt[b*x^2 + a*x^3])]/(2*Sqrt[a]*b^(3/4))","A",1
2097,1,78,152,0.6070796,"\int \frac{x (-4 a+3 x)}{\sqrt[3]{x^2 (-a+x)} \left(a d-d x+x^4\right)} \, dx","IntegrateAlgebraic[(x*(-4*a + 3*x))/((x^2*(-a + x))^(1/3)*(a*d - d*x + x^4)),x]","-\frac{a \text{RootSum}\left[a^3-d \text{$\#$1}^3+3 d \text{$\#$1}^6-3 d \text{$\#$1}^9+d \text{$\#$1}^{12}\&,\frac{-\log (x)+\log \left(\sqrt[3]{-a x^2+x^3}-x \text{$\#$1}\right)}{-\text{$\#$1}+\text{$\#$1}^4}\&\right]}{d}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{d} \sqrt[3]{-a x^2+x^3}}\right)}{d^{2/3}}+\frac{\log \left(a x^2-a \sqrt[3]{d} \sqrt[3]{-a x^2+x^3}\right)}{d^{2/3}}-\frac{\log \left(a^2 x^4+a^2 \sqrt[3]{d} x^2 \sqrt[3]{-a x^2+x^3}+a^2 d^{2/3} \left(-a x^2+x^3\right)^{2/3}\right)}{2 d^{2/3}}",1,"-((a*RootSum[a^3 - d*#1^3 + 3*d*#1^6 - 3*d*#1^9 + d*#1^12 & , (-Log[x] + Log[(-(a*x^2) + x^3)^(1/3) - x*#1])/(-#1 + #1^4) & ])/d)","C",1
2098,1,152,152,1.1796343,"\int \frac{\left(-b+a x^5\right)^{3/4} \left(4 b+a x^5\right)}{x^4 \left(-b+c x^4+a x^5\right)} \, dx","IntegrateAlgebraic[((-b + a*x^5)^(3/4)*(4*b + a*x^5))/(x^4*(-b + c*x^4 + a*x^5)),x]","\frac{4 \left(-b+a x^5\right)^{3/4}}{3 x^3}+\sqrt{2} c^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x \sqrt[4]{-b+a x^5}}{-\sqrt{c} x^2+\sqrt{-b+a x^5}}\right)+\sqrt{2} c^{3/4} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{c} x^2}{\sqrt{2}}+\frac{\sqrt{-b+a x^5}}{\sqrt{2} \sqrt[4]{c}}}{x \sqrt[4]{-b+a x^5}}\right)","\frac{4 \left(-b+a x^5\right)^{3/4}}{3 x^3}+\sqrt{2} c^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x \sqrt[4]{-b+a x^5}}{-\sqrt{c} x^2+\sqrt{-b+a x^5}}\right)+\sqrt{2} c^{3/4} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{c} x^2}{\sqrt{2}}+\frac{\sqrt{-b+a x^5}}{\sqrt{2} \sqrt[4]{c}}}{x \sqrt[4]{-b+a x^5}}\right)",1,"(4*(-b + a*x^5)^(3/4))/(3*x^3) + Sqrt[2]*c^(3/4)*ArcTan[(Sqrt[2]*c^(1/4)*x*(-b + a*x^5)^(1/4))/(-(Sqrt[c]*x^2) + Sqrt[-b + a*x^5])] + Sqrt[2]*c^(3/4)*ArcTanh[((c^(1/4)*x^2)/Sqrt[2] + Sqrt[-b + a*x^5]/(Sqrt[2]*c^(1/4)))/(x*(-b + a*x^5)^(1/4))]","A",1
2099,1,152,152,0.5189338,"\int \frac{\left(1+x^8\right)^{3/4}}{-1+x^8} \, dx","IntegrateAlgebraic[(1 + x^8)^(3/4)/(-1 + x^8),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{1+x^8}}\right)}{4 \sqrt[4]{2}}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{1+x^8}}{\sqrt{2} x^2-\sqrt{1+x^8}}\right)}{4\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{1+x^8}}\right)}{4 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{2 \sqrt[4]{2} x \sqrt[4]{1+x^8}}{2 x^2+\sqrt{2} \sqrt{1+x^8}}\right)}{4\ 2^{3/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{1+x^8}}\right)}{4 \sqrt[4]{2}}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{1+x^8}}{\sqrt{2} x^2-\sqrt{1+x^8}}\right)}{4\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{1+x^8}}\right)}{4 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{2 \sqrt[4]{2} x \sqrt[4]{1+x^8}}{2 x^2+\sqrt{2} \sqrt{1+x^8}}\right)}{4\ 2^{3/4}}",1,"-1/4*ArcTan[(2^(1/4)*x)/(1 + x^8)^(1/4)]/2^(1/4) + ArcTan[(2^(3/4)*x*(1 + x^8)^(1/4))/(Sqrt[2]*x^2 - Sqrt[1 + x^8])]/(4*2^(3/4)) - ArcTanh[(2^(1/4)*x)/(1 + x^8)^(1/4)]/(4*2^(1/4)) - ArcTanh[(2*2^(1/4)*x*(1 + x^8)^(1/4))/(2*x^2 + Sqrt[2]*Sqrt[1 + x^8])]/(4*2^(3/4))","A",1
2100,1,152,152,0.5033498,"\int \frac{x^4}{\sqrt[4]{-1+x^4} \left(-1+2 x^4+x^8\right)} \, dx","IntegrateAlgebraic[x^4/((-1 + x^4)^(1/4)*(-1 + 2*x^4 + x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{-1+x^4}}\right)}{4\ 2^{5/8}}+\frac{\tan ^{-1}\left(\frac{2^{5/8} x \sqrt[4]{-1+x^4}}{\sqrt[4]{2} x^2-\sqrt{-1+x^4}}\right)}{8 \sqrt[8]{2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{-1+x^4}}\right)}{4\ 2^{5/8}}-\frac{\tanh ^{-1}\left(\frac{2\ 2^{3/8} x \sqrt[4]{-1+x^4}}{2 x^2+2^{3/4} \sqrt{-1+x^4}}\right)}{8 \sqrt[8]{2}}","\frac{\tan ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{-1+x^4}}\right)}{4\ 2^{5/8}}+\frac{\tan ^{-1}\left(\frac{2^{5/8} x \sqrt[4]{-1+x^4}}{\sqrt[4]{2} x^2-\sqrt{-1+x^4}}\right)}{8 \sqrt[8]{2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{-1+x^4}}\right)}{4\ 2^{5/8}}-\frac{\tanh ^{-1}\left(\frac{2\ 2^{3/8} x \sqrt[4]{-1+x^4}}{2 x^2+2^{3/4} \sqrt{-1+x^4}}\right)}{8 \sqrt[8]{2}}",1,"ArcTan[(2^(1/8)*x)/(-1 + x^4)^(1/4)]/(4*2^(5/8)) + ArcTan[(2^(5/8)*x*(-1 + x^4)^(1/4))/(2^(1/4)*x^2 - Sqrt[-1 + x^4])]/(8*2^(1/8)) + ArcTanh[(2^(1/8)*x)/(-1 + x^4)^(1/4)]/(4*2^(5/8)) - ArcTanh[(2*2^(3/8)*x*(-1 + x^4)^(1/4))/(2*x^2 + 2^(3/4)*Sqrt[-1 + x^4])]/(8*2^(1/8))","A",1
2101,1,152,152,0.4519024,"\int \frac{\left(-1+x^4\right)^{3/4}}{-1+2 x^4+x^8} \, dx","IntegrateAlgebraic[(-1 + x^4)^(3/4)/(-1 + 2*x^4 + x^8),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{-1+x^4}}\right)}{4 \sqrt[8]{2}}-\frac{\tan ^{-1}\left(\frac{2^{5/8} x \sqrt[4]{-1+x^4}}{\sqrt[4]{2} x^2-\sqrt{-1+x^4}}\right)}{4\ 2^{5/8}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{-1+x^4}}\right)}{4 \sqrt[8]{2}}+\frac{\tanh ^{-1}\left(\frac{2\ 2^{3/8} x \sqrt[4]{-1+x^4}}{2 x^2+2^{3/4} \sqrt{-1+x^4}}\right)}{4\ 2^{5/8}}","\frac{\tan ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{-1+x^4}}\right)}{4 \sqrt[8]{2}}-\frac{\tan ^{-1}\left(\frac{2^{5/8} x \sqrt[4]{-1+x^4}}{\sqrt[4]{2} x^2-\sqrt{-1+x^4}}\right)}{4\ 2^{5/8}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{-1+x^4}}\right)}{4 \sqrt[8]{2}}+\frac{\tanh ^{-1}\left(\frac{2\ 2^{3/8} x \sqrt[4]{-1+x^4}}{2 x^2+2^{3/4} \sqrt{-1+x^4}}\right)}{4\ 2^{5/8}}",1,"ArcTan[(2^(1/8)*x)/(-1 + x^4)^(1/4)]/(4*2^(1/8)) - ArcTan[(2^(5/8)*x*(-1 + x^4)^(1/4))/(2^(1/4)*x^2 - Sqrt[-1 + x^4])]/(4*2^(5/8)) + ArcTanh[(2^(1/8)*x)/(-1 + x^4)^(1/4)]/(4*2^(1/8)) + ArcTanh[(2*2^(3/8)*x*(-1 + x^4)^(1/4))/(2*x^2 + 2^(3/4)*Sqrt[-1 + x^4])]/(4*2^(5/8))","A",1
2102,1,152,152,0.4929558,"\int \frac{1+x^4}{\left(-1+x^4\right) \sqrt{1+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[(1 + x^4)/((-1 + x^4)*Sqrt[1 + Sqrt[1 + x^2]]),x]","\frac{2 x}{\sqrt{1+\sqrt{1+x^2}}}-2 \tan ^{-1}\left(\frac{x}{\sqrt{1+\sqrt{1+x^2}}}\right)+\sqrt{2} \tan ^{-1}\left(\frac{x}{\sqrt{2} \sqrt{1+\sqrt{1+x^2}}}\right)-\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{-1+\sqrt{2}} x}{\sqrt{1+\sqrt{1+x^2}}}\right)-\sqrt{-1+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{1+\sqrt{2}} x}{\sqrt{1+\sqrt{1+x^2}}}\right)","\frac{2 x}{\sqrt{1+\sqrt{1+x^2}}}-2 \tan ^{-1}\left(\frac{x}{\sqrt{1+\sqrt{1+x^2}}}\right)+\sqrt{2} \tan ^{-1}\left(\frac{x}{\sqrt{2} \sqrt{1+\sqrt{1+x^2}}}\right)-\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{x}{\sqrt{1+\sqrt{2}} \sqrt{1+\sqrt{1+x^2}}}\right)-\sqrt{-1+\sqrt{2}} \tanh ^{-1}\left(\frac{x}{\sqrt{-1+\sqrt{2}} \sqrt{1+\sqrt{1+x^2}}}\right)",1,"(2*x)/Sqrt[1 + Sqrt[1 + x^2]] - 2*ArcTan[x/Sqrt[1 + Sqrt[1 + x^2]]] + Sqrt[2]*ArcTan[x/(Sqrt[2]*Sqrt[1 + Sqrt[1 + x^2]])] - Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[-1 + Sqrt[2]]*x)/Sqrt[1 + Sqrt[1 + x^2]]] - Sqrt[-1 + Sqrt[2]]*ArcTanh[(Sqrt[1 + Sqrt[2]]*x)/Sqrt[1 + Sqrt[1 + x^2]]]","A",1
2103,1,152,153,0.2799569,"\int \frac{1}{x^3 \left(-b+a x^2\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x^3*(-b + a*x^2)^(3/4)),x]","\frac{\sqrt[4]{-b+a x^2}}{2 b x^2}+\frac{3 a \tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^2}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^2}}\right)}{4 \sqrt{2} b^{7/4}}+\frac{3 a \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^2}}{\sqrt{b}+\sqrt{-b+a x^2}}\right)}{4 \sqrt{2} b^{7/4}}","\frac{\sqrt[4]{-b+a x^2}}{2 b x^2}-\frac{3 a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^2}}{-\sqrt{b}+\sqrt{-b+a x^2}}\right)}{4 \sqrt{2} b^{7/4}}+\frac{3 a \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^2}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^2}}\right)}{4 \sqrt{2} b^{7/4}}",1,"(-b + a*x^2)^(1/4)/(2*b*x^2) + (3*a*ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^2]/(Sqrt[2]*b^(1/4)))/(-b + a*x^2)^(1/4)])/(4*Sqrt[2]*b^(7/4)) + (3*a*ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^2)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^2])])/(4*Sqrt[2]*b^(7/4))","A",1
2104,1,153,153,0.3955981,"\int \frac{a b-2 b x+x^2}{\sqrt[4]{x (-a+x) (-b+x)^3} \left(b-(1+a d) x+d x^2\right)} \, dx","IntegrateAlgebraic[(a*b - 2*b*x + x^2)/((x*(-a + x)*(-b + x)^3)^(1/4)*(b - (1 + a*d)*x + d*x^2)),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b^3 x+\left(-3 a b^2-b^3\right) x^2+\left(3 a b+3 b^2\right) x^3+(-a-3 b) x^4+x^5}}{b-x}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b^3 x+\left(-3 a b^2-b^3\right) x^2+\left(3 a b+3 b^2\right) x^3+(-a-3 b) x^4+x^5}}{b-x}\right)}{d^{3/4}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b^3 x+\left(-3 a b^2-b^3\right) x^2+\left(3 a b+3 b^2\right) x^3+(-a-3 b) x^4+x^5}}{b-x}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b^3 x+\left(-3 a b^2-b^3\right) x^2+\left(3 a b+3 b^2\right) x^3+(-a-3 b) x^4+x^5}}{b-x}\right)}{d^{3/4}}",1,"(-2*ArcTan[(d^(1/4)*(a*b^3*x + (-3*a*b^2 - b^3)*x^2 + (3*a*b + 3*b^2)*x^3 + (-a - 3*b)*x^4 + x^5)^(1/4))/(b - x)])/d^(3/4) + (2*ArcTanh[(d^(1/4)*(a*b^3*x + (-3*a*b^2 - b^3)*x^2 + (3*a*b + 3*b^2)*x^3 + (-a - 3*b)*x^4 + x^5)^(1/4))/(b - x)])/d^(3/4)","A",1
2105,1,152,153,0.1910881,"\int \frac{1}{x^4 \left(-b+a x^3\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x^4*(-b + a*x^3)^(3/4)),x]","\frac{\sqrt[4]{-b+a x^3}}{3 b x^3}+\frac{a \tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^3}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^3}}\right)}{2 \sqrt{2} b^{7/4}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^3}}{\sqrt{b}+\sqrt{-b+a x^3}}\right)}{2 \sqrt{2} b^{7/4}}","\frac{\sqrt[4]{-b+a x^3}}{3 b x^3}-\frac{a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^3}}{-\sqrt{b}+\sqrt{-b+a x^3}}\right)}{2 \sqrt{2} b^{7/4}}+\frac{a \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^3}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^3}}\right)}{2 \sqrt{2} b^{7/4}}",1,"(-b + a*x^3)^(1/4)/(3*b*x^3) + (a*ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^3]/(Sqrt[2]*b^(1/4)))/(-b + a*x^3)^(1/4)])/(2*Sqrt[2]*b^(7/4)) + (a*ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^3)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^3])])/(2*Sqrt[2]*b^(7/4))","A",1
2106,1,152,153,0.3250333,"\int \frac{1}{x^4 \sqrt[4]{-b+a x^3}} \, dx","IntegrateAlgebraic[1/(x^4*(-b + a*x^3)^(1/4)),x]","\frac{\left(-b+a x^3\right)^{3/4}}{3 b x^3}+\frac{a \tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^3}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^3}}\right)}{6 \sqrt{2} b^{5/4}}-\frac{a \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^3}}{\sqrt{b}+\sqrt{-b+a x^3}}\right)}{6 \sqrt{2} b^{5/4}}","\frac{\left(-b+a x^3\right)^{3/4}}{3 b x^3}-\frac{a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^3}}{-\sqrt{b}+\sqrt{-b+a x^3}}\right)}{6 \sqrt{2} b^{5/4}}-\frac{a \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^3}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^3}}\right)}{6 \sqrt{2} b^{5/4}}",1,"(-b + a*x^3)^(3/4)/(3*b*x^3) + (a*ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^3]/(Sqrt[2]*b^(1/4)))/(-b + a*x^3)^(1/4)])/(6*Sqrt[2]*b^(5/4)) - (a*ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^3)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^3])])/(6*Sqrt[2]*b^(5/4))","A",1
2107,1,153,153,0.3682692,"\int \frac{(2+x) \left(-1-x+x^2\right) \sqrt[4]{\frac{1+x-2 x^2}{1+x+4 x^2}}}{2 x \left(1+2 x+x^2+x^4\right)} \, dx","IntegrateAlgebraic[((2 + x)*(-1 - x + x^2)*((1 + x - 2*x^2)/(1 + x + 4*x^2))^(1/4))/(2*x*(1 + 2*x + x^2 + x^4)),x]","\tan ^{-1}\left(\sqrt[4]{\frac{1+x-2 x^2}{1+x+4 x^2}}\right)+\tanh ^{-1}\left(\sqrt[4]{\frac{1+x-2 x^2}{1+x+4 x^2}}\right)-\frac{1}{4} \text{RootSum}\left[5+14 \text{$\#$1}^4+17 \text{$\#$1}^8\&,\frac{5 \log \left(\sqrt[4]{\frac{1+x-2 x^2}{1+x+4 x^2}}-\text{$\#$1}\right)+13 \log \left(\sqrt[4]{\frac{1+x-2 x^2}{1+x+4 x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{7 \text{$\#$1}^3+17 \text{$\#$1}^7}\&\right]","\tan ^{-1}\left(\sqrt[4]{\frac{1+x-2 x^2}{1+x+4 x^2}}\right)+\tanh ^{-1}\left(\sqrt[4]{\frac{1+x-2 x^2}{1+x+4 x^2}}\right)-\frac{1}{4} \text{RootSum}\left[5+14 \text{$\#$1}^4+17 \text{$\#$1}^8\&,\frac{5 \log \left(\sqrt[4]{\frac{1+x-2 x^2}{1+x+4 x^2}}-\text{$\#$1}\right)+13 \log \left(\sqrt[4]{\frac{1+x-2 x^2}{1+x+4 x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{7 \text{$\#$1}^3+17 \text{$\#$1}^7}\&\right]",1,"ArcTan[((1 + x - 2*x^2)/(1 + x + 4*x^2))^(1/4)] + ArcTanh[((1 + x - 2*x^2)/(1 + x + 4*x^2))^(1/4)] - RootSum[5 + 14*#1^4 + 17*#1^8 & , (5*Log[((1 + x - 2*x^2)/(1 + x + 4*x^2))^(1/4) - #1] + 13*Log[((1 + x - 2*x^2)/(1 + x + 4*x^2))^(1/4) - #1]*#1^4)/(7*#1^3 + 17*#1^7) & ]/4","A",1
2108,1,153,153,0.3677818,"\int \frac{1+x^4}{\left(-1+x^4\right) \sqrt{1-x-x^2+x^3+x^4}} \, dx","IntegrateAlgebraic[(1 + x^4)/((-1 + x^4)*Sqrt[1 - x - x^2 + x^3 + x^4]),x]","\tanh ^{-1}\left(\frac{x}{-1+x^2-\sqrt{1-x-x^2+x^3+x^4}}\right)+\frac{1}{4} \text{RootSum}\left[5-16 \text{$\#$1}+14 \text{$\#$1}^2+\text{$\#$1}^4\&,\frac{\log (x)-\log \left(1-x^2+\sqrt{1-x-x^2+x^3+x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^2+\log \left(1-x^2+\sqrt{1-x-x^2+x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^2}{-4+7 \text{$\#$1}+\text{$\#$1}^3}\&\right]","\tanh ^{-1}\left(\frac{x}{-1+x^2-\sqrt{1-x-x^2+x^3+x^4}}\right)+\frac{1}{4} \text{RootSum}\left[5-16 \text{$\#$1}+14 \text{$\#$1}^2+\text{$\#$1}^4\&,\frac{\log (x)-\log \left(1-x^2+\sqrt{1-x-x^2+x^3+x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^2+\log \left(1-x^2+\sqrt{1-x-x^2+x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^2}{-4+7 \text{$\#$1}+\text{$\#$1}^3}\&\right]",1,"ArcTanh[x/(-1 + x^2 - Sqrt[1 - x - x^2 + x^3 + x^4])] + RootSum[5 - 16*#1 + 14*#1^2 + #1^4 & , (Log[x] - Log[1 - x^2 + Sqrt[1 - x - x^2 + x^3 + x^4] - x*#1] - Log[x]*#1^2 + Log[1 - x^2 + Sqrt[1 - x - x^2 + x^3 + x^4] - x*#1]*#1^2)/(-4 + 7*#1 + #1^3) & ]/4","A",1
2109,1,152,153,0.1991768,"\int \frac{1}{x^5 \left(-b+a x^4\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x^5*(-b + a*x^4)^(3/4)),x]","\frac{\sqrt[4]{-b+a x^4}}{4 b x^4}+\frac{3 a \tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^4}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^4}}\right)}{8 \sqrt{2} b^{7/4}}+\frac{3 a \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^4}}{\sqrt{b}+\sqrt{-b+a x^4}}\right)}{8 \sqrt{2} b^{7/4}}","\frac{\sqrt[4]{-b+a x^4}}{4 b x^4}-\frac{3 a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^4}}{-\sqrt{b}+\sqrt{-b+a x^4}}\right)}{8 \sqrt{2} b^{7/4}}+\frac{3 a \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^4}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^4}}\right)}{8 \sqrt{2} b^{7/4}}",1,"(-b + a*x^4)^(1/4)/(4*b*x^4) + (3*a*ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^4]/(Sqrt[2]*b^(1/4)))/(-b + a*x^4)^(1/4)])/(8*Sqrt[2]*b^(7/4)) + (3*a*ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^4)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^4])])/(8*Sqrt[2]*b^(7/4))","A",1
2110,1,153,153,0.4959071,"\int \frac{a b^3-2 (3 a-b) b^2 x+3 (3 a-b) b x^2-4 a x^3+x^4}{\sqrt[4]{x (-a+x) (-b+x)^3} \left(a^3-\left(3 a^2+b^3 d\right) x+3 \left(a+b^2 d\right) x^2-(1+3 b d) x^3+d x^4\right)} \, dx","IntegrateAlgebraic[(a*b^3 - 2*(3*a - b)*b^2*x + 3*(3*a - b)*b*x^2 - 4*a*x^3 + x^4)/((x*(-a + x)*(-b + x)^3)^(1/4)*(a^3 - (3*a^2 + b^3*d)*x + 3*(a + b^2*d)*x^2 - (1 + 3*b*d)*x^3 + d*x^4)),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b^3 x+\left(-3 a b^2-b^3\right) x^2+\left(3 a b+3 b^2\right) x^3+(-a-3 b) x^4+x^5}}{a-x}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b^3 x+\left(-3 a b^2-b^3\right) x^2+\left(3 a b+3 b^2\right) x^3+(-a-3 b) x^4+x^5}}{a-x}\right)}{d^{3/4}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b^3 x+\left(-3 a b^2-b^3\right) x^2+\left(3 a b+3 b^2\right) x^3+(-a-3 b) x^4+x^5}}{a-x}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{a b^3 x+\left(-3 a b^2-b^3\right) x^2+\left(3 a b+3 b^2\right) x^3+(-a-3 b) x^4+x^5}}{a-x}\right)}{d^{3/4}}",1,"(-2*ArcTan[(d^(1/4)*(a*b^3*x + (-3*a*b^2 - b^3)*x^2 + (3*a*b + 3*b^2)*x^3 + (-a - 3*b)*x^4 + x^5)^(1/4))/(a - x)])/d^(3/4) + (2*ArcTanh[(d^(1/4)*(a*b^3*x + (-3*a*b^2 - b^3)*x^2 + (3*a*b + 3*b^2)*x^3 + (-a - 3*b)*x^4 + x^5)^(1/4))/(a - x)])/d^(3/4)","A",1
2111,1,152,153,0.187566,"\int \frac{1}{x^6 \left(-b+a x^5\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x^6*(-b + a*x^5)^(3/4)),x]","\frac{\sqrt[4]{-b+a x^5}}{5 b x^5}+\frac{3 a \tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^5}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^5}}\right)}{10 \sqrt{2} b^{7/4}}+\frac{3 a \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^5}}{\sqrt{b}+\sqrt{-b+a x^5}}\right)}{10 \sqrt{2} b^{7/4}}","\frac{\sqrt[4]{-b+a x^5}}{5 b x^5}-\frac{3 a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^5}}{-\sqrt{b}+\sqrt{-b+a x^5}}\right)}{10 \sqrt{2} b^{7/4}}+\frac{3 a \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^5}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^5}}\right)}{10 \sqrt{2} b^{7/4}}",1,"(-b + a*x^5)^(1/4)/(5*b*x^5) + (3*a*ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^5]/(Sqrt[2]*b^(1/4)))/(-b + a*x^5)^(1/4)])/(10*Sqrt[2]*b^(7/4)) + (3*a*ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^5)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^5])])/(10*Sqrt[2]*b^(7/4))","A",1
2112,1,152,153,0.183577,"\int \frac{1}{x^7 \left(-b+a x^6\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x^7*(-b + a*x^6)^(3/4)),x]","\frac{\sqrt[4]{-b+a x^6}}{6 b x^6}+\frac{a \tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^6}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^6}}\right)}{4 \sqrt{2} b^{7/4}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^6}}{\sqrt{b}+\sqrt{-b+a x^6}}\right)}{4 \sqrt{2} b^{7/4}}","\frac{\sqrt[4]{-b+a x^6}}{6 b x^6}-\frac{a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^6}}{-\sqrt{b}+\sqrt{-b+a x^6}}\right)}{4 \sqrt{2} b^{7/4}}+\frac{a \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^6}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^6}}\right)}{4 \sqrt{2} b^{7/4}}",1,"(-b + a*x^6)^(1/4)/(6*b*x^6) + (a*ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^6]/(Sqrt[2]*b^(1/4)))/(-b + a*x^6)^(1/4)])/(4*Sqrt[2]*b^(7/4)) + (a*ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^6)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^6])])/(4*Sqrt[2]*b^(7/4))","A",1
2113,1,71,153,1.4205765,"\int \frac{b^6+a^6 x^6}{\sqrt{b^4+a^4 x^4} \left(-b^6+a^6 x^6\right)} \, dx","IntegrateAlgebraic[(b^6 + a^6*x^6)/(Sqrt[b^4 + a^4*x^4]*(-b^6 + a^6*x^6)),x]","-\frac{2 \tan ^{-1}\left(\frac{a b x}{\sqrt{b^4+a^4 x^4}}\right)}{3 a b}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} a b x}{\sqrt{b^4+a^4 x^4}}\right)}{3 \sqrt{2} a b}","-\frac{2 \tan ^{-1}\left(\frac{a b x}{\sqrt{b^4+a^4 x^4}}\right)}{3 a b}+\frac{\tanh ^{-1}\left(\frac{\sqrt{6-4 \sqrt{2}} a b x}{b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}}\right)}{3 \sqrt{2} a b}-\frac{\tanh ^{-1}\left(\frac{\sqrt{6+4 \sqrt{2}} a b x}{b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}}\right)}{3 \sqrt{2} a b}",1,"(-2*ArcTan[(a*b*x)/Sqrt[b^4 + a^4*x^4]])/(3*a*b) - ArcTanh[(Sqrt[2]*a*b*x)/Sqrt[b^4 + a^4*x^4]]/(3*Sqrt[2]*a*b)","A",1
2114,1,153,153,2.4375505,"\int \frac{\sqrt{-1+x^2} \sqrt{x^2+x \sqrt{-1+x^2}}}{1+x} \, dx","IntegrateAlgebraic[(Sqrt[-1 + x^2]*Sqrt[x^2 + x*Sqrt[-1 + x^2]])/(1 + x),x]","\frac{1}{4} (-4-x) \sqrt{x^2+x \sqrt{-1+x^2}}+\frac{3}{4} \sqrt{-1+x^2} \sqrt{x^2+x \sqrt{-1+x^2}}-\sqrt{2} \tanh ^{-1}\left(x+\sqrt{-1+x^2}-\sqrt{2} \sqrt{x^2+x \sqrt{-1+x^2}}\right)-\frac{3 \log \left(x+\sqrt{-1+x^2}-\sqrt{2} \sqrt{x^2+x \sqrt{-1+x^2}}\right)}{4 \sqrt{2}}","\frac{1}{4} (-4-x) \sqrt{x^2+x \sqrt{-1+x^2}}+\frac{3}{4} \sqrt{-1+x^2} \sqrt{x^2+x \sqrt{-1+x^2}}-\sqrt{2} \tanh ^{-1}\left(x+\sqrt{-1+x^2}-\sqrt{2} \sqrt{x^2+x \sqrt{-1+x^2}}\right)-\frac{3 \log \left(x+\sqrt{-1+x^2}-\sqrt{2} \sqrt{x^2+x \sqrt{-1+x^2}}\right)}{4 \sqrt{2}}",1,"((-4 - x)*Sqrt[x^2 + x*Sqrt[-1 + x^2]])/4 + (3*Sqrt[-1 + x^2]*Sqrt[x^2 + x*Sqrt[-1 + x^2]])/4 - Sqrt[2]*ArcTanh[x + Sqrt[-1 + x^2] - Sqrt[2]*Sqrt[x^2 + x*Sqrt[-1 + x^2]]] - (3*Log[x + Sqrt[-1 + x^2] - Sqrt[2]*Sqrt[x^2 + x*Sqrt[-1 + x^2]]])/(4*Sqrt[2])","A",1
2115,1,153,153,0.2720664,"\int \frac{\left(1+x^4\right) \sqrt{1+\sqrt{1+x^2}}}{-1+x^4} \, dx","IntegrateAlgebraic[((1 + x^4)*Sqrt[1 + Sqrt[1 + x^2]])/(-1 + x^4),x]","\frac{4 x}{3 \sqrt{1+\sqrt{1+x^2}}}+\frac{2 x \sqrt{1+x^2}}{3 \sqrt{1+\sqrt{1+x^2}}}-2 \tan ^{-1}\left(\frac{x}{\sqrt{1+\sqrt{1+x^2}}}\right)+\sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{-1+\sqrt{2}} x}{\sqrt{1+\sqrt{1+x^2}}}\right)-\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{1+\sqrt{2}} x}{\sqrt{1+\sqrt{1+x^2}}}\right)","\frac{4 x}{3 \sqrt{1+\sqrt{1+x^2}}}+\frac{2 x \sqrt{1+x^2}}{3 \sqrt{1+\sqrt{1+x^2}}}-2 \tan ^{-1}\left(\frac{x}{\sqrt{1+\sqrt{1+x^2}}}\right)+\sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\frac{x}{\sqrt{1+\sqrt{2}} \sqrt{1+\sqrt{1+x^2}}}\right)-\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{x}{\sqrt{-1+\sqrt{2}} \sqrt{1+\sqrt{1+x^2}}}\right)",1,"(4*x)/(3*Sqrt[1 + Sqrt[1 + x^2]]) + (2*x*Sqrt[1 + x^2])/(3*Sqrt[1 + Sqrt[1 + x^2]]) - 2*ArcTan[x/Sqrt[1 + Sqrt[1 + x^2]]] + Sqrt[-1 + Sqrt[2]]*ArcTan[(Sqrt[-1 + Sqrt[2]]*x)/Sqrt[1 + Sqrt[1 + x^2]]] - Sqrt[1 + Sqrt[2]]*ArcTanh[(Sqrt[1 + Sqrt[2]]*x)/Sqrt[1 + Sqrt[1 + x^2]]]","A",1
2116,1,120,153,0.2627917,"\int \frac{-b^2+a x^2}{\left(b^2+a x^2\right) \sqrt{b+\sqrt{b^2+a x^2}}} \, dx","IntegrateAlgebraic[(-b^2 + a*x^2)/((b^2 + a*x^2)*Sqrt[b + Sqrt[b^2 + a*x^2]]),x]","\frac{2 x}{\sqrt{b+\sqrt{b^2+a x^2}}}-\frac{4 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{\sqrt{a}}+\frac{\sqrt{2} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{\sqrt{a}}","\frac{2 x}{\sqrt{b+\sqrt{b^2+a x^2}}}-\frac{4 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{\sqrt{a}}+\frac{2 \sqrt{2} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2} \sqrt{b}}\right)}{\sqrt{a}}",1,"(2*x)/Sqrt[b + Sqrt[b^2 + a*x^2]] - (4*Sqrt[b]*ArcTan[(Sqrt[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/Sqrt[a] + (Sqrt[2]*Sqrt[b]*ArcTan[(Sqrt[a]*x)/(Sqrt[2]*Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/Sqrt[a]","A",1
2117,1,153,153,0.2708377,"\int \frac{x}{1+\sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[x/(1 + Sqrt[x + Sqrt[1 + x^2]]),x]","\frac{-6 x-3 x^2-12 x^3+\left(4+12 x^2+8 x^3\right) \sqrt{x+\sqrt{1+x^2}}+\sqrt{1+x^2} \left(-3 x-12 x^2+\left(-4+12 x+8 x^2\right) \sqrt{x+\sqrt{1+x^2}}\right)}{24 x \sqrt{1+x^2}+12 \left(1+2 x^2\right)}-\tanh ^{-1}\left(\frac{-1+\sqrt{x+\sqrt{1+x^2}}}{1+\sqrt{x+\sqrt{1+x^2}}}\right)","\frac{-6 x-3 x^2-12 x^3+\left(4+12 x^2+8 x^3\right) \sqrt{x+\sqrt{1+x^2}}+\sqrt{1+x^2} \left(-3 x-12 x^2+\left(-4+12 x+8 x^2\right) \sqrt{x+\sqrt{1+x^2}}\right)}{24 x \sqrt{1+x^2}+12 \left(1+2 x^2\right)}-\tanh ^{-1}\left(\frac{-1+\sqrt{x+\sqrt{1+x^2}}}{1+\sqrt{x+\sqrt{1+x^2}}}\right)",1,"(-6*x - 3*x^2 - 12*x^3 + (4 + 12*x^2 + 8*x^3)*Sqrt[x + Sqrt[1 + x^2]] + Sqrt[1 + x^2]*(-3*x - 12*x^2 + (-4 + 12*x + 8*x^2)*Sqrt[x + Sqrt[1 + x^2]]))/(24*x*Sqrt[1 + x^2] + 12*(1 + 2*x^2)) - ArcTanh[(-1 + Sqrt[x + Sqrt[1 + x^2]])/(1 + Sqrt[x + Sqrt[1 + x^2]])]","A",1
2118,1,154,154,0.408553,"\int \frac{-2 x+(1+k) x^2}{((1-x) x (1-k x))^{2/3} \left(b-b (1+k) x+(-1+b k) x^2\right)} \, dx","IntegrateAlgebraic[(-2*x + (1 + k)*x^2)/(((1 - x)*x*(1 - k*x))^(2/3)*(b - b*(1 + k)*x + (-1 + b*k)*x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{\sqrt[3]{b}}+\frac{\log \left(x-\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{\sqrt[3]{b}}-\frac{\log \left(x^2+\sqrt[3]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 \sqrt[3]{b}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{\sqrt[3]{b}}+\frac{\log \left(x-\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{\sqrt[3]{b}}-\frac{\log \left(x^2+\sqrt[3]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 \sqrt[3]{b}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/b^(1/3) + Log[x - b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3)]/b^(1/3) - Log[x^2 + b^(1/3)*x*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + b^(2/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3)]/(2*b^(1/3))","A",1
2119,1,165,154,0.4632268,"\int \frac{\left(-b+a x^2\right) \sqrt{-b x+a x^3}}{x^2 \left(b+a x^2\right)} \, dx","IntegrateAlgebraic[((-b + a*x^2)*Sqrt[-(b*x) + a*x^3])/(x^2*(b + a*x^2)),x]","\frac{2 \sqrt{-b x+a x^3}}{x}-\sqrt[4]{a} \sqrt[4]{b} \tan ^{-1}\left(\frac{-\frac{b^{3/4}}{2 \sqrt[4]{a}}-\sqrt[4]{a} \sqrt[4]{b} x+\frac{a^{3/4} x^2}{2 \sqrt[4]{b}}}{\sqrt{-b x+a x^3}}\right)-\sqrt[4]{a} \sqrt[4]{b} \tanh ^{-1}\left(\frac{-\frac{b^{3/4}}{2 \sqrt[4]{a}}+\sqrt[4]{a} \sqrt[4]{b} x+\frac{a^{3/4} x^2}{2 \sqrt[4]{b}}}{\sqrt{-b x+a x^3}}\right)","\frac{2 \sqrt{-b x+a x^3}}{x}+\sqrt[4]{a} \sqrt[4]{b} \tan ^{-1}\left(\frac{2 \sqrt[4]{a} \sqrt[4]{b} \sqrt{-b x+a x^3}}{-b-2 \sqrt{a} \sqrt{b} x+a x^2}\right)-\sqrt[4]{a} \sqrt[4]{b} \tanh ^{-1}\left(\frac{-\frac{b^{3/4}}{2 \sqrt[4]{a}}+\sqrt[4]{a} \sqrt[4]{b} x+\frac{a^{3/4} x^2}{2 \sqrt[4]{b}}}{\sqrt{-b x+a x^3}}\right)",1,"(2*Sqrt[-(b*x) + a*x^3])/x - a^(1/4)*b^(1/4)*ArcTan[(-1/2*b^(3/4)/a^(1/4) - a^(1/4)*b^(1/4)*x + (a^(3/4)*x^2)/(2*b^(1/4)))/Sqrt[-(b*x) + a*x^3]] - a^(1/4)*b^(1/4)*ArcTanh[(-1/2*b^(3/4)/a^(1/4) + a^(1/4)*b^(1/4)*x + (a^(3/4)*x^2)/(2*b^(1/4)))/Sqrt[-(b*x) + a*x^3]]","A",1
2120,1,154,154,0.3937406,"\int \frac{\sqrt[4]{-x^3+x^4}}{x \left(1+x^3\right)} \, dx","IntegrateAlgebraic[(-x^3 + x^4)^(1/4)/(x*(1 + x^3)),x]","\frac{2}{3} \sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)-\frac{2}{3} \sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)-\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{\log (x)-\log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]","\frac{2}{3} \sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)-\frac{2}{3} \sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)-\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{\log (x)-\log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]",1,"(2*2^(1/4)*ArcTan[(2^(1/4)*x)/(-x^3 + x^4)^(1/4)])/3 - (2*2^(1/4)*ArcTanh[(2^(1/4)*x)/(-x^3 + x^4)^(1/4)])/3 - RootSum[1 - #1^4 + #1^8 & , (Log[x] - Log[(-x^3 + x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-x^3 + x^4)^(1/4) - x*#1]*#1^4)/(-#1^3 + 2*#1^7) & ]/3","A",1
2121,1,154,154,0.0008137,"\int \frac{\sqrt[4]{-x^3+x^4}}{x \left(1+x^3\right)} \, dx","IntegrateAlgebraic[(-x^3 + x^4)^(1/4)/(x*(1 + x^3)),x]","\frac{2}{3} \sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)-\frac{2}{3} \sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)-\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{\log (x)-\log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]","\frac{2}{3} \sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)-\frac{2}{3} \sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)-\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{\log (x)-\log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]",1,"(2*2^(1/4)*ArcTan[(2^(1/4)*x)/(-x^3 + x^4)^(1/4)])/3 - (2*2^(1/4)*ArcTanh[(2^(1/4)*x)/(-x^3 + x^4)^(1/4)])/3 - RootSum[1 - #1^4 + #1^8 & , (Log[x] - Log[(-x^3 + x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-x^3 + x^4)^(1/4) - x*#1]*#1^4)/(-#1^3 + 2*#1^7) & ]/3","A",1
2122,1,154,154,0.7963565,"\int \frac{\left(-3+x^4\right) \left(1-x^3+x^4\right)^{2/3}}{x^3 \left(1+x^3+x^4\right)} \, dx","IntegrateAlgebraic[((-3 + x^4)*(1 - x^3 + x^4)^(2/3))/(x^3*(1 + x^3 + x^4)),x]","\frac{3 \left(1-x^3+x^4\right)^{2/3}}{2 x^2}+2^{2/3} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2^{2/3} \sqrt[3]{1-x^3+x^4}}\right)+2^{2/3} \log \left(2 x+2^{2/3} \sqrt[3]{1-x^3+x^4}\right)-\frac{\log \left(-2 x^2+2^{2/3} x \sqrt[3]{1-x^3+x^4}-\sqrt[3]{2} \left(1-x^3+x^4\right)^{2/3}\right)}{\sqrt[3]{2}}","\frac{3 \left(1-x^3+x^4\right)^{2/3}}{2 x^2}+2^{2/3} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2^{2/3} \sqrt[3]{1-x^3+x^4}}\right)+2^{2/3} \log \left(2 x+2^{2/3} \sqrt[3]{1-x^3+x^4}\right)-\frac{\log \left(-2 x^2+2^{2/3} x \sqrt[3]{1-x^3+x^4}-\sqrt[3]{2} \left(1-x^3+x^4\right)^{2/3}\right)}{\sqrt[3]{2}}",1,"(3*(1 - x^3 + x^4)^(2/3))/(2*x^2) + 2^(2/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2^(2/3)*(1 - x^3 + x^4)^(1/3))] + 2^(2/3)*Log[2*x + 2^(2/3)*(1 - x^3 + x^4)^(1/3)] - Log[-2*x^2 + 2^(2/3)*x*(1 - x^3 + x^4)^(1/3) - 2^(1/3)*(1 - x^3 + x^4)^(2/3)]/2^(1/3)","A",1
2123,1,154,154,1.3830707,"\int \frac{\left(-4 b+a x^5\right) \left(b+a x^5\right)^{3/4}}{x^4 \left(2 b+c x^4+2 a x^5\right)} \, dx","IntegrateAlgebraic[((-4*b + a*x^5)*(b + a*x^5)^(3/4))/(x^4*(2*b + c*x^4 + 2*a*x^5)),x]","\frac{2 \left(b+a x^5\right)^{3/4}}{3 x^3}-\frac{c^{3/4} \tan ^{-1}\left(\frac{-\frac{\sqrt[4]{c} x^2}{2^{3/4}}+\frac{\sqrt{b+a x^5}}{\sqrt[4]{2} \sqrt[4]{c}}}{x \sqrt[4]{b+a x^5}}\right)}{2 \sqrt[4]{2}}+\frac{c^{3/4} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{c} x^2}{2^{3/4}}+\frac{\sqrt{b+a x^5}}{\sqrt[4]{2} \sqrt[4]{c}}}{x \sqrt[4]{b+a x^5}}\right)}{2 \sqrt[4]{2}}","\frac{2 \left(b+a x^5\right)^{3/4}}{3 x^3}+\frac{c^{3/4} \tan ^{-1}\left(\frac{2^{3/4} \sqrt[4]{c} x \sqrt[4]{b+a x^5}}{-\sqrt{c} x^2+\sqrt{2} \sqrt{b+a x^5}}\right)}{2 \sqrt[4]{2}}+\frac{c^{3/4} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{c} x^2}{2^{3/4}}+\frac{\sqrt{b+a x^5}}{\sqrt[4]{2} \sqrt[4]{c}}}{x \sqrt[4]{b+a x^5}}\right)}{2 \sqrt[4]{2}}",1,"(2*(b + a*x^5)^(3/4))/(3*x^3) - (c^(3/4)*ArcTan[(-((c^(1/4)*x^2)/2^(3/4)) + Sqrt[b + a*x^5]/(2^(1/4)*c^(1/4)))/(x*(b + a*x^5)^(1/4))])/(2*2^(1/4)) + (c^(3/4)*ArcTanh[((c^(1/4)*x^2)/2^(3/4) + Sqrt[b + a*x^5]/(2^(1/4)*c^(1/4)))/(x*(b + a*x^5)^(1/4))])/(2*2^(1/4))","A",1
2124,1,154,154,0.5614741,"\int \frac{-1+x^6}{\sqrt[3]{x^2+x^4} \left(1+x^6\right)} \, dx","IntegrateAlgebraic[(-1 + x^6)/((x^2 + x^4)^(1/3)*(1 + x^6)),x]","-\frac{\left(x^2+x^4\right)^{2/3}}{x \left(1+x^2\right)}+\frac{\tan ^{-1}\left(\frac{3^{2/3} x \sqrt[3]{x^2+x^4}}{\sqrt[3]{3} x^2-\left(x^2+x^4\right)^{2/3}}\right)}{3^{2/3}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[6]{3} x}{\sqrt[3]{x^2+x^4}}\right)}{3 \sqrt[6]{3}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{3} x^2+\frac{\left(x^2+x^4\right)^{2/3}}{\sqrt[6]{3}}}{x \sqrt[3]{x^2+x^4}}\right)}{3 \sqrt[6]{3}}","-\frac{\left(x^2+x^4\right)^{2/3}}{x \left(1+x^2\right)}+\frac{\tan ^{-1}\left(\frac{3^{2/3} x \sqrt[3]{x^2+x^4}}{\sqrt[3]{3} x^2-\left(x^2+x^4\right)^{2/3}}\right)}{3^{2/3}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[6]{3} x}{\sqrt[3]{x^2+x^4}}\right)}{3 \sqrt[6]{3}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{3} x^2+\frac{\left(x^2+x^4\right)^{2/3}}{\sqrt[6]{3}}}{x \sqrt[3]{x^2+x^4}}\right)}{3 \sqrt[6]{3}}",1,"-((x^2 + x^4)^(2/3)/(x*(1 + x^2))) + ArcTan[(3^(2/3)*x*(x^2 + x^4)^(1/3))/(3^(1/3)*x^2 - (x^2 + x^4)^(2/3))]/3^(2/3) - (2*ArcTanh[(3^(1/6)*x)/(x^2 + x^4)^(1/3)])/(3*3^(1/6)) - ArcTanh[(3^(1/6)*x^2 + (x^2 + x^4)^(2/3)/3^(1/6))/(x*(x^2 + x^4)^(1/3))]/(3*3^(1/6))","A",1
2125,1,154,154,0.4253195,"\int \frac{\sqrt[4]{2+3 x^4} \left(4+6 x^4+x^8\right)}{x^6 \left(1+x^4\right) \left(1+2 x^4\right)} \, dx","IntegrateAlgebraic[((2 + 3*x^4)^(1/4)*(4 + 6*x^4 + x^8))/(x^6*(1 + x^4)*(1 + 2*x^4)),x]","\frac{4 \sqrt[4]{2+3 x^4} \left(-1+6 x^4\right)}{5 x^5}+\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{2+3 x^4}}\right)+\frac{5 \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{2+3 x^4}}{-x^2+\sqrt{2+3 x^4}}\right)}{2 \sqrt{2}}-\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{2+3 x^4}}\right)-\frac{5 \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{2+3 x^4}}{x^2+\sqrt{2+3 x^4}}\right)}{2 \sqrt{2}}","\frac{4 \sqrt[4]{2+3 x^4} \left(-1+6 x^4\right)}{5 x^5}+\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt[4]{2+3 x^4}}\right)+\frac{5 \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{2+3 x^4}}{-x^2+\sqrt{2+3 x^4}}\right)}{2 \sqrt{2}}-\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{2+3 x^4}}\right)-\frac{5 \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{2+3 x^4}}{x^2+\sqrt{2+3 x^4}}\right)}{2 \sqrt{2}}",1,"(4*(2 + 3*x^4)^(1/4)*(-1 + 6*x^4))/(5*x^5) + ArcTan[x/(2 + 3*x^4)^(1/4)]/2 + (5*ArcTan[(Sqrt[2]*x*(2 + 3*x^4)^(1/4))/(-x^2 + Sqrt[2 + 3*x^4])])/(2*Sqrt[2]) - ArcTanh[x/(2 + 3*x^4)^(1/4)]/2 - (5*ArcTanh[(Sqrt[2]*x*(2 + 3*x^4)^(1/4))/(x^2 + Sqrt[2 + 3*x^4])])/(2*Sqrt[2])","A",1
2126,1,154,154,20.5612927,"\int \frac{\sqrt[8]{256-256 x^2+96 x^4-16 x^6+x^8}}{-1+x^3} \, dx","IntegrateAlgebraic[(256 - 256*x^2 + 96*x^4 - 16*x^6 + x^8)^(1/8)/(-1 + x^3),x]","\frac{\sqrt[8]{\left(-4+x^2\right)^4} \left(\frac{2 \tan ^{-1}\left(\frac{\sqrt{-4+x^2}}{\sqrt{3} (-2+x)}\right)}{\sqrt{3}}-\frac{1}{3} \sqrt{2 \left(-3-5 i \sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{1-\frac{2 i}{\sqrt{3}}} \sqrt{-4+x^2}}{-2+x}\right)-\frac{1}{3} \sqrt{2 \left(-3+5 i \sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{1+\frac{2 i}{\sqrt{3}}} \sqrt{-4+x^2}}{-2+x}\right)\right)}{\sqrt{-4+x^2}}","\frac{\sqrt[8]{\left(-4+x^2\right)^4} \left(\frac{2 \tan ^{-1}\left(\frac{\sqrt{-4+x^2}}{\sqrt{3} (-2+x)}\right)}{\sqrt{3}}-\frac{1}{3} \sqrt{2 \left(-3-5 i \sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{1-\frac{2 i}{\sqrt{3}}} \sqrt{-4+x^2}}{-2+x}\right)-\frac{1}{3} \sqrt{2 \left(-3+5 i \sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{1+\frac{2 i}{\sqrt{3}}} \sqrt{-4+x^2}}{-2+x}\right)\right)}{\sqrt{-4+x^2}}",1,"(((-4 + x^2)^4)^(1/8)*((2*ArcTan[Sqrt[-4 + x^2]/(Sqrt[3]*(-2 + x))])/Sqrt[3] - (Sqrt[2*(-3 - (5*I)*Sqrt[3])]*ArcTan[(Sqrt[1 - (2*I)/Sqrt[3]]*Sqrt[-4 + x^2])/(-2 + x)])/3 - (Sqrt[2*(-3 + (5*I)*Sqrt[3])]*ArcTan[(Sqrt[1 + (2*I)/Sqrt[3]]*Sqrt[-4 + x^2])/(-2 + x)])/3))/Sqrt[-4 + x^2]","A",1
2127,1,154,154,3.1081311,"\int \frac{-36-6 x^2+6 x^3+x^6}{x \left(-6+x^3\right) \sqrt[6]{\frac{6+x^3}{-6+x^3}} \left(36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right)} \, dx","IntegrateAlgebraic[(-36 - 6*x^2 + 6*x^3 + x^6)/(x*(-6 + x^3)*((6 + x^3)/(-6 + x^3))^(1/6)*(36 - 90*x + 122*x^2 - 96*x^3 + 51*x^4 - 26*x^5 + 15*x^6 - 6*x^7 + x^8)),x]","-\frac{1}{3} \tan ^{-1}\left(\frac{\sqrt[6]{\frac{6+x^3}{-6+x^3}}}{-1+x}\right)-\frac{1}{6} \tan ^{-1}\left(\frac{-1+2 x-x^2+\sqrt[3]{\frac{6+x^3}{-6+x^3}}}{(-1+x) \sqrt[6]{\frac{6+x^3}{-6+x^3}}}\right)+\frac{\tanh ^{-1}\left(\frac{\left(-\sqrt{3}+\sqrt{3} x\right) \sqrt[6]{\frac{6+x^3}{-6+x^3}}}{1-2 x+x^2+\sqrt[3]{\frac{6+x^3}{-6+x^3}}}\right)}{2 \sqrt{3}}","-\frac{1}{3} \tan ^{-1}\left(\frac{\sqrt[6]{\frac{6+x^3}{-6+x^3}}}{-1+x}\right)-\frac{1}{6} \tan ^{-1}\left(\frac{-1+2 x-x^2+\sqrt[3]{\frac{6+x^3}{-6+x^3}}}{(-1+x) \sqrt[6]{\frac{6+x^3}{-6+x^3}}}\right)+\frac{\tanh ^{-1}\left(\frac{\left(-\sqrt{3}+\sqrt{3} x\right) \sqrt[6]{\frac{6+x^3}{-6+x^3}}}{1-2 x+x^2+\sqrt[3]{\frac{6+x^3}{-6+x^3}}}\right)}{2 \sqrt{3}}",1,"-1/3*ArcTan[((6 + x^3)/(-6 + x^3))^(1/6)/(-1 + x)] - ArcTan[(-1 + 2*x - x^2 + ((6 + x^3)/(-6 + x^3))^(1/3))/((-1 + x)*((6 + x^3)/(-6 + x^3))^(1/6))]/6 + ArcTanh[((-Sqrt[3] + Sqrt[3]*x)*((6 + x^3)/(-6 + x^3))^(1/6))/(1 - 2*x + x^2 + ((6 + x^3)/(-6 + x^3))^(1/3))]/(2*Sqrt[3])","A",1
2128,1,154,154,1.239622,"\int \frac{2 b-a x^4+2 x^8}{\sqrt[4]{b+a x^4} \left(-2 b+a x^4+x^8\right)} \, dx","IntegrateAlgebraic[(2*b - a*x^4 + 2*x^8)/((b + a*x^4)^(1/4)*(-2*b + a*x^4 + x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}-\frac{3}{4} \text{RootSum}\left[3 a^2-b-5 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-3 a \log (x)+3 a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)+2 \log (x) \text{$\#$1}^4-2 \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-5 a \text{$\#$1}+4 \text{$\#$1}^5}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}-\frac{3}{4} \text{RootSum}\left[3 a^2-b-5 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{3 a \log (x)-3 a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)-2 \log (x) \text{$\#$1}^4+2 \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{5 a \text{$\#$1}-4 \text{$\#$1}^5}\&\right]",1,"ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(1/4) + ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(1/4) - (3*RootSum[3*a^2 - b - 5*a*#1^4 + 2*#1^8 & , (-3*a*Log[x] + 3*a*Log[(b + a*x^4)^(1/4) - x*#1] + 2*Log[x]*#1^4 - 2*Log[(b + a*x^4)^(1/4) - x*#1]*#1^4)/(-5*a*#1 + 4*#1^5) & ])/4","A",1
2129,1,154,154,0.0016966,"\int \frac{2 b-a x^4+2 x^8}{\sqrt[4]{b+a x^4} \left(-2 b+a x^4+x^8\right)} \, dx","IntegrateAlgebraic[(2*b - a*x^4 + 2*x^8)/((b + a*x^4)^(1/4)*(-2*b + a*x^4 + x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}-\frac{3}{4} \text{RootSum}\left[3 a^2-b-5 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-3 a \log (x)+3 a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)+2 \log (x) \text{$\#$1}^4-2 \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-5 a \text{$\#$1}+4 \text{$\#$1}^5}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}-\frac{3}{4} \text{RootSum}\left[3 a^2-b-5 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{3 a \log (x)-3 a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)-2 \log (x) \text{$\#$1}^4+2 \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{5 a \text{$\#$1}-4 \text{$\#$1}^5}\&\right]",1,"ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(1/4) + ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(1/4) - (3*RootSum[3*a^2 - b - 5*a*#1^4 + 2*#1^8 & , (-3*a*Log[x] + 3*a*Log[(b + a*x^4)^(1/4) - x*#1] + 2*Log[x]*#1^4 - 2*Log[(b + a*x^4)^(1/4) - x*#1]*#1^4)/(-5*a*#1 + 4*#1^5) & ])/4","A",1
2130,1,154,154,3.0892179,"\int \frac{-b+a x^8}{\left(b+a x^8\right) \sqrt[4]{b-c x^4+a x^8}} \, dx","IntegrateAlgebraic[(-b + a*x^8)/((b + a*x^8)*(b - c*x^4 + a*x^8)^(1/4)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x \sqrt[4]{b-c x^4+a x^8}}{-\sqrt{c} x^2+\sqrt{b-c x^4+a x^8}}\right)}{2 \sqrt{2} \sqrt[4]{c}}-\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[4]{c} x^2}{\sqrt{2}}+\frac{\sqrt{b-c x^4+a x^8}}{\sqrt{2} \sqrt[4]{c}}}{x \sqrt[4]{b-c x^4+a x^8}}\right)}{2 \sqrt{2} \sqrt[4]{c}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x \sqrt[4]{b-c x^4+a x^8}}{-\sqrt{c} x^2+\sqrt{b-c x^4+a x^8}}\right)}{2 \sqrt{2} \sqrt[4]{c}}-\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[4]{c} x^2}{\sqrt{2}}+\frac{\sqrt{b-c x^4+a x^8}}{\sqrt{2} \sqrt[4]{c}}}{x \sqrt[4]{b-c x^4+a x^8}}\right)}{2 \sqrt{2} \sqrt[4]{c}}",1,"-1/2*ArcTan[(Sqrt[2]*c^(1/4)*x*(b - c*x^4 + a*x^8)^(1/4))/(-(Sqrt[c]*x^2) + Sqrt[b - c*x^4 + a*x^8])]/(Sqrt[2]*c^(1/4)) - ArcTanh[((c^(1/4)*x^2)/Sqrt[2] + Sqrt[b - c*x^4 + a*x^8]/(Sqrt[2]*c^(1/4)))/(x*(b - c*x^4 + a*x^8)^(1/4))]/(2*Sqrt[2]*c^(1/4))","A",1
2131,1,154,154,5.5533492,"\int \frac{1}{x^2 \sqrt{-b x+a^2 x^2} \left(a x^2+x \sqrt{-b x+a^2 x^2}\right)^{3/2}} \, dx","IntegrateAlgebraic[1/(x^2*Sqrt[-(b*x) + a^2*x^2]*(a*x^2 + x*Sqrt[-(b*x) + a^2*x^2])^(3/2)),x]","-\frac{4 \sqrt{-b x+a^2 x^2} \left(3003 b^3-4032 a^2 b^2 x-5120 a^4 b x^2-8192 a^6 x^3\right) \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{45045 b^5 x^5}+\frac{4 \left(6699 a b^3+448 a^3 b^2 x+1024 a^5 b x^2+8192 a^7 x^3\right) \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{45045 b^5 x^4}","-\frac{4 \sqrt{-b x+a^2 x^2} \left(3003 b^3-4032 a^2 b^2 x-5120 a^4 b x^2-8192 a^6 x^3\right) \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{45045 b^5 x^5}+\frac{4 \left(6699 a b^3+448 a^3 b^2 x+1024 a^5 b x^2+8192 a^7 x^3\right) \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{45045 b^5 x^4}",1,"(-4*Sqrt[-(b*x) + a^2*x^2]*(3003*b^3 - 4032*a^2*b^2*x - 5120*a^4*b*x^2 - 8192*a^6*x^3)*Sqrt[x*(a*x + Sqrt[-(b*x) + a^2*x^2])])/(45045*b^5*x^5) + (4*(6699*a*b^3 + 448*a^3*b^2*x + 1024*a^5*b*x^2 + 8192*a^7*x^3)*Sqrt[x*(a*x + Sqrt[-(b*x) + a^2*x^2])])/(45045*b^5*x^4)","A",1
2132,1,154,154,0.2435101,"\int \frac{1+\sqrt{1+x^2}}{1+\sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[(1 + Sqrt[1 + x^2])/(1 + Sqrt[x + Sqrt[1 + x^2]]),x]","\frac{104-36 x+214 x^2-24 x^3+\left(40+48 x+72 x^2+16 x^3\right) \sqrt{x+\sqrt{1+x^2}}+\sqrt{1+x^2} \left(-24+214 x-24 x^2+\left(40+72 x+16 x^2\right) \sqrt{x+\sqrt{1+x^2}}\right)}{48 x \sqrt{1+x^2}+24 \left(1+2 x^2\right)}+\frac{5}{4} \log \left(x+\sqrt{1+x^2}\right)-4 \log \left(1+\sqrt{x+\sqrt{1+x^2}}\right)","\frac{104-36 x+214 x^2-24 x^3+\left(40+48 x+72 x^2+16 x^3\right) \sqrt{x+\sqrt{1+x^2}}+\sqrt{1+x^2} \left(-24+214 x-24 x^2+\left(40+72 x+16 x^2\right) \sqrt{x+\sqrt{1+x^2}}\right)}{48 x \sqrt{1+x^2}+24 \left(1+2 x^2\right)}+\frac{5}{4} \log \left(x+\sqrt{1+x^2}\right)-4 \log \left(1+\sqrt{x+\sqrt{1+x^2}}\right)",1,"(104 - 36*x + 214*x^2 - 24*x^3 + (40 + 48*x + 72*x^2 + 16*x^3)*Sqrt[x + Sqrt[1 + x^2]] + Sqrt[1 + x^2]*(-24 + 214*x - 24*x^2 + (40 + 72*x + 16*x^2)*Sqrt[x + Sqrt[1 + x^2]]))/(48*x*Sqrt[1 + x^2] + 24*(1 + 2*x^2)) + (5*Log[x + Sqrt[1 + x^2]])/4 - 4*Log[1 + Sqrt[x + Sqrt[1 + x^2]]]","A",1
2133,1,155,155,0.294406,"\int \frac{-2+(1+k) x}{\sqrt[3]{(1-x) x (1-k x)} \left(b-b (1+k) x+(-1+b k) x^2\right)} \, dx","IntegrateAlgebraic[(-2 + (1 + k)*x)/(((1 - x)*x*(1 - k*x))^(1/3)*(b - b*(1 + k)*x + (-1 + b*k)*x^2)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{b^{2/3}}+\frac{\log \left(x-\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{b^{2/3}}-\frac{\log \left(x^2+\sqrt[3]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 b^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{b^{2/3}}+\frac{\log \left(x-\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{b^{2/3}}-\frac{\log \left(x^2+\sqrt[3]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 b^{2/3}}",1,"-((Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/b^(2/3)) + Log[x - b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3)]/b^(2/3) - Log[x^2 + b^(1/3)*x*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + b^(2/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3)]/(2*b^(2/3))","A",1
2134,1,155,155,0.4975358,"\int \frac{\sqrt[4]{-b+a x^4} \left(-8 b+a x^8\right)}{x^{10} \left(b+a x^4\right)} \, dx","IntegrateAlgebraic[((-b + a*x^4)^(1/4)*(-8*b + a*x^8))/(x^10*(b + a*x^4)),x]","\frac{\sqrt[4]{-b+a x^4} \left(8 b^2-16 a b x^4+80 a^2 x^8-9 a b x^8\right)}{9 b^2 x^9}+\frac{\left(8 a^{9/4}-a^{5/4} b\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2^{3/4} b^2}-\frac{\left(8 a^{9/4}-a^{5/4} b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2^{3/4} b^2}","\frac{\sqrt[4]{-b+a x^4} \left(8 b^2-16 a b x^4+80 a^2 x^8-9 a b x^8\right)}{9 b^2 x^9}+\frac{\left(8 a^{9/4}-a^{5/4} b\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2^{3/4} b^2}-\frac{\left(8 a^{9/4}-a^{5/4} b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2^{3/4} b^2}",1,"((-b + a*x^4)^(1/4)*(8*b^2 - 16*a*b*x^4 + 80*a^2*x^8 - 9*a*b*x^8))/(9*b^2*x^9) + ((8*a^(9/4) - a^(5/4)*b)*ArcTan[(2^(1/4)*a^(1/4)*x)/(-b + a*x^4)^(1/4)])/(2^(3/4)*b^2) - ((8*a^(9/4) - a^(5/4)*b)*ArcTanh[(2^(1/4)*a^(1/4)*x)/(-b + a*x^4)^(1/4)])/(2^(3/4)*b^2)","A",1
2135,1,160,155,0.371502,"\int \frac{x}{x+\sqrt{x+\sqrt{1+x}}} \, dx","IntegrateAlgebraic[x/(x + Sqrt[x + Sqrt[1 + x]]),x]","1+x-2 \sqrt{x+\sqrt{1+x}}-\log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}\right)+4 \text{RootSum}\left[1+3 \text{$\#$1}-5 \text{$\#$1}^2+2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{\log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}+\log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^3}{3-10 \text{$\#$1}+6 \text{$\#$1}^2+4 \text{$\#$1}^3}\&\right]","x-2 \sqrt{x+\sqrt{1+x}}-\log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}\right)+4 \text{RootSum}\left[1+3 \text{$\#$1}-5 \text{$\#$1}^2+2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{\log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}+\log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^3}{3-10 \text{$\#$1}+6 \text{$\#$1}^2+4 \text{$\#$1}^3}\&\right]",1,"1 + x - 2*Sqrt[x + Sqrt[1 + x]] - Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]] + 4*RootSum[1 + 3*#1 - 5*#1^2 + 2*#1^3 + #1^4 & , (Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1 + Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^3)/(3 - 10*#1 + 6*#1^2 + 4*#1^3) & ]","A",1
2136,1,160,155,0.0011513,"\int \frac{x}{x+\sqrt{x+\sqrt{1+x}}} \, dx","IntegrateAlgebraic[x/(x + Sqrt[x + Sqrt[1 + x]]),x]","1+x-2 \sqrt{x+\sqrt{1+x}}-\log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}\right)+4 \text{RootSum}\left[1+3 \text{$\#$1}-5 \text{$\#$1}^2+2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{\log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}+\log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^3}{3-10 \text{$\#$1}+6 \text{$\#$1}^2+4 \text{$\#$1}^3}\&\right]","x-2 \sqrt{x+\sqrt{1+x}}-\log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}\right)+4 \text{RootSum}\left[1+3 \text{$\#$1}-5 \text{$\#$1}^2+2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{\log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}+\log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^3}{3-10 \text{$\#$1}+6 \text{$\#$1}^2+4 \text{$\#$1}^3}\&\right]",1,"1 + x - 2*Sqrt[x + Sqrt[1 + x]] - Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]] + 4*RootSum[1 + 3*#1 - 5*#1^2 + 2*#1^3 + #1^4 & , (Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1 + Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^3)/(3 - 10*#1 + 6*#1^2 + 4*#1^3) & ]","A",1
2137,1,156,156,1.0110047,"\int \frac{\left(-6+x^2\right) \left(-2+x^2\right) \left(2-x^2+x^3\right) \sqrt[3]{-2+x^2+2 x^3}}{x^5 \left(-2+x^2+x^3\right)^2} \, dx","IntegrateAlgebraic[((-6 + x^2)*(-2 + x^2)*(2 - x^2 + x^3)*(-2 + x^2 + 2*x^3)^(1/3))/(x^5*(-2 + x^2 + x^3)^2),x]","\frac{\sqrt[3]{-2+x^2+2 x^3} \left(12-12 x^2+54 x^3+3 x^4-27 x^5-38 x^6\right)}{4 x^4 \left(-2+x^2+x^3\right)}-\frac{7 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-2+x^2+2 x^3}}\right)}{\sqrt{3}}-\frac{7}{3} \log \left(-x+\sqrt[3]{-2+x^2+2 x^3}\right)+\frac{7}{6} \log \left(x^2+x \sqrt[3]{-2+x^2+2 x^3}+\left(-2+x^2+2 x^3\right)^{2/3}\right)","\frac{\sqrt[3]{-2+x^2+2 x^3} \left(12-12 x^2+54 x^3+3 x^4-27 x^5-38 x^6\right)}{4 x^4 \left(-2+x^2+x^3\right)}-\frac{7 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-2+x^2+2 x^3}}\right)}{\sqrt{3}}-\frac{7}{3} \log \left(-x+\sqrt[3]{-2+x^2+2 x^3}\right)+\frac{7}{6} \log \left(x^2+x \sqrt[3]{-2+x^2+2 x^3}+\left(-2+x^2+2 x^3\right)^{2/3}\right)",1,"((-2 + x^2 + 2*x^3)^(1/3)*(12 - 12*x^2 + 54*x^3 + 3*x^4 - 27*x^5 - 38*x^6))/(4*x^4*(-2 + x^2 + x^3)) - (7*ArcTan[(Sqrt[3]*x)/(x + 2*(-2 + x^2 + 2*x^3)^(1/3))])/Sqrt[3] - (7*Log[-x + (-2 + x^2 + 2*x^3)^(1/3)])/3 + (7*Log[x^2 + x*(-2 + x^2 + 2*x^3)^(1/3) + (-2 + x^2 + 2*x^3)^(2/3)])/6","A",1
2138,1,156,156,0.407746,"\int \frac{(-1+x) \sqrt[4]{-x^3+x^4}}{x \left(1+x^3\right)} \, dx","IntegrateAlgebraic[((-1 + x)*(-x^3 + x^4)^(1/4))/(x*(1 + x^3)),x]","-\frac{4}{3} \sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)+\frac{4}{3} \sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)-\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 \log (x)+2 \log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]","-\frac{4}{3} \sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)+\frac{4}{3} \sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)-\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 \log (x)+2 \log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]",1,"(-4*2^(1/4)*ArcTan[(2^(1/4)*x)/(-x^3 + x^4)^(1/4)])/3 + (4*2^(1/4)*ArcTanh[(2^(1/4)*x)/(-x^3 + x^4)^(1/4)])/3 - RootSum[1 - #1^4 + #1^8 & , (-2*Log[x] + 2*Log[(-x^3 + x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-x^3 + x^4)^(1/4) - x*#1]*#1^4)/(-#1^3 + 2*#1^7) & ]/3","A",1
2139,1,156,156,0.0008895,"\int \frac{(-1+x) \sqrt[4]{-x^3+x^4}}{x \left(1+x^3\right)} \, dx","IntegrateAlgebraic[((-1 + x)*(-x^3 + x^4)^(1/4))/(x*(1 + x^3)),x]","-\frac{4}{3} \sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)+\frac{4}{3} \sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)-\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 \log (x)+2 \log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]","-\frac{4}{3} \sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)+\frac{4}{3} \sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)-\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 \log (x)+2 \log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]",1,"(-4*2^(1/4)*ArcTan[(2^(1/4)*x)/(-x^3 + x^4)^(1/4)])/3 + (4*2^(1/4)*ArcTanh[(2^(1/4)*x)/(-x^3 + x^4)^(1/4)])/3 - RootSum[1 - #1^4 + #1^8 & , (-2*Log[x] + 2*Log[(-x^3 + x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-x^3 + x^4)^(1/4) - x*#1]*#1^4)/(-#1^3 + 2*#1^7) & ]/3","A",1
2140,1,156,156,0.5323475,"\int \frac{(1+x) \sqrt[4]{x^3+x^5}}{x \left(-1+x^3\right)} \, dx","IntegrateAlgebraic[((1 + x)*(x^3 + x^5)^(1/4))/(x*(-1 + x^3)),x]","\frac{2}{3} \sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^5}}\right)-\frac{1}{3} \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{x^3+x^5}}{-x^2+\sqrt{x^3+x^5}}\right)-\frac{2}{3} \sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^5}}\right)+\frac{1}{3} \sqrt{2} \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{x^3+x^5}}{\sqrt{2}}}{x \sqrt[4]{x^3+x^5}}\right)","\frac{2}{3} \sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^5}}\right)-\frac{1}{3} \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{x^3+x^5}}{-x^2+\sqrt{x^3+x^5}}\right)-\frac{2}{3} \sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^5}}\right)+\frac{1}{3} \sqrt{2} \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{x^3+x^5}}{\sqrt{2}}}{x \sqrt[4]{x^3+x^5}}\right)",1,"(2*2^(1/4)*ArcTan[(2^(1/4)*x)/(x^3 + x^5)^(1/4)])/3 - (Sqrt[2]*ArcTan[(Sqrt[2]*x*(x^3 + x^5)^(1/4))/(-x^2 + Sqrt[x^3 + x^5])])/3 - (2*2^(1/4)*ArcTanh[(2^(1/4)*x)/(x^3 + x^5)^(1/4)])/3 + (Sqrt[2]*ArcTanh[(x^2/Sqrt[2] + Sqrt[x^3 + x^5]/Sqrt[2])/(x*(x^3 + x^5)^(1/4))])/3","A",1
2141,1,156,156,0.5389901,"\int \frac{x^3}{\sqrt[3]{-b x^2+a x^3} \left(-b^2+a^2 x^6\right)} \, dx","IntegrateAlgebraic[x^3/((-(b*x^2) + a*x^3)^(1/3)*(-b^2 + a^2*x^6)),x]","\frac{\text{RootSum}\left[a^3-a b^2-3 a^2 \text{$\#$1}^3+3 a \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 a b}-\frac{\text{RootSum}\left[a^3+a b^2-3 a^2 \text{$\#$1}^3+3 a \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 a b}","\frac{\text{RootSum}\left[a^3-a b^2-3 a^2 \text{$\#$1}^3+3 a \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 a b}-\frac{\text{RootSum}\left[a^3+a b^2-3 a^2 \text{$\#$1}^3+3 a \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 a b}",1,"RootSum[a^3 - a*b^2 - 3*a^2*#1^3 + 3*a*#1^6 - #1^9 & , (-Log[x] + Log[(-(b*x^2) + a*x^3)^(1/3) - x*#1])/#1 & ]/(6*a*b) - RootSum[a^3 + a*b^2 - 3*a^2*#1^3 + 3*a*#1^6 - #1^9 & , (-Log[x] + Log[(-(b*x^2) + a*x^3)^(1/3) - x*#1])/#1 & ]/(6*a*b)","A",1
2142,1,153,156,0.2110571,"\int \frac{1+x^2}{\left(-1+x^2\right) \sqrt{1+\sqrt{1+\sqrt{1+x}}}} \, dx","IntegrateAlgebraic[(1 + x^2)/((-1 + x^2)*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]),x]","\frac{-4 \sqrt{1+\sqrt{1+x}} \left(-16+6 \sqrt{1+x}\right)-4 \left(-121-2 \sqrt{1+x}-30 (1+x)\right)}{105 \sqrt{1+\sqrt{1+\sqrt{1+x}}}}+2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{1+\sqrt{1+\sqrt{1+x}}}}{\sqrt{2}}\right)+\text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{-4 (-151-30 x)+64 \sqrt{1+\sqrt{1+x}}+\sqrt{1+x} \left(8-24 \sqrt{1+\sqrt{1+x}}\right)}{105 \sqrt{1+\sqrt{1+\sqrt{1+x}}}}+2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{1+\sqrt{1+\sqrt{1+x}}}}{\sqrt{2}}\right)+\text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"(-4*Sqrt[1 + Sqrt[1 + x]]*(-16 + 6*Sqrt[1 + x]) - 4*(-121 - 2*Sqrt[1 + x] - 30*(1 + x)))/(105*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]) + 2*Sqrt[2]*ArcTanh[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]/Sqrt[2]] + RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]] - #1]/#1 & ]","A",1
2143,1,153,156,0.0017115,"\int \frac{1+x^2}{\left(-1+x^2\right) \sqrt{1+\sqrt{1+\sqrt{1+x}}}} \, dx","IntegrateAlgebraic[(1 + x^2)/((-1 + x^2)*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]),x]","\frac{-4 \sqrt{1+\sqrt{1+x}} \left(-16+6 \sqrt{1+x}\right)-4 \left(-121-2 \sqrt{1+x}-30 (1+x)\right)}{105 \sqrt{1+\sqrt{1+\sqrt{1+x}}}}+2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{1+\sqrt{1+\sqrt{1+x}}}}{\sqrt{2}}\right)+\text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{-4 (-151-30 x)+64 \sqrt{1+\sqrt{1+x}}+\sqrt{1+x} \left(8-24 \sqrt{1+\sqrt{1+x}}\right)}{105 \sqrt{1+\sqrt{1+\sqrt{1+x}}}}+2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{1+\sqrt{1+\sqrt{1+x}}}}{\sqrt{2}}\right)+\text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"(-4*Sqrt[1 + Sqrt[1 + x]]*(-16 + 6*Sqrt[1 + x]) - 4*(-121 - 2*Sqrt[1 + x] - 30*(1 + x)))/(105*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]) + 2*Sqrt[2]*ArcTanh[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]/Sqrt[2]] + RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]] - #1]/#1 & ]","A",1
2144,1,157,157,0.4322189,"\int \frac{-1+x+x^3}{\left(1-x+x^3\right) \sqrt[3]{-x^2+x^3}} \, dx","IntegrateAlgebraic[(-1 + x + x^3)/((1 - x + x^3)*(-x^2 + x^3)^(1/3)),x]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{-x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)+2 \text{RootSum}\left[-1+\text{$\#$1}^3-2 \text{$\#$1}^6+\text{$\#$1}^9\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-1+3 \text{$\#$1}^3}\&\right]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{-x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)+2 \text{RootSum}\left[-1+\text{$\#$1}^3-2 \text{$\#$1}^6+\text{$\#$1}^9\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-1+3 \text{$\#$1}^3}\&\right]",1,"Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-x^2 + x^3)^(1/3))] - Log[-x + (-x^2 + x^3)^(1/3)] + Log[x^2 + x*(-x^2 + x^3)^(1/3) + (-x^2 + x^3)^(2/3)]/2 + 2*RootSum[-1 + #1^3 - 2*#1^6 + #1^9 & , (-(Log[x]*#1^2) + Log[(-x^2 + x^3)^(1/3) - x*#1]*#1^2)/(-1 + 3*#1^3) & ]","A",1
2145,1,157,157,0.001108,"\int \frac{-1+x+x^3}{\left(1-x+x^3\right) \sqrt[3]{-x^2+x^3}} \, dx","IntegrateAlgebraic[(-1 + x + x^3)/((1 - x + x^3)*(-x^2 + x^3)^(1/3)),x]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{-x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)+2 \text{RootSum}\left[-1+\text{$\#$1}^3-2 \text{$\#$1}^6+\text{$\#$1}^9\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-1+3 \text{$\#$1}^3}\&\right]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{-x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)+2 \text{RootSum}\left[-1+\text{$\#$1}^3-2 \text{$\#$1}^6+\text{$\#$1}^9\&,\frac{-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2}{-1+3 \text{$\#$1}^3}\&\right]",1,"Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-x^2 + x^3)^(1/3))] - Log[-x + (-x^2 + x^3)^(1/3)] + Log[x^2 + x*(-x^2 + x^3)^(1/3) + (-x^2 + x^3)^(2/3)]/2 + 2*RootSum[-1 + #1^3 - 2*#1^6 + #1^9 & , (-(Log[x]*#1^2) + Log[(-x^2 + x^3)^(1/3) - x*#1]*#1^2)/(-1 + 3*#1^3) & ]","A",1
2146,1,157,157,0.3569454,"\int \frac{\left(1+2 x^3\right)^{4/3} \left(1+3 x^3\right)}{x^8 \left(1+4 x^3\right)} \, dx","IntegrateAlgebraic[((1 + 2*x^3)^(4/3)*(1 + 3*x^3))/(x^8*(1 + 4*x^3)),x]","\frac{\sqrt[3]{1+2 x^3} \left(-4-9 x^3-58 x^6\right)}{28 x^7}-\frac{2 \sqrt[3]{2} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2^{2/3} \sqrt[3]{1+2 x^3}}\right)}{\sqrt{3}}+\frac{2}{3} \sqrt[3]{2} \log \left(2 x+2^{2/3} \sqrt[3]{1+2 x^3}\right)-\frac{1}{3} \sqrt[3]{2} \log \left(-2 x^2+2^{2/3} x \sqrt[3]{1+2 x^3}-\sqrt[3]{2} \left(1+2 x^3\right)^{2/3}\right)","\frac{\sqrt[3]{1+2 x^3} \left(-4-9 x^3-58 x^6\right)}{28 x^7}-\frac{2 \sqrt[3]{2} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2^{2/3} \sqrt[3]{1+2 x^3}}\right)}{\sqrt{3}}+\frac{2}{3} \sqrt[3]{2} \log \left(2 x+2^{2/3} \sqrt[3]{1+2 x^3}\right)-\frac{1}{3} \sqrt[3]{2} \log \left(-2 x^2+2^{2/3} x \sqrt[3]{1+2 x^3}-\sqrt[3]{2} \left(1+2 x^3\right)^{2/3}\right)",1,"((1 + 2*x^3)^(1/3)*(-4 - 9*x^3 - 58*x^6))/(28*x^7) - (2*2^(1/3)*ArcTan[(Sqrt[3]*x)/(-x + 2^(2/3)*(1 + 2*x^3)^(1/3))])/Sqrt[3] + (2*2^(1/3)*Log[2*x + 2^(2/3)*(1 + 2*x^3)^(1/3)])/3 - (2^(1/3)*Log[-2*x^2 + 2^(2/3)*x*(1 + 2*x^3)^(1/3) - 2^(1/3)*(1 + 2*x^3)^(2/3)])/3","A",1
2147,1,157,157,0.4491455,"\int \frac{-1+x^2}{\left(1+x^2\right) \sqrt[3]{-x^2+x^4}} \, dx","IntegrateAlgebraic[(-1 + x^2)/((1 + x^2)*(-x^2 + x^4)^(1/3)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{2} x}{\sqrt[3]{-x^2+x^4}}\right)}{\sqrt[3]{2}}-\frac{\tan ^{-1}\left(\frac{2^{2/3} x \sqrt[3]{-x^2+x^4}}{-2 x^2+\sqrt[3]{2} \left(-x^2+x^4\right)^{2/3}}\right)}{2 \sqrt[3]{2}}-\frac{\sqrt{3} \tanh ^{-1}\left(\frac{\frac{\sqrt[3]{2} x^2}{\sqrt{3}}+\frac{\left(-x^2+x^4\right)^{2/3}}{\sqrt[3]{2} \sqrt{3}}}{x \sqrt[3]{-x^2+x^4}}\right)}{2 \sqrt[3]{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{2} x}{\sqrt[3]{-x^2+x^4}}\right)}{\sqrt[3]{2}}-\frac{\tan ^{-1}\left(\frac{2^{2/3} x \sqrt[3]{-x^2+x^4}}{-2 x^2+\sqrt[3]{2} \left(-x^2+x^4\right)^{2/3}}\right)}{2 \sqrt[3]{2}}-\frac{\sqrt{3} \tanh ^{-1}\left(\frac{\frac{\sqrt[3]{2} x^2}{\sqrt{3}}+\frac{\left(-x^2+x^4\right)^{2/3}}{\sqrt[3]{2} \sqrt{3}}}{x \sqrt[3]{-x^2+x^4}}\right)}{2 \sqrt[3]{2}}",1,"-(ArcTan[(2^(1/3)*x)/(-x^2 + x^4)^(1/3)]/2^(1/3)) - ArcTan[(2^(2/3)*x*(-x^2 + x^4)^(1/3))/(-2*x^2 + 2^(1/3)*(-x^2 + x^4)^(2/3))]/(2*2^(1/3)) - (Sqrt[3]*ArcTanh[((2^(1/3)*x^2)/Sqrt[3] + (-x^2 + x^4)^(2/3)/(2^(1/3)*Sqrt[3]))/(x*(-x^2 + x^4)^(1/3))])/(2*2^(1/3))","A",1
2148,1,157,157,0.6083984,"\int \frac{\sqrt[4]{-b x^2+a x^4} \left(-b+2 a x^4\right)}{b+a x^4} \, dx","IntegrateAlgebraic[((-(b*x^2) + a*x^4)^(1/4)*(-b + 2*a*x^4))/(b + a*x^4),x]","x \sqrt[4]{-b x^2+a x^4}+\frac{b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{2 a^{3/4}}-\frac{b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{2 a^{3/4}}-\frac{3}{4} b \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]","x \sqrt[4]{-b x^2+a x^4}+\frac{b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{2 a^{3/4}}-\frac{b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{2 a^{3/4}}-\frac{3}{4} b \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]",1,"x*(-(b*x^2) + a*x^4)^(1/4) + (b*ArcTan[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/(2*a^(3/4)) - (b*ArcTanh[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/(2*a^(3/4)) - (3*b*RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (-(Log[x]*#1) + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1)/(-a + #1^4) & ])/4","A",1
2149,1,157,157,0.0016454,"\int \frac{\sqrt[4]{-b x^2+a x^4} \left(-b+2 a x^4\right)}{b+a x^4} \, dx","IntegrateAlgebraic[((-(b*x^2) + a*x^4)^(1/4)*(-b + 2*a*x^4))/(b + a*x^4),x]","x \sqrt[4]{-b x^2+a x^4}+\frac{b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{2 a^{3/4}}-\frac{b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{2 a^{3/4}}-\frac{3}{4} b \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]","x \sqrt[4]{-b x^2+a x^4}+\frac{b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{2 a^{3/4}}-\frac{b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{2 a^{3/4}}-\frac{3}{4} b \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]",1,"x*(-(b*x^2) + a*x^4)^(1/4) + (b*ArcTan[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/(2*a^(3/4)) - (b*ArcTanh[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/(2*a^(3/4)) - (3*b*RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (-(Log[x]*#1) + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1)/(-a + #1^4) & ])/4","A",1
2150,1,157,157,0.5701831,"\int \frac{x^3 (-4 a+3 x)}{\left(x^2 (-a+x)\right)^{2/3} \left(a-x+d x^4\right)} \, dx","IntegrateAlgebraic[(x^3*(-4*a + 3*x))/((x^2*(-a + x))^(2/3)*(a - x + d*x^4)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} x^2}{\sqrt[3]{d} x^2+2 \sqrt[3]{-a x^2+x^3}}\right)}{d^{2/3}}+\frac{\log \left(-a \sqrt[3]{d} x^2+a \sqrt[3]{-a x^2+x^3}\right)}{d^{2/3}}-\frac{\log \left(a^2 d^{2/3} x^4+a^2 \sqrt[3]{d} x^2 \sqrt[3]{-a x^2+x^3}+a^2 \left(-a x^2+x^3\right)^{2/3}\right)}{2 d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} x^2}{\sqrt[3]{d} x^2+2 \sqrt[3]{-a x^2+x^3}}\right)}{d^{2/3}}+\frac{\log \left(-a \sqrt[3]{d} x^2+a \sqrt[3]{-a x^2+x^3}\right)}{d^{2/3}}-\frac{\log \left(a^2 d^{2/3} x^4+a^2 \sqrt[3]{d} x^2 \sqrt[3]{-a x^2+x^3}+a^2 \left(-a x^2+x^3\right)^{2/3}\right)}{2 d^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*x^2)/(d^(1/3)*x^2 + 2*(-(a*x^2) + x^3)^(1/3))])/d^(2/3) + Log[-(a*d^(1/3)*x^2) + a*(-(a*x^2) + x^3)^(1/3)]/d^(2/3) - Log[a^2*d^(2/3)*x^4 + a^2*d^(1/3)*x^2*(-(a*x^2) + x^3)^(1/3) + a^2*(-(a*x^2) + x^3)^(2/3)]/(2*d^(2/3))","A",1
2151,1,157,157,0.938087,"\int \frac{b+a x^6}{x^6 \left(b+a x^3\right) \sqrt[4]{-b x+a x^4}} \, dx","IntegrateAlgebraic[(b + a*x^6)/(x^6*(b + a*x^3)*(-(b*x) + a*x^4)^(1/4)),x]","\frac{4 \left(b-a x^3\right) \left(-b x+a x^4\right)^{3/4}}{21 b^2 x^6}+\frac{2^{3/4} \left(a^{7/4}+a^{3/4} b\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)}{3 b^2}+\frac{2^{3/4} \left(a^{7/4}+a^{3/4} b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)}{3 b^2}","\frac{4 \left(b-a x^3\right) \left(-b x+a x^4\right)^{3/4}}{21 b^2 x^6}+\frac{2^{3/4} \left(a^{7/4}+a^{3/4} b\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)}{3 b^2}+\frac{2^{3/4} \left(a^{7/4}+a^{3/4} b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(-b x+a x^4\right)^{3/4}}{-b+a x^3}\right)}{3 b^2}",1,"(4*(b - a*x^3)*(-(b*x) + a*x^4)^(3/4))/(21*b^2*x^6) + (2^(3/4)*(a^(7/4) + a^(3/4)*b)*ArcTan[(2^(1/4)*a^(1/4)*(-(b*x) + a*x^4)^(3/4))/(-b + a*x^3)])/(3*b^2) + (2^(3/4)*(a^(7/4) + a^(3/4)*b)*ArcTanh[(2^(1/4)*a^(1/4)*(-(b*x) + a*x^4)^(3/4))/(-b + a*x^3)])/(3*b^2)","A",1
2152,1,159,157,0.3536945,"\int \frac{-1+x^2}{\sqrt{1+x} \left(1+x^2\right) \sqrt{x+\sqrt{1+x}}} \, dx","IntegrateAlgebraic[(-1 + x^2)/(Sqrt[1 + x]*(1 + x^2)*Sqrt[x + Sqrt[1 + x]]),x]","-2 \log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}\right)+128 \text{RootSum}\left[625+1000 \text{$\#$1}+300 \text{$\#$1}^2+120 \text{$\#$1}^3+470 \text{$\#$1}^4+24 \text{$\#$1}^5+12 \text{$\#$1}^6+8 \text{$\#$1}^7+\text{$\#$1}^8\&,\frac{\log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^3}{125+75 \text{$\#$1}+45 \text{$\#$1}^2+235 \text{$\#$1}^3+15 \text{$\#$1}^4+9 \text{$\#$1}^5+7 \text{$\#$1}^6+\text{$\#$1}^7}\&\right]","-2 \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}\right)+128 \text{RootSum}\left[625+1000 \text{$\#$1}+300 \text{$\#$1}^2+120 \text{$\#$1}^3+470 \text{$\#$1}^4+24 \text{$\#$1}^5+12 \text{$\#$1}^6+8 \text{$\#$1}^7+\text{$\#$1}^8\&,\frac{\log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^3}{125+75 \text{$\#$1}+45 \text{$\#$1}^2+235 \text{$\#$1}^3+15 \text{$\#$1}^4+9 \text{$\#$1}^5+7 \text{$\#$1}^6+\text{$\#$1}^7}\&\right]",1,"-2*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]] + 128*RootSum[625 + 1000*#1 + 300*#1^2 + 120*#1^3 + 470*#1^4 + 24*#1^5 + 12*#1^6 + 8*#1^7 + #1^8 & , (Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]] - #1]*#1^3)/(125 + 75*#1 + 45*#1^2 + 235*#1^3 + 15*#1^4 + 9*#1^5 + 7*#1^6 + #1^7) & ]","A",1
2153,1,159,157,0.0014012,"\int \frac{-1+x^2}{\sqrt{1+x} \left(1+x^2\right) \sqrt{x+\sqrt{1+x}}} \, dx","IntegrateAlgebraic[(-1 + x^2)/(Sqrt[1 + x]*(1 + x^2)*Sqrt[x + Sqrt[1 + x]]),x]","-2 \log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}\right)+128 \text{RootSum}\left[625+1000 \text{$\#$1}+300 \text{$\#$1}^2+120 \text{$\#$1}^3+470 \text{$\#$1}^4+24 \text{$\#$1}^5+12 \text{$\#$1}^6+8 \text{$\#$1}^7+\text{$\#$1}^8\&,\frac{\log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^3}{125+75 \text{$\#$1}+45 \text{$\#$1}^2+235 \text{$\#$1}^3+15 \text{$\#$1}^4+9 \text{$\#$1}^5+7 \text{$\#$1}^6+\text{$\#$1}^7}\&\right]","-2 \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}\right)+128 \text{RootSum}\left[625+1000 \text{$\#$1}+300 \text{$\#$1}^2+120 \text{$\#$1}^3+470 \text{$\#$1}^4+24 \text{$\#$1}^5+12 \text{$\#$1}^6+8 \text{$\#$1}^7+\text{$\#$1}^8\&,\frac{\log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^3}{125+75 \text{$\#$1}+45 \text{$\#$1}^2+235 \text{$\#$1}^3+15 \text{$\#$1}^4+9 \text{$\#$1}^5+7 \text{$\#$1}^6+\text{$\#$1}^7}\&\right]",1,"-2*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]] + 128*RootSum[625 + 1000*#1 + 300*#1^2 + 120*#1^3 + 470*#1^4 + 24*#1^5 + 12*#1^6 + 8*#1^7 + #1^8 & , (Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]] - #1]*#1^3)/(125 + 75*#1 + 45*#1^2 + 235*#1^3 + 15*#1^4 + 9*#1^5 + 7*#1^6 + #1^7) & ]","A",1
2154,1,157,157,0.5573599,"\int x^2 \sqrt{a x^2+\sqrt{b+a^2 x^4}} \, dx","IntegrateAlgebraic[x^2*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]],x]","-\frac{i \left(2 i a x^4 \sqrt{b+a^2 x^4}+i x^2 \left(-b+2 a^2 x^4\right)\right)}{8 a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}+\frac{b \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{8 \sqrt{2} a^{3/2}}","-\frac{i \left(2 i a x^4 \sqrt{b+a^2 x^4}+i x^2 \left(-b+2 a^2 x^4\right)\right)}{8 a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}+\frac{b \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{8 \sqrt{2} a^{3/2}}",1,"((-1/8*I)*((2*I)*a*x^4*Sqrt[b + a^2*x^4] + I*x^2*(-b + 2*a^2*x^4)))/(a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]) + (b*Log[I*a*x^2 + I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]])/(8*Sqrt[2]*a^(3/2))","A",1
2155,1,157,157,0.4111848,"\int \frac{1}{\sqrt{-b+a^2 x^2} \sqrt[3]{a x+\sqrt{-b+a^2 x^2}} \sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}} \, dx","IntegrateAlgebraic[1/(Sqrt[-b + a^2*x^2]*(a*x + Sqrt[-b + a^2*x^2])^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)),x]","-\frac{3 \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}}{a c \sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}-\frac{3 \tan ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{2 a c^{5/4}}+\frac{3 \tanh ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{2 a c^{5/4}}","-\frac{3 \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}}{a c \sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}-\frac{3 \tan ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{2 a c^{5/4}}+\frac{3 \tanh ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{2 a c^{5/4}}",1,"(-3*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4))/(a*c*(a*x + Sqrt[-b + a^2*x^2])^(1/3)) - (3*ArcTan[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)])/(2*a*c^(5/4)) + (3*ArcTanh[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)])/(2*a*c^(5/4))","A",1
2156,-1,0,158,180.0029302,"\int \frac{(2 a-3 b+x) \left(-a^3+3 a^2 x-3 a x^2+x^3\right)}{(-b+x) \sqrt[4]{(-a+x) (-b+x)^2} \left(-a^3-b^2 d+\left(3 a^2+2 b d\right) x-(3 a+d) x^2+x^3\right)} \, dx","IntegrateAlgebraic[((2*a - 3*b + x)*(-a^3 + 3*a^2*x - 3*a*x^2 + x^3))/((-b + x)*((-a + x)*(-b + x)^2)^(1/4)*(-a^3 - b^2*d + (3*a^2 + 2*b*d)*x - (3*a + d)*x^2 + x^3)),x]","\text{\$Aborted}","\frac{4 \left(-a b^2+2 a b x+b^2 x-a x^2-2 b x^2+x^3\right)^{3/4}}{(b-x)^2}-2 \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{a-x}\right)+2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{a-x}\right)",1,"$Aborted","F",-1
2157,1,158,158,0.3857536,"\int \frac{\left(2-x+2 x^2\right) \sqrt[4]{-x^3+x^4}}{-2-2 x+x^2} \, dx","IntegrateAlgebraic[((2 - x + 2*x^2)*(-x^3 + x^4)^(1/4))/(-2 - 2*x + x^2),x]","\frac{1}{4} (11+4 x) \sqrt[4]{-x^3+x^4}-\frac{177}{8} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)+\frac{177}{8} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)-9 \text{RootSum}\left[3-6 \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-2 \log (x)+2 \log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 \text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]","\frac{1}{4} (11+4 x) \sqrt[4]{-x^3+x^4}-\frac{177}{8} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)+\frac{177}{8} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)-9 \text{RootSum}\left[3-6 \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-2 \log (x)+2 \log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 \text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]",1,"((11 + 4*x)*(-x^3 + x^4)^(1/4))/4 - (177*ArcTan[x/(-x^3 + x^4)^(1/4)])/8 + (177*ArcTanh[x/(-x^3 + x^4)^(1/4)])/8 - 9*RootSum[3 - 6*#1^4 + 2*#1^8 & , (-2*Log[x] + 2*Log[(-x^3 + x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-x^3 + x^4)^(1/4) - x*#1]*#1^4)/(-3*#1^3 + 2*#1^7) & ]","A",1
2158,1,158,158,0.0011129,"\int \frac{\left(2-x+2 x^2\right) \sqrt[4]{-x^3+x^4}}{-2-2 x+x^2} \, dx","IntegrateAlgebraic[((2 - x + 2*x^2)*(-x^3 + x^4)^(1/4))/(-2 - 2*x + x^2),x]","\frac{1}{4} (11+4 x) \sqrt[4]{-x^3+x^4}-\frac{177}{8} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)+\frac{177}{8} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)-9 \text{RootSum}\left[3-6 \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-2 \log (x)+2 \log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 \text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]","\frac{1}{4} (11+4 x) \sqrt[4]{-x^3+x^4}-\frac{177}{8} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)+\frac{177}{8} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)-9 \text{RootSum}\left[3-6 \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-2 \log (x)+2 \log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 \text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]",1,"((11 + 4*x)*(-x^3 + x^4)^(1/4))/4 - (177*ArcTan[x/(-x^3 + x^4)^(1/4)])/8 + (177*ArcTanh[x/(-x^3 + x^4)^(1/4)])/8 - 9*RootSum[3 - 6*#1^4 + 2*#1^8 & , (-2*Log[x] + 2*Log[(-x^3 + x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-x^3 + x^4)^(1/4) - x*#1]*#1^4)/(-3*#1^3 + 2*#1^7) & ]","A",1
2159,1,79,158,0.4946055,"\int \frac{\sqrt{1-3 x^2-2 x^4} \left(1+2 x^4\right)}{\left(-1+x^2+2 x^4\right) \left(-1+2 x^2+2 x^4\right)} \, dx","IntegrateAlgebraic[(Sqrt[1 - 3*x^2 - 2*x^4]*(1 + 2*x^4))/((-1 + x^2 + 2*x^4)*(-1 + 2*x^2 + 2*x^4)),x]","\tan ^{-1}\left(\frac{x \sqrt{1-3 x^2-2 x^4}}{-1+3 x^2+2 x^4}\right)-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt{1-3 x^2-2 x^4}}{-1+3 x^2+2 x^4}\right)","\tan ^{-1}\left(\frac{x \sqrt{1-3 x^2-2 x^4}}{-1+3 x^2+2 x^4}\right)-i \sqrt{2} \tanh ^{-1}\left(\frac{-2 i x+2 \sqrt{2} x^3-2 i x \sqrt{1-3 x^2-2 x^4}}{-\sqrt{2}+3 \sqrt{2} x^2+2 \sqrt{2} x^4-\sqrt{2} \sqrt{1-3 x^2-2 x^4}-2 i x^2 \sqrt{1-3 x^2-2 x^4}}\right)",1,"ArcTan[(x*Sqrt[1 - 3*x^2 - 2*x^4])/(-1 + 3*x^2 + 2*x^4)] - Sqrt[2]*ArcTan[(Sqrt[2]*x*Sqrt[1 - 3*x^2 - 2*x^4])/(-1 + 3*x^2 + 2*x^4)]","A",1
2160,1,158,158,0.6275788,"\int \frac{\sqrt[4]{b x^2+a x^4} \left(-b+2 a x^4\right)}{-2 b+a x^4} \, dx","IntegrateAlgebraic[((b*x^2 + a*x^4)^(1/4)*(-b + 2*a*x^4))/(-2*b + a*x^4),x]","x \sqrt[4]{b x^2+a x^4}-\frac{b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{2 a^{3/4}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{2 a^{3/4}}+\frac{3}{8} b \text{RootSum}\left[2 a^2-a b-4 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]","x \sqrt[4]{b x^2+a x^4}-\frac{b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{2 a^{3/4}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{2 a^{3/4}}+\frac{3}{8} b \text{RootSum}\left[2 a^2-a b-4 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]",1,"x*(b*x^2 + a*x^4)^(1/4) - (b*ArcTan[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/(2*a^(3/4)) + (b*ArcTanh[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/(2*a^(3/4)) + (3*b*RootSum[2*a^2 - a*b - 4*a*#1^4 + 2*#1^8 & , (-(Log[x]*#1) + Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1)/(-a + #1^4) & ])/8","A",1
2161,1,158,158,0.0016428,"\int \frac{\sqrt[4]{b x^2+a x^4} \left(-b+2 a x^4\right)}{-2 b+a x^4} \, dx","IntegrateAlgebraic[((b*x^2 + a*x^4)^(1/4)*(-b + 2*a*x^4))/(-2*b + a*x^4),x]","x \sqrt[4]{b x^2+a x^4}-\frac{b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{2 a^{3/4}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{2 a^{3/4}}+\frac{3}{8} b \text{RootSum}\left[2 a^2-a b-4 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]","x \sqrt[4]{b x^2+a x^4}-\frac{b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{2 a^{3/4}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{2 a^{3/4}}+\frac{3}{8} b \text{RootSum}\left[2 a^2-a b-4 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]",1,"x*(b*x^2 + a*x^4)^(1/4) - (b*ArcTan[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/(2*a^(3/4)) + (b*ArcTanh[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/(2*a^(3/4)) + (3*b*RootSum[2*a^2 - a*b - 4*a*#1^4 + 2*#1^8 & , (-(Log[x]*#1) + Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1)/(-a + #1^4) & ])/8","A",1
2162,1,72,158,0.4713548,"\int \frac{x (-4 a+3 x)}{\sqrt[3]{x^2 (-a+x)} \left(a-x+d x^4\right)} \, dx","IntegrateAlgebraic[(x*(-4*a + 3*x))/((x^2*(-a + x))^(1/3)*(a - x + d*x^4)),x]","-a \text{RootSum}\left[a^3 d-\text{$\#$1}^3+3 \text{$\#$1}^6-3 \text{$\#$1}^9+\text{$\#$1}^{12}\&,\frac{-\log (x)+\log \left(\sqrt[3]{-a x^2+x^3}-x \text{$\#$1}\right)}{-\text{$\#$1}+\text{$\#$1}^4}\&\right]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} x^2}{\sqrt[3]{d} x^2+2 \sqrt[3]{-a x^2+x^3}}\right)}{\sqrt[3]{d}}+\frac{\log \left(-a \sqrt[3]{d} x^2+a \sqrt[3]{-a x^2+x^3}\right)}{\sqrt[3]{d}}-\frac{\log \left(a^2 d^{2/3} x^4+a^2 \sqrt[3]{d} x^2 \sqrt[3]{-a x^2+x^3}+a^2 \left(-a x^2+x^3\right)^{2/3}\right)}{2 \sqrt[3]{d}}",1,"-(a*RootSum[a^3*d - #1^3 + 3*#1^6 - 3*#1^9 + #1^12 & , (-Log[x] + Log[(-(a*x^2) + x^3)^(1/3) - x*#1])/(-#1 + #1^4) & ])","C",1
2163,1,158,158,1.2209617,"\int \frac{-b-2 a x^4+2 x^8}{\sqrt[4]{-b+a x^4} \left(-b-a x^4+x^8\right)} \, dx","IntegrateAlgebraic[(-b - 2*a*x^4 + 2*x^8)/((-b + a*x^4)^(1/4)*(-b - a*x^4 + x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{\sqrt[4]{a}}-\frac{1}{4} \text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a \log (x)+a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{\sqrt[4]{a}}-\frac{1}{4} \text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a \log (x)-a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{3 a \text{$\#$1}-2 \text{$\#$1}^5}\&\right]",1,"ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/a^(1/4) + ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/a^(1/4) - RootSum[2*a^2 - b - 3*a*#1^4 + #1^8 & , (-(a*Log[x]) + a*Log[(-b + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1 + 2*#1^5) & ]/4","A",1
2164,1,158,158,0.0017597,"\int \frac{-b-2 a x^4+2 x^8}{\sqrt[4]{-b+a x^4} \left(-b-a x^4+x^8\right)} \, dx","IntegrateAlgebraic[(-b - 2*a*x^4 + 2*x^8)/((-b + a*x^4)^(1/4)*(-b - a*x^4 + x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{\sqrt[4]{a}}-\frac{1}{4} \text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a \log (x)+a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{\sqrt[4]{a}}-\frac{1}{4} \text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a \log (x)-a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{3 a \text{$\#$1}-2 \text{$\#$1}^5}\&\right]",1,"ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/a^(1/4) + ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/a^(1/4) - RootSum[2*a^2 - b - 3*a*#1^4 + #1^8 & , (-(a*Log[x]) + a*Log[(-b + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1 + 2*#1^5) & ]/4","A",1
2165,1,159,159,0.4350246,"\int \frac{1}{\left(-1+x^2\right) \sqrt[3]{-x^2+x^3}} \, dx","IntegrateAlgebraic[1/((-1 + x^2)*(-x^2 + x^3)^(1/3)),x]","-\frac{3 \left(-x^2+x^3\right)^{2/3}}{2 (-1+x) x}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-x^2+x^3}}\right)}{2 \sqrt[3]{2}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{-x^2+x^3}\right)}{2 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-x^2+x^3}+\sqrt[3]{2} \left(-x^2+x^3\right)^{2/3}\right)}{4 \sqrt[3]{2}}","-\frac{3 \left(-x^2+x^3\right)^{2/3}}{2 (-1+x) x}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-x^2+x^3}}\right)}{2 \sqrt[3]{2}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{-x^2+x^3}\right)}{2 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-x^2+x^3}+\sqrt[3]{2} \left(-x^2+x^3\right)^{2/3}\right)}{4 \sqrt[3]{2}}",1,"(-3*(-x^2 + x^3)^(2/3))/(2*(-1 + x)*x) - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-x^2 + x^3)^(1/3))])/(2*2^(1/3)) + Log[-2*x + 2^(2/3)*(-x^2 + x^3)^(1/3)]/(2*2^(1/3)) - Log[2*x^2 + 2^(2/3)*x*(-x^2 + x^3)^(1/3) + 2^(1/3)*(-x^2 + x^3)^(2/3)]/(4*2^(1/3))","A",1
2166,1,159,159,0.99346,"\int \frac{-b^2+a^2 x^4}{\sqrt{-b x+a x^3} \left(b^2+c x^2+a^2 x^4\right)} \, dx","IntegrateAlgebraic[(-b^2 + a^2*x^4)/(Sqrt[-(b*x) + a*x^3]*(b^2 + c*x^2 + a^2*x^4)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{2 a b+c} \sqrt{-b x+a x^3}}{b+\sqrt{2 a b+c} x-a x^2}\right)}{\sqrt{2} \sqrt[4]{2 a b+c}}-\frac{\tanh ^{-1}\left(\frac{-\frac{b}{\sqrt{2} \sqrt[4]{2 a b+c}}+\frac{\sqrt[4]{2 a b+c} x}{\sqrt{2}}+\frac{a x^2}{\sqrt{2} \sqrt[4]{2 a b+c}}}{\sqrt{-b x+a x^3}}\right)}{\sqrt{2} \sqrt[4]{2 a b+c}}","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{2 a b+c} \sqrt{-b x+a x^3}}{b+\sqrt{2 a b+c} x-a x^2}\right)}{\sqrt{2} \sqrt[4]{2 a b+c}}-\frac{\tanh ^{-1}\left(\frac{-\frac{b}{\sqrt{2} \sqrt[4]{2 a b+c}}+\frac{\sqrt[4]{2 a b+c} x}{\sqrt{2}}+\frac{a x^2}{\sqrt{2} \sqrt[4]{2 a b+c}}}{\sqrt{-b x+a x^3}}\right)}{\sqrt{2} \sqrt[4]{2 a b+c}}",1,"ArcTan[(Sqrt[2]*(2*a*b + c)^(1/4)*Sqrt[-(b*x) + a*x^3])/(b + Sqrt[2*a*b + c]*x - a*x^2)]/(Sqrt[2]*(2*a*b + c)^(1/4)) - ArcTanh[(-(b/(Sqrt[2]*(2*a*b + c)^(1/4))) + ((2*a*b + c)^(1/4)*x)/Sqrt[2] + (a*x^2)/(Sqrt[2]*(2*a*b + c)^(1/4)))/Sqrt[-(b*x) + a*x^3]]/(Sqrt[2]*(2*a*b + c)^(1/4))","A",1
2167,1,159,159,0.7361397,"\int \frac{\left(-q+2 p x^3\right) \sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x \left(b x^2+a \left(q+p x^3\right)^2\right)} \, dx","IntegrateAlgebraic[((-q + 2*p*x^3)*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6])/(x*(b*x^2 + a*(q + p*x^3)^2)),x]","\frac{\sqrt{b+2 a p q} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{b+2 a p q} x^2}{a q^2+b x^2+2 a p q x^3+a p^2 x^6+\left(a q+a p x^3\right) \sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}}\right)}{a \sqrt{b}}-\frac{\log (x)}{a}+\frac{\log \left(q+p x^3+\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right)}{a}","\frac{\sqrt{b+2 a p q} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{b+2 a p q} x^2}{a q^2+b x^2+2 a p q x^3+a p^2 x^6+\left(a q+a p x^3\right) \sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}}\right)}{a \sqrt{b}}-\frac{\log (x)}{a}+\frac{\log \left(q+p x^3+\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right)}{a}",1,"(Sqrt[b + 2*a*p*q]*ArcTanh[(Sqrt[b]*Sqrt[b + 2*a*p*q]*x^2)/(a*q^2 + b*x^2 + 2*a*p*q*x^3 + a*p^2*x^6 + (a*q + a*p*x^3)*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6])])/(a*Sqrt[b]) - Log[x]/a + Log[q + p*x^3 + Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]]/a","A",1
2168,1,159,159,0.8020388,"\int \frac{\left(-2 q+p x^3\right) \sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x \left(b x^4+a \left(q+p x^3\right)^2\right)} \, dx","IntegrateAlgebraic[((-2*q + p*x^3)*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6])/(x*(b*x^4 + a*(q + p*x^3)^2)),x]","\frac{\sqrt{b+2 a p q} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{b+2 a p q} x^4}{a q^2+2 a p q x^3+b x^4+a p^2 x^6+\left(a q+a p x^3\right) \sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}}\right)}{a \sqrt{b}}-\frac{2 \log (x)}{a}+\frac{\log \left(q+p x^3+\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}\right)}{a}","\frac{\sqrt{b+2 a p q} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{b+2 a p q} x^4}{a q^2+2 a p q x^3+b x^4+a p^2 x^6+\left(a q+a p x^3\right) \sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}}\right)}{a \sqrt{b}}-\frac{2 \log (x)}{a}+\frac{\log \left(q+p x^3+\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}\right)}{a}",1,"(Sqrt[b + 2*a*p*q]*ArcTanh[(Sqrt[b]*Sqrt[b + 2*a*p*q]*x^4)/(a*q^2 + 2*a*p*q*x^3 + b*x^4 + a*p^2*x^6 + (a*q + a*p*x^3)*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6])])/(a*Sqrt[b]) - (2*Log[x])/a + Log[q + p*x^3 + Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]]/a","A",1
2169,1,159,159,0.943864,"\int \frac{x^2-\sqrt{1+x^2}}{x^2+\sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[(x^2 - Sqrt[1 + x^2])/(x^2 + Sqrt[x + Sqrt[1 + x^2]]),x]","x-\log \left(x+\sqrt{1+x^2}\right)-\text{RootSum}\left[1-2 \text{$\#$1}^4+4 \text{$\#$1}^5+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right)+2 \log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}-2 \log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{-2 \text{$\#$1}+5 \text{$\#$1}^2+2 \text{$\#$1}^5}\&\right]","x-\log \left(x+\sqrt{1+x^2}\right)-\text{RootSum}\left[1-2 \text{$\#$1}^4+4 \text{$\#$1}^5+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right)+2 \log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}-2 \log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{-2 \text{$\#$1}+5 \text{$\#$1}^2+2 \text{$\#$1}^5}\&\right]",1,"x - Log[x + Sqrt[1 + x^2]] - RootSum[1 - 2*#1^4 + 4*#1^5 + #1^8 & , (Log[Sqrt[x + Sqrt[1 + x^2]] - #1] + 2*Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1 - 2*Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1^2 + Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1^4)/(-2*#1 + 5*#1^2 + 2*#1^5) & ]","A",1
2170,1,159,159,0.0012267,"\int \frac{x^2-\sqrt{1+x^2}}{x^2+\sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[(x^2 - Sqrt[1 + x^2])/(x^2 + Sqrt[x + Sqrt[1 + x^2]]),x]","x-\log \left(x+\sqrt{1+x^2}\right)-\text{RootSum}\left[1-2 \text{$\#$1}^4+4 \text{$\#$1}^5+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right)+2 \log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}-2 \log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{-2 \text{$\#$1}+5 \text{$\#$1}^2+2 \text{$\#$1}^5}\&\right]","x-\log \left(x+\sqrt{1+x^2}\right)-\text{RootSum}\left[1-2 \text{$\#$1}^4+4 \text{$\#$1}^5+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right)+2 \log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}-2 \log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{x+\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{-2 \text{$\#$1}+5 \text{$\#$1}^2+2 \text{$\#$1}^5}\&\right]",1,"x - Log[x + Sqrt[1 + x^2]] - RootSum[1 - 2*#1^4 + 4*#1^5 + #1^8 & , (Log[Sqrt[x + Sqrt[1 + x^2]] - #1] + 2*Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1 - 2*Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1^2 + Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1^4)/(-2*#1 + 5*#1^2 + 2*#1^5) & ]","A",1
2171,1,160,160,0.8862365,"\int \frac{\sqrt{1+\sqrt{1-\sqrt{1+\frac{1}{x^2}}}}}{x} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[1 - Sqrt[1 + x^(-2)]]]/x,x]","-4 \sqrt{1+\sqrt{1-\sqrt{1+\frac{1}{x^2}}}}+\sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\sqrt{1+\sqrt{2}} \sqrt{1+\sqrt{1-\sqrt{\frac{1+x^2}{x^2}}}}\right)+2 \tanh ^{-1}\left(\sqrt{1+\sqrt{1-\sqrt{\frac{1+x^2}{x^2}}}}\right)+\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\sqrt{-1+\sqrt{2}} \sqrt{1+\sqrt{1-\sqrt{\frac{1+x^2}{x^2}}}}\right)","-4 \sqrt{1+\sqrt{1-\sqrt{1+\frac{1}{x^2}}}}+\sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{1+\sqrt{1-\sqrt{\frac{1+x^2}{x^2}}}}}{\sqrt{-1+\sqrt{2}}}\right)+2 \tanh ^{-1}\left(\sqrt{1+\sqrt{1-\sqrt{\frac{1+x^2}{x^2}}}}\right)+\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{1+\sqrt{1-\sqrt{\frac{1+x^2}{x^2}}}}}{\sqrt{1+\sqrt{2}}}\right)",1,"-4*Sqrt[1 + Sqrt[1 - Sqrt[1 + x^(-2)]]] + Sqrt[-1 + Sqrt[2]]*ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[1 + Sqrt[1 - Sqrt[(1 + x^2)/x^2]]]] + 2*ArcTanh[Sqrt[1 + Sqrt[1 - Sqrt[(1 + x^2)/x^2]]]] + Sqrt[1 + Sqrt[2]]*ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[1 + Sqrt[1 - Sqrt[(1 + x^2)/x^2]]]]","A",1
2172,1,160,160,0.3640665,"\int \frac{-2 x+(1+k) x^2}{((1-x) x (1-k x))^{2/3} \left(1-(1+k) x+(-b+k) x^2\right)} \, dx","IntegrateAlgebraic[(-2*x + (1 + k)*x^2)/(((1 - x)*x*(1 - k*x))^(2/3)*(1 - (1 + k)*x + (-b + k)*x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} x}{\sqrt[3]{b} x+2 \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{b^{2/3}}+\frac{\log \left(-\sqrt[3]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{b^{2/3}}-\frac{\log \left(b^{2/3} x^2+\sqrt[3]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 b^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} x}{\sqrt[3]{b} x+2 \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{b^{2/3}}+\frac{\log \left(-\sqrt[3]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{b^{2/3}}-\frac{\log \left(b^{2/3} x^2+\sqrt[3]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 b^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*b^(1/3)*x)/(b^(1/3)*x + 2*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/b^(2/3) + Log[-(b^(1/3)*x) + (x + (-1 - k)*x^2 + k*x^3)^(1/3)]/b^(2/3) - Log[b^(2/3)*x^2 + b^(1/3)*x*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + (x + (-1 - k)*x^2 + k*x^3)^(2/3)]/(2*b^(2/3))","A",1
2173,1,161,161,0.2462216,"\int \frac{2-(1+k) x}{\sqrt[3]{(1-x) x (1-k x)} (1-(1+k) x)} \, dx","IntegrateAlgebraic[(2 - (1 + k)*x)/(((1 - x)*x*(1 - k*x))^(1/3)*(1 - (1 + k)*x)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{k} x}{\sqrt[3]{k} x+2 \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{\sqrt[3]{k}}-\frac{\log \left(-\sqrt[3]{k} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{\sqrt[3]{k}}+\frac{\log \left(k^{2/3} x^2+\sqrt[3]{k} x \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 \sqrt[3]{k}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{k} x}{\sqrt[3]{k} x+2 \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{\sqrt[3]{k}}-\frac{\log \left(-\sqrt[3]{k} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{\sqrt[3]{k}}+\frac{\log \left(k^{2/3} x^2+\sqrt[3]{k} x \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 \sqrt[3]{k}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*k^(1/3)*x)/(k^(1/3)*x + 2*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/k^(1/3) - Log[-(k^(1/3)*x) + (x + (-1 - k)*x^2 + k*x^3)^(1/3)]/k^(1/3) + Log[k^(2/3)*x^2 + k^(1/3)*x*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + (x + (-1 - k)*x^2 + k*x^3)^(2/3)]/(2*k^(1/3))","A",1
2174,1,161,161,0.2849125,"\int \frac{-2+(1+k) x}{\sqrt[3]{(1-x) x (1-k x)} \left(1-(1+k) x+(-b+k) x^2\right)} \, dx","IntegrateAlgebraic[(-2 + (1 + k)*x)/(((1 - x)*x*(1 - k*x))^(1/3)*(1 - (1 + k)*x + (-b + k)*x^2)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} x}{\sqrt[3]{b} x+2 \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{\sqrt[3]{b}}+\frac{\log \left(-\sqrt[3]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{\sqrt[3]{b}}-\frac{\log \left(b^{2/3} x^2+\sqrt[3]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 \sqrt[3]{b}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} x}{\sqrt[3]{b} x+2 \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{\sqrt[3]{b}}+\frac{\log \left(-\sqrt[3]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{\sqrt[3]{b}}-\frac{\log \left(b^{2/3} x^2+\sqrt[3]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 \sqrt[3]{b}}",1,"-((Sqrt[3]*ArcTan[(Sqrt[3]*b^(1/3)*x)/(b^(1/3)*x + 2*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/b^(1/3)) + Log[-(b^(1/3)*x) + (x + (-1 - k)*x^2 + k*x^3)^(1/3)]/b^(1/3) - Log[b^(2/3)*x^2 + b^(1/3)*x*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + (x + (-1 - k)*x^2 + k*x^3)^(2/3)]/(2*b^(1/3))","A",1
2175,1,161,161,0.4539399,"\int \frac{\left(3+2 x^2\right) \left(1+2 x^2+2 x^3\right)^{2/3}}{x^3 \left(-1-2 x^2+x^3\right)} \, dx","IntegrateAlgebraic[((3 + 2*x^2)*(1 + 2*x^2 + 2*x^3)^(2/3))/(x^3*(-1 - 2*x^2 + x^3)),x]","\frac{3 \left(1+2 x^2+2 x^3\right)^{2/3}}{2 x^2}-3 \sqrt[6]{3} \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2 \sqrt[3]{1+2 x^2+2 x^3}}\right)+3^{2/3} \log \left(-3 x+3^{2/3} \sqrt[3]{1+2 x^2+2 x^3}\right)-\frac{1}{2} 3^{2/3} \log \left(3 x^2+3^{2/3} x \sqrt[3]{1+2 x^2+2 x^3}+\sqrt[3]{3} \left(1+2 x^2+2 x^3\right)^{2/3}\right)","\frac{3 \left(1+2 x^2+2 x^3\right)^{2/3}}{2 x^2}-3 \sqrt[6]{3} \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2 \sqrt[3]{1+2 x^2+2 x^3}}\right)+3^{2/3} \log \left(-3 x+3^{2/3} \sqrt[3]{1+2 x^2+2 x^3}\right)-\frac{1}{2} 3^{2/3} \log \left(3 x^2+3^{2/3} x \sqrt[3]{1+2 x^2+2 x^3}+\sqrt[3]{3} \left(1+2 x^2+2 x^3\right)^{2/3}\right)",1,"(3*(1 + 2*x^2 + 2*x^3)^(2/3))/(2*x^2) - 3*3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*(1 + 2*x^2 + 2*x^3)^(1/3))] + 3^(2/3)*Log[-3*x + 3^(2/3)*(1 + 2*x^2 + 2*x^3)^(1/3)] - (3^(2/3)*Log[3*x^2 + 3^(2/3)*x*(1 + 2*x^2 + 2*x^3)^(1/3) + 3^(1/3)*(1 + 2*x^2 + 2*x^3)^(2/3)])/2","A",1
2176,1,159,161,0.5347222,"\int \frac{-b+c x+a x^2}{\left(b+a x^2\right) \sqrt{-b x+a x^3}} \, dx","IntegrateAlgebraic[(-b + c*x + a*x^2)/((b + a*x^2)*Sqrt[-(b*x) + a*x^3]),x]","\frac{\left(2 \sqrt{a} \sqrt{b}+c\right) \tan ^{-1}\left(\frac{-b-2 \sqrt{a} \sqrt{b} x+a x^2}{2 \sqrt[4]{a} \sqrt[4]{b} \sqrt{-b x+a x^3}}\right)}{4 a^{3/4} b^{3/4}}+\frac{\left(-2 \sqrt{a} \sqrt{b}+c\right) \tanh ^{-1}\left(\frac{-b+2 \sqrt{a} \sqrt{b} x+a x^2}{2 \sqrt[4]{a} \sqrt[4]{b} \sqrt{-b x+a x^3}}\right)}{4 a^{3/4} b^{3/4}}","\frac{\left(-2 \sqrt{a} \sqrt{b}-c\right) \tan ^{-1}\left(\frac{2 \sqrt[4]{a} \sqrt[4]{b} \sqrt{-b x+a x^3}}{-b-2 \sqrt{a} \sqrt{b} x+a x^2}\right)}{4 a^{3/4} b^{3/4}}+\frac{\left(-2 \sqrt{a} \sqrt{b}+c\right) \tanh ^{-1}\left(\frac{-b+2 \sqrt{a} \sqrt{b} x+a x^2}{2 \sqrt[4]{a} \sqrt[4]{b} \sqrt{-b x+a x^3}}\right)}{4 a^{3/4} b^{3/4}}",1,"((2*Sqrt[a]*Sqrt[b] + c)*ArcTan[(-b - 2*Sqrt[a]*Sqrt[b]*x + a*x^2)/(2*a^(1/4)*b^(1/4)*Sqrt[-(b*x) + a*x^3])])/(4*a^(3/4)*b^(3/4)) + ((-2*Sqrt[a]*Sqrt[b] + c)*ArcTanh[(-b + 2*Sqrt[a]*Sqrt[b]*x + a*x^2)/(2*a^(1/4)*b^(1/4)*Sqrt[-(b*x) + a*x^3])])/(4*a^(3/4)*b^(3/4))","A",1
2177,1,161,161,0.5176106,"\int \frac{-4-2 x+2 x^2+x^4}{x \left(-2+x^2\right) \sqrt[4]{\frac{2+x^2}{-2+x^2}} \left(8-10 x+4 x^2+4 x^3-4 x^4+x^5\right)} \, dx","IntegrateAlgebraic[(-4 - 2*x + 2*x^2 + x^4)/(x*(-2 + x^2)*((2 + x^2)/(-2 + x^2))^(1/4)*(8 - 10*x + 4*x^2 + 4*x^3 - 4*x^4 + x^5)),x]","-\frac{\tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\sqrt{2} x-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{\frac{2+x^2}{-2+x^2}}}{\sqrt{2}}}{(-1+x) \sqrt[4]{\frac{2+x^2}{-2+x^2}}}\right)}{2 \sqrt{2}}+\frac{\tanh ^{-1}\left(\frac{\frac{1}{\sqrt{2}}-\sqrt{2} x+\frac{x^2}{\sqrt{2}}+\frac{\sqrt{\frac{2+x^2}{-2+x^2}}}{\sqrt{2}}}{(-1+x) \sqrt[4]{\frac{2+x^2}{-2+x^2}}}\right)}{2 \sqrt{2}}","-\frac{\tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\sqrt{2} x-\frac{x^2}{\sqrt{2}}+\frac{\sqrt{\frac{2+x^2}{-2+x^2}}}{\sqrt{2}}}{(-1+x) \sqrt[4]{\frac{2+x^2}{-2+x^2}}}\right)}{2 \sqrt{2}}+\frac{\tanh ^{-1}\left(\frac{\frac{1}{\sqrt{2}}-\sqrt{2} x+\frac{x^2}{\sqrt{2}}+\frac{\sqrt{\frac{2+x^2}{-2+x^2}}}{\sqrt{2}}}{(-1+x) \sqrt[4]{\frac{2+x^2}{-2+x^2}}}\right)}{2 \sqrt{2}}",1,"-1/2*ArcTan[(-(1/Sqrt[2]) + Sqrt[2]*x - x^2/Sqrt[2] + Sqrt[(2 + x^2)/(-2 + x^2)]/Sqrt[2])/((-1 + x)*((2 + x^2)/(-2 + x^2))^(1/4))]/Sqrt[2] + ArcTanh[(1/Sqrt[2] - Sqrt[2]*x + x^2/Sqrt[2] + Sqrt[(2 + x^2)/(-2 + x^2)]/Sqrt[2])/((-1 + x)*((2 + x^2)/(-2 + x^2))^(1/4))]/(2*Sqrt[2])","A",1
2178,1,152,161,0.1977844,"\int \frac{\sqrt{1+x} \left(-1+x^4\right) \sqrt{1+\sqrt{1+x}}}{1+x^4} \, dx","IntegrateAlgebraic[(Sqrt[1 + x]*(-1 + x^4)*Sqrt[1 + Sqrt[1 + x]])/(1 + x^4),x]","\frac{4}{105} \sqrt{1+\sqrt{1+x}} \left(8-4 \sqrt{1+x}+3 (1+x)+15 (1+x)^{3/2}\right)-\frac{1}{2} \text{RootSum}\left[1+16 \text{$\#$1}^8-32 \text{$\#$1}^{10}+24 \text{$\#$1}^{12}-8 \text{$\#$1}^{14}+\text{$\#$1}^{16}\&,\frac{-\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)+\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2}{-8 \text{$\#$1}^5+12 \text{$\#$1}^7-6 \text{$\#$1}^9+\text{$\#$1}^{11}}\&\right]","\frac{4}{105} (11+3 x) \sqrt{1+\sqrt{1+x}}+\frac{4}{105} \sqrt{1+x} (11+15 x) \sqrt{1+\sqrt{1+x}}-\frac{1}{2} \text{RootSum}\left[1+16 \text{$\#$1}^8-32 \text{$\#$1}^{10}+24 \text{$\#$1}^{12}-8 \text{$\#$1}^{14}+\text{$\#$1}^{16}\&,\frac{-\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)+\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2}{-8 \text{$\#$1}^5+12 \text{$\#$1}^7-6 \text{$\#$1}^9+\text{$\#$1}^{11}}\&\right]",1,"(4*Sqrt[1 + Sqrt[1 + x]]*(8 - 4*Sqrt[1 + x] + 3*(1 + x) + 15*(1 + x)^(3/2)))/105 - RootSum[1 + 16*#1^8 - 32*#1^10 + 24*#1^12 - 8*#1^14 + #1^16 & , (-Log[Sqrt[1 + Sqrt[1 + x]] - #1] + Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^2)/(-8*#1^5 + 12*#1^7 - 6*#1^9 + #1^11) & ]/2","A",1
2179,1,152,161,0.0011475,"\int \frac{\sqrt{1+x} \left(-1+x^4\right) \sqrt{1+\sqrt{1+x}}}{1+x^4} \, dx","IntegrateAlgebraic[(Sqrt[1 + x]*(-1 + x^4)*Sqrt[1 + Sqrt[1 + x]])/(1 + x^4),x]","\frac{4}{105} \sqrt{1+\sqrt{1+x}} \left(8-4 \sqrt{1+x}+3 (1+x)+15 (1+x)^{3/2}\right)-\frac{1}{2} \text{RootSum}\left[1+16 \text{$\#$1}^8-32 \text{$\#$1}^{10}+24 \text{$\#$1}^{12}-8 \text{$\#$1}^{14}+\text{$\#$1}^{16}\&,\frac{-\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)+\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2}{-8 \text{$\#$1}^5+12 \text{$\#$1}^7-6 \text{$\#$1}^9+\text{$\#$1}^{11}}\&\right]","\frac{4}{105} (11+3 x) \sqrt{1+\sqrt{1+x}}+\frac{4}{105} \sqrt{1+x} (11+15 x) \sqrt{1+\sqrt{1+x}}-\frac{1}{2} \text{RootSum}\left[1+16 \text{$\#$1}^8-32 \text{$\#$1}^{10}+24 \text{$\#$1}^{12}-8 \text{$\#$1}^{14}+\text{$\#$1}^{16}\&,\frac{-\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)+\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2}{-8 \text{$\#$1}^5+12 \text{$\#$1}^7-6 \text{$\#$1}^9+\text{$\#$1}^{11}}\&\right]",1,"(4*Sqrt[1 + Sqrt[1 + x]]*(8 - 4*Sqrt[1 + x] + 3*(1 + x) + 15*(1 + x)^(3/2)))/105 - RootSum[1 + 16*#1^8 - 32*#1^10 + 24*#1^12 - 8*#1^14 + #1^16 & , (-Log[Sqrt[1 + Sqrt[1 + x]] - #1] + Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^2)/(-8*#1^5 + 12*#1^7 - 6*#1^9 + #1^11) & ]/2","A",1
2180,1,162,162,9.4439988,"\int \frac{-d+c x}{x^4 \sqrt[3]{-b+a x^3}} \, dx","IntegrateAlgebraic[(-d + c*x)/(x^4*(-b + a*x^3)^(1/3)),x]","\frac{(-2 d+3 c x) \left(-b+a x^3\right)^{2/3}}{6 b x^3}+\frac{a d \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-b+a x^3}}{\sqrt{3} \sqrt[3]{b}}\right)}{3 \sqrt{3} b^{4/3}}+\frac{a d \log \left(\sqrt[3]{b}+\sqrt[3]{-b+a x^3}\right)}{9 b^{4/3}}-\frac{a d \log \left(b^{2/3}-\sqrt[3]{b} \sqrt[3]{-b+a x^3}+\left(-b+a x^3\right)^{2/3}\right)}{18 b^{4/3}}","\frac{(-2 d+3 c x) \left(-b+a x^3\right)^{2/3}}{6 b x^3}+\frac{a d \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-b+a x^3}}{\sqrt{3} \sqrt[3]{b}}\right)}{3 \sqrt{3} b^{4/3}}+\frac{a d \log \left(\sqrt[3]{b}+\sqrt[3]{-b+a x^3}\right)}{9 b^{4/3}}-\frac{a d \log \left(b^{2/3}-\sqrt[3]{b} \sqrt[3]{-b+a x^3}+\left(-b+a x^3\right)^{2/3}\right)}{18 b^{4/3}}",1,"((-2*d + 3*c*x)*(-b + a*x^3)^(2/3))/(6*b*x^3) + (a*d*ArcTan[1/Sqrt[3] - (2*(-b + a*x^3)^(1/3))/(Sqrt[3]*b^(1/3))])/(3*Sqrt[3]*b^(4/3)) + (a*d*Log[b^(1/3) + (-b + a*x^3)^(1/3)])/(9*b^(4/3)) - (a*d*Log[b^(2/3) - b^(1/3)*(-b + a*x^3)^(1/3) + (-b + a*x^3)^(2/3)])/(18*b^(4/3))","A",1
2181,1,162,162,0.7268126,"\int \frac{\sqrt[4]{-b x^3+a x^4} \left(-d+c x^4\right)}{x^2} \, dx","IntegrateAlgebraic[((-(b*x^3) + a*x^4)^(1/4)*(-d + c*x^4))/x^2,x]","\frac{\sqrt[4]{-b x^3+a x^4} \left(6144 a^3 d-77 b^3 c x-44 a b^2 c x^2-32 a^2 b c x^3+384 a^3 c x^4\right)}{1536 a^3 x}+\frac{\left(77 b^4 c+2048 a^4 d\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)}{1024 a^{15/4}}+\frac{\left(-77 b^4 c-2048 a^4 d\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)}{1024 a^{15/4}}","\frac{\sqrt[4]{-b x^3+a x^4} \left(6144 a^3 d-77 b^3 c x-44 a b^2 c x^2-32 a^2 b c x^3+384 a^3 c x^4\right)}{1536 a^3 x}+\frac{\left(77 b^4 c+2048 a^4 d\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)}{1024 a^{15/4}}+\frac{\left(-77 b^4 c-2048 a^4 d\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)}{1024 a^{15/4}}",1,"((-(b*x^3) + a*x^4)^(1/4)*(6144*a^3*d - 77*b^3*c*x - 44*a*b^2*c*x^2 - 32*a^2*b*c*x^3 + 384*a^3*c*x^4))/(1536*a^3*x) + ((77*b^4*c + 2048*a^4*d)*ArcTan[(a^(1/4)*x)/(-(b*x^3) + a*x^4)^(1/4)])/(1024*a^(15/4)) + ((-77*b^4*c - 2048*a^4*d)*ArcTanh[(a^(1/4)*x)/(-(b*x^3) + a*x^4)^(1/4)])/(1024*a^(15/4))","A",1
2182,1,162,162,6.5386907,"\int \frac{x^3 \left(5-4 (1+k) x+3 k x^2\right)}{((1-x) x (1-k x))^{2/3} \left(-b+b (1+k) x-b k x^2+x^5\right)} \, dx","IntegrateAlgebraic[(x^3*(5 - 4*(1 + k)*x + 3*k*x^2))/(((1 - x)*x*(1 - k*x))^(2/3)*(-b + b*(1 + k)*x - b*k*x^2 + x^5)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{\sqrt[3]{b}}+\frac{\log \left(x^2-\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{\sqrt[3]{b}}-\frac{\log \left(x^4+\sqrt[3]{b} x^2 \sqrt[3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 \sqrt[3]{b}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{\sqrt[3]{b}}+\frac{\log \left(x^2-\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{\sqrt[3]{b}}-\frac{\log \left(x^4+\sqrt[3]{b} x^2 \sqrt[3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 \sqrt[3]{b}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/b^(1/3) + Log[x^2 - b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3)]/b^(1/3) - Log[x^4 + b^(1/3)*x^2*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + b^(2/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3)]/(2*b^(1/3))","A",1
2183,1,167,162,2.9821878,"\int \frac{\left(-8+x^5\right) \left(2+x^5\right) \sqrt[4]{2-3 x^4+x^5}}{x^6 \left(4-3 x^4+2 x^5\right)} \, dx","IntegrateAlgebraic[((-8 + x^5)*(2 + x^5)*(2 - 3*x^4 + x^5)^(1/4))/(x^6*(4 - 3*x^4 + 2*x^5)),x]","\frac{\sqrt[4]{2-3 x^4+x^5} \left(4+9 x^4+2 x^5\right)}{5 x^5}-\frac{3 \sqrt[4]{3} \tan ^{-1}\left(\frac{-\frac{\sqrt[4]{3} x^2}{2^{3/4}}+\frac{\sqrt{2-3 x^4+x^5}}{\sqrt[4]{6}}}{x \sqrt[4]{2-3 x^4+x^5}}\right)}{2\ 2^{3/4}}-\frac{3 \sqrt[4]{3} \tanh ^{-1}\left(\frac{6^{3/4} x \sqrt[4]{2-3 x^4+x^5}}{3 x^2+\sqrt{6} \sqrt{2-3 x^4+x^5}}\right)}{2\ 2^{3/4}}","\frac{\sqrt[4]{2-3 x^4+x^5} \left(4+9 x^4+2 x^5\right)}{5 x^5}+\frac{3 \sqrt[4]{3} \tan ^{-1}\left(\frac{6^{3/4} x \sqrt[4]{2-3 x^4+x^5}}{-3 x^2+\sqrt{6} \sqrt{2-3 x^4+x^5}}\right)}{2\ 2^{3/4}}-\frac{3 \sqrt[4]{3} \tanh ^{-1}\left(\frac{6^{3/4} x \sqrt[4]{2-3 x^4+x^5}}{3 x^2+\sqrt{6} \sqrt{2-3 x^4+x^5}}\right)}{2\ 2^{3/4}}",1,"((2 - 3*x^4 + x^5)^(1/4)*(4 + 9*x^4 + 2*x^5))/(5*x^5) - (3*3^(1/4)*ArcTan[(-((3^(1/4)*x^2)/2^(3/4)) + Sqrt[2 - 3*x^4 + x^5]/6^(1/4))/(x*(2 - 3*x^4 + x^5)^(1/4))])/(2*2^(3/4)) - (3*3^(1/4)*ArcTanh[(6^(3/4)*x*(2 - 3*x^4 + x^5)^(1/4))/(3*x^2 + Sqrt[6]*Sqrt[2 - 3*x^4 + x^5])])/(2*2^(3/4))","A",1
2184,1,162,162,0.5212295,"\int \frac{x^2}{\left(-1+x^4\right) \sqrt[4]{x^2+x^6}} \, dx","IntegrateAlgebraic[x^2/((-1 + x^4)*(x^2 + x^6)^(1/4)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}-\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{4\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}+\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{4\ 2^{3/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}-\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{4\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}+\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{4\ 2^{3/4}}",1,"-1/4*ArcTan[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/2^(1/4) - ArcTan[(2^(3/4)*x*(x^2 + x^6)^(1/4))/(Sqrt[2]*x^2 - Sqrt[x^2 + x^6])]/(4*2^(3/4)) - ArcTanh[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/(4*2^(1/4)) + ArcTanh[(x^2/2^(1/4) + Sqrt[x^2 + x^6]/2^(3/4))/(x*(x^2 + x^6)^(1/4))]/(4*2^(3/4))","A",1
2185,1,162,162,0.0011898,"\int \frac{x^2}{\left(-1+x^4\right) \sqrt[4]{x^2+x^6}} \, dx","IntegrateAlgebraic[x^2/((-1 + x^4)*(x^2 + x^6)^(1/4)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}-\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{4\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}+\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{4\ 2^{3/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}-\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{4\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}+\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{4\ 2^{3/4}}",1,"-1/4*ArcTan[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/2^(1/4) - ArcTan[(2^(3/4)*x*(x^2 + x^6)^(1/4))/(Sqrt[2]*x^2 - Sqrt[x^2 + x^6])]/(4*2^(3/4)) - ArcTanh[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/(4*2^(1/4)) + ArcTanh[(x^2/2^(1/4) + Sqrt[x^2 + x^6]/2^(3/4))/(x*(x^2 + x^6)^(1/4))]/(4*2^(3/4))","A",1
2186,1,162,162,0.5260015,"\int \frac{1+x^4}{\left(-1+x^4\right) \sqrt[4]{x^2+x^6}} \, dx","IntegrateAlgebraic[(1 + x^4)/((-1 + x^4)*(x^2 + x^6)^(1/4)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2 \sqrt[4]{2}}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{2\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{2\ 2^{3/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2 \sqrt[4]{2}}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{2\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{2\ 2^{3/4}}",1,"-1/2*ArcTan[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/2^(1/4) + ArcTan[(2^(3/4)*x*(x^2 + x^6)^(1/4))/(Sqrt[2]*x^2 - Sqrt[x^2 + x^6])]/(2*2^(3/4)) - ArcTanh[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/(2*2^(1/4)) - ArcTanh[(x^2/2^(1/4) + Sqrt[x^2 + x^6]/2^(3/4))/(x*(x^2 + x^6)^(1/4))]/(2*2^(3/4))","A",1
2187,1,162,162,0.0013176,"\int \frac{1+x^4}{\left(-1+x^4\right) \sqrt[4]{x^2+x^6}} \, dx","IntegrateAlgebraic[(1 + x^4)/((-1 + x^4)*(x^2 + x^6)^(1/4)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2 \sqrt[4]{2}}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{2\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{2\ 2^{3/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2 \sqrt[4]{2}}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{2\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{2\ 2^{3/4}}",1,"-1/2*ArcTan[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/2^(1/4) + ArcTan[(2^(3/4)*x*(x^2 + x^6)^(1/4))/(Sqrt[2]*x^2 - Sqrt[x^2 + x^6])]/(2*2^(3/4)) - ArcTanh[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/(2*2^(1/4)) - ArcTanh[(x^2/2^(1/4) + Sqrt[x^2 + x^6]/2^(3/4))/(x*(x^2 + x^6)^(1/4))]/(2*2^(3/4))","A",1
2188,1,162,162,0.7962487,"\int \frac{1-x^2+x^4}{\left(-1+x^4\right) \sqrt[4]{x^2+x^6}} \, dx","IntegrateAlgebraic[(1 - x^2 + x^4)/((-1 + x^4)*(x^2 + x^6)^(1/4)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}+\frac{3 \tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{4\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}-\frac{3 \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{4\ 2^{3/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}+\frac{3 \tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{4\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}-\frac{3 \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{4\ 2^{3/4}}",1,"-1/4*ArcTan[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/2^(1/4) + (3*ArcTan[(2^(3/4)*x*(x^2 + x^6)^(1/4))/(Sqrt[2]*x^2 - Sqrt[x^2 + x^6])])/(4*2^(3/4)) - ArcTanh[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/(4*2^(1/4)) - (3*ArcTanh[(x^2/2^(1/4) + Sqrt[x^2 + x^6]/2^(3/4))/(x*(x^2 + x^6)^(1/4))])/(4*2^(3/4))","A",1
2189,1,162,162,0.0012671,"\int \frac{1-x^2+x^4}{\left(-1+x^4\right) \sqrt[4]{x^2+x^6}} \, dx","IntegrateAlgebraic[(1 - x^2 + x^4)/((-1 + x^4)*(x^2 + x^6)^(1/4)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}+\frac{3 \tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{4\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}-\frac{3 \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{4\ 2^{3/4}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}+\frac{3 \tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{4\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}-\frac{3 \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{4\ 2^{3/4}}",1,"-1/4*ArcTan[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/2^(1/4) + (3*ArcTan[(2^(3/4)*x*(x^2 + x^6)^(1/4))/(Sqrt[2]*x^2 - Sqrt[x^2 + x^6])])/(4*2^(3/4)) - ArcTanh[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/(4*2^(1/4)) - (3*ArcTanh[(x^2/2^(1/4) + Sqrt[x^2 + x^6]/2^(3/4))/(x*(x^2 + x^6)^(1/4))])/(4*2^(3/4))","A",1
2190,1,162,162,0.804359,"\int \frac{1+x^2+x^4}{\left(-1+x^4\right) \sqrt[4]{x^2+x^6}} \, dx","IntegrateAlgebraic[(1 + x^2 + x^4)/((-1 + x^4)*(x^2 + x^6)^(1/4)),x]","-\frac{3 \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{4\ 2^{3/4}}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{4\ 2^{3/4}}","-\frac{3 \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{4\ 2^{3/4}}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{4\ 2^{3/4}}",1,"(-3*ArcTan[(2^(1/4)*x)/(x^2 + x^6)^(1/4)])/(4*2^(1/4)) + ArcTan[(2^(3/4)*x*(x^2 + x^6)^(1/4))/(Sqrt[2]*x^2 - Sqrt[x^2 + x^6])]/(4*2^(3/4)) - (3*ArcTanh[(2^(1/4)*x)/(x^2 + x^6)^(1/4)])/(4*2^(1/4)) - ArcTanh[(x^2/2^(1/4) + Sqrt[x^2 + x^6]/2^(3/4))/(x*(x^2 + x^6)^(1/4))]/(4*2^(3/4))","A",1
2191,1,162,162,0.0012339,"\int \frac{1+x^2+x^4}{\left(-1+x^4\right) \sqrt[4]{x^2+x^6}} \, dx","IntegrateAlgebraic[(1 + x^2 + x^4)/((-1 + x^4)*(x^2 + x^6)^(1/4)),x]","-\frac{3 \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{4\ 2^{3/4}}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{4\ 2^{3/4}}","-\frac{3 \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{4\ 2^{3/4}}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{4\ 2^{3/4}}",1,"(-3*ArcTan[(2^(1/4)*x)/(x^2 + x^6)^(1/4)])/(4*2^(1/4)) + ArcTan[(2^(3/4)*x*(x^2 + x^6)^(1/4))/(Sqrt[2]*x^2 - Sqrt[x^2 + x^6])]/(4*2^(3/4)) - (3*ArcTanh[(2^(1/4)*x)/(x^2 + x^6)^(1/4)])/(4*2^(1/4)) - ArcTanh[(x^2/2^(1/4) + Sqrt[x^2 + x^6]/2^(3/4))/(x*(x^2 + x^6)^(1/4))]/(4*2^(3/4))","A",1
2192,1,162,162,0.4495744,"\int \frac{\left(-4+x^3\right) \left(-2+x^3\right) \left(-1+x^3\right)^{2/3}}{x^6 \left(-2+x^3+x^6\right)} \, dx","IntegrateAlgebraic[((-4 + x^3)*(-2 + x^3)*(-1 + x^3)^(2/3))/(x^6*(-2 + x^3 + x^6)),x]","\frac{\left(8-13 x^3\right) \left(-1+x^3\right)^{2/3}}{10 x^5}+\sqrt[3]{2} \sqrt[6]{3} \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2 \sqrt[3]{2} \sqrt[3]{-1+x^3}}\right)-\sqrt[3]{\frac{2}{3}} \log \left(-3 x+\sqrt[3]{2} 3^{2/3} \sqrt[3]{-1+x^3}\right)+\frac{\log \left(3 x^2+\sqrt[3]{2} 3^{2/3} x \sqrt[3]{-1+x^3}+2^{2/3} \sqrt[3]{3} \left(-1+x^3\right)^{2/3}\right)}{2^{2/3} \sqrt[3]{3}}","\frac{\left(8-13 x^3\right) \left(-1+x^3\right)^{2/3}}{10 x^5}+\sqrt[3]{2} \sqrt[6]{3} \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2 \sqrt[3]{2} \sqrt[3]{-1+x^3}}\right)-\sqrt[3]{\frac{2}{3}} \log \left(-3 x+\sqrt[3]{2} 3^{2/3} \sqrt[3]{-1+x^3}\right)+\frac{\log \left(3 x^2+\sqrt[3]{2} 3^{2/3} x \sqrt[3]{-1+x^3}+2^{2/3} \sqrt[3]{3} \left(-1+x^3\right)^{2/3}\right)}{2^{2/3} \sqrt[3]{3}}",1,"((8 - 13*x^3)*(-1 + x^3)^(2/3))/(10*x^5) + 2^(1/3)*3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*2^(1/3)*(-1 + x^3)^(1/3))] - (2/3)^(1/3)*Log[-3*x + 2^(1/3)*3^(2/3)*(-1 + x^3)^(1/3)] + Log[3*x^2 + 2^(1/3)*3^(2/3)*x*(-1 + x^3)^(1/3) + 2^(2/3)*3^(1/3)*(-1 + x^3)^(2/3)]/(2^(2/3)*3^(1/3))","A",1
2193,1,162,162,0.6312384,"\int \frac{\left(-1+x^4\right) \sqrt[4]{x^2+x^6}}{1+x^8} \, dx","IntegrateAlgebraic[((-1 + x^4)*(x^2 + x^6)^(1/4))/(1 + x^8),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2\ 2^{3/8}}+\frac{\tan ^{-1}\left(\frac{2^{5/8} x \sqrt[4]{x^2+x^6}}{\sqrt[4]{2} x^2-\sqrt{x^2+x^6}}\right)}{2\ 2^{7/8}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2\ 2^{3/8}}+\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{2^{3/8}}+\frac{\sqrt{x^2+x^6}}{2^{5/8}}}{x \sqrt[4]{x^2+x^6}}\right)}{2\ 2^{7/8}}","\frac{\tan ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2\ 2^{3/8}}+\frac{\tan ^{-1}\left(\frac{2^{5/8} x \sqrt[4]{x^2+x^6}}{\sqrt[4]{2} x^2-\sqrt{x^2+x^6}}\right)}{2\ 2^{7/8}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2\ 2^{3/8}}+\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{2^{3/8}}+\frac{\sqrt{x^2+x^6}}{2^{5/8}}}{x \sqrt[4]{x^2+x^6}}\right)}{2\ 2^{7/8}}",1,"ArcTan[(2^(1/8)*x)/(x^2 + x^6)^(1/4)]/(2*2^(3/8)) + ArcTan[(2^(5/8)*x*(x^2 + x^6)^(1/4))/(2^(1/4)*x^2 - Sqrt[x^2 + x^6])]/(2*2^(7/8)) - ArcTanh[(2^(1/8)*x)/(x^2 + x^6)^(1/4)]/(2*2^(3/8)) + ArcTanh[(x^2/2^(3/8) + Sqrt[x^2 + x^6]/2^(5/8))/(x*(x^2 + x^6)^(1/4))]/(2*2^(7/8))","A",1
2194,1,162,162,0.0010208,"\int \frac{\left(-1+x^4\right) \sqrt[4]{x^2+x^6}}{1+x^8} \, dx","IntegrateAlgebraic[((-1 + x^4)*(x^2 + x^6)^(1/4))/(1 + x^8),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2\ 2^{3/8}}+\frac{\tan ^{-1}\left(\frac{2^{5/8} x \sqrt[4]{x^2+x^6}}{\sqrt[4]{2} x^2-\sqrt{x^2+x^6}}\right)}{2\ 2^{7/8}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2\ 2^{3/8}}+\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{2^{3/8}}+\frac{\sqrt{x^2+x^6}}{2^{5/8}}}{x \sqrt[4]{x^2+x^6}}\right)}{2\ 2^{7/8}}","\frac{\tan ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2\ 2^{3/8}}+\frac{\tan ^{-1}\left(\frac{2^{5/8} x \sqrt[4]{x^2+x^6}}{\sqrt[4]{2} x^2-\sqrt{x^2+x^6}}\right)}{2\ 2^{7/8}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2\ 2^{3/8}}+\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{2^{3/8}}+\frac{\sqrt{x^2+x^6}}{2^{5/8}}}{x \sqrt[4]{x^2+x^6}}\right)}{2\ 2^{7/8}}",1,"ArcTan[(2^(1/8)*x)/(x^2 + x^6)^(1/4)]/(2*2^(3/8)) + ArcTan[(2^(5/8)*x*(x^2 + x^6)^(1/4))/(2^(1/4)*x^2 - Sqrt[x^2 + x^6])]/(2*2^(7/8)) - ArcTanh[(2^(1/8)*x)/(x^2 + x^6)^(1/4)]/(2*2^(3/8)) + ArcTanh[(x^2/2^(3/8) + Sqrt[x^2 + x^6]/2^(5/8))/(x*(x^2 + x^6)^(1/4))]/(2*2^(7/8))","A",1
2195,1,162,162,0.6949353,"\int \frac{-1+x^8}{\sqrt[4]{x^2+x^6} \left(1+x^8\right)} \, dx","IntegrateAlgebraic[(-1 + x^8)/((x^2 + x^6)^(1/4)*(1 + x^8)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2 \sqrt[8]{2}}+\frac{\tan ^{-1}\left(\frac{2^{5/8} x \sqrt[4]{x^2+x^6}}{\sqrt[4]{2} x^2-\sqrt{x^2+x^6}}\right)}{2\ 2^{5/8}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2 \sqrt[8]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{2^{3/8}}+\frac{\sqrt{x^2+x^6}}{2^{5/8}}}{x \sqrt[4]{x^2+x^6}}\right)}{2\ 2^{5/8}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2 \sqrt[8]{2}}+\frac{\tan ^{-1}\left(\frac{2^{5/8} x \sqrt[4]{x^2+x^6}}{\sqrt[4]{2} x^2-\sqrt{x^2+x^6}}\right)}{2\ 2^{5/8}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2 \sqrt[8]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{2^{3/8}}+\frac{\sqrt{x^2+x^6}}{2^{5/8}}}{x \sqrt[4]{x^2+x^6}}\right)}{2\ 2^{5/8}}",1,"-1/2*ArcTan[(2^(1/8)*x)/(x^2 + x^6)^(1/4)]/2^(1/8) + ArcTan[(2^(5/8)*x*(x^2 + x^6)^(1/4))/(2^(1/4)*x^2 - Sqrt[x^2 + x^6])]/(2*2^(5/8)) - ArcTanh[(2^(1/8)*x)/(x^2 + x^6)^(1/4)]/(2*2^(1/8)) - ArcTanh[(x^2/2^(3/8) + Sqrt[x^2 + x^6]/2^(5/8))/(x*(x^2 + x^6)^(1/4))]/(2*2^(5/8))","A",1
2196,1,162,162,0.0015781,"\int \frac{-1+x^8}{\sqrt[4]{x^2+x^6} \left(1+x^8\right)} \, dx","IntegrateAlgebraic[(-1 + x^8)/((x^2 + x^6)^(1/4)*(1 + x^8)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2 \sqrt[8]{2}}+\frac{\tan ^{-1}\left(\frac{2^{5/8} x \sqrt[4]{x^2+x^6}}{\sqrt[4]{2} x^2-\sqrt{x^2+x^6}}\right)}{2\ 2^{5/8}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2 \sqrt[8]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{2^{3/8}}+\frac{\sqrt{x^2+x^6}}{2^{5/8}}}{x \sqrt[4]{x^2+x^6}}\right)}{2\ 2^{5/8}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2 \sqrt[8]{2}}+\frac{\tan ^{-1}\left(\frac{2^{5/8} x \sqrt[4]{x^2+x^6}}{\sqrt[4]{2} x^2-\sqrt{x^2+x^6}}\right)}{2\ 2^{5/8}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2 \sqrt[8]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{2^{3/8}}+\frac{\sqrt{x^2+x^6}}{2^{5/8}}}{x \sqrt[4]{x^2+x^6}}\right)}{2\ 2^{5/8}}",1,"-1/2*ArcTan[(2^(1/8)*x)/(x^2 + x^6)^(1/4)]/2^(1/8) + ArcTan[(2^(5/8)*x*(x^2 + x^6)^(1/4))/(2^(1/4)*x^2 - Sqrt[x^2 + x^6])]/(2*2^(5/8)) - ArcTanh[(2^(1/8)*x)/(x^2 + x^6)^(1/4)]/(2*2^(1/8)) - ArcTanh[(x^2/2^(3/8) + Sqrt[x^2 + x^6]/2^(5/8))/(x*(x^2 + x^6)^(1/4))]/(2*2^(5/8))","A",1
2197,1,162,162,8.2982281,"\int \frac{x^5 \left(8-7 (1+k) x+6 k x^2\right)}{((1-x) x (1-k x))^{2/3} \left(-b+b (1+k) x-b k x^2+x^8\right)} \, dx","IntegrateAlgebraic[(x^5*(8 - 7*(1 + k)*x + 6*k*x^2))/(((1 - x)*x*(1 - k*x))^(2/3)*(-b + b*(1 + k)*x - b*k*x^2 + x^8)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x^3}{x^3+2 \sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{\sqrt[3]{b}}+\frac{\log \left(x^3-\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{\sqrt[3]{b}}-\frac{\log \left(x^6+\sqrt[3]{b} x^3 \sqrt[3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 \sqrt[3]{b}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x^3}{x^3+2 \sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{\sqrt[3]{b}}+\frac{\log \left(x^3-\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{\sqrt[3]{b}}-\frac{\log \left(x^6+\sqrt[3]{b} x^3 \sqrt[3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 \sqrt[3]{b}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*x^3)/(x^3 + 2*b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/b^(1/3) + Log[x^3 - b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3)]/b^(1/3) - Log[x^6 + b^(1/3)*x^3*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + b^(2/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3)]/(2*b^(1/3))","A",1
2198,1,162,162,1.270143,"\int \frac{-2 b-a x^4+2 x^8}{\sqrt[4]{-b+a x^4} \left(-2 b-a x^4+x^8\right)} \, dx","IntegrateAlgebraic[(-2*b - a*x^4 + 2*x^8)/((-b + a*x^4)^(1/4)*(-2*b - a*x^4 + x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{\sqrt[4]{a}}-\frac{1}{4} \text{RootSum}\left[3 a^2-b-5 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-3 a \log (x)+3 a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+2 \log (x) \text{$\#$1}^4-2 \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-5 a \text{$\#$1}+4 \text{$\#$1}^5}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{\sqrt[4]{a}}-\frac{1}{4} \text{RootSum}\left[3 a^2-b-5 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{3 a \log (x)-3 a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-2 \log (x) \text{$\#$1}^4+2 \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{5 a \text{$\#$1}-4 \text{$\#$1}^5}\&\right]",1,"ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/a^(1/4) + ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/a^(1/4) - RootSum[3*a^2 - b - 5*a*#1^4 + 2*#1^8 & , (-3*a*Log[x] + 3*a*Log[(-b + a*x^4)^(1/4) - x*#1] + 2*Log[x]*#1^4 - 2*Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4)/(-5*a*#1 + 4*#1^5) & ]/4","A",1
2199,1,130,162,0.2772816,"\int \frac{\sqrt{b+\sqrt{b^2+a x^2}}}{x^4} \, dx","IntegrateAlgebraic[Sqrt[b + Sqrt[b^2 + a*x^2]]/x^4,x]","\frac{\sqrt{b+\sqrt{b^2+a x^2}} \left(-3 a^{3/2} x^2+\sqrt{a} \left(-10 b^2+2 b \sqrt{b^2+a x^2}\right)\right)}{24 \sqrt{a} b^2 x^3}-\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{8 \sqrt{2} b^{5/2}}","\frac{\sqrt{b+\sqrt{b^2+a x^2}} \left(-3 a^{3/2} x^2+\sqrt{a} \left(-10 b^2+2 b \sqrt{b^2+a x^2}\right)\right)}{24 \sqrt{a} b^2 x^3}-\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2} \sqrt{b}}\right)}{4 \sqrt{2} b^{5/2}}",1,"(Sqrt[b + Sqrt[b^2 + a*x^2]]*(-3*a^(3/2)*x^2 + Sqrt[a]*(-10*b^2 + 2*b*Sqrt[b^2 + a*x^2])))/(24*Sqrt[a]*b^2*x^3) - (a^(3/2)*ArcTan[(Sqrt[a]*x)/(Sqrt[2]*Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/(8*Sqrt[2]*b^(5/2))","A",1
2200,1,130,162,0.3036512,"\int \frac{\sqrt{b+\sqrt{b^2+a x^2}}}{x^4 \sqrt{b^2+a x^2}} \, dx","IntegrateAlgebraic[Sqrt[b + Sqrt[b^2 + a*x^2]]/(x^4*Sqrt[b^2 + a*x^2]),x]","\frac{\sqrt{b+\sqrt{b^2+a x^2}} \left(15 a^{3/2} x^2+\sqrt{a} \left(2 b^2-10 b \sqrt{b^2+a x^2}\right)\right)}{24 \sqrt{a} b^3 x^3}+\frac{5 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{8 \sqrt{2} b^{7/2}}","\frac{\sqrt{b+\sqrt{b^2+a x^2}} \left(15 a^{3/2} x^2+\sqrt{a} \left(2 b^2-10 b \sqrt{b^2+a x^2}\right)\right)}{24 \sqrt{a} b^3 x^3}+\frac{5 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2} \sqrt{b}}\right)}{4 \sqrt{2} b^{7/2}}",1,"(Sqrt[b + Sqrt[b^2 + a*x^2]]*(15*a^(3/2)*x^2 + Sqrt[a]*(2*b^2 - 10*b*Sqrt[b^2 + a*x^2])))/(24*Sqrt[a]*b^3*x^3) + (5*a^(3/2)*ArcTan[(Sqrt[a]*x)/(Sqrt[2]*Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/(8*Sqrt[2]*b^(7/2))","A",1
2201,1,163,163,0.5046627,"\int \frac{1}{\sqrt[3]{x+x^3} \left(-1+x^6\right)} \, dx","IntegrateAlgebraic[1/((x + x^3)^(1/3)*(-1 + x^6)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x+x^3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x+x^3}\right)}{6 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x+x^3}+\sqrt[3]{2} \left(x+x^3\right)^{2/3}\right)}{12 \sqrt[3]{2}}+\frac{1}{6} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{x+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x+x^3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x+x^3}\right)}{6 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x+x^3}+\sqrt[3]{2} \left(x+x^3\right)^{2/3}\right)}{12 \sqrt[3]{2}}+\frac{1}{6} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{x+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"-1/2*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(x + x^3)^(1/3))]/(2^(1/3)*Sqrt[3]) + Log[-2*x + 2^(2/3)*(x + x^3)^(1/3)]/(6*2^(1/3)) - Log[2*x^2 + 2^(2/3)*x*(x + x^3)^(1/3) + 2^(1/3)*(x + x^3)^(2/3)]/(12*2^(1/3)) + RootSum[1 - #1^3 + #1^6 & , (-Log[x] + Log[(x + x^3)^(1/3) - x*#1])/#1 & ]/6","A",1
2202,1,163,163,0.0012984,"\int \frac{1}{\sqrt[3]{x+x^3} \left(-1+x^6\right)} \, dx","IntegrateAlgebraic[1/((x + x^3)^(1/3)*(-1 + x^6)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x+x^3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x+x^3}\right)}{6 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x+x^3}+\sqrt[3]{2} \left(x+x^3\right)^{2/3}\right)}{12 \sqrt[3]{2}}+\frac{1}{6} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{x+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x+x^3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x+x^3}\right)}{6 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x+x^3}+\sqrt[3]{2} \left(x+x^3\right)^{2/3}\right)}{12 \sqrt[3]{2}}+\frac{1}{6} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{x+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"-1/2*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(x + x^3)^(1/3))]/(2^(1/3)*Sqrt[3]) + Log[-2*x + 2^(2/3)*(x + x^3)^(1/3)]/(6*2^(1/3)) - Log[2*x^2 + 2^(2/3)*x*(x + x^3)^(1/3) + 2^(1/3)*(x + x^3)^(2/3)]/(12*2^(1/3)) + RootSum[1 - #1^3 + #1^6 & , (-Log[x] + Log[(x + x^3)^(1/3) - x*#1])/#1 & ]/6","A",1
2203,1,162,163,0.5479469,"\int \frac{\left(1+x^3\right) \sqrt{-2-x^3+x^6}}{x^4 \left(-1-2 x^3+x^6\right)} \, dx","IntegrateAlgebraic[((1 + x^3)*Sqrt[-2 - x^3 + x^6])/(x^4*(-1 - 2*x^3 + x^6)),x]","\frac{\sqrt{-2-x^3+x^6}}{3 x^3}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{-2-x^3+x^6}}{-2+x^3}\right)}{\sqrt{2}}-\frac{1}{3} \sqrt{4+3 \sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2+\frac{3}{\sqrt{2}}} \sqrt{-2-x^3+x^6}}{-2+x^3}\right)-\frac{1}{3} \sqrt{-4+3 \sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{-2+\frac{3}{\sqrt{2}}} \sqrt{-2-x^3+x^6}}{-2+x^3}\right)","\frac{\sqrt{-2-x^3+x^6}}{3 x^3}-\frac{\tan ^{-1}\left(\frac{\sqrt{-2-x^3+x^6}}{\sqrt{2} \left(1+x^3\right)}\right)}{\sqrt{2}}+\frac{1}{3} \sqrt{4+3 \sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{-4+3 \sqrt{2}} \sqrt{-2-x^3+x^6}}{1+x^3}\right)-\frac{1}{3} \sqrt{-4+3 \sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{4+3 \sqrt{2}} \sqrt{-2-x^3+x^6}}{1+x^3}\right)",1,"Sqrt[-2 - x^3 + x^6]/(3*x^3) + ArcTan[(Sqrt[2]*Sqrt[-2 - x^3 + x^6])/(-2 + x^3)]/Sqrt[2] - (Sqrt[4 + 3*Sqrt[2]]*ArcTan[(Sqrt[2 + 3/Sqrt[2]]*Sqrt[-2 - x^3 + x^6])/(-2 + x^3)])/3 - (Sqrt[-4 + 3*Sqrt[2]]*ArcTanh[(Sqrt[-2 + 3/Sqrt[2]]*Sqrt[-2 - x^3 + x^6])/(-2 + x^3)])/3","A",1
2204,1,163,163,10.1461221,"\int \frac{x^2 \left(-4+7 x^3\right)}{\sqrt[3]{-x+x^4} \left(-1-x^4+x^7\right)} \, dx","IntegrateAlgebraic[(x^2*(-4 + 7*x^3))/((-x + x^4)^(1/3)*(-1 - x^4 + x^7)),x]","-\sqrt{3} \tan ^{-1}\left(\frac{3 \sqrt{3} x \sqrt[3]{-x+x^4}-3 x^2 \sqrt[3]{-x+x^4}}{-6+2 \sqrt{3} x-3 x \sqrt[3]{-x+x^4}+\sqrt{3} x^2 \sqrt[3]{-x+x^4}}\right)+2 \tanh ^{-1}\left(1-2 x \sqrt[3]{-x+x^4}\right)-\tanh ^{-1}\left(\frac{1+x \sqrt[3]{-x+x^4}}{1+x \sqrt[3]{-x+x^4}+2 x^2 \left(-x+x^4\right)^{2/3}}\right)","-\sqrt{3} \tan ^{-1}\left(\frac{3 \sqrt{3} x \sqrt[3]{-x+x^4}-3 x^2 \sqrt[3]{-x+x^4}}{-6+2 \sqrt{3} x-3 x \sqrt[3]{-x+x^4}+\sqrt{3} x^2 \sqrt[3]{-x+x^4}}\right)+2 \tanh ^{-1}\left(1-2 x \sqrt[3]{-x+x^4}\right)-\tanh ^{-1}\left(\frac{1+x \sqrt[3]{-x+x^4}}{1+x \sqrt[3]{-x+x^4}+2 x^2 \left(-x+x^4\right)^{2/3}}\right)",1,"-(Sqrt[3]*ArcTan[(3*Sqrt[3]*x*(-x + x^4)^(1/3) - 3*x^2*(-x + x^4)^(1/3))/(-6 + 2*Sqrt[3]*x - 3*x*(-x + x^4)^(1/3) + Sqrt[3]*x^2*(-x + x^4)^(1/3))]) + 2*ArcTanh[1 - 2*x*(-x + x^4)^(1/3)] - ArcTanh[(1 + x*(-x + x^4)^(1/3))/(1 + x*(-x + x^4)^(1/3) + 2*x^2*(-x + x^4)^(2/3))]","A",1
2205,1,163,163,12.3490513,"\int \frac{a x^4+x^8}{\sqrt[4]{-b x^2+a x^4} \left(-b+2 a x^4+x^8\right)} \, dx","IntegrateAlgebraic[(a*x^4 + x^8)/((-(b*x^2) + a*x^4)^(1/4)*(-b + 2*a*x^4 + x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{1}{8} \text{RootSum}\left[a^4-2 a^3 b-b^3-4 a^3 \text{$\#$1}^4+4 a^2 b \text{$\#$1}^4+6 a^2 \text{$\#$1}^8-2 a b \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{1}{8} \text{RootSum}\left[a^4-2 a^3 b-b^3-4 a^3 \text{$\#$1}^4+4 a^2 b \text{$\#$1}^4+6 a^2 \text{$\#$1}^8-2 a b \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"ArcTan[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)]/a^(1/4) + ArcTanh[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)]/a^(1/4) + RootSum[a^4 - 2*a^3*b - b^3 - 4*a^3*#1^4 + 4*a^2*b*#1^4 + 6*a^2*#1^8 - 2*a*b*#1^8 - 4*a*#1^12 + #1^16 & , (-Log[x] + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1])/#1 & ]/8","A",1
2206,1,163,163,0.0021684,"\int \frac{a x^4+x^8}{\sqrt[4]{-b x^2+a x^4} \left(-b+2 a x^4+x^8\right)} \, dx","IntegrateAlgebraic[(a*x^4 + x^8)/((-(b*x^2) + a*x^4)^(1/4)*(-b + 2*a*x^4 + x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{1}{8} \text{RootSum}\left[a^4-2 a^3 b-b^3-4 a^3 \text{$\#$1}^4+4 a^2 b \text{$\#$1}^4+6 a^2 \text{$\#$1}^8-2 a b \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{\sqrt[4]{a}}+\frac{1}{8} \text{RootSum}\left[a^4-2 a^3 b-b^3-4 a^3 \text{$\#$1}^4+4 a^2 b \text{$\#$1}^4+6 a^2 \text{$\#$1}^8-2 a b \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"ArcTan[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)]/a^(1/4) + ArcTanh[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)]/a^(1/4) + RootSum[a^4 - 2*a^3*b - b^3 - 4*a^3*#1^4 + 4*a^2*b*#1^4 + 6*a^2*#1^8 - 2*a*b*#1^8 - 4*a*#1^12 + #1^16 & , (-Log[x] + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1])/#1 & ]/8","A",1
2207,1,163,163,1.1025689,"\int \frac{\sqrt{1+2 x^2-x^4} \left(-1+x^4\right) \left(1+x^4\right)}{\left(-1-x^2+x^4\right) \left(1+3 x^2-x^4-3 x^6+x^8\right)} \, dx","IntegrateAlgebraic[(Sqrt[1 + 2*x^2 - x^4]*(-1 + x^4)*(1 + x^4))/((-1 - x^2 + x^4)*(1 + 3*x^2 - x^4 - 3*x^6 + x^8)),x]","-\sqrt{\frac{1}{10} \left(1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x \sqrt{1+2 x^2-x^4}}{-1-2 x^2+x^4}\right)-\tanh ^{-1}\left(\frac{x \sqrt{1+2 x^2-x^4}}{-1-2 x^2+x^4}\right)+\sqrt{\frac{1}{10} \left(-1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x \sqrt{1+2 x^2-x^4}}{-1-2 x^2+x^4}\right)","-\sqrt{\frac{1}{10} \left(1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x \sqrt{1+2 x^2-x^4}}{-1-2 x^2+x^4}\right)-\tanh ^{-1}\left(\frac{x \sqrt{1+2 x^2-x^4}}{-1-2 x^2+x^4}\right)+\sqrt{\frac{1}{10} \left(-1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x \sqrt{1+2 x^2-x^4}}{-1-2 x^2+x^4}\right)",1,"-(Sqrt[(1 + Sqrt[5])/10]*ArcTan[(Sqrt[-1/2 + Sqrt[5]/2]*x*Sqrt[1 + 2*x^2 - x^4])/(-1 - 2*x^2 + x^4)]) - ArcTanh[(x*Sqrt[1 + 2*x^2 - x^4])/(-1 - 2*x^2 + x^4)] + Sqrt[(-1 + Sqrt[5])/10]*ArcTanh[(Sqrt[1/2 + Sqrt[5]/2]*x*Sqrt[1 + 2*x^2 - x^4])/(-1 - 2*x^2 + x^4)]","A",1
2208,1,163,163,0.2851444,"\int \frac{1}{\left(-1+x^2\right) \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[1/((-1 + x^2)*Sqrt[x + Sqrt[1 + x^2]]),x]","-\sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\sqrt{-1+\sqrt{2}} \sqrt{x+\sqrt{1+x^2}}\right)+\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\sqrt{1+\sqrt{2}} \sqrt{x+\sqrt{1+x^2}}\right)-\sqrt{-1+\sqrt{2}} \tanh ^{-1}\left(\sqrt{-1+\sqrt{2}} \sqrt{x+\sqrt{1+x^2}}\right)+\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\sqrt{1+\sqrt{2}} \sqrt{x+\sqrt{1+x^2}}\right)","\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{x+\sqrt{1+x^2}}}{\sqrt{-1+\sqrt{2}}}\right)-\sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{x+\sqrt{1+x^2}}}{\sqrt{1+\sqrt{2}}}\right)+\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{x+\sqrt{1+x^2}}}{\sqrt{-1+\sqrt{2}}}\right)-\sqrt{-1+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{x+\sqrt{1+x^2}}}{\sqrt{1+\sqrt{2}}}\right)",1,"-(Sqrt[-1 + Sqrt[2]]*ArcTan[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]]) + Sqrt[1 + Sqrt[2]]*ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] - Sqrt[-1 + Sqrt[2]]*ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + Sqrt[2]]*ArcTanh[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]]","A",1
2209,1,163,163,3.1266144,"\int \frac{\sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}{\sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}} \, dx","IntegrateAlgebraic[Sqrt[-(a/b^2) + (a^2*x^2)/b^2]/Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]],x]","\frac{\left(3-2 a x^2\right) \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{4 b}+\frac{1}{2} x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}} \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}+\frac{5 \tanh ^{-1}\left(\sqrt{2} \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}\right)}{4 \sqrt{2} b}","\frac{\left(3-2 a x^2\right) \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{4 b}+\frac{1}{2} x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}} \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}+\frac{5 \tanh ^{-1}\left(\sqrt{2} \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}\right)}{4 \sqrt{2} b}",1,"((3 - 2*a*x^2)*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]])/(4*b) + (x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]])/2 + (5*ArcTanh[Sqrt[2]*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]]])/(4*Sqrt[2]*b)","A",1
2210,1,223,163,3.2746858,"\int \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}} \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}} \, dx","IntegrateAlgebraic[Sqrt[-(a/b^2) + (a^2*x^2)/b^2]*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]],x]","\frac{\left(-9-2 a x^2\right) \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{24 b}+\frac{5}{12} x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}} \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}+\frac{3 \log \left(1+\sqrt{2} \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}\right)}{16 \sqrt{2} b}-\frac{3 \log \left(-b+\sqrt{2} b \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}\right)}{16 \sqrt{2} b}","\frac{\left(-9-2 a x^2\right) \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{24 b}+\frac{5}{12} x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}} \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}+\frac{3 \tanh ^{-1}\left(\sqrt{2} \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}\right)}{8 \sqrt{2} b}",1,"((-9 - 2*a*x^2)*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]])/(24*b) + (5*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]])/12 + (3*Log[1 + Sqrt[2]*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]]])/(16*Sqrt[2]*b) - (3*Log[-b + Sqrt[2]*b*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]]])/(16*Sqrt[2]*b)","A",1
2211,1,164,164,0.5657998,"\int \frac{(-3+x) (-2+x) \left(2-x+2 x^3\right)^{2/3}}{x^6 \left(-2+x+2 x^3\right)} \, dx","IntegrateAlgebraic[((-3 + x)*(-2 + x)*(2 - x + 2*x^3)^(2/3))/(x^6*(-2 + x + 2*x^3)),x]","\frac{3 \left(2-x+2 x^3\right)^{2/3} \left(2-x+7 x^3\right)}{10 x^5}-2 \sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+\sqrt[3]{2} \sqrt[3]{2-x+2 x^3}}\right)+2 \sqrt[3]{2} \log \left(-2 x+\sqrt[3]{2} \sqrt[3]{2-x+2 x^3}\right)-\sqrt[3]{2} \log \left(4 x^2+2 \sqrt[3]{2} x \sqrt[3]{2-x+2 x^3}+2^{2/3} \left(2-x+2 x^3\right)^{2/3}\right)","\frac{3 \left(2-x+2 x^3\right)^{2/3} \left(2-x+7 x^3\right)}{10 x^5}-2 \sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+\sqrt[3]{2} \sqrt[3]{2-x+2 x^3}}\right)+2 \sqrt[3]{2} \log \left(-2 x+\sqrt[3]{2} \sqrt[3]{2-x+2 x^3}\right)-\sqrt[3]{2} \log \left(4 x^2+2 \sqrt[3]{2} x \sqrt[3]{2-x+2 x^3}+2^{2/3} \left(2-x+2 x^3\right)^{2/3}\right)",1,"(3*(2 - x + 2*x^3)^(2/3)*(2 - x + 7*x^3))/(10*x^5) - 2*2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(1/3)*(2 - x + 2*x^3)^(1/3))] + 2*2^(1/3)*Log[-2*x + 2^(1/3)*(2 - x + 2*x^3)^(1/3)] - 2^(1/3)*Log[4*x^2 + 2*2^(1/3)*x*(2 - x + 2*x^3)^(1/3) + 2^(2/3)*(2 - x + 2*x^3)^(2/3)]","A",1
2212,1,164,164,0.4210232,"\int \frac{\left(1+x^3\right)^{2/3} \left(2+x^3\right)}{x^6 \left(-2-x^3+x^6\right)} \, dx","IntegrateAlgebraic[((1 + x^3)^(2/3)*(2 + x^3))/(x^6*(-2 - x^3 + x^6)),x]","\frac{\left(1+x^3\right)^{5/3}}{5 x^5}-\frac{\tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2 \sqrt[3]{2} \sqrt[3]{1+x^3}}\right)}{2^{2/3} 3^{5/6}}+\frac{\log \left(-3 x+\sqrt[3]{2} 3^{2/3} \sqrt[3]{1+x^3}\right)}{3\ 2^{2/3} \sqrt[3]{3}}-\frac{\log \left(3 x^2+\sqrt[3]{2} 3^{2/3} x \sqrt[3]{1+x^3}+2^{2/3} \sqrt[3]{3} \left(1+x^3\right)^{2/3}\right)}{6\ 2^{2/3} \sqrt[3]{3}}","\frac{\left(1+x^3\right)^{5/3}}{5 x^5}-\frac{\tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2 \sqrt[3]{2} \sqrt[3]{1+x^3}}\right)}{2^{2/3} 3^{5/6}}+\frac{\log \left(-3 x+\sqrt[3]{2} 3^{2/3} \sqrt[3]{1+x^3}\right)}{3\ 2^{2/3} \sqrt[3]{3}}-\frac{\log \left(3 x^2+\sqrt[3]{2} 3^{2/3} x \sqrt[3]{1+x^3}+2^{2/3} \sqrt[3]{3} \left(1+x^3\right)^{2/3}\right)}{6\ 2^{2/3} \sqrt[3]{3}}",1,"(1 + x^3)^(5/3)/(5*x^5) - ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*2^(1/3)*(1 + x^3)^(1/3))]/(2^(2/3)*3^(5/6)) + Log[-3*x + 2^(1/3)*3^(2/3)*(1 + x^3)^(1/3)]/(3*2^(2/3)*3^(1/3)) - Log[3*x^2 + 2^(1/3)*3^(2/3)*x*(1 + x^3)^(1/3) + 2^(2/3)*3^(1/3)*(1 + x^3)^(2/3)]/(6*2^(2/3)*3^(1/3))","A",1
2213,1,167,164,2.8011427,"\int \frac{b+a x^6}{\sqrt[3]{x+x^3} \left(d+c x^6\right)} \, dx","IntegrateAlgebraic[(b + a*x^6)/((x + x^3)^(1/3)*(d + c*x^6)),x]","\frac{\sqrt{3} a \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)}{2 c}-\frac{a \log \left(-c x+c \sqrt[3]{x+x^3}\right)}{2 c}+\frac{a \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)}{4 c}+\frac{(-b c+a d) \text{RootSum}\left[c-d+3 d \text{$\#$1}^3-3 d \text{$\#$1}^6+d \text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{x+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 c d}","\frac{\sqrt{3} a \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)}{2 c}-\frac{a \log \left(-x+\sqrt[3]{x+x^3}\right)}{2 c}+\frac{a \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)}{4 c}+\frac{(-b c+a d) \text{RootSum}\left[c-d+3 d \text{$\#$1}^3-3 d \text{$\#$1}^6+d \text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{x+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 c d}",1,"(Sqrt[3]*a*ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^3)^(1/3))])/(2*c) - (a*Log[-(c*x) + c*(x + x^3)^(1/3)])/(2*c) + (a*Log[x^2 + x*(x + x^3)^(1/3) + (x + x^3)^(2/3)])/(4*c) + ((-(b*c) + a*d)*RootSum[c - d + 3*d*#1^3 - 3*d*#1^6 + d*#1^9 & , (-Log[x] + Log[(x + x^3)^(1/3) - x*#1])/#1 & ])/(6*c*d)","A",1
2214,1,167,164,0.0014625,"\int \frac{b+a x^6}{\sqrt[3]{x+x^3} \left(d+c x^6\right)} \, dx","IntegrateAlgebraic[(b + a*x^6)/((x + x^3)^(1/3)*(d + c*x^6)),x]","\frac{\sqrt{3} a \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)}{2 c}-\frac{a \log \left(-c x+c \sqrt[3]{x+x^3}\right)}{2 c}+\frac{a \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)}{4 c}+\frac{(-b c+a d) \text{RootSum}\left[c-d+3 d \text{$\#$1}^3-3 d \text{$\#$1}^6+d \text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{x+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 c d}","\frac{\sqrt{3} a \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)}{2 c}-\frac{a \log \left(-x+\sqrt[3]{x+x^3}\right)}{2 c}+\frac{a \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)}{4 c}+\frac{(-b c+a d) \text{RootSum}\left[c-d+3 d \text{$\#$1}^3-3 d \text{$\#$1}^6+d \text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{x+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 c d}",1,"(Sqrt[3]*a*ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^3)^(1/3))])/(2*c) - (a*Log[-(c*x) + c*(x + x^3)^(1/3)])/(2*c) + (a*Log[x^2 + x*(x + x^3)^(1/3) + (x + x^3)^(2/3)])/(4*c) + ((-(b*c) + a*d)*RootSum[c - d + 3*d*#1^3 - 3*d*#1^6 + d*#1^9 & , (-Log[x] + Log[(x + x^3)^(1/3) - x*#1])/#1 & ])/(6*c*d)","A",1
2215,1,165,165,0.2069089,"\int \frac{1}{(2+x) \sqrt[3]{1+x+x^2}} \, dx","IntegrateAlgebraic[1/((2 + x)*(1 + x + x^2)^(1/3)),x]","-\frac{\tan ^{-1}\left(\frac{\frac{2}{3 \sqrt[6]{3}}-\frac{2 x}{3 \sqrt[6]{3}}+\frac{\sqrt[3]{1+x+x^2}}{\sqrt{3}}}{\sqrt[3]{1+x+x^2}}\right)}{3^{5/6}}+\frac{\log \left(-\sqrt[3]{3}+\sqrt[3]{3} x+3 \sqrt[3]{1+x+x^2}\right)}{3 \sqrt[3]{3}}-\frac{\log \left(3^{2/3}-2\ 3^{2/3} x+3^{2/3} x^2+\left(3 \sqrt[3]{3}-3 \sqrt[3]{3} x\right) \sqrt[3]{1+x+x^2}+9 \left(1+x+x^2\right)^{2/3}\right)}{6 \sqrt[3]{3}}","-\frac{\tan ^{-1}\left(\frac{\frac{2}{3 \sqrt[6]{3}}-\frac{2 x}{3 \sqrt[6]{3}}+\frac{\sqrt[3]{1+x+x^2}}{\sqrt{3}}}{\sqrt[3]{1+x+x^2}}\right)}{3^{5/6}}+\frac{\log \left(-\sqrt[3]{3}+\sqrt[3]{3} x+3 \sqrt[3]{1+x+x^2}\right)}{3 \sqrt[3]{3}}-\frac{\log \left(3^{2/3}-2\ 3^{2/3} x+3^{2/3} x^2+\left(3 \sqrt[3]{3}-3 \sqrt[3]{3} x\right) \sqrt[3]{1+x+x^2}+9 \left(1+x+x^2\right)^{2/3}\right)}{6 \sqrt[3]{3}}",1,"-(ArcTan[(2/(3*3^(1/6)) - (2*x)/(3*3^(1/6)) + (1 + x + x^2)^(1/3)/Sqrt[3])/(1 + x + x^2)^(1/3)]/3^(5/6)) + Log[-3^(1/3) + 3^(1/3)*x + 3*(1 + x + x^2)^(1/3)]/(3*3^(1/3)) - Log[3^(2/3) - 2*3^(2/3)*x + 3^(2/3)*x^2 + (3*3^(1/3) - 3*3^(1/3)*x)*(1 + x + x^2)^(1/3) + 9*(1 + x + x^2)^(2/3)]/(6*3^(1/3))","A",1
2216,1,165,165,0.5426947,"\int \frac{(-3+2 x) \sqrt[3]{-1+x+x^3}}{x^2 \left(2-2 x+x^3\right)} \, dx","IntegrateAlgebraic[((-3 + 2*x)*(-1 + x + x^3)^(1/3))/(x^2*(2 - 2*x + x^3)),x]","\frac{3 \sqrt[3]{-1+x+x^3}}{2 x}+\frac{3^{5/6} \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2 \sqrt[3]{2} \sqrt[3]{-1+x+x^3}}\right)}{2 \sqrt[3]{2}}+\frac{1}{2} \sqrt[3]{\frac{3}{2}} \log \left(-3 x+\sqrt[3]{2} 3^{2/3} \sqrt[3]{-1+x+x^3}\right)-\frac{1}{4} \sqrt[3]{\frac{3}{2}} \log \left(3 x^2+\sqrt[3]{2} 3^{2/3} x \sqrt[3]{-1+x+x^3}+2^{2/3} \sqrt[3]{3} \left(-1+x+x^3\right)^{2/3}\right)","\frac{3 \sqrt[3]{-1+x+x^3}}{2 x}+\frac{3^{5/6} \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2 \sqrt[3]{2} \sqrt[3]{-1+x+x^3}}\right)}{2 \sqrt[3]{2}}+\frac{1}{2} \sqrt[3]{\frac{3}{2}} \log \left(-3 x+\sqrt[3]{2} 3^{2/3} \sqrt[3]{-1+x+x^3}\right)-\frac{1}{4} \sqrt[3]{\frac{3}{2}} \log \left(3 x^2+\sqrt[3]{2} 3^{2/3} x \sqrt[3]{-1+x+x^3}+2^{2/3} \sqrt[3]{3} \left(-1+x+x^3\right)^{2/3}\right)",1,"(3*(-1 + x + x^3)^(1/3))/(2*x) + (3^(5/6)*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*2^(1/3)*(-1 + x + x^3)^(1/3))])/(2*2^(1/3)) + ((3/2)^(1/3)*Log[-3*x + 2^(1/3)*3^(2/3)*(-1 + x + x^3)^(1/3)])/2 - ((3/2)^(1/3)*Log[3*x^2 + 2^(1/3)*3^(2/3)*x*(-1 + x + x^3)^(1/3) + 2^(2/3)*3^(1/3)*(-1 + x + x^3)^(2/3)])/4","A",1
2217,1,165,165,0.9246098,"\int \frac{-\left((2 a-b) b^2\right)+(4 a-b) b x-(2 a+b) x^2+x^3}{(-a+x) \sqrt[4]{(-a+x) (-b+x)^2} \left(b^2+a d-(2 b+d) x+x^2\right)} \, dx","IntegrateAlgebraic[(-((2*a - b)*b^2) + (4*a - b)*b*x - (2*a + b)*x^2 + x^3)/((-a + x)*((-a + x)*(-b + x)^2)^(1/4)*(b^2 + a*d - (2*b + d)*x + x^2)),x]","-\frac{4 \left(-a b^2+2 a b x+b^2 x-a x^2-2 b x^2+x^3\right)^{3/4}}{(b-x) (-a+x)}-2 \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{b-x}\right)+2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{b-x}\right)","-\frac{4 \left(-a b^2+2 a b x+b^2 x-a x^2-2 b x^2+x^3\right)^{3/4}}{(b-x) (-a+x)}-2 \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{b-x}\right)+2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{b-x}\right)",1,"(-4*(-(a*b^2) + 2*a*b*x + b^2*x - a*x^2 - 2*b*x^2 + x^3)^(3/4))/((b - x)*(-a + x)) - 2*d^(1/4)*ArcTan[(d^(1/4)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/4))/(b - x)] + 2*d^(1/4)*ArcTanh[(d^(1/4)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/4))/(b - x)]","A",1
2218,1,164,165,1.0373624,"\int \frac{x^4}{\left(b+a x^4\right)^2 \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[x^4/((b + a*x^4)^2*(b*x^2 + a*x^4)^(1/4)),x]","\frac{\left(-b+a x^2\right) \left(b x^2+a x^4\right)^{3/4}}{4 a b (a+b) x \left(b+a x^4\right)}+\frac{\text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 a \log (x)+2 a \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}+\text{$\#$1}^5}\&\right]}{32 a (a+b)}","\frac{\left(-b+a x^2\right) \left(b x^2+a x^4\right)^{3/4}}{4 a b (a+b) x \left(b+a x^4\right)}+\frac{\text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 a \log (x)-2 a \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}-\text{$\#$1}^5}\&\right]}{32 a (a+b)}",1,"((-b + a*x^2)*(b*x^2 + a*x^4)^(3/4))/(4*a*b*(a + b)*x*(b + a*x^4)) + RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (-2*a*Log[x] + 2*a*Log[(b*x^2 + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1) + #1^5) & ]/(32*a*(a + b))","A",1
2219,1,164,165,0.0017888,"\int \frac{x^4}{\left(b+a x^4\right)^2 \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[x^4/((b + a*x^4)^2*(b*x^2 + a*x^4)^(1/4)),x]","\frac{\left(-b+a x^2\right) \left(b x^2+a x^4\right)^{3/4}}{4 a b (a+b) x \left(b+a x^4\right)}+\frac{\text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 a \log (x)+2 a \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}+\text{$\#$1}^5}\&\right]}{32 a (a+b)}","\frac{\left(-b+a x^2\right) \left(b x^2+a x^4\right)^{3/4}}{4 a b (a+b) x \left(b+a x^4\right)}+\frac{\text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 a \log (x)-2 a \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}-\text{$\#$1}^5}\&\right]}{32 a (a+b)}",1,"((-b + a*x^2)*(b*x^2 + a*x^4)^(3/4))/(4*a*b*(a + b)*x*(b + a*x^4)) + RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (-2*a*Log[x] + 2*a*Log[(b*x^2 + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1) + #1^5) & ]/(32*a*(a + b))","A",1
2220,1,164,165,0.0019789,"\int \frac{x^4}{\left(b+a x^4\right)^2 \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[x^4/((b + a*x^4)^2*(b*x^2 + a*x^4)^(1/4)),x]","\frac{\left(-b+a x^2\right) \left(b x^2+a x^4\right)^{3/4}}{4 a b (a+b) x \left(b+a x^4\right)}+\frac{\text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 a \log (x)+2 a \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}+\text{$\#$1}^5}\&\right]}{32 a (a+b)}","\frac{\left(-b+a x^2\right) \left(b x^2+a x^4\right)^{3/4}}{4 a b (a+b) x \left(b+a x^4\right)}+\frac{\text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 a \log (x)-2 a \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}-\text{$\#$1}^5}\&\right]}{32 a (a+b)}",1,"((-b + a*x^2)*(b*x^2 + a*x^4)^(3/4))/(4*a*b*(a + b)*x*(b + a*x^4)) + RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (-2*a*Log[x] + 2*a*Log[(b*x^2 + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1) + #1^5) & ]/(32*a*(a + b))","A",1
2221,1,165,165,0.5196242,"\int \frac{\left(-b+a x^4\right) \left(b+a x^4\right)^{3/4}}{x^8 \left(b+2 a x^4\right)} \, dx","IntegrateAlgebraic[((-b + a*x^4)*(b + a*x^4)^(3/4))/(x^8*(b + 2*a*x^4)),x]","\frac{\left(b-6 a x^4\right) \left(b+a x^4\right)^{3/4}}{7 b x^7}-\frac{3 a^{7/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} x \sqrt[4]{b+a x^4}}{-\sqrt{a} x^2+\sqrt{b+a x^4}}\right)}{2 \sqrt{2} b}-\frac{3 a^{7/4} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{a} x^2}{\sqrt{2}}+\frac{\sqrt{b+a x^4}}{\sqrt{2} \sqrt[4]{a}}}{x \sqrt[4]{b+a x^4}}\right)}{2 \sqrt{2} b}","\frac{\left(b-6 a x^4\right) \left(b+a x^4\right)^{3/4}}{7 b x^7}-\frac{3 a^{7/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} x \sqrt[4]{b+a x^4}}{-\sqrt{a} x^2+\sqrt{b+a x^4}}\right)}{2 \sqrt{2} b}-\frac{3 a^{7/4} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{a} x^2}{\sqrt{2}}+\frac{\sqrt{b+a x^4}}{\sqrt{2} \sqrt[4]{a}}}{x \sqrt[4]{b+a x^4}}\right)}{2 \sqrt{2} b}",1,"((b - 6*a*x^4)*(b + a*x^4)^(3/4))/(7*b*x^7) - (3*a^(7/4)*ArcTan[(Sqrt[2]*a^(1/4)*x*(b + a*x^4)^(1/4))/(-(Sqrt[a]*x^2) + Sqrt[b + a*x^4])])/(2*Sqrt[2]*b) - (3*a^(7/4)*ArcTanh[((a^(1/4)*x^2)/Sqrt[2] + Sqrt[b + a*x^4]/(Sqrt[2]*a^(1/4)))/(x*(b + a*x^4)^(1/4))])/(2*Sqrt[2]*b)","A",1
2222,1,142,165,0.166916,"\int \frac{x \sqrt{b+a x}}{x+\sqrt{c+\sqrt{b+a x}}} \, dx","IntegrateAlgebraic[(x*Sqrt[b + a*x])/(x + Sqrt[c + Sqrt[b + a*x]]),x]","-\frac{4}{3} \left(c+\sqrt{b+a x}\right)^{3/2}+\frac{2 \left(c^3+(b+a x)^{3/2}\right)}{3 a}+4 \text{RootSum}\left[b-c^2-a \text{$\#$1}+2 c \text{$\#$1}^2-\text{$\#$1}^4\&,\frac{-b \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2+a \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3}{a-4 c \text{$\#$1}+4 \text{$\#$1}^3}\&\right]","-\frac{4}{3} c \sqrt{c+\sqrt{b+a x}}+\sqrt{b+a x} \left(\frac{2 (b+a x)}{3 a}-\frac{4}{3} \sqrt{c+\sqrt{b+a x}}\right)+4 \text{RootSum}\left[b-c^2-a \text{$\#$1}+2 c \text{$\#$1}^2-\text{$\#$1}^4\&,\frac{-b \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2+a \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3}{a-4 c \text{$\#$1}+4 \text{$\#$1}^3}\&\right]",1,"(-4*(c + Sqrt[b + a*x])^(3/2))/3 + (2*(c^3 + (b + a*x)^(3/2)))/(3*a) + 4*RootSum[b - c^2 - a*#1 + 2*c*#1^2 - #1^4 & , (-(b*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^2) + a*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^3)/(a - 4*c*#1 + 4*#1^3) & ]","A",1
2223,1,142,165,0.0017804,"\int \frac{x \sqrt{b+a x}}{x+\sqrt{c+\sqrt{b+a x}}} \, dx","IntegrateAlgebraic[(x*Sqrt[b + a*x])/(x + Sqrt[c + Sqrt[b + a*x]]),x]","-\frac{4}{3} \left(c+\sqrt{b+a x}\right)^{3/2}+\frac{2 \left(c^3+(b+a x)^{3/2}\right)}{3 a}+4 \text{RootSum}\left[b-c^2-a \text{$\#$1}+2 c \text{$\#$1}^2-\text{$\#$1}^4\&,\frac{-b \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2+a \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3}{a-4 c \text{$\#$1}+4 \text{$\#$1}^3}\&\right]","-\frac{4}{3} c \sqrt{c+\sqrt{b+a x}}+\sqrt{b+a x} \left(\frac{2 (b+a x)}{3 a}-\frac{4}{3} \sqrt{c+\sqrt{b+a x}}\right)+4 \text{RootSum}\left[b-c^2-a \text{$\#$1}+2 c \text{$\#$1}^2-\text{$\#$1}^4\&,\frac{-b \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2+a \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3}{a-4 c \text{$\#$1}+4 \text{$\#$1}^3}\&\right]",1,"(-4*(c + Sqrt[b + a*x])^(3/2))/3 + (2*(c^3 + (b + a*x)^(3/2)))/(3*a) + 4*RootSum[b - c^2 - a*#1 + 2*c*#1^2 - #1^4 & , (-(b*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^2) + a*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^3)/(a - 4*c*#1 + 4*#1^3) & ]","A",1
2224,1,166,166,1.2623739,"\int \frac{\sqrt{1-\sqrt{1-\sqrt{1-\frac{1}{x}}}}}{x} \, dx","IntegrateAlgebraic[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]/x,x]","-8 \sqrt{1-\sqrt{1-\sqrt{\frac{-1+x}{x}}}}+2 \sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\sqrt{1+\sqrt{2}} \sqrt{1-\sqrt{1-\sqrt{\frac{-1+x}{x}}}}\right)+4 \tanh ^{-1}\left(\sqrt{1-\sqrt{1-\sqrt{\frac{-1+x}{x}}}}\right)+2 \sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\sqrt{-1+\sqrt{2}} \sqrt{1-\sqrt{1-\sqrt{\frac{-1+x}{x}}}}\right)","-8 \sqrt{1-\sqrt{1-\sqrt{\frac{-1+x}{x}}}}+2 \sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{1-\sqrt{1-\sqrt{\frac{-1+x}{x}}}}}{\sqrt{-1+\sqrt{2}}}\right)+4 \tanh ^{-1}\left(\sqrt{1-\sqrt{1-\sqrt{\frac{-1+x}{x}}}}\right)+2 \sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{1-\sqrt{1-\sqrt{\frac{-1+x}{x}}}}}{\sqrt{1+\sqrt{2}}}\right)",1,"-8*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]] + 2*Sqrt[-1 + Sqrt[2]]*ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]] + 4*ArcTanh[Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]] + 2*Sqrt[1 + Sqrt[2]]*ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]]","A",1
2225,1,166,166,0.7924386,"\int \frac{-2 a b x+(a+b) x^2}{(x (-a+x) (-b+x))^{2/3} \left(a b d-(a+b) d x+(-1+d) x^2\right)} \, dx","IntegrateAlgebraic[(-2*a*b*x + (a + b)*x^2)/((x*(-a + x)*(-b + x))^(2/3)*(a*b*d - (a + b)*d*x + (-1 + d)*x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}}\right)}{\sqrt[3]{d}}+\frac{\log \left(x-\sqrt[3]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}\right)}{\sqrt[3]{d}}-\frac{\log \left(x^2+\sqrt[3]{d} x \sqrt[3]{a b x+(-a-b) x^2+x^3}+d^{2/3} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}\right)}{2 \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}}\right)}{\sqrt[3]{d}}+\frac{\log \left(x-\sqrt[3]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}\right)}{\sqrt[3]{d}}-\frac{\log \left(x^2+\sqrt[3]{d} x \sqrt[3]{a b x+(-a-b) x^2+x^3}+d^{2/3} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}\right)}{2 \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*d^(1/3)*(a*b*x + (-a - b)*x^2 + x^3)^(1/3))])/d^(1/3) + Log[x - d^(1/3)*(a*b*x + (-a - b)*x^2 + x^3)^(1/3)]/d^(1/3) - Log[x^2 + d^(1/3)*x*(a*b*x + (-a - b)*x^2 + x^3)^(1/3) + d^(2/3)*(a*b*x + (-a - b)*x^2 + x^3)^(2/3)]/(2*d^(1/3))","A",1
2226,1,165,166,0.3866829,"\int \frac{\sqrt[4]{-b+a x^3}}{x^7} \, dx","IntegrateAlgebraic[(-b + a*x^3)^(1/4)/x^7,x]","\frac{\left(-4 b+a x^3\right) \sqrt[4]{-b+a x^3}}{24 b x^6}+\frac{a^2 \tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^3}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^3}}\right)}{16 \sqrt{2} b^{7/4}}+\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^3}}{\sqrt{b}+\sqrt{-b+a x^3}}\right)}{16 \sqrt{2} b^{7/4}}","\frac{\left(-4 b+a x^3\right) \sqrt[4]{-b+a x^3}}{24 b x^6}-\frac{a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^3}}{-\sqrt{b}+\sqrt{-b+a x^3}}\right)}{16 \sqrt{2} b^{7/4}}+\frac{a^2 \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^3}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^3}}\right)}{16 \sqrt{2} b^{7/4}}",1,"((-4*b + a*x^3)*(-b + a*x^3)^(1/4))/(24*b*x^6) + (a^2*ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^3]/(Sqrt[2]*b^(1/4)))/(-b + a*x^3)^(1/4)])/(16*Sqrt[2]*b^(7/4)) + (a^2*ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^3)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^3])])/(16*Sqrt[2]*b^(7/4))","A",1
2227,1,166,166,0.5970029,"\int \frac{\left(b+2 a x^2\right) \sqrt[4]{b x^2+a x^4}}{-b+a x^2} \, dx","IntegrateAlgebraic[((b + 2*a*x^2)*(b*x^2 + a*x^4)^(1/4))/(-b + a*x^2),x]","x \sqrt[4]{b x^2+a x^4}-\frac{7 b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{2 a^{3/4}}+\frac{3 \sqrt[4]{2} b \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{a^{3/4}}+\frac{7 b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{2 a^{3/4}}-\frac{3 \sqrt[4]{2} b \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{a^{3/4}}","x \sqrt[4]{b x^2+a x^4}-\frac{7 b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{2 a^{3/4}}+\frac{3 \sqrt[4]{2} b \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{a^{3/4}}+\frac{7 b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{2 a^{3/4}}-\frac{3 \sqrt[4]{2} b \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{a^{3/4}}",1,"x*(b*x^2 + a*x^4)^(1/4) - (7*b*ArcTan[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/(2*a^(3/4)) + (3*2^(1/4)*b*ArcTan[(2^(1/4)*a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/a^(3/4) + (7*b*ArcTanh[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/(2*a^(3/4)) - (3*2^(1/4)*b*ArcTanh[(2^(1/4)*a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/a^(3/4)","A",1
2228,1,166,166,0.5850407,"\int \frac{-b^4+a^4 x^4}{\sqrt{-b^2 x+a^2 x^3} \left(b^4+a^4 x^4\right)} \, dx","IntegrateAlgebraic[(-b^4 + a^4*x^4)/(Sqrt[-(b^2*x) + a^2*x^3]*(b^4 + a^4*x^4)),x]","\frac{\tan ^{-1}\left(\frac{2^{3/4} \sqrt{a} \sqrt{b} \sqrt{-b^2 x+a^2 x^3}}{b^2+\sqrt{2} a b x-a^2 x^2}\right)}{2^{3/4} \sqrt{a} \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{-\frac{b^{3/2}}{2^{3/4} \sqrt{a}}+\frac{\sqrt{a} \sqrt{b} x}{\sqrt[4]{2}}+\frac{a^{3/2} x^2}{2^{3/4} \sqrt{b}}}{\sqrt{-b^2 x+a^2 x^3}}\right)}{2^{3/4} \sqrt{a} \sqrt{b}}","\frac{\tan ^{-1}\left(\frac{2^{3/4} \sqrt{a} \sqrt{b} \sqrt{-b^2 x+a^2 x^3}}{b^2+\sqrt{2} a b x-a^2 x^2}\right)}{2^{3/4} \sqrt{a} \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{-\frac{b^{3/2}}{2^{3/4} \sqrt{a}}+\frac{\sqrt{a} \sqrt{b} x}{\sqrt[4]{2}}+\frac{a^{3/2} x^2}{2^{3/4} \sqrt{b}}}{\sqrt{-b^2 x+a^2 x^3}}\right)}{2^{3/4} \sqrt{a} \sqrt{b}}",1,"ArcTan[(2^(3/4)*Sqrt[a]*Sqrt[b]*Sqrt[-(b^2*x) + a^2*x^3])/(b^2 + Sqrt[2]*a*b*x - a^2*x^2)]/(2^(3/4)*Sqrt[a]*Sqrt[b]) - ArcTanh[(-(b^(3/2)/(2^(3/4)*Sqrt[a])) + (Sqrt[a]*Sqrt[b]*x)/2^(1/4) + (a^(3/2)*x^2)/(2^(3/4)*Sqrt[b]))/Sqrt[-(b^2*x) + a^2*x^3]]/(2^(3/4)*Sqrt[a]*Sqrt[b])","A",1
2229,1,166,166,2.0870833,"\int \frac{-1+x^5}{\sqrt{1+x^4} \left(1+x^5\right)} \, dx","IntegrateAlgebraic[(-1 + x^5)/(Sqrt[1 + x^4]*(1 + x^5)),x]","-\frac{1}{5} \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{1+2 x+x^2+\sqrt{1+x^4}}\right)+\frac{4}{5} \text{RootSum}\left[4-4 \text{$\#$1}-2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{-2 \log (x)+2 \log \left(1+x^2+\sqrt{1+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}-\log \left(1+x^2+\sqrt{1+x^4}-x \text{$\#$1}\right) \text{$\#$1}-\log (x) \text{$\#$1}^2+\log \left(1+x^2+\sqrt{1+x^4}-x \text{$\#$1}\right) \text{$\#$1}^2}{-2-3 \text{$\#$1}^2+2 \text{$\#$1}^3}\&\right]","-\frac{1}{5} \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{1+2 x+x^2+\sqrt{1+x^4}}\right)+\frac{4}{5} \text{RootSum}\left[4-4 \text{$\#$1}-2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{-2 \log (x)+2 \log \left(1+x^2+\sqrt{1+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}-\log \left(1+x^2+\sqrt{1+x^4}-x \text{$\#$1}\right) \text{$\#$1}-\log (x) \text{$\#$1}^2+\log \left(1+x^2+\sqrt{1+x^4}-x \text{$\#$1}\right) \text{$\#$1}^2}{-2-3 \text{$\#$1}^2+2 \text{$\#$1}^3}\&\right]",1,"-1/5*(Sqrt[2]*ArcTanh[(Sqrt[2]*x)/(1 + 2*x + x^2 + Sqrt[1 + x^4])]) + (4*RootSum[4 - 4*#1 - 2*#1^3 + #1^4 & , (-2*Log[x] + 2*Log[1 + x^2 + Sqrt[1 + x^4] - x*#1] + Log[x]*#1 - Log[1 + x^2 + Sqrt[1 + x^4] - x*#1]*#1 - Log[x]*#1^2 + Log[1 + x^2 + Sqrt[1 + x^4] - x*#1]*#1^2)/(-2 - 3*#1^2 + 2*#1^3) & ])/5","A",1
2230,1,166,166,0.0011026,"\int \frac{-1+x^5}{\sqrt{1+x^4} \left(1+x^5\right)} \, dx","IntegrateAlgebraic[(-1 + x^5)/(Sqrt[1 + x^4]*(1 + x^5)),x]","-\frac{1}{5} \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{1+2 x+x^2+\sqrt{1+x^4}}\right)+\frac{4}{5} \text{RootSum}\left[4-4 \text{$\#$1}-2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{-2 \log (x)+2 \log \left(1+x^2+\sqrt{1+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}-\log \left(1+x^2+\sqrt{1+x^4}-x \text{$\#$1}\right) \text{$\#$1}-\log (x) \text{$\#$1}^2+\log \left(1+x^2+\sqrt{1+x^4}-x \text{$\#$1}\right) \text{$\#$1}^2}{-2-3 \text{$\#$1}^2+2 \text{$\#$1}^3}\&\right]","-\frac{1}{5} \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{1+2 x+x^2+\sqrt{1+x^4}}\right)+\frac{4}{5} \text{RootSum}\left[4-4 \text{$\#$1}-2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{-2 \log (x)+2 \log \left(1+x^2+\sqrt{1+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}-\log \left(1+x^2+\sqrt{1+x^4}-x \text{$\#$1}\right) \text{$\#$1}-\log (x) \text{$\#$1}^2+\log \left(1+x^2+\sqrt{1+x^4}-x \text{$\#$1}\right) \text{$\#$1}^2}{-2-3 \text{$\#$1}^2+2 \text{$\#$1}^3}\&\right]",1,"-1/5*(Sqrt[2]*ArcTanh[(Sqrt[2]*x)/(1 + 2*x + x^2 + Sqrt[1 + x^4])]) + (4*RootSum[4 - 4*#1 - 2*#1^3 + #1^4 & , (-2*Log[x] + 2*Log[1 + x^2 + Sqrt[1 + x^4] - x*#1] + Log[x]*#1 - Log[1 + x^2 + Sqrt[1 + x^4] - x*#1]*#1 - Log[x]*#1^2 + Log[1 + x^2 + Sqrt[1 + x^4] - x*#1]*#1^2)/(-2 - 3*#1^2 + 2*#1^3) & ])/5","A",1
2231,1,166,166,2.2169858,"\int \frac{1+x^5}{\sqrt{1+x^4} \left(-1+x^5\right)} \, dx","IntegrateAlgebraic[(1 + x^5)/(Sqrt[1 + x^4]*(-1 + x^5)),x]","-\frac{1}{5} \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{1-2 x+x^2+\sqrt{1+x^4}}\right)+\frac{4}{5} \text{RootSum}\left[4+4 \text{$\#$1}+2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{-2 \log (x)+2 \log \left(1+x^2+\sqrt{1+x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}+\log \left(1+x^2+\sqrt{1+x^4}-x \text{$\#$1}\right) \text{$\#$1}-\log (x) \text{$\#$1}^2+\log \left(1+x^2+\sqrt{1+x^4}-x \text{$\#$1}\right) \text{$\#$1}^2}{2+3 \text{$\#$1}^2+2 \text{$\#$1}^3}\&\right]","-\frac{1}{5} \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{1-2 x+x^2+\sqrt{1+x^4}}\right)+\frac{4}{5} \text{RootSum}\left[4+4 \text{$\#$1}+2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{-2 \log (x)+2 \log \left(1+x^2+\sqrt{1+x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}+\log \left(1+x^2+\sqrt{1+x^4}-x \text{$\#$1}\right) \text{$\#$1}-\log (x) \text{$\#$1}^2+\log \left(1+x^2+\sqrt{1+x^4}-x \text{$\#$1}\right) \text{$\#$1}^2}{2+3 \text{$\#$1}^2+2 \text{$\#$1}^3}\&\right]",1,"-1/5*(Sqrt[2]*ArcTanh[(Sqrt[2]*x)/(1 - 2*x + x^2 + Sqrt[1 + x^4])]) + (4*RootSum[4 + 4*#1 + 2*#1^3 + #1^4 & , (-2*Log[x] + 2*Log[1 + x^2 + Sqrt[1 + x^4] - x*#1] - Log[x]*#1 + Log[1 + x^2 + Sqrt[1 + x^4] - x*#1]*#1 - Log[x]*#1^2 + Log[1 + x^2 + Sqrt[1 + x^4] - x*#1]*#1^2)/(2 + 3*#1^2 + 2*#1^3) & ])/5","A",1
2232,1,166,166,0.0009821,"\int \frac{1+x^5}{\sqrt{1+x^4} \left(-1+x^5\right)} \, dx","IntegrateAlgebraic[(1 + x^5)/(Sqrt[1 + x^4]*(-1 + x^5)),x]","-\frac{1}{5} \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{1-2 x+x^2+\sqrt{1+x^4}}\right)+\frac{4}{5} \text{RootSum}\left[4+4 \text{$\#$1}+2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{-2 \log (x)+2 \log \left(1+x^2+\sqrt{1+x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}+\log \left(1+x^2+\sqrt{1+x^4}-x \text{$\#$1}\right) \text{$\#$1}-\log (x) \text{$\#$1}^2+\log \left(1+x^2+\sqrt{1+x^4}-x \text{$\#$1}\right) \text{$\#$1}^2}{2+3 \text{$\#$1}^2+2 \text{$\#$1}^3}\&\right]","-\frac{1}{5} \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x}{1-2 x+x^2+\sqrt{1+x^4}}\right)+\frac{4}{5} \text{RootSum}\left[4+4 \text{$\#$1}+2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{-2 \log (x)+2 \log \left(1+x^2+\sqrt{1+x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}+\log \left(1+x^2+\sqrt{1+x^4}-x \text{$\#$1}\right) \text{$\#$1}-\log (x) \text{$\#$1}^2+\log \left(1+x^2+\sqrt{1+x^4}-x \text{$\#$1}\right) \text{$\#$1}^2}{2+3 \text{$\#$1}^2+2 \text{$\#$1}^3}\&\right]",1,"-1/5*(Sqrt[2]*ArcTanh[(Sqrt[2]*x)/(1 - 2*x + x^2 + Sqrt[1 + x^4])]) + (4*RootSum[4 + 4*#1 + 2*#1^3 + #1^4 & , (-2*Log[x] + 2*Log[1 + x^2 + Sqrt[1 + x^4] - x*#1] - Log[x]*#1 + Log[1 + x^2 + Sqrt[1 + x^4] - x*#1]*#1 - Log[x]*#1^2 + Log[1 + x^2 + Sqrt[1 + x^4] - x*#1]*#1^2)/(2 + 3*#1^2 + 2*#1^3) & ])/5","A",1
2233,1,84,166,1.2139224,"\int \frac{-b^6+a^6 x^6}{\sqrt{b^4+a^4 x^4} \left(b^6+a^6 x^6\right)} \, dx","IntegrateAlgebraic[(-b^6 + a^6*x^6)/(Sqrt[b^4 + a^4*x^4]*(b^6 + a^6*x^6)),x]","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} a b x}{b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}}\right)}{3 a b}-\frac{2 \tanh ^{-1}\left(\frac{a b x}{\sqrt{b^4+a^4 x^4}}\right)}{3 a b}","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} a b x}{b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}}\right)}{3 a b}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{4-2 \sqrt{3}} a b x}{b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}}\right)}{3 a b}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{4+2 \sqrt{3}} a b x}{b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}}\right)}{3 a b}",1,"-1/3*(Sqrt[2]*ArcTan[(Sqrt[2]*a*b*x)/(b^2 + a^2*x^2 + Sqrt[b^4 + a^4*x^4])])/(a*b) - (2*ArcTanh[(a*b*x)/Sqrt[b^4 + a^4*x^4]])/(3*a*b)","A",1
2234,1,167,167,0.4352372,"\int \frac{-2 a b+(a+b) x}{\sqrt[3]{x (-a+x) (-b+x)} \left(a b d-(a+b) d x+(-1+d) x^2\right)} \, dx","IntegrateAlgebraic[(-2*a*b + (a + b)*x)/((x*(-a + x)*(-b + x))^(1/3)*(a*b*d - (a + b)*d*x + (-1 + d)*x^2)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}}\right)}{d^{2/3}}+\frac{\log \left(x-\sqrt[3]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}\right)}{d^{2/3}}-\frac{\log \left(x^2+\sqrt[3]{d} x \sqrt[3]{a b x+(-a-b) x^2+x^3}+d^{2/3} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}\right)}{2 d^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}}\right)}{d^{2/3}}+\frac{\log \left(x-\sqrt[3]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}\right)}{d^{2/3}}-\frac{\log \left(x^2+\sqrt[3]{d} x \sqrt[3]{a b x+(-a-b) x^2+x^3}+d^{2/3} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}\right)}{2 d^{2/3}}",1,"-((Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*d^(1/3)*(a*b*x + (-a - b)*x^2 + x^3)^(1/3))])/d^(2/3)) + Log[x - d^(1/3)*(a*b*x + (-a - b)*x^2 + x^3)^(1/3)]/d^(2/3) - Log[x^2 + d^(1/3)*x*(a*b*x + (-a - b)*x^2 + x^3)^(1/3) + d^(2/3)*(a*b*x + (-a - b)*x^2 + x^3)^(2/3)]/(2*d^(2/3))","A",1
2235,1,166,167,0.245004,"\int \frac{1}{x^7 \left(-b+a x^3\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x^7*(-b + a*x^3)^(3/4)),x]","\frac{\sqrt[4]{-b+a x^3} \left(4 b+7 a x^3\right)}{24 b^2 x^6}+\frac{7 a^2 \tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^3}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^3}}\right)}{16 \sqrt{2} b^{11/4}}+\frac{7 a^2 \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^3}}{\sqrt{b}+\sqrt{-b+a x^3}}\right)}{16 \sqrt{2} b^{11/4}}","\frac{\sqrt[4]{-b+a x^3} \left(4 b+7 a x^3\right)}{24 b^2 x^6}-\frac{7 a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^3}}{-\sqrt{b}+\sqrt{-b+a x^3}}\right)}{16 \sqrt{2} b^{11/4}}+\frac{7 a^2 \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^3}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^3}}\right)}{16 \sqrt{2} b^{11/4}}",1,"((-b + a*x^3)^(1/4)*(4*b + 7*a*x^3))/(24*b^2*x^6) + (7*a^2*ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^3]/(Sqrt[2]*b^(1/4)))/(-b + a*x^3)^(1/4)])/(16*Sqrt[2]*b^(11/4)) + (7*a^2*ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^3)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^3])])/(16*Sqrt[2]*b^(11/4))","A",1
2236,1,166,167,0.2893687,"\int \frac{1}{x^7 \sqrt[4]{-b+a x^3}} \, dx","IntegrateAlgebraic[1/(x^7*(-b + a*x^3)^(1/4)),x]","\frac{\left(-b+a x^3\right)^{3/4} \left(4 b+5 a x^3\right)}{24 b^2 x^6}+\frac{5 a^2 \tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^3}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^3}}\right)}{48 \sqrt{2} b^{9/4}}-\frac{5 a^2 \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^3}}{\sqrt{b}+\sqrt{-b+a x^3}}\right)}{48 \sqrt{2} b^{9/4}}","\frac{\left(-b+a x^3\right)^{3/4} \left(4 b+5 a x^3\right)}{24 b^2 x^6}-\frac{5 a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^3}}{-\sqrt{b}+\sqrt{-b+a x^3}}\right)}{48 \sqrt{2} b^{9/4}}-\frac{5 a^2 \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^3}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^3}}\right)}{48 \sqrt{2} b^{9/4}}",1,"((-b + a*x^3)^(3/4)*(4*b + 5*a*x^3))/(24*b^2*x^6) + (5*a^2*ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^3]/(Sqrt[2]*b^(1/4)))/(-b + a*x^3)^(1/4)])/(48*Sqrt[2]*b^(9/4)) - (5*a^2*ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^3)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^3])])/(48*Sqrt[2]*b^(9/4))","A",1
2237,1,167,167,0.5292482,"\int \frac{1+x^2}{\left(-1+x^2\right) \sqrt[3]{x^2+x^4}} \, dx","IntegrateAlgebraic[(1 + x^2)/((-1 + x^2)*(x^2 + x^4)^(1/3)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2^{2/3} \sqrt[3]{x^2+x^4}}\right)}{2 \sqrt[3]{2}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x^2+x^4}}\right)}{2 \sqrt[3]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[3]{2} x}{\sqrt[3]{x^2+x^4}}\right)}{\sqrt[3]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[3]{2} x^2+\frac{\left(x^2+x^4\right)^{2/3}}{\sqrt[3]{2}}}{x \sqrt[3]{x^2+x^4}}\right)}{2 \sqrt[3]{2}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2^{2/3} \sqrt[3]{x^2+x^4}}\right)}{2 \sqrt[3]{2}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x^2+x^4}}\right)}{2 \sqrt[3]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[3]{2} x}{\sqrt[3]{x^2+x^4}}\right)}{\sqrt[3]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[3]{2} x^2+\frac{\left(x^2+x^4\right)^{2/3}}{\sqrt[3]{2}}}{x \sqrt[3]{x^2+x^4}}\right)}{2 \sqrt[3]{2}}",1,"-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2^(2/3)*(x^2 + x^4)^(1/3))])/2^(1/3) - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(x^2 + x^4)^(1/3))])/(2*2^(1/3)) - ArcTanh[(2^(1/3)*x)/(x^2 + x^4)^(1/3)]/2^(1/3) - ArcTanh[(2^(1/3)*x^2 + (x^2 + x^4)^(2/3)/2^(1/3))/(x*(x^2 + x^4)^(1/3))]/(2*2^(1/3))","A",1
2238,1,167,167,3.5926744,"\int \frac{\left(-1+x^4\right)^{2/3} \left(3+x^4\right) \left(-2-x^3+2 x^4\right)}{x^6 \left(-2+3 x^3+2 x^4\right)} \, dx","IntegrateAlgebraic[((-1 + x^4)^(2/3)*(3 + x^4)*(-2 - x^3 + 2*x^4))/(x^6*(-2 + 3*x^3 + 2*x^4)),x]","\frac{3 \left(-1+x^4\right)^{2/3} \left(-1-5 x^3+x^4\right)}{5 x^5}+3 \sqrt[3]{2} \sqrt[6]{3} \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x-2 \sqrt[3]{2} \sqrt[3]{-1+x^4}}\right)-\sqrt[3]{2} 3^{2/3} \log \left(3 x+\sqrt[3]{2} 3^{2/3} \sqrt[3]{-1+x^4}\right)+\left(\frac{3}{2}\right)^{2/3} \log \left(3 x^2-\sqrt[3]{2} 3^{2/3} x \sqrt[3]{-1+x^4}+2^{2/3} \sqrt[3]{3} \left(-1+x^4\right)^{2/3}\right)","\frac{3 \left(-1+x^4\right)^{2/3} \left(-1-5 x^3+x^4\right)}{5 x^5}+3 \sqrt[3]{2} \sqrt[6]{3} \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x-2 \sqrt[3]{2} \sqrt[3]{-1+x^4}}\right)-\sqrt[3]{2} 3^{2/3} \log \left(3 x+\sqrt[3]{2} 3^{2/3} \sqrt[3]{-1+x^4}\right)+\left(\frac{3}{2}\right)^{2/3} \log \left(3 x^2-\sqrt[3]{2} 3^{2/3} x \sqrt[3]{-1+x^4}+2^{2/3} \sqrt[3]{3} \left(-1+x^4\right)^{2/3}\right)",1,"(3*(-1 + x^4)^(2/3)*(-1 - 5*x^3 + x^4))/(5*x^5) + 3*2^(1/3)*3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/3)*x - 2*2^(1/3)*(-1 + x^4)^(1/3))] - 2^(1/3)*3^(2/3)*Log[3*x + 2^(1/3)*3^(2/3)*(-1 + x^4)^(1/3)] + (3/2)^(2/3)*Log[3*x^2 - 2^(1/3)*3^(2/3)*x*(-1 + x^4)^(1/3) + 2^(2/3)*3^(1/3)*(-1 + x^4)^(2/3)]","A",1
2239,1,166,167,0.2088981,"\int \frac{1}{x^9 \left(-b+a x^4\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x^9*(-b + a*x^4)^(3/4)),x]","\frac{\sqrt[4]{-b+a x^4} \left(4 b+7 a x^4\right)}{32 b^2 x^8}+\frac{21 a^2 \tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^4}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^4}}\right)}{64 \sqrt{2} b^{11/4}}+\frac{21 a^2 \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^4}}{\sqrt{b}+\sqrt{-b+a x^4}}\right)}{64 \sqrt{2} b^{11/4}}","\frac{\sqrt[4]{-b+a x^4} \left(4 b+7 a x^4\right)}{32 b^2 x^8}-\frac{21 a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^4}}{-\sqrt{b}+\sqrt{-b+a x^4}}\right)}{64 \sqrt{2} b^{11/4}}+\frac{21 a^2 \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^4}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^4}}\right)}{64 \sqrt{2} b^{11/4}}",1,"((-b + a*x^4)^(1/4)*(4*b + 7*a*x^4))/(32*b^2*x^8) + (21*a^2*ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^4]/(Sqrt[2]*b^(1/4)))/(-b + a*x^4)^(1/4)])/(64*Sqrt[2]*b^(11/4)) + (21*a^2*ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^4)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^4])])/(64*Sqrt[2]*b^(11/4))","A",1
2240,1,167,167,0.6552006,"\int \frac{b^2+c x^2+a^2 x^4}{\sqrt{b x+a x^3} \left(-b^2+a^2 x^4\right)} \, dx","IntegrateAlgebraic[(b^2 + c*x^2 + a^2*x^4)/(Sqrt[b*x + a*x^3]*(-b^2 + a^2*x^4)),x]","\frac{(-2 a b+c) \sqrt{b x+a x^3}}{2 a b \left(b+a x^2\right)}-\frac{(2 a b+c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)}{4 \sqrt{2} a^{5/4} b^{5/4}}-\frac{(2 a b+c) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)}{4 \sqrt{2} a^{5/4} b^{5/4}}","\frac{(-2 a b+c) \sqrt{b x+a x^3}}{2 a b \left(b+a x^2\right)}-\frac{(2 a b+c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)}{4 \sqrt{2} a^{5/4} b^{5/4}}-\frac{(2 a b+c) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{b x+a x^3}}{b+a x^2}\right)}{4 \sqrt{2} a^{5/4} b^{5/4}}",1,"((-2*a*b + c)*Sqrt[b*x + a*x^3])/(2*a*b*(b + a*x^2)) - ((2*a*b + c)*ArcTan[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[b*x + a*x^3])/(b + a*x^2)])/(4*Sqrt[2]*a^(5/4)*b^(5/4)) - ((2*a*b + c)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[b*x + a*x^3])/(b + a*x^2)])/(4*Sqrt[2]*a^(5/4)*b^(5/4))","A",1
2241,1,166,167,0.1933267,"\int \frac{1}{x^{11} \left(-b+a x^5\right)^{3/4}} \, dx","IntegrateAlgebraic[1/(x^11*(-b + a*x^5)^(3/4)),x]","\frac{\sqrt[4]{-b+a x^5} \left(4 b+7 a x^5\right)}{40 b^2 x^{10}}+\frac{21 a^2 \tan ^{-1}\left(\frac{-\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^5}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^5}}\right)}{80 \sqrt{2} b^{11/4}}+\frac{21 a^2 \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^5}}{\sqrt{b}+\sqrt{-b+a x^5}}\right)}{80 \sqrt{2} b^{11/4}}","\frac{\sqrt[4]{-b+a x^5} \left(4 b+7 a x^5\right)}{40 b^2 x^{10}}-\frac{21 a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{-b+a x^5}}{-\sqrt{b}+\sqrt{-b+a x^5}}\right)}{80 \sqrt{2} b^{11/4}}+\frac{21 a^2 \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{\sqrt{-b+a x^5}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt[4]{-b+a x^5}}\right)}{80 \sqrt{2} b^{11/4}}",1,"((-b + a*x^5)^(1/4)*(4*b + 7*a*x^5))/(40*b^2*x^10) + (21*a^2*ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + a*x^5]/(Sqrt[2]*b^(1/4)))/(-b + a*x^5)^(1/4)])/(80*Sqrt[2]*b^(11/4)) + (21*a^2*ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^5)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^5])])/(80*Sqrt[2]*b^(11/4))","A",1
2242,1,167,167,0.5380112,"\int \frac{1+x^6}{\sqrt[3]{-x^2+x^4} \left(-1+x^6\right)} \, dx","IntegrateAlgebraic[(1 + x^6)/((-x^2 + x^4)^(1/3)*(-1 + x^6)),x]","-\frac{\left(-x^2+x^4\right)^{2/3}}{x \left(-1+x^2\right)}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[6]{3} x}{\sqrt[3]{-x^2+x^4}}\right)}{3 \sqrt[6]{3}}-\frac{\tan ^{-1}\left(\frac{3^{5/6} x \sqrt[3]{-x^2+x^4}}{-3 x^2+3^{2/3} \left(-x^2+x^4\right)^{2/3}}\right)}{3 \sqrt[6]{3}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[3]{3}}+\frac{\left(-x^2+x^4\right)^{2/3}}{3^{2/3}}}{x \sqrt[3]{-x^2+x^4}}\right)}{3^{2/3}}","-\frac{\left(-x^2+x^4\right)^{2/3}}{x \left(-1+x^2\right)}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[6]{3} x}{\sqrt[3]{-x^2+x^4}}\right)}{3 \sqrt[6]{3}}-\frac{\tan ^{-1}\left(\frac{3^{5/6} x \sqrt[3]{-x^2+x^4}}{-3 x^2+3^{2/3} \left(-x^2+x^4\right)^{2/3}}\right)}{3 \sqrt[6]{3}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[3]{3}}+\frac{\left(-x^2+x^4\right)^{2/3}}{3^{2/3}}}{x \sqrt[3]{-x^2+x^4}}\right)}{3^{2/3}}",1,"-((-x^2 + x^4)^(2/3)/(x*(-1 + x^2))) - (2*ArcTan[(3^(1/6)*x)/(-x^2 + x^4)^(1/3)])/(3*3^(1/6)) - ArcTan[(3^(5/6)*x*(-x^2 + x^4)^(1/3))/(-3*x^2 + 3^(2/3)*(-x^2 + x^4)^(2/3))]/(3*3^(1/6)) - ArcTanh[(x^2/3^(1/3) + (-x^2 + x^4)^(2/3)/3^(2/3))/(x*(-x^2 + x^4)^(1/3))]/3^(2/3)","A",1
2243,1,167,167,0.5463025,"\int \frac{-b^6+a^6 x^6}{\sqrt{b^2 x+a^2 x^3} \left(b^6+a^6 x^6\right)} \, dx","IntegrateAlgebraic[(-b^6 + a^6*x^6)/(Sqrt[b^2*x + a^2*x^3]*(b^6 + a^6*x^6)),x]","-\frac{2 \sqrt{b^2 x+a^2 x^3}}{3 \left(b^2+a^2 x^2\right)}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{3} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{3 \sqrt[4]{3} \sqrt{a} \sqrt{b}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{3} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{3 \sqrt[4]{3} \sqrt{a} \sqrt{b}}","-\frac{2 \sqrt{b^2 x+a^2 x^3}}{3 \left(b^2+a^2 x^2\right)}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{3} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{3 \sqrt[4]{3} \sqrt{a} \sqrt{b}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{3} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{3 \sqrt[4]{3} \sqrt{a} \sqrt{b}}",1,"(-2*Sqrt[b^2*x + a^2*x^3])/(3*(b^2 + a^2*x^2)) - (2*ArcTan[(3^(1/4)*Sqrt[a]*Sqrt[b]*Sqrt[b^2*x + a^2*x^3])/(b^2 + a^2*x^2)])/(3*3^(1/4)*Sqrt[a]*Sqrt[b]) - (2*ArcTanh[(3^(1/4)*Sqrt[a]*Sqrt[b]*Sqrt[b^2*x + a^2*x^3])/(b^2 + a^2*x^2)])/(3*3^(1/4)*Sqrt[a]*Sqrt[b])","A",1
2244,1,167,167,0.2549454,"\int \frac{1+a x^2}{\left(-1+a x^2\right) \sqrt{a x+\sqrt{b+a^2 x^2}}} \, dx","IntegrateAlgebraic[(1 + a*x^2)/((-1 + a*x^2)*Sqrt[a*x + Sqrt[b + a^2*x^2]]),x]","-\frac{b}{3 a \left(a x+\sqrt{b+a^2 x^2}\right)^{3/2}}+\frac{\sqrt{a x+\sqrt{b+a^2 x^2}}}{a}+\text{RootSum}\left[b^2-4 a \text{$\#$1}^4-2 b \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-b \log \left(\sqrt{a x+\sqrt{b+a^2 x^2}}-\text{$\#$1}\right)-\log \left(\sqrt{a x+\sqrt{b+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 a \text{$\#$1}^3+b \text{$\#$1}^3-\text{$\#$1}^7}\&\right]","-\frac{b}{3 a \left(a x+\sqrt{b+a^2 x^2}\right)^{3/2}}+\frac{\sqrt{a x+\sqrt{b+a^2 x^2}}}{a}+\text{RootSum}\left[b^2-4 a \text{$\#$1}^4-2 b \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-b \log \left(\sqrt{a x+\sqrt{b+a^2 x^2}}-\text{$\#$1}\right)-\log \left(\sqrt{a x+\sqrt{b+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 a \text{$\#$1}^3+b \text{$\#$1}^3-\text{$\#$1}^7}\&\right]",1,"-1/3*b/(a*(a*x + Sqrt[b + a^2*x^2])^(3/2)) + Sqrt[a*x + Sqrt[b + a^2*x^2]]/a + RootSum[b^2 - 4*a*#1^4 - 2*b*#1^4 + #1^8 & , (-(b*Log[Sqrt[a*x + Sqrt[b + a^2*x^2]] - #1]) - Log[Sqrt[a*x + Sqrt[b + a^2*x^2]] - #1]*#1^4)/(2*a*#1^3 + b*#1^3 - #1^7) & ]","A",1
2245,1,223,167,3.2952577,"\int \frac{\sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}{x^2 \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}} \, dx","IntegrateAlgebraic[Sqrt[-(a/b^2) + (a^2*x^2)/b^2]/(x^2*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]]),x]","\frac{2 \left(-1+2 a x^2\right) \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{3 b x^2}-\frac{4 \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}} \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{3 x}+\frac{a \log \left(-1+\sqrt{2} \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}\right)}{\sqrt{2} b}-\frac{a \log \left(b+\sqrt{2} b \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}\right)}{\sqrt{2} b}","\frac{2 \left(-1+2 a x^2\right) \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{3 b x^2}-\frac{4 \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}} \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{3 x}-\frac{\sqrt{2} a \tanh ^{-1}\left(\sqrt{2} \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}\right)}{b}",1,"(2*(-1 + 2*a*x^2)*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]])/(3*b*x^2) - (4*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]])/(3*x) + (a*Log[-1 + Sqrt[2]*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]]])/(Sqrt[2]*b) - (a*Log[b + Sqrt[2]*b*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]]])/(Sqrt[2]*b)","A",1
2246,1,167,167,1.066467,"\int \frac{\left(-2 q+p x^3\right) \sqrt{q+p x^3}}{b x^4+a \left(q+p x^3\right)^2} \, dx","IntegrateAlgebraic[((-2*q + p*x^3)*Sqrt[q + p*x^3])/(b*x^4 + a*(q + p*x^3)^2),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x \sqrt{q+p x^3}}{\sqrt{a} q-\sqrt{b} x^2+\sqrt{a} p x^3}\right)}{\sqrt{2} a^{3/4} \sqrt[4]{b}}-\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[4]{a} q}{\sqrt{2} \sqrt[4]{b}}+\frac{\sqrt[4]{b} x^2}{\sqrt{2} \sqrt[4]{a}}+\frac{\sqrt[4]{a} p x^3}{\sqrt{2} \sqrt[4]{b}}}{x \sqrt{q+p x^3}}\right)}{\sqrt{2} a^{3/4} \sqrt[4]{b}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x \sqrt{q+p x^3}}{\sqrt{a} q-\sqrt{b} x^2+\sqrt{a} p x^3}\right)}{\sqrt{2} a^{3/4} \sqrt[4]{b}}-\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[4]{a} q}{\sqrt{2} \sqrt[4]{b}}+\frac{\sqrt[4]{b} x^2}{\sqrt{2} \sqrt[4]{a}}+\frac{\sqrt[4]{a} p x^3}{\sqrt{2} \sqrt[4]{b}}}{x \sqrt{q+p x^3}}\right)}{\sqrt{2} a^{3/4} \sqrt[4]{b}}",1,"-(ArcTan[(Sqrt[2]*a^(1/4)*b^(1/4)*x*Sqrt[q + p*x^3])/(Sqrt[a]*q - Sqrt[b]*x^2 + Sqrt[a]*p*x^3)]/(Sqrt[2]*a^(3/4)*b^(1/4))) - ArcTanh[((a^(1/4)*q)/(Sqrt[2]*b^(1/4)) + (b^(1/4)*x^2)/(Sqrt[2]*a^(1/4)) + (a^(1/4)*p*x^3)/(Sqrt[2]*b^(1/4)))/(x*Sqrt[q + p*x^3])]/(Sqrt[2]*a^(3/4)*b^(1/4))","A",1
2247,1,167,167,0.5923009,"\int x^4 \sqrt{1+x^4} \sqrt{x^2+\sqrt{1+x^4}} \, dx","IntegrateAlgebraic[x^4*Sqrt[1 + x^4]*Sqrt[x^2 + Sqrt[1 + x^4]],x]","\frac{x \sqrt{1+x^4} \left(104 x^2+264 x^6+192 x^{10}\right) \sqrt{x^2+\sqrt{1+x^4}}+x \left(39+212 x^4+360 x^8+192 x^{12}\right) \sqrt{x^2+\sqrt{1+x^4}}}{384 \sqrt{1+x^4} \left(1+4 x^4\right)+384 \left(3 x^2+4 x^6\right)}-\frac{13 \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)}{64 \sqrt{2}}","\frac{x \sqrt{1+x^4} \left(104 x^2+264 x^6+192 x^{10}\right) \sqrt{x^2+\sqrt{1+x^4}}+x \left(39+212 x^4+360 x^8+192 x^{12}\right) \sqrt{x^2+\sqrt{1+x^4}}}{384 \sqrt{1+x^4} \left(1+4 x^4\right)+384 \left(3 x^2+4 x^6\right)}-\frac{13 \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)}{64 \sqrt{2}}",1,"(x*Sqrt[1 + x^4]*(104*x^2 + 264*x^6 + 192*x^10)*Sqrt[x^2 + Sqrt[1 + x^4]] + x*(39 + 212*x^4 + 360*x^8 + 192*x^12)*Sqrt[x^2 + Sqrt[1 + x^4]])/(384*Sqrt[1 + x^4]*(1 + 4*x^4) + 384*(3*x^2 + 4*x^6)) - (13*ArcTanh[(Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2 + Sqrt[1 + x^4])])/(64*Sqrt[2])","A",1
2248,1,167,167,2.1284668,"\int \frac{\sqrt{q+p x^5} \left(-2 q+3 p x^5\right)}{b x^4+a \left(q+p x^5\right)^2} \, dx","IntegrateAlgebraic[(Sqrt[q + p*x^5]*(-2*q + 3*p*x^5))/(b*x^4 + a*(q + p*x^5)^2),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x \sqrt{q+p x^5}}{\sqrt{a} q-\sqrt{b} x^2+\sqrt{a} p x^5}\right)}{\sqrt{2} a^{3/4} \sqrt[4]{b}}-\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[4]{a} q}{\sqrt{2} \sqrt[4]{b}}+\frac{\sqrt[4]{b} x^2}{\sqrt{2} \sqrt[4]{a}}+\frac{\sqrt[4]{a} p x^5}{\sqrt{2} \sqrt[4]{b}}}{x \sqrt{q+p x^5}}\right)}{\sqrt{2} a^{3/4} \sqrt[4]{b}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x \sqrt{q+p x^5}}{\sqrt{a} q-\sqrt{b} x^2+\sqrt{a} p x^5}\right)}{\sqrt{2} a^{3/4} \sqrt[4]{b}}-\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[4]{a} q}{\sqrt{2} \sqrt[4]{b}}+\frac{\sqrt[4]{b} x^2}{\sqrt{2} \sqrt[4]{a}}+\frac{\sqrt[4]{a} p x^5}{\sqrt{2} \sqrt[4]{b}}}{x \sqrt{q+p x^5}}\right)}{\sqrt{2} a^{3/4} \sqrt[4]{b}}",1,"-(ArcTan[(Sqrt[2]*a^(1/4)*b^(1/4)*x*Sqrt[q + p*x^5])/(Sqrt[a]*q - Sqrt[b]*x^2 + Sqrt[a]*p*x^5)]/(Sqrt[2]*a^(3/4)*b^(1/4))) - ArcTanh[((a^(1/4)*q)/(Sqrt[2]*b^(1/4)) + (b^(1/4)*x^2)/(Sqrt[2]*a^(1/4)) + (a^(1/4)*p*x^5)/(Sqrt[2]*b^(1/4)))/(x*Sqrt[q + p*x^5])]/(Sqrt[2]*a^(3/4)*b^(1/4))","A",1
2249,1,168,168,0.245943,"\int \frac{-3+x}{\sqrt[3]{1-x^2} \left(3+x^2\right)} \, dx","IntegrateAlgebraic[(-3 + x)/((1 - x^2)^(1/3)*(3 + x^2)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{1-x^2}}{-2^{2/3}-2^{2/3} x+\sqrt[3]{1-x^2}}\right)}{2^{2/3}}-\frac{\log \left(2^{2/3}+2^{2/3} x+2 \sqrt[3]{1-x^2}\right)}{2^{2/3}}+\frac{\log \left(-\sqrt[3]{2}-2 \sqrt[3]{2} x-\sqrt[3]{2} x^2+\left(2^{2/3}+2^{2/3} x\right) \sqrt[3]{1-x^2}-2 \left(1-x^2\right)^{2/3}\right)}{2\ 2^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{1-x^2}}{-2^{2/3}-2^{2/3} x+\sqrt[3]{1-x^2}}\right)}{2^{2/3}}-\frac{\log \left(2^{2/3}+2^{2/3} x+2 \sqrt[3]{1-x^2}\right)}{2^{2/3}}+\frac{\log \left(-\sqrt[3]{2}-2 \sqrt[3]{2} x-\sqrt[3]{2} x^2+\left(2^{2/3}+2^{2/3} x\right) \sqrt[3]{1-x^2}-2 \left(1-x^2\right)^{2/3}\right)}{2\ 2^{2/3}}",1,"-((Sqrt[3]*ArcTan[(Sqrt[3]*(1 - x^2)^(1/3))/(-2^(2/3) - 2^(2/3)*x + (1 - x^2)^(1/3))])/2^(2/3)) - Log[2^(2/3) + 2^(2/3)*x + 2*(1 - x^2)^(1/3)]/2^(2/3) + Log[-2^(1/3) - 2*2^(1/3)*x - 2^(1/3)*x^2 + (2^(2/3) + 2^(2/3)*x)*(1 - x^2)^(1/3) - 2*(1 - x^2)^(2/3)]/(2*2^(2/3))","A",1
2250,1,168,168,1.3359224,"\int \frac{x^3 (-3 a b+(a+2 b) x)}{(-a+x) (-b+x) \sqrt[4]{x (-a+x) (-b+x)^2} \left(-a b^2 d+b (2 a+b) d x-(a+2 b) d x^2+(-1+d) x^3\right)} \, dx","IntegrateAlgebraic[(x^3*(-3*a*b + (a + 2*b)*x))/((-a + x)*(-b + x)*(x*(-a + x)*(-b + x)^2)^(1/4)*(-(a*b^2*d) + b*(2*a + b)*d*x - (a + 2*b)*d*x^2 + (-1 + d)*x^3)),x]","\frac{4 \left(-a b^2 x+2 a b x^2+b^2 x^2-a x^3-2 b x^3+x^4\right)^{3/4}}{(-a+x) (-b+x)^2}+2 \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{x}\right)-2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{x}\right)","\frac{4 \left(-a b^2 x+2 a b x^2+b^2 x^2-a x^3-2 b x^3+x^4\right)^{3/4}}{(-a+x) (-b+x)^2}+2 \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{x}\right)-2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{x}\right)",1,"(4*(-(a*b^2*x) + 2*a*b*x^2 + b^2*x^2 - a*x^3 - 2*b*x^3 + x^4)^(3/4))/((-a + x)*(-b + x)^2) + 2*d^(1/4)*ArcTan[(d^(1/4)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/4))/x] - 2*d^(1/4)*ArcTanh[(d^(1/4)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/4))/x]","A",1
2251,1,168,168,0.3050199,"\int \frac{a x+\sqrt{b^2+a^2 x^2}}{b+\sqrt{a x+\sqrt{b^2+a^2 x^2}}} \, dx","IntegrateAlgebraic[(a*x + Sqrt[b^2 + a^2*x^2])/(b + Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]),x]","-\frac{b x}{2}+\frac{\left(3 b^2+a x\right) \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{3 a}+\sqrt{b^2+a^2 x^2} \left(-\frac{b}{2 a}+\frac{\sqrt{a x+\sqrt{b^2+a^2 x^2}}}{3 a}\right)+\frac{b \log \left(a x+\sqrt{b^2+a^2 x^2}\right)}{2 a}+\frac{\left(-b-b^3\right) \log \left(b+\sqrt{a x+\sqrt{b^2+a^2 x^2}}\right)}{a}","-\frac{b x}{2}+\frac{\left(3 b^2+a x\right) \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{3 a}+\sqrt{b^2+a^2 x^2} \left(-\frac{b}{2 a}+\frac{\sqrt{a x+\sqrt{b^2+a^2 x^2}}}{3 a}\right)+\frac{b \log \left(a x+\sqrt{b^2+a^2 x^2}\right)}{2 a}+\frac{\left(-b-b^3\right) \log \left(b+\sqrt{a x+\sqrt{b^2+a^2 x^2}}\right)}{a}",1,"-1/2*(b*x) + ((3*b^2 + a*x)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/(3*a) + Sqrt[b^2 + a^2*x^2]*(-1/2*b/a + Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]/(3*a)) + (b*Log[a*x + Sqrt[b^2 + a^2*x^2]])/(2*a) + ((-b - b^3)*Log[b + Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]])/a","A",1
2252,0,0,169,4.0748054,"\int \frac{1}{\left(3+x^2\right) \sqrt[3]{1+3 x^2}} \, dx","IntegrateAlgebraic[1/((3 + x^2)*(1 + 3*x^2)^(1/3)),x]","\int \frac{1}{\left(3+x^2\right) \sqrt[3]{1+3 x^2}} \, dx","\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{1+2 \sqrt[3]{1+3 x^2}}\right)}{4 \sqrt{3}}-\frac{i \tanh ^{-1}\left(\frac{2 i \sqrt{3} x-2 i \sqrt{3} x \sqrt[3]{1+3 x^2}}{-1+3 x^2+2 \sqrt[3]{1+3 x^2}-4 \left(1+3 x^2\right)^{2/3}}\right)}{8 \sqrt{3}}+\frac{1}{8} \tanh ^{-1}\left(\frac{6 x \sqrt[3]{1+3 x^2}}{1+3 x^2-2 \sqrt[3]{1+3 x^2}+4 \left(1+3 x^2\right)^{2/3}}\right)",1,"Could not integrate","F",-1
2253,1,169,169,0.4299763,"\int \frac{\left(1+x^2\right) \sqrt[4]{x^3+x^5}}{x^2 \left(-1+x^2\right)} \, dx","IntegrateAlgebraic[((1 + x^2)*(x^3 + x^5)^(1/4))/(x^2*(-1 + x^2)),x]","\frac{4 \sqrt[4]{x^3+x^5}}{x}+\sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^5}}\right)-\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^3+x^5}}{\sqrt{2} x^2-\sqrt{x^3+x^5}}\right)}{\sqrt[4]{2}}-\sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^5}}\right)-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^3+x^5}}{2^{3/4}}}{x \sqrt[4]{x^3+x^5}}\right)}{\sqrt[4]{2}}","\frac{4 \sqrt[4]{x^3+x^5}}{x}+\sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^5}}\right)-\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^3+x^5}}{\sqrt{2} x^2-\sqrt{x^3+x^5}}\right)}{\sqrt[4]{2}}-\sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^5}}\right)-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^3+x^5}}{2^{3/4}}}{x \sqrt[4]{x^3+x^5}}\right)}{\sqrt[4]{2}}",1,"(4*(x^3 + x^5)^(1/4))/x + 2^(1/4)*ArcTan[(2^(1/4)*x)/(x^3 + x^5)^(1/4)] - ArcTan[(2^(3/4)*x*(x^3 + x^5)^(1/4))/(Sqrt[2]*x^2 - Sqrt[x^3 + x^5])]/2^(1/4) - 2^(1/4)*ArcTanh[(2^(1/4)*x)/(x^3 + x^5)^(1/4)] - ArcTanh[(x^2/2^(1/4) + Sqrt[x^3 + x^5]/2^(3/4))/(x*(x^3 + x^5)^(1/4))]/2^(1/4)","A",1
2254,1,170,170,0.4051165,"\int \frac{\left(4+x^2+x^5\right) \sqrt[4]{-2-x^2-2 x^4+2 x^5}}{x^2 \left(-2-x^2+2 x^5\right)} \, dx","IntegrateAlgebraic[((4 + x^2 + x^5)*(-2 - x^2 - 2*x^4 + 2*x^5)^(1/4))/(x^2*(-2 - x^2 + 2*x^5)),x]","\frac{2 \sqrt[4]{-2-x^2-2 x^4+2 x^5}}{x}-\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{-2-x^2-2 x^4+2 x^5}}{\sqrt{2} x^2-\sqrt{-2-x^2-2 x^4+2 x^5}}\right)}{\sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{2 \sqrt[4]{2} x \sqrt[4]{-2-x^2-2 x^4+2 x^5}}{2 x^2+\sqrt{2} \sqrt{-2-x^2-2 x^4+2 x^5}}\right)}{\sqrt[4]{2}}","\frac{2 \sqrt[4]{-2-x^2-2 x^4+2 x^5}}{x}-\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{-2-x^2-2 x^4+2 x^5}}{\sqrt{2} x^2-\sqrt{-2-x^2-2 x^4+2 x^5}}\right)}{\sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{2 \sqrt[4]{2} x \sqrt[4]{-2-x^2-2 x^4+2 x^5}}{2 x^2+\sqrt{2} \sqrt{-2-x^2-2 x^4+2 x^5}}\right)}{\sqrt[4]{2}}",1,"(2*(-2 - x^2 - 2*x^4 + 2*x^5)^(1/4))/x - ArcTan[(2^(3/4)*x*(-2 - x^2 - 2*x^4 + 2*x^5)^(1/4))/(Sqrt[2]*x^2 - Sqrt[-2 - x^2 - 2*x^4 + 2*x^5])]/2^(1/4) - ArcTanh[(2*2^(1/4)*x*(-2 - x^2 - 2*x^4 + 2*x^5)^(1/4))/(2*x^2 + Sqrt[2]*Sqrt[-2 - x^2 - 2*x^4 + 2*x^5])]/2^(1/4)","A",1
2255,1,170,170,0.4416467,"\int \frac{\left(-1+x^4\right)^2 \sqrt{x^2+\sqrt{1+x^4}}}{\left(1+x^4\right)^2} \, dx","IntegrateAlgebraic[((-1 + x^4)^2*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^4)^2,x]","\frac{x \sqrt{1+x^4} \left(7 x^2+2 x^6\right) \sqrt{x^2+\sqrt{1+x^4}}+x \left(4+8 x^4+2 x^8\right) \sqrt{x^2+\sqrt{1+x^4}}}{2 \sqrt{1+x^4} \left(2 x^2+2 x^6\right)+2 \left(1+3 x^4+2 x^8\right)}-\frac{3}{2} \tan ^{-1}\left(x \sqrt{x^2+\sqrt{1+x^4}}\right)+\frac{\tan ^{-1}\left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}\right)}{2 \sqrt{2}}","\frac{x \sqrt{1+x^4} \left(7 x^2+2 x^6\right) \sqrt{x^2+\sqrt{1+x^4}}+x \left(4+8 x^4+2 x^8\right) \sqrt{x^2+\sqrt{1+x^4}}}{2 \sqrt{1+x^4} \left(2 x^2+2 x^6\right)+2 \left(1+3 x^4+2 x^8\right)}-\frac{3}{2} \tan ^{-1}\left(x \sqrt{x^2+\sqrt{1+x^4}}\right)+\frac{\tan ^{-1}\left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}\right)}{2 \sqrt{2}}",1,"(x*Sqrt[1 + x^4]*(7*x^2 + 2*x^6)*Sqrt[x^2 + Sqrt[1 + x^4]] + x*(4 + 8*x^4 + 2*x^8)*Sqrt[x^2 + Sqrt[1 + x^4]])/(2*Sqrt[1 + x^4]*(2*x^2 + 2*x^6) + 2*(1 + 3*x^4 + 2*x^8)) - (3*ArcTan[x*Sqrt[x^2 + Sqrt[1 + x^4]]])/2 + ArcTan[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]]]/(2*Sqrt[2])","A",1
2256,1,170,170,0.3079315,"\int \frac{-1+x^2}{\sqrt{1+x} \left(1+x^2\right) \sqrt{1+\sqrt{1+\sqrt{1+x}}}} \, dx","IntegrateAlgebraic[(-1 + x^2)/(Sqrt[1 + x]*(1 + x^2)*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]),x]","-\frac{16}{3} \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\frac{8}{3} \sqrt{1+\sqrt{1+x}} \sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{RootSum}\left[2-8 \text{$\#$1}^4+8 \text{$\#$1}^6+14 \text{$\#$1}^8-32 \text{$\#$1}^{10}+24 \text{$\#$1}^{12}-8 \text{$\#$1}^{14}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{$\#$1}\right)}{2 \text{$\#$1}^3-\text{$\#$1}^5-8 \text{$\#$1}^7+12 \text{$\#$1}^9-6 \text{$\#$1}^{11}+\text{$\#$1}^{13}}\&\right]","-\frac{16}{3} \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\frac{8}{3} \sqrt{1+\sqrt{1+x}} \sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{RootSum}\left[2-8 \text{$\#$1}^4+8 \text{$\#$1}^6+14 \text{$\#$1}^8-32 \text{$\#$1}^{10}+24 \text{$\#$1}^{12}-8 \text{$\#$1}^{14}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{$\#$1}\right)}{2 \text{$\#$1}^3-\text{$\#$1}^5-8 \text{$\#$1}^7+12 \text{$\#$1}^9-6 \text{$\#$1}^{11}+\text{$\#$1}^{13}}\&\right]",1,"(-16*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/3 + (8*Sqrt[1 + Sqrt[1 + x]]*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/3 - RootSum[2 - 8*#1^4 + 8*#1^6 + 14*#1^8 - 32*#1^10 + 24*#1^12 - 8*#1^14 + #1^16 & , Log[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]] - #1]/(2*#1^3 - #1^5 - 8*#1^7 + 12*#1^9 - 6*#1^11 + #1^13) & ]","A",1
2257,1,170,170,0.0021964,"\int \frac{-1+x^2}{\sqrt{1+x} \left(1+x^2\right) \sqrt{1+\sqrt{1+\sqrt{1+x}}}} \, dx","IntegrateAlgebraic[(-1 + x^2)/(Sqrt[1 + x]*(1 + x^2)*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]),x]","-\frac{16}{3} \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\frac{8}{3} \sqrt{1+\sqrt{1+x}} \sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{RootSum}\left[2-8 \text{$\#$1}^4+8 \text{$\#$1}^6+14 \text{$\#$1}^8-32 \text{$\#$1}^{10}+24 \text{$\#$1}^{12}-8 \text{$\#$1}^{14}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{$\#$1}\right)}{2 \text{$\#$1}^3-\text{$\#$1}^5-8 \text{$\#$1}^7+12 \text{$\#$1}^9-6 \text{$\#$1}^{11}+\text{$\#$1}^{13}}\&\right]","-\frac{16}{3} \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\frac{8}{3} \sqrt{1+\sqrt{1+x}} \sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{RootSum}\left[2-8 \text{$\#$1}^4+8 \text{$\#$1}^6+14 \text{$\#$1}^8-32 \text{$\#$1}^{10}+24 \text{$\#$1}^{12}-8 \text{$\#$1}^{14}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{$\#$1}\right)}{2 \text{$\#$1}^3-\text{$\#$1}^5-8 \text{$\#$1}^7+12 \text{$\#$1}^9-6 \text{$\#$1}^{11}+\text{$\#$1}^{13}}\&\right]",1,"(-16*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/3 + (8*Sqrt[1 + Sqrt[1 + x]]*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/3 - RootSum[2 - 8*#1^4 + 8*#1^6 + 14*#1^8 - 32*#1^10 + 24*#1^12 - 8*#1^14 + #1^16 & , Log[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]] - #1]/(2*#1^3 - #1^5 - 8*#1^7 + 12*#1^9 - 6*#1^11 + #1^13) & ]","A",1
2258,1,86,171,0.6160826,"\int \left(\frac{1}{\sqrt{1-\sqrt{x}}}-\sqrt{1-\sqrt{x}-x}\right) \, dx","IntegrateAlgebraic[1/Sqrt[1 - Sqrt[x]] - Sqrt[1 - Sqrt[x] - x],x]","-\frac{4}{3} \sqrt{1-\sqrt{x}} \left(2+\sqrt{x}\right)+\frac{1}{12} \left(11-2 \sqrt{x}-8 x\right) \sqrt{1-\sqrt{x}-x}-\frac{5}{4} \tan ^{-1}\left(\frac{-1+\sqrt{1-\sqrt{x}-x}}{\sqrt{x}}\right)","-\frac{8}{3} \sqrt{1-\sqrt{x}}+\left(\frac{2}{3}+\frac{1}{4} \sqrt{\left(1+2 \sqrt{x}\right)^2}\right) \sqrt{1-\sqrt{x}-x}-\frac{4}{3} \sqrt{1-\sqrt{x}} \sqrt{x}-\frac{2}{3} \sqrt{1-\sqrt{x}-x} \sqrt{x}-\frac{2}{3} \sqrt{1-\sqrt{x}-x} x+\frac{5}{8} i \log \left(i \sqrt{\left(1+2 \sqrt{x}\right)^2}-2 \sqrt{1-\sqrt{x}-x}\right)",1,"(-4*Sqrt[1 - Sqrt[x]]*(2 + Sqrt[x]))/3 + ((11 - 2*Sqrt[x] - 8*x)*Sqrt[1 - Sqrt[x] - x])/12 - (5*ArcTan[(-1 + Sqrt[1 - Sqrt[x] - x])/Sqrt[x]])/4","A",1
2259,1,171,171,0.2190492,"\int \frac{-5+x}{\sqrt[3]{-2-x+x^2} \left(-3+4 x+x^2\right)} \, dx","IntegrateAlgebraic[(-5 + x)/((-2 - x + x^2)^(1/3)*(-3 + 4*x + x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-2-x+x^2}}{-2^{2/3}-2^{2/3} x+\sqrt[3]{-2-x+x^2}}\right)}{2^{2/3}}+\frac{\log \left(2^{2/3}+2^{2/3} x+2 \sqrt[3]{-2-x+x^2}\right)}{2^{2/3}}-\frac{\log \left(-\sqrt[3]{2}-2 \sqrt[3]{2} x-\sqrt[3]{2} x^2+\left(2^{2/3}+2^{2/3} x\right) \sqrt[3]{-2-x+x^2}-2 \left(-2-x+x^2\right)^{2/3}\right)}{2\ 2^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-2-x+x^2}}{-2^{2/3}-2^{2/3} x+\sqrt[3]{-2-x+x^2}}\right)}{2^{2/3}}+\frac{\log \left(2^{2/3}+2^{2/3} x+2 \sqrt[3]{-2-x+x^2}\right)}{2^{2/3}}-\frac{\log \left(-\sqrt[3]{2}-2 \sqrt[3]{2} x-\sqrt[3]{2} x^2+\left(2^{2/3}+2^{2/3} x\right) \sqrt[3]{-2-x+x^2}-2 \left(-2-x+x^2\right)^{2/3}\right)}{2\ 2^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(-2 - x + x^2)^(1/3))/(-2^(2/3) - 2^(2/3)*x + (-2 - x + x^2)^(1/3))])/2^(2/3) + Log[2^(2/3) + 2^(2/3)*x + 2*(-2 - x + x^2)^(1/3)]/2^(2/3) - Log[-2^(1/3) - 2*2^(1/3)*x - 2^(1/3)*x^2 + (2^(2/3) + 2^(2/3)*x)*(-2 - x + x^2)^(1/3) - 2*(-2 - x + x^2)^(2/3)]/(2*2^(2/3))","A",1
2260,1,171,171,4.0722392,"\int \frac{(-a+x) (-2 a+b+x)}{\left((-a+x) (-b+x)^2\right)^{3/4} \left(a+b^2 d-(1+2 b d) x+d x^2\right)} \, dx","IntegrateAlgebraic[((-a + x)*(-2*a + b + x))/(((-a + x)*(-b + x)^2)^(3/4)*(a + b^2*d - (1 + 2*b*d)*x + d*x^2)),x]","-\frac{4 \sqrt[4]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{b-x}-2 \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{3/4}}{(b-x) (-a+x)}\right)+2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt[4]{d} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{3/4}}{(b-x) (-a+x)}\right)","-\frac{4 \sqrt[4]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{b-x}-2 \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{3/4}}{(b-x) (-a+x)}\right)+2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt[4]{d} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{3/4}}{(b-x) (-a+x)}\right)",1,"(-4*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/4))/(b - x) - 2*d^(1/4)*ArcTan[(d^(1/4)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(3/4))/((b - x)*(-a + x))] + 2*d^(1/4)*ArcTanh[(d^(1/4)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(3/4))/((b - x)*(-a + x))]","A",1
2261,1,171,171,1.1483008,"\int \frac{1+x}{\left(1+4 x+x^2\right) \sqrt[3]{1-x^3}} \, dx","IntegrateAlgebraic[(1 + x)/((1 + 4*x + x^2)*(1 - x^3)^(1/3)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{1-x^3}}{2^{2/3}-2^{2/3} x+\sqrt[3]{1-x^3}}\right)}{2^{2/3} \sqrt{3}}+\frac{\log \left(-2^{2/3}+2^{2/3} x+2 \sqrt[3]{1-x^3}\right)}{3\ 2^{2/3}}-\frac{\log \left(-\sqrt[3]{2}+2 \sqrt[3]{2} x-\sqrt[3]{2} x^2+\left(-2^{2/3}+2^{2/3} x\right) \sqrt[3]{1-x^3}-2 \left(1-x^3\right)^{2/3}\right)}{6\ 2^{2/3}}","\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{1-x^3}}{2^{2/3}-2^{2/3} x+\sqrt[3]{1-x^3}}\right)}{2^{2/3} \sqrt{3}}+\frac{\log \left(-2^{2/3}+2^{2/3} x+2 \sqrt[3]{1-x^3}\right)}{3\ 2^{2/3}}-\frac{\log \left(-\sqrt[3]{2}+2 \sqrt[3]{2} x-\sqrt[3]{2} x^2+\left(-2^{2/3}+2^{2/3} x\right) \sqrt[3]{1-x^3}-2 \left(1-x^3\right)^{2/3}\right)}{6\ 2^{2/3}}",1,"ArcTan[(Sqrt[3]*(1 - x^3)^(1/3))/(2^(2/3) - 2^(2/3)*x + (1 - x^3)^(1/3))]/(2^(2/3)*Sqrt[3]) + Log[-2^(2/3) + 2^(2/3)*x + 2*(1 - x^3)^(1/3)]/(3*2^(2/3)) - Log[-2^(1/3) + 2*2^(1/3)*x - 2^(1/3)*x^2 + (-2^(2/3) + 2^(2/3)*x)*(1 - x^3)^(1/3) - 2*(1 - x^3)^(2/3)]/(6*2^(2/3))","A",1
2262,1,170,171,1.0043297,"\int \frac{x^4}{\left(-b+a x^4\right)^2 \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[x^4/((-b + a*x^4)^2*(b*x^2 + a*x^4)^(1/4)),x]","\frac{\left(b-a x^2\right) \left(b x^2+a x^4\right)^{3/4}}{4 a (a-b) b x \left(-b+a x^4\right)}+\frac{\text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 a \log (x)+2 a \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}+\text{$\#$1}^5}\&\right]}{32 a (a-b)}","\frac{\left(b-a x^2\right) \left(b x^2+a x^4\right)^{3/4}}{4 a (a-b) b x \left(-b+a x^4\right)}+\frac{\text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 a \log (x)-2 a \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}-\text{$\#$1}^5}\&\right]}{32 a (a-b)}",1,"((b - a*x^2)*(b*x^2 + a*x^4)^(3/4))/(4*a*(a - b)*b*x*(-b + a*x^4)) + RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (-2*a*Log[x] + 2*a*Log[(b*x^2 + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1) + #1^5) & ]/(32*a*(a - b))","A",1
2263,1,170,171,0.0021156,"\int \frac{x^4}{\left(-b+a x^4\right)^2 \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[x^4/((-b + a*x^4)^2*(b*x^2 + a*x^4)^(1/4)),x]","\frac{\left(b-a x^2\right) \left(b x^2+a x^4\right)^{3/4}}{4 a (a-b) b x \left(-b+a x^4\right)}+\frac{\text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 a \log (x)+2 a \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}+\text{$\#$1}^5}\&\right]}{32 a (a-b)}","\frac{\left(b-a x^2\right) \left(b x^2+a x^4\right)^{3/4}}{4 a (a-b) b x \left(-b+a x^4\right)}+\frac{\text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 a \log (x)-2 a \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}-\text{$\#$1}^5}\&\right]}{32 a (a-b)}",1,"((b - a*x^2)*(b*x^2 + a*x^4)^(3/4))/(4*a*(a - b)*b*x*(-b + a*x^4)) + RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (-2*a*Log[x] + 2*a*Log[(b*x^2 + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1) + #1^5) & ]/(32*a*(a - b))","A",1
2264,1,170,171,0.0020587,"\int \frac{x^4}{\left(-b+a x^4\right)^2 \sqrt[4]{b x^2+a x^4}} \, dx","IntegrateAlgebraic[x^4/((-b + a*x^4)^2*(b*x^2 + a*x^4)^(1/4)),x]","\frac{\left(b-a x^2\right) \left(b x^2+a x^4\right)^{3/4}}{4 a (a-b) b x \left(-b+a x^4\right)}+\frac{\text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 a \log (x)+2 a \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}+\text{$\#$1}^5}\&\right]}{32 a (a-b)}","\frac{\left(b-a x^2\right) \left(b x^2+a x^4\right)^{3/4}}{4 a (a-b) b x \left(-b+a x^4\right)}+\frac{\text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 a \log (x)-2 a \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}-\text{$\#$1}^5}\&\right]}{32 a (a-b)}",1,"((b - a*x^2)*(b*x^2 + a*x^4)^(3/4))/(4*a*(a - b)*b*x*(-b + a*x^4)) + RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (-2*a*Log[x] + 2*a*Log[(b*x^2 + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1) + #1^5) & ]/(32*a*(a - b))","A",1
2265,1,153,171,0.7904923,"\int \frac{x^8}{\sqrt{-1+x^4} \left(-1+x^{16}\right)} \, dx","IntegrateAlgebraic[x^8/(Sqrt[-1 + x^4]*(-1 + x^16)),x]","-\frac{x}{8 \sqrt{-1+x^4}}-\left(\frac{1}{32}-\frac{i}{32}\right) \tan ^{-1}\left(\frac{(1+i) x}{\sqrt{-1+x^4}}\right)+\left(\frac{1}{32}+\frac{i}{32}\right) \tan ^{-1}\left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{-1+x^4}}{x}\right)-\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt{-1+x^4}}{1+\sqrt{2} x^2-x^4}\right)}{8\ 2^{3/4}}+\frac{\tanh ^{-1}\left(\frac{2^{3/4} x \sqrt{-1+x^4}}{-1+\sqrt{2} x^2+x^4}\right)}{8\ 2^{3/4}}","-\frac{x}{8 \sqrt{-1+x^4}}+\frac{1}{32} \tan ^{-1}\left(\frac{-\frac{1}{2}-x^2+\frac{x^4}{2}}{x \sqrt{-1+x^4}}\right)+\frac{\tan ^{-1}\left(\frac{\frac{1}{2^{3/4}}+\frac{x^2}{\sqrt[4]{2}}-\frac{x^4}{2^{3/4}}}{x \sqrt{-1+x^4}}\right)}{8\ 2^{3/4}}-\frac{1}{32} \tanh ^{-1}\left(\frac{-\frac{1}{2}+x^2+\frac{x^4}{2}}{x \sqrt{-1+x^4}}\right)+\frac{\tanh ^{-1}\left(\frac{2^{3/4} x \sqrt{-1+x^4}}{-1+\sqrt{2} x^2+x^4}\right)}{8\ 2^{3/4}}",1,"-1/8*x/Sqrt[-1 + x^4] - (1/32 - I/32)*ArcTan[((1 + I)*x)/Sqrt[-1 + x^4]] + (1/32 + I/32)*ArcTan[((1/2 + I/2)*Sqrt[-1 + x^4])/x] - ArcTan[(2^(3/4)*x*Sqrt[-1 + x^4])/(1 + Sqrt[2]*x^2 - x^4)]/(8*2^(3/4)) + ArcTanh[(2^(3/4)*x*Sqrt[-1 + x^4])/(-1 + Sqrt[2]*x^2 + x^4)]/(8*2^(3/4))","C",1
2266,1,171,171,0.3746327,"\int \frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\left(b^2+a x^2\right)^4} \, dx","IntegrateAlgebraic[Sqrt[b + Sqrt[b^2 + a*x^2]]/(b^2 + a*x^2)^4,x]","\frac{11 x \left(611 b^4+798 a b^2 x^2+315 a^2 x^4\right)}{7680 b^6 \left(b^2+a x^2\right)^{5/2} \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{x \left(2587 b^4+3102 a b^2 x^2+1155 a^2 x^4\right)}{3840 b^5 \left(b^2+a x^2\right)^3 \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{231 \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{512 \sqrt{a} b^{13/2}}","\frac{11 x \left(611 b^4+798 a b^2 x^2+315 a^2 x^4\right)}{7680 b^6 \left(b^2+a x^2\right)^{5/2} \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{x \left(2587 b^4+3102 a b^2 x^2+1155 a^2 x^4\right)}{3840 b^5 \left(b^2+a x^2\right)^3 \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{231 \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{512 \sqrt{a} b^{13/2}}",1,"(11*x*(611*b^4 + 798*a*b^2*x^2 + 315*a^2*x^4))/(7680*b^6*(b^2 + a*x^2)^(5/2)*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (x*(2587*b^4 + 3102*a*b^2*x^2 + 1155*a^2*x^4))/(3840*b^5*(b^2 + a*x^2)^3*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (231*ArcTan[(Sqrt[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/(512*Sqrt[a]*b^(13/2))","A",1
2267,1,242,171,0.8278318,"\int \frac{\sqrt{x^2+\sqrt{1+x^4}}}{1+x^2} \, dx","IntegrateAlgebraic[Sqrt[x^2 + Sqrt[1 + x^4]]/(1 + x^2),x]","\sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\frac{-\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^4}}{\sqrt{2}}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)-\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{-\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)","\sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2 \left(-1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)",1,"Sqrt[-1 + Sqrt[2]]*ArcTan[(-Sqrt[1/2 + 1/Sqrt[2]] + Sqrt[1/2 + 1/Sqrt[2]]*x^2 + Sqrt[1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] + Sqrt[2]*ArcTanh[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] - Sqrt[1 + Sqrt[2]]*ArcTanh[(-Sqrt[-1/2 + 1/Sqrt[2]] + Sqrt[-1/2 + 1/Sqrt[2]]*x^2 + Sqrt[-1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])]","A",1
2268,1,172,172,1.2104345,"\int \frac{\sqrt{1-\sqrt{1-\sqrt{1-\frac{1}{x^2}}}}}{x} \, dx","IntegrateAlgebraic[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-2)]]]/x,x]","-4 \sqrt{1-\sqrt{1-\sqrt{\frac{-1+x^2}{x^2}}}}+\sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\sqrt{1+\sqrt{2}} \sqrt{1-\sqrt{1-\sqrt{\frac{-1+x^2}{x^2}}}}\right)+2 \tanh ^{-1}\left(\sqrt{1-\sqrt{1-\sqrt{\frac{-1+x^2}{x^2}}}}\right)+\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\sqrt{-1+\sqrt{2}} \sqrt{1-\sqrt{1-\sqrt{\frac{-1+x^2}{x^2}}}}\right)","-4 \sqrt{1-\sqrt{1-\sqrt{\frac{-1+x^2}{x^2}}}}+\sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{1-\sqrt{1-\sqrt{\frac{-1+x^2}{x^2}}}}}{\sqrt{-1+\sqrt{2}}}\right)+2 \tanh ^{-1}\left(\sqrt{1-\sqrt{1-\sqrt{\frac{-1+x^2}{x^2}}}}\right)+\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{1-\sqrt{1-\sqrt{\frac{-1+x^2}{x^2}}}}}{\sqrt{1+\sqrt{2}}}\right)",1,"-4*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x^2)/x^2]]] + Sqrt[-1 + Sqrt[2]]*ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x^2)/x^2]]]] + 2*ArcTanh[Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x^2)/x^2]]]] + Sqrt[1 + Sqrt[2]]*ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x^2)/x^2]]]]","A",1
2269,1,172,172,0.9862009,"\int \frac{1}{(1+x) \sqrt[3]{1-x^3}} \, dx","IntegrateAlgebraic[1/((1 + x)*(1 - x^3)^(1/3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{1-x^3}}{\sqrt[3]{2}-\sqrt[3]{2} x+\sqrt[3]{1-x^3}}\right)}{2 \sqrt[3]{2}}+\frac{\log \left(-\sqrt[3]{2}+\sqrt[3]{2} x+2 \sqrt[3]{1-x^3}\right)}{2 \sqrt[3]{2}}-\frac{\log \left(2^{2/3}-2\ 2^{2/3} x+2^{2/3} x^2+\left(2 \sqrt[3]{2}-2 \sqrt[3]{2} x\right) \sqrt[3]{1-x^3}+4 \left(1-x^3\right)^{2/3}\right)}{4 \sqrt[3]{2}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{1-x^3}}{\sqrt[3]{2}-\sqrt[3]{2} x+\sqrt[3]{1-x^3}}\right)}{2 \sqrt[3]{2}}+\frac{\log \left(-\sqrt[3]{2}+\sqrt[3]{2} x+2 \sqrt[3]{1-x^3}\right)}{2 \sqrt[3]{2}}-\frac{\log \left(2^{2/3}-2\ 2^{2/3} x+2^{2/3} x^2+\left(2 \sqrt[3]{2}-2 \sqrt[3]{2} x\right) \sqrt[3]{1-x^3}+4 \left(1-x^3\right)^{2/3}\right)}{4 \sqrt[3]{2}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(1 - x^3)^(1/3))/(2^(1/3) - 2^(1/3)*x + (1 - x^3)^(1/3))])/(2*2^(1/3)) + Log[-2^(1/3) + 2^(1/3)*x + 2*(1 - x^3)^(1/3)]/(2*2^(1/3)) - Log[2^(2/3) - 2*2^(2/3)*x + 2^(2/3)*x^2 + (2*2^(1/3) - 2*2^(1/3)*x)*(1 - x^3)^(1/3) + 4*(1 - x^3)^(2/3)]/(4*2^(1/3))","A",1
2270,1,173,173,3.8713634,"\int \frac{\left(a b-2 b x+x^2\right) \left(b^2-2 b x+x^2\right)}{\left(x (-a+x) (-b+x)^3\right)^{3/4} \left(b d-(a+d) x+x^2\right)} \, dx","IntegrateAlgebraic[((a*b - 2*b*x + x^2)*(b^2 - 2*b*x + x^2))/((x*(-a + x)*(-b + x)^3)^(3/4)*(b*d - (a + d)*x + x^2)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(a b^3 x+\left(-3 a b^2-b^3\right) x^2+\left(3 a b+3 b^2\right) x^3+(-a-3 b) x^4+x^5\right)^{3/4}}{(b-x)^2 x (-a+x)}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \left(a b^3 x+\left(-3 a b^2-b^3\right) x^2+\left(3 a b+3 b^2\right) x^3+(-a-3 b) x^4+x^5\right)^{3/4}}{(b-x)^2 x (-a+x)}\right)}{d^{3/4}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(a b^3 x+\left(-3 a b^2-b^3\right) x^2+\left(3 a b+3 b^2\right) x^3+(-a-3 b) x^4+x^5\right)^{3/4}}{(b-x)^2 x (-a+x)}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \left(a b^3 x+\left(-3 a b^2-b^3\right) x^2+\left(3 a b+3 b^2\right) x^3+(-a-3 b) x^4+x^5\right)^{3/4}}{(b-x)^2 x (-a+x)}\right)}{d^{3/4}}",1,"(2*ArcTan[(d^(1/4)*(a*b^3*x + (-3*a*b^2 - b^3)*x^2 + (3*a*b + 3*b^2)*x^3 + (-a - 3*b)*x^4 + x^5)^(3/4))/((b - x)^2*x*(-a + x))])/d^(3/4) - (2*ArcTanh[(d^(1/4)*(a*b^3*x + (-3*a*b^2 - b^3)*x^2 + (3*a*b + 3*b^2)*x^3 + (-a - 3*b)*x^4 + x^5)^(3/4))/((b - x)^2*x*(-a + x))])/d^(3/4)","A",1
2271,1,173,173,0.4863329,"\int \frac{-2 a b+(a+b) x}{\sqrt[3]{x (-a+x) (-b+x)} \left(-a b+(a+b) x+(-1+d) x^2\right)} \, dx","IntegrateAlgebraic[(-2*a*b + (a + b)*x)/((x*(-a + x)*(-b + x))^(1/3)*(-(a*b) + (a + b)*x + (-1 + d)*x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} x}{\sqrt[3]{d} x+2 \sqrt[3]{a b x+(-a-b) x^2+x^3}}\right)}{\sqrt[3]{d}}-\frac{\log \left(-\sqrt[3]{d} x+\sqrt[3]{a b x+(-a-b) x^2+x^3}\right)}{\sqrt[3]{d}}+\frac{\log \left(d^{2/3} x^2+\sqrt[3]{d} x \sqrt[3]{a b x+(-a-b) x^2+x^3}+\left(a b x+(-a-b) x^2+x^3\right)^{2/3}\right)}{2 \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} x}{\sqrt[3]{d} x+2 \sqrt[3]{a b x+(-a-b) x^2+x^3}}\right)}{\sqrt[3]{d}}-\frac{\log \left(-\sqrt[3]{d} x+\sqrt[3]{a b x+(-a-b) x^2+x^3}\right)}{\sqrt[3]{d}}+\frac{\log \left(d^{2/3} x^2+\sqrt[3]{d} x \sqrt[3]{a b x+(-a-b) x^2+x^3}+\left(a b x+(-a-b) x^2+x^3\right)^{2/3}\right)}{2 \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*x)/(d^(1/3)*x + 2*(a*b*x + (-a - b)*x^2 + x^3)^(1/3))])/d^(1/3) - Log[-(d^(1/3)*x) + (a*b*x + (-a - b)*x^2 + x^3)^(1/3)]/d^(1/3) + Log[d^(2/3)*x^2 + d^(1/3)*x*(a*b*x + (-a - b)*x^2 + x^3)^(1/3) + (a*b*x + (-a - b)*x^2 + x^3)^(2/3)]/(2*d^(1/3))","A",1
2272,1,173,173,0.3928082,"\int \frac{\left(-1+x^3\right)^{2/3} \left(1+x^3\right)}{x^6 \left(2+x^3\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(1 + x^3))/(x^6*(2 + x^3)),x]","\frac{\left(-4-x^3\right) \left(-1+x^3\right)^{2/3}}{40 x^5}+\frac{\sqrt[6]{3} \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2 \sqrt[3]{2} \sqrt[3]{-1+x^3}}\right)}{4\ 2^{2/3}}-\frac{\log \left(-3 x+\sqrt[3]{2} 3^{2/3} \sqrt[3]{-1+x^3}\right)}{4\ 2^{2/3} \sqrt[3]{3}}+\frac{\log \left(3 x^2+\sqrt[3]{2} 3^{2/3} x \sqrt[3]{-1+x^3}+2^{2/3} \sqrt[3]{3} \left(-1+x^3\right)^{2/3}\right)}{8\ 2^{2/3} \sqrt[3]{3}}","\frac{\left(-4-x^3\right) \left(-1+x^3\right)^{2/3}}{40 x^5}+\frac{\sqrt[6]{3} \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2 \sqrt[3]{2} \sqrt[3]{-1+x^3}}\right)}{4\ 2^{2/3}}-\frac{\log \left(-3 x+\sqrt[3]{2} 3^{2/3} \sqrt[3]{-1+x^3}\right)}{4\ 2^{2/3} \sqrt[3]{3}}+\frac{\log \left(3 x^2+\sqrt[3]{2} 3^{2/3} x \sqrt[3]{-1+x^3}+2^{2/3} \sqrt[3]{3} \left(-1+x^3\right)^{2/3}\right)}{8\ 2^{2/3} \sqrt[3]{3}}",1,"((-4 - x^3)*(-1 + x^3)^(2/3))/(40*x^5) + (3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*2^(1/3)*(-1 + x^3)^(1/3))])/(4*2^(2/3)) - Log[-3*x + 2^(1/3)*3^(2/3)*(-1 + x^3)^(1/3)]/(4*2^(2/3)*3^(1/3)) + Log[3*x^2 + 2^(1/3)*3^(2/3)*x*(-1 + x^3)^(1/3) + 2^(2/3)*3^(1/3)*(-1 + x^3)^(2/3)]/(8*2^(2/3)*3^(1/3))","A",1
2273,1,173,173,10.3068144,"\int \frac{1}{(1+x) \left(-2+3 x-2 x^2+3 x^3-2 x^4\right)^{3/2}} \, dx","IntegrateAlgebraic[1/((1 + x)*(-2 + 3*x - 2*x^2 + 3*x^3 - 2*x^4)^(3/2)),x]","\frac{i (-1+x) \sqrt{2+x+2 x^2} \left(\frac{i \left(-18-107 x+73 x^2+22 x^3\right)}{600 (-1+x)^2 \sqrt{2+x+2 x^2}}-\frac{i \tanh ^{-1}\left(\sqrt{\frac{2}{3}}+\sqrt{\frac{2}{3}} x-\frac{\sqrt{2+x+2 x^2}}{\sqrt{3}}\right)}{12 \sqrt{3}}-\frac{91 i \tanh ^{-1}\left(\sqrt{\frac{2}{5}}-\sqrt{\frac{2}{5}} x+\frac{\sqrt{2+x+2 x^2}}{\sqrt{5}}\right)}{200 \sqrt{5}}\right)}{\sqrt{-(-1+x)^2 \left(2+x+2 x^2\right)}}","\frac{i (-1+x) \sqrt{2+x+2 x^2} \left(\frac{i \left(-18-107 x+73 x^2+22 x^3\right)}{600 (-1+x)^2 \sqrt{2+x+2 x^2}}-\frac{i \tanh ^{-1}\left(\sqrt{\frac{2}{3}}+\sqrt{\frac{2}{3}} x-\frac{\sqrt{2+x+2 x^2}}{\sqrt{3}}\right)}{12 \sqrt{3}}-\frac{91 i \tanh ^{-1}\left(\sqrt{\frac{2}{5}}-\sqrt{\frac{2}{5}} x+\frac{\sqrt{2+x+2 x^2}}{\sqrt{5}}\right)}{200 \sqrt{5}}\right)}{\sqrt{-(-1+x)^2 \left(2+x+2 x^2\right)}}",1,"(I*(-1 + x)*Sqrt[2 + x + 2*x^2]*(((I/600)*(-18 - 107*x + 73*x^2 + 22*x^3))/((-1 + x)^2*Sqrt[2 + x + 2*x^2]) - ((I/12)*ArcTanh[Sqrt[2/3] + Sqrt[2/3]*x - Sqrt[2 + x + 2*x^2]/Sqrt[3]])/Sqrt[3] - (((91*I)/200)*ArcTanh[Sqrt[2/5] - Sqrt[2/5]*x + Sqrt[2 + x + 2*x^2]/Sqrt[5]])/Sqrt[5]))/Sqrt[-((-1 + x)^2*(2 + x + 2*x^2))]","A",1
2274,1,173,173,0.4332845,"\int \frac{x^2 \sqrt[4]{x^3+x^4}}{-1+x^4} \, dx","IntegrateAlgebraic[(x^2*(x^3 + x^4)^(1/4))/(-1 + x^4),x]","-2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^4}}\right)}{2^{3/4}}+2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^4}}\right)}{2^{3/4}}-\frac{1}{4} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 \log (x)+2 \log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+\text{$\#$1}^7}\&\right]","-2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^4}}\right)}{2^{3/4}}+2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^4}}\right)}{2^{3/4}}-\frac{1}{4} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 \log (x)+2 \log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+\text{$\#$1}^7}\&\right]",1,"-2*ArcTan[x/(x^3 + x^4)^(1/4)] + ArcTan[(2^(1/4)*x)/(x^3 + x^4)^(1/4)]/2^(3/4) + 2*ArcTanh[x/(x^3 + x^4)^(1/4)] - ArcTanh[(2^(1/4)*x)/(x^3 + x^4)^(1/4)]/2^(3/4) - RootSum[2 - 2*#1^4 + #1^8 & , (-2*Log[x] + 2*Log[(x^3 + x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(x^3 + x^4)^(1/4) - x*#1]*#1^4)/(-#1^3 + #1^7) & ]/4","A",1
2275,1,173,173,0.0009918,"\int \frac{x^2 \sqrt[4]{x^3+x^4}}{-1+x^4} \, dx","IntegrateAlgebraic[(x^2*(x^3 + x^4)^(1/4))/(-1 + x^4),x]","-2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^4}}\right)}{2^{3/4}}+2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^4}}\right)}{2^{3/4}}-\frac{1}{4} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 \log (x)+2 \log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+\text{$\#$1}^7}\&\right]","-2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^4}}\right)}{2^{3/4}}+2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^4}}\right)}{2^{3/4}}-\frac{1}{4} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 \log (x)+2 \log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+\text{$\#$1}^7}\&\right]",1,"-2*ArcTan[x/(x^3 + x^4)^(1/4)] + ArcTan[(2^(1/4)*x)/(x^3 + x^4)^(1/4)]/2^(3/4) + 2*ArcTanh[x/(x^3 + x^4)^(1/4)] - ArcTanh[(2^(1/4)*x)/(x^3 + x^4)^(1/4)]/2^(3/4) - RootSum[2 - 2*#1^4 + #1^8 & , (-2*Log[x] + 2*Log[(x^3 + x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(x^3 + x^4)^(1/4) - x*#1]*#1^4)/(-#1^3 + #1^7) & ]/4","A",1
2276,1,173,173,0.5058934,"\int \frac{1}{\sqrt[3]{-x+x^3} \left(1+x^6\right)} \, dx","IntegrateAlgebraic[1/((-x + x^3)^(1/3)*(1 + x^6)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-x+x^3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}-\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{-x+x^3}\right)}{6 \sqrt[3]{2}}+\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-x+x^3}+\sqrt[3]{2} \left(-x+x^3\right)^{2/3}\right)}{12 \sqrt[3]{2}}-\frac{1}{6} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-x+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-x+x^3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}-\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{-x+x^3}\right)}{6 \sqrt[3]{2}}+\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-x+x^3}+\sqrt[3]{2} \left(-x+x^3\right)^{2/3}\right)}{12 \sqrt[3]{2}}-\frac{1}{6} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-x+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-x + x^3)^(1/3))]/(2*2^(1/3)*Sqrt[3]) - Log[-2*x + 2^(2/3)*(-x + x^3)^(1/3)]/(6*2^(1/3)) + Log[2*x^2 + 2^(2/3)*x*(-x + x^3)^(1/3) + 2^(1/3)*(-x + x^3)^(2/3)]/(12*2^(1/3)) - RootSum[1 - #1^3 + #1^6 & , (-Log[x] + Log[(-x + x^3)^(1/3) - x*#1])/#1 & ]/6","A",1
2277,1,173,173,0.0010001,"\int \frac{1}{\sqrt[3]{-x+x^3} \left(1+x^6\right)} \, dx","IntegrateAlgebraic[1/((-x + x^3)^(1/3)*(1 + x^6)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-x+x^3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}-\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{-x+x^3}\right)}{6 \sqrt[3]{2}}+\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-x+x^3}+\sqrt[3]{2} \left(-x+x^3\right)^{2/3}\right)}{12 \sqrt[3]{2}}-\frac{1}{6} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-x+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-x+x^3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}-\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{-x+x^3}\right)}{6 \sqrt[3]{2}}+\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-x+x^3}+\sqrt[3]{2} \left(-x+x^3\right)^{2/3}\right)}{12 \sqrt[3]{2}}-\frac{1}{6} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-x+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-x + x^3)^(1/3))]/(2*2^(1/3)*Sqrt[3]) - Log[-2*x + 2^(2/3)*(-x + x^3)^(1/3)]/(6*2^(1/3)) + Log[2*x^2 + 2^(2/3)*x*(-x + x^3)^(1/3) + 2^(1/3)*(-x + x^3)^(2/3)]/(12*2^(1/3)) - RootSum[1 - #1^3 + #1^6 & , (-Log[x] + Log[(-x + x^3)^(1/3) - x*#1])/#1 & ]/6","A",1
2278,1,173,173,0.7885325,"\int \frac{\sqrt{-x+x^4}}{-b+a x^6} \, dx","IntegrateAlgebraic[Sqrt[-x + x^4]/(-b + a*x^6),x]","-\frac{\sqrt{-\left(\left(\sqrt{a}+\sqrt{b}\right) \sqrt{b}\right)} \tan ^{-1}\left(\frac{\sqrt{-\sqrt{a} \sqrt{b}-b} x \sqrt{-x+x^4}}{\sqrt{b} (-1+x) \left(1+x+x^2\right)}\right)}{3 \sqrt{a} b}+\frac{\sqrt{\left(\sqrt{a}-\sqrt{b}\right) \sqrt{b}} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a} \sqrt{b}-b} x \sqrt{-x+x^4}}{\sqrt{b} (-1+x) \left(1+x+x^2\right)}\right)}{3 \sqrt{a} b}","-\frac{\sqrt{-\left(\left(\sqrt{a}+\sqrt{b}\right) \sqrt{b}\right)} \tan ^{-1}\left(\frac{\sqrt{-\sqrt{a} \sqrt{b}-b} x \sqrt{-x+x^4}}{\sqrt{b} (-1+x) \left(1+x+x^2\right)}\right)}{3 \sqrt{a} b}+\frac{\sqrt{\left(\sqrt{a}-\sqrt{b}\right) \sqrt{b}} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a} \sqrt{b}-b} x \sqrt{-x+x^4}}{\sqrt{b} (-1+x) \left(1+x+x^2\right)}\right)}{3 \sqrt{a} b}",1,"-1/3*(Sqrt[-((Sqrt[a] + Sqrt[b])*Sqrt[b])]*ArcTan[(Sqrt[-(Sqrt[a]*Sqrt[b]) - b]*x*Sqrt[-x + x^4])/(Sqrt[b]*(-1 + x)*(1 + x + x^2))])/(Sqrt[a]*b) + (Sqrt[(Sqrt[a] - Sqrt[b])*Sqrt[b]]*ArcTan[(Sqrt[Sqrt[a]*Sqrt[b] - b]*x*Sqrt[-x + x^4])/(Sqrt[b]*(-1 + x)*(1 + x + x^2))])/(3*Sqrt[a]*b)","A",1
2279,1,173,173,2.8824318,"\int \frac{\left(1+2 x^8\right) \sqrt[4]{-1-2 x^4+2 x^8} \left(1-3 x^8+4 x^{16}\right)}{x^{10} \left(-1+2 x^8\right)} \, dx","IntegrateAlgebraic[((1 + 2*x^8)*(-1 - 2*x^4 + 2*x^8)^(1/4)*(1 - 3*x^8 + 4*x^16))/(x^10*(-1 + 2*x^8)),x]","\frac{\sqrt[4]{-1-2 x^4+2 x^8} \left(5+2 x^4+9 x^8-4 x^{12}+20 x^{16}\right)}{45 x^9}-\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{-1-2 x^4+2 x^8}}{\sqrt{2} x^2-\sqrt{-1-2 x^4+2 x^8}}\right)}{2 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{2 \sqrt[4]{2} x \sqrt[4]{-1-2 x^4+2 x^8}}{2 x^2+\sqrt{2} \sqrt{-1-2 x^4+2 x^8}}\right)}{2 \sqrt[4]{2}}","\frac{\sqrt[4]{-1-2 x^4+2 x^8} \left(5+2 x^4+9 x^8-4 x^{12}+20 x^{16}\right)}{45 x^9}-\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{-1-2 x^4+2 x^8}}{\sqrt{2} x^2-\sqrt{-1-2 x^4+2 x^8}}\right)}{2 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{2 \sqrt[4]{2} x \sqrt[4]{-1-2 x^4+2 x^8}}{2 x^2+\sqrt{2} \sqrt{-1-2 x^4+2 x^8}}\right)}{2 \sqrt[4]{2}}",1,"((-1 - 2*x^4 + 2*x^8)^(1/4)*(5 + 2*x^4 + 9*x^8 - 4*x^12 + 20*x^16))/(45*x^9) - ArcTan[(2^(3/4)*x*(-1 - 2*x^4 + 2*x^8)^(1/4))/(Sqrt[2]*x^2 - Sqrt[-1 - 2*x^4 + 2*x^8])]/(2*2^(1/4)) - ArcTanh[(2*2^(1/4)*x*(-1 - 2*x^4 + 2*x^8)^(1/4))/(2*x^2 + Sqrt[2]*Sqrt[-1 - 2*x^4 + 2*x^8])]/(2*2^(1/4))","A",1
2280,1,140,173,0.256105,"\int \frac{\left(b^2+a x^2\right)^{3/2}}{\sqrt{b+\sqrt{b^2+a x^2}}} \, dx","IntegrateAlgebraic[(b^2 + a*x^2)^(3/2)/Sqrt[b + Sqrt[b^2 + a*x^2]],x]","\frac{2 x \sqrt{b^2+a x^2} \left(46 b^2+15 a x^2\right)}{105 \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{2 x \left(46 b^3+3 a b x^2\right)}{105 \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{\sqrt{2} b^{7/2} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{\sqrt{a}}","\frac{2 x \sqrt{b^2+a x^2} \left(46 b^2+15 a x^2\right)}{105 \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{2 x \left(46 b^3+3 a b x^2\right)}{105 \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{2 \sqrt{2} b^{7/2} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2} \sqrt{b}}\right)}{\sqrt{a}}",1,"(2*x*Sqrt[b^2 + a*x^2]*(46*b^2 + 15*a*x^2))/(105*Sqrt[b + Sqrt[b^2 + a*x^2]]) - (2*x*(46*b^3 + 3*a*b*x^2))/(105*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (Sqrt[2]*b^(7/2)*ArcTan[(Sqrt[a]*x)/(Sqrt[2]*Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/Sqrt[a]","A",1
2281,1,173,173,27.692522,"\int \frac{x \left(-2 q+p x^6\right) \sqrt{q+p x^6}}{b x^8+a \left(q+p x^6\right)^2} \, dx","IntegrateAlgebraic[(x*(-2*q + p*x^6)*Sqrt[q + p*x^6])/(b*x^8 + a*(q + p*x^6)^2),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x^2 \sqrt{q+p x^6}}{\sqrt{a} q-\sqrt{b} x^4+\sqrt{a} p x^6}\right)}{2 \sqrt{2} a^{3/4} \sqrt[4]{b}}-\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[4]{a} q}{\sqrt{2} \sqrt[4]{b}}+\frac{\sqrt[4]{b} x^4}{\sqrt{2} \sqrt[4]{a}}+\frac{\sqrt[4]{a} p x^6}{\sqrt{2} \sqrt[4]{b}}}{x^2 \sqrt{q+p x^6}}\right)}{2 \sqrt{2} a^{3/4} \sqrt[4]{b}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x^2 \sqrt{q+p x^6}}{\sqrt{a} q-\sqrt{b} x^4+\sqrt{a} p x^6}\right)}{2 \sqrt{2} a^{3/4} \sqrt[4]{b}}-\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[4]{a} q}{\sqrt{2} \sqrt[4]{b}}+\frac{\sqrt[4]{b} x^4}{\sqrt{2} \sqrt[4]{a}}+\frac{\sqrt[4]{a} p x^6}{\sqrt{2} \sqrt[4]{b}}}{x^2 \sqrt{q+p x^6}}\right)}{2 \sqrt{2} a^{3/4} \sqrt[4]{b}}",1,"-1/2*ArcTan[(Sqrt[2]*a^(1/4)*b^(1/4)*x^2*Sqrt[q + p*x^6])/(Sqrt[a]*q - Sqrt[b]*x^4 + Sqrt[a]*p*x^6)]/(Sqrt[2]*a^(3/4)*b^(1/4)) - ArcTanh[((a^(1/4)*q)/(Sqrt[2]*b^(1/4)) + (b^(1/4)*x^4)/(Sqrt[2]*a^(1/4)) + (a^(1/4)*p*x^6)/(Sqrt[2]*b^(1/4)))/(x^2*Sqrt[q + p*x^6])]/(2*Sqrt[2]*a^(3/4)*b^(1/4))","A",1
2282,1,174,174,0.426671,"\int \frac{\sqrt[3]{-1-2 x+6 x^2}}{-1+6 x} \, dx","IntegrateAlgebraic[(-1 - 2*x + 6*x^2)^(1/3)/(-1 + 6*x),x]","\frac{1}{4} \sqrt[3]{-1-2 x+6 x^2}+\frac{\sqrt[3]{\frac{7}{2}} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{\frac{2}{7}} \sqrt[3]{-1-2 x+6 x^2}}{\sqrt[6]{3}}\right)}{4\ 3^{5/6}}-\frac{1}{12} \sqrt[3]{\frac{7}{6}} \log \left(7+\sqrt[3]{6} 7^{2/3} \sqrt[3]{-1-2 x+6 x^2}\right)+\frac{1}{24} \sqrt[3]{\frac{7}{6}} \log \left(7-\sqrt[3]{6} 7^{2/3} \sqrt[3]{-1-2 x+6 x^2}+6^{2/3} \sqrt[3]{7} \left(-1-2 x+6 x^2\right)^{2/3}\right)","\frac{1}{4} \sqrt[3]{-1-2 x+6 x^2}+\frac{\sqrt[3]{\frac{7}{2}} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{\frac{2}{7}} \sqrt[3]{-1-2 x+6 x^2}}{\sqrt[6]{3}}\right)}{4\ 3^{5/6}}-\frac{1}{12} \sqrt[3]{\frac{7}{6}} \log \left(7+\sqrt[3]{6} 7^{2/3} \sqrt[3]{-1-2 x+6 x^2}\right)+\frac{1}{24} \sqrt[3]{\frac{7}{6}} \log \left(7-\sqrt[3]{6} 7^{2/3} \sqrt[3]{-1-2 x+6 x^2}+6^{2/3} \sqrt[3]{7} \left(-1-2 x+6 x^2\right)^{2/3}\right)",1,"(-1 - 2*x + 6*x^2)^(1/3)/4 + ((7/2)^(1/3)*ArcTan[1/Sqrt[3] - (2*(2/7)^(1/3)*(-1 - 2*x + 6*x^2)^(1/3))/3^(1/6)])/(4*3^(5/6)) - ((7/6)^(1/3)*Log[7 + 6^(1/3)*7^(2/3)*(-1 - 2*x + 6*x^2)^(1/3)])/12 + ((7/6)^(1/3)*Log[7 - 6^(1/3)*7^(2/3)*(-1 - 2*x + 6*x^2)^(1/3) + 6^(2/3)*7^(1/3)*(-1 - 2*x + 6*x^2)^(2/3)])/24","A",1
2283,1,174,174,0.9255751,"\int \frac{-2 a b x+(a+b) x^2}{(x (-a+x) (-b+x))^{2/3} \left(-a b+(a+b) x+(-1+d) x^2\right)} \, dx","IntegrateAlgebraic[(-2*a*b*x + (a + b)*x^2)/((x*(-a + x)*(-b + x))^(2/3)*(-(a*b) + (a + b)*x + (-1 + d)*x^2)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} x}{\sqrt[3]{d} x+2 \sqrt[3]{a b x+(-a-b) x^2+x^3}}\right)}{d^{2/3}}-\frac{\log \left(-\sqrt[3]{d} x+\sqrt[3]{a b x+(-a-b) x^2+x^3}\right)}{d^{2/3}}+\frac{\log \left(d^{2/3} x^2+\sqrt[3]{d} x \sqrt[3]{a b x+(-a-b) x^2+x^3}+\left(a b x+(-a-b) x^2+x^3\right)^{2/3}\right)}{2 d^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} x}{\sqrt[3]{d} x+2 \sqrt[3]{a b x+(-a-b) x^2+x^3}}\right)}{d^{2/3}}-\frac{\log \left(-\sqrt[3]{d} x+\sqrt[3]{a b x+(-a-b) x^2+x^3}\right)}{d^{2/3}}+\frac{\log \left(d^{2/3} x^2+\sqrt[3]{d} x \sqrt[3]{a b x+(-a-b) x^2+x^3}+\left(a b x+(-a-b) x^2+x^3\right)^{2/3}\right)}{2 d^{2/3}}",1,"-((Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*x)/(d^(1/3)*x + 2*(a*b*x + (-a - b)*x^2 + x^3)^(1/3))])/d^(2/3)) - Log[-(d^(1/3)*x) + (a*b*x + (-a - b)*x^2 + x^3)^(1/3)]/d^(2/3) + Log[d^(2/3)*x^2 + d^(1/3)*x*(a*b*x + (-a - b)*x^2 + x^3)^(1/3) + (a*b*x + (-a - b)*x^2 + x^3)^(2/3)]/(2*d^(2/3))","A",1
2284,1,174,174,0.5106401,"\int \frac{x \left(-a b+x^2\right)}{\left(x^2 (-a+x) (-b+x)\right)^{2/3} \left(a b d-(1+a d+b d) x+d x^2\right)} \, dx","IntegrateAlgebraic[(x*(-(a*b) + x^2))/((x^2*(-a + x)*(-b + x))^(2/3)*(a*b*d - (1 + a*d + b*d)*x + d*x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{\sqrt[3]{d}}+\frac{\log \left(x-\sqrt[3]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}\right)}{\sqrt[3]{d}}-\frac{\log \left(x^2+\sqrt[3]{d} x \sqrt[3]{a b x^2+(-a-b) x^3+x^4}+d^{2/3} \left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{\sqrt[3]{d}}+\frac{\log \left(x-\sqrt[3]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}\right)}{\sqrt[3]{d}}-\frac{\log \left(x^2+\sqrt[3]{d} x \sqrt[3]{a b x^2+(-a-b) x^3+x^4}+d^{2/3} \left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*d^(1/3)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))])/d^(1/3) + Log[x - d^(1/3)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3)]/d^(1/3) - Log[x^2 + d^(1/3)*x*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3) + d^(2/3)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(2/3)]/(2*d^(1/3))","A",1
2285,1,174,174,0.2530436,"\int \frac{\left(-2+x^3\right) \left(-1+x^3\right)^{2/3}}{x^3 \left(-1+2 x^3\right)} \, dx","IntegrateAlgebraic[((-2 + x^3)*(-1 + x^3)^(2/3))/(x^3*(-1 + 2*x^3)),x]","-\frac{\left(-1+x^3\right)^{2/3}}{x^2}-\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-1+x^3}}\right)+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{2 \sqrt{3}}-\frac{1}{6} \log \left(-x+\sqrt[3]{-1+x^3}\right)-\frac{1}{2} \log \left(x+\sqrt[3]{-1+x^3}\right)+\frac{1}{4} \log \left(x^2-x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)+\frac{1}{12} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)","-\frac{\left(-1+x^3\right)^{2/3}}{x^2}-\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-1+x^3}}\right)+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{2 \sqrt{3}}-\frac{1}{6} \log \left(-x+\sqrt[3]{-1+x^3}\right)-\frac{1}{2} \log \left(x+\sqrt[3]{-1+x^3}\right)+\frac{1}{4} \log \left(x^2-x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)+\frac{1}{12} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)",1,"-((-1 + x^3)^(2/3)/x^2) - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(-1 + x^3)^(1/3))])/2 + ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^3)^(1/3))]/(2*Sqrt[3]) - Log[-x + (-1 + x^3)^(1/3)]/6 - Log[x + (-1 + x^3)^(1/3)]/2 + Log[x^2 - x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)]/4 + Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)]/12","A",1
2286,1,174,174,0.637431,"\int \frac{\left(-b+a x^2\right) \sqrt[4]{-b x^2+a x^4}}{b+a x^2} \, dx","IntegrateAlgebraic[((-b + a*x^2)*(-(b*x^2) + a*x^4)^(1/4))/(b + a*x^2),x]","\frac{1}{2} x \sqrt[4]{-b x^2+a x^4}+\frac{9 b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{4 a^{3/4}}-\frac{2 \sqrt[4]{2} b \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{a^{3/4}}-\frac{9 b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{4 a^{3/4}}+\frac{2 \sqrt[4]{2} b \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{a^{3/4}}","\frac{1}{2} x \sqrt[4]{-b x^2+a x^4}+\frac{9 b \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{4 a^{3/4}}-\frac{2 \sqrt[4]{2} b \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{a^{3/4}}-\frac{9 b \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{4 a^{3/4}}+\frac{2 \sqrt[4]{2} b \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{a^{3/4}}",1,"(x*(-(b*x^2) + a*x^4)^(1/4))/2 + (9*b*ArcTan[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/(4*a^(3/4)) - (2*2^(1/4)*b*ArcTan[(2^(1/4)*a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/a^(3/4) - (9*b*ArcTanh[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/(4*a^(3/4)) + (2*2^(1/4)*b*ArcTanh[(2^(1/4)*a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/a^(3/4)","A",1
2287,1,174,174,1.304141,"\int \frac{1+x^3}{\left(-1+x^3\right) \sqrt{a+b x+c x^2+b x^3+a x^4}} \, dx","IntegrateAlgebraic[(1 + x^3)/((-1 + x^3)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]),x]","-\frac{2 \sqrt{-2 a-2 b-c} \tan ^{-1}\left(\frac{\sqrt{-2 a-2 b-c} x}{\sqrt{a}-2 \sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{3 (2 a+2 b+c)}+\frac{4 \tan ^{-1}\left(\frac{\sqrt{a+b-c} x}{\sqrt{a}+\sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{3 \sqrt{a+b-c}}","-\frac{2 \sqrt{-2 a-2 b-c} \tan ^{-1}\left(\frac{\sqrt{-2 a-2 b-c} x}{\sqrt{a}-2 \sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{3 (2 a+2 b+c)}+\frac{4 \tan ^{-1}\left(\frac{\sqrt{a+b-c} x}{\sqrt{a}+\sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{3 \sqrt{a+b-c}}",1,"(-2*Sqrt[-2*a - 2*b - c]*ArcTan[(Sqrt[-2*a - 2*b - c]*x)/(Sqrt[a] - 2*Sqrt[a]*x + Sqrt[a]*x^2 - Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])])/(3*(2*a + 2*b + c)) + (4*ArcTan[(Sqrt[a + b - c]*x)/(Sqrt[a] + Sqrt[a]*x + Sqrt[a]*x^2 - Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])])/(3*Sqrt[a + b - c])","A",1
2288,1,177,174,2.7899579,"\int \frac{-b+a x^6}{\sqrt[3]{-x+x^3} \left(-d+c x^6\right)} \, dx","IntegrateAlgebraic[(-b + a*x^6)/((-x + x^3)^(1/3)*(-d + c*x^6)),x]","\frac{\sqrt{3} a \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)}{2 c}-\frac{a \log \left(-c x+c \sqrt[3]{-x+x^3}\right)}{2 c}+\frac{a \log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)}{4 c}+\frac{(-b c+a d) \text{RootSum}\left[c-d+3 d \text{$\#$1}^3-3 d \text{$\#$1}^6+d \text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{-x+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 c d}","\frac{\sqrt{3} a \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)}{2 c}-\frac{a \log \left(-x+\sqrt[3]{-x+x^3}\right)}{2 c}+\frac{a \log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)}{4 c}+\frac{(-b c+a d) \text{RootSum}\left[c-d+3 d \text{$\#$1}^3-3 d \text{$\#$1}^6+d \text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{-x+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 c d}",1,"(Sqrt[3]*a*ArcTan[(Sqrt[3]*x)/(x + 2*(-x + x^3)^(1/3))])/(2*c) - (a*Log[-(c*x) + c*(-x + x^3)^(1/3)])/(2*c) + (a*Log[x^2 + x*(-x + x^3)^(1/3) + (-x + x^3)^(2/3)])/(4*c) + ((-(b*c) + a*d)*RootSum[c - d + 3*d*#1^3 - 3*d*#1^6 + d*#1^9 & , (-Log[x] + Log[(-x + x^3)^(1/3) - x*#1])/#1 & ])/(6*c*d)","A",1
2289,1,177,174,0.0013179,"\int \frac{-b+a x^6}{\sqrt[3]{-x+x^3} \left(-d+c x^6\right)} \, dx","IntegrateAlgebraic[(-b + a*x^6)/((-x + x^3)^(1/3)*(-d + c*x^6)),x]","\frac{\sqrt{3} a \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)}{2 c}-\frac{a \log \left(-c x+c \sqrt[3]{-x+x^3}\right)}{2 c}+\frac{a \log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)}{4 c}+\frac{(-b c+a d) \text{RootSum}\left[c-d+3 d \text{$\#$1}^3-3 d \text{$\#$1}^6+d \text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{-x+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 c d}","\frac{\sqrt{3} a \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)}{2 c}-\frac{a \log \left(-x+\sqrt[3]{-x+x^3}\right)}{2 c}+\frac{a \log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)}{4 c}+\frac{(-b c+a d) \text{RootSum}\left[c-d+3 d \text{$\#$1}^3-3 d \text{$\#$1}^6+d \text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{-x+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{6 c d}",1,"(Sqrt[3]*a*ArcTan[(Sqrt[3]*x)/(x + 2*(-x + x^3)^(1/3))])/(2*c) - (a*Log[-(c*x) + c*(-x + x^3)^(1/3)])/(2*c) + (a*Log[x^2 + x*(-x + x^3)^(1/3) + (-x + x^3)^(2/3)])/(4*c) + ((-(b*c) + a*d)*RootSum[c - d + 3*d*#1^3 - 3*d*#1^6 + d*#1^9 & , (-Log[x] + Log[(-x + x^3)^(1/3) - x*#1])/#1 & ])/(6*c*d)","A",1
2290,1,114,174,0.118993,"\int \frac{1}{\sqrt{d+\sqrt{c+\sqrt{b+a x}}}} \, dx","IntegrateAlgebraic[1/Sqrt[d + Sqrt[c + Sqrt[b + a*x]]],x]","-\frac{8 \left(20 c-24 d^2-15 \sqrt{b+a x}\right) \sqrt{c+\sqrt{b+a x}} \sqrt{d+\sqrt{c+\sqrt{b+a x}}}}{105 a}+\frac{16 \left(26 c d-24 d^3-9 d \sqrt{b+a x}\right) \sqrt{d+\sqrt{c+\sqrt{b+a x}}}}{105 a}","\frac{32 \left(13 c d-12 d^3\right) \sqrt{d+\sqrt{c+\sqrt{b+a x}}}}{105 a}-\frac{32 \left(5 c-6 d^2\right) \sqrt{c+\sqrt{b+a x}} \sqrt{d+\sqrt{c+\sqrt{b+a x}}}}{105 a}+\sqrt{b+a x} \left(-\frac{48 d \sqrt{d+\sqrt{c+\sqrt{b+a x}}}}{35 a}+\frac{8 \sqrt{c+\sqrt{b+a x}} \sqrt{d+\sqrt{c+\sqrt{b+a x}}}}{7 a}\right)",1,"(-8*(20*c - 24*d^2 - 15*Sqrt[b + a*x])*Sqrt[c + Sqrt[b + a*x]]*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]])/(105*a) + (16*(26*c*d - 24*d^3 - 9*d*Sqrt[b + a*x])*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]])/(105*a)","A",1
2291,1,175,175,0.2065438,"\int \frac{1}{x \sqrt[3]{3+3 x+x^2}} \, dx","IntegrateAlgebraic[1/(x*(3 + 3*x + x^2)^(1/3)),x]","-\frac{\tan ^{-1}\left(\frac{\frac{2}{\sqrt[6]{3}}+\frac{2 x}{3 \sqrt[6]{3}}+\frac{\sqrt[3]{3+3 x+x^2}}{\sqrt{3}}}{\sqrt[3]{3+3 x+x^2}}\right)}{3^{5/6}}+\frac{\log \left(3 \sqrt[3]{3}+\sqrt[3]{3} x-3 \sqrt[3]{3+3 x+x^2}\right)}{3 \sqrt[3]{3}}-\frac{\log \left(9\ 3^{2/3}+6\ 3^{2/3} x+3^{2/3} x^2+\left(9 \sqrt[3]{3}+3 \sqrt[3]{3} x\right) \sqrt[3]{3+3 x+x^2}+9 \left(3+3 x+x^2\right)^{2/3}\right)}{6 \sqrt[3]{3}}","-\frac{\tan ^{-1}\left(\frac{\frac{2}{\sqrt[6]{3}}+\frac{2 x}{3 \sqrt[6]{3}}+\frac{\sqrt[3]{3+3 x+x^2}}{\sqrt{3}}}{\sqrt[3]{3+3 x+x^2}}\right)}{3^{5/6}}+\frac{\log \left(3 \sqrt[3]{3}+\sqrt[3]{3} x-3 \sqrt[3]{3+3 x+x^2}\right)}{3 \sqrt[3]{3}}-\frac{\log \left(9\ 3^{2/3}+6\ 3^{2/3} x+3^{2/3} x^2+\left(9 \sqrt[3]{3}+3 \sqrt[3]{3} x\right) \sqrt[3]{3+3 x+x^2}+9 \left(3+3 x+x^2\right)^{2/3}\right)}{6 \sqrt[3]{3}}",1,"-(ArcTan[(2/3^(1/6) + (2*x)/(3*3^(1/6)) + (3 + 3*x + x^2)^(1/3)/Sqrt[3])/(3 + 3*x + x^2)^(1/3)]/3^(5/6)) + Log[3*3^(1/3) + 3^(1/3)*x - 3*(3 + 3*x + x^2)^(1/3)]/(3*3^(1/3)) - Log[9*3^(2/3) + 6*3^(2/3)*x + 3^(2/3)*x^2 + (9*3^(1/3) + 3*3^(1/3)*x)*(3 + 3*x + x^2)^(1/3) + 9*(3 + 3*x + x^2)^(2/3)]/(6*3^(1/3))","A",1
2292,1,175,175,0.4151353,"\int \frac{-a b+x^2}{\sqrt[3]{x^2 (-a+x) (-b+x)} \left(a b d-(1+a d+b d) x+d x^2\right)} \, dx","IntegrateAlgebraic[(-(a*b) + x^2)/((x^2*(-a + x)*(-b + x))^(1/3)*(a*b*d - (1 + a*d + b*d)*x + d*x^2)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{d^{2/3}}+\frac{\log \left(x-\sqrt[3]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}\right)}{d^{2/3}}-\frac{\log \left(x^2+\sqrt[3]{d} x \sqrt[3]{a b x^2+(-a-b) x^3+x^4}+d^{2/3} \left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}\right)}{2 d^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{d^{2/3}}+\frac{\log \left(x-\sqrt[3]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}\right)}{d^{2/3}}-\frac{\log \left(x^2+\sqrt[3]{d} x \sqrt[3]{a b x^2+(-a-b) x^3+x^4}+d^{2/3} \left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}\right)}{2 d^{2/3}}",1,"-((Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*d^(1/3)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))])/d^(2/3)) + Log[x - d^(1/3)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3)]/d^(2/3) - Log[x^2 + d^(1/3)*x*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3) + d^(2/3)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(2/3)]/(2*d^(2/3))","A",1
2293,1,175,175,0.5759459,"\int \frac{(-3+2 x) \left(1-x+x^3\right)^{2/3}}{x^3 \left(-2+2 x+x^3\right)} \, dx","IntegrateAlgebraic[((-3 + 2*x)*(1 - x + x^3)^(2/3))/(x^3*(-2 + 2*x + x^3)),x]","-\frac{3 \left(1-x+x^3\right)^{2/3}}{4 x^2}+\frac{3 \sqrt[6]{3} \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2 \sqrt[3]{2} \sqrt[3]{1-x+x^3}}\right)}{2\ 2^{2/3}}-\frac{1}{2} \left(\frac{3}{2}\right)^{2/3} \log \left(-3 x+\sqrt[3]{2} 3^{2/3} \sqrt[3]{1-x+x^3}\right)+\frac{1}{4} \left(\frac{3}{2}\right)^{2/3} \log \left(3 x^2+\sqrt[3]{2} 3^{2/3} x \sqrt[3]{1-x+x^3}+2^{2/3} \sqrt[3]{3} \left(1-x+x^3\right)^{2/3}\right)","-\frac{3 \left(1-x+x^3\right)^{2/3}}{4 x^2}+\frac{3 \sqrt[6]{3} \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2 \sqrt[3]{2} \sqrt[3]{1-x+x^3}}\right)}{2\ 2^{2/3}}-\frac{1}{2} \left(\frac{3}{2}\right)^{2/3} \log \left(-3 x+\sqrt[3]{2} 3^{2/3} \sqrt[3]{1-x+x^3}\right)+\frac{1}{4} \left(\frac{3}{2}\right)^{2/3} \log \left(3 x^2+\sqrt[3]{2} 3^{2/3} x \sqrt[3]{1-x+x^3}+2^{2/3} \sqrt[3]{3} \left(1-x+x^3\right)^{2/3}\right)",1,"(-3*(1 - x + x^3)^(2/3))/(4*x^2) + (3*3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*2^(1/3)*(1 - x + x^3)^(1/3))])/(2*2^(2/3)) - ((3/2)^(2/3)*Log[-3*x + 2^(1/3)*3^(2/3)*(1 - x + x^3)^(1/3)])/2 + ((3/2)^(2/3)*Log[3*x^2 + 2^(1/3)*3^(2/3)*x*(1 - x + x^3)^(1/3) + 2^(2/3)*3^(1/3)*(1 - x + x^3)^(2/3)])/4","A",1
2294,1,175,175,0.6095196,"\int \frac{(-3+4 x) \left(-1+2 x+x^3\right)^{2/3}}{x^3 \left(2-4 x+x^3\right)} \, dx","IntegrateAlgebraic[((-3 + 4*x)*(-1 + 2*x + x^3)^(2/3))/(x^3*(2 - 4*x + x^3)),x]","\frac{3 \left(-1+2 x+x^3\right)^{2/3}}{4 x^2}-\frac{3 \sqrt[6]{3} \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2 \sqrt[3]{2} \sqrt[3]{-1+2 x+x^3}}\right)}{2\ 2^{2/3}}+\frac{1}{2} \left(\frac{3}{2}\right)^{2/3} \log \left(-3 x+\sqrt[3]{2} 3^{2/3} \sqrt[3]{-1+2 x+x^3}\right)-\frac{1}{4} \left(\frac{3}{2}\right)^{2/3} \log \left(3 x^2+\sqrt[3]{2} 3^{2/3} x \sqrt[3]{-1+2 x+x^3}+2^{2/3} \sqrt[3]{3} \left(-1+2 x+x^3\right)^{2/3}\right)","\frac{3 \left(-1+2 x+x^3\right)^{2/3}}{4 x^2}-\frac{3 \sqrt[6]{3} \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2 \sqrt[3]{2} \sqrt[3]{-1+2 x+x^3}}\right)}{2\ 2^{2/3}}+\frac{1}{2} \left(\frac{3}{2}\right)^{2/3} \log \left(-3 x+\sqrt[3]{2} 3^{2/3} \sqrt[3]{-1+2 x+x^3}\right)-\frac{1}{4} \left(\frac{3}{2}\right)^{2/3} \log \left(3 x^2+\sqrt[3]{2} 3^{2/3} x \sqrt[3]{-1+2 x+x^3}+2^{2/3} \sqrt[3]{3} \left(-1+2 x+x^3\right)^{2/3}\right)",1,"(3*(-1 + 2*x + x^3)^(2/3))/(4*x^2) - (3*3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*2^(1/3)*(-1 + 2*x + x^3)^(1/3))])/(2*2^(2/3)) + ((3/2)^(2/3)*Log[-3*x + 2^(1/3)*3^(2/3)*(-1 + 2*x + x^3)^(1/3)])/2 - ((3/2)^(2/3)*Log[3*x^2 + 2^(1/3)*3^(2/3)*x*(-1 + 2*x + x^3)^(1/3) + 2^(2/3)*3^(1/3)*(-1 + 2*x + x^3)^(2/3)])/4","A",1
2295,1,175,175,0.4540071,"\int \frac{\sqrt{-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6}}{-1+x} \, dx","IntegrateAlgebraic[Sqrt[-81 + 27*x + 135*x^2 - 150*x^3 + 65*x^4 - 13*x^5 + x^6]/(-1 + x),x]","\frac{\left(115-62 x+8 x^2\right) \sqrt{-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6}}{24 (-3+x)^2}-8 \tan ^{-1}\left(\frac{9-6 x+x^2}{-9+15 x-7 x^2+x^3-\sqrt{-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6}}\right)+\frac{77}{8} \log (-3+x)-\frac{77}{16} \log \left(9-24 x+13 x^2-2 x^3+2 \sqrt{-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6}\right)","\frac{\left(115-62 x+8 x^2\right) \sqrt{-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6}}{24 (-3+x)^2}-8 \tan ^{-1}\left(\frac{9-6 x+x^2}{-9+15 x-7 x^2+x^3-\sqrt{-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6}}\right)+\frac{77}{8} \log (-3+x)-\frac{77}{16} \log \left(9-24 x+13 x^2-2 x^3+2 \sqrt{-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6}\right)",1,"((115 - 62*x + 8*x^2)*Sqrt[-81 + 27*x + 135*x^2 - 150*x^3 + 65*x^4 - 13*x^5 + x^6])/(24*(-3 + x)^2) - 8*ArcTan[(9 - 6*x + x^2)/(-9 + 15*x - 7*x^2 + x^3 - Sqrt[-81 + 27*x + 135*x^2 - 150*x^3 + 65*x^4 - 13*x^5 + x^6])] + (77*Log[-3 + x])/8 - (77*Log[9 - 24*x + 13*x^2 - 2*x^3 + 2*Sqrt[-81 + 27*x + 135*x^2 - 150*x^3 + 65*x^4 - 13*x^5 + x^6]])/16","A",1
2296,1,175,175,0.9951104,"\int \frac{\left(-q+2 p x^3\right) \sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^2 \left(a q+b x+a p x^3\right)} \, dx","IntegrateAlgebraic[((-q + 2*p*x^3)*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6])/(x^2*(a*q + b*x + a*p*x^3)),x]","\frac{\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{a x}+\frac{2 \sqrt{-b^2+2 a^2 p q} \tan ^{-1}\left(\frac{\sqrt{-b^2+2 a^2 p q} x}{a q+b x+a p x^3+a \sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}}\right)}{a^2}+\frac{b \log (x)}{a^2}-\frac{b \log \left(q+p x^3+\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right)}{a^2}","\frac{\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{a x}+\frac{2 \sqrt{-b^2+2 a^2 p q} \tan ^{-1}\left(\frac{\sqrt{-b^2+2 a^2 p q} x}{a q+b x+a p x^3+a \sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}}\right)}{a^2}+\frac{b \log (x)}{a^2}-\frac{b \log \left(q+p x^3+\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right)}{a^2}",1,"Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]/(a*x) + (2*Sqrt[-b^2 + 2*a^2*p*q]*ArcTan[(Sqrt[-b^2 + 2*a^2*p*q]*x)/(a*q + b*x + a*p*x^3 + a*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6])])/a^2 + (b*Log[x])/a^2 - (b*Log[q + p*x^3 + Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]])/a^2","A",1
2297,1,175,175,27.0598805,"\int \frac{1}{x^2 \sqrt[3]{-1+5 x-7 x^2-2 x^3+10 x^4-2 x^5-5 x^6+x^7+x^8}} \, dx","IntegrateAlgebraic[1/(x^2*(-1 + 5*x - 7*x^2 - 2*x^3 + 10*x^4 - 2*x^5 - 5*x^6 + x^7 + x^8)^(1/3)),x]","\frac{(-1+x)^{2/3} \left(-1+x+x^2\right) \left(-\frac{\sqrt[3]{-1+x}}{x}+\frac{5 \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x}}{\sqrt{3}}\right)}{\sqrt{3}}-\frac{5}{3} \log \left(1+\sqrt[3]{-1+x}\right)+\frac{5}{6} \log \left(1-\sqrt[3]{-1+x}+(-1+x)^{2/3}\right)+\text{RootSum}\left[1+3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{3 \log \left(\sqrt[3]{-1+x}-\text{$\#$1}\right)+\log \left(\sqrt[3]{-1+x}-\text{$\#$1}\right) \text{$\#$1}^3}{3 \text{$\#$1}^2+2 \text{$\#$1}^5}\&\right]\right)}{\sqrt[3]{(-1+x)^2 \left(-1+x+x^2\right)^3}}","\frac{(-1+x)^{2/3} \left(-1+x+x^2\right) \left(-\frac{\sqrt[3]{-1+x}}{x}+\frac{5 \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x}}{\sqrt{3}}\right)}{\sqrt{3}}-\frac{5}{3} \log \left(1+\sqrt[3]{-1+x}\right)+\frac{5}{6} \log \left(1-\sqrt[3]{-1+x}+(-1+x)^{2/3}\right)+\text{RootSum}\left[1+3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{3 \log \left(\sqrt[3]{-1+x}-\text{$\#$1}\right)+\log \left(\sqrt[3]{-1+x}-\text{$\#$1}\right) \text{$\#$1}^3}{3 \text{$\#$1}^2+2 \text{$\#$1}^5}\&\right]\right)}{\sqrt[3]{(-1+x)^2 \left(-1+x+x^2\right)^3}}",1,"((-1 + x)^(2/3)*(-1 + x + x^2)*(-((-1 + x)^(1/3)/x) + (5*ArcTan[1/Sqrt[3] - (2*(-1 + x)^(1/3))/Sqrt[3]])/Sqrt[3] - (5*Log[1 + (-1 + x)^(1/3)])/3 + (5*Log[1 - (-1 + x)^(1/3) + (-1 + x)^(2/3)])/6 + RootSum[1 + 3*#1^3 + #1^6 & , (3*Log[(-1 + x)^(1/3) - #1] + Log[(-1 + x)^(1/3) - #1]*#1^3)/(3*#1^2 + 2*#1^5) & ]))/((-1 + x)^2*(-1 + x + x^2)^3)^(1/3)","A",1
2298,1,246,175,0.765284,"\int \frac{\left(1+x^2\right) \sqrt{x^2+\sqrt{1+x^4}}}{\left(-1+x^2\right) \sqrt{1+x^4}} \, dx","IntegrateAlgebraic[((1 + x^2)*Sqrt[x^2 + Sqrt[1 + x^4]])/((-1 + x^2)*Sqrt[1 + x^4]),x]","\sqrt{2 \left(-1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{-\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^4}}{\sqrt{2}}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)-\sqrt{2 \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{-\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)","\sqrt{2 \left(-1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(-1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-\sqrt{2 \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)",1,"Sqrt[2*(-1 + Sqrt[2])]*ArcTan[(-Sqrt[-1/2 + 1/Sqrt[2]] + Sqrt[-1/2 + 1/Sqrt[2]]*x^2 + Sqrt[-1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] + Sqrt[2]*ArcTanh[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] - Sqrt[2*(1 + Sqrt[2])]*ArcTanh[(-Sqrt[1/2 + 1/Sqrt[2]] + Sqrt[1/2 + 1/Sqrt[2]]*x^2 + Sqrt[1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])]","A",1
2299,1,176,176,0.3251209,"\int \frac{2+x}{(-3+x) \sqrt[4]{1-x^2} \left(1+x^2\right)} \, dx","IntegrateAlgebraic[(2 + x)/((-3 + x)*(1 - x^2)^(1/4)*(1 + x^2)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{1-x^2}}{\sqrt[4]{2}-\sqrt[4]{2} x+\sqrt[4]{1-x^2}}\right)}{2 \sqrt[4]{2}}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{1-x^2}}{-\sqrt[4]{2}+\sqrt[4]{2} x+\sqrt[4]{1-x^2}}\right)}{2 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\left(-2 \sqrt[4]{2}+2 \sqrt[4]{2} x\right) \sqrt[4]{1-x^2}}{\sqrt{2}-2 \sqrt{2} x+\sqrt{2} x^2+2 \sqrt{1-x^2}}\right)}{2 \sqrt[4]{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{1-x^2}}{\sqrt[4]{2}-\sqrt[4]{2} x+\sqrt[4]{1-x^2}}\right)}{2 \sqrt[4]{2}}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{1-x^2}}{-\sqrt[4]{2}+\sqrt[4]{2} x+\sqrt[4]{1-x^2}}\right)}{2 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\left(-2 \sqrt[4]{2}+2 \sqrt[4]{2} x\right) \sqrt[4]{1-x^2}}{\sqrt{2}-2 \sqrt{2} x+\sqrt{2} x^2+2 \sqrt{1-x^2}}\right)}{2 \sqrt[4]{2}}",1,"-1/2*ArcTan[(1 - x^2)^(1/4)/(2^(1/4) - 2^(1/4)*x + (1 - x^2)^(1/4))]/2^(1/4) + ArcTan[(1 - x^2)^(1/4)/(-2^(1/4) + 2^(1/4)*x + (1 - x^2)^(1/4))]/(2*2^(1/4)) - ArcTanh[((-2*2^(1/4) + 2*2^(1/4)*x)*(1 - x^2)^(1/4))/(Sqrt[2] - 2*Sqrt[2]*x + Sqrt[2]*x^2 + 2*Sqrt[1 - x^2])]/(2*2^(1/4))","A",1
2300,1,176,176,0.2428519,"\int \frac{\sqrt{b^2+a^2 x^3} \left(2 b^2+c x^3+a^2 x^6\right)}{x \left(-b^2+a^2 x^6\right)} \, dx","IntegrateAlgebraic[(Sqrt[b^2 + a^2*x^3]*(2*b^2 + c*x^3 + a^2*x^6))/(x*(-b^2 + a^2*x^6)),x]","\frac{2}{3} \sqrt{b^2+a^2 x^3}+\frac{\left(-3 a \sqrt{a-b} b+\sqrt{a-b} c\right) \tan ^{-1}\left(\frac{\sqrt{b^2+a^2 x^3}}{\sqrt{a-b} \sqrt{b}}\right)}{3 a \sqrt{b}}+\frac{4}{3} b \tanh ^{-1}\left(\frac{\sqrt{b^2+a^2 x^3}}{b}\right)+\frac{\left(-3 a b \sqrt{a+b}-\sqrt{a+b} c\right) \tanh ^{-1}\left(\frac{\sqrt{b^2+a^2 x^3}}{\sqrt{b} \sqrt{a+b}}\right)}{3 a \sqrt{b}}","\frac{2}{3} \sqrt{b^2+a^2 x^3}+\frac{4}{3} b \tanh ^{-1}\left(\frac{\sqrt{b^2+a^2 x^3}}{b}\right)+\frac{\left(-3 a b \sqrt{-a+b}+\sqrt{-a+b} c\right) \tanh ^{-1}\left(\frac{\sqrt{b^2+a^2 x^3}}{\sqrt{b} \sqrt{-a+b}}\right)}{3 a \sqrt{b}}+\frac{\left(-3 a b \sqrt{a+b}-\sqrt{a+b} c\right) \tanh ^{-1}\left(\frac{\sqrt{b^2+a^2 x^3}}{\sqrt{b} \sqrt{a+b}}\right)}{3 a \sqrt{b}}",1,"(2*Sqrt[b^2 + a^2*x^3])/3 + ((-3*a*Sqrt[a - b]*b + Sqrt[a - b]*c)*ArcTan[Sqrt[b^2 + a^2*x^3]/(Sqrt[a - b]*Sqrt[b])])/(3*a*Sqrt[b]) + (4*b*ArcTanh[Sqrt[b^2 + a^2*x^3]/b])/3 + ((-3*a*b*Sqrt[a + b] - Sqrt[a + b]*c)*ArcTanh[Sqrt[b^2 + a^2*x^3]/(Sqrt[b]*Sqrt[a + b])])/(3*a*Sqrt[b])","A",1
2301,1,176,176,0.6498244,"\int \frac{1-x^4+2 x^8}{\sqrt[4]{1+x^4} \left(-1-2 x^4+x^8\right)} \, dx","IntegrateAlgebraic[(1 - x^4 + 2*x^8)/((1 + x^4)^(1/4)*(-1 - 2*x^4 + x^8)),x]","\tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\frac{3 \tan ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{1+x^4}}\right)}{4 \sqrt[8]{2}}+\frac{3 \tan ^{-1}\left(\frac{2^{5/8} x \sqrt[4]{1+x^4}}{\sqrt[4]{2} x^2-\sqrt{1+x^4}}\right)}{4\ 2^{5/8}}+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\frac{3 \tanh ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{1+x^4}}\right)}{4 \sqrt[8]{2}}-\frac{3 \tanh ^{-1}\left(\frac{2\ 2^{3/8} x \sqrt[4]{1+x^4}}{2 x^2+2^{3/4} \sqrt{1+x^4}}\right)}{4\ 2^{5/8}}","\tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\frac{3 \tan ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{1+x^4}}\right)}{4 \sqrt[8]{2}}+\frac{3 \tan ^{-1}\left(\frac{2^{5/8} x \sqrt[4]{1+x^4}}{\sqrt[4]{2} x^2-\sqrt{1+x^4}}\right)}{4\ 2^{5/8}}+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\frac{3 \tanh ^{-1}\left(\frac{\sqrt[8]{2} x}{\sqrt[4]{1+x^4}}\right)}{4 \sqrt[8]{2}}-\frac{3 \tanh ^{-1}\left(\frac{2\ 2^{3/8} x \sqrt[4]{1+x^4}}{2 x^2+2^{3/4} \sqrt{1+x^4}}\right)}{4\ 2^{5/8}}",1,"ArcTan[x/(1 + x^4)^(1/4)] - (3*ArcTan[(2^(1/8)*x)/(1 + x^4)^(1/4)])/(4*2^(1/8)) + (3*ArcTan[(2^(5/8)*x*(1 + x^4)^(1/4))/(2^(1/4)*x^2 - Sqrt[1 + x^4])])/(4*2^(5/8)) + ArcTanh[x/(1 + x^4)^(1/4)] - (3*ArcTanh[(2^(1/8)*x)/(1 + x^4)^(1/4)])/(4*2^(1/8)) - (3*ArcTanh[(2*2^(3/8)*x*(1 + x^4)^(1/4))/(2*x^2 + 2^(3/4)*Sqrt[1 + x^4])])/(4*2^(5/8))","A",1
2302,1,247,176,0.9178631,"\int \frac{\left(-1+x^2\right) \sqrt{x^2+\sqrt{1+x^4}}}{\left(1+x^2\right) \sqrt{1+x^4}} \, dx","IntegrateAlgebraic[((-1 + x^2)*Sqrt[x^2 + Sqrt[1 + x^4]])/((1 + x^2)*Sqrt[1 + x^4]),x]","-\sqrt{2 \left(-1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{-\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^4}}{\sqrt{2}}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)-\sqrt{2 \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{-\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)","-\sqrt{2 \left(-1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-\sqrt{2 \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2 \left(-1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)",1,"-(Sqrt[2*(-1 + Sqrt[2])]*ArcTan[(-Sqrt[1/2 + 1/Sqrt[2]] + Sqrt[1/2 + 1/Sqrt[2]]*x^2 + Sqrt[1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])]) + Sqrt[2]*ArcTanh[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] - Sqrt[2*(1 + Sqrt[2])]*ArcTanh[(-Sqrt[-1/2 + 1/Sqrt[2]] + Sqrt[-1/2 + 1/Sqrt[2]]*x^2 + Sqrt[-1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])]","A",1
2303,1,183,176,0.1509684,"\int \frac{x}{x+\sqrt{c+\sqrt{b+a x}}} \, dx","IntegrateAlgebraic[x/(x + Sqrt[c + Sqrt[b + a*x]]),x]","\frac{b-c^2+a x}{a}-4 \sqrt{c+\sqrt{b+a x}}+4 \text{RootSum}\left[b-c^2-a \text{$\#$1}+2 c \text{$\#$1}^2-\text{$\#$1}^4\&,\frac{b \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right)-c^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right)-a \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}+c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2}{-a+4 c \text{$\#$1}-4 \text{$\#$1}^3}\&\right]","x-4 \sqrt{c+\sqrt{b+a x}}+\frac{4 \text{RootSum}\left[b-c^2-a \text{$\#$1}+2 c \text{$\#$1}^2-\text{$\#$1}^4\&,\frac{-a b \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right)+a c^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right)+a^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}-a c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2}{a-4 c \text{$\#$1}+4 \text{$\#$1}^3}\&\right]}{a}",1,"(b - c^2 + a*x)/a - 4*Sqrt[c + Sqrt[b + a*x]] + 4*RootSum[b - c^2 - a*#1 + 2*c*#1^2 - #1^4 & , (b*Log[Sqrt[c + Sqrt[b + a*x]] - #1] - c^2*Log[Sqrt[c + Sqrt[b + a*x]] - #1] - a*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1 + c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^2)/(-a + 4*c*#1 - 4*#1^3) & ]","A",1
2304,1,183,176,0.0014173,"\int \frac{x}{x+\sqrt{c+\sqrt{b+a x}}} \, dx","IntegrateAlgebraic[x/(x + Sqrt[c + Sqrt[b + a*x]]),x]","\frac{b-c^2+a x}{a}-4 \sqrt{c+\sqrt{b+a x}}+4 \text{RootSum}\left[b-c^2-a \text{$\#$1}+2 c \text{$\#$1}^2-\text{$\#$1}^4\&,\frac{b \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right)-c^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right)-a \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}+c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2}{-a+4 c \text{$\#$1}-4 \text{$\#$1}^3}\&\right]","x-4 \sqrt{c+\sqrt{b+a x}}+\frac{4 \text{RootSum}\left[b-c^2-a \text{$\#$1}+2 c \text{$\#$1}^2-\text{$\#$1}^4\&,\frac{-a b \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right)+a c^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right)+a^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}-a c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2}{a-4 c \text{$\#$1}+4 \text{$\#$1}^3}\&\right]}{a}",1,"(b - c^2 + a*x)/a - 4*Sqrt[c + Sqrt[b + a*x]] + 4*RootSum[b - c^2 - a*#1 + 2*c*#1^2 - #1^4 & , (b*Log[Sqrt[c + Sqrt[b + a*x]] - #1] - c^2*Log[Sqrt[c + Sqrt[b + a*x]] - #1] - a*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1 + c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^2)/(-a + 4*c*#1 - 4*#1^3) & ]","A",1
2305,1,177,177,0.2073058,"\int \frac{1}{(1+x) \sqrt[3]{1-x+x^2}} \, dx","IntegrateAlgebraic[1/((1 + x)*(1 - x + x^2)^(1/3)),x]","-\frac{\tan ^{-1}\left(\frac{\frac{4}{3 \sqrt[6]{3}}-\frac{2 x}{3 \sqrt[6]{3}}+\frac{\sqrt[3]{1-x+x^2}}{\sqrt{3}}}{\sqrt[3]{1-x+x^2}}\right)}{3^{5/6}}+\frac{\log \left(-2 \sqrt[3]{3}+\sqrt[3]{3} x+3 \sqrt[3]{1-x+x^2}\right)}{3 \sqrt[3]{3}}-\frac{\log \left(4\ 3^{2/3}-4\ 3^{2/3} x+3^{2/3} x^2+\left(6 \sqrt[3]{3}-3 \sqrt[3]{3} x\right) \sqrt[3]{1-x+x^2}+9 \left(1-x+x^2\right)^{2/3}\right)}{6 \sqrt[3]{3}}","-\frac{\tan ^{-1}\left(\frac{\frac{4}{3 \sqrt[6]{3}}-\frac{2 x}{3 \sqrt[6]{3}}+\frac{\sqrt[3]{1-x+x^2}}{\sqrt{3}}}{\sqrt[3]{1-x+x^2}}\right)}{3^{5/6}}+\frac{\log \left(-2 \sqrt[3]{3}+\sqrt[3]{3} x+3 \sqrt[3]{1-x+x^2}\right)}{3 \sqrt[3]{3}}-\frac{\log \left(4\ 3^{2/3}-4\ 3^{2/3} x+3^{2/3} x^2+\left(6 \sqrt[3]{3}-3 \sqrt[3]{3} x\right) \sqrt[3]{1-x+x^2}+9 \left(1-x+x^2\right)^{2/3}\right)}{6 \sqrt[3]{3}}",1,"-(ArcTan[(4/(3*3^(1/6)) - (2*x)/(3*3^(1/6)) + (1 - x + x^2)^(1/3)/Sqrt[3])/(1 - x + x^2)^(1/3)]/3^(5/6)) + Log[-2*3^(1/3) + 3^(1/3)*x + 3*(1 - x + x^2)^(1/3)]/(3*3^(1/3)) - Log[4*3^(2/3) - 4*3^(2/3)*x + 3^(2/3)*x^2 + (6*3^(1/3) - 3*3^(1/3)*x)*(1 - x + x^2)^(1/3) + 9*(1 - x + x^2)^(2/3)]/(6*3^(1/3))","A",1
2306,1,177,177,0.7360749,"\int \frac{\left(1+x^4\right) \sqrt[4]{-x^3+x^4}}{x^4 \left(-1+x^4\right)} \, dx","IntegrateAlgebraic[((1 + x^4)*(-x^3 + x^4)^(1/4))/(x^4*(-1 + x^4)),x]","-\frac{4 \left(-5+x+4 x^2\right) \sqrt[4]{-x^3+x^4}}{45 x^3}+\sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)-\sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)+\frac{1}{2} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 \log (x)+2 \log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+\text{$\#$1}^7}\&\right]","-\frac{4 \left(-5+x+4 x^2\right) \sqrt[4]{-x^3+x^4}}{45 x^3}+\sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)-\sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)+\frac{1}{2} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 \log (x)+2 \log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+\text{$\#$1}^7}\&\right]",1,"(-4*(-5 + x + 4*x^2)*(-x^3 + x^4)^(1/4))/(45*x^3) + 2^(1/4)*ArcTan[(2^(1/4)*x)/(-x^3 + x^4)^(1/4)] - 2^(1/4)*ArcTanh[(2^(1/4)*x)/(-x^3 + x^4)^(1/4)] + RootSum[2 - 2*#1^4 + #1^8 & , (-2*Log[x] + 2*Log[(-x^3 + x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-x^3 + x^4)^(1/4) - x*#1]*#1^4)/(-#1^3 + #1^7) & ]/2","A",1
2307,1,177,177,0.0011486,"\int \frac{\left(1+x^4\right) \sqrt[4]{-x^3+x^4}}{x^4 \left(-1+x^4\right)} \, dx","IntegrateAlgebraic[((1 + x^4)*(-x^3 + x^4)^(1/4))/(x^4*(-1 + x^4)),x]","-\frac{4 \left(-5+x+4 x^2\right) \sqrt[4]{-x^3+x^4}}{45 x^3}+\sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)-\sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)+\frac{1}{2} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 \log (x)+2 \log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+\text{$\#$1}^7}\&\right]","-\frac{4 \left(-5+x+4 x^2\right) \sqrt[4]{-x^3+x^4}}{45 x^3}+\sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)-\sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)+\frac{1}{2} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 \log (x)+2 \log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+\text{$\#$1}^7}\&\right]",1,"(-4*(-5 + x + 4*x^2)*(-x^3 + x^4)^(1/4))/(45*x^3) + 2^(1/4)*ArcTan[(2^(1/4)*x)/(-x^3 + x^4)^(1/4)] - 2^(1/4)*ArcTanh[(2^(1/4)*x)/(-x^3 + x^4)^(1/4)] + RootSum[2 - 2*#1^4 + #1^8 & , (-2*Log[x] + 2*Log[(-x^3 + x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-x^3 + x^4)^(1/4) - x*#1]*#1^4)/(-#1^3 + #1^7) & ]/2","A",1
2308,1,177,177,0.567594,"\int \frac{\sqrt[3]{b x+a x^3} \left(b+a x^4\right)}{x^4} \, dx","IntegrateAlgebraic[((b*x + a*x^3)^(1/3)*(b + a*x^4))/x^4,x]","\frac{\sqrt[3]{b x+a x^3} \left(-3 b-3 a x^2+4 a x^4\right)}{8 x^3}-\frac{\sqrt[3]{a} b \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a} x}{\sqrt[3]{a} x+2 \sqrt[3]{b x+a x^3}}\right)}{2 \sqrt{3}}-\frac{1}{6} \sqrt[3]{a} b \log \left(-\sqrt[3]{a} x+\sqrt[3]{b x+a x^3}\right)+\frac{1}{12} \sqrt[3]{a} b \log \left(a^{2/3} x^2+\sqrt[3]{a} x \sqrt[3]{b x+a x^3}+\left(b x+a x^3\right)^{2/3}\right)","\frac{\sqrt[3]{b x+a x^3} \left(-3 b-3 a x^2+4 a x^4\right)}{8 x^3}-\frac{\sqrt[3]{a} b \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a} x}{\sqrt[3]{a} x+2 \sqrt[3]{b x+a x^3}}\right)}{2 \sqrt{3}}-\frac{1}{6} \sqrt[3]{a} b \log \left(-\sqrt[3]{a} x+\sqrt[3]{b x+a x^3}\right)+\frac{1}{12} \sqrt[3]{a} b \log \left(a^{2/3} x^2+\sqrt[3]{a} x \sqrt[3]{b x+a x^3}+\left(b x+a x^3\right)^{2/3}\right)",1,"((b*x + a*x^3)^(1/3)*(-3*b - 3*a*x^2 + 4*a*x^4))/(8*x^3) - (a^(1/3)*b*ArcTan[(Sqrt[3]*a^(1/3)*x)/(a^(1/3)*x + 2*(b*x + a*x^3)^(1/3))])/(2*Sqrt[3]) - (a^(1/3)*b*Log[-(a^(1/3)*x) + (b*x + a*x^3)^(1/3)])/6 + (a^(1/3)*b*Log[a^(2/3)*x^2 + a^(1/3)*x*(b*x + a*x^3)^(1/3) + (b*x + a*x^3)^(2/3)])/12","A",1
2309,1,177,177,0.5000104,"\int \frac{b^4+a^4 x^4}{\sqrt{-b^2 x+a^2 x^3} \left(-b^4+a^4 x^4\right)} \, dx","IntegrateAlgebraic[(b^4 + a^4*x^4)/(Sqrt[-(b^2*x) + a^2*x^3]*(-b^4 + a^4*x^4)),x]","\frac{\sqrt{-b^2 x+a^2 x^3}}{b^2-a^2 x^2}-\frac{\tan ^{-1}\left(\frac{2 \sqrt{a} \sqrt{b} \sqrt{-b^2 x+a^2 x^3}}{-b^2-2 a b x+a^2 x^2}\right)}{4 \sqrt{a} \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{-\frac{b^{3/2}}{2 \sqrt{a}}+\sqrt{a} \sqrt{b} x+\frac{a^{3/2} x^2}{2 \sqrt{b}}}{\sqrt{-b^2 x+a^2 x^3}}\right)}{4 \sqrt{a} \sqrt{b}}","\frac{\sqrt{-b^2 x+a^2 x^3}}{b^2-a^2 x^2}-\frac{\tan ^{-1}\left(\frac{2 \sqrt{a} \sqrt{b} \sqrt{-b^2 x+a^2 x^3}}{-b^2-2 a b x+a^2 x^2}\right)}{4 \sqrt{a} \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{-\frac{b^{3/2}}{2 \sqrt{a}}+\sqrt{a} \sqrt{b} x+\frac{a^{3/2} x^2}{2 \sqrt{b}}}{\sqrt{-b^2 x+a^2 x^3}}\right)}{4 \sqrt{a} \sqrt{b}}",1,"Sqrt[-(b^2*x) + a^2*x^3]/(b^2 - a^2*x^2) - ArcTan[(2*Sqrt[a]*Sqrt[b]*Sqrt[-(b^2*x) + a^2*x^3])/(-b^2 - 2*a*b*x + a^2*x^2)]/(4*Sqrt[a]*Sqrt[b]) - ArcTanh[(-1/2*b^(3/2)/Sqrt[a] + Sqrt[a]*Sqrt[b]*x + (a^(3/2)*x^2)/(2*Sqrt[b]))/Sqrt[-(b^2*x) + a^2*x^3]]/(4*Sqrt[a]*Sqrt[b])","A",1
2310,1,177,177,19.506743,"\int \frac{1+x}{\left(-1+x^3\right) \sqrt[8]{256-256 x^2+96 x^4-16 x^6+x^8}} \, dx","IntegrateAlgebraic[(1 + x)/((-1 + x^3)*(256 - 256*x^2 + 96*x^4 - 16*x^6 + x^8)^(1/8)),x]","\frac{\left(\left(-4+x^2\right)^4\right)^{7/8} \left(\frac{4 \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{x}{\sqrt{3}}+\frac{\sqrt{-4+x^2}}{\sqrt{3}}\right)}{3 \sqrt{3}}-\frac{2}{3} \text{RootSum}\left[16-8 \text{$\#$1}+12 \text{$\#$1}^2-2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{4 \log \left(-x+\sqrt{-4+x^2}-\text{$\#$1}\right)-\log \left(-x+\sqrt{-4+x^2}-\text{$\#$1}\right) \text{$\#$1}+\log \left(-x+\sqrt{-4+x^2}-\text{$\#$1}\right) \text{$\#$1}^2}{-4+12 \text{$\#$1}-3 \text{$\#$1}^2+2 \text{$\#$1}^3}\&\right]\right)}{\left(-4+x^2\right)^{7/2}}","\frac{\left(\left(-4+x^2\right)^4\right)^{7/8} \left(\frac{4 \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{x}{\sqrt{3}}+\frac{\sqrt{-4+x^2}}{\sqrt{3}}\right)}{3 \sqrt{3}}-\frac{2}{3} \text{RootSum}\left[16-8 \text{$\#$1}+12 \text{$\#$1}^2-2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{4 \log \left(-x+\sqrt{-4+x^2}-\text{$\#$1}\right)-\log \left(-x+\sqrt{-4+x^2}-\text{$\#$1}\right) \text{$\#$1}+\log \left(-x+\sqrt{-4+x^2}-\text{$\#$1}\right) \text{$\#$1}^2}{-4+12 \text{$\#$1}-3 \text{$\#$1}^2+2 \text{$\#$1}^3}\&\right]\right)}{\left(-4+x^2\right)^{7/2}}",1,"(((-4 + x^2)^4)^(7/8)*((4*ArcTan[1/Sqrt[3] - x/Sqrt[3] + Sqrt[-4 + x^2]/Sqrt[3]])/(3*Sqrt[3]) - (2*RootSum[16 - 8*#1 + 12*#1^2 - 2*#1^3 + #1^4 & , (4*Log[-x + Sqrt[-4 + x^2] - #1] - Log[-x + Sqrt[-4 + x^2] - #1]*#1 + Log[-x + Sqrt[-4 + x^2] - #1]*#1^2)/(-4 + 12*#1 - 3*#1^2 + 2*#1^3) & ])/3))/(-4 + x^2)^(7/2)","A",1
2311,1,177,177,27.2113366,"\int \frac{1}{x^2 \sqrt[3]{-1-x+5 x^2+2 x^3-10 x^4+2 x^5+7 x^6-5 x^7+x^8}} \, dx","IntegrateAlgebraic[1/(x^2*(-1 - x + 5*x^2 + 2*x^3 - 10*x^4 + 2*x^5 + 7*x^6 - 5*x^7 + x^8)^(1/3)),x]","\frac{(-1+x)^{2/3} \left(-1-x+x^2\right) \left(-\frac{\sqrt[3]{-1+x}}{x}-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x}}{\sqrt{3}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(1+\sqrt[3]{-1+x}\right)-\frac{1}{6} \log \left(1-\sqrt[3]{-1+x}+(-1+x)^{2/3}\right)-\text{RootSum}\left[-1+\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log \left(\sqrt[3]{-1+x}-\text{$\#$1}\right)+\log \left(\sqrt[3]{-1+x}-\text{$\#$1}\right) \text{$\#$1}^3}{\text{$\#$1}^2+2 \text{$\#$1}^5}\&\right]\right)}{\sqrt[3]{(-1+x)^2 \left(-1-x+x^2\right)^3}}","\frac{(-1+x)^{2/3} \left(-1-x+x^2\right) \left(-\frac{\sqrt[3]{-1+x}}{x}-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x}}{\sqrt{3}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(1+\sqrt[3]{-1+x}\right)-\frac{1}{6} \log \left(1-\sqrt[3]{-1+x}+(-1+x)^{2/3}\right)-\text{RootSum}\left[-1+\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log \left(\sqrt[3]{-1+x}-\text{$\#$1}\right)+\log \left(\sqrt[3]{-1+x}-\text{$\#$1}\right) \text{$\#$1}^3}{\text{$\#$1}^2+2 \text{$\#$1}^5}\&\right]\right)}{\sqrt[3]{(-1+x)^2 \left(-1-x+x^2\right)^3}}",1,"((-1 + x)^(2/3)*(-1 - x + x^2)*(-((-1 + x)^(1/3)/x) - ArcTan[1/Sqrt[3] - (2*(-1 + x)^(1/3))/Sqrt[3]]/Sqrt[3] + Log[1 + (-1 + x)^(1/3)]/3 - Log[1 - (-1 + x)^(1/3) + (-1 + x)^(2/3)]/6 - RootSum[-1 + #1^3 + #1^6 & , (-Log[(-1 + x)^(1/3) - #1] + Log[(-1 + x)^(1/3) - #1]*#1^3)/(#1^2 + 2*#1^5) & ]))/((-1 + x)^2*(-1 - x + x^2)^3)^(1/3)","A",1
2312,1,176,177,14.2654863,"\int \frac{c x^6 \left(-4 b+a x^5\right)}{\left(b+a x^5\right)^{3/4} \left(b^2+2 a b x^5-c^2 x^8+a^2 x^{10}\right)} \, dx","IntegrateAlgebraic[(c*x^6*(-4*b + a*x^5))/((b + a*x^5)^(3/4)*(b^2 + 2*a*b*x^5 - c^2*x^8 + a^2*x^10)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{b+a x^5}}\right)}{c^{3/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x \sqrt[4]{b+a x^5}}{-\sqrt{c} x^2+\sqrt{b+a x^5}}\right)}{\sqrt{2} c^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{b+a x^5}}\right)}{c^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[4]{c} x^2}{\sqrt{2}}+\frac{\sqrt{b+a x^5}}{\sqrt{2} \sqrt[4]{c}}}{x \sqrt[4]{b+a x^5}}\right)}{\sqrt{2} c^{3/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{b+a x^5}}\right)}{c^{3/4}}-\frac{\tan ^{-1}\left(\frac{-\frac{\sqrt[4]{c} x^2}{\sqrt{2}}+\frac{\sqrt{b+a x^5}}{\sqrt{2} \sqrt[4]{c}}}{x \sqrt[4]{b+a x^5}}\right)}{\sqrt{2} c^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{b+a x^5}}\right)}{c^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x \sqrt[4]{b+a x^5}}{\sqrt{c} x^2+\sqrt{b+a x^5}}\right)}{\sqrt{2} c^{3/4}}",1,"ArcTan[(c^(1/4)*x)/(b + a*x^5)^(1/4)]/c^(3/4) + ArcTan[(Sqrt[2]*c^(1/4)*x*(b + a*x^5)^(1/4))/(-(Sqrt[c]*x^2) + Sqrt[b + a*x^5])]/(Sqrt[2]*c^(3/4)) - ArcTanh[(c^(1/4)*x)/(b + a*x^5)^(1/4)]/c^(3/4) - ArcTanh[((c^(1/4)*x^2)/Sqrt[2] + Sqrt[b + a*x^5]/(Sqrt[2]*c^(1/4)))/(x*(b + a*x^5)^(1/4))]/(Sqrt[2]*c^(3/4))","A",1
2313,1,209,177,0.5333768,"\int \frac{x^2 \sqrt{a x+\sqrt{-b+a x}}}{\sqrt{-b+a x}} \, dx","IntegrateAlgebraic[(x^2*Sqrt[a*x + Sqrt[-b + a*x]])/Sqrt[-b + a*x],x]","\frac{\sqrt{a x+\sqrt{-b+a x}} \left(315-480 b-432 b^2-210 \sqrt{-b+a x}+96 b \sqrt{-b+a x}+5280 b^2 \sqrt{-b+a x}+168 (-b+a x)+96 b (-b+a x)-144 (-b+a x)^{3/2}+4160 b (-b+a x)^{3/2}+128 (-b+a x)^2+1280 (-b+a x)^{5/2}\right)}{3840 a^3}+\frac{\left(21-60 b-16 b^2-320 b^3\right) \log \left(-1-2 \sqrt{-b+a x}+2 \sqrt{a x+\sqrt{-b+a x}}\right)}{512 a^3}","\frac{\left(315-648 b-400 b^2+168 a x-160 a b x+128 a^2 x^2\right) \sqrt{a x+\sqrt{-b+a x}}}{3840 a^3}+\frac{\sqrt{-b+a x} \left(-105+120 b+1200 b^2-72 a x+800 a b x+640 a^2 x^2\right) \sqrt{a x+\sqrt{-b+a x}}}{1920 a^3}+\frac{\left(21-60 b-16 b^2-320 b^3\right) \log \left(1+2 \sqrt{-b+a x}-2 \sqrt{a x+\sqrt{-b+a x}}\right)}{512 a^3}",1,"(Sqrt[a*x + Sqrt[-b + a*x]]*(315 - 480*b - 432*b^2 - 210*Sqrt[-b + a*x] + 96*b*Sqrt[-b + a*x] + 5280*b^2*Sqrt[-b + a*x] + 168*(-b + a*x) + 96*b*(-b + a*x) - 144*(-b + a*x)^(3/2) + 4160*b*(-b + a*x)^(3/2) + 128*(-b + a*x)^2 + 1280*(-b + a*x)^(5/2)))/(3840*a^3) + ((21 - 60*b - 16*b^2 - 320*b^3)*Log[-1 - 2*Sqrt[-b + a*x] + 2*Sqrt[a*x + Sqrt[-b + a*x]]])/(512*a^3)","A",1
2314,1,178,178,0.4100169,"\int \frac{a+b x^2+a k^4 x^4}{\sqrt{(1-x) x \left(1-k^2 x\right)} \left(-1+k^4 x^4\right)} \, dx","IntegrateAlgebraic[(a + b*x^2 + a*k^4*x^4)/(Sqrt[(1 - x)*x*(1 - k^2*x)]*(-1 + k^4*x^4)),x]","\frac{\left(-b-2 a k^2\right) \tan ^{-1}\left(\frac{(-1+k) x}{\sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}\right)}{4 (-1+k) k^2}+\frac{\left(-b-2 a k^2\right) \tan ^{-1}\left(\frac{(1+k) x}{\sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}\right)}{4 k^2 (1+k)}+\frac{\left(b-2 a k^2\right) \tan ^{-1}\left(\frac{\sqrt{1+k^2} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{(-1+x) \left(-1+k^2 x\right)}\right)}{2 k^2 \sqrt{1+k^2}}","\frac{\left(-b-2 a k^2\right) \tan ^{-1}\left(\frac{(-1+k) x}{\sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}\right)}{4 (-1+k) k^2}+\frac{\left(-b-2 a k^2\right) \tan ^{-1}\left(\frac{(1+k) x}{\sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}\right)}{4 k^2 (1+k)}+\frac{\left(b-2 a k^2\right) \tan ^{-1}\left(\frac{\sqrt{1+k^2} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{(-1+x) \left(-1+k^2 x\right)}\right)}{2 k^2 \sqrt{1+k^2}}",1,"((-b - 2*a*k^2)*ArcTan[((-1 + k)*x)/Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3]])/(4*(-1 + k)*k^2) + ((-b - 2*a*k^2)*ArcTan[((1 + k)*x)/Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3]])/(4*k^2*(1 + k)) + ((b - 2*a*k^2)*ArcTan[(Sqrt[1 + k^2]*Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3])/((-1 + x)*(-1 + k^2*x))])/(2*k^2*Sqrt[1 + k^2])","A",1
2315,1,178,178,0.400109,"\int \frac{\left(-2 q+p x^3\right) \sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left(a q^2+2 a p q x^3+b x^4+a p^2 x^6\right)}{x^9} \, dx","IntegrateAlgebraic[((-2*q + p*x^3)*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]*(a*q^2 + 2*a*p*q*x^3 + b*x^4 + a*p^2*x^6))/x^9,x]","\frac{\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left(a q^3+3 a p q^2 x^3+2 b q x^4-a p q^2 x^4+3 a p^2 q x^6+2 b p x^7-a p^2 q x^7+a p^3 x^9\right)}{4 x^8}+\left(2 b p q+a p^2 q^2\right) \log (x)+\frac{1}{2} \left(-2 b p q-a p^2 q^2\right) \log \left(q+p x^3+\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}\right)","\frac{\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left(a q^3+3 a p q^2 x^3+2 b q x^4-a p q^2 x^4+3 a p^2 q x^6+2 b p x^7-a p^2 q x^7+a p^3 x^9\right)}{4 x^8}+\left(2 b p q+a p^2 q^2\right) \log (x)+\frac{1}{2} \left(-2 b p q-a p^2 q^2\right) \log \left(q+p x^3+\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}\right)",1,"(Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]*(a*q^3 + 3*a*p*q^2*x^3 + 2*b*q*x^4 - a*p*q^2*x^4 + 3*a*p^2*q*x^6 + 2*b*p*x^7 - a*p^2*q*x^7 + a*p^3*x^9))/(4*x^8) + (2*b*p*q + a*p^2*q^2)*Log[x] + ((-2*b*p*q - a*p^2*q^2)*Log[q + p*x^3 + Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]])/2","A",1
2316,1,179,179,1.282497,"\int \frac{-1+x^3}{\left(1+x^3\right) \sqrt{a+b x+c x^2+b x^3+a x^4}} \, dx","IntegrateAlgebraic[(-1 + x^3)/((1 + x^3)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]),x]","\frac{4 \tan ^{-1}\left(\frac{\sqrt{a-b-c} x}{\sqrt{a}-\sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{3 \sqrt{a-b-c}}-\frac{2 \sqrt{-2 a+2 b-c} \tan ^{-1}\left(\frac{\sqrt{-2 a+2 b-c} x}{\sqrt{a}+2 \sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{3 (2 a-2 b+c)}","\frac{4 \tan ^{-1}\left(\frac{\sqrt{a-b-c} x}{\sqrt{a}-\sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{3 \sqrt{a-b-c}}-\frac{2 \sqrt{-2 a+2 b-c} \tan ^{-1}\left(\frac{\sqrt{-2 a+2 b-c} x}{\sqrt{a}+2 \sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{3 (2 a-2 b+c)}",1,"(4*ArcTan[(Sqrt[a - b - c]*x)/(Sqrt[a] - Sqrt[a]*x + Sqrt[a]*x^2 - Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])])/(3*Sqrt[a - b - c]) - (2*Sqrt[-2*a + 2*b - c]*ArcTan[(Sqrt[-2*a + 2*b - c]*x)/(Sqrt[a] + 2*Sqrt[a]*x + Sqrt[a]*x^2 - Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])])/(3*(2*a - 2*b + c))","A",1
2317,1,179,179,1.1344865,"\int \frac{5 x-4 (1+k) x^2+3 k x^3}{\sqrt[3]{(1-x) x (1-k x)} \left(-1+(1+k) x-k x^2+b x^5\right)} \, dx","IntegrateAlgebraic[(5*x - 4*(1 + k)*x^2 + 3*k*x^3)/(((1 - x)*x*(1 - k*x))^(1/3)*(-1 + (1 + k)*x - k*x^2 + b*x^5)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{x+(-1-k) x^2+k x^3}}{2 \sqrt[3]{b} x^2+\sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{\sqrt[3]{b}}+\frac{\log \left(-\sqrt[3]{b} x^2+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{\sqrt[3]{b}}-\frac{\log \left(b^{2/3} x^4+\sqrt[3]{b} x^2 \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 \sqrt[3]{b}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{x+(-1-k) x^2+k x^3}}{2 \sqrt[3]{b} x^2+\sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{\sqrt[3]{b}}+\frac{\log \left(-\sqrt[3]{b} x^2+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{\sqrt[3]{b}}-\frac{\log \left(b^{2/3} x^4+\sqrt[3]{b} x^2 \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 \sqrt[3]{b}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(x + (-1 - k)*x^2 + k*x^3)^(1/3))/(2*b^(1/3)*x^2 + (x + (-1 - k)*x^2 + k*x^3)^(1/3))])/b^(1/3) + Log[-(b^(1/3)*x^2) + (x + (-1 - k)*x^2 + k*x^3)^(1/3)]/b^(1/3) - Log[b^(2/3)*x^4 + b^(1/3)*x^2*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + (x + (-1 - k)*x^2 + k*x^3)^(2/3)]/(2*b^(1/3))","A",1
2318,1,179,179,2.9439301,"\int \frac{x^2 \left(8-7 (1+k) x+6 k x^2\right)}{\sqrt[3]{(1-x) x (1-k x)} \left(-1+(1+k) x-k x^2+b x^8\right)} \, dx","IntegrateAlgebraic[(x^2*(8 - 7*(1 + k)*x + 6*k*x^2))/(((1 - x)*x*(1 - k*x))^(1/3)*(-1 + (1 + k)*x - k*x^2 + b*x^8)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{x+(-1-k) x^2+k x^3}}{2 \sqrt[3]{b} x^3+\sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{\sqrt[3]{b}}+\frac{\log \left(-\sqrt[3]{b} x^3+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{\sqrt[3]{b}}-\frac{\log \left(b^{2/3} x^6+\sqrt[3]{b} x^3 \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 \sqrt[3]{b}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{x+(-1-k) x^2+k x^3}}{2 \sqrt[3]{b} x^3+\sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{\sqrt[3]{b}}+\frac{\log \left(-\sqrt[3]{b} x^3+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{\sqrt[3]{b}}-\frac{\log \left(b^{2/3} x^6+\sqrt[3]{b} x^3 \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 \sqrt[3]{b}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(x + (-1 - k)*x^2 + k*x^3)^(1/3))/(2*b^(1/3)*x^3 + (x + (-1 - k)*x^2 + k*x^3)^(1/3))])/b^(1/3) + Log[-(b^(1/3)*x^3) + (x + (-1 - k)*x^2 + k*x^3)^(1/3)]/b^(1/3) - Log[b^(2/3)*x^6 + b^(1/3)*x^3*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + (x + (-1 - k)*x^2 + k*x^3)^(2/3)]/(2*b^(1/3))","A",1
2319,1,180,180,0.5035605,"\int \frac{x \left(-a b+x^2\right)}{\left(x^2 (-a+x) (-b+x)\right)^{2/3} \left(a b-(a+b+d) x+x^2\right)} \, dx","IntegrateAlgebraic[(x*(-(a*b) + x^2))/((x^2*(-a + x)*(-b + x))^(2/3)*(a*b - (a + b + d)*x + x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} x}{\sqrt[3]{d} x+2 \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{d^{2/3}}+\frac{\log \left(-\sqrt[3]{d} x+\sqrt[3]{a b x^2+(-a-b) x^3+x^4}\right)}{d^{2/3}}-\frac{\log \left(d^{2/3} x^2+\sqrt[3]{d} x \sqrt[3]{a b x^2+(-a-b) x^3+x^4}+\left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}\right)}{2 d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} x}{\sqrt[3]{d} x+2 \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{d^{2/3}}+\frac{\log \left(-\sqrt[3]{d} x+\sqrt[3]{a b x^2+(-a-b) x^3+x^4}\right)}{d^{2/3}}-\frac{\log \left(d^{2/3} x^2+\sqrt[3]{d} x \sqrt[3]{a b x^2+(-a-b) x^3+x^4}+\left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}\right)}{2 d^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*x)/(d^(1/3)*x + 2*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))])/d^(2/3) + Log[-(d^(1/3)*x) + (a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3)]/d^(2/3) - Log[d^(2/3)*x^2 + d^(1/3)*x*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3) + (a*b*x^2 + (-a - b)*x^3 + x^4)^(2/3)]/(2*d^(2/3))","A",1
2320,1,180,180,0.7056168,"\int \frac{\left(b+x^3\right) \left(c+x^3\right)}{\sqrt[3]{a+x^3}} \, dx","IntegrateAlgebraic[((b + x^3)*(c + x^3))/(a + x^3)^(1/3),x]","\frac{1}{18} \left(a+x^3\right)^{2/3} \left(-4 a x+6 b x+6 c x+3 x^4\right)+\frac{1}{27} \left(2 \sqrt{3} a^2-3 \sqrt{3} a b-3 \sqrt{3} a c+9 \sqrt{3} b c\right) \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{a+x^3}}\right)+\frac{1}{27} \left(-2 a^2+3 a b+3 a c-9 b c\right) \log \left(-x+\sqrt[3]{a+x^3}\right)+\frac{1}{54} \left(2 a^2-3 a b-3 a c+9 b c\right) \log \left(x^2+x \sqrt[3]{a+x^3}+\left(a+x^3\right)^{2/3}\right)","\frac{1}{18} \left(a+x^3\right)^{2/3} \left(-4 a x+6 b x+6 c x+3 x^4\right)+\frac{1}{27} \left(2 \sqrt{3} a^2-3 \sqrt{3} a b-3 \sqrt{3} a c+9 \sqrt{3} b c\right) \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{a+x^3}}\right)+\frac{1}{27} \left(-2 a^2+3 a b+3 a c-9 b c\right) \log \left(-x+\sqrt[3]{a+x^3}\right)+\frac{1}{54} \left(2 a^2-3 a b-3 a c+9 b c\right) \log \left(x^2+x \sqrt[3]{a+x^3}+\left(a+x^3\right)^{2/3}\right)",1,"((a + x^3)^(2/3)*(-4*a*x + 6*b*x + 6*c*x + 3*x^4))/18 + ((2*Sqrt[3]*a^2 - 3*Sqrt[3]*a*b - 3*Sqrt[3]*a*c + 9*Sqrt[3]*b*c)*ArcTan[(Sqrt[3]*x)/(x + 2*(a + x^3)^(1/3))])/27 + ((-2*a^2 + 3*a*b + 3*a*c - 9*b*c)*Log[-x + (a + x^3)^(1/3)])/27 + ((2*a^2 - 3*a*b - 3*a*c + 9*b*c)*Log[x^2 + x*(a + x^3)^(1/3) + (a + x^3)^(2/3)])/54","A",1
2321,1,180,180,0.4812172,"\int \frac{1}{\left(1+x^3\right) \sqrt[3]{-x^2+x^3}} \, dx","IntegrateAlgebraic[1/((1 + x^3)*(-x^2 + x^3)^(1/3)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-x^2+x^3}}\right)}{\sqrt[3]{2} \sqrt{3}}-\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{-x^2+x^3}\right)}{3 \sqrt[3]{2}}+\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-x^2+x^3}+\sqrt[3]{2} \left(-x^2+x^3\right)^{2/3}\right)}{6 \sqrt[3]{2}}-\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-x^2+x^3}}\right)}{\sqrt[3]{2} \sqrt{3}}-\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{-x^2+x^3}\right)}{3 \sqrt[3]{2}}+\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-x^2+x^3}+\sqrt[3]{2} \left(-x^2+x^3\right)^{2/3}\right)}{6 \sqrt[3]{2}}-\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-x^2 + x^3)^(1/3))]/(2^(1/3)*Sqrt[3]) - Log[-2*x + 2^(2/3)*(-x^2 + x^3)^(1/3)]/(3*2^(1/3)) + Log[2*x^2 + 2^(2/3)*x*(-x^2 + x^3)^(1/3) + 2^(1/3)*(-x^2 + x^3)^(2/3)]/(6*2^(1/3)) - RootSum[1 - #1^3 + #1^6 & , (-Log[x] + Log[(-x^2 + x^3)^(1/3) - x*#1])/#1 & ]/3","A",1
2322,1,180,180,0.0009308,"\int \frac{1}{\left(1+x^3\right) \sqrt[3]{-x^2+x^3}} \, dx","IntegrateAlgebraic[1/((1 + x^3)*(-x^2 + x^3)^(1/3)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-x^2+x^3}}\right)}{\sqrt[3]{2} \sqrt{3}}-\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{-x^2+x^3}\right)}{3 \sqrt[3]{2}}+\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-x^2+x^3}+\sqrt[3]{2} \left(-x^2+x^3\right)^{2/3}\right)}{6 \sqrt[3]{2}}-\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-x^2+x^3}}\right)}{\sqrt[3]{2} \sqrt{3}}-\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{-x^2+x^3}\right)}{3 \sqrt[3]{2}}+\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-x^2+x^3}+\sqrt[3]{2} \left(-x^2+x^3\right)^{2/3}\right)}{6 \sqrt[3]{2}}-\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-x^2 + x^3)^(1/3))]/(2^(1/3)*Sqrt[3]) - Log[-2*x + 2^(2/3)*(-x^2 + x^3)^(1/3)]/(3*2^(1/3)) + Log[2*x^2 + 2^(2/3)*x*(-x^2 + x^3)^(1/3) + 2^(1/3)*(-x^2 + x^3)^(2/3)]/(6*2^(1/3)) - RootSum[1 - #1^3 + #1^6 & , (-Log[x] + Log[(-x^2 + x^3)^(1/3) - x*#1])/#1 & ]/3","A",1
2323,1,180,180,0.9910898,"\int \frac{\left(-2 q+p x^3\right) \sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^3 \left(a q+b x^2+a p x^3\right)} \, dx","IntegrateAlgebraic[((-2*q + p*x^3)*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6])/(x^3*(a*q + b*x^2 + a*p*x^3)),x]","\frac{\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{a x^2}+\frac{2 \sqrt{-b^2+2 a^2 p q} \tan ^{-1}\left(\frac{\sqrt{-b^2+2 a^2 p q} x^2}{a q+b x^2+a p x^3+a \sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}}\right)}{a^2}+\frac{2 b \log (x)}{a^2}-\frac{b \log \left(q+p x^3+\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}\right)}{a^2}","\frac{\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{a x^2}+\frac{2 \sqrt{-b^2+2 a^2 p q} \tan ^{-1}\left(\frac{\sqrt{-b^2+2 a^2 p q} x^2}{a q+b x^2+a p x^3+a \sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}}\right)}{a^2}+\frac{2 b \log (x)}{a^2}-\frac{b \log \left(q+p x^3+\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}\right)}{a^2}",1,"Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]/(a*x^2) + (2*Sqrt[-b^2 + 2*a^2*p*q]*ArcTan[(Sqrt[-b^2 + 2*a^2*p*q]*x^2)/(a*q + b*x^2 + a*p*x^3 + a*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6])])/a^2 + (2*b*Log[x])/a^2 - (b*Log[q + p*x^3 + Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]])/a^2","A",1
2324,1,180,180,0.7334335,"\int \frac{\sqrt[4]{x^2+x^6} \left(1+x^8\right)}{x^4 \left(-1+x^4\right)} \, dx","IntegrateAlgebraic[((x^2 + x^6)^(1/4)*(1 + x^8))/(x^4*(-1 + x^4)),x]","\frac{2 \left(1+x^4\right) \sqrt[4]{x^2+x^6}}{5 x^3}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2^{3/4}}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{2 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{2 \sqrt[4]{2}}","\frac{2 \left(1+x^4\right) \sqrt[4]{x^2+x^6}}{5 x^3}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2^{3/4}}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{2 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{2 \sqrt[4]{2}}",1,"(2*(1 + x^4)*(x^2 + x^6)^(1/4))/(5*x^3) + ArcTan[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/2^(3/4) + ArcTan[(2^(3/4)*x*(x^2 + x^6)^(1/4))/(Sqrt[2]*x^2 - Sqrt[x^2 + x^6])]/(2*2^(1/4)) - ArcTanh[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/2^(3/4) + ArcTanh[(x^2/2^(1/4) + Sqrt[x^2 + x^6]/2^(3/4))/(x*(x^2 + x^6)^(1/4))]/(2*2^(1/4))","A",1
2325,1,180,180,0.0010427,"\int \frac{\sqrt[4]{x^2+x^6} \left(1+x^8\right)}{x^4 \left(-1+x^4\right)} \, dx","IntegrateAlgebraic[((x^2 + x^6)^(1/4)*(1 + x^8))/(x^4*(-1 + x^4)),x]","\frac{2 \left(1+x^4\right) \sqrt[4]{x^2+x^6}}{5 x^3}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2^{3/4}}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{2 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{2 \sqrt[4]{2}}","\frac{2 \left(1+x^4\right) \sqrt[4]{x^2+x^6}}{5 x^3}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2^{3/4}}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{2 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{2^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{2 \sqrt[4]{2}}",1,"(2*(1 + x^4)*(x^2 + x^6)^(1/4))/(5*x^3) + ArcTan[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/2^(3/4) + ArcTan[(2^(3/4)*x*(x^2 + x^6)^(1/4))/(Sqrt[2]*x^2 - Sqrt[x^2 + x^6])]/(2*2^(1/4)) - ArcTanh[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/2^(3/4) + ArcTanh[(x^2/2^(1/4) + Sqrt[x^2 + x^6]/2^(3/4))/(x*(x^2 + x^6)^(1/4))]/(2*2^(1/4))","A",1
2326,1,181,181,0.2257082,"\int \frac{1}{x \sqrt[3]{2-3 x+x^2}} \, dx","IntegrateAlgebraic[1/(x*(2 - 3*x + x^2)^(1/3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{2-3 x+x^2}}{2 \sqrt[3]{2}-\sqrt[3]{2} x+\sqrt[3]{2-3 x+x^2}}\right)}{2 \sqrt[3]{2}}+\frac{\log \left(-2 \sqrt[3]{2}+\sqrt[3]{2} x+2 \sqrt[3]{2-3 x+x^2}\right)}{2 \sqrt[3]{2}}-\frac{\log \left(4\ 2^{2/3}-4\ 2^{2/3} x+2^{2/3} x^2+\left(4 \sqrt[3]{2}-2 \sqrt[3]{2} x\right) \sqrt[3]{2-3 x+x^2}+4 \left(2-3 x+x^2\right)^{2/3}\right)}{4 \sqrt[3]{2}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{2-3 x+x^2}}{2 \sqrt[3]{2}-\sqrt[3]{2} x+\sqrt[3]{2-3 x+x^2}}\right)}{2 \sqrt[3]{2}}+\frac{\log \left(-2 \sqrt[3]{2}+\sqrt[3]{2} x+2 \sqrt[3]{2-3 x+x^2}\right)}{2 \sqrt[3]{2}}-\frac{\log \left(4\ 2^{2/3}-4\ 2^{2/3} x+2^{2/3} x^2+\left(4 \sqrt[3]{2}-2 \sqrt[3]{2} x\right) \sqrt[3]{2-3 x+x^2}+4 \left(2-3 x+x^2\right)^{2/3}\right)}{4 \sqrt[3]{2}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(2 - 3*x + x^2)^(1/3))/(2*2^(1/3) - 2^(1/3)*x + (2 - 3*x + x^2)^(1/3))])/(2*2^(1/3)) + Log[-2*2^(1/3) + 2^(1/3)*x + 2*(2 - 3*x + x^2)^(1/3)]/(2*2^(1/3)) - Log[4*2^(2/3) - 4*2^(2/3)*x + 2^(2/3)*x^2 + (4*2^(1/3) - 2*2^(1/3)*x)*(2 - 3*x + x^2)^(1/3) + 4*(2 - 3*x + x^2)^(2/3)]/(4*2^(1/3))","A",1
2327,1,181,181,0.3968878,"\int \frac{a b-x^2}{\sqrt[3]{x^2 (-a+x) (-b+x)} \left(a b-(a+b+d) x+x^2\right)} \, dx","IntegrateAlgebraic[(a*b - x^2)/((x^2*(-a + x)*(-b + x))^(1/3)*(a*b - (a + b + d)*x + x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} x}{\sqrt[3]{d} x+2 \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{\sqrt[3]{d}}-\frac{\log \left(-\sqrt[3]{d} x+\sqrt[3]{a b x^2+(-a-b) x^3+x^4}\right)}{\sqrt[3]{d}}+\frac{\log \left(d^{2/3} x^2+\sqrt[3]{d} x \sqrt[3]{a b x^2+(-a-b) x^3+x^4}+\left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} x}{\sqrt[3]{d} x+2 \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{\sqrt[3]{d}}-\frac{\log \left(-\sqrt[3]{d} x+\sqrt[3]{a b x^2+(-a-b) x^3+x^4}\right)}{\sqrt[3]{d}}+\frac{\log \left(d^{2/3} x^2+\sqrt[3]{d} x \sqrt[3]{a b x^2+(-a-b) x^3+x^4}+\left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*x)/(d^(1/3)*x + 2*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))])/d^(1/3) - Log[-(d^(1/3)*x) + (a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3)]/d^(1/3) + Log[d^(2/3)*x^2 + d^(1/3)*x*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3) + (a*b*x^2 + (-a - b)*x^3 + x^4)^(2/3)]/(2*d^(1/3))","A",1
2328,1,181,181,0.3388045,"\int \frac{\left(-q+2 p x^3\right) \sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left(b x^2+a \left(q+p x^3\right)^2\right)}{x^5} \, dx","IntegrateAlgebraic[((-q + 2*p*x^3)*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]*(b*x^2 + a*(q + p*x^3)^2))/x^5,x]","\frac{\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left(a q^3+2 b q x^2-a p q^2 x^2+3 a p q^2 x^3+2 b p x^5-a p^2 q x^5+3 a p^2 q x^6+a p^3 x^9\right)}{4 x^4}+\frac{1}{2} \left(2 b p q+a p^2 q^2\right) \log (x)+\frac{1}{2} \left(-2 b p q-a p^2 q^2\right) \log \left(q+p x^3+\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right)","\frac{\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left(a q^3+2 b q x^2-a p q^2 x^2+3 a p q^2 x^3+2 b p x^5-a p^2 q x^5+3 a p^2 q x^6+a p^3 x^9\right)}{4 x^4}+\frac{1}{2} \left(2 b p q+a p^2 q^2\right) \log (x)+\frac{1}{2} \left(-2 b p q-a p^2 q^2\right) \log \left(q+p x^3+\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right)",1,"(Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]*(a*q^3 + 2*b*q*x^2 - a*p*q^2*x^2 + 3*a*p*q^2*x^3 + 2*b*p*x^5 - a*p^2*q*x^5 + 3*a*p^2*q*x^6 + a*p^3*x^9))/(4*x^4) + ((2*b*p*q + a*p^2*q^2)*Log[x])/2 + ((-2*b*p*q - a*p^2*q^2)*Log[q + p*x^3 + Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]])/2","A",1
2329,1,135,181,0.1355886,"\int \frac{1+x}{(-1+x) \sqrt{1+\sqrt{1+\sqrt{1+x}}}} \, dx","IntegrateAlgebraic[(1 + x)/((-1 + x)*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]),x]","-\frac{16}{105} \left(-2+9 \sqrt{1+x}\right) \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\frac{8}{105} \sqrt{1+\sqrt{1+x}} \left(4+15 \sqrt{1+x}\right) \sqrt{1+\sqrt{1+\sqrt{1+x}}}+2 \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{32}{105} \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\frac{32}{105} \sqrt{1+\sqrt{1+x}} \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\sqrt{1+x} \left(-\frac{48}{35} \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\frac{8}{7} \sqrt{1+\sqrt{1+x}} \sqrt{1+\sqrt{1+\sqrt{1+x}}}\right)+2 \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"(-16*(-2 + 9*Sqrt[1 + x])*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/105 + (8*Sqrt[1 + Sqrt[1 + x]]*(4 + 15*Sqrt[1 + x])*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/105 + 2*RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]] - #1]/#1 & ]","A",1
2330,1,135,181,0.0013059,"\int \frac{1+x}{(-1+x) \sqrt{1+\sqrt{1+\sqrt{1+x}}}} \, dx","IntegrateAlgebraic[(1 + x)/((-1 + x)*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]),x]","-\frac{16}{105} \left(-2+9 \sqrt{1+x}\right) \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\frac{8}{105} \sqrt{1+\sqrt{1+x}} \left(4+15 \sqrt{1+x}\right) \sqrt{1+\sqrt{1+\sqrt{1+x}}}+2 \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{32}{105} \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\frac{32}{105} \sqrt{1+\sqrt{1+x}} \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\sqrt{1+x} \left(-\frac{48}{35} \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\frac{8}{7} \sqrt{1+\sqrt{1+x}} \sqrt{1+\sqrt{1+\sqrt{1+x}}}\right)+2 \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"(-16*(-2 + 9*Sqrt[1 + x])*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/105 + (8*Sqrt[1 + Sqrt[1 + x]]*(4 + 15*Sqrt[1 + x])*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/105 + 2*RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]] - #1]/#1 & ]","A",1
2331,1,183,183,0.8161056,"\int \frac{1}{d+c x+\sqrt{a x+\sqrt{b^2+a^2 x^2}}} \, dx","IntegrateAlgebraic[(d + c*x + Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])^(-1),x]","-\frac{\log \left(a x+\sqrt{b^2+a^2 x^2}\right)}{c}+\frac{2 \text{RootSum}\left[b^2 c-2 a d \text{$\#$1}^2-2 a \text{$\#$1}^3-c \text{$\#$1}^4\&,\frac{a d \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)+a \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}+c \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^2}{2 a d+3 a \text{$\#$1}+2 c \text{$\#$1}^2}\&\right]}{c}","-\frac{\log \left(a x+\sqrt{b^2+a^2 x^2}\right)}{c}+\frac{2 \text{RootSum}\left[b^2 c-2 a d \text{$\#$1}^2-2 a \text{$\#$1}^3-c \text{$\#$1}^4\&,\frac{a d \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)+a \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}+c \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^2}{2 a d+3 a \text{$\#$1}+2 c \text{$\#$1}^2}\&\right]}{c}",1,"-(Log[a*x + Sqrt[b^2 + a^2*x^2]]/c) + (2*RootSum[b^2*c - 2*a*d*#1^2 - 2*a*#1^3 - c*#1^4 & , (a*d*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1] + a*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]*#1 + c*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]*#1^2)/(2*a*d + 3*a*#1 + 2*c*#1^2) & ])/c","A",1
2332,1,183,183,0.0016816,"\int \frac{1}{d+c x+\sqrt{a x+\sqrt{b^2+a^2 x^2}}} \, dx","IntegrateAlgebraic[(d + c*x + Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])^(-1),x]","-\frac{\log \left(a x+\sqrt{b^2+a^2 x^2}\right)}{c}+\frac{2 \text{RootSum}\left[b^2 c-2 a d \text{$\#$1}^2-2 a \text{$\#$1}^3-c \text{$\#$1}^4\&,\frac{a d \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)+a \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}+c \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^2}{2 a d+3 a \text{$\#$1}+2 c \text{$\#$1}^2}\&\right]}{c}","-\frac{\log \left(a x+\sqrt{b^2+a^2 x^2}\right)}{c}+\frac{2 \text{RootSum}\left[b^2 c-2 a d \text{$\#$1}^2-2 a \text{$\#$1}^3-c \text{$\#$1}^4\&,\frac{a d \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)+a \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}+c \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^2}{2 a d+3 a \text{$\#$1}+2 c \text{$\#$1}^2}\&\right]}{c}",1,"-(Log[a*x + Sqrt[b^2 + a^2*x^2]]/c) + (2*RootSum[b^2*c - 2*a*d*#1^2 - 2*a*#1^3 - c*#1^4 & , (a*d*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1] + a*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]*#1 + c*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]*#1^2)/(2*a*d + 3*a*#1 + 2*c*#1^2) & ])/c","A",1
2333,1,184,184,17.9108348,"\int \frac{-d+c x}{x^7 \sqrt[3]{-b+a x^3}} \, dx","IntegrateAlgebraic[(-d + c*x)/(x^7*(-b + a*x^3)^(1/3)),x]","\frac{\left(-b+a x^3\right)^{2/3} \left(-15 b d+18 b c x-20 a d x^3+27 a c x^4\right)}{90 b^2 x^6}+\frac{2 a^2 d \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-b+a x^3}}{\sqrt{3} \sqrt[3]{b}}\right)}{9 \sqrt{3} b^{7/3}}+\frac{2 a^2 d \log \left(\sqrt[3]{b}+\sqrt[3]{-b+a x^3}\right)}{27 b^{7/3}}-\frac{a^2 d \log \left(b^{2/3}-\sqrt[3]{b} \sqrt[3]{-b+a x^3}+\left(-b+a x^3\right)^{2/3}\right)}{27 b^{7/3}}","\frac{\left(-b+a x^3\right)^{2/3} \left(-15 b d+18 b c x-20 a d x^3+27 a c x^4\right)}{90 b^2 x^6}+\frac{2 a^2 d \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-b+a x^3}}{\sqrt{3} \sqrt[3]{b}}\right)}{9 \sqrt{3} b^{7/3}}+\frac{2 a^2 d \log \left(\sqrt[3]{b}+\sqrt[3]{-b+a x^3}\right)}{27 b^{7/3}}-\frac{a^2 d \log \left(b^{2/3}-\sqrt[3]{b} \sqrt[3]{-b+a x^3}+\left(-b+a x^3\right)^{2/3}\right)}{27 b^{7/3}}",1,"((-b + a*x^3)^(2/3)*(-15*b*d + 18*b*c*x - 20*a*d*x^3 + 27*a*c*x^4))/(90*b^2*x^6) + (2*a^2*d*ArcTan[1/Sqrt[3] - (2*(-b + a*x^3)^(1/3))/(Sqrt[3]*b^(1/3))])/(9*Sqrt[3]*b^(7/3)) + (2*a^2*d*Log[b^(1/3) + (-b + a*x^3)^(1/3)])/(27*b^(7/3)) - (a^2*d*Log[b^(2/3) - b^(1/3)*(-b + a*x^3)^(1/3) + (-b + a*x^3)^(2/3)])/(27*b^(7/3))","A",1
2334,1,184,184,0.6232249,"\int \frac{1+x^2+x^4}{\left(1-x^4\right) \sqrt[4]{x^3+x^5}} \, dx","IntegrateAlgebraic[(1 + x^2 + x^4)/((1 - x^4)*(x^3 + x^5)^(1/4)),x]","\frac{\left(x^3+x^5\right)^{3/4}}{x^2 \left(1+x^2\right)}+\frac{3 \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^5}}\right)}{4 \sqrt[4]{2}}-\frac{3 \tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^3+x^5}}{\sqrt{2} x^2-\sqrt{x^3+x^5}}\right)}{4\ 2^{3/4}}+\frac{3 \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^5}}\right)}{4 \sqrt[4]{2}}+\frac{3 \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^3+x^5}}{2^{3/4}}}{x \sqrt[4]{x^3+x^5}}\right)}{4\ 2^{3/4}}","\frac{\left(x^3+x^5\right)^{3/4}}{x^2 \left(1+x^2\right)}+\frac{3 \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^5}}\right)}{4 \sqrt[4]{2}}-\frac{3 \tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^3+x^5}}{\sqrt{2} x^2-\sqrt{x^3+x^5}}\right)}{4\ 2^{3/4}}+\frac{3 \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^5}}\right)}{4 \sqrt[4]{2}}+\frac{3 \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^3+x^5}}{2^{3/4}}}{x \sqrt[4]{x^3+x^5}}\right)}{4\ 2^{3/4}}",1,"(x^3 + x^5)^(3/4)/(x^2*(1 + x^2)) + (3*ArcTan[(2^(1/4)*x)/(x^3 + x^5)^(1/4)])/(4*2^(1/4)) - (3*ArcTan[(2^(3/4)*x*(x^3 + x^5)^(1/4))/(Sqrt[2]*x^2 - Sqrt[x^3 + x^5])])/(4*2^(3/4)) + (3*ArcTanh[(2^(1/4)*x)/(x^3 + x^5)^(1/4)])/(4*2^(1/4)) + (3*ArcTanh[(x^2/2^(1/4) + Sqrt[x^3 + x^5]/2^(3/4))/(x*(x^3 + x^5)^(1/4))])/(4*2^(3/4))","A",1
2335,1,138,184,0.4406285,"\int \frac{\sqrt{-1+2 x^4} \left(-1+2 x^8\right)}{x^7 \left(-1+2 x^4+x^8\right)} \, dx","IntegrateAlgebraic[(Sqrt[-1 + 2*x^4]*(-1 + 2*x^8))/(x^7*(-1 + 2*x^4 + x^8)),x]","\frac{\left(-1-4 x^4\right) \sqrt{-1+2 x^4}}{6 x^6}+\sqrt{2} \text{RootSum}\left[1+20 \text{$\#$1}^2-26 \text{$\#$1}^4+20 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{2} x^2+\sqrt{-1+2 x^4}-\text{$\#$1}\right)+\log \left(\sqrt{2} x^2+\sqrt{-1+2 x^4}-\text{$\#$1}\right) \text{$\#$1}^4}{5-13 \text{$\#$1}^2+15 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]","\frac{\left(-1-4 x^4\right) \sqrt{-1+2 x^4}}{6 x^6}+\sqrt{2} \text{RootSum}\left[-64+128 \text{$\#$1}-80 \text{$\#$1}^2+16 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{2 \log \left(4 x^4+2 \sqrt{2} x^2 \sqrt{-1+2 x^4}-\text{$\#$1}\right)-2 \log \left(4 x^4+2 \sqrt{2} x^2 \sqrt{-1+2 x^4}-\text{$\#$1}\right) \text{$\#$1}+\log \left(4 x^4+2 \sqrt{2} x^2 \sqrt{-1+2 x^4}-\text{$\#$1}\right) \text{$\#$1}^2}{32-40 \text{$\#$1}+12 \text{$\#$1}^2+\text{$\#$1}^3}\&\right]",1,"((-1 - 4*x^4)*Sqrt[-1 + 2*x^4])/(6*x^6) + Sqrt[2]*RootSum[1 + 20*#1^2 - 26*#1^4 + 20*#1^6 + #1^8 & , (Log[Sqrt[2]*x^2 + Sqrt[-1 + 2*x^4] - #1] + Log[Sqrt[2]*x^2 + Sqrt[-1 + 2*x^4] - #1]*#1^4)/(5 - 13*#1^2 + 15*#1^4 + #1^6) & ]","A",1
2336,1,165,184,0.2789753,"\int \frac{\sqrt{1+x} \sqrt{1+\sqrt{1+x}}}{x \sqrt{1+\sqrt{1+\sqrt{1+x}}}} \, dx","IntegrateAlgebraic[(Sqrt[1 + x]*Sqrt[1 + Sqrt[1 + x]])/(x*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]),x]","-\frac{32}{15} \sqrt{1+\sqrt{1+x}} \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\frac{8}{15} \left(11+3 \sqrt{1+x}\right) \sqrt{1+\sqrt{1+\sqrt{1+x}}}-2 \sqrt{2 \left(1+\sqrt{2}\right)} \tan ^{-1}\left(\sqrt{1+\sqrt{2}} \sqrt{1+\sqrt{1+\sqrt{1+x}}}\right)-2 \sqrt{2 \left(-1+\sqrt{2}\right)} \tanh ^{-1}\left(\sqrt{-1+\sqrt{2}} \sqrt{1+\sqrt{1+\sqrt{1+x}}}\right)","\frac{88}{15} \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\frac{8}{5} \sqrt{1+x} \sqrt{1+\sqrt{1+\sqrt{1+x}}}-\frac{32}{15} \sqrt{1+\sqrt{1+x}} \sqrt{1+\sqrt{1+\sqrt{1+x}}}-2 \sqrt{2 \left(1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{1+\sqrt{1+\sqrt{1+x}}}}{\sqrt{-1+\sqrt{2}}}\right)-2 \sqrt{2 \left(-1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{1+\sqrt{1+\sqrt{1+x}}}}{\sqrt{1+\sqrt{2}}}\right)",1,"(-32*Sqrt[1 + Sqrt[1 + x]]*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/15 + (8*(11 + 3*Sqrt[1 + x])*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/15 - 2*Sqrt[2*(1 + Sqrt[2])]*ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]] - 2*Sqrt[2*(-1 + Sqrt[2])]*ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]]","A",1
2337,1,185,185,0.2585069,"\int \frac{1+x}{(3+x) (1+2 x) \sqrt[3]{1+x^2}} \, dx","IntegrateAlgebraic[(1 + x)/((3 + x)*(1 + 2*x)*(1 + x^2)^(1/3)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\frac{4 \sqrt[3]{2}}{\sqrt{3} 5^{2/3}}-\frac{2 \sqrt[3]{2} x}{\sqrt{3} 5^{2/3}}+\frac{\sqrt[3]{1+x^2}}{\sqrt{3}}}{\sqrt[3]{1+x^2}}\right)}{5 \sqrt[3]{10}}+\frac{\log \left(-2 \sqrt[3]{10}+\sqrt[3]{10} x+5 \sqrt[3]{1+x^2}\right)}{5 \sqrt[3]{10}}-\frac{\log \left(4\ 10^{2/3}-4\ 10^{2/3} x+10^{2/3} x^2+\left(10 \sqrt[3]{10}-5 \sqrt[3]{10} x\right) \sqrt[3]{1+x^2}+25 \left(1+x^2\right)^{2/3}\right)}{10 \sqrt[3]{10}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\frac{4 \sqrt[3]{2}}{\sqrt{3} 5^{2/3}}-\frac{2 \sqrt[3]{2} x}{\sqrt{3} 5^{2/3}}+\frac{\sqrt[3]{1+x^2}}{\sqrt{3}}}{\sqrt[3]{1+x^2}}\right)}{5 \sqrt[3]{10}}+\frac{\log \left(-2 \sqrt[3]{10}+\sqrt[3]{10} x+5 \sqrt[3]{1+x^2}\right)}{5 \sqrt[3]{10}}-\frac{\log \left(4\ 10^{2/3}-4\ 10^{2/3} x+10^{2/3} x^2+\left(10 \sqrt[3]{10}-5 \sqrt[3]{10} x\right) \sqrt[3]{1+x^2}+25 \left(1+x^2\right)^{2/3}\right)}{10 \sqrt[3]{10}}",1,"-1/5*(Sqrt[3]*ArcTan[((4*2^(1/3))/(Sqrt[3]*5^(2/3)) - (2*2^(1/3)*x)/(Sqrt[3]*5^(2/3)) + (1 + x^2)^(1/3)/Sqrt[3])/(1 + x^2)^(1/3)])/10^(1/3) + Log[-2*10^(1/3) + 10^(1/3)*x + 5*(1 + x^2)^(1/3)]/(5*10^(1/3)) - Log[4*10^(2/3) - 4*10^(2/3)*x + 10^(2/3)*x^2 + (10*10^(1/3) - 5*10^(1/3)*x)*(1 + x^2)^(1/3) + 25*(1 + x^2)^(2/3)]/(10*10^(1/3))","A",1
2338,1,185,185,0.281294,"\int \frac{\sqrt[3]{-x^2+x^3}}{1+x+x^2} \, dx","IntegrateAlgebraic[(-x^2 + x^3)^(1/3)/(1 + x + x^2),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{-x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)-\text{RootSum}\left[3-3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-3 \log (x)+3 \log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-3 \text{$\#$1}^2+2 \text{$\#$1}^5}\&\right]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{-x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)-\text{RootSum}\left[3-3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-3 \log (x)+3 \log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-3 \text{$\#$1}^2+2 \text{$\#$1}^5}\&\right]",1,"-(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-x^2 + x^3)^(1/3))]) - Log[-x + (-x^2 + x^3)^(1/3)] + Log[x^2 + x*(-x^2 + x^3)^(1/3) + (-x^2 + x^3)^(2/3)]/2 - RootSum[3 - 3*#1^3 + #1^6 & , (-3*Log[x] + 3*Log[(-x^2 + x^3)^(1/3) - x*#1] + Log[x]*#1^3 - Log[(-x^2 + x^3)^(1/3) - x*#1]*#1^3)/(-3*#1^2 + 2*#1^5) & ]","A",1
2339,1,185,185,0.0007813,"\int \frac{\sqrt[3]{-x^2+x^3}}{1+x+x^2} \, dx","IntegrateAlgebraic[(-x^2 + x^3)^(1/3)/(1 + x + x^2),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{-x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)-\text{RootSum}\left[3-3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-3 \log (x)+3 \log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-3 \text{$\#$1}^2+2 \text{$\#$1}^5}\&\right]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{-x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)-\text{RootSum}\left[3-3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-3 \log (x)+3 \log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-3 \text{$\#$1}^2+2 \text{$\#$1}^5}\&\right]",1,"-(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-x^2 + x^3)^(1/3))]) - Log[-x + (-x^2 + x^3)^(1/3)] + Log[x^2 + x*(-x^2 + x^3)^(1/3) + (-x^2 + x^3)^(2/3)]/2 - RootSum[3 - 3*#1^3 + #1^6 & , (-3*Log[x] + 3*Log[(-x^2 + x^3)^(1/3) - x*#1] + Log[x]*#1^3 - Log[(-x^2 + x^3)^(1/3) - x*#1]*#1^3)/(-3*#1^2 + 2*#1^5) & ]","A",1
2340,1,185,185,2.868332,"\int \frac{x \left(5-4 (1+k) x+3 k x^2\right)}{\sqrt[3]{(1-x) x (1-k x)} \left(-b+(b+b k) x-b k x^2+x^5\right)} \, dx","IntegrateAlgebraic[(x*(5 - 4*(1 + k)*x + 3*k*x^2))/(((1 - x)*x*(1 - k*x))^(1/3)*(-b + (b + b*k)*x - b*k*x^2 + x^5)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}{2 x^2+\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{b^{2/3}}+\frac{\log \left(x^2-\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{b^{2/3}}-\frac{\log \left(x^4+\sqrt[3]{b} x^2 \sqrt[3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 b^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}{2 x^2+\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{b^{2/3}}+\frac{\log \left(x^2-\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{b^{2/3}}-\frac{\log \left(x^4+\sqrt[3]{b} x^2 \sqrt[3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 b^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))/(2*x^2 + b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/b^(2/3) + Log[x^2 - b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3)]/b^(2/3) - Log[x^4 + b^(1/3)*x^2*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + b^(2/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3)]/(2*b^(2/3))","A",1
2341,1,185,185,0.5815047,"\int \frac{1+x^4}{\left(1-x^4\right) \sqrt[4]{x^3+x^5}} \, dx","IntegrateAlgebraic[(1 + x^4)/((1 - x^4)*(x^3 + x^5)^(1/4)),x]","\frac{2 \left(x^3+x^5\right)^{3/4}}{x^2 \left(1+x^2\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^5}}\right)}{2 \sqrt[4]{2}}-\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^3+x^5}}{\sqrt{2} x^2-\sqrt{x^3+x^5}}\right)}{2\ 2^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^5}}\right)}{2 \sqrt[4]{2}}+\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^3+x^5}}{2^{3/4}}}{x \sqrt[4]{x^3+x^5}}\right)}{2\ 2^{3/4}}","\frac{2 \left(x^3+x^5\right)^{3/4}}{x^2 \left(1+x^2\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^5}}\right)}{2 \sqrt[4]{2}}-\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^3+x^5}}{\sqrt{2} x^2-\sqrt{x^3+x^5}}\right)}{2\ 2^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^5}}\right)}{2 \sqrt[4]{2}}+\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^3+x^5}}{2^{3/4}}}{x \sqrt[4]{x^3+x^5}}\right)}{2\ 2^{3/4}}",1,"(2*(x^3 + x^5)^(3/4))/(x^2*(1 + x^2)) + ArcTan[(2^(1/4)*x)/(x^3 + x^5)^(1/4)]/(2*2^(1/4)) - ArcTan[(2^(3/4)*x*(x^3 + x^5)^(1/4))/(Sqrt[2]*x^2 - Sqrt[x^3 + x^5])]/(2*2^(3/4)) + ArcTanh[(2^(1/4)*x)/(x^3 + x^5)^(1/4)]/(2*2^(1/4)) + ArcTanh[(x^2/2^(1/4) + Sqrt[x^3 + x^5]/2^(3/4))/(x*(x^3 + x^5)^(1/4))]/(2*2^(3/4))","A",1
2342,1,185,185,0.5496935,"\int \frac{\left(1+x^3\right)^{2/3} \left(2+x^3+x^6\right)}{x^6 \left(-2+x^3\right)^2} \, dx","IntegrateAlgebraic[((1 + x^3)^(2/3)*(2 + x^3 + x^6))/(x^6*(-2 + x^3)^2),x]","\frac{\left(1+x^3\right)^{2/3} \left(24+102 x^3-97 x^6\right)}{120 x^5 \left(-2+x^3\right)}+\frac{35 \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2 \sqrt[3]{2} \sqrt[3]{1+x^3}}\right)}{12\ 2^{2/3} 3^{5/6}}-\frac{35 \log \left(-3 x+\sqrt[3]{2} 3^{2/3} \sqrt[3]{1+x^3}\right)}{36\ 2^{2/3} \sqrt[3]{3}}+\frac{35 \log \left(3 x^2+\sqrt[3]{2} 3^{2/3} x \sqrt[3]{1+x^3}+2^{2/3} \sqrt[3]{3} \left(1+x^3\right)^{2/3}\right)}{72\ 2^{2/3} \sqrt[3]{3}}","\frac{\left(1+x^3\right)^{2/3} \left(24+102 x^3-97 x^6\right)}{120 x^5 \left(-2+x^3\right)}+\frac{35 \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2 \sqrt[3]{2} \sqrt[3]{1+x^3}}\right)}{12\ 2^{2/3} 3^{5/6}}-\frac{35 \log \left(-3 x+\sqrt[3]{2} 3^{2/3} \sqrt[3]{1+x^3}\right)}{36\ 2^{2/3} \sqrt[3]{3}}+\frac{35 \log \left(3 x^2+\sqrt[3]{2} 3^{2/3} x \sqrt[3]{1+x^3}+2^{2/3} \sqrt[3]{3} \left(1+x^3\right)^{2/3}\right)}{72\ 2^{2/3} \sqrt[3]{3}}",1,"((1 + x^3)^(2/3)*(24 + 102*x^3 - 97*x^6))/(120*x^5*(-2 + x^3)) + (35*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*2^(1/3)*(1 + x^3)^(1/3))])/(12*2^(2/3)*3^(5/6)) - (35*Log[-3*x + 2^(1/3)*3^(2/3)*(1 + x^3)^(1/3)])/(36*2^(2/3)*3^(1/3)) + (35*Log[3*x^2 + 2^(1/3)*3^(2/3)*x*(1 + x^3)^(1/3) + 2^(2/3)*3^(1/3)*(1 + x^3)^(2/3)])/(72*2^(2/3)*3^(1/3))","A",1
2343,1,185,185,0.7250467,"\int \frac{1+x^8}{\sqrt[4]{x^2+x^6} \left(-1+x^8\right)} \, dx","IntegrateAlgebraic[(1 + x^8)/((x^2 + x^6)^(1/4)*(-1 + x^8)),x]","-\frac{\left(x^2+x^6\right)^{3/4}}{x \left(1+x^4\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{4\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{4\ 2^{3/4}}","-\frac{\left(x^2+x^6\right)^{3/4}}{x \left(1+x^4\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{4\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{4\ 2^{3/4}}",1,"-((x^2 + x^6)^(3/4)/(x*(1 + x^4))) - ArcTan[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/(4*2^(1/4)) + ArcTan[(2^(3/4)*x*(x^2 + x^6)^(1/4))/(Sqrt[2]*x^2 - Sqrt[x^2 + x^6])]/(4*2^(3/4)) - ArcTanh[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/(4*2^(1/4)) - ArcTanh[(x^2/2^(1/4) + Sqrt[x^2 + x^6]/2^(3/4))/(x*(x^2 + x^6)^(1/4))]/(4*2^(3/4))","A",1
2344,1,185,185,0.0010355,"\int \frac{1+x^8}{\sqrt[4]{x^2+x^6} \left(-1+x^8\right)} \, dx","IntegrateAlgebraic[(1 + x^8)/((x^2 + x^6)^(1/4)*(-1 + x^8)),x]","-\frac{\left(x^2+x^6\right)^{3/4}}{x \left(1+x^4\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{4\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{4\ 2^{3/4}}","-\frac{\left(x^2+x^6\right)^{3/4}}{x \left(1+x^4\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{4\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{4 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{4\ 2^{3/4}}",1,"-((x^2 + x^6)^(3/4)/(x*(1 + x^4))) - ArcTan[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/(4*2^(1/4)) + ArcTan[(2^(3/4)*x*(x^2 + x^6)^(1/4))/(Sqrt[2]*x^2 - Sqrt[x^2 + x^6])]/(4*2^(3/4)) - ArcTanh[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/(4*2^(1/4)) - ArcTanh[(x^2/2^(1/4) + Sqrt[x^2 + x^6]/2^(3/4))/(x*(x^2 + x^6)^(1/4))]/(4*2^(3/4))","A",1
2345,1,185,185,2.9802845,"\int \frac{x^2 \left(8-7 (1+k) x+6 k x^2\right)}{\sqrt[3]{(1-x) x (1-k x)} \left(-b+b (1+k) x-b k x^2+x^8\right)} \, dx","IntegrateAlgebraic[(x^2*(8 - 7*(1 + k)*x + 6*k*x^2))/(((1 - x)*x*(1 - k*x))^(1/3)*(-b + b*(1 + k)*x - b*k*x^2 + x^8)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}{2 x^3+\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{b^{2/3}}+\frac{\log \left(x^3-\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{b^{2/3}}-\frac{\log \left(x^6+\sqrt[3]{b} x^3 \sqrt[3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 b^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}{2 x^3+\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{b^{2/3}}+\frac{\log \left(x^3-\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{b^{2/3}}-\frac{\log \left(x^6+\sqrt[3]{b} x^3 \sqrt[3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 b^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))/(2*x^3 + b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/b^(2/3) + Log[x^3 - b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3)]/b^(2/3) - Log[x^6 + b^(1/3)*x^3*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + b^(2/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3)]/(2*b^(2/3))","A",1
2346,1,185,185,0.3124737,"\int \frac{1}{\left(1+x^2\right)^2 \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[1/((1 + x^2)^2*Sqrt[x + Sqrt[1 + x^2]]),x]","-\frac{3 \sqrt{x+\sqrt{1+x^2}}}{8 \left(1+x^2+x \sqrt{1+x^2}\right)^2}+\frac{\left(x+\sqrt{1+x^2}\right)^{5/2}}{8 \left(1+x^2+x \sqrt{1+x^2}\right)^2}+\frac{3 \tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x}{\sqrt{2}}+\frac{\sqrt{1+x^2}}{\sqrt{2}}}{\sqrt{x+\sqrt{1+x^2}}}\right)}{4 \sqrt{2}}+\frac{3 \tanh ^{-1}\left(\frac{\frac{1}{\sqrt{2}}+\frac{x}{\sqrt{2}}+\frac{\sqrt{1+x^2}}{\sqrt{2}}}{\sqrt{x+\sqrt{1+x^2}}}\right)}{4 \sqrt{2}}","-\frac{3 \sqrt{x+\sqrt{1+x^2}}}{8 \left(1+x^2+x \sqrt{1+x^2}\right)^2}+\frac{\left(x+\sqrt{1+x^2}\right)^{5/2}}{8 \left(1+x^2+x \sqrt{1+x^2}\right)^2}+\frac{3 \tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x}{\sqrt{2}}+\frac{\sqrt{1+x^2}}{\sqrt{2}}}{\sqrt{x+\sqrt{1+x^2}}}\right)}{4 \sqrt{2}}+\frac{3 \tanh ^{-1}\left(\frac{\frac{1}{\sqrt{2}}+\frac{x}{\sqrt{2}}+\frac{\sqrt{1+x^2}}{\sqrt{2}}}{\sqrt{x+\sqrt{1+x^2}}}\right)}{4 \sqrt{2}}",1,"(-3*Sqrt[x + Sqrt[1 + x^2]])/(8*(1 + x^2 + x*Sqrt[1 + x^2])^2) + (x + Sqrt[1 + x^2])^(5/2)/(8*(1 + x^2 + x*Sqrt[1 + x^2])^2) + (3*ArcTan[(-(1/Sqrt[2]) + x/Sqrt[2] + Sqrt[1 + x^2]/Sqrt[2])/Sqrt[x + Sqrt[1 + x^2]]])/(4*Sqrt[2]) + (3*ArcTanh[(1/Sqrt[2] + x/Sqrt[2] + Sqrt[1 + x^2]/Sqrt[2])/Sqrt[x + Sqrt[1 + x^2]]])/(4*Sqrt[2])","A",1
2347,1,186,186,4.1359325,"\int \frac{x (-a+x) (a b+(a-2 b) x)}{\left(x (-a+x) (-b+x)^2\right)^{3/4} \left(b^2 d+(a-2 b d) x+(-1+d) x^2\right)} \, dx","IntegrateAlgebraic[(x*(-a + x)*(a*b + (a - 2*b)*x))/((x*(-a + x)*(-b + x)^2)^(3/4)*(b^2*d + (a - 2*b*d)*x + (-1 + d)*x^2)),x]","\frac{4 \sqrt[4]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{b-x}+2 \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{3/4}}{(b-x) x (-a+x)}\right)-2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt[4]{d} \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{3/4}}{(b-x) x (-a+x)}\right)","\frac{4 \sqrt[4]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{b-x}+2 \sqrt[4]{d} \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{3/4}}{(b-x) x (-a+x)}\right)-2 \sqrt[4]{d} \tanh ^{-1}\left(\frac{\sqrt[4]{d} \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{3/4}}{(b-x) x (-a+x)}\right)",1,"(4*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/4))/(b - x) + 2*d^(1/4)*ArcTan[(d^(1/4)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(3/4))/((b - x)*x*(-a + x))] - 2*d^(1/4)*ArcTanh[(d^(1/4)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(3/4))/((b - x)*x*(-a + x))]","A",1
2348,1,188,186,0.692431,"\int \frac{\left(1+x^2\right) \left(1-3 x^2+x^4\right)}{x^2 \sqrt{\frac{-2+x+2 x^2}{-1+x+x^2}} \left(1-x-3 x^2+x^3+x^4\right)} \, dx","IntegrateAlgebraic[((1 + x^2)*(1 - 3*x^2 + x^4))/(x^2*Sqrt[(-2 + x + 2*x^2)/(-1 + x + x^2)]*(1 - x - 3*x^2 + x^3 + x^4)),x]","\frac{\left(-1+x+x^2\right) \sqrt{\frac{-2+x+2 x^2}{-1+x+x^2}}}{2 x}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{\frac{-2+x+2 x^2}{-1+x+x^2}}}{\sqrt{2}}\right)}{2 \sqrt{2}}+\frac{1}{5} \sqrt{2 \left(5+\sqrt{5}\right)} \tanh ^{-1}\left(\sqrt{\frac{1}{2}-\frac{1}{2 \sqrt{5}}} \sqrt{\frac{-2+x+2 x^2}{-1+x+x^2}}\right)+\frac{1}{5} \sqrt{2 \left(5-\sqrt{5}\right)} \tanh ^{-1}\left(\sqrt{\frac{1}{2}+\frac{1}{2 \sqrt{5}}} \sqrt{\frac{-2+x+2 x^2}{-1+x+x^2}}\right)","\frac{\left(-1+x+x^2\right) \sqrt{\frac{-2+x+2 x^2}{-1+x+x^2}}}{2 x}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{\frac{-2+x+2 x^2}{-1+x+x^2}}}{\sqrt{2}}\right)}{2 \sqrt{2}}+\frac{1}{5} \sqrt{2 \left(5-\sqrt{5}\right)} \tanh ^{-1}\left(\sqrt{\frac{1}{2}+\frac{1}{2 \sqrt{5}}} \sqrt{\frac{-2+x+2 x^2}{-1+x+x^2}}\right)+\frac{1}{5} \sqrt{2 \left(5+\sqrt{5}\right)} \tanh ^{-1}\left(\sqrt{\frac{2}{5+\sqrt{5}}} \sqrt{\frac{-2+x+2 x^2}{-1+x+x^2}}\right)",1,"((-1 + x + x^2)*Sqrt[(-2 + x + 2*x^2)/(-1 + x + x^2)])/(2*x) - (3*ArcTanh[Sqrt[(-2 + x + 2*x^2)/(-1 + x + x^2)]/Sqrt[2]])/(2*Sqrt[2]) + (Sqrt[2*(5 + Sqrt[5])]*ArcTanh[Sqrt[1/2 - 1/(2*Sqrt[5])]*Sqrt[(-2 + x + 2*x^2)/(-1 + x + x^2)]])/5 + (Sqrt[2*(5 - Sqrt[5])]*ArcTanh[Sqrt[1/2 + 1/(2*Sqrt[5])]*Sqrt[(-2 + x + 2*x^2)/(-1 + x + x^2)]])/5","A",1
2349,1,186,186,8.7129119,"\int \frac{x^3 \left(5-4 (1+k) x+3 k x^2\right)}{((1-x) x (1-k x))^{2/3} \left(-1+(1+k) x-k x^2+b x^5\right)} \, dx","IntegrateAlgebraic[(x^3*(5 - 4*(1 + k)*x + 3*k*x^2))/(((1 - x)*x*(1 - k*x))^(2/3)*(-1 + (1 + k)*x - k*x^2 + b*x^5)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{x+(-1-k) x^2+k x^3}}{2 \sqrt[3]{b} x^2+\sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{b^{2/3}}+\frac{\log \left(b x^2-b^{2/3} \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{b^{2/3}}-\frac{\log \left(b^2 x^4+b^{5/3} x^2 \sqrt[3]{x+(-1-k) x^2+k x^3}+b^{4/3} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 b^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{x+(-1-k) x^2+k x^3}}{2 \sqrt[3]{b} x^2+\sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{b^{2/3}}+\frac{\log \left(b x^2-b^{2/3} \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{b^{2/3}}-\frac{\log \left(b^2 x^4+b^{5/3} x^2 \sqrt[3]{x+(-1-k) x^2+k x^3}+b^{4/3} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 b^{2/3}}",1,"-((Sqrt[3]*ArcTan[(Sqrt[3]*(x + (-1 - k)*x^2 + k*x^3)^(1/3))/(2*b^(1/3)*x^2 + (x + (-1 - k)*x^2 + k*x^3)^(1/3))])/b^(2/3)) + Log[b*x^2 - b^(2/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3)]/b^(2/3) - Log[b^2*x^4 + b^(5/3)*x^2*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + b^(4/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3)]/(2*b^(2/3))","A",1
2350,1,33,186,0.4961072,"\int \frac{\left(-2+x^4\right) \sqrt{2+x^4}}{4+3 x^4+x^8} \, dx","IntegrateAlgebraic[((-2 + x^4)*Sqrt[2 + x^4])/(4 + 3*x^4 + x^8),x]","-\frac{1}{2} \tan ^{-1}\left(\frac{x}{\sqrt{2+x^4}}\right)-\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt{2+x^4}}\right)","-4 \sqrt[4]{2} \text{RootSum}\left[1-4 \text{$\#$1}^2-122 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(-\sqrt{2}+2^{3/4} x-x^2\right) \text{$\#$1}+\log \left(\sqrt{2+x^4}+\sqrt{2} \text{$\#$1}-2^{3/4} x \text{$\#$1}+x^2 \text{$\#$1}\right) \text{$\#$1}-\log \left(-\sqrt{2}+2^{3/4} x-x^2\right) \text{$\#$1}^3+\log \left(\sqrt{2+x^4}+\sqrt{2} \text{$\#$1}-2^{3/4} x \text{$\#$1}+x^2 \text{$\#$1}\right) \text{$\#$1}^3}{-1-61 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]",1,"-1/2*ArcTan[x/Sqrt[2 + x^4]] - ArcTanh[x/Sqrt[2 + x^4]]/2","A",1
2351,1,190,186,23.2556766,"\int \frac{\sqrt[3]{-1-x+5 x^2+2 x^3-10 x^4+2 x^5+7 x^6-5 x^7+x^8}}{x^2} \, dx","IntegrateAlgebraic[(-1 - x + 5*x^2 + 2*x^3 - 10*x^4 + 2*x^5 + 7*x^6 - 5*x^7 + x^8)^(1/3)/x^2,x]","-\frac{\left(1+\sqrt[3]{-1+x}\right) \left(1-\sqrt[3]{-1+x}+(-1+x)^{2/3}\right)^2 (-1+x)^{2/3} \sqrt[3]{(-1+x)^2 \left(-1-x+x^2\right)^3} \left(\frac{\left(-5-9 (-1+x)+6 (-1+x)^2\right) (-1+x)^{2/3}}{10 x}-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x}}{\sqrt{3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(1+\sqrt[3]{-1+x}\right)+\frac{1}{6} \log \left(1-\sqrt[3]{-1+x}+(-1+x)^{2/3}\right)\right)}{\left(1-\sqrt[3]{-1+x}+(-1+x)^{2/3}-x\right) x \left(1-2 x^2+x^3\right)}","-\frac{\left(1+\sqrt[3]{-1+x}\right) \left(1-\sqrt[3]{-1+x}+(-1+x)^{2/3}\right)^2 (-1+x)^{2/3} \sqrt[3]{(-1+x)^2 \left(-1-x+x^2\right)^3} \left(\frac{(-1+x)^{2/3} \left(10-21 x+6 x^2\right)}{10 x}-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+x}}{\sqrt{3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(1+\sqrt[3]{-1+x}\right)+\frac{1}{6} \log \left(1-\sqrt[3]{-1+x}+(-1+x)^{2/3}\right)\right)}{\left(1-\sqrt[3]{-1+x}+(-1+x)^{2/3}-x\right) x \left(1-2 x^2+x^3\right)}",1,"-(((1 + (-1 + x)^(1/3))*(1 - (-1 + x)^(1/3) + (-1 + x)^(2/3))^2*(-1 + x)^(2/3)*((-1 + x)^2*(-1 - x + x^2)^3)^(1/3)*(((-5 - 9*(-1 + x) + 6*(-1 + x)^2)*(-1 + x)^(2/3))/(10*x) - ArcTan[1/Sqrt[3] - (2*(-1 + x)^(1/3))/Sqrt[3]]/Sqrt[3] - Log[1 + (-1 + x)^(1/3)]/3 + Log[1 - (-1 + x)^(1/3) + (-1 + x)^(2/3)]/6))/((1 - (-1 + x)^(1/3) + (-1 + x)^(2/3) - x)*x*(1 - 2*x^2 + x^3)))","A",1
2352,1,186,186,0.5421619,"\int \frac{\left(-b+a^2 x^2\right) \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{b+a^2 x^4}} \, dx","IntegrateAlgebraic[((-b + a^2*x^2)*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[b + a^2*x^4],x]","\frac{1}{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\frac{\sqrt{a} \sqrt{b} \tan ^{-1}\left(\frac{a x^2}{\sqrt{b}}+\frac{\sqrt{b+a^2 x^4}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{b}}\right)}{\sqrt{2}}+\frac{b \log \left(a x^2+\sqrt{b+a^2 x^4}-\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{\sqrt{2} \sqrt{a}}","\frac{1}{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\frac{\sqrt{a} \sqrt{b} \tan ^{-1}\left(\frac{a x^2}{\sqrt{b}}+\frac{\sqrt{b+a^2 x^4}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{b}}\right)}{\sqrt{2}}-\frac{b \log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{\sqrt{2} \sqrt{a}}",1,"(a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/2 + (Sqrt[a]*Sqrt[b]*ArcTan[(a*x^2)/Sqrt[b] + Sqrt[b + a^2*x^4]/Sqrt[b] - (Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[b]])/Sqrt[2] + (b*Log[a*x^2 + Sqrt[b + a^2*x^4] - Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]])/(Sqrt[2]*Sqrt[a])","A",1
2353,1,184,186,0.3583929,"\int \frac{\sqrt{1+x}}{x+\sqrt{x+\sqrt{1+x}}} \, dx","IntegrateAlgebraic[Sqrt[1 + x]/(x + Sqrt[x + Sqrt[1 + x]]),x]","2 \sqrt{1+x}+2 \log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}\right)-4 \text{RootSum}\left[1+3 \text{$\#$1}-5 \text{$\#$1}^2+2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{-\log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right)-2 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}+3 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2}{3-10 \text{$\#$1}+6 \text{$\#$1}^2+4 \text{$\#$1}^3}\&\right]","2 \sqrt{1+x}+2 \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}\right)+2 \text{RootSum}\left[25-2 \text{$\#$1}^2-8 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{-5 \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right)-2 \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}+3 \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2}{-\text{$\#$1}-6 \text{$\#$1}^2+\text{$\#$1}^3}\&\right]",1,"2*Sqrt[1 + x] + 2*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]] - 4*RootSum[1 + 3*#1 - 5*#1^2 + 2*#1^3 + #1^4 & , (-Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1] - 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1 + 3*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^2)/(3 - 10*#1 + 6*#1^2 + 4*#1^3) & ]","A",1
2354,1,184,186,0.0009071,"\int \frac{\sqrt{1+x}}{x+\sqrt{x+\sqrt{1+x}}} \, dx","IntegrateAlgebraic[Sqrt[1 + x]/(x + Sqrt[x + Sqrt[1 + x]]),x]","2 \sqrt{1+x}+2 \log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}\right)-4 \text{RootSum}\left[1+3 \text{$\#$1}-5 \text{$\#$1}^2+2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{-\log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right)-2 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}+3 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2}{3-10 \text{$\#$1}+6 \text{$\#$1}^2+4 \text{$\#$1}^3}\&\right]","2 \sqrt{1+x}+2 \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}\right)+2 \text{RootSum}\left[25-2 \text{$\#$1}^2-8 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{-5 \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right)-2 \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}+3 \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2}{-\text{$\#$1}-6 \text{$\#$1}^2+\text{$\#$1}^3}\&\right]",1,"2*Sqrt[1 + x] + 2*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]] - 4*RootSum[1 + 3*#1 - 5*#1^2 + 2*#1^3 + #1^4 & , (-Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1] - 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1 + 3*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^2)/(3 - 10*#1 + 6*#1^2 + 4*#1^3) & ]","A",1
2355,1,56,187,3.626094,"\int \frac{-i+\sqrt{k} x}{\left(i+\sqrt{k} x\right) \sqrt{\left(1-x^2\right) \left(1-k^2 x^2\right)}} \, dx","IntegrateAlgebraic[(-I + Sqrt[k]*x)/((I + Sqrt[k]*x)*Sqrt[(1 - x^2)*(1 - k^2*x^2)]),x]","-\frac{2 \tan ^{-1}\left(\frac{(1+k) x}{-1+2 i \sqrt{k} x+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{1+k}","\frac{\tan ^{-1}\left(\frac{\left(-1-2 \sqrt{k}-k\right) x}{-1+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{1+k}+\frac{\tan ^{-1}\left(\frac{\left(-1+2 \sqrt{k}-k\right) x}{-1+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{1+k}+\frac{i \tanh ^{-1}\left(\frac{\left(2 \sqrt{k}+2 k^{3/2}\right) x^2}{1+2 k x^2+k^2 x^4+\left(-1+k x^2\right) \sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{1+k}",1,"(-2*ArcTan[((1 + k)*x)/(-1 + (2*I)*Sqrt[k]*x + k*x^2 + Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4])])/(1 + k)","A",1
2356,1,56,187,3.628235,"\int \frac{i+\sqrt{k} x}{\left(-i+\sqrt{k} x\right) \sqrt{\left(1-x^2\right) \left(1-k^2 x^2\right)}} \, dx","IntegrateAlgebraic[(I + Sqrt[k]*x)/((-I + Sqrt[k]*x)*Sqrt[(1 - x^2)*(1 - k^2*x^2)]),x]","-\frac{2 \tan ^{-1}\left(\frac{(1+k) x}{-1-2 i \sqrt{k} x+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{1+k}","\frac{\tan ^{-1}\left(\frac{\left(-1-2 \sqrt{k}-k\right) x}{-1+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{1+k}+\frac{\tan ^{-1}\left(\frac{\left(-1+2 \sqrt{k}-k\right) x}{-1+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{1+k}-\frac{i \tanh ^{-1}\left(\frac{\left(2 \sqrt{k}+2 k^{3/2}\right) x^2}{1+2 k x^2+k^2 x^4+\left(-1+k x^2\right) \sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{1+k}",1,"(-2*ArcTan[((1 + k)*x)/(-1 - (2*I)*Sqrt[k]*x + k*x^2 + Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4])])/(1 + k)","A",1
2357,1,211,187,18.7996008,"\int \frac{1}{\sqrt[4]{-1+3 x-3 x^2+x^3} \left(-1-2 x+x^2+3 x^3\right)^4} \, dx","IntegrateAlgebraic[1/((-1 + 3*x - 3*x^2 + x^3)^(1/4)*(-1 - 2*x + x^2 + 3*x^3)^4),x]","\frac{\left((-1+x)^3\right)^{3/4} \left(-\frac{\left(112513275+3037822236 (-1+x)+29822004289 (-1+x)^2+130678971480 (-1+x)^3+257277180684 (-1+x)^4+263070561900 (-1+x)^5+146937971619 (-1+x)^6+42764872266 (-1+x)^7+5094769914 (-1+x)^8\right) \sqrt[4]{-1+x}}{2859936 \left(1+9 (-1+x)+10 (-1+x)^2+3 (-1+x)^3\right)^3}-\frac{\text{RootSum}\left[1+9 \text{$\#$1}^4+10 \text{$\#$1}^8+3 \text{$\#$1}^{12}\&,\frac{-41317673 \log \left(\sqrt[4]{-1+x}-\text{$\#$1}\right)-234521814 \log \left(\sqrt[4]{-1+x}-\text{$\#$1}\right) \text{$\#$1}^4+566085546 \log \left(\sqrt[4]{-1+x}-\text{$\#$1}\right) \text{$\#$1}^8}{9 \text{$\#$1}^3+20 \text{$\#$1}^7+9 \text{$\#$1}^{11}}\&\right]}{3813248}\right)}{(-1+x)^{9/4}}","\frac{\left((-1+x)^3\right)^{3/4} \left(\frac{\sqrt[4]{-1+x} \left(307788101+1357068302 x+596630756 x^2-4979849490 x^3-5857310139 x^4+5802065412 x^5+9762576651 x^6-2006712954 x^7-5094769914 x^8\right)}{2859936 \left(-1-2 x+x^2+3 x^3\right)^3}-\frac{\text{RootSum}\left[1+9 \text{$\#$1}^4+10 \text{$\#$1}^8+3 \text{$\#$1}^{12}\&,\frac{-41317673 \log \left(\sqrt[4]{-1+x}-\text{$\#$1}\right)-234521814 \log \left(\sqrt[4]{-1+x}-\text{$\#$1}\right) \text{$\#$1}^4+566085546 \log \left(\sqrt[4]{-1+x}-\text{$\#$1}\right) \text{$\#$1}^8}{9 \text{$\#$1}^3+20 \text{$\#$1}^7+9 \text{$\#$1}^{11}}\&\right]}{3813248}\right)}{(-1+x)^{9/4}}",1,"(((-1 + x)^3)^(3/4)*(-1/2859936*((112513275 + 3037822236*(-1 + x) + 29822004289*(-1 + x)^2 + 130678971480*(-1 + x)^3 + 257277180684*(-1 + x)^4 + 263070561900*(-1 + x)^5 + 146937971619*(-1 + x)^6 + 42764872266*(-1 + x)^7 + 5094769914*(-1 + x)^8)*(-1 + x)^(1/4))/(1 + 9*(-1 + x) + 10*(-1 + x)^2 + 3*(-1 + x)^3)^3 - RootSum[1 + 9*#1^4 + 10*#1^8 + 3*#1^12 & , (-41317673*Log[(-1 + x)^(1/4) - #1] - 234521814*Log[(-1 + x)^(1/4) - #1]*#1^4 + 566085546*Log[(-1 + x)^(1/4) - #1]*#1^8)/(9*#1^3 + 20*#1^7 + 9*#1^11) & ]/3813248))/(-1 + x)^(9/4)","A",1
2358,1,186,187,0.6167295,"\int \frac{\sqrt[4]{b x^3+a x^4}}{x^2 \left(-d+c x^2\right)} \, dx","IntegrateAlgebraic[(b*x^3 + a*x^4)^(1/4)/(x^2*(-d + c*x^2)),x]","\frac{4 \sqrt[4]{b x^3+a x^4}}{d x}+\frac{\text{RootSum}\left[b^2 c-a^2 d+2 a d \text{$\#$1}^4-d \text{$\#$1}^8\&,\frac{-b^2 c \log (x)+a^2 d \log (x)+b^2 c \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)-a^2 d \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)-a d \log (x) \text{$\#$1}^4+a d \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}^3+\text{$\#$1}^7}\&\right]}{2 d^2}","\frac{4 \sqrt[4]{b x^3+a x^4}}{d x}+\frac{\text{RootSum}\left[b^2 c-a^2 d+2 a d \text{$\#$1}^4-d \text{$\#$1}^8\&,\frac{b^2 c \log (x)-a^2 d \log (x)-b^2 c \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)+a^2 d \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)+a d \log (x) \text{$\#$1}^4-a d \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}^3-\text{$\#$1}^7}\&\right]}{2 d^2}",1,"(4*(b*x^3 + a*x^4)^(1/4))/(d*x) + RootSum[b^2*c - a^2*d + 2*a*d*#1^4 - d*#1^8 & , (-(b^2*c*Log[x]) + a^2*d*Log[x] + b^2*c*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] - a^2*d*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] - a*d*Log[x]*#1^4 + a*d*Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3) + #1^7) & ]/(2*d^2)","A",1
2359,1,186,187,0.0016053,"\int \frac{\sqrt[4]{b x^3+a x^4}}{x^2 \left(-d+c x^2\right)} \, dx","IntegrateAlgebraic[(b*x^3 + a*x^4)^(1/4)/(x^2*(-d + c*x^2)),x]","\frac{4 \sqrt[4]{b x^3+a x^4}}{d x}+\frac{\text{RootSum}\left[b^2 c-a^2 d+2 a d \text{$\#$1}^4-d \text{$\#$1}^8\&,\frac{-b^2 c \log (x)+a^2 d \log (x)+b^2 c \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)-a^2 d \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)-a d \log (x) \text{$\#$1}^4+a d \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}^3+\text{$\#$1}^7}\&\right]}{2 d^2}","\frac{4 \sqrt[4]{b x^3+a x^4}}{d x}+\frac{\text{RootSum}\left[b^2 c-a^2 d+2 a d \text{$\#$1}^4-d \text{$\#$1}^8\&,\frac{b^2 c \log (x)-a^2 d \log (x)-b^2 c \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)+a^2 d \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)+a d \log (x) \text{$\#$1}^4-a d \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}^3-\text{$\#$1}^7}\&\right]}{2 d^2}",1,"(4*(b*x^3 + a*x^4)^(1/4))/(d*x) + RootSum[b^2*c - a^2*d + 2*a*d*#1^4 - d*#1^8 & , (-(b^2*c*Log[x]) + a^2*d*Log[x] + b^2*c*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] - a^2*d*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] - a*d*Log[x]*#1^4 + a*d*Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3) + #1^7) & ]/(2*d^2)","A",1
2360,1,187,187,0.6203954,"\int \frac{x^4}{\sqrt[4]{x^2+x^6} \left(-1+x^8\right)} \, dx","IntegrateAlgebraic[x^4/((x^2 + x^6)^(1/4)*(-1 + x^8)),x]","\frac{\left(x^2+x^6\right)^{3/4}}{2 x \left(1+x^4\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{8 \sqrt[4]{2}}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{8\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{8 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{8\ 2^{3/4}}","\frac{\left(x^2+x^6\right)^{3/4}}{2 x \left(1+x^4\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{8 \sqrt[4]{2}}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{8\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{8 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{8\ 2^{3/4}}",1,"(x^2 + x^6)^(3/4)/(2*x*(1 + x^4)) - ArcTan[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/(8*2^(1/4)) + ArcTan[(2^(3/4)*x*(x^2 + x^6)^(1/4))/(Sqrt[2]*x^2 - Sqrt[x^2 + x^6])]/(8*2^(3/4)) - ArcTanh[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/(8*2^(1/4)) - ArcTanh[(x^2/2^(1/4) + Sqrt[x^2 + x^6]/2^(3/4))/(x*(x^2 + x^6)^(1/4))]/(8*2^(3/4))","A",1
2361,1,187,187,0.0019038,"\int \frac{x^4}{\sqrt[4]{x^2+x^6} \left(-1+x^8\right)} \, dx","IntegrateAlgebraic[x^4/((x^2 + x^6)^(1/4)*(-1 + x^8)),x]","\frac{\left(x^2+x^6\right)^{3/4}}{2 x \left(1+x^4\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{8 \sqrt[4]{2}}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{8\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{8 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{8\ 2^{3/4}}","\frac{\left(x^2+x^6\right)^{3/4}}{2 x \left(1+x^4\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{8 \sqrt[4]{2}}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{8\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{8 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{8\ 2^{3/4}}",1,"(x^2 + x^6)^(3/4)/(2*x*(1 + x^4)) - ArcTan[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/(8*2^(1/4)) + ArcTan[(2^(3/4)*x*(x^2 + x^6)^(1/4))/(Sqrt[2]*x^2 - Sqrt[x^2 + x^6])]/(8*2^(3/4)) - ArcTanh[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/(8*2^(1/4)) - ArcTanh[(x^2/2^(1/4) + Sqrt[x^2 + x^6]/2^(3/4))/(x*(x^2 + x^6)^(1/4))]/(8*2^(3/4))","A",1
2362,1,187,187,0.8919217,"\int \frac{\left(-1+x^4\right) \sqrt[4]{x^2+x^6}}{1-x^4+x^8} \, dx","IntegrateAlgebraic[((-1 + x^4)*(x^2 + x^6)^(1/4))/(1 - x^4 + x^8),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[8]{3} x}{\sqrt[4]{x^2+x^6}}\right)}{2\ 3^{3/8}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} 3^{7/8} x \sqrt[4]{x^2+x^6}}{-3 x^2+3^{3/4} \sqrt{x^2+x^6}}\right)}{2 \sqrt{2} 3^{3/8}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[8]{3} x}{\sqrt[4]{x^2+x^6}}\right)}{2\ 3^{3/8}}+\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[8]{3} x^2}{\sqrt{2}}+\frac{\sqrt{x^2+x^6}}{\sqrt{2} \sqrt[8]{3}}}{x \sqrt[4]{x^2+x^6}}\right)}{2 \sqrt{2} 3^{3/8}}","\frac{\tan ^{-1}\left(\frac{\sqrt[8]{3} x}{\sqrt[4]{x^2+x^6}}\right)}{2\ 3^{3/8}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} 3^{7/8} x \sqrt[4]{x^2+x^6}}{-3 x^2+3^{3/4} \sqrt{x^2+x^6}}\right)}{2 \sqrt{2} 3^{3/8}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[8]{3} x}{\sqrt[4]{x^2+x^6}}\right)}{2\ 3^{3/8}}+\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[8]{3} x^2}{\sqrt{2}}+\frac{\sqrt{x^2+x^6}}{\sqrt{2} \sqrt[8]{3}}}{x \sqrt[4]{x^2+x^6}}\right)}{2 \sqrt{2} 3^{3/8}}",1,"ArcTan[(3^(1/8)*x)/(x^2 + x^6)^(1/4)]/(2*3^(3/8)) - ArcTan[(Sqrt[2]*3^(7/8)*x*(x^2 + x^6)^(1/4))/(-3*x^2 + 3^(3/4)*Sqrt[x^2 + x^6])]/(2*Sqrt[2]*3^(3/8)) - ArcTanh[(3^(1/8)*x)/(x^2 + x^6)^(1/4)]/(2*3^(3/8)) + ArcTanh[((3^(1/8)*x^2)/Sqrt[2] + Sqrt[x^2 + x^6]/(Sqrt[2]*3^(1/8)))/(x*(x^2 + x^6)^(1/4))]/(2*Sqrt[2]*3^(3/8))","A",1
2363,1,187,187,0.0012081,"\int \frac{\left(-1+x^4\right) \sqrt[4]{x^2+x^6}}{1-x^4+x^8} \, dx","IntegrateAlgebraic[((-1 + x^4)*(x^2 + x^6)^(1/4))/(1 - x^4 + x^8),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[8]{3} x}{\sqrt[4]{x^2+x^6}}\right)}{2\ 3^{3/8}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} 3^{7/8} x \sqrt[4]{x^2+x^6}}{-3 x^2+3^{3/4} \sqrt{x^2+x^6}}\right)}{2 \sqrt{2} 3^{3/8}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[8]{3} x}{\sqrt[4]{x^2+x^6}}\right)}{2\ 3^{3/8}}+\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[8]{3} x^2}{\sqrt{2}}+\frac{\sqrt{x^2+x^6}}{\sqrt{2} \sqrt[8]{3}}}{x \sqrt[4]{x^2+x^6}}\right)}{2 \sqrt{2} 3^{3/8}}","\frac{\tan ^{-1}\left(\frac{\sqrt[8]{3} x}{\sqrt[4]{x^2+x^6}}\right)}{2\ 3^{3/8}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} 3^{7/8} x \sqrt[4]{x^2+x^6}}{-3 x^2+3^{3/4} \sqrt{x^2+x^6}}\right)}{2 \sqrt{2} 3^{3/8}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[8]{3} x}{\sqrt[4]{x^2+x^6}}\right)}{2\ 3^{3/8}}+\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[8]{3} x^2}{\sqrt{2}}+\frac{\sqrt{x^2+x^6}}{\sqrt{2} \sqrt[8]{3}}}{x \sqrt[4]{x^2+x^6}}\right)}{2 \sqrt{2} 3^{3/8}}",1,"ArcTan[(3^(1/8)*x)/(x^2 + x^6)^(1/4)]/(2*3^(3/8)) - ArcTan[(Sqrt[2]*3^(7/8)*x*(x^2 + x^6)^(1/4))/(-3*x^2 + 3^(3/4)*Sqrt[x^2 + x^6])]/(2*Sqrt[2]*3^(3/8)) - ArcTanh[(3^(1/8)*x)/(x^2 + x^6)^(1/4)]/(2*3^(3/8)) + ArcTanh[((3^(1/8)*x^2)/Sqrt[2] + Sqrt[x^2 + x^6]/(Sqrt[2]*3^(1/8)))/(x*(x^2 + x^6)^(1/4))]/(2*Sqrt[2]*3^(3/8))","A",1
2364,1,187,187,0.8643348,"\int \frac{1-x^4+x^8}{\sqrt[4]{x^2+x^6} \left(-1+x^8\right)} \, dx","IntegrateAlgebraic[(1 - x^4 + x^8)/((x^2 + x^6)^(1/4)*(-1 + x^8)),x]","-\frac{3 \left(x^2+x^6\right)^{3/4}}{2 x \left(1+x^4\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{8 \sqrt[4]{2}}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{8\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{8 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{8\ 2^{3/4}}","-\frac{3 \left(x^2+x^6\right)^{3/4}}{2 x \left(1+x^4\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{8 \sqrt[4]{2}}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{8\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{8 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{8\ 2^{3/4}}",1,"(-3*(x^2 + x^6)^(3/4))/(2*x*(1 + x^4)) - ArcTan[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/(8*2^(1/4)) + ArcTan[(2^(3/4)*x*(x^2 + x^6)^(1/4))/(Sqrt[2]*x^2 - Sqrt[x^2 + x^6])]/(8*2^(3/4)) - ArcTanh[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/(8*2^(1/4)) - ArcTanh[(x^2/2^(1/4) + Sqrt[x^2 + x^6]/2^(3/4))/(x*(x^2 + x^6)^(1/4))]/(8*2^(3/4))","A",1
2365,1,187,187,0.0011828,"\int \frac{1-x^4+x^8}{\sqrt[4]{x^2+x^6} \left(-1+x^8\right)} \, dx","IntegrateAlgebraic[(1 - x^4 + x^8)/((x^2 + x^6)^(1/4)*(-1 + x^8)),x]","-\frac{3 \left(x^2+x^6\right)^{3/4}}{2 x \left(1+x^4\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{8 \sqrt[4]{2}}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{8\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{8 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{8\ 2^{3/4}}","-\frac{3 \left(x^2+x^6\right)^{3/4}}{2 x \left(1+x^4\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{8 \sqrt[4]{2}}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^2+x^6}}{\sqrt{2} x^2-\sqrt{x^2+x^6}}\right)}{8\ 2^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^2+x^6}}\right)}{8 \sqrt[4]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^2+x^6}}{2^{3/4}}}{x \sqrt[4]{x^2+x^6}}\right)}{8\ 2^{3/4}}",1,"(-3*(x^2 + x^6)^(3/4))/(2*x*(1 + x^4)) - ArcTan[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/(8*2^(1/4)) + ArcTan[(2^(3/4)*x*(x^2 + x^6)^(1/4))/(Sqrt[2]*x^2 - Sqrt[x^2 + x^6])]/(8*2^(3/4)) - ArcTanh[(2^(1/4)*x)/(x^2 + x^6)^(1/4)]/(8*2^(1/4)) - ArcTanh[(x^2/2^(1/4) + Sqrt[x^2 + x^6]/2^(3/4))/(x*(x^2 + x^6)^(1/4))]/(8*2^(3/4))","A",1
2366,1,188,188,1.2961183,"\int \frac{1}{(-b+a x) \sqrt[4]{-x^3+x^4}} \, dx","IntegrateAlgebraic[1/((-b + a*x)*(-x^3 + x^4)^(1/4)),x]","\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a-b} \sqrt[4]{b} x \sqrt[4]{-x^3+x^4}}{\sqrt{a-b} x^2-\sqrt{b} \sqrt{-x^3+x^4}}\right)}{\sqrt[4]{a-b} b^{3/4}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{a-b} x^2}{\sqrt{2} \sqrt[4]{b}}+\frac{\sqrt[4]{b} \sqrt{-x^3+x^4}}{\sqrt{2} \sqrt[4]{a-b}}}{x \sqrt[4]{-x^3+x^4}}\right)}{\sqrt[4]{a-b} b^{3/4}}","\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a-b} \sqrt[4]{b} x \sqrt[4]{-x^3+x^4}}{\sqrt{a-b} x^2-\sqrt{b} \sqrt{-x^3+x^4}}\right)}{\sqrt[4]{a-b} b^{3/4}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{a-b} x^2}{\sqrt{2} \sqrt[4]{b}}+\frac{\sqrt[4]{b} \sqrt{-x^3+x^4}}{\sqrt{2} \sqrt[4]{a-b}}}{x \sqrt[4]{-x^3+x^4}}\right)}{\sqrt[4]{a-b} b^{3/4}}",1,"(Sqrt[2]*ArcTan[(Sqrt[2]*(a - b)^(1/4)*b^(1/4)*x*(-x^3 + x^4)^(1/4))/(Sqrt[a - b]*x^2 - Sqrt[b]*Sqrt[-x^3 + x^4])])/((a - b)^(1/4)*b^(3/4)) - (Sqrt[2]*ArcTanh[(((a - b)^(1/4)*x^2)/(Sqrt[2]*b^(1/4)) + (b^(1/4)*Sqrt[-x^3 + x^4])/(Sqrt[2]*(a - b)^(1/4)))/(x*(-x^3 + x^4)^(1/4))])/((a - b)^(1/4)*b^(3/4))","A",1
2367,1,189,189,0.5244217,"\int \frac{-x+x^2}{\sqrt{(1-x) x \left(1-k^2 x\right)} \left(1-2 x+k^2 x^2\right)} \, dx","IntegrateAlgebraic[(-x + x^2)/(Sqrt[(1 - x)*x*(1 - k^2*x)]*(1 - 2*x + k^2*x^2)),x]","-\frac{i \tan ^{-1}\left(\frac{\sqrt{-2+k^2-2 i \sqrt{-1+k^2}} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{-1+k^2 x}\right)}{2 \sqrt{-1+k^2} \sqrt{-2+k^2-2 i \sqrt{-1+k^2}}}+\frac{i \tan ^{-1}\left(\frac{\sqrt{-2+k^2+2 i \sqrt{-1+k^2}} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{-1+k^2 x}\right)}{2 \sqrt{-1+k^2} \sqrt{-2+k^2+2 i \sqrt{-1+k^2}}}","-\frac{i \tan ^{-1}\left(\frac{\sqrt{-2+k^2-2 i \sqrt{-1+k^2}} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{-1+k^2 x}\right)}{2 \sqrt{-1+k^2} \sqrt{-2+k^2-2 i \sqrt{-1+k^2}}}+\frac{i \tan ^{-1}\left(\frac{\sqrt{-2+k^2+2 i \sqrt{-1+k^2}} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{-1+k^2 x}\right)}{2 \sqrt{-1+k^2} \sqrt{-2+k^2+2 i \sqrt{-1+k^2}}}",1,"((-1/2*I)*ArcTan[(Sqrt[-2 + k^2 - (2*I)*Sqrt[-1 + k^2]]*Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3])/(-1 + k^2*x)])/(Sqrt[-1 + k^2]*Sqrt[-2 + k^2 - (2*I)*Sqrt[-1 + k^2]]) + ((I/2)*ArcTan[(Sqrt[-2 + k^2 + (2*I)*Sqrt[-1 + k^2]]*Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3])/(-1 + k^2*x)])/(Sqrt[-1 + k^2]*Sqrt[-2 + k^2 + (2*I)*Sqrt[-1 + k^2]])","A",1
2368,1,193,189,0.555289,"\int \frac{-b+a^3 x^2}{\left(b+a^3 x^2\right) \sqrt[3]{-b x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[(-b + a^3*x^2)/((b + a^3*x^2)*(-(b*x^2) + a^3*x^3)^(1/3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{-b x^2+a^3 x^3}}\right)}{a}-\frac{\log \left(a^2 x-a \sqrt[3]{-b x^2+a^3 x^3}\right)}{a}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{-b x^2+a^3 x^3}+\left(-b x^2+a^3 x^3\right)^{2/3}\right)}{2 a}+\text{RootSum}\left[a^6+a^3 b-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{-b x^2+a^3 x^3}}\right)}{a}-\frac{\log \left(-a x+\sqrt[3]{-b x^2+a^3 x^3}\right)}{a}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{-b x^2+a^3 x^3}+\left(-b x^2+a^3 x^3\right)^{2/3}\right)}{2 a}+\text{RootSum}\left[a^6+a^3 b-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*a*x)/(a*x + 2*(-(b*x^2) + a^3*x^3)^(1/3))])/a - Log[a^2*x - a*(-(b*x^2) + a^3*x^3)^(1/3)]/a + Log[a^2*x^2 + a*x*(-(b*x^2) + a^3*x^3)^(1/3) + (-(b*x^2) + a^3*x^3)^(2/3)]/(2*a) + RootSum[a^6 + a^3*b - 2*a^3*#1^3 + #1^6 & , (-Log[x] + Log[(-(b*x^2) + a^3*x^3)^(1/3) - x*#1])/#1 & ]","A",1
2369,1,193,189,0.0098323,"\int \frac{-b+a^3 x^2}{\left(b+a^3 x^2\right) \sqrt[3]{-b x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[(-b + a^3*x^2)/((b + a^3*x^2)*(-(b*x^2) + a^3*x^3)^(1/3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{-b x^2+a^3 x^3}}\right)}{a}-\frac{\log \left(a^2 x-a \sqrt[3]{-b x^2+a^3 x^3}\right)}{a}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{-b x^2+a^3 x^3}+\left(-b x^2+a^3 x^3\right)^{2/3}\right)}{2 a}+\text{RootSum}\left[a^6+a^3 b-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{-b x^2+a^3 x^3}}\right)}{a}-\frac{\log \left(-a x+\sqrt[3]{-b x^2+a^3 x^3}\right)}{a}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{-b x^2+a^3 x^3}+\left(-b x^2+a^3 x^3\right)^{2/3}\right)}{2 a}+\text{RootSum}\left[a^6+a^3 b-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*a*x)/(a*x + 2*(-(b*x^2) + a^3*x^3)^(1/3))])/a - Log[a^2*x - a*(-(b*x^2) + a^3*x^3)^(1/3)]/a + Log[a^2*x^2 + a*x*(-(b*x^2) + a^3*x^3)^(1/3) + (-(b*x^2) + a^3*x^3)^(2/3)]/(2*a) + RootSum[a^6 + a^3*b - 2*a^3*#1^3 + #1^6 & , (-Log[x] + Log[(-(b*x^2) + a^3*x^3)^(1/3) - x*#1])/#1 & ]","A",1
2370,1,189,189,1.298943,"\int \frac{1+x^2}{\left(-1+x^2\right) \sqrt{a+b x+c x^2+b x^3+a x^4}} \, dx","IntegrateAlgebraic[(1 + x^2)/((-1 + x^2)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]),x]","-\frac{\sqrt{-2 a-2 b-c} \tan ^{-1}\left(\frac{\sqrt{-2 a-2 b-c} x}{\sqrt{a}-2 \sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{2 a+2 b+c}-\frac{\sqrt{-2 a+2 b-c} \tan ^{-1}\left(\frac{\sqrt{-2 a+2 b-c} x}{\sqrt{a}+2 \sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{2 a-2 b+c}","-\frac{\sqrt{-2 a-2 b-c} \tan ^{-1}\left(\frac{\sqrt{-2 a-2 b-c} x}{\sqrt{a}-2 \sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{2 a+2 b+c}-\frac{\sqrt{-2 a+2 b-c} \tan ^{-1}\left(\frac{\sqrt{-2 a+2 b-c} x}{\sqrt{a}+2 \sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{2 a-2 b+c}",1,"-((Sqrt[-2*a - 2*b - c]*ArcTan[(Sqrt[-2*a - 2*b - c]*x)/(Sqrt[a] - 2*Sqrt[a]*x + Sqrt[a]*x^2 - Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])])/(2*a + 2*b + c)) - (Sqrt[-2*a + 2*b - c]*ArcTan[(Sqrt[-2*a + 2*b - c]*x)/(Sqrt[a] + 2*Sqrt[a]*x + Sqrt[a]*x^2 - Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])])/(2*a - 2*b + c)","A",1
2371,1,189,189,0.2339229,"\int \frac{x^4 \sqrt[4]{-x^2+x^4}}{1+x^4+x^8} \, dx","IntegrateAlgebraic[(x^4*(-x^2 + x^4)^(1/4))/(1 + x^4 + x^8),x]","\frac{1}{4} \text{RootSum}\left[3-3 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-3 \log (x)+3 \log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 \text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]-\frac{1}{4} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]","\frac{1}{4} \text{RootSum}\left[3-3 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-3 \log (x)+3 \log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 \text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]-\frac{1}{4} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]",1,"RootSum[3 - 3*#1^4 + #1^8 & , (-3*Log[x] + 3*Log[(-x^2 + x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-x^2 + x^4)^(1/4) - x*#1]*#1^4)/(-3*#1^3 + 2*#1^7) & ]/4 - RootSum[1 - #1^4 + #1^8 & , (-Log[x] + Log[(-x^2 + x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-x^2 + x^4)^(1/4) - x*#1]*#1^4)/(-#1^3 + 2*#1^7) & ]/4","A",1
2372,1,189,189,0.0011076,"\int \frac{x^4 \sqrt[4]{-x^2+x^4}}{1+x^4+x^8} \, dx","IntegrateAlgebraic[(x^4*(-x^2 + x^4)^(1/4))/(1 + x^4 + x^8),x]","\frac{1}{4} \text{RootSum}\left[3-3 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-3 \log (x)+3 \log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 \text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]-\frac{1}{4} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]","\frac{1}{4} \text{RootSum}\left[3-3 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-3 \log (x)+3 \log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 \text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]-\frac{1}{4} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]",1,"RootSum[3 - 3*#1^4 + #1^8 & , (-3*Log[x] + 3*Log[(-x^2 + x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-x^2 + x^4)^(1/4) - x*#1]*#1^4)/(-3*#1^3 + 2*#1^7) & ]/4 - RootSum[1 - #1^4 + #1^8 & , (-Log[x] + Log[(-x^2 + x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-x^2 + x^4)^(1/4) - x*#1]*#1^4)/(-#1^3 + 2*#1^7) & ]/4","A",1
2373,1,189,189,1.5853216,"\int \frac{\left(-2 q+p x^3\right) \sqrt{q+p x^3}}{c x^4+b x^2 \left(q+p x^3\right)+a \left(q+p x^3\right)^2} \, dx","IntegrateAlgebraic[((-2*q + p*x^3)*Sqrt[q + p*x^3])/(c*x^4 + b*x^2*(q + p*x^3) + a*(q + p*x^3)^2),x]","-\frac{\sqrt{2} \left(-b+\sqrt{b^2-4 a c}\right) \tan ^{-1}\left(\frac{\sqrt{b-\sqrt{b^2-4 a c}} x}{\sqrt{2} \sqrt{a} \sqrt{q+p x^3}}\right)}{\sqrt{a} \sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \sqrt{b+\sqrt{b^2-4 a c}} \tan ^{-1}\left(\frac{\sqrt{b+\sqrt{b^2-4 a c}} x}{\sqrt{2} \sqrt{a} \sqrt{q+p x^3}}\right)}{\sqrt{a} \sqrt{b^2-4 a c}}","-\frac{\sqrt{2} \left(-b+\sqrt{b^2-4 a c}\right) \tan ^{-1}\left(\frac{\sqrt{b-\sqrt{b^2-4 a c}} x}{\sqrt{2} \sqrt{a} \sqrt{q+p x^3}}\right)}{\sqrt{a} \sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \sqrt{b+\sqrt{b^2-4 a c}} \tan ^{-1}\left(\frac{\sqrt{b+\sqrt{b^2-4 a c}} x}{\sqrt{2} \sqrt{a} \sqrt{q+p x^3}}\right)}{\sqrt{a} \sqrt{b^2-4 a c}}",1,"-((Sqrt[2]*(-b + Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[b - Sqrt[b^2 - 4*a*c]]*x)/(Sqrt[2]*Sqrt[a]*Sqrt[q + p*x^3])])/(Sqrt[a]*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]])) - (Sqrt[2]*Sqrt[b + Sqrt[b^2 - 4*a*c]]*ArcTan[(Sqrt[b + Sqrt[b^2 - 4*a*c]]*x)/(Sqrt[2]*Sqrt[a]*Sqrt[q + p*x^3])])/(Sqrt[a]*Sqrt[b^2 - 4*a*c])","A",1
2374,1,202,189,0.7021185,"\int \frac{\left(-1+x^4\right)^2}{\left(1+x^4\right)^2 \sqrt{x^2+\sqrt{1+x^4}}} \, dx","IntegrateAlgebraic[(-1 + x^4)^2/((1 + x^4)^2*Sqrt[x^2 + Sqrt[1 + x^4]]),x]","\frac{x^2 \sqrt{1+x^4} \left(9 x^2+6 x^6\right)+x^2 \left(4+12 x^4+6 x^8\right)}{2 x \sqrt{1+x^4} \left(2 x^2+2 x^6\right) \sqrt{x^2+\sqrt{1+x^4}}+2 x \left(1+3 x^4+2 x^8\right) \sqrt{x^2+\sqrt{1+x^4}}}-\frac{3}{2} \tanh ^{-1}\left(\frac{x}{\sqrt{x^2+\sqrt{1+x^4}}}\right)+\frac{\tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^4}}{\sqrt{2}}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)}{\sqrt{2}}","\frac{x^2 \sqrt{1+x^4} \left(9 x^2+6 x^6\right)+x^2 \left(4+12 x^4+6 x^8\right)}{2 x \sqrt{1+x^4} \left(2 x^2+2 x^6\right) \sqrt{x^2+\sqrt{1+x^4}}+2 x \left(1+3 x^4+2 x^8\right) \sqrt{x^2+\sqrt{1+x^4}}}-\frac{3}{2} \tanh ^{-1}\left(\frac{x}{\sqrt{x^2+\sqrt{1+x^4}}}\right)+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)}{\sqrt{2}}",1,"(x^2*Sqrt[1 + x^4]*(9*x^2 + 6*x^6) + x^2*(4 + 12*x^4 + 6*x^8))/(2*x*Sqrt[1 + x^4]*(2*x^2 + 2*x^6)*Sqrt[x^2 + Sqrt[1 + x^4]] + 2*x*(1 + 3*x^4 + 2*x^8)*Sqrt[x^2 + Sqrt[1 + x^4]]) - (3*ArcTanh[x/Sqrt[x^2 + Sqrt[1 + x^4]]])/2 + ArcTanh[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])]/Sqrt[2]","A",1
2375,1,247,189,1.3782051,"\int \frac{\sqrt{x^2+\sqrt{1+x^4}}}{\left(1+x^2\right)^2 \sqrt{1+x^4}} \, dx","IntegrateAlgebraic[Sqrt[x^2 + Sqrt[1 + x^4]]/((1 + x^2)^2*Sqrt[1 + x^4]),x]","\frac{-x^2 \left(-1+x^2\right)-x^2 \sqrt{1+x^4}}{4 x \left(1+x^2\right) \sqrt{x^2+\sqrt{1+x^4}}}+\frac{1}{4} \sqrt{-1+5 \sqrt{2}} \tan ^{-1}\left(\frac{-\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)+\frac{1}{4} \sqrt{1+5 \sqrt{2}} \tanh ^{-1}\left(\frac{-\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)","\frac{-x^2 \left(-1+x^2\right)-x^2 \sqrt{1+x^4}}{4 x \left(1+x^2\right) \sqrt{x^2+\sqrt{1+x^4}}}+\frac{1}{4} \sqrt{-1+5 \sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)+\frac{1}{4} \sqrt{1+5 \sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2 \left(-1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)",1,"(-(x^2*(-1 + x^2)) - x^2*Sqrt[1 + x^4])/(4*x*(1 + x^2)*Sqrt[x^2 + Sqrt[1 + x^4]]) + (Sqrt[-1 + 5*Sqrt[2]]*ArcTan[(-Sqrt[1/2 + 1/Sqrt[2]] + Sqrt[1/2 + 1/Sqrt[2]]*x^2 + Sqrt[1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])])/4 + (Sqrt[1 + 5*Sqrt[2]]*ArcTanh[(-Sqrt[-1/2 + 1/Sqrt[2]] + Sqrt[-1/2 + 1/Sqrt[2]]*x^2 + Sqrt[-1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])])/4","A",1
2376,1,189,189,13.364154,"\int \frac{\sqrt{q+p x^5} \left(-2 q+3 p x^5\right)}{c x^4+b x^2 \left(q+p x^5\right)+a \left(q+p x^5\right)^2} \, dx","IntegrateAlgebraic[(Sqrt[q + p*x^5]*(-2*q + 3*p*x^5))/(c*x^4 + b*x^2*(q + p*x^5) + a*(q + p*x^5)^2),x]","-\frac{\sqrt{2} \left(-b+\sqrt{b^2-4 a c}\right) \tan ^{-1}\left(\frac{\sqrt{b-\sqrt{b^2-4 a c}} x}{\sqrt{2} \sqrt{a} \sqrt{q+p x^5}}\right)}{\sqrt{a} \sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \sqrt{b+\sqrt{b^2-4 a c}} \tan ^{-1}\left(\frac{\sqrt{b+\sqrt{b^2-4 a c}} x}{\sqrt{2} \sqrt{a} \sqrt{q+p x^5}}\right)}{\sqrt{a} \sqrt{b^2-4 a c}}","-\frac{\sqrt{2} \left(-b+\sqrt{b^2-4 a c}\right) \tan ^{-1}\left(\frac{\sqrt{b-\sqrt{b^2-4 a c}} x}{\sqrt{2} \sqrt{a} \sqrt{q+p x^5}}\right)}{\sqrt{a} \sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \sqrt{b+\sqrt{b^2-4 a c}} \tan ^{-1}\left(\frac{\sqrt{b+\sqrt{b^2-4 a c}} x}{\sqrt{2} \sqrt{a} \sqrt{q+p x^5}}\right)}{\sqrt{a} \sqrt{b^2-4 a c}}",1,"-((Sqrt[2]*(-b + Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[b - Sqrt[b^2 - 4*a*c]]*x)/(Sqrt[2]*Sqrt[a]*Sqrt[q + p*x^5])])/(Sqrt[a]*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]])) - (Sqrt[2]*Sqrt[b + Sqrt[b^2 - 4*a*c]]*ArcTan[(Sqrt[b + Sqrt[b^2 - 4*a*c]]*x)/(Sqrt[2]*Sqrt[a]*Sqrt[q + p*x^5])])/(Sqrt[a]*Sqrt[b^2 - 4*a*c])","A",1
2377,1,189,189,0.4461087,"\int \frac{1}{x \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}} \, dx","IntegrateAlgebraic[1/(x*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]),x]","\frac{2 \sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}}{c \sqrt{a x+\sqrt{-b+a^2 x^2}}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{c}}\right)}{c^{3/2}}-\text{RootSum}\left[b+c^4-4 c^3 \text{$\#$1}^2+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}-\text{$\#$1}\right)}{c \text{$\#$1}-\text{$\#$1}^3}\&\right]","\frac{2 \sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}}{c \sqrt{a x+\sqrt{-b+a^2 x^2}}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{c}}\right)}{c^{3/2}}-\text{RootSum}\left[b+c^4-4 c^3 \text{$\#$1}^2+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}-\text{$\#$1}\right)}{c \text{$\#$1}-\text{$\#$1}^3}\&\right]",1,"(2*Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]])/(c*Sqrt[a*x + Sqrt[-b + a^2*x^2]]) - (2*ArcTanh[Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]/Sqrt[c]])/c^(3/2) - RootSum[b + c^4 - 4*c^3*#1^2 + 6*c^2*#1^4 - 4*c*#1^6 + #1^8 & , Log[Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]] - #1]/(c*#1 - #1^3) & ]","A",1
2378,0,0,190,3.8563729,"\int \frac{1}{\sqrt[3]{1+x^2} \left(9+x^2\right)} \, dx","IntegrateAlgebraic[1/((1 + x^2)^(1/3)*(9 + x^2)),x]","\int \frac{1}{\sqrt[3]{1+x^2} \left(9+x^2\right)} \, dx","\frac{1}{12} \tan ^{-1}\left(\frac{x}{1+2 \sqrt[3]{1+x^2}}\right)+\frac{i \tan ^{-1}\left(\frac{-\frac{1}{\sqrt{3}}-\frac{i x}{\sqrt{3}}+\frac{\sqrt[3]{1+x^2}}{\sqrt{3}}}{\sqrt[3]{1+x^2}}\right)}{8 \sqrt{3}}-\frac{i \tan ^{-1}\left(\frac{-\frac{1}{\sqrt{3}}+\frac{i x}{\sqrt{3}}+\frac{\sqrt[3]{1+x^2}}{\sqrt{3}}}{\sqrt[3]{1+x^2}}\right)}{8 \sqrt{3}}-\frac{1}{24} i \tanh ^{-1}\left(\frac{2 i x-2 i x \sqrt[3]{1+x^2}}{-1+x^2+2 \sqrt[3]{1+x^2}-4 \left(1+x^2\right)^{2/3}}\right)",1,"Could not integrate","F",-1
2379,1,194,190,0.5291603,"\int \frac{b+a^3 x^2}{\left(-b+a^3 x^2\right) \sqrt[3]{-b x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[(b + a^3*x^2)/((-b + a^3*x^2)*(-(b*x^2) + a^3*x^3)^(1/3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{-b x^2+a^3 x^3}}\right)}{a}-\frac{\log \left(a^2 x-a \sqrt[3]{-b x^2+a^3 x^3}\right)}{a}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{-b x^2+a^3 x^3}+\left(-b x^2+a^3 x^3\right)^{2/3}\right)}{2 a}+\text{RootSum}\left[a^6-a^3 b-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{-b x^2+a^3 x^3}}\right)}{a}-\frac{\log \left(-a x+\sqrt[3]{-b x^2+a^3 x^3}\right)}{a}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{-b x^2+a^3 x^3}+\left(-b x^2+a^3 x^3\right)^{2/3}\right)}{2 a}+\text{RootSum}\left[a^6-a^3 b-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*a*x)/(a*x + 2*(-(b*x^2) + a^3*x^3)^(1/3))])/a - Log[a^2*x - a*(-(b*x^2) + a^3*x^3)^(1/3)]/a + Log[a^2*x^2 + a*x*(-(b*x^2) + a^3*x^3)^(1/3) + (-(b*x^2) + a^3*x^3)^(2/3)]/(2*a) + RootSum[a^6 - a^3*b - 2*a^3*#1^3 + #1^6 & , (-Log[x] + Log[(-(b*x^2) + a^3*x^3)^(1/3) - x*#1])/#1 & ]","A",1
2380,1,194,190,0.0021085,"\int \frac{b+a^3 x^2}{\left(-b+a^3 x^2\right) \sqrt[3]{-b x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[(b + a^3*x^2)/((-b + a^3*x^2)*(-(b*x^2) + a^3*x^3)^(1/3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{-b x^2+a^3 x^3}}\right)}{a}-\frac{\log \left(a^2 x-a \sqrt[3]{-b x^2+a^3 x^3}\right)}{a}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{-b x^2+a^3 x^3}+\left(-b x^2+a^3 x^3\right)^{2/3}\right)}{2 a}+\text{RootSum}\left[a^6-a^3 b-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{-b x^2+a^3 x^3}}\right)}{a}-\frac{\log \left(-a x+\sqrt[3]{-b x^2+a^3 x^3}\right)}{a}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{-b x^2+a^3 x^3}+\left(-b x^2+a^3 x^3\right)^{2/3}\right)}{2 a}+\text{RootSum}\left[a^6-a^3 b-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-b x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*a*x)/(a*x + 2*(-(b*x^2) + a^3*x^3)^(1/3))])/a - Log[a^2*x - a*(-(b*x^2) + a^3*x^3)^(1/3)]/a + Log[a^2*x^2 + a*x*(-(b*x^2) + a^3*x^3)^(1/3) + (-(b*x^2) + a^3*x^3)^(2/3)]/(2*a) + RootSum[a^6 - a^3*b - 2*a^3*#1^3 + #1^6 & , (-Log[x] + Log[(-(b*x^2) + a^3*x^3)^(1/3) - x*#1])/#1 & ]","A",1
2381,1,190,190,8.1047528,"\int \frac{-3-2 \left(1+k^2\right) x+\left(1+k^2\right) x^2+4 k^2 x^3+k^2 x^4}{\left(\left(1-x^2\right) \left(1-k^2 x^2\right)\right)^{2/3} \left(-1+d-(2+d) x-\left(1+d k^2\right) x^2+d k^2 x^3\right)} \, dx","IntegrateAlgebraic[(-3 - 2*(1 + k^2)*x + (1 + k^2)*x^2 + 4*k^2*x^3 + k^2*x^4)/(((1 - x^2)*(1 - k^2*x^2))^(2/3)*(-1 + d - (2 + d)*x - (1 + d*k^2)*x^2 + d*k^2*x^3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3}+\sqrt{3} x}{1+x+2 \sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{\sqrt[3]{d}}+\frac{\log \left(1+x-\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{\sqrt[3]{d}}-\frac{\log \left(1+2 x+x^2+\left(\sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+d^{2/3} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3}+\sqrt{3} x}{1+x+2 \sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{\sqrt[3]{d}}+\frac{\log \left(1+x-\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{\sqrt[3]{d}}-\frac{\log \left(1+2 x+x^2+\left(\sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+d^{2/3} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3] + Sqrt[3]*x)/(1 + x + 2*d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))])/d^(1/3) + Log[1 + x - d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3)]/d^(1/3) - Log[1 + 2*x + x^2 + (d^(1/3) + d^(1/3)*x)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3) + d^(2/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3)]/(2*d^(1/3))","A",1
2382,1,190,190,0.6646412,"\int \frac{\sqrt[4]{x^3+x^5} \left(1+x^4+x^8\right)}{x^4 \left(-1+x^4\right)} \, dx","IntegrateAlgebraic[((x^3 + x^5)^(1/4)*(1 + x^4 + x^8))/(x^4*(-1 + x^4)),x]","\frac{4 \left(1+2 x^2+x^4\right) \sqrt[4]{x^3+x^5}}{9 x^3}+\frac{3 \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^5}}\right)}{2\ 2^{3/4}}-\frac{3 \tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^3+x^5}}{\sqrt{2} x^2-\sqrt{x^3+x^5}}\right)}{4 \sqrt[4]{2}}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^5}}\right)}{2\ 2^{3/4}}-\frac{3 \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^3+x^5}}{2^{3/4}}}{x \sqrt[4]{x^3+x^5}}\right)}{4 \sqrt[4]{2}}","\frac{4 \left(1+2 x^2+x^4\right) \sqrt[4]{x^3+x^5}}{9 x^3}+\frac{3 \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^5}}\right)}{2\ 2^{3/4}}-\frac{3 \tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^3+x^5}}{\sqrt{2} x^2-\sqrt{x^3+x^5}}\right)}{4 \sqrt[4]{2}}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^5}}\right)}{2\ 2^{3/4}}-\frac{3 \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^3+x^5}}{2^{3/4}}}{x \sqrt[4]{x^3+x^5}}\right)}{4 \sqrt[4]{2}}",1,"(4*(1 + 2*x^2 + x^4)*(x^3 + x^5)^(1/4))/(9*x^3) + (3*ArcTan[(2^(1/4)*x)/(x^3 + x^5)^(1/4)])/(2*2^(3/4)) - (3*ArcTan[(2^(3/4)*x*(x^3 + x^5)^(1/4))/(Sqrt[2]*x^2 - Sqrt[x^3 + x^5])])/(4*2^(1/4)) - (3*ArcTanh[(2^(1/4)*x)/(x^3 + x^5)^(1/4)])/(2*2^(3/4)) - (3*ArcTanh[(x^2/2^(1/4) + Sqrt[x^3 + x^5]/2^(3/4))/(x*(x^3 + x^5)^(1/4))])/(4*2^(1/4))","A",1
2383,1,191,191,0.4946992,"\int \frac{1}{\left(b+a x^2\right) \sqrt[3]{x+x^3}} \, dx","IntegrateAlgebraic[1/((b + a*x^2)*(x + x^3)^(1/3)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a-b} x}{\sqrt[3]{a-b} x-2 \sqrt[3]{b} \sqrt[3]{x+x^3}}\right)}{2 \sqrt[3]{a-b} b^{2/3}}+\frac{\log \left(\sqrt[3]{a-b} x+\sqrt[3]{b} \sqrt[3]{x+x^3}\right)}{2 \sqrt[3]{a-b} b^{2/3}}-\frac{\log \left((a-b)^{2/3} x^2-\sqrt[3]{a-b} \sqrt[3]{b} x \sqrt[3]{x+x^3}+b^{2/3} \left(x+x^3\right)^{2/3}\right)}{4 \sqrt[3]{a-b} b^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a-b} x}{\sqrt[3]{a-b} x-2 \sqrt[3]{b} \sqrt[3]{x+x^3}}\right)}{2 \sqrt[3]{a-b} b^{2/3}}+\frac{\log \left(\sqrt[3]{a-b} x+\sqrt[3]{b} \sqrt[3]{x+x^3}\right)}{2 \sqrt[3]{a-b} b^{2/3}}-\frac{\log \left((a-b)^{2/3} x^2-\sqrt[3]{a-b} \sqrt[3]{b} x \sqrt[3]{x+x^3}+b^{2/3} \left(x+x^3\right)^{2/3}\right)}{4 \sqrt[3]{a-b} b^{2/3}}",1,"-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*(a - b)^(1/3)*x)/((a - b)^(1/3)*x - 2*b^(1/3)*(x + x^3)^(1/3))])/((a - b)^(1/3)*b^(2/3)) + Log[(a - b)^(1/3)*x + b^(1/3)*(x + x^3)^(1/3)]/(2*(a - b)^(1/3)*b^(2/3)) - Log[(a - b)^(2/3)*x^2 - (a - b)^(1/3)*b^(1/3)*x*(x + x^3)^(1/3) + b^(2/3)*(x + x^3)^(2/3)]/(4*(a - b)^(1/3)*b^(2/3))","A",1
2384,1,190,191,0.6308788,"\int \frac{\sqrt[4]{-b x^3+a x^4}}{x^2 \left(-d+c x^2\right)} \, dx","IntegrateAlgebraic[(-(b*x^3) + a*x^4)^(1/4)/(x^2*(-d + c*x^2)),x]","\frac{4 \sqrt[4]{-b x^3+a x^4}}{d x}+\frac{\text{RootSum}\left[b^2 c-a^2 d+2 a d \text{$\#$1}^4-d \text{$\#$1}^8\&,\frac{-b^2 c \log (x)+a^2 d \log (x)+b^2 c \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right)-a^2 d \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right)-a d \log (x) \text{$\#$1}^4+a d \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}^3+\text{$\#$1}^7}\&\right]}{2 d^2}","\frac{4 \sqrt[4]{-b x^3+a x^4}}{d x}+\frac{\text{RootSum}\left[b^2 c-a^2 d+2 a d \text{$\#$1}^4-d \text{$\#$1}^8\&,\frac{b^2 c \log (x)-a^2 d \log (x)-b^2 c \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right)+a^2 d \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right)+a d \log (x) \text{$\#$1}^4-a d \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}^3-\text{$\#$1}^7}\&\right]}{2 d^2}",1,"(4*(-(b*x^3) + a*x^4)^(1/4))/(d*x) + RootSum[b^2*c - a^2*d + 2*a*d*#1^4 - d*#1^8 & , (-(b^2*c*Log[x]) + a^2*d*Log[x] + b^2*c*Log[(-(b*x^3) + a*x^4)^(1/4) - x*#1] - a^2*d*Log[(-(b*x^3) + a*x^4)^(1/4) - x*#1] - a*d*Log[x]*#1^4 + a*d*Log[(-(b*x^3) + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3) + #1^7) & ]/(2*d^2)","A",1
2385,1,190,191,0.0018124,"\int \frac{\sqrt[4]{-b x^3+a x^4}}{x^2 \left(-d+c x^2\right)} \, dx","IntegrateAlgebraic[(-(b*x^3) + a*x^4)^(1/4)/(x^2*(-d + c*x^2)),x]","\frac{4 \sqrt[4]{-b x^3+a x^4}}{d x}+\frac{\text{RootSum}\left[b^2 c-a^2 d+2 a d \text{$\#$1}^4-d \text{$\#$1}^8\&,\frac{-b^2 c \log (x)+a^2 d \log (x)+b^2 c \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right)-a^2 d \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right)-a d \log (x) \text{$\#$1}^4+a d \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}^3+\text{$\#$1}^7}\&\right]}{2 d^2}","\frac{4 \sqrt[4]{-b x^3+a x^4}}{d x}+\frac{\text{RootSum}\left[b^2 c-a^2 d+2 a d \text{$\#$1}^4-d \text{$\#$1}^8\&,\frac{b^2 c \log (x)-a^2 d \log (x)-b^2 c \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right)+a^2 d \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right)+a d \log (x) \text{$\#$1}^4-a d \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}^3-\text{$\#$1}^7}\&\right]}{2 d^2}",1,"(4*(-(b*x^3) + a*x^4)^(1/4))/(d*x) + RootSum[b^2*c - a^2*d + 2*a*d*#1^4 - d*#1^8 & , (-(b^2*c*Log[x]) + a^2*d*Log[x] + b^2*c*Log[(-(b*x^3) + a*x^4)^(1/4) - x*#1] - a^2*d*Log[(-(b*x^3) + a*x^4)^(1/4) - x*#1] - a*d*Log[x]*#1^4 + a*d*Log[(-(b*x^3) + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3) + #1^7) & ]/(2*d^2)","A",1
2386,1,194,191,2.9278539,"\int \frac{\sqrt[3]{x+x^3} \left(b+a x^6\right)}{d+c x^6} \, dx","IntegrateAlgebraic[((x + x^3)^(1/3)*(b + a*x^6))/(d + c*x^6),x]","\frac{a x \sqrt[3]{x+x^3}}{2 c}-\frac{a \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)}{2 \sqrt{3} c}-\frac{a \log \left(-c x+c \sqrt[3]{x+x^3}\right)}{6 c}+\frac{a \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)}{12 c}+\frac{(-b c+a d) \text{RootSum}\left[c-d+3 d \text{$\#$1}^3-3 d \text{$\#$1}^6+d \text{$\#$1}^9\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{x+x^3}-x \text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^3}\&\right]}{6 c d}","\frac{a x \sqrt[3]{x+x^3}}{2 c}-\frac{a \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)}{2 \sqrt{3} c}-\frac{a \log \left(-x+\sqrt[3]{x+x^3}\right)}{6 c}+\frac{a \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)}{12 c}+\frac{(-b c+a d) \text{RootSum}\left[c-d+3 d \text{$\#$1}^3-3 d \text{$\#$1}^6+d \text{$\#$1}^9\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{x+x^3}-x \text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^3}\&\right]}{6 c d}",1,"(a*x*(x + x^3)^(1/3))/(2*c) - (a*ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^3)^(1/3))])/(2*Sqrt[3]*c) - (a*Log[-(c*x) + c*(x + x^3)^(1/3)])/(6*c) + (a*Log[x^2 + x*(x + x^3)^(1/3) + (x + x^3)^(2/3)])/(12*c) + ((-(b*c) + a*d)*RootSum[c - d + 3*d*#1^3 - 3*d*#1^6 + d*#1^9 & , (-(Log[x]*#1) + Log[(x + x^3)^(1/3) - x*#1]*#1)/(-1 + #1^3) & ])/(6*c*d)","A",1
2387,1,194,191,0.0012232,"\int \frac{\sqrt[3]{x+x^3} \left(b+a x^6\right)}{d+c x^6} \, dx","IntegrateAlgebraic[((x + x^3)^(1/3)*(b + a*x^6))/(d + c*x^6),x]","\frac{a x \sqrt[3]{x+x^3}}{2 c}-\frac{a \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)}{2 \sqrt{3} c}-\frac{a \log \left(-c x+c \sqrt[3]{x+x^3}\right)}{6 c}+\frac{a \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)}{12 c}+\frac{(-b c+a d) \text{RootSum}\left[c-d+3 d \text{$\#$1}^3-3 d \text{$\#$1}^6+d \text{$\#$1}^9\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{x+x^3}-x \text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^3}\&\right]}{6 c d}","\frac{a x \sqrt[3]{x+x^3}}{2 c}-\frac{a \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)}{2 \sqrt{3} c}-\frac{a \log \left(-x+\sqrt[3]{x+x^3}\right)}{6 c}+\frac{a \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)}{12 c}+\frac{(-b c+a d) \text{RootSum}\left[c-d+3 d \text{$\#$1}^3-3 d \text{$\#$1}^6+d \text{$\#$1}^9\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{x+x^3}-x \text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^3}\&\right]}{6 c d}",1,"(a*x*(x + x^3)^(1/3))/(2*c) - (a*ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^3)^(1/3))])/(2*Sqrt[3]*c) - (a*Log[-(c*x) + c*(x + x^3)^(1/3)])/(6*c) + (a*Log[x^2 + x*(x + x^3)^(1/3) + (x + x^3)^(2/3)])/(12*c) + ((-(b*c) + a*d)*RootSum[c - d + 3*d*#1^3 - 3*d*#1^6 + d*#1^9 & , (-(Log[x]*#1) + Log[(x + x^3)^(1/3) - x*#1]*#1)/(-1 + #1^3) & ])/(6*c*d)","A",1
2388,1,191,191,2.9916732,"\int \frac{\left(-1+2 x^6\right) \sqrt[3]{x+x^7}}{\left(1-2 x^2+x^6\right) \left(1-x^2+x^6\right)} \, dx","IntegrateAlgebraic[((-1 + 2*x^6)*(x + x^7)^(1/3))/((1 - 2*x^2 + x^6)*(1 - x^2 + x^6)),x]","-\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^7}}\right)+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x+x^7}}\right)}{2^{2/3}}-\frac{1}{2} \log \left(-x+\sqrt[3]{x+x^7}\right)+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x+x^7}\right)}{2^{2/3}}+\frac{1}{4} \log \left(x^2+x \sqrt[3]{x+x^7}+\left(x+x^7\right)^{2/3}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x+x^7}+\sqrt[3]{2} \left(x+x^7\right)^{2/3}\right)}{2\ 2^{2/3}}","-\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^7}}\right)+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x+x^7}}\right)}{2^{2/3}}-\frac{1}{2} \log \left(-x+\sqrt[3]{x+x^7}\right)+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x+x^7}\right)}{2^{2/3}}+\frac{1}{4} \log \left(x^2+x \sqrt[3]{x+x^7}+\left(x+x^7\right)^{2/3}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x+x^7}+\sqrt[3]{2} \left(x+x^7\right)^{2/3}\right)}{2\ 2^{2/3}}",1,"-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^7)^(1/3))]) + (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(x + x^7)^(1/3))])/2^(2/3) - Log[-x + (x + x^7)^(1/3)]/2 + Log[-2*x + 2^(2/3)*(x + x^7)^(1/3)]/2^(2/3) + Log[x^2 + x*(x + x^7)^(1/3) + (x + x^7)^(2/3)]/4 - Log[2*x^2 + 2^(2/3)*x*(x + x^7)^(1/3) + 2^(1/3)*(x + x^7)^(2/3)]/(2*2^(2/3))","A",1
2389,1,191,191,0.4910311,"\int \frac{\left(-q+2 p x^3\right) \sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left(b x^3+a \left(q+p x^3\right)^3\right)}{x^6} \, dx","IntegrateAlgebraic[((-q + 2*p*x^3)*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]*(b*x^3 + a*(q + p*x^3)^3))/x^6,x]","\frac{\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left(6 a q^4-4 a p q^3 x^2+15 b q x^3+24 a p q^3 x^3-16 a p^2 q^2 x^4-8 a p^2 q^2 x^5+15 b p x^6+36 a p^2 q^2 x^6-4 a p^3 q x^8+24 a p^3 q x^9+6 a p^4 x^{12}\right)}{30 x^5}+b p q \log (x)-b p q \log \left(q+p x^3+\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right)","\frac{\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left(6 a q^4-4 a p q^3 x^2+15 b q x^3+24 a p q^3 x^3-16 a p^2 q^2 x^4-8 a p^2 q^2 x^5+15 b p x^6+36 a p^2 q^2 x^6-4 a p^3 q x^8+24 a p^3 q x^9+6 a p^4 x^{12}\right)}{30 x^5}+b p q \log (x)-b p q \log \left(q+p x^3+\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right)",1,"(Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]*(6*a*q^4 - 4*a*p*q^3*x^2 + 15*b*q*x^3 + 24*a*p*q^3*x^3 - 16*a*p^2*q^2*x^4 - 8*a*p^2*q^2*x^5 + 15*b*p*x^6 + 36*a*p^2*q^2*x^6 - 4*a*p^3*q*x^8 + 24*a*p^3*q*x^9 + 6*a*p^4*x^12))/(30*x^5) + b*p*q*Log[x] - b*p*q*Log[q + p*x^3 + Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]]","A",1
2390,1,192,192,0.6409109,"\int \frac{-1+(-1+2 k) x}{\sqrt[3]{(1-x) x (1-k x)} \left(1-(2+b) x+(1+b k) x^2\right)} \, dx","IntegrateAlgebraic[(-1 + (-1 + 2*k)*x)/(((1 - x)*x*(1 - k*x))^(1/3)*(1 - (2 + b)*x + (1 + b*k)*x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}{2-2 x+\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{b^{2/3}}+\frac{\log \left(-1+x+\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{b^{2/3}}-\frac{\log \left(1-2 x+x^2+\left(\sqrt[3]{b}-\sqrt[3]{b} x\right) \sqrt[3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 b^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}{2-2 x+\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{b^{2/3}}+\frac{\log \left(-1+x+\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{b^{2/3}}-\frac{\log \left(1-2 x+x^2+\left(\sqrt[3]{b}-\sqrt[3]{b} x\right) \sqrt[3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 b^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))/(2 - 2*x + b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/b^(2/3) + Log[-1 + x + b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3)]/b^(2/3) - Log[1 - 2*x + x^2 + (b^(1/3) - b^(1/3)*x)*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + b^(2/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3)]/(2*b^(2/3))","A",1
2391,1,192,192,8.1370886,"\int \frac{-3+2 \left(1+k^2\right) x+\left(1+k^2\right) x^2-4 k^2 x^3+k^2 x^4}{\left(\left(1-x^2\right) \left(1-k^2 x^2\right)\right)^{2/3} \left(1-d-(2+d) x+\left(1+d k^2\right) x^2+d k^2 x^3\right)} \, dx","IntegrateAlgebraic[(-3 + 2*(1 + k^2)*x + (1 + k^2)*x^2 - 4*k^2*x^3 + k^2*x^4)/(((1 - x^2)*(1 - k^2*x^2))^(2/3)*(1 - d - (2 + d)*x + (1 + d*k^2)*x^2 + d*k^2*x^3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{-\sqrt{3}+\sqrt{3} x}{-1+x-2 \sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{\sqrt[3]{d}}+\frac{\log \left(-1+x+\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{\sqrt[3]{d}}-\frac{\log \left(1-2 x+x^2+\left(\sqrt[3]{d}-\sqrt[3]{d} x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+d^{2/3} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{-\sqrt{3}+\sqrt{3} x}{-1+x-2 \sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{\sqrt[3]{d}}+\frac{\log \left(-1+x+\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{\sqrt[3]{d}}-\frac{\log \left(1-2 x+x^2+\left(\sqrt[3]{d}-\sqrt[3]{d} x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+d^{2/3} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(-Sqrt[3] + Sqrt[3]*x)/(-1 + x - 2*d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))])/d^(1/3) + Log[-1 + x + d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3)]/d^(1/3) - Log[1 - 2*x + x^2 + (d^(1/3) - d^(1/3)*x)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3) + d^(2/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3)]/(2*d^(1/3))","A",1
2392,1,192,192,1.0608239,"\int \frac{b-3 a x^3+3 x^6}{x^6 \left(-b+2 a x^3\right) \sqrt[4]{-b x+a x^4}} \, dx","IntegrateAlgebraic[(b - 3*a*x^3 + 3*x^6)/(x^6*(-b + 2*a*x^3)*(-(b*x) + a*x^4)^(1/4)),x]","-\frac{4 \left(b-a x^3\right) \left(-b x+a x^4\right)^{3/4}}{21 b^2 x^6}+\frac{\sqrt{2} \left(2 a^2-3 b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} x \sqrt[4]{-b x+a x^4}}{-\sqrt{a} x^2+\sqrt{-b x+a x^4}}\right)}{3 \sqrt[4]{a} b^2}+\frac{\sqrt{2} \left(2 a^2-3 b\right) \tanh ^{-1}\left(\frac{\sqrt{a} x^2+\sqrt{-b x+a x^4}}{\sqrt{2} \sqrt[4]{a} x \sqrt[4]{-b x+a x^4}}\right)}{3 \sqrt[4]{a} b^2}","-\frac{4 \left(b-a x^3\right) \left(-b x+a x^4\right)^{3/4}}{21 b^2 x^6}+\frac{\sqrt{2} \left(2 a^2-3 b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} x \sqrt[4]{-b x+a x^4}}{-\sqrt{a} x^2+\sqrt{-b x+a x^4}}\right)}{3 \sqrt[4]{a} b^2}+\frac{\sqrt{2} \left(2 a^2-3 b\right) \tanh ^{-1}\left(\frac{\sqrt{a} x^2+\sqrt{-b x+a x^4}}{\sqrt{2} \sqrt[4]{a} x \sqrt[4]{-b x+a x^4}}\right)}{3 \sqrt[4]{a} b^2}",1,"(-4*(b - a*x^3)*(-(b*x) + a*x^4)^(3/4))/(21*b^2*x^6) + (Sqrt[2]*(2*a^2 - 3*b)*ArcTan[(Sqrt[2]*a^(1/4)*x*(-(b*x) + a*x^4)^(1/4))/(-(Sqrt[a]*x^2) + Sqrt[-(b*x) + a*x^4])])/(3*a^(1/4)*b^2) + (Sqrt[2]*(2*a^2 - 3*b)*ArcTanh[(Sqrt[a]*x^2 + Sqrt[-(b*x) + a*x^4])/(Sqrt[2]*a^(1/4)*x*(-(b*x) + a*x^4)^(1/4))])/(3*a^(1/4)*b^2)","A",1
2393,1,192,192,0.5826869,"\int \frac{\left(-2 q+p x^3\right) \sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left(b x^6+a \left(q+p x^3\right)^3\right)}{x^{11}} \, dx","IntegrateAlgebraic[((-2*q + p*x^3)*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]*(b*x^6 + a*(q + p*x^3)^3))/x^11,x]","\frac{\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left(6 a q^4+24 a p q^3 x^3-4 a p q^3 x^4+15 b q x^6+36 a p^2 q^2 x^6-8 a p^2 q^2 x^7-16 a p^2 q^2 x^8+15 b p x^9+24 a p^3 q x^9-4 a p^3 q x^{10}+6 a p^4 x^{12}\right)}{30 x^{10}}+2 b p q \log (x)-b p q \log \left(q+p x^3+\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}\right)","\frac{\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left(6 a q^4+24 a p q^3 x^3-4 a p q^3 x^4+15 b q x^6+36 a p^2 q^2 x^6-8 a p^2 q^2 x^7-16 a p^2 q^2 x^8+15 b p x^9+24 a p^3 q x^9-4 a p^3 q x^{10}+6 a p^4 x^{12}\right)}{30 x^{10}}+2 b p q \log (x)-b p q \log \left(q+p x^3+\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}\right)",1,"(Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]*(6*a*q^4 + 24*a*p*q^3*x^3 - 4*a*p*q^3*x^4 + 15*b*q*x^6 + 36*a*p^2*q^2*x^6 - 8*a*p^2*q^2*x^7 - 16*a*p^2*q^2*x^8 + 15*b*p*x^9 + 24*a*p^3*q*x^9 - 4*a*p^3*q*x^10 + 6*a*p^4*x^12))/(30*x^10) + 2*b*p*q*Log[x] - b*p*q*Log[q + p*x^3 + Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]]","A",1
2394,1,197,192,0.4168475,"\int \frac{x \sqrt{1+x}}{x+\sqrt{x+\sqrt{1+x}}} \, dx","IntegrateAlgebraic[(x*Sqrt[1 + x])/(x + Sqrt[x + Sqrt[1 + x]]),x]","\frac{1}{2} \left(-1-2 \sqrt{1+x}\right) \sqrt{x+\sqrt{1+x}}+\frac{2}{3} \left(3 \sqrt{1+x}+(1+x)^{3/2}\right)+\frac{3}{4} \log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}\right)+4 \text{RootSum}\left[1+3 \text{$\#$1}-5 \text{$\#$1}^2+2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{-3 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^3}{3-10 \text{$\#$1}+6 \text{$\#$1}^2+4 \text{$\#$1}^3}\&\right]","-\frac{1}{2} \sqrt{x+\sqrt{1+x}}+\sqrt{1+x} \left(\frac{2 (4+x)}{3}-\sqrt{x+\sqrt{1+x}}\right)+\frac{3}{4} \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}\right)+4 \text{RootSum}\left[1+3 \text{$\#$1}-5 \text{$\#$1}^2+2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{-3 \log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^3}{3-10 \text{$\#$1}+6 \text{$\#$1}^2+4 \text{$\#$1}^3}\&\right]",1,"((-1 - 2*Sqrt[1 + x])*Sqrt[x + Sqrt[1 + x]])/2 + (2*(3*Sqrt[1 + x] + (1 + x)^(3/2)))/3 + (3*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]])/4 + 4*RootSum[1 + 3*#1 - 5*#1^2 + 2*#1^3 + #1^4 & , (-3*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^2 + Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^3)/(3 - 10*#1 + 6*#1^2 + 4*#1^3) & ]","A",1
2395,1,197,192,0.0009112,"\int \frac{x \sqrt{1+x}}{x+\sqrt{x+\sqrt{1+x}}} \, dx","IntegrateAlgebraic[(x*Sqrt[1 + x])/(x + Sqrt[x + Sqrt[1 + x]]),x]","\frac{1}{2} \left(-1-2 \sqrt{1+x}\right) \sqrt{x+\sqrt{1+x}}+\frac{2}{3} \left(3 \sqrt{1+x}+(1+x)^{3/2}\right)+\frac{3}{4} \log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}\right)+4 \text{RootSum}\left[1+3 \text{$\#$1}-5 \text{$\#$1}^2+2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{-3 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^3}{3-10 \text{$\#$1}+6 \text{$\#$1}^2+4 \text{$\#$1}^3}\&\right]","-\frac{1}{2} \sqrt{x+\sqrt{1+x}}+\sqrt{1+x} \left(\frac{2 (4+x)}{3}-\sqrt{x+\sqrt{1+x}}\right)+\frac{3}{4} \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}\right)+4 \text{RootSum}\left[1+3 \text{$\#$1}-5 \text{$\#$1}^2+2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{-3 \log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^3}{3-10 \text{$\#$1}+6 \text{$\#$1}^2+4 \text{$\#$1}^3}\&\right]",1,"((-1 - 2*Sqrt[1 + x])*Sqrt[x + Sqrt[1 + x]])/2 + (2*(3*Sqrt[1 + x] + (1 + x)^(3/2)))/3 + (3*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]])/4 + 4*RootSum[1 + 3*#1 - 5*#1^2 + 2*#1^3 + #1^4 & , (-3*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^2 + Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^3)/(3 - 10*#1 + 6*#1^2 + 4*#1^3) & ]","A",1
2396,1,193,193,0.304142,"\int \frac{x^2}{\sqrt{\frac{b+a x}{d+c x}}} \, dx","IntegrateAlgebraic[x^2/Sqrt[(b + a*x)/(d + c*x)],x]","\frac{\sqrt{\frac{b+a x}{d+c x}} \left(15 b^2 c^2 d-4 a b c d^2-3 a^2 d^3+15 b^2 c^3 x-14 a b c^2 d x-a^2 c d^2 x-10 a b c^3 x^2+10 a^2 c^2 d x^2+8 a^2 c^3 x^3\right)}{24 a^3 c^2}+\frac{\left(-5 b^3 c^3+3 a b^2 c^2 d+a^2 b c d^2+a^3 d^3\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{b+a x}{d+c x}}}{\sqrt{a}}\right)}{8 a^{7/2} c^{5/2}}","\frac{\sqrt{\frac{b+a x}{d+c x}} \left(15 b^2 c^2 d-4 a b c d^2-3 a^2 d^3+15 b^2 c^3 x-14 a b c^2 d x-a^2 c d^2 x-10 a b c^3 x^2+10 a^2 c^2 d x^2+8 a^2 c^3 x^3\right)}{24 a^3 c^2}+\frac{\left(-5 b^3 c^3+3 a b^2 c^2 d+a^2 b c d^2+a^3 d^3\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{b+a x}{d+c x}}}{\sqrt{a}}\right)}{8 a^{7/2} c^{5/2}}",1,"(Sqrt[(b + a*x)/(d + c*x)]*(15*b^2*c^2*d - 4*a*b*c*d^2 - 3*a^2*d^3 + 15*b^2*c^3*x - 14*a*b*c^2*d*x - a^2*c*d^2*x - 10*a*b*c^3*x^2 + 10*a^2*c^2*d*x^2 + 8*a^2*c^3*x^3))/(24*a^3*c^2) + ((-5*b^3*c^3 + 3*a*b^2*c^2*d + a^2*b*c*d^2 + a^3*d^3)*ArcTanh[(Sqrt[c]*Sqrt[(b + a*x)/(d + c*x)])/Sqrt[a]])/(8*a^(7/2)*c^(5/2))","A",1
2397,1,193,193,0.5792896,"\int \frac{b+2 a x}{(-b+a x) (2 b+a x) \sqrt[4]{-1+b x+a x^2}} \, dx","IntegrateAlgebraic[(b + 2*a*x)/((-b + a*x)*(2*b + a*x)*(-1 + b*x + a*x^2)^(1/4)),x]","\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{a-2 b^2} \sqrt[4]{-1+b x+a x^2}}{\sqrt{a-2 b^2}-\sqrt{a} \sqrt{-1+b x+a x^2}}\right)}{a^{3/4} \sqrt[4]{a-2 b^2}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{a-2 b^2}}{\sqrt{2} \sqrt[4]{a}}+\frac{\sqrt[4]{a} \sqrt{-1+b x+a x^2}}{\sqrt{2} \sqrt[4]{a-2 b^2}}}{\sqrt[4]{-1+b x+a x^2}}\right)}{a^{3/4} \sqrt[4]{a-2 b^2}}","\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{a-2 b^2} \sqrt[4]{-1+b x+a x^2}}{\sqrt{a-2 b^2}-\sqrt{a} \sqrt{-1+b x+a x^2}}\right)}{a^{3/4} \sqrt[4]{a-2 b^2}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{a-2 b^2}}{\sqrt{2} \sqrt[4]{a}}+\frac{\sqrt[4]{a} \sqrt{-1+b x+a x^2}}{\sqrt{2} \sqrt[4]{a-2 b^2}}}{\sqrt[4]{-1+b x+a x^2}}\right)}{a^{3/4} \sqrt[4]{a-2 b^2}}",1,"(Sqrt[2]*ArcTan[(Sqrt[2]*a^(1/4)*(a - 2*b^2)^(1/4)*(-1 + b*x + a*x^2)^(1/4))/(Sqrt[a - 2*b^2] - Sqrt[a]*Sqrt[-1 + b*x + a*x^2])])/(a^(3/4)*(a - 2*b^2)^(1/4)) - (Sqrt[2]*ArcTanh[((a - 2*b^2)^(1/4)/(Sqrt[2]*a^(1/4)) + (a^(1/4)*Sqrt[-1 + b*x + a*x^2])/(Sqrt[2]*(a - 2*b^2)^(1/4)))/(-1 + b*x + a*x^2)^(1/4)])/(a^(3/4)*(a - 2*b^2)^(1/4))","A",1
2398,1,193,193,3.995629,"\int \frac{-1+a k x+k x^2}{\left(1+k x^2\right) \sqrt{\left(1-x^2\right) \left(1-k^2 x^2\right)}} \, dx","IntegrateAlgebraic[(-1 + a*k*x + k*x^2)/((1 + k*x^2)*Sqrt[(1 - x^2)*(1 - k^2*x^2)]),x]","\frac{\tan ^{-1}\left(\frac{\left(-1-2 \sqrt{k}-k\right) x}{-1+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{1+k}+\frac{\tan ^{-1}\left(\frac{\left(-1+2 \sqrt{k}-k\right) x}{-1+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{1+k}-\frac{a \sqrt{k} \tanh ^{-1}\left(\frac{\left(2 \sqrt{k}+2 k^{3/2}\right) x^2}{1+2 k x^2+k^2 x^4+\left(-1+k x^2\right) \sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{2 (1+k)}","\frac{\tan ^{-1}\left(\frac{\left(-1-2 \sqrt{k}-k\right) x}{-1+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{1+k}+\frac{\tan ^{-1}\left(\frac{\left(-1+2 \sqrt{k}-k\right) x}{-1+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{1+k}-\frac{a \sqrt{k} \tanh ^{-1}\left(\frac{\left(2 \sqrt{k}+2 k^{3/2}\right) x^2}{1+2 k x^2+k^2 x^4+\left(-1+k x^2\right) \sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{2 (1+k)}",1,"ArcTan[((-1 - 2*Sqrt[k] - k)*x)/(-1 + k*x^2 + Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4])]/(1 + k) + ArcTan[((-1 + 2*Sqrt[k] - k)*x)/(-1 + k*x^2 + Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4])]/(1 + k) - (a*Sqrt[k]*ArcTanh[((2*Sqrt[k] + 2*k^(3/2))*x^2)/(1 + 2*k*x^2 + k^2*x^4 + (-1 + k*x^2)*Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4])])/(2*(1 + k))","A",1
2399,1,193,193,0.7786451,"\int \frac{\left(-b+x^3\right) \left(b+x^3\right)}{\sqrt[3]{a x^2+x^3}} \, dx","IntegrateAlgebraic[((-b + x^3)*(b + x^3))/(a*x^2 + x^3)^(1/3),x]","\frac{\left(a x^2+x^3\right)^{2/3} \left(-7280 a^5+5460 a^4 x-4680 a^3 x^2+4212 a^2 x^3-3888 a x^4+3645 x^5\right)}{21870 x}+\frac{\left(728 \sqrt{3} a^6-6561 \sqrt{3} b^2\right) \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{a x^2+x^3}}\right)}{6561}+\frac{\left(-728 a^6+6561 b^2\right) \log \left(-x+\sqrt[3]{a x^2+x^3}\right)}{6561}+\frac{\left(728 a^6-6561 b^2\right) \log \left(x^2+x \sqrt[3]{a x^2+x^3}+\left(a x^2+x^3\right)^{2/3}\right)}{13122}","\frac{\left(a x^2+x^3\right)^{2/3} \left(-7280 a^5+5460 a^4 x-4680 a^3 x^2+4212 a^2 x^3-3888 a x^4+3645 x^5\right)}{21870 x}+\frac{\left(728 \sqrt{3} a^6-6561 \sqrt{3} b^2\right) \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{a x^2+x^3}}\right)}{6561}+\frac{\left(-728 a^6+6561 b^2\right) \log \left(-x+\sqrt[3]{a x^2+x^3}\right)}{6561}+\frac{\left(728 a^6-6561 b^2\right) \log \left(x^2+x \sqrt[3]{a x^2+x^3}+\left(a x^2+x^3\right)^{2/3}\right)}{13122}",1,"((a*x^2 + x^3)^(2/3)*(-7280*a^5 + 5460*a^4*x - 4680*a^3*x^2 + 4212*a^2*x^3 - 3888*a*x^4 + 3645*x^5))/(21870*x) + ((728*Sqrt[3]*a^6 - 6561*Sqrt[3]*b^2)*ArcTan[(Sqrt[3]*x)/(x + 2*(a*x^2 + x^3)^(1/3))])/6561 + ((-728*a^6 + 6561*b^2)*Log[-x + (a*x^2 + x^3)^(1/3)])/6561 + ((728*a^6 - 6561*b^2)*Log[x^2 + x*(a*x^2 + x^3)^(1/3) + (a*x^2 + x^3)^(2/3)])/13122","A",1
2400,1,193,193,1.3168879,"\int \frac{x}{\left(1-x^2\right) \sqrt{a+b x+c x^2+b x^3+a x^4}} \, dx","IntegrateAlgebraic[x/((1 - x^2)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]),x]","\frac{\sqrt{-2 a-2 b-c} \tan ^{-1}\left(\frac{\sqrt{-2 a-2 b-c} x}{\sqrt{a}-2 \sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{2 (2 a+2 b+c)}-\frac{\sqrt{-2 a+2 b-c} \tan ^{-1}\left(\frac{\sqrt{-2 a+2 b-c} x}{\sqrt{a}+2 \sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{2 (2 a-2 b+c)}","\frac{\sqrt{-2 a-2 b-c} \tan ^{-1}\left(\frac{\sqrt{-2 a-2 b-c} x}{\sqrt{a}-2 \sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{2 (2 a+2 b+c)}-\frac{\sqrt{-2 a+2 b-c} \tan ^{-1}\left(\frac{\sqrt{-2 a+2 b-c} x}{\sqrt{a}+2 \sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{2 (2 a-2 b+c)}",1,"(Sqrt[-2*a - 2*b - c]*ArcTan[(Sqrt[-2*a - 2*b - c]*x)/(Sqrt[a] - 2*Sqrt[a]*x + Sqrt[a]*x^2 - Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])])/(2*(2*a + 2*b + c)) - (Sqrt[-2*a + 2*b - c]*ArcTan[(Sqrt[-2*a + 2*b - c]*x)/(Sqrt[a] + 2*Sqrt[a]*x + Sqrt[a]*x^2 - Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])])/(2*(2*a - 2*b + c))","A",1
2401,1,193,193,0.6460374,"\int \frac{1}{\sqrt[4]{-1+x^4} \left(-1-x^4+x^8\right)} \, dx","IntegrateAlgebraic[1/((-1 + x^4)^(1/4)*(-1 - x^4 + x^8)),x]","-\frac{1}{2} \sqrt{\frac{1}{10} \left(-1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(-1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)","-\frac{1}{2} \sqrt{\frac{1}{10} \left(-1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(-1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)",1,"-1/2*(Sqrt[(-1 + Sqrt[5])/10]*ArcTan[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(-1 + x^4)^(1/4)]) - (Sqrt[(1 + Sqrt[5])/10]*ArcTan[(Sqrt[1/2 + Sqrt[5]/2]*x)/(-1 + x^4)^(1/4)])/2 - (Sqrt[(-1 + Sqrt[5])/10]*ArcTanh[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(-1 + x^4)^(1/4)])/2 - (Sqrt[(1 + Sqrt[5])/10]*ArcTanh[(Sqrt[1/2 + Sqrt[5]/2]*x)/(-1 + x^4)^(1/4)])/2","A",1
2402,1,193,193,0.7457978,"\int \frac{-1+2 x^4}{\sqrt[4]{-1+x^4} \left(-1-x^4+x^8\right)} \, dx","IntegrateAlgebraic[(-1 + 2*x^4)/((-1 + x^4)^(1/4)*(-1 - x^4 + x^8)),x]","-\frac{1}{2} \sqrt{\frac{1}{2} \left(-1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)+\frac{1}{2} \sqrt{\frac{1}{2} \left(1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{2} \left(-1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)+\frac{1}{2} \sqrt{\frac{1}{2} \left(1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)","-\frac{1}{2} \sqrt{\frac{1}{2} \left(-1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)+\frac{1}{2} \sqrt{\frac{1}{2} \left(1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{2} \left(-1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)+\frac{1}{2} \sqrt{\frac{1}{2} \left(1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)",1,"-1/2*(Sqrt[(-1 + Sqrt[5])/2]*ArcTan[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(-1 + x^4)^(1/4)]) + (Sqrt[(1 + Sqrt[5])/2]*ArcTan[(Sqrt[1/2 + Sqrt[5]/2]*x)/(-1 + x^4)^(1/4)])/2 - (Sqrt[(-1 + Sqrt[5])/2]*ArcTanh[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(-1 + x^4)^(1/4)])/2 + (Sqrt[(1 + Sqrt[5])/2]*ArcTanh[(Sqrt[1/2 + Sqrt[5]/2]*x)/(-1 + x^4)^(1/4)])/2","A",1
2403,1,193,193,0.5846759,"\int \frac{1}{\sqrt[4]{1+x^4} \left(-1+x^4+x^8\right)} \, dx","IntegrateAlgebraic[1/((1 + x^4)^(1/4)*(-1 + x^4 + x^8)),x]","-\frac{1}{2} \sqrt{\frac{1}{10} \left(-1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(-1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)","-\frac{1}{2} \sqrt{\frac{1}{10} \left(-1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(-1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)",1,"-1/2*(Sqrt[(-1 + Sqrt[5])/10]*ArcTan[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(1 + x^4)^(1/4)]) - (Sqrt[(1 + Sqrt[5])/10]*ArcTan[(Sqrt[1/2 + Sqrt[5]/2]*x)/(1 + x^4)^(1/4)])/2 - (Sqrt[(-1 + Sqrt[5])/10]*ArcTanh[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(1 + x^4)^(1/4)])/2 - (Sqrt[(1 + Sqrt[5])/10]*ArcTanh[(Sqrt[1/2 + Sqrt[5]/2]*x)/(1 + x^4)^(1/4)])/2","A",1
2404,1,193,193,0.6682373,"\int \frac{-2+x^4}{\sqrt[4]{1+x^4} \left(-1+x^4+x^8\right)} \, dx","IntegrateAlgebraic[(-2 + x^4)/((1 + x^4)^(1/4)*(-1 + x^4 + x^8)),x]","\frac{1}{2} \sqrt{\frac{1}{2} \left(1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)+\frac{1}{2} \sqrt{\frac{1}{2} \left(-1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)+\frac{1}{2} \sqrt{\frac{1}{2} \left(1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)+\frac{1}{2} \sqrt{\frac{1}{2} \left(-1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)","\frac{1}{2} \sqrt{\frac{1}{2} \left(1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)+\frac{1}{2} \sqrt{\frac{1}{2} \left(-1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)+\frac{1}{2} \sqrt{\frac{1}{2} \left(1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)+\frac{1}{2} \sqrt{\frac{1}{2} \left(-1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)",1,"(Sqrt[(1 + Sqrt[5])/2]*ArcTan[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(1 + x^4)^(1/4)])/2 + (Sqrt[(-1 + Sqrt[5])/2]*ArcTan[(Sqrt[1/2 + Sqrt[5]/2]*x)/(1 + x^4)^(1/4)])/2 + (Sqrt[(1 + Sqrt[5])/2]*ArcTanh[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(1 + x^4)^(1/4)])/2 + (Sqrt[(-1 + Sqrt[5])/2]*ArcTanh[(Sqrt[1/2 + Sqrt[5]/2]*x)/(1 + x^4)^(1/4)])/2","A",1
2405,1,193,193,2.6242601,"\int \frac{\sqrt[4]{b x^3+a x^4} \left(-d+c x^8\right)}{x^8} \, dx","IntegrateAlgebraic[((b*x^3 + a*x^4)^(1/4)*(-d + c*x^8))/x^8,x]","\frac{\sqrt[4]{b x^3+a x^4} \left(445536 a b^6 d+21216 a^2 b^5 d x-24960 a^3 b^4 d x^2+30720 a^4 b^3 d x^3-40960 a^5 b^2 d x^4+65536 a^6 b d x^5-262144 a^7 d x^6+348075 b^7 c x^7+1392300 a b^6 c x^8\right)}{2784600 a b^6 x^7}+\frac{3 b^2 c \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{16 a^{7/4}}-\frac{3 b^2 c \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{16 a^{7/4}}","\frac{\sqrt[4]{b x^3+a x^4} \left(445536 a b^6 d+21216 a^2 b^5 d x-24960 a^3 b^4 d x^2+30720 a^4 b^3 d x^3-40960 a^5 b^2 d x^4+65536 a^6 b d x^5-262144 a^7 d x^6+348075 b^7 c x^7+1392300 a b^6 c x^8\right)}{2784600 a b^6 x^7}+\frac{3 b^2 c \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{16 a^{7/4}}-\frac{3 b^2 c \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{16 a^{7/4}}",1,"((b*x^3 + a*x^4)^(1/4)*(445536*a*b^6*d + 21216*a^2*b^5*d*x - 24960*a^3*b^4*d*x^2 + 30720*a^4*b^3*d*x^3 - 40960*a^5*b^2*d*x^4 + 65536*a^6*b*d*x^5 - 262144*a^7*d*x^6 + 348075*b^7*c*x^7 + 1392300*a*b^6*c*x^8))/(2784600*a*b^6*x^7) + (3*b^2*c*ArcTan[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(16*a^(7/4)) - (3*b^2*c*ArcTanh[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(16*a^(7/4))","A",1
2406,1,193,193,0.3172878,"\int \frac{x^4 \left(2+x^5\right)}{\sqrt{1+x^5} \left(-1-x^5+a x^{10}\right)} \, dx","IntegrateAlgebraic[(x^4*(2 + x^5))/(Sqrt[1 + x^5]*(-1 - x^5 + a*x^10)),x]","\frac{\sqrt{2} \left(1+4 a+\sqrt{1+4 a}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{1+x^5}}{\sqrt{-1-2 a-\sqrt{1+4 a}}}\right)}{5 \sqrt{a} \sqrt{1+4 a} \sqrt{-1-2 a-\sqrt{1+4 a}}}+\frac{\sqrt{2} \left(-1-4 a+\sqrt{1+4 a}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{1+x^5}}{\sqrt{-1-2 a+\sqrt{1+4 a}}}\right)}{5 \sqrt{a} \sqrt{1+4 a} \sqrt{-1-2 a+\sqrt{1+4 a}}}","\frac{\sqrt{2} \left(1+4 a+\sqrt{1+4 a}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{1+x^5}}{\sqrt{-1-2 a-\sqrt{1+4 a}}}\right)}{5 \sqrt{a} \sqrt{1+4 a} \sqrt{-1-2 a-\sqrt{1+4 a}}}+\frac{\sqrt{2} \left(-1-4 a+\sqrt{1+4 a}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{1+x^5}}{\sqrt{-1-2 a+\sqrt{1+4 a}}}\right)}{5 \sqrt{a} \sqrt{1+4 a} \sqrt{-1-2 a+\sqrt{1+4 a}}}",1,"(Sqrt[2]*(1 + 4*a + Sqrt[1 + 4*a])*ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[1 + x^5])/Sqrt[-1 - 2*a - Sqrt[1 + 4*a]]])/(5*Sqrt[a]*Sqrt[1 + 4*a]*Sqrt[-1 - 2*a - Sqrt[1 + 4*a]]) + (Sqrt[2]*(-1 - 4*a + Sqrt[1 + 4*a])*ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[1 + x^5])/Sqrt[-1 - 2*a + Sqrt[1 + 4*a]]])/(5*Sqrt[a]*Sqrt[1 + 4*a]*Sqrt[-1 - 2*a + Sqrt[1 + 4*a]])","A",1
2407,1,193,193,0.312608,"\int \frac{x^4 \left(-2+x^5\right)}{\sqrt{-1+x^5} \left(1-x^5+a x^{10}\right)} \, dx","IntegrateAlgebraic[(x^4*(-2 + x^5))/(Sqrt[-1 + x^5]*(1 - x^5 + a*x^10)),x]","\frac{\sqrt{2} \left(1+\sqrt{1-4 a}-4 a\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{-1+x^5}}{\sqrt{-1-\sqrt{1-4 a}+2 a}}\right)}{5 \sqrt{1-4 a} \sqrt{a} \sqrt{-1-\sqrt{1-4 a}+2 a}}+\frac{\sqrt{2} \left(-1+\sqrt{1-4 a}+4 a\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{-1+x^5}}{\sqrt{-1+\sqrt{1-4 a}+2 a}}\right)}{5 \sqrt{1-4 a} \sqrt{a} \sqrt{-1+\sqrt{1-4 a}+2 a}}","\frac{\sqrt{2} \left(1+\sqrt{1-4 a}-4 a\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{-1+x^5}}{\sqrt{-1-\sqrt{1-4 a}+2 a}}\right)}{5 \sqrt{1-4 a} \sqrt{a} \sqrt{-1-\sqrt{1-4 a}+2 a}}+\frac{\sqrt{2} \left(-1+\sqrt{1-4 a}+4 a\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{-1+x^5}}{\sqrt{-1+\sqrt{1-4 a}+2 a}}\right)}{5 \sqrt{1-4 a} \sqrt{a} \sqrt{-1+\sqrt{1-4 a}+2 a}}",1,"(Sqrt[2]*(1 + Sqrt[1 - 4*a] - 4*a)*ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[-1 + x^5])/Sqrt[-1 - Sqrt[1 - 4*a] + 2*a]])/(5*Sqrt[1 - 4*a]*Sqrt[a]*Sqrt[-1 - Sqrt[1 - 4*a] + 2*a]) + (Sqrt[2]*(-1 + Sqrt[1 - 4*a] + 4*a)*ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[-1 + x^5])/Sqrt[-1 + Sqrt[1 - 4*a] + 2*a]])/(5*Sqrt[1 - 4*a]*Sqrt[a]*Sqrt[-1 + Sqrt[1 - 4*a] + 2*a])","A",1
2408,1,193,193,0.3193078,"\int \frac{x^4 \left(3+x^5\right)}{\sqrt{1+x^5} \left(-1+a-(1+2 a) x^5+a x^{10}\right)} \, dx","IntegrateAlgebraic[(x^4*(3 + x^5))/(Sqrt[1 + x^5]*(-1 + a - (1 + 2*a)*x^5 + a*x^10)),x]","\frac{\sqrt{2} \left(1+8 a+\sqrt{1+8 a}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{1+x^5}}{\sqrt{-1-4 a-\sqrt{1+8 a}}}\right)}{5 \sqrt{a} \sqrt{1+8 a} \sqrt{-1-4 a-\sqrt{1+8 a}}}+\frac{\sqrt{2} \left(-1-8 a+\sqrt{1+8 a}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{1+x^5}}{\sqrt{-1-4 a+\sqrt{1+8 a}}}\right)}{5 \sqrt{a} \sqrt{1+8 a} \sqrt{-1-4 a+\sqrt{1+8 a}}}","\frac{\sqrt{2} \left(1+8 a+\sqrt{1+8 a}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{1+x^5}}{\sqrt{-1-4 a-\sqrt{1+8 a}}}\right)}{5 \sqrt{a} \sqrt{1+8 a} \sqrt{-1-4 a-\sqrt{1+8 a}}}+\frac{\sqrt{2} \left(-1-8 a+\sqrt{1+8 a}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{1+x^5}}{\sqrt{-1-4 a+\sqrt{1+8 a}}}\right)}{5 \sqrt{a} \sqrt{1+8 a} \sqrt{-1-4 a+\sqrt{1+8 a}}}",1,"(Sqrt[2]*(1 + 8*a + Sqrt[1 + 8*a])*ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[1 + x^5])/Sqrt[-1 - 4*a - Sqrt[1 + 8*a]]])/(5*Sqrt[a]*Sqrt[1 + 8*a]*Sqrt[-1 - 4*a - Sqrt[1 + 8*a]]) + (Sqrt[2]*(-1 - 8*a + Sqrt[1 + 8*a])*ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[1 + x^5])/Sqrt[-1 - 4*a + Sqrt[1 + 8*a]]])/(5*Sqrt[a]*Sqrt[1 + 8*a]*Sqrt[-1 - 4*a + Sqrt[1 + 8*a]])","A",1
2409,1,193,193,0.317551,"\int \frac{1}{x^2-\sqrt{b+a x} \sqrt{c+\sqrt{b+a x}}} \, dx","IntegrateAlgebraic[(x^2 - Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]])^(-1),x]","4 a \text{RootSum}\left[b^2-2 b c^2+c^4+a^2 c \text{$\#$1}+4 b c \text{$\#$1}^2-4 c^3 \text{$\#$1}^2-a^2 \text{$\#$1}^3-2 b \text{$\#$1}^4+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}-\log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3}{-a^2 c-8 b c \text{$\#$1}+8 c^3 \text{$\#$1}+3 a^2 \text{$\#$1}^2+8 b \text{$\#$1}^3-24 c^2 \text{$\#$1}^3+24 c \text{$\#$1}^5-8 \text{$\#$1}^7}\&\right]","4 a \text{RootSum}\left[b^2-2 b c^2+c^4+a^2 c \text{$\#$1}+4 b c \text{$\#$1}^2-4 c^3 \text{$\#$1}^2-a^2 \text{$\#$1}^3-2 b \text{$\#$1}^4+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}-\log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3}{-a^2 c-8 b c \text{$\#$1}+8 c^3 \text{$\#$1}+3 a^2 \text{$\#$1}^2+8 b \text{$\#$1}^3-24 c^2 \text{$\#$1}^3+24 c \text{$\#$1}^5-8 \text{$\#$1}^7}\&\right]",1,"4*a*RootSum[b^2 - 2*b*c^2 + c^4 + a^2*c*#1 + 4*b*c*#1^2 - 4*c^3*#1^2 - a^2*#1^3 - 2*b*#1^4 + 6*c^2*#1^4 - 4*c*#1^6 + #1^8 & , (c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1 - Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^3)/(-(a^2*c) - 8*b*c*#1 + 8*c^3*#1 + 3*a^2*#1^2 + 8*b*#1^3 - 24*c^2*#1^3 + 24*c*#1^5 - 8*#1^7) & ]","A",1
2410,1,193,193,0.0020846,"\int \frac{1}{x^2-\sqrt{b+a x} \sqrt{c+\sqrt{b+a x}}} \, dx","IntegrateAlgebraic[(x^2 - Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]])^(-1),x]","4 a \text{RootSum}\left[b^2-2 b c^2+c^4+a^2 c \text{$\#$1}+4 b c \text{$\#$1}^2-4 c^3 \text{$\#$1}^2-a^2 \text{$\#$1}^3-2 b \text{$\#$1}^4+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}-\log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3}{-a^2 c-8 b c \text{$\#$1}+8 c^3 \text{$\#$1}+3 a^2 \text{$\#$1}^2+8 b \text{$\#$1}^3-24 c^2 \text{$\#$1}^3+24 c \text{$\#$1}^5-8 \text{$\#$1}^7}\&\right]","4 a \text{RootSum}\left[b^2-2 b c^2+c^4+a^2 c \text{$\#$1}+4 b c \text{$\#$1}^2-4 c^3 \text{$\#$1}^2-a^2 \text{$\#$1}^3-2 b \text{$\#$1}^4+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}-\log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3}{-a^2 c-8 b c \text{$\#$1}+8 c^3 \text{$\#$1}+3 a^2 \text{$\#$1}^2+8 b \text{$\#$1}^3-24 c^2 \text{$\#$1}^3+24 c \text{$\#$1}^5-8 \text{$\#$1}^7}\&\right]",1,"4*a*RootSum[b^2 - 2*b*c^2 + c^4 + a^2*c*#1 + 4*b*c*#1^2 - 4*c^3*#1^2 - a^2*#1^3 - 2*b*#1^4 + 6*c^2*#1^4 - 4*c*#1^6 + #1^8 & , (c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1 - Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^3)/(-(a^2*c) - 8*b*c*#1 + 8*c^3*#1 + 3*a^2*#1^2 + 8*b*#1^3 - 24*c^2*#1^3 + 24*c*#1^5 - 8*#1^7) & ]","A",1
2411,1,193,193,0.7156447,"\int \frac{1}{\sqrt{-b+a^2 x^2} \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}} \, dx","IntegrateAlgebraic[1/(Sqrt[-b + a^2*x^2]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3)),x]","-\frac{4 \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[3]{c}}\right)}{a c^{2/3}}+\frac{4 \log \left(-\sqrt[3]{c}+\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{a c^{2/3}}-\frac{2 \log \left(c^{2/3}+\sqrt[3]{c} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}\right)}{a c^{2/3}}","-\frac{4 \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[3]{c}}\right)}{a c^{2/3}}+\frac{4 \log \left(-\sqrt[3]{c}+\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{a c^{2/3}}-\frac{2 \log \left(c^{2/3}+\sqrt[3]{c} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}\right)}{a c^{2/3}}",1,"(-4*Sqrt[3]*ArcTan[1/Sqrt[3] + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3))/(Sqrt[3]*c^(1/3))])/(a*c^(2/3)) + (4*Log[-c^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)])/(a*c^(2/3)) - (2*Log[c^(2/3) + c^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3)])/(a*c^(2/3))","A",1
2412,1,193,193,0.6281975,"\int \frac{1}{\sqrt{-b+a^2 x^2} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}} \, dx","IntegrateAlgebraic[1/(Sqrt[-b + a^2*x^2]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)),x]","\frac{4 \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[3]{c}}\right)}{a \sqrt[3]{c}}+\frac{4 \log \left(-\sqrt[3]{c}+\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{a \sqrt[3]{c}}-\frac{2 \log \left(c^{2/3}+\sqrt[3]{c} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}\right)}{a \sqrt[3]{c}}","\frac{4 \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[3]{c}}\right)}{a \sqrt[3]{c}}+\frac{4 \log \left(-\sqrt[3]{c}+\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{a \sqrt[3]{c}}-\frac{2 \log \left(c^{2/3}+\sqrt[3]{c} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}\right)}{a \sqrt[3]{c}}",1,"(4*Sqrt[3]*ArcTan[1/Sqrt[3] + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3))/(Sqrt[3]*c^(1/3))])/(a*c^(1/3)) + (4*Log[-c^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)])/(a*c^(1/3)) - (2*Log[c^(2/3) + c^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3)])/(a*c^(1/3))","A",1
2413,0,0,194,3.9695495,"\int \frac{1}{\sqrt[3]{1-3 x^2} \left(-3+x^2\right)} \, dx","IntegrateAlgebraic[1/((1 - 3*x^2)^(1/3)*(-3 + x^2)),x]","\int \frac{1}{\sqrt[3]{1-3 x^2} \left(-3+x^2\right)} \, dx","-\frac{1}{8} \tan ^{-1}\left(\frac{\frac{1}{\sqrt{3}}+x-\frac{\sqrt[3]{1-3 x^2}}{\sqrt{3}}}{\sqrt[3]{1-3 x^2}}\right)-\frac{1}{8} \tan ^{-1}\left(\frac{-\frac{1}{\sqrt{3}}+x+\frac{\sqrt[3]{1-3 x^2}}{\sqrt{3}}}{\sqrt[3]{1-3 x^2}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt{3} x}{1+2 \sqrt[3]{1-3 x^2}}\right)}{4 \sqrt{3}}-\frac{\tanh ^{-1}\left(\frac{-2 \sqrt{3} x+2 \sqrt{3} x \sqrt[3]{1-3 x^2}}{1+3 x^2-2 \sqrt[3]{1-3 x^2}+4 \left(1-3 x^2\right)^{2/3}}\right)}{8 \sqrt{3}}",1,"Could not integrate","F",-1
2414,1,194,194,0.5747034,"\int \frac{-a+x}{\sqrt[3]{x^2 (-a+x)} \left(a^2 d-2 a d x+(-1+d) x^2\right)} \, dx","IntegrateAlgebraic[(-a + x)/((x^2*(-a + x))^(1/3)*(a^2*d - 2*a*d*x + (-1 + d)*x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x-2 \sqrt[6]{d} \sqrt[3]{-a x^2+x^3}}\right)}{2 a d^{5/6}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[6]{d} \sqrt[3]{-a x^2+x^3}}\right)}{2 a d^{5/6}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[6]{d} \sqrt[3]{-a x^2+x^3}}\right)}{a d^{5/6}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[6]{d}}+\sqrt[6]{d} \left(-a x^2+x^3\right)^{2/3}}{x \sqrt[3]{-a x^2+x^3}}\right)}{2 a d^{5/6}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x-2 \sqrt[6]{d} \sqrt[3]{-a x^2+x^3}}\right)}{2 a d^{5/6}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[6]{d} \sqrt[3]{-a x^2+x^3}}\right)}{2 a d^{5/6}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[6]{d} \sqrt[3]{-a x^2+x^3}}\right)}{a d^{5/6}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[6]{d}}+\sqrt[6]{d} \left(-a x^2+x^3\right)^{2/3}}{x \sqrt[3]{-a x^2+x^3}}\right)}{2 a d^{5/6}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x - 2*d^(1/6)*(-(a*x^2) + x^3)^(1/3))])/(2*a*d^(5/6)) - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*d^(1/6)*(-(a*x^2) + x^3)^(1/3))])/(2*a*d^(5/6)) - ArcTanh[x/(d^(1/6)*(-(a*x^2) + x^3)^(1/3))]/(a*d^(5/6)) - ArcTanh[(x^2/d^(1/6) + d^(1/6)*(-(a*x^2) + x^3)^(2/3))/(x*(-(a*x^2) + x^3)^(1/3))]/(2*a*d^(5/6))","A",1
2415,1,194,194,1.0672249,"\int \frac{b-a x^4+2 x^8}{\sqrt[4]{-b+a x^4} \left(b+3 a x^4\right)} \, dx","IntegrateAlgebraic[(b - a*x^4 + 2*x^8)/((-b + a*x^4)^(1/4)*(b + 3*a*x^4)),x]","\frac{x \left(-b+a x^4\right)^{3/4}}{6 a^2}+\frac{\left(-6 a^2-b\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{36 a^{9/4}}+\frac{\left(6 a^2+b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{9 \sqrt{2} a^{9/4}}+\frac{\left(-6 a^2-b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{36 a^{9/4}}+\frac{\left(6 a^2+b\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{9 \sqrt{2} a^{9/4}}","\frac{x \left(-b+a x^4\right)^{3/4}}{6 a^2}+\frac{\left(-6 a^2-b\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{36 a^{9/4}}+\frac{\left(6 a^2+b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{9 \sqrt{2} a^{9/4}}+\frac{\left(-6 a^2-b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{36 a^{9/4}}+\frac{\left(6 a^2+b\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{9 \sqrt{2} a^{9/4}}",1,"(x*(-b + a*x^4)^(3/4))/(6*a^2) + ((-6*a^2 - b)*ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)])/(36*a^(9/4)) + ((6*a^2 + b)*ArcTan[(Sqrt[2]*a^(1/4)*x)/(-b + a*x^4)^(1/4)])/(9*Sqrt[2]*a^(9/4)) + ((-6*a^2 - b)*ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)])/(36*a^(9/4)) + ((6*a^2 + b)*ArcTanh[(Sqrt[2]*a^(1/4)*x)/(-b + a*x^4)^(1/4)])/(9*Sqrt[2]*a^(9/4))","A",1
2416,1,194,194,0.2606098,"\int \frac{d+c x^2}{\sqrt{a x+\sqrt{b^2+a^2 x^2}}} \, dx","IntegrateAlgebraic[(d + c*x^2)/Sqrt[a*x + Sqrt[b^2 + a^2*x^2]],x]","\frac{2 \sqrt{b^2+a^2 x^2} \left(-4 b^4 c x+25 a^2 b^2 d x-3 a^2 b^2 c x^3+60 a^4 d x^3+12 a^4 c x^5\right)}{15 a^2 \left(a x+\sqrt{b^2+a^2 x^2}\right)^{7/2}}+\frac{2 \left(-8 b^6 c+35 a^2 b^4 d-49 a^2 b^4 c x^2+385 a^4 b^2 d x^2+21 a^4 b^2 c x^4+420 a^6 d x^4+84 a^6 c x^6\right)}{105 a^3 \left(a x+\sqrt{b^2+a^2 x^2}\right)^{7/2}}","\frac{2 \sqrt{b^2+a^2 x^2} \left(-4 b^4 c x+25 a^2 b^2 d x-3 a^2 b^2 c x^3+60 a^4 d x^3+12 a^4 c x^5\right)}{15 a^2 \left(a x+\sqrt{b^2+a^2 x^2}\right)^{7/2}}+\frac{2 \left(-8 b^6 c+35 a^2 b^4 d-49 a^2 b^4 c x^2+385 a^4 b^2 d x^2+21 a^4 b^2 c x^4+420 a^6 d x^4+84 a^6 c x^6\right)}{105 a^3 \left(a x+\sqrt{b^2+a^2 x^2}\right)^{7/2}}",1,"(2*Sqrt[b^2 + a^2*x^2]*(-4*b^4*c*x + 25*a^2*b^2*d*x - 3*a^2*b^2*c*x^3 + 60*a^4*d*x^3 + 12*a^4*c*x^5))/(15*a^2*(a*x + Sqrt[b^2 + a^2*x^2])^(7/2)) + (2*(-8*b^6*c + 35*a^2*b^4*d - 49*a^2*b^4*c*x^2 + 385*a^4*b^2*d*x^2 + 21*a^4*b^2*c*x^4 + 420*a^6*d*x^4 + 84*a^6*c*x^6))/(105*a^3*(a*x + Sqrt[b^2 + a^2*x^2])^(7/2))","A",1
2417,1,195,195,1.9655073,"\int \frac{\sqrt[4]{b x^3+a x^4} \left(-c+d x^8\right)}{x^4} \, dx","IntegrateAlgebraic[((b*x^3 + a*x^4)^(1/4)*(-c + d*x^8))/x^4,x]","\frac{\sqrt[4]{b x^3+a x^4} \left(327680 a^5 b^2 c+65536 a^6 b c x-262144 a^7 c x^2+21945 b^7 d x^3-12540 a b^6 d x^4+9120 a^2 b^5 d x^5-7296 a^3 b^4 d x^6+6144 a^4 b^3 d x^7+122880 a^5 b^2 d x^8\right)}{737280 a^5 b^2 x^3}+\frac{1463 b^6 d \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{32768 a^{23/4}}-\frac{1463 b^6 d \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{32768 a^{23/4}}","\frac{\sqrt[4]{b x^3+a x^4} \left(327680 a^5 b^2 c+65536 a^6 b c x-262144 a^7 c x^2+21945 b^7 d x^3-12540 a b^6 d x^4+9120 a^2 b^5 d x^5-7296 a^3 b^4 d x^6+6144 a^4 b^3 d x^7+122880 a^5 b^2 d x^8\right)}{737280 a^5 b^2 x^3}+\frac{1463 b^6 d \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{32768 a^{23/4}}-\frac{1463 b^6 d \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{32768 a^{23/4}}",1,"((b*x^3 + a*x^4)^(1/4)*(327680*a^5*b^2*c + 65536*a^6*b*c*x - 262144*a^7*c*x^2 + 21945*b^7*d*x^3 - 12540*a*b^6*d*x^4 + 9120*a^2*b^5*d*x^5 - 7296*a^3*b^4*d*x^6 + 6144*a^4*b^3*d*x^7 + 122880*a^5*b^2*d*x^8))/(737280*a^5*b^2*x^3) + (1463*b^6*d*ArcTan[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(32768*a^(23/4)) - (1463*b^6*d*ArcTanh[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(32768*a^(23/4))","A",1
2418,1,162,195,0.2826767,"\int \frac{\left(b^2+a x^2\right)^{5/2}}{\sqrt{b+\sqrt{b^2+a x^2}}} \, dx","IntegrateAlgebraic[(b^2 + a*x^2)^(5/2)/Sqrt[b + Sqrt[b^2 + a*x^2]],x]","\frac{2 x \sqrt{b^2+a x^2} \left(247 b^4+155 a b^2 x^2+45 a^2 x^4\right)}{495 \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{2 x \left(247 b^5+31 a b^3 x^2+5 a^2 b x^4\right)}{495 \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{\sqrt{2} b^{11/2} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{\sqrt{a}}","\frac{2 x \sqrt{b^2+a x^2} \left(247 b^4+155 a b^2 x^2+45 a^2 x^4\right)}{495 \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{2 x \left(247 b^5+31 a b^3 x^2+5 a^2 b x^4\right)}{495 \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{2 \sqrt{2} b^{11/2} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2} \sqrt{b}}\right)}{\sqrt{a}}",1,"(2*x*Sqrt[b^2 + a*x^2]*(247*b^4 + 155*a*b^2*x^2 + 45*a^2*x^4))/(495*Sqrt[b + Sqrt[b^2 + a*x^2]]) - (2*x*(247*b^5 + 31*a*b^3*x^2 + 5*a^2*b*x^4))/(495*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (Sqrt[2]*b^(11/2)*ArcTan[(Sqrt[a]*x)/(Sqrt[2]*Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/Sqrt[a]","A",1
2419,1,195,195,0.2099789,"\int \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]],x]","\frac{\left(-75-8 x+60 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+(16+6 x) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left((-8+60 x) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+6 \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{105 \sqrt{x+\sqrt{1+x^2}}}-\tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)","\frac{\left(-75-8 x+60 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+(16+6 x) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left((-8+60 x) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+6 \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{105 \sqrt{x+\sqrt{1+x^2}}}-\tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)",1,"((-75 - 8*x + 60*x^2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (16 + 6*x)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((-8 + 60*x)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + 6*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(105*Sqrt[x + Sqrt[1 + x^2]]) - ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]]","A",1
2420,1,195,195,1.0159265,"\int \frac{\left(x^2 c_3-c_4\right) \left(x+3 x^2 c_3+3 c_4\right)}{x \sqrt{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \left(-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right)} \, dx","IntegrateAlgebraic[((x^2*C[3] - C[4])*(x + 3*x^2*C[3] + 3*C[4]))/(x*Sqrt[(x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4])]*(-x^2 + x^4*C[3]^2 + 2*x^2*C[3]*C[4] + C[4]^2)),x]","6 \tanh ^{-1}\left(\sqrt{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}\right)-\frac{2 \tan ^{-1}\left(\frac{\sqrt{1-c_0} \sqrt{-1+c_1} \sqrt{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{-1+c_0}\right) \sqrt{-1+c_1}}{\sqrt{1-c_0}}-\frac{4 \tan ^{-1}\left(\frac{\sqrt{-1-c_0} \sqrt{1+c_1} \sqrt{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{1+c_0}\right) \sqrt{1+c_1}}{\sqrt{-1-c_0}}","6 \tanh ^{-1}\left(\sqrt{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}\right)-\frac{2 \tan ^{-1}\left(\frac{\sqrt{1-c_0} \sqrt{-1+c_1} \sqrt{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{-1+c_0}\right) \sqrt{-1+c_1}}{\sqrt{1-c_0}}-\frac{4 \tan ^{-1}\left(\frac{\sqrt{-1-c_0} \sqrt{1+c_1} \sqrt{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{1+c_0}\right) \sqrt{1+c_1}}{\sqrt{-1-c_0}}",1,"6*ArcTanh[Sqrt[(x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4])]] - (2*ArcTan[(Sqrt[1 - C[0]]*Sqrt[-1 + C[1]]*Sqrt[(x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4])])/(-1 + C[0])]*Sqrt[-1 + C[1]])/Sqrt[1 - C[0]] - (4*ArcTan[(Sqrt[-1 - C[0]]*Sqrt[1 + C[1]]*Sqrt[(x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4])])/(1 + C[0])]*Sqrt[1 + C[1]])/Sqrt[-1 - C[0]]","A",1
2421,1,196,196,1.1253153,"\int \frac{1+x^6}{\sqrt[3]{x+x^5} \left(-1+x^6\right)} \, dx","IntegrateAlgebraic[(1 + x^6)/((x + x^5)^(1/3)*(-1 + x^6)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{x+x^5}}\right)}{\sqrt{3}}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x+x^5}}\right)}{2 \sqrt[3]{2} \sqrt{3}}-\frac{1}{3} \log \left(x+\sqrt[3]{x+x^5}\right)+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x+x^5}\right)}{6 \sqrt[3]{2}}+\frac{1}{6} \log \left(x^2-x \sqrt[3]{x+x^5}+\left(x+x^5\right)^{2/3}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x+x^5}+\sqrt[3]{2} \left(x+x^5\right)^{2/3}\right)}{12 \sqrt[3]{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{x+x^5}}\right)}{\sqrt{3}}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x+x^5}}\right)}{2 \sqrt[3]{2} \sqrt{3}}-\frac{1}{3} \log \left(x+\sqrt[3]{x+x^5}\right)+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x+x^5}\right)}{6 \sqrt[3]{2}}+\frac{1}{6} \log \left(x^2-x \sqrt[3]{x+x^5}+\left(x+x^5\right)^{2/3}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x+x^5}+\sqrt[3]{2} \left(x+x^5\right)^{2/3}\right)}{12 \sqrt[3]{2}}",1,"-(ArcTan[(Sqrt[3]*x)/(-x + 2*(x + x^5)^(1/3))]/Sqrt[3]) - ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(x + x^5)^(1/3))]/(2*2^(1/3)*Sqrt[3]) - Log[x + (x + x^5)^(1/3)]/3 + Log[-2*x + 2^(2/3)*(x + x^5)^(1/3)]/(6*2^(1/3)) + Log[x^2 - x*(x + x^5)^(1/3) + (x + x^5)^(2/3)]/6 - Log[2*x^2 + 2^(2/3)*x*(x + x^5)^(1/3) + 2^(1/3)*(x + x^5)^(2/3)]/(12*2^(1/3))","A",1
2422,1,196,196,27.2967839,"\int \frac{1}{x^2 \sqrt[3]{-4-8 x+11 x^2+17 x^3-20 x^4-7 x^5+16 x^6-7 x^7+x^8}} \, dx","IntegrateAlgebraic[1/(x^2*(-4 - 8*x + 11*x^2 + 17*x^3 - 20*x^4 - 7*x^5 + 16*x^6 - 7*x^7 + x^8)^(1/3)),x]","\frac{(-2+x)^{2/3} \left(-1-x+x^2\right) \left(-\frac{\sqrt[3]{-2+x}}{2 x}-\frac{\sqrt[3]{2} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2^{2/3} \sqrt[3]{-2+x}}{\sqrt{3}}\right)}{\sqrt{3}}+\frac{1}{3} \sqrt[3]{2} \log \left(2+2^{2/3} \sqrt[3]{-2+x}\right)-\frac{\log \left(-2+2^{2/3} \sqrt[3]{-2+x}-\sqrt[3]{2} (-2+x)^{2/3}\right)}{3\ 2^{2/3}}-\text{RootSum}\left[1+3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{\log \left(\sqrt[3]{-2+x}-\text{$\#$1}\right) \text{$\#$1}}{3+2 \text{$\#$1}^3}\&\right]\right)}{\sqrt[3]{(-2+x)^2 \left(-1-x+x^2\right)^3}}","\frac{(-2+x)^{2/3} \left(-1-x+x^2\right) \left(-\frac{\sqrt[3]{-2+x}}{2 x}-\frac{\sqrt[3]{2} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2^{2/3} \sqrt[3]{-2+x}}{\sqrt{3}}\right)}{\sqrt{3}}+\frac{1}{3} \sqrt[3]{2} \log \left(2+2^{2/3} \sqrt[3]{-2+x}\right)-\frac{\log \left(-2+2^{2/3} \sqrt[3]{-2+x}-\sqrt[3]{2} (-2+x)^{2/3}\right)}{3\ 2^{2/3}}-\text{RootSum}\left[1+3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{\log \left(\sqrt[3]{-2+x}-\text{$\#$1}\right) \text{$\#$1}}{3+2 \text{$\#$1}^3}\&\right]\right)}{\sqrt[3]{(-2+x)^2 \left(-1-x+x^2\right)^3}}",1,"((-2 + x)^(2/3)*(-1 - x + x^2)*(-1/2*(-2 + x)^(1/3)/x - (2^(1/3)*ArcTan[1/Sqrt[3] - (2^(2/3)*(-2 + x)^(1/3))/Sqrt[3]])/Sqrt[3] + (2^(1/3)*Log[2 + 2^(2/3)*(-2 + x)^(1/3)])/3 - Log[-2 + 2^(2/3)*(-2 + x)^(1/3) - 2^(1/3)*(-2 + x)^(2/3)]/(3*2^(2/3)) - RootSum[1 + 3*#1^3 + #1^6 & , (Log[(-2 + x)^(1/3) - #1]*#1)/(3 + 2*#1^3) & ]))/((-2 + x)^2*(-1 - x + x^2)^3)^(1/3)","A",1
2423,1,114,196,0.845221,"\int \frac{b^8+a^8 x^8}{\sqrt{b^4+a^4 x^4} \left(-b^8+a^8 x^8\right)} \, dx","IntegrateAlgebraic[(b^8 + a^8*x^8)/(Sqrt[b^4 + a^4*x^4]*(-b^8 + a^8*x^8)),x]","-\frac{x}{2 \sqrt{b^4+a^4 x^4}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} a b x}{b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}}\right)}{2 \sqrt{2} a b}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} a b x}{\sqrt{b^4+a^4 x^4}}\right)}{4 \sqrt{2} a b}","-\frac{x}{2 \sqrt{b^4+a^4 x^4}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} a b x}{b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}}\right)}{2 \sqrt{2} a b}+\frac{\tanh ^{-1}\left(\frac{\sqrt{6-4 \sqrt{2}} a b x}{b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}}\right)}{4 \sqrt{2} a b}-\frac{\tanh ^{-1}\left(\frac{\sqrt{6+4 \sqrt{2}} a b x}{b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}}\right)}{4 \sqrt{2} a b}",1,"-1/2*x/Sqrt[b^4 + a^4*x^4] - ArcTan[(Sqrt[2]*a*b*x)/(b^2 + a^2*x^2 + Sqrt[b^4 + a^4*x^4])]/(2*Sqrt[2]*a*b) - ArcTanh[(Sqrt[2]*a*b*x)/Sqrt[b^4 + a^4*x^4]]/(4*Sqrt[2]*a*b)","A",1
2424,1,189,196,0.2375233,"\int \frac{\sqrt{x+\sqrt{1+x}}}{x-\sqrt{1+x}} \, dx","IntegrateAlgebraic[Sqrt[x + Sqrt[1 + x]]/(x - Sqrt[1 + x]),x]","2 \sqrt{x+\sqrt{1+x}}-3 \log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}\right)+4 \text{RootSum}\left[-1+6 \text{$\#$1}-3 \text{$\#$1}^2+2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{2 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right)-2 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}+\log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2}{3-3 \text{$\#$1}+3 \text{$\#$1}^2+2 \text{$\#$1}^3}\&\right]","2 \sqrt{x+\sqrt{1+x}}-3 \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}\right)-4 \text{RootSum}\left[25-40 \text{$\#$1}+6 \text{$\#$1}^2-8 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{5 \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right)+2 \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}+\log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2}{-10+3 \text{$\#$1}-6 \text{$\#$1}^2+\text{$\#$1}^3}\&\right]",1,"2*Sqrt[x + Sqrt[1 + x]] - 3*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]] + 4*RootSum[-1 + 6*#1 - 3*#1^2 + 2*#1^3 + #1^4 & , (2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1] - 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1 + Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^2)/(3 - 3*#1 + 3*#1^2 + 2*#1^3) & ]","A",1
2425,1,189,196,0.001112,"\int \frac{\sqrt{x+\sqrt{1+x}}}{x-\sqrt{1+x}} \, dx","IntegrateAlgebraic[Sqrt[x + Sqrt[1 + x]]/(x - Sqrt[1 + x]),x]","2 \sqrt{x+\sqrt{1+x}}-3 \log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}\right)+4 \text{RootSum}\left[-1+6 \text{$\#$1}-3 \text{$\#$1}^2+2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{2 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right)-2 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}+\log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2}{3-3 \text{$\#$1}+3 \text{$\#$1}^2+2 \text{$\#$1}^3}\&\right]","2 \sqrt{x+\sqrt{1+x}}-3 \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}\right)-4 \text{RootSum}\left[25-40 \text{$\#$1}+6 \text{$\#$1}^2-8 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{5 \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right)+2 \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}+\log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2}{-10+3 \text{$\#$1}-6 \text{$\#$1}^2+\text{$\#$1}^3}\&\right]",1,"2*Sqrt[x + Sqrt[1 + x]] - 3*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]] + 4*RootSum[-1 + 6*#1 - 3*#1^2 + 2*#1^3 + #1^4 & , (2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1] - 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1 + Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^2)/(3 - 3*#1 + 3*#1^2 + 2*#1^3) & ]","A",1
2426,1,196,196,0.3778079,"\int \frac{\left(d+c x^2\right) \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{-d+c x^2} \, dx","IntegrateAlgebraic[((d + c*x^2)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/(-d + c*x^2),x]","-\frac{b^2}{a \sqrt{a x+\sqrt{b^2+a^2 x^2}}}+\frac{\left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}{3 a}+a d \text{RootSum}\left[b^4 c-2 b^2 c \text{$\#$1}^4-4 a^2 d \text{$\#$1}^4+c \text{$\#$1}^8\&,\frac{-b^2 \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-\log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{b^2 c \text{$\#$1}+2 a^2 d \text{$\#$1}-c \text{$\#$1}^5}\&\right]","-\frac{b^2}{a \sqrt{a x+\sqrt{b^2+a^2 x^2}}}+\frac{\left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}{3 a}+a d \text{RootSum}\left[b^4 c-2 b^2 c \text{$\#$1}^4-4 a^2 d \text{$\#$1}^4+c \text{$\#$1}^8\&,\frac{-b^2 \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-\log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{b^2 c \text{$\#$1}+2 a^2 d \text{$\#$1}-c \text{$\#$1}^5}\&\right]",1,"-(b^2/(a*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])) + (a*x + Sqrt[b^2 + a^2*x^2])^(3/2)/(3*a) + a*d*RootSum[b^4*c - 2*b^2*c*#1^4 - 4*a^2*d*#1^4 + c*#1^8 & , (-(b^2*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]) - Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]*#1^4)/(b^2*c*#1 + 2*a^2*d*#1 - c*#1^5) & ]","A",1
2427,1,196,196,0.0019101,"\int \frac{\left(d+c x^2\right) \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{-d+c x^2} \, dx","IntegrateAlgebraic[((d + c*x^2)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/(-d + c*x^2),x]","-\frac{b^2}{a \sqrt{a x+\sqrt{b^2+a^2 x^2}}}+\frac{\left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}{3 a}+a d \text{RootSum}\left[b^4 c-2 b^2 c \text{$\#$1}^4-4 a^2 d \text{$\#$1}^4+c \text{$\#$1}^8\&,\frac{-b^2 \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-\log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{b^2 c \text{$\#$1}+2 a^2 d \text{$\#$1}-c \text{$\#$1}^5}\&\right]","-\frac{b^2}{a \sqrt{a x+\sqrt{b^2+a^2 x^2}}}+\frac{\left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}{3 a}+a d \text{RootSum}\left[b^4 c-2 b^2 c \text{$\#$1}^4-4 a^2 d \text{$\#$1}^4+c \text{$\#$1}^8\&,\frac{-b^2 \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-\log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{b^2 c \text{$\#$1}+2 a^2 d \text{$\#$1}-c \text{$\#$1}^5}\&\right]",1,"-(b^2/(a*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])) + (a*x + Sqrt[b^2 + a^2*x^2])^(3/2)/(3*a) + a*d*RootSum[b^4*c - 2*b^2*c*#1^4 - 4*a^2*d*#1^4 + c*#1^8 & , (-(b^2*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]) - Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]*#1^4)/(b^2*c*#1 + 2*a^2*d*#1 - c*#1^5) & ]","A",1
2428,1,196,196,3.2338924,"\int \frac{\sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}{x \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}} \, dx","IntegrateAlgebraic[Sqrt[-(a/b^2) + (a^2*x^2)/b^2]/(x*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]]),x]","\frac{\left(-2-a x^2\right) \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{b x}+\sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}} \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}-\frac{\sqrt{a} \log \left(-a x-b \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}+\sqrt{2} \sqrt{a} \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}\right)}{\sqrt{2} b}","\frac{\left(-2-a x^2\right) \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{b x}+\sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}} \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}-\frac{\sqrt{a} \log \left(-a x-b \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}+\sqrt{2} \sqrt{a} \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}\right)}{\sqrt{2} b}",1,"((-2 - a*x^2)*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]])/(b*x) + Sqrt[-(a/b^2) + (a^2*x^2)/b^2]*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]] - (Sqrt[a]*Log[-(a*x) - b*Sqrt[-(a/b^2) + (a^2*x^2)/b^2] + Sqrt[2]*Sqrt[a]*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]]])/(Sqrt[2]*b)","A",1
2429,1,177,196,0.2979166,"\int \frac{-1+x^2}{\left(1+x^2\right) \sqrt{1+\sqrt{1+x}} \sqrt{1+\sqrt{1+\sqrt{1+x}}}} \, dx","IntegrateAlgebraic[(-1 + x^2)/((1 + x^2)*Sqrt[1 + Sqrt[1 + x]]*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]),x]","-\frac{32}{15} \sqrt{1+\sqrt{1+x}} \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\frac{8}{15} \left(-4+3 \sqrt{1+x}\right) \sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{RootSum}\left[2-8 \text{$\#$1}^4+8 \text{$\#$1}^6+14 \text{$\#$1}^8-32 \text{$\#$1}^{10}+24 \text{$\#$1}^{12}-8 \text{$\#$1}^{14}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{$\#$1}\right)}{\text{$\#$1}-\text{$\#$1}^3-4 \text{$\#$1}^5+8 \text{$\#$1}^7-5 \text{$\#$1}^9+\text{$\#$1}^{11}}\&\right]","-\frac{32}{15} \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\frac{8}{5} \sqrt{1+x} \sqrt{1+\sqrt{1+\sqrt{1+x}}}-\frac{32}{15} \sqrt{1+\sqrt{1+x}} \sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{RootSum}\left[2-8 \text{$\#$1}^4+8 \text{$\#$1}^6+14 \text{$\#$1}^8-32 \text{$\#$1}^{10}+24 \text{$\#$1}^{12}-8 \text{$\#$1}^{14}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{$\#$1}\right)}{\text{$\#$1}-\text{$\#$1}^3-4 \text{$\#$1}^5+8 \text{$\#$1}^7-5 \text{$\#$1}^9+\text{$\#$1}^{11}}\&\right]",1,"(-32*Sqrt[1 + Sqrt[1 + x]]*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/15 + (8*(-4 + 3*Sqrt[1 + x])*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/15 - RootSum[2 - 8*#1^4 + 8*#1^6 + 14*#1^8 - 32*#1^10 + 24*#1^12 - 8*#1^14 + #1^16 & , Log[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]] - #1]/(#1 - #1^3 - 4*#1^5 + 8*#1^7 - 5*#1^9 + #1^11) & ]","A",1
2430,1,177,196,0.0020608,"\int \frac{-1+x^2}{\left(1+x^2\right) \sqrt{1+\sqrt{1+x}} \sqrt{1+\sqrt{1+\sqrt{1+x}}}} \, dx","IntegrateAlgebraic[(-1 + x^2)/((1 + x^2)*Sqrt[1 + Sqrt[1 + x]]*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]),x]","-\frac{32}{15} \sqrt{1+\sqrt{1+x}} \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\frac{8}{15} \left(-4+3 \sqrt{1+x}\right) \sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{RootSum}\left[2-8 \text{$\#$1}^4+8 \text{$\#$1}^6+14 \text{$\#$1}^8-32 \text{$\#$1}^{10}+24 \text{$\#$1}^{12}-8 \text{$\#$1}^{14}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{$\#$1}\right)}{\text{$\#$1}-\text{$\#$1}^3-4 \text{$\#$1}^5+8 \text{$\#$1}^7-5 \text{$\#$1}^9+\text{$\#$1}^{11}}\&\right]","-\frac{32}{15} \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\frac{8}{5} \sqrt{1+x} \sqrt{1+\sqrt{1+\sqrt{1+x}}}-\frac{32}{15} \sqrt{1+\sqrt{1+x}} \sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{RootSum}\left[2-8 \text{$\#$1}^4+8 \text{$\#$1}^6+14 \text{$\#$1}^8-32 \text{$\#$1}^{10}+24 \text{$\#$1}^{12}-8 \text{$\#$1}^{14}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{$\#$1}\right)}{\text{$\#$1}-\text{$\#$1}^3-4 \text{$\#$1}^5+8 \text{$\#$1}^7-5 \text{$\#$1}^9+\text{$\#$1}^{11}}\&\right]",1,"(-32*Sqrt[1 + Sqrt[1 + x]]*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/15 + (8*(-4 + 3*Sqrt[1 + x])*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/15 - RootSum[2 - 8*#1^4 + 8*#1^6 + 14*#1^8 - 32*#1^10 + 24*#1^12 - 8*#1^14 + #1^16 & , Log[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]] - #1]/(#1 - #1^3 - 4*#1^5 + 8*#1^7 - 5*#1^9 + #1^11) & ]","A",1
2431,1,196,196,0.1940871,"\int \sqrt{1+\sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]],x]","\frac{\left(-6-16 x+48 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+(-15+8 x) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left((-16+48 x) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+8 \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{60 x+60 \sqrt{1+x^2}}+\frac{1}{4} \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)","\frac{\left(-6-16 x+48 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+(-15+8 x) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left((-16+48 x) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+8 \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{60 x+60 \sqrt{1+x^2}}+\frac{1}{4} \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)",1,"((-6 - 16*x + 48*x^2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-15 + 8*x)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((-16 + 48*x)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + 8*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(60*x + 60*Sqrt[1 + x^2]) + ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]]/4","A",1
2432,1,197,197,0.2898016,"\int \frac{2+3 x}{\sqrt[3]{4+3 x^2} \left(-12+52 x+9 x^2\right)} \, dx","IntegrateAlgebraic[(2 + 3*x)/((4 + 3*x^2)^(1/3)*(-12 + 52*x + 9*x^2)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\frac{10 \sqrt[3]{2}}{\sqrt{3} 7^{2/3}}-\frac{\sqrt[3]{2} \sqrt{3} x}{7^{2/3}}+\frac{\sqrt[3]{4+3 x^2}}{\sqrt{3}}}{\sqrt[3]{4+3 x^2}}\right)}{14 \sqrt[3]{14}}+\frac{\log \left(-10 \sqrt[3]{14}+3 \sqrt[3]{14} x+14 \sqrt[3]{4+3 x^2}\right)}{14 \sqrt[3]{14}}-\frac{\log \left(100\ 14^{2/3}-60\ 14^{2/3} x+9\ 14^{2/3} x^2+\left(140 \sqrt[3]{14}-42 \sqrt[3]{14} x\right) \sqrt[3]{4+3 x^2}+196 \left(4+3 x^2\right)^{2/3}\right)}{28 \sqrt[3]{14}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\frac{10 \sqrt[3]{2}}{\sqrt{3} 7^{2/3}}-\frac{\sqrt[3]{2} \sqrt{3} x}{7^{2/3}}+\frac{\sqrt[3]{4+3 x^2}}{\sqrt{3}}}{\sqrt[3]{4+3 x^2}}\right)}{14 \sqrt[3]{14}}+\frac{\log \left(-10 \sqrt[3]{14}+3 \sqrt[3]{14} x+14 \sqrt[3]{4+3 x^2}\right)}{14 \sqrt[3]{14}}-\frac{\log \left(100\ 14^{2/3}-60\ 14^{2/3} x+9\ 14^{2/3} x^2+\left(140 \sqrt[3]{14}-42 \sqrt[3]{14} x\right) \sqrt[3]{4+3 x^2}+196 \left(4+3 x^2\right)^{2/3}\right)}{28 \sqrt[3]{14}}",1,"-1/14*(Sqrt[3]*ArcTan[((10*2^(1/3))/(Sqrt[3]*7^(2/3)) - (2^(1/3)*Sqrt[3]*x)/7^(2/3) + (4 + 3*x^2)^(1/3)/Sqrt[3])/(4 + 3*x^2)^(1/3)])/14^(1/3) + Log[-10*14^(1/3) + 3*14^(1/3)*x + 14*(4 + 3*x^2)^(1/3)]/(14*14^(1/3)) - Log[100*14^(2/3) - 60*14^(2/3)*x + 9*14^(2/3)*x^2 + (140*14^(1/3) - 42*14^(1/3)*x)*(4 + 3*x^2)^(1/3) + 196*(4 + 3*x^2)^(2/3)]/(28*14^(1/3))","A",1
2433,1,197,197,0.6577049,"\int \left(c+b x+a x^2\right)^{5/2} \, dx","IntegrateAlgebraic[(c + b*x + a*x^2)^(5/2),x]","\frac{\sqrt{c+b x+a x^2} \left(15 b^5-160 a b^3 c+528 a^2 b c^2-10 a b^4 x+96 a^2 b^2 c x+1056 a^3 c^2 x+8 a^2 b^3 x^2+1248 a^3 b c x^2+432 a^3 b^2 x^3+832 a^4 c x^3+640 a^4 b x^4+256 a^5 x^5\right)}{1536 a^3}-\frac{5 \left(-b^6+12 a b^4 c-48 a^2 b^2 c^2+64 a^3 c^3\right) \log \left(b+2 a x-2 \sqrt{a} \sqrt{c+b x+a x^2}\right)}{1024 a^{7/2}}","\frac{\sqrt{c+b x+a x^2} \left(15 b^5-160 a b^3 c+528 a^2 b c^2-10 a b^4 x+96 a^2 b^2 c x+1056 a^3 c^2 x+8 a^2 b^3 x^2+1248 a^3 b c x^2+432 a^3 b^2 x^3+832 a^4 c x^3+640 a^4 b x^4+256 a^5 x^5\right)}{1536 a^3}-\frac{5 \left(-b^6+12 a b^4 c-48 a^2 b^2 c^2+64 a^3 c^3\right) \log \left(b+2 a x-2 \sqrt{a} \sqrt{c+b x+a x^2}\right)}{1024 a^{7/2}}",1,"(Sqrt[c + b*x + a*x^2]*(15*b^5 - 160*a*b^3*c + 528*a^2*b*c^2 - 10*a*b^4*x + 96*a^2*b^2*c*x + 1056*a^3*c^2*x + 8*a^2*b^3*x^2 + 1248*a^3*b*c*x^2 + 432*a^3*b^2*x^3 + 832*a^4*c*x^3 + 640*a^4*b*x^4 + 256*a^5*x^5))/(1536*a^3) - (5*(-b^6 + 12*a*b^4*c - 48*a^2*b^2*c^2 + 64*a^3*c^3)*Log[b + 2*a*x - 2*Sqrt[a]*Sqrt[c + b*x + a*x^2]])/(1024*a^(7/2))","A",1
2434,1,197,197,0.8781148,"\int \frac{1-x^4+x^8}{x^2 \left(-1+x^4\right)^{3/4} \left(-1-x^4+x^8\right)} \, dx","IntegrateAlgebraic[(1 - x^4 + x^8)/(x^2*(-1 + x^4)^(3/4)*(-1 - x^4 + x^8)),x]","-\frac{\sqrt[4]{-1+x^4}}{x}+\sqrt{\frac{1}{10} \left(-1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)-\sqrt{\frac{1}{10} \left(1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)-\sqrt{\frac{1}{10} \left(-1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)+\sqrt{\frac{1}{10} \left(1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)","-\frac{\sqrt[4]{-1+x^4}}{x}+\sqrt{\frac{1}{10} \left(-1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)-\sqrt{\frac{1}{10} \left(1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)-\sqrt{\frac{1}{10} \left(-1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)+\sqrt{\frac{1}{10} \left(1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)",1,"-((-1 + x^4)^(1/4)/x) + Sqrt[(-1 + Sqrt[5])/10]*ArcTan[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(-1 + x^4)^(1/4)] - Sqrt[(1 + Sqrt[5])/10]*ArcTan[(Sqrt[1/2 + Sqrt[5]/2]*x)/(-1 + x^4)^(1/4)] - Sqrt[(-1 + Sqrt[5])/10]*ArcTanh[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(-1 + x^4)^(1/4)] + Sqrt[(1 + Sqrt[5])/10]*ArcTanh[(Sqrt[1/2 + Sqrt[5]/2]*x)/(-1 + x^4)^(1/4)]","A",1
2435,1,197,197,0.3595213,"\int \frac{\left(-d+c x^2\right) \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{d+c x^2} \, dx","IntegrateAlgebraic[((-d + c*x^2)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/(d + c*x^2),x]","-\frac{b^2}{a \sqrt{a x+\sqrt{b^2+a^2 x^2}}}+\frac{\left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}{3 a}-a d \text{RootSum}\left[b^4 c-2 b^2 c \text{$\#$1}^4+4 a^2 d \text{$\#$1}^4+c \text{$\#$1}^8\&,\frac{-b^2 \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-\log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{b^2 c \text{$\#$1}-2 a^2 d \text{$\#$1}-c \text{$\#$1}^5}\&\right]","-\frac{b^2}{a \sqrt{a x+\sqrt{b^2+a^2 x^2}}}+\frac{\left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}{3 a}-a d \text{RootSum}\left[b^4 c-2 b^2 c \text{$\#$1}^4+4 a^2 d \text{$\#$1}^4+c \text{$\#$1}^8\&,\frac{-b^2 \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-\log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{b^2 c \text{$\#$1}-2 a^2 d \text{$\#$1}-c \text{$\#$1}^5}\&\right]",1,"-(b^2/(a*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])) + (a*x + Sqrt[b^2 + a^2*x^2])^(3/2)/(3*a) - a*d*RootSum[b^4*c - 2*b^2*c*#1^4 + 4*a^2*d*#1^4 + c*#1^8 & , (-(b^2*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]) - Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]*#1^4)/(b^2*c*#1 - 2*a^2*d*#1 - c*#1^5) & ]","A",1
2436,1,197,197,0.0019887,"\int \frac{\left(-d+c x^2\right) \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{d+c x^2} \, dx","IntegrateAlgebraic[((-d + c*x^2)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/(d + c*x^2),x]","-\frac{b^2}{a \sqrt{a x+\sqrt{b^2+a^2 x^2}}}+\frac{\left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}{3 a}-a d \text{RootSum}\left[b^4 c-2 b^2 c \text{$\#$1}^4+4 a^2 d \text{$\#$1}^4+c \text{$\#$1}^8\&,\frac{-b^2 \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-\log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{b^2 c \text{$\#$1}-2 a^2 d \text{$\#$1}-c \text{$\#$1}^5}\&\right]","-\frac{b^2}{a \sqrt{a x+\sqrt{b^2+a^2 x^2}}}+\frac{\left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}{3 a}-a d \text{RootSum}\left[b^4 c-2 b^2 c \text{$\#$1}^4+4 a^2 d \text{$\#$1}^4+c \text{$\#$1}^8\&,\frac{-b^2 \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-\log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{b^2 c \text{$\#$1}-2 a^2 d \text{$\#$1}-c \text{$\#$1}^5}\&\right]",1,"-(b^2/(a*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])) + (a*x + Sqrt[b^2 + a^2*x^2])^(3/2)/(3*a) - a*d*RootSum[b^4*c - 2*b^2*c*#1^4 + 4*a^2*d*#1^4 + c*#1^8 & , (-(b^2*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]) - Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]*#1^4)/(b^2*c*#1 - 2*a^2*d*#1 - c*#1^5) & ]","A",1
2437,1,197,197,0.2284262,"\int \frac{\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/Sqrt[x + Sqrt[1 + x^2]],x]","\frac{\left(8+3 x+32 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+(-2+16 x) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left((3+32 x) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+16 \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{24 \left(x+\sqrt{1+x^2}\right)^{3/2}}-\frac{1}{8} \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)","\frac{\left(8+3 x+32 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+(-2+16 x) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left((3+32 x) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+16 \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{24 \left(x+\sqrt{1+x^2}\right)^{3/2}}-\frac{1}{8} \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)",1,"((8 + 3*x + 32*x^2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-2 + 16*x)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((3 + 32*x)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + 16*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(24*(x + Sqrt[1 + x^2])^(3/2)) - ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]]/8","A",1
2438,1,198,198,0.2235394,"\int \frac{1}{x \sqrt[3]{4-6 x+3 x^2}} \, dx","IntegrateAlgebraic[1/(x*(4 - 6*x + 3*x^2)^(1/3)),x]","-\frac{\tan ^{-1}\left(\frac{\frac{2\ 2^{2/3}}{\sqrt{3}}-\frac{2^{2/3} x}{\sqrt{3}}+\frac{\sqrt[3]{4-6 x+3 x^2}}{\sqrt{3}}}{\sqrt[3]{4-6 x+3 x^2}}\right)}{2^{2/3} \sqrt{3}}+\frac{\log \left(-2 2^{2/3}+2^{2/3} x+2 \sqrt[3]{4-6 x+3 x^2}\right)}{3\ 2^{2/3}}-\frac{\log \left(-4 \sqrt[3]{2}+4 \sqrt[3]{2} x-\sqrt[3]{2} x^2+\left(-2 2^{2/3}+2^{2/3} x\right) \sqrt[3]{4-6 x+3 x^2}-2 \left(4-6 x+3 x^2\right)^{2/3}\right)}{6\ 2^{2/3}}","-\frac{\tan ^{-1}\left(\frac{\frac{2\ 2^{2/3}}{\sqrt{3}}-\frac{2^{2/3} x}{\sqrt{3}}+\frac{\sqrt[3]{4-6 x+3 x^2}}{\sqrt{3}}}{\sqrt[3]{4-6 x+3 x^2}}\right)}{2^{2/3} \sqrt{3}}+\frac{\log \left(-2 2^{2/3}+2^{2/3} x+2 \sqrt[3]{4-6 x+3 x^2}\right)}{3\ 2^{2/3}}-\frac{\log \left(-4 \sqrt[3]{2}+4 \sqrt[3]{2} x-\sqrt[3]{2} x^2+\left(-2 2^{2/3}+2^{2/3} x\right) \sqrt[3]{4-6 x+3 x^2}-2 \left(4-6 x+3 x^2\right)^{2/3}\right)}{6\ 2^{2/3}}",1,"-(ArcTan[((2*2^(2/3))/Sqrt[3] - (2^(2/3)*x)/Sqrt[3] + (4 - 6*x + 3*x^2)^(1/3)/Sqrt[3])/(4 - 6*x + 3*x^2)^(1/3)]/(2^(2/3)*Sqrt[3])) + Log[-2*2^(2/3) + 2^(2/3)*x + 2*(4 - 6*x + 3*x^2)^(1/3)]/(3*2^(2/3)) - Log[-4*2^(1/3) + 4*2^(1/3)*x - 2^(1/3)*x^2 + (-2*2^(2/3) + 2^(2/3)*x)*(4 - 6*x + 3*x^2)^(1/3) - 2*(4 - 6*x + 3*x^2)^(2/3)]/(6*2^(2/3))","A",1
2439,1,198,198,0.8204857,"\int \frac{x^2 \sqrt[4]{b x^3+a x^4}}{-b+a x} \, dx","IntegrateAlgebraic[(x^2*(b*x^3 + a*x^4)^(1/4))/(-b + a*x),x]","\frac{\left(101 b^2+52 a b x+32 a^2 x^2\right) \sqrt[4]{b x^3+a x^4}}{96 a^3}-\frac{155 b^3 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{64 a^{15/4}}+\frac{2 \sqrt[4]{2} b^3 \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{a^{15/4}}+\frac{155 b^3 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{64 a^{15/4}}-\frac{2 \sqrt[4]{2} b^3 \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{a^{15/4}}","\frac{\left(101 b^2+52 a b x+32 a^2 x^2\right) \sqrt[4]{b x^3+a x^4}}{96 a^3}-\frac{155 b^3 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{64 a^{15/4}}+\frac{2 \sqrt[4]{2} b^3 \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{a^{15/4}}+\frac{155 b^3 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{64 a^{15/4}}-\frac{2 \sqrt[4]{2} b^3 \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{a^{15/4}}",1,"((101*b^2 + 52*a*b*x + 32*a^2*x^2)*(b*x^3 + a*x^4)^(1/4))/(96*a^3) - (155*b^3*ArcTan[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(64*a^(15/4)) + (2*2^(1/4)*b^3*ArcTan[(2^(1/4)*a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/a^(15/4) + (155*b^3*ArcTanh[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(64*a^(15/4)) - (2*2^(1/4)*b^3*ArcTanh[(2^(1/4)*a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/a^(15/4)","A",1
2440,1,198,198,1.1271273,"\int \frac{-1+x^6}{\sqrt[3]{x+x^5} \left(1+x^6\right)} \, dx","IntegrateAlgebraic[(-1 + x^6)/((x + x^5)^(1/3)*(1 + x^6)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^5}}\right)}{\sqrt{3}}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2^{2/3} \sqrt[3]{x+x^5}}\right)}{2 \sqrt[3]{2} \sqrt{3}}+\frac{1}{3} \log \left(-x+\sqrt[3]{x+x^5}\right)-\frac{\log \left(2 x+2^{2/3} \sqrt[3]{x+x^5}\right)}{6 \sqrt[3]{2}}-\frac{1}{6} \log \left(x^2+x \sqrt[3]{x+x^5}+\left(x+x^5\right)^{2/3}\right)+\frac{\log \left(-2 x^2+2^{2/3} x \sqrt[3]{x+x^5}-\sqrt[3]{2} \left(x+x^5\right)^{2/3}\right)}{12 \sqrt[3]{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^5}}\right)}{\sqrt{3}}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2^{2/3} \sqrt[3]{x+x^5}}\right)}{2 \sqrt[3]{2} \sqrt{3}}+\frac{1}{3} \log \left(-x+\sqrt[3]{x+x^5}\right)-\frac{\log \left(2 x+2^{2/3} \sqrt[3]{x+x^5}\right)}{6 \sqrt[3]{2}}-\frac{1}{6} \log \left(x^2+x \sqrt[3]{x+x^5}+\left(x+x^5\right)^{2/3}\right)+\frac{\log \left(-2 x^2+2^{2/3} x \sqrt[3]{x+x^5}-\sqrt[3]{2} \left(x+x^5\right)^{2/3}\right)}{12 \sqrt[3]{2}}",1,"-(ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^5)^(1/3))]/Sqrt[3]) - ArcTan[(Sqrt[3]*x)/(-x + 2^(2/3)*(x + x^5)^(1/3))]/(2*2^(1/3)*Sqrt[3]) + Log[-x + (x + x^5)^(1/3)]/3 - Log[2*x + 2^(2/3)*(x + x^5)^(1/3)]/(6*2^(1/3)) - Log[x^2 + x*(x + x^5)^(1/3) + (x + x^5)^(2/3)]/6 + Log[-2*x^2 + 2^(2/3)*x*(x + x^5)^(1/3) - 2^(1/3)*(x + x^5)^(2/3)]/(12*2^(1/3))","A",1
2441,1,198,198,6.8726028,"\int \frac{b-2 x^4+a x^8}{\sqrt[4]{-b+a x^4} \left(b-2 x^4+2 a x^8\right)} \, dx","IntegrateAlgebraic[(b - 2*x^4 + a*x^8)/((-b + a*x^4)^(1/4)*(b - 2*x^4 + 2*a*x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{4 \sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{4 \sqrt[4]{a}}+\frac{1}{16} \text{RootSum}\left[-2 a+a^2+2 a b+2 \text{$\#$1}^4-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 \log (x)-a \log (x)-2 \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{\text{$\#$1}-a \text{$\#$1}+\text{$\#$1}^5}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{4 \sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{4 \sqrt[4]{a}}+\frac{1}{16} \text{RootSum}\left[-2 a+a^2+2 a b+2 \text{$\#$1}^4-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 \log (x)-a \log (x)-2 \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{\text{$\#$1}-a \text{$\#$1}+\text{$\#$1}^5}\&\right]",1,"ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(4*a^(1/4)) + ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(4*a^(1/4)) + RootSum[-2*a + a^2 + 2*a*b + 2*#1^4 - 2*a*#1^4 + #1^8 & , (2*Log[x] - a*Log[x] - 2*Log[(-b + a*x^4)^(1/4) - x*#1] + a*Log[(-b + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4)/(#1 - a*#1 + #1^5) & ]/16","A",1
2442,1,198,198,0.0017133,"\int \frac{b-2 x^4+a x^8}{\sqrt[4]{-b+a x^4} \left(b-2 x^4+2 a x^8\right)} \, dx","IntegrateAlgebraic[(b - 2*x^4 + a*x^8)/((-b + a*x^4)^(1/4)*(b - 2*x^4 + 2*a*x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{4 \sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{4 \sqrt[4]{a}}+\frac{1}{16} \text{RootSum}\left[-2 a+a^2+2 a b+2 \text{$\#$1}^4-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 \log (x)-a \log (x)-2 \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{\text{$\#$1}-a \text{$\#$1}+\text{$\#$1}^5}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{4 \sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{4 \sqrt[4]{a}}+\frac{1}{16} \text{RootSum}\left[-2 a+a^2+2 a b+2 \text{$\#$1}^4-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 \log (x)-a \log (x)-2 \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{\text{$\#$1}-a \text{$\#$1}+\text{$\#$1}^5}\&\right]",1,"ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(4*a^(1/4)) + ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(4*a^(1/4)) + RootSum[-2*a + a^2 + 2*a*b + 2*#1^4 - 2*a*#1^4 + #1^8 & , (2*Log[x] - a*Log[x] - 2*Log[(-b + a*x^4)^(1/4) - x*#1] + a*Log[(-b + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4)/(#1 - a*#1 + #1^5) & ]/16","A",1
2443,1,197,198,5.1810258,"\int \frac{-b+c x^4+2 a x^8}{\sqrt[4]{-b+a x^4} \left(b-c x^4+a x^8\right)} \, dx","IntegrateAlgebraic[(-b + c*x^4 + 2*a*x^8)/((-b + a*x^4)^(1/4)*(b - c*x^4 + a*x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{3}{4} \text{RootSum}\left[a^2+a b-a c-2 a \text{$\#$1}^4+c \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a \log (x)-c \log (x)-a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+c \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-2 a \text{$\#$1}+c \text{$\#$1}+2 \text{$\#$1}^5}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{3}{4} \text{RootSum}\left[a^2+a b-a c-2 a \text{$\#$1}^4+c \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a \log (x)+c \log (x)+a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-c \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{2 a \text{$\#$1}-c \text{$\#$1}-2 \text{$\#$1}^5}\&\right]",1,"ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/a^(1/4) + ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/a^(1/4) + (3*RootSum[a^2 + a*b - a*c - 2*a*#1^4 + c*#1^4 + #1^8 & , (a*Log[x] - c*Log[x] - a*Log[(-b + a*x^4)^(1/4) - x*#1] + c*Log[(-b + a*x^4)^(1/4) - x*#1] - Log[x]*#1^4 + Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4)/(-2*a*#1 + c*#1 + 2*#1^5) & ])/4","A",1
2444,1,197,198,0.0019028,"\int \frac{-b+c x^4+2 a x^8}{\sqrt[4]{-b+a x^4} \left(b-c x^4+a x^8\right)} \, dx","IntegrateAlgebraic[(-b + c*x^4 + 2*a*x^8)/((-b + a*x^4)^(1/4)*(b - c*x^4 + a*x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{3}{4} \text{RootSum}\left[a^2+a b-a c-2 a \text{$\#$1}^4+c \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a \log (x)-c \log (x)-a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+c \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-2 a \text{$\#$1}+c \text{$\#$1}+2 \text{$\#$1}^5}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{3}{4} \text{RootSum}\left[a^2+a b-a c-2 a \text{$\#$1}^4+c \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a \log (x)+c \log (x)+a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-c \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{2 a \text{$\#$1}-c \text{$\#$1}-2 \text{$\#$1}^5}\&\right]",1,"ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/a^(1/4) + ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/a^(1/4) + (3*RootSum[a^2 + a*b - a*c - 2*a*#1^4 + c*#1^4 + #1^8 & , (a*Log[x] - c*Log[x] - a*Log[(-b + a*x^4)^(1/4) - x*#1] + c*Log[(-b + a*x^4)^(1/4) - x*#1] - Log[x]*#1^4 + Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4)/(-2*a*#1 + c*#1 + 2*#1^5) & ])/4","A",1
2445,1,198,198,1.387302,"\int \frac{\sqrt[4]{-b x^3+a x^4} \left(-d+c x^8\right)}{x^4} \, dx","IntegrateAlgebraic[((-(b*x^3) + a*x^4)^(1/4)*(-d + c*x^8))/x^4,x]","\frac{\sqrt[4]{-b x^3+a x^4} \left(327680 a^5 b^2 d-65536 a^6 b d x-262144 a^7 d x^2-21945 b^7 c x^3-12540 a b^6 c x^4-9120 a^2 b^5 c x^5-7296 a^3 b^4 c x^6-6144 a^4 b^3 c x^7+122880 a^5 b^2 c x^8\right)}{737280 a^5 b^2 x^3}+\frac{1463 b^6 c \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)}{32768 a^{23/4}}-\frac{1463 b^6 c \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)}{32768 a^{23/4}}","\frac{\sqrt[4]{-b x^3+a x^4} \left(327680 a^5 b^2 d-65536 a^6 b d x-262144 a^7 d x^2-21945 b^7 c x^3-12540 a b^6 c x^4-9120 a^2 b^5 c x^5-7296 a^3 b^4 c x^6-6144 a^4 b^3 c x^7+122880 a^5 b^2 c x^8\right)}{737280 a^5 b^2 x^3}+\frac{1463 b^6 c \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)}{32768 a^{23/4}}-\frac{1463 b^6 c \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)}{32768 a^{23/4}}",1,"((-(b*x^3) + a*x^4)^(1/4)*(327680*a^5*b^2*d - 65536*a^6*b*d*x - 262144*a^7*d*x^2 - 21945*b^7*c*x^3 - 12540*a*b^6*c*x^4 - 9120*a^2*b^5*c*x^5 - 7296*a^3*b^4*c*x^6 - 6144*a^4*b^3*c*x^7 + 122880*a^5*b^2*c*x^8))/(737280*a^5*b^2*x^3) + (1463*b^6*c*ArcTan[(a^(1/4)*x)/(-(b*x^3) + a*x^4)^(1/4)])/(32768*a^(23/4)) - (1463*b^6*c*ArcTanh[(a^(1/4)*x)/(-(b*x^3) + a*x^4)^(1/4)])/(32768*a^(23/4))","A",1
2446,1,166,198,0.4001862,"\int \frac{\sqrt{b+\sqrt{b^2+a x^2}}}{x^6} \, dx","IntegrateAlgebraic[Sqrt[b + Sqrt[b^2 + a*x^2]]/x^6,x]","\frac{\sqrt{b+\sqrt{b^2+a x^2}} \left(105 a^{5/2} x^4+\sqrt{a} \left(-432 b^4+48 b^3 \sqrt{b^2+a x^2}\right)+a^{3/2} \left(14 b^2 x^2-70 b x^2 \sqrt{b^2+a x^2}\right)\right)}{1920 \sqrt{a} b^4 x^5}+\frac{7 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{128 \sqrt{2} b^{9/2}}","\frac{\sqrt{b+\sqrt{b^2+a x^2}} \left(105 a^{5/2} x^4+\sqrt{a} \left(-432 b^4+48 b^3 \sqrt{b^2+a x^2}\right)+a^{3/2} \left(14 b^2 x^2-70 b x^2 \sqrt{b^2+a x^2}\right)\right)}{1920 \sqrt{a} b^4 x^5}+\frac{7 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2} \sqrt{b}}\right)}{64 \sqrt{2} b^{9/2}}",1,"(Sqrt[b + Sqrt[b^2 + a*x^2]]*(105*a^(5/2)*x^4 + Sqrt[a]*(-432*b^4 + 48*b^3*Sqrt[b^2 + a*x^2]) + a^(3/2)*(14*b^2*x^2 - 70*b*x^2*Sqrt[b^2 + a*x^2])))/(1920*Sqrt[a]*b^4*x^5) + (7*a^(5/2)*ArcTan[(Sqrt[a]*x)/(Sqrt[2]*Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/(128*Sqrt[2]*b^(9/2))","A",1
2447,1,166,198,0.3699456,"\int \frac{\sqrt{b+\sqrt{b^2+a x^2}}}{x^6 \sqrt{b^2+a x^2}} \, dx","IntegrateAlgebraic[Sqrt[b + Sqrt[b^2 + a*x^2]]/(x^6*Sqrt[b^2 + a*x^2]),x]","\frac{\sqrt{b+\sqrt{b^2+a x^2}} \left(-315 a^{5/2} x^4+\sqrt{a} \left(16 b^4-144 b^3 \sqrt{b^2+a x^2}\right)+a^{3/2} \left(-42 b^2 x^2+210 b x^2 \sqrt{b^2+a x^2}\right)\right)}{640 \sqrt{a} b^5 x^5}-\frac{63 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{128 \sqrt{2} b^{11/2}}","\frac{\sqrt{b+\sqrt{b^2+a x^2}} \left(-315 a^{5/2} x^4+\sqrt{a} \left(16 b^4-144 b^3 \sqrt{b^2+a x^2}\right)+a^{3/2} \left(-42 b^2 x^2+210 b x^2 \sqrt{b^2+a x^2}\right)\right)}{640 \sqrt{a} b^5 x^5}-\frac{63 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2} \sqrt{b}}\right)}{64 \sqrt{2} b^{11/2}}",1,"(Sqrt[b + Sqrt[b^2 + a*x^2]]*(-315*a^(5/2)*x^4 + Sqrt[a]*(16*b^4 - 144*b^3*Sqrt[b^2 + a*x^2]) + a^(3/2)*(-42*b^2*x^2 + 210*b*x^2*Sqrt[b^2 + a*x^2])))/(640*Sqrt[a]*b^5*x^5) - (63*a^(5/2)*ArcTan[(Sqrt[a]*x)/(Sqrt[2]*Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/(128*Sqrt[2]*b^(11/2))","A",1
2448,1,144,198,0.1245901,"\int \sqrt{d+\sqrt{c+\sqrt{b+a x}}} \, dx","IntegrateAlgebraic[Sqrt[d + Sqrt[c + Sqrt[b + a*x]]],x]","-\frac{8 \sqrt{c+\sqrt{b+a x}} \left(16 c d-8 d^3-5 d \sqrt{b+a x}\right) \sqrt{d+\sqrt{c+\sqrt{b+a x}}}}{315 a}-\frac{8 \left(28 c^2-36 c d^2+16 d^4-7 c \sqrt{b+a x}+6 d^2 \sqrt{b+a x}-35 (b+a x)\right) \sqrt{d+\sqrt{c+\sqrt{b+a x}}}}{315 a}","\frac{8 \left(35 b-28 c^2+36 c d^2-16 d^4+35 a x\right) \sqrt{d+\sqrt{c+\sqrt{b+a x}}}}{315 a}-\frac{64 \left(2 c d-d^3\right) \sqrt{c+\sqrt{b+a x}} \sqrt{d+\sqrt{c+\sqrt{b+a x}}}}{315 a}+\sqrt{b+a x} \left(\frac{8 \left(7 c-6 d^2\right) \sqrt{d+\sqrt{c+\sqrt{b+a x}}}}{315 a}+\frac{8 d \sqrt{c+\sqrt{b+a x}} \sqrt{d+\sqrt{c+\sqrt{b+a x}}}}{63 a}\right)",1,"(-8*Sqrt[c + Sqrt[b + a*x]]*(16*c*d - 8*d^3 - 5*d*Sqrt[b + a*x])*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]])/(315*a) - (8*(28*c^2 - 36*c*d^2 + 16*d^4 - 7*c*Sqrt[b + a*x] + 6*d^2*Sqrt[b + a*x] - 35*(b + a*x))*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]])/(315*a)","A",1
2449,1,199,199,0.4962618,"\int \frac{1}{\left(-b+a x^2\right) \sqrt[3]{-x+x^3}} \, dx","IntegrateAlgebraic[1/((-b + a*x^2)*(-x + x^3)^(1/3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a-b} x}{\sqrt[3]{a-b} x-2 \sqrt[3]{b} \sqrt[3]{-x+x^3}}\right)}{2 \sqrt[3]{a-b} b^{2/3}}-\frac{\log \left(\sqrt[3]{a-b} x+\sqrt[3]{b} \sqrt[3]{-x+x^3}\right)}{2 \sqrt[3]{a-b} b^{2/3}}+\frac{\log \left((a-b)^{2/3} x^2-\sqrt[3]{a-b} \sqrt[3]{b} x \sqrt[3]{-x+x^3}+b^{2/3} \left(-x+x^3\right)^{2/3}\right)}{4 \sqrt[3]{a-b} b^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a-b} x}{\sqrt[3]{a-b} x-2 \sqrt[3]{b} \sqrt[3]{-x+x^3}}\right)}{2 \sqrt[3]{a-b} b^{2/3}}-\frac{\log \left(\sqrt[3]{a-b} x+\sqrt[3]{b} \sqrt[3]{-x+x^3}\right)}{2 \sqrt[3]{a-b} b^{2/3}}+\frac{\log \left((a-b)^{2/3} x^2-\sqrt[3]{a-b} \sqrt[3]{b} x \sqrt[3]{-x+x^3}+b^{2/3} \left(-x+x^3\right)^{2/3}\right)}{4 \sqrt[3]{a-b} b^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(a - b)^(1/3)*x)/((a - b)^(1/3)*x - 2*b^(1/3)*(-x + x^3)^(1/3))])/(2*(a - b)^(1/3)*b^(2/3)) - Log[(a - b)^(1/3)*x + b^(1/3)*(-x + x^3)^(1/3)]/(2*(a - b)^(1/3)*b^(2/3)) + Log[(a - b)^(2/3)*x^2 - (a - b)^(1/3)*b^(1/3)*x*(-x + x^3)^(1/3) + b^(2/3)*(-x + x^3)^(2/3)]/(4*(a - b)^(1/3)*b^(2/3))","A",1
2450,1,199,199,0.5375238,"\int \frac{1}{x^3 \left(-1+x^3\right) \sqrt[3]{x^2+x^3}} \, dx","IntegrateAlgebraic[1/(x^3*(-1 + x^3)*(x^2 + x^3)^(1/3)),x]","\frac{3 \left(5-6 x+9 x^2\right) \left(x^2+x^3\right)^{2/3}}{40 x^4}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x^2+x^3}}\right)}{\sqrt[3]{2} \sqrt{3}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x^2+x^3}\right)}{3 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x^2+x^3}+\sqrt[3]{2} \left(x^2+x^3\right)^{2/3}\right)}{6 \sqrt[3]{2}}+\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{3 \left(5-6 x+9 x^2\right) \left(x^2+x^3\right)^{2/3}}{40 x^4}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x^2+x^3}}\right)}{\sqrt[3]{2} \sqrt{3}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x^2+x^3}\right)}{3 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x^2+x^3}+\sqrt[3]{2} \left(x^2+x^3\right)^{2/3}\right)}{6 \sqrt[3]{2}}+\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"(3*(5 - 6*x + 9*x^2)*(x^2 + x^3)^(2/3))/(40*x^4) - ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(x^2 + x^3)^(1/3))]/(2^(1/3)*Sqrt[3]) + Log[-2*x + 2^(2/3)*(x^2 + x^3)^(1/3)]/(3*2^(1/3)) - Log[2*x^2 + 2^(2/3)*x*(x^2 + x^3)^(1/3) + 2^(1/3)*(x^2 + x^3)^(2/3)]/(6*2^(1/3)) + RootSum[1 - #1^3 + #1^6 & , (-Log[x] + Log[(x^2 + x^3)^(1/3) - x*#1])/#1 & ]/3","A",1
2451,1,199,199,0.0011683,"\int \frac{1}{x^3 \left(-1+x^3\right) \sqrt[3]{x^2+x^3}} \, dx","IntegrateAlgebraic[1/(x^3*(-1 + x^3)*(x^2 + x^3)^(1/3)),x]","\frac{3 \left(5-6 x+9 x^2\right) \left(x^2+x^3\right)^{2/3}}{40 x^4}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x^2+x^3}}\right)}{\sqrt[3]{2} \sqrt{3}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x^2+x^3}\right)}{3 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x^2+x^3}+\sqrt[3]{2} \left(x^2+x^3\right)^{2/3}\right)}{6 \sqrt[3]{2}}+\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{3 \left(5-6 x+9 x^2\right) \left(x^2+x^3\right)^{2/3}}{40 x^4}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x^2+x^3}}\right)}{\sqrt[3]{2} \sqrt{3}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x^2+x^3}\right)}{3 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x^2+x^3}+\sqrt[3]{2} \left(x^2+x^3\right)^{2/3}\right)}{6 \sqrt[3]{2}}+\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"(3*(5 - 6*x + 9*x^2)*(x^2 + x^3)^(2/3))/(40*x^4) - ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(x^2 + x^3)^(1/3))]/(2^(1/3)*Sqrt[3]) + Log[-2*x + 2^(2/3)*(x^2 + x^3)^(1/3)]/(3*2^(1/3)) - Log[2*x^2 + 2^(2/3)*x*(x^2 + x^3)^(1/3) + 2^(1/3)*(x^2 + x^3)^(2/3)]/(6*2^(1/3)) + RootSum[1 - #1^3 + #1^6 & , (-Log[x] + Log[(x^2 + x^3)^(1/3) - x*#1])/#1 & ]/3","A",1
2452,1,199,199,1.2644004,"\int \frac{1}{x^6 \left(-b+a x^3\right) \sqrt[3]{b^2 x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[1/(x^6*(-b + a*x^3)*(b^2*x^2 + a^3*x^3)^(1/3)),x]","\frac{3 \left(b^2 x^2+a^3 x^3\right)^{2/3} \left(3080 b^{10}-3300 a^3 b^8 x+3600 a^6 b^6 x^2-4050 a^9 b^4 x^3+6545 a b^9 x^3+4860 a^{12} b^2 x^4-7854 a^4 b^7 x^4-7290 a^{15} x^5+11781 a^7 b^5 x^5\right)}{52360 b^{13} x^7}+\frac{a^2 \text{RootSum}\left[a^9+a b^5-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{3 b^3}","\frac{3 \left(b^2 x^2+a^3 x^3\right)^{2/3} \left(3080 b^{10}-3300 a^3 b^8 x+3600 a^6 b^6 x^2-4050 a^9 b^4 x^3+6545 a b^9 x^3+4860 a^{12} b^2 x^4-7854 a^4 b^7 x^4-7290 a^{15} x^5+11781 a^7 b^5 x^5\right)}{52360 b^{13} x^7}+\frac{a^2 \text{RootSum}\left[a^9+a b^5-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{3 b^3}",1,"(3*(b^2*x^2 + a^3*x^3)^(2/3)*(3080*b^10 - 3300*a^3*b^8*x + 3600*a^6*b^6*x^2 - 4050*a^9*b^4*x^3 + 6545*a*b^9*x^3 + 4860*a^12*b^2*x^4 - 7854*a^4*b^7*x^4 - 7290*a^15*x^5 + 11781*a^7*b^5*x^5))/(52360*b^13*x^7) + (a^2*RootSum[a^9 + a*b^5 - 3*a^6*#1^3 + 3*a^3*#1^6 - #1^9 & , (-Log[x] + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1])/#1 & ])/(3*b^3)","A",1
2453,1,199,199,0.0024271,"\int \frac{1}{x^6 \left(-b+a x^3\right) \sqrt[3]{b^2 x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[1/(x^6*(-b + a*x^3)*(b^2*x^2 + a^3*x^3)^(1/3)),x]","\frac{3 \left(b^2 x^2+a^3 x^3\right)^{2/3} \left(3080 b^{10}-3300 a^3 b^8 x+3600 a^6 b^6 x^2-4050 a^9 b^4 x^3+6545 a b^9 x^3+4860 a^{12} b^2 x^4-7854 a^4 b^7 x^4-7290 a^{15} x^5+11781 a^7 b^5 x^5\right)}{52360 b^{13} x^7}+\frac{a^2 \text{RootSum}\left[a^9+a b^5-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{3 b^3}","\frac{3 \left(b^2 x^2+a^3 x^3\right)^{2/3} \left(3080 b^{10}-3300 a^3 b^8 x+3600 a^6 b^6 x^2-4050 a^9 b^4 x^3+6545 a b^9 x^3+4860 a^{12} b^2 x^4-7854 a^4 b^7 x^4-7290 a^{15} x^5+11781 a^7 b^5 x^5\right)}{52360 b^{13} x^7}+\frac{a^2 \text{RootSum}\left[a^9+a b^5-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{3 b^3}",1,"(3*(b^2*x^2 + a^3*x^3)^(2/3)*(3080*b^10 - 3300*a^3*b^8*x + 3600*a^6*b^6*x^2 - 4050*a^9*b^4*x^3 + 6545*a*b^9*x^3 + 4860*a^12*b^2*x^4 - 7854*a^4*b^7*x^4 - 7290*a^15*x^5 + 11781*a^7*b^5*x^5))/(52360*b^13*x^7) + (a^2*RootSum[a^9 + a*b^5 - 3*a^6*#1^3 + 3*a^3*#1^6 - #1^9 & , (-Log[x] + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1])/#1 & ])/(3*b^3)","A",1
2454,1,313,199,1.5707566,"\int \frac{\left(-1+x^2\right) \left(1+x+x^2\right) \sqrt{1+3 x^2+x^4}}{\left(1+x+x^2+x^3+x^4\right)^2} \, dx","IntegrateAlgebraic[((-1 + x^2)*(1 + x + x^2)*Sqrt[1 + 3*x^2 + x^4])/(1 + x + x^2 + x^3 + x^4)^2,x]","\frac{\left(-1-3 x-x^2\right) \sqrt{1+3 x^2+x^4}}{5 \left(1+x+x^2+x^3+x^4\right)}+\text{RootSum}\left[1-2 \text{$\#$1}-6 \text{$\#$1}^2+2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{-7 \log (x)+7 \log \left(1+x^2+\sqrt{1+3 x^2+x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^2+\log \left(1+x^2+\sqrt{1+3 x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}^2}{-1-6 \text{$\#$1}+3 \text{$\#$1}^2+2 \text{$\#$1}^3}\&\right]-\frac{2}{5} \text{RootSum}\left[1-2 \text{$\#$1}-6 \text{$\#$1}^2+2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{-19 \log (x)+19 \log \left(1+x^2+\sqrt{1+3 x^2+x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}+\log \left(1+x^2+\sqrt{1+3 x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}-\log (x) \text{$\#$1}^2+\log \left(1+x^2+\sqrt{1+3 x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}^2}{-1-6 \text{$\#$1}+3 \text{$\#$1}^2+2 \text{$\#$1}^3}\&\right]","\frac{\left(-1-3 x-x^2\right) \sqrt{1+3 x^2+x^4}}{5 \left(1+x+x^2+x^3+x^4\right)}+\frac{1}{5} \text{RootSum}\left[1-2 \text{$\#$1}-6 \text{$\#$1}^2+2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{3 \log (x)-3 \log \left(1+x^2+\sqrt{1+3 x^2+x^4}-x \text{$\#$1}\right)+2 \log (x) \text{$\#$1}-2 \log \left(1+x^2+\sqrt{1+3 x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}-3 \log (x) \text{$\#$1}^2+3 \log \left(1+x^2+\sqrt{1+3 x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}^2}{-1-6 \text{$\#$1}+3 \text{$\#$1}^2+2 \text{$\#$1}^3}\&\right]",1,"((-1 - 3*x - x^2)*Sqrt[1 + 3*x^2 + x^4])/(5*(1 + x + x^2 + x^3 + x^4)) + RootSum[1 - 2*#1 - 6*#1^2 + 2*#1^3 + #1^4 & , (-7*Log[x] + 7*Log[1 + x^2 + Sqrt[1 + 3*x^2 + x^4] - x*#1] - Log[x]*#1^2 + Log[1 + x^2 + Sqrt[1 + 3*x^2 + x^4] - x*#1]*#1^2)/(-1 - 6*#1 + 3*#1^2 + 2*#1^3) & ] - (2*RootSum[1 - 2*#1 - 6*#1^2 + 2*#1^3 + #1^4 & , (-19*Log[x] + 19*Log[1 + x^2 + Sqrt[1 + 3*x^2 + x^4] - x*#1] - Log[x]*#1 + Log[1 + x^2 + Sqrt[1 + 3*x^2 + x^4] - x*#1]*#1 - Log[x]*#1^2 + Log[1 + x^2 + Sqrt[1 + 3*x^2 + x^4] - x*#1]*#1^2)/(-1 - 6*#1 + 3*#1^2 + 2*#1^3) & ])/5","A",1
2455,1,207,199,3.1392335,"\int \frac{b+a x^4}{\left(-b+a x^4\right) \sqrt[4]{b^2+c x^4+a^2 x^8}} \, dx","IntegrateAlgebraic[(b + a*x^4)/((-b + a*x^4)*(b^2 + c*x^4 + a^2*x^8)^(1/4)),x]","-\frac{\left(\frac{1}{4}+\frac{i}{4}\right) \tan ^{-1}\left(\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \sqrt[4]{2 a b+c} x^2-\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{b^2+c x^4+a^2 x^8}}{\sqrt[4]{2 a b+c}}}{x \sqrt[4]{b^2+c x^4+a^2 x^8}}\right)}{\sqrt[4]{2 a b+c}}-\frac{\left(\frac{1}{4}+\frac{i}{4}\right) \tanh ^{-1}\left(\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \sqrt[4]{2 a b+c} x^2+\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{b^2+c x^4+a^2 x^8}}{\sqrt[4]{2 a b+c}}}{x \sqrt[4]{b^2+c x^4+a^2 x^8}}\right)}{\sqrt[4]{2 a b+c}}","\frac{\left(\frac{1}{4}+\frac{i}{4}\right) \tan ^{-1}\left(\frac{(1+i) \sqrt[4]{2 a b+c} x \sqrt[4]{b^2+c x^4+a^2 x^8}}{\sqrt{2 a b+c} x^2-i \sqrt{b^2+c x^4+a^2 x^8}}\right)}{\sqrt[4]{2 a b+c}}-\frac{\left(\frac{1}{4}+\frac{i}{4}\right) \tanh ^{-1}\left(\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \sqrt[4]{2 a b+c} x^2+\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{b^2+c x^4+a^2 x^8}}{\sqrt[4]{2 a b+c}}}{x \sqrt[4]{b^2+c x^4+a^2 x^8}}\right)}{\sqrt[4]{2 a b+c}}",1,"((-1/4 - I/4)*ArcTan[((1/2 - I/2)*(2*a*b + c)^(1/4)*x^2 - ((1/2 + I/2)*Sqrt[b^2 + c*x^4 + a^2*x^8])/(2*a*b + c)^(1/4))/(x*(b^2 + c*x^4 + a^2*x^8)^(1/4))])/(2*a*b + c)^(1/4) - ((1/4 + I/4)*ArcTanh[((1/2 - I/2)*(2*a*b + c)^(1/4)*x^2 + ((1/2 + I/2)*Sqrt[b^2 + c*x^4 + a^2*x^8])/(2*a*b + c)^(1/4))/(x*(b^2 + c*x^4 + a^2*x^8)^(1/4))])/(2*a*b + c)^(1/4)","A",1
2456,1,199,199,0.378105,"\int \frac{2 x^4-x^9}{\sqrt{-1+x^5} \left(a-a x^5+x^{10}\right)} \, dx","IntegrateAlgebraic[(2*x^4 - x^9)/(Sqrt[-1 + x^5]*(a - a*x^5 + x^10)),x]","-\frac{\sqrt{2} \left(-4+\sqrt{-4+a} \sqrt{a}+a\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{-1+x^5}}{\sqrt{2-\sqrt{-4+a} \sqrt{a}-a}}\right)}{5 \sqrt{2-\sqrt{-4+a} \sqrt{a}-a} \sqrt{-4+a} \sqrt{a}}-\frac{\sqrt{2} \left(4+\sqrt{-4+a} \sqrt{a}-a\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{-1+x^5}}{\sqrt{2+\sqrt{-4+a} \sqrt{a}-a}}\right)}{5 \sqrt{2+\sqrt{-4+a} \sqrt{a}-a} \sqrt{-4+a} \sqrt{a}}","-\frac{\sqrt{2} \left(-4+\sqrt{-4+a} \sqrt{a}+a\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{-1+x^5}}{\sqrt{2-\sqrt{-4+a} \sqrt{a}-a}}\right)}{5 \sqrt{2-\sqrt{-4+a} \sqrt{a}-a} \sqrt{-4+a} \sqrt{a}}-\frac{\sqrt{2} \left(4+\sqrt{-4+a} \sqrt{a}-a\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{-1+x^5}}{\sqrt{2+\sqrt{-4+a} \sqrt{a}-a}}\right)}{5 \sqrt{2+\sqrt{-4+a} \sqrt{a}-a} \sqrt{-4+a} \sqrt{a}}",1,"-1/5*(Sqrt[2]*(-4 + Sqrt[-4 + a]*Sqrt[a] + a)*ArcTan[(Sqrt[2]*Sqrt[-1 + x^5])/Sqrt[2 - Sqrt[-4 + a]*Sqrt[a] - a]])/(Sqrt[2 - Sqrt[-4 + a]*Sqrt[a] - a]*Sqrt[-4 + a]*Sqrt[a]) - (Sqrt[2]*(4 + Sqrt[-4 + a]*Sqrt[a] - a)*ArcTan[(Sqrt[2]*Sqrt[-1 + x^5])/Sqrt[2 + Sqrt[-4 + a]*Sqrt[a] - a]])/(5*Sqrt[2 + Sqrt[-4 + a]*Sqrt[a] - a]*Sqrt[-4 + a]*Sqrt[a])","A",1
2457,1,199,199,0.2874056,"\int \frac{1+x^2}{\left(-1+x^2\right) \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[(1 + x^2)/((-1 + x^2)*Sqrt[x + Sqrt[1 + x^2]]),x]","-\frac{1}{3 \left(x+\sqrt{1+x^2}\right)^{3/2}}+\sqrt{x+\sqrt{1+x^2}}-2 \sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\sqrt{-1+\sqrt{2}} \sqrt{x+\sqrt{1+x^2}}\right)+2 \sqrt{1+\sqrt{2}} \tan ^{-1}\left(\sqrt{1+\sqrt{2}} \sqrt{x+\sqrt{1+x^2}}\right)-2 \sqrt{-1+\sqrt{2}} \tanh ^{-1}\left(\sqrt{-1+\sqrt{2}} \sqrt{x+\sqrt{1+x^2}}\right)+2 \sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\sqrt{1+\sqrt{2}} \sqrt{x+\sqrt{1+x^2}}\right)","-\frac{1}{3 \left(x+\sqrt{1+x^2}\right)^{3/2}}+\sqrt{x+\sqrt{1+x^2}}+2 \sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{x+\sqrt{1+x^2}}}{\sqrt{-1+\sqrt{2}}}\right)-2 \sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{x+\sqrt{1+x^2}}}{\sqrt{1+\sqrt{2}}}\right)+2 \sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{x+\sqrt{1+x^2}}}{\sqrt{-1+\sqrt{2}}}\right)-2 \sqrt{-1+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{x+\sqrt{1+x^2}}}{\sqrt{1+\sqrt{2}}}\right)",1,"-1/3*1/(x + Sqrt[1 + x^2])^(3/2) + Sqrt[x + Sqrt[1 + x^2]] - 2*Sqrt[-1 + Sqrt[2]]*ArcTan[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] + 2*Sqrt[1 + Sqrt[2]]*ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] - 2*Sqrt[-1 + Sqrt[2]]*ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] + 2*Sqrt[1 + Sqrt[2]]*ArcTanh[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]]","A",1
2458,1,199,199,0.2918481,"\int \frac{d+c x^2}{\left(-d+c x^2\right) \sqrt{a x+\sqrt{b^2+a^2 x^2}}} \, dx","IntegrateAlgebraic[(d + c*x^2)/((-d + c*x^2)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]),x]","-\frac{b^2}{3 a \left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}+\frac{\sqrt{a x+\sqrt{b^2+a^2 x^2}}}{a}+a d \text{RootSum}\left[b^4 c-2 b^2 c \text{$\#$1}^4-4 a^2 d \text{$\#$1}^4+c \text{$\#$1}^8\&,\frac{-b^2 \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-\log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{b^2 c \text{$\#$1}^3+2 a^2 d \text{$\#$1}^3-c \text{$\#$1}^7}\&\right]","-\frac{b^2}{3 a \left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}+\frac{\sqrt{a x+\sqrt{b^2+a^2 x^2}}}{a}+a d \text{RootSum}\left[b^4 c-2 b^2 c \text{$\#$1}^4-4 a^2 d \text{$\#$1}^4+c \text{$\#$1}^8\&,\frac{-b^2 \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-\log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{b^2 c \text{$\#$1}^3+2 a^2 d \text{$\#$1}^3-c \text{$\#$1}^7}\&\right]",1,"-1/3*b^2/(a*(a*x + Sqrt[b^2 + a^2*x^2])^(3/2)) + Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]/a + a*d*RootSum[b^4*c - 2*b^2*c*#1^4 - 4*a^2*d*#1^4 + c*#1^8 & , (-(b^2*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]) - Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]*#1^4)/(b^2*c*#1^3 + 2*a^2*d*#1^3 - c*#1^7) & ]","A",1
2459,1,199,199,0.0023369,"\int \frac{d+c x^2}{\left(-d+c x^2\right) \sqrt{a x+\sqrt{b^2+a^2 x^2}}} \, dx","IntegrateAlgebraic[(d + c*x^2)/((-d + c*x^2)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]),x]","-\frac{b^2}{3 a \left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}+\frac{\sqrt{a x+\sqrt{b^2+a^2 x^2}}}{a}+a d \text{RootSum}\left[b^4 c-2 b^2 c \text{$\#$1}^4-4 a^2 d \text{$\#$1}^4+c \text{$\#$1}^8\&,\frac{-b^2 \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-\log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{b^2 c \text{$\#$1}^3+2 a^2 d \text{$\#$1}^3-c \text{$\#$1}^7}\&\right]","-\frac{b^2}{3 a \left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}+\frac{\sqrt{a x+\sqrt{b^2+a^2 x^2}}}{a}+a d \text{RootSum}\left[b^4 c-2 b^2 c \text{$\#$1}^4-4 a^2 d \text{$\#$1}^4+c \text{$\#$1}^8\&,\frac{-b^2 \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-\log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{b^2 c \text{$\#$1}^3+2 a^2 d \text{$\#$1}^3-c \text{$\#$1}^7}\&\right]",1,"-1/3*b^2/(a*(a*x + Sqrt[b^2 + a^2*x^2])^(3/2)) + Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]/a + a*d*RootSum[b^4*c - 2*b^2*c*#1^4 - 4*a^2*d*#1^4 + c*#1^8 & , (-(b^2*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]) - Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]*#1^4)/(b^2*c*#1^3 + 2*a^2*d*#1^3 - c*#1^7) & ]","A",1
2460,1,200,200,1.2279091,"\int \frac{1}{x^6 \left(b+a x^3\right) \sqrt[3]{b^2 x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[1/(x^6*(b + a*x^3)*(b^2*x^2 + a^3*x^3)^(1/3)),x]","-\frac{3 \left(b^2 x^2+a^3 x^3\right)^{2/3} \left(3080 b^{10}-3300 a^3 b^8 x+3600 a^6 b^6 x^2-4050 a^9 b^4 x^3-6545 a b^9 x^3+4860 a^{12} b^2 x^4+7854 a^4 b^7 x^4-7290 a^{15} x^5-11781 a^7 b^5 x^5\right)}{52360 b^{13} x^7}-\frac{a^2 \text{RootSum}\left[a^9-a b^5-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{3 b^3}","-\frac{3 \left(b^2 x^2+a^3 x^3\right)^{2/3} \left(3080 b^{10}-3300 a^3 b^8 x+3600 a^6 b^6 x^2-4050 a^9 b^4 x^3-6545 a b^9 x^3+4860 a^{12} b^2 x^4+7854 a^4 b^7 x^4-7290 a^{15} x^5-11781 a^7 b^5 x^5\right)}{52360 b^{13} x^7}-\frac{a^2 \text{RootSum}\left[a^9-a b^5-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{3 b^3}",1,"(-3*(b^2*x^2 + a^3*x^3)^(2/3)*(3080*b^10 - 3300*a^3*b^8*x + 3600*a^6*b^6*x^2 - 4050*a^9*b^4*x^3 - 6545*a*b^9*x^3 + 4860*a^12*b^2*x^4 + 7854*a^4*b^7*x^4 - 7290*a^15*x^5 - 11781*a^7*b^5*x^5))/(52360*b^13*x^7) - (a^2*RootSum[a^9 - a*b^5 - 3*a^6*#1^3 + 3*a^3*#1^6 - #1^9 & , (-Log[x] + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1])/#1 & ])/(3*b^3)","A",1
2461,1,200,200,0.002314,"\int \frac{1}{x^6 \left(b+a x^3\right) \sqrt[3]{b^2 x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[1/(x^6*(b + a*x^3)*(b^2*x^2 + a^3*x^3)^(1/3)),x]","-\frac{3 \left(b^2 x^2+a^3 x^3\right)^{2/3} \left(3080 b^{10}-3300 a^3 b^8 x+3600 a^6 b^6 x^2-4050 a^9 b^4 x^3-6545 a b^9 x^3+4860 a^{12} b^2 x^4+7854 a^4 b^7 x^4-7290 a^{15} x^5-11781 a^7 b^5 x^5\right)}{52360 b^{13} x^7}-\frac{a^2 \text{RootSum}\left[a^9-a b^5-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{3 b^3}","-\frac{3 \left(b^2 x^2+a^3 x^3\right)^{2/3} \left(3080 b^{10}-3300 a^3 b^8 x+3600 a^6 b^6 x^2-4050 a^9 b^4 x^3-6545 a b^9 x^3+4860 a^{12} b^2 x^4+7854 a^4 b^7 x^4-7290 a^{15} x^5-11781 a^7 b^5 x^5\right)}{52360 b^{13} x^7}-\frac{a^2 \text{RootSum}\left[a^9-a b^5-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{3 b^3}",1,"(-3*(b^2*x^2 + a^3*x^3)^(2/3)*(3080*b^10 - 3300*a^3*b^8*x + 3600*a^6*b^6*x^2 - 4050*a^9*b^4*x^3 - 6545*a*b^9*x^3 + 4860*a^12*b^2*x^4 + 7854*a^4*b^7*x^4 - 7290*a^15*x^5 - 11781*a^7*b^5*x^5))/(52360*b^13*x^7) - (a^2*RootSum[a^9 - a*b^5 - 3*a^6*#1^3 + 3*a^3*#1^6 - #1^9 & , (-Log[x] + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1])/#1 & ])/(3*b^3)","A",1
2462,1,200,200,3.9422535,"\int \frac{\left(-q+p x^2\right) \left(a q+b x+a p x^2\right) \sqrt{q^2+p^2 x^4}}{x^3 \left(c q+d x+c p x^2\right)} \, dx","IntegrateAlgebraic[((-q + p*x^2)*(a*q + b*x + a*p*x^2)*Sqrt[q^2 + p^2*x^4])/(x^3*(c*q + d*x + c*p*x^2)),x]","\frac{\left(a c q+2 b c x-2 a d x+a c p x^2\right) \sqrt{q^2+p^2 x^4}}{2 c^2 x^2}-\frac{2 (-b c+a d) \sqrt{-d^2+2 c^2 p q} \tan ^{-1}\left(\frac{\sqrt{-d^2+2 c^2 p q} x}{c q+d x+c p x^2+c \sqrt{q^2+p^2 x^4}}\right)}{c^3}+\frac{\left(b c d-a d^2+a c^2 p q\right) \log (x)}{c^3}+\frac{\left(-b c d+a d^2-a c^2 p q\right) \log \left(q+p x^2+\sqrt{q^2+p^2 x^4}\right)}{c^3}","\frac{\left(a c q+2 b c x-2 a d x+a c p x^2\right) \sqrt{q^2+p^2 x^4}}{2 c^2 x^2}-\frac{2 (-b c+a d) \sqrt{-d^2+2 c^2 p q} \tan ^{-1}\left(\frac{\sqrt{-d^2+2 c^2 p q} x}{c q+d x+c p x^2+c \sqrt{q^2+p^2 x^4}}\right)}{c^3}+\frac{\left(b c d-a d^2+a c^2 p q\right) \log (x)}{c^3}+\frac{\left(-b c d+a d^2-a c^2 p q\right) \log \left(q+p x^2+\sqrt{q^2+p^2 x^4}\right)}{c^3}",1,"((a*c*q + 2*b*c*x - 2*a*d*x + a*c*p*x^2)*Sqrt[q^2 + p^2*x^4])/(2*c^2*x^2) - (2*(-(b*c) + a*d)*Sqrt[-d^2 + 2*c^2*p*q]*ArcTan[(Sqrt[-d^2 + 2*c^2*p*q]*x)/(c*q + d*x + c*p*x^2 + c*Sqrt[q^2 + p^2*x^4])])/c^3 + ((b*c*d - a*d^2 + a*c^2*p*q)*Log[x])/c^3 + ((-(b*c*d) + a*d^2 - a*c^2*p*q)*Log[q + p*x^2 + Sqrt[q^2 + p^2*x^4]])/c^3","A",1
2463,1,200,200,19.1979204,"\int \frac{x \left(3+7 x^4\right)}{\sqrt[3]{1+x^4} \left(-4+x^3+x^7\right)} \, dx","IntegrateAlgebraic[(x*(3 + 7*x^4))/((1 + x^4)^(1/3)*(-4 + x^3 + x^7)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{3 \sqrt{3} x \sqrt[3]{1+x^4}-6 x^2 \sqrt[3]{1+x^4}}{-6 2^{2/3}+4\ 2^{2/3} \sqrt{3} x-3 x \sqrt[3]{1+x^4}+2 \sqrt{3} x^2 \sqrt[3]{1+x^4}}\right)}{2^{2/3}}+\sqrt[3]{2} \tanh ^{-1}\left(1-\sqrt[3]{2} x \sqrt[3]{1+x^4}\right)-\frac{\tanh ^{-1}\left(\frac{2 \sqrt[3]{2}+2^{2/3} x \sqrt[3]{1+x^4}}{2 \sqrt[3]{2}+2^{2/3} x \sqrt[3]{1+x^4}+2 x^2 \left(1+x^4\right)^{2/3}}\right)}{2^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{3 \sqrt{3} x \sqrt[3]{1+x^4}-6 x^2 \sqrt[3]{1+x^4}}{-6 2^{2/3}+4\ 2^{2/3} \sqrt{3} x-3 x \sqrt[3]{1+x^4}+2 \sqrt{3} x^2 \sqrt[3]{1+x^4}}\right)}{2^{2/3}}+\sqrt[3]{2} \tanh ^{-1}\left(1-\sqrt[3]{2} x \sqrt[3]{1+x^4}\right)-\frac{\tanh ^{-1}\left(\frac{2 \sqrt[3]{2}+2^{2/3} x \sqrt[3]{1+x^4}}{2 \sqrt[3]{2}+2^{2/3} x \sqrt[3]{1+x^4}+2 x^2 \left(1+x^4\right)^{2/3}}\right)}{2^{2/3}}",1,"-((Sqrt[3]*ArcTan[(3*Sqrt[3]*x*(1 + x^4)^(1/3) - 6*x^2*(1 + x^4)^(1/3))/(-6*2^(2/3) + 4*2^(2/3)*Sqrt[3]*x - 3*x*(1 + x^4)^(1/3) + 2*Sqrt[3]*x^2*(1 + x^4)^(1/3))])/2^(2/3)) + 2^(1/3)*ArcTanh[1 - 2^(1/3)*x*(1 + x^4)^(1/3)] - ArcTanh[(2*2^(1/3) + 2^(2/3)*x*(1 + x^4)^(1/3))/(2*2^(1/3) + 2^(2/3)*x*(1 + x^4)^(1/3) + 2*x^2*(1 + x^4)^(2/3))]/2^(2/3)","A",1
2464,1,200,200,0.2639973,"\int \frac{-d+c x^2}{\left(d+c x^2\right) \sqrt{a x+\sqrt{b^2+a^2 x^2}}} \, dx","IntegrateAlgebraic[(-d + c*x^2)/((d + c*x^2)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]),x]","-\frac{b^2}{3 a \left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}+\frac{\sqrt{a x+\sqrt{b^2+a^2 x^2}}}{a}-a d \text{RootSum}\left[b^4 c-2 b^2 c \text{$\#$1}^4+4 a^2 d \text{$\#$1}^4+c \text{$\#$1}^8\&,\frac{-b^2 \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-\log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{b^2 c \text{$\#$1}^3-2 a^2 d \text{$\#$1}^3-c \text{$\#$1}^7}\&\right]","-\frac{b^2}{3 a \left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}+\frac{\sqrt{a x+\sqrt{b^2+a^2 x^2}}}{a}-a d \text{RootSum}\left[b^4 c-2 b^2 c \text{$\#$1}^4+4 a^2 d \text{$\#$1}^4+c \text{$\#$1}^8\&,\frac{-b^2 \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-\log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{b^2 c \text{$\#$1}^3-2 a^2 d \text{$\#$1}^3-c \text{$\#$1}^7}\&\right]",1,"-1/3*b^2/(a*(a*x + Sqrt[b^2 + a^2*x^2])^(3/2)) + Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]/a - a*d*RootSum[b^4*c - 2*b^2*c*#1^4 + 4*a^2*d*#1^4 + c*#1^8 & , (-(b^2*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]) - Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]*#1^4)/(b^2*c*#1^3 - 2*a^2*d*#1^3 - c*#1^7) & ]","A",1
2465,1,200,200,0.0024534,"\int \frac{-d+c x^2}{\left(d+c x^2\right) \sqrt{a x+\sqrt{b^2+a^2 x^2}}} \, dx","IntegrateAlgebraic[(-d + c*x^2)/((d + c*x^2)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]),x]","-\frac{b^2}{3 a \left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}+\frac{\sqrt{a x+\sqrt{b^2+a^2 x^2}}}{a}-a d \text{RootSum}\left[b^4 c-2 b^2 c \text{$\#$1}^4+4 a^2 d \text{$\#$1}^4+c \text{$\#$1}^8\&,\frac{-b^2 \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-\log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{b^2 c \text{$\#$1}^3-2 a^2 d \text{$\#$1}^3-c \text{$\#$1}^7}\&\right]","-\frac{b^2}{3 a \left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}+\frac{\sqrt{a x+\sqrt{b^2+a^2 x^2}}}{a}-a d \text{RootSum}\left[b^4 c-2 b^2 c \text{$\#$1}^4+4 a^2 d \text{$\#$1}^4+c \text{$\#$1}^8\&,\frac{-b^2 \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-\log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{b^2 c \text{$\#$1}^3-2 a^2 d \text{$\#$1}^3-c \text{$\#$1}^7}\&\right]",1,"-1/3*b^2/(a*(a*x + Sqrt[b^2 + a^2*x^2])^(3/2)) + Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]/a - a*d*RootSum[b^4*c - 2*b^2*c*#1^4 + 4*a^2*d*#1^4 + c*#1^8 & , (-(b^2*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]) - Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]*#1^4)/(b^2*c*#1^3 - 2*a^2*d*#1^3 - c*#1^7) & ]","A",1
2466,1,200,200,5.398374,"\int \frac{\sqrt{-b x+a^2 x^2}}{\sqrt{a x^2+x \sqrt{-b x+a^2 x^2}}} \, dx","IntegrateAlgebraic[Sqrt[-(b*x) + a^2*x^2]/Sqrt[a*x^2 + x*Sqrt[-(b*x) + a^2*x^2]],x]","\frac{\left(-9 b+8 a^2 x\right) \sqrt{-b x+a^2 x^2} \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{12 a b x}+\sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)} \left(\frac{19 b-8 a^2 x}{12 b}+\frac{3 \sqrt{b} \sqrt{-a x+\sqrt{-b x+a^2 x^2}} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{-a x+\sqrt{-b x+a^2 x^2}}}{\sqrt{b}}\right)}{4 \sqrt{2} a^{3/2} x}\right)","\frac{\left(-9 b+8 a^2 x\right) \sqrt{-b x+a^2 x^2} \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{12 a b x}+\sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)} \left(\frac{19 b-8 a^2 x}{12 b}+\frac{3 \sqrt{b} \sqrt{-a x+\sqrt{-b x+a^2 x^2}} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{-a x+\sqrt{-b x+a^2 x^2}}}{\sqrt{b}}\right)}{4 \sqrt{2} a^{3/2} x}\right)",1,"((-9*b + 8*a^2*x)*Sqrt[-(b*x) + a^2*x^2]*Sqrt[x*(a*x + Sqrt[-(b*x) + a^2*x^2])])/(12*a*b*x) + Sqrt[x*(a*x + Sqrt[-(b*x) + a^2*x^2])]*((19*b - 8*a^2*x)/(12*b) + (3*Sqrt[b]*Sqrt[-(a*x) + Sqrt[-(b*x) + a^2*x^2]]*ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[-(a*x) + Sqrt[-(b*x) + a^2*x^2]])/Sqrt[b]])/(4*Sqrt[2]*a^(3/2)*x))","A",1
2467,1,201,201,0.4299386,"\int \frac{a-2 b+x}{\sqrt[3]{(-a+x) (-b+x)} \left(a^2+b d-(2 a+d) x+x^2\right)} \, dx","IntegrateAlgebraic[(a - 2*b + x)/(((-a + x)*(-b + x))^(1/3)*(a^2 + b*d - (2*a + d)*x + x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{a b+(-a-b) x+x^2}}{-2 a+2 x+\sqrt[3]{d} \sqrt[3]{a b+(-a-b) x+x^2}}\right)}{d^{2/3}}+\frac{\log \left(a-x+\sqrt[3]{d} \sqrt[3]{a b+(-a-b) x+x^2}\right)}{d^{2/3}}-\frac{\log \left(a^2-2 a x+x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{a b+(-a-b) x+x^2}+d^{2/3} \left(a b+(-a-b) x+x^2\right)^{2/3}\right)}{2 d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{a b+(-a-b) x+x^2}}{-2 a+2 x+\sqrt[3]{d} \sqrt[3]{a b+(-a-b) x+x^2}}\right)}{d^{2/3}}+\frac{\log \left(a-x+\sqrt[3]{d} \sqrt[3]{a b+(-a-b) x+x^2}\right)}{d^{2/3}}-\frac{\log \left(a^2-2 a x+x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{a b+(-a-b) x+x^2}+d^{2/3} \left(a b+(-a-b) x+x^2\right)^{2/3}\right)}{2 d^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(a*b + (-a - b)*x + x^2)^(1/3))/(-2*a + 2*x + d^(1/3)*(a*b + (-a - b)*x + x^2)^(1/3))])/d^(2/3) + Log[a - x + d^(1/3)*(a*b + (-a - b)*x + x^2)^(1/3)]/d^(2/3) - Log[a^2 - 2*a*x + x^2 + (-(a*d^(1/3)) + d^(1/3)*x)*(a*b + (-a - b)*x + x^2)^(1/3) + d^(2/3)*(a*b + (-a - b)*x + x^2)^(2/3)]/(2*d^(2/3))","A",1
2468,1,201,201,0.7025435,"\int \frac{-a (a-2 b)-2 b x+x^2}{((-a+x) (-b+x))^{2/3} \left(b+a^2 d-(1+2 a d) x+d x^2\right)} \, dx","IntegrateAlgebraic[(-(a*(a - 2*b)) - 2*b*x + x^2)/(((-a + x)*(-b + x))^(2/3)*(b + a^2*d - (1 + 2*a*d)*x + d*x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \left(a b+(-a-b) x+x^2\right)^{2/3}}{-2 b+2 x+\sqrt[3]{d} \left(a b+(-a-b) x+x^2\right)^{2/3}}\right)}{d^{2/3}}+\frac{\log \left(b-x+\sqrt[3]{d} \left(a b+(-a-b) x+x^2\right)^{2/3}\right)}{d^{2/3}}-\frac{\log \left(b^2-2 b x+x^2+\left(-b \sqrt[3]{d}+\sqrt[3]{d} x\right) \left(a b+(-a-b) x+x^2\right)^{2/3}+d^{2/3} \left(a b+(-a-b) x+x^2\right)^{4/3}\right)}{2 d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \left(a b+(-a-b) x+x^2\right)^{2/3}}{-2 b+2 x+\sqrt[3]{d} \left(a b+(-a-b) x+x^2\right)^{2/3}}\right)}{d^{2/3}}+\frac{\log \left(b-x+\sqrt[3]{d} \left(a b+(-a-b) x+x^2\right)^{2/3}\right)}{d^{2/3}}-\frac{\log \left(b^2-2 b x+x^2+\left(-b \sqrt[3]{d}+\sqrt[3]{d} x\right) \left(a b+(-a-b) x+x^2\right)^{2/3}+d^{2/3} \left(a b+(-a-b) x+x^2\right)^{4/3}\right)}{2 d^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(a*b + (-a - b)*x + x^2)^(2/3))/(-2*b + 2*x + d^(1/3)*(a*b + (-a - b)*x + x^2)^(2/3))])/d^(2/3) + Log[b - x + d^(1/3)*(a*b + (-a - b)*x + x^2)^(2/3)]/d^(2/3) - Log[b^2 - 2*b*x + x^2 + (-(b*d^(1/3)) + d^(1/3)*x)*(a*b + (-a - b)*x + x^2)^(2/3) + d^(2/3)*(a*b + (-a - b)*x + x^2)^(4/3)]/(2*d^(2/3))","A",1
2469,1,201,201,0.6069486,"\int \frac{-1+(2-k) x}{\sqrt[3]{(1-x) x (1-k x)} \left(1-(b+2 k) x+\left(b+k^2\right) x^2\right)} \, dx","IntegrateAlgebraic[(-1 + (2 - k)*x)/(((1 - x)*x*(1 - k*x))^(1/3)*(1 - (b + 2*k)*x + (b + k^2)*x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}{2-2 k x+\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{b^{2/3}}+\frac{\log \left(-1+k x+\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{b^{2/3}}-\frac{\log \left(1-2 k x+k^2 x^2+\left(\sqrt[3]{b}-\sqrt[3]{b} k x\right) \sqrt[3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 b^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}{2-2 k x+\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{b^{2/3}}+\frac{\log \left(-1+k x+\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{b^{2/3}}-\frac{\log \left(1-2 k x+k^2 x^2+\left(\sqrt[3]{b}-\sqrt[3]{b} k x\right) \sqrt[3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 b^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))/(2 - 2*k*x + b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/b^(2/3) + Log[-1 + k*x + b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3)]/b^(2/3) - Log[1 - 2*k*x + k^2*x^2 + (b^(1/3) - b^(1/3)*k*x)*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + b^(2/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3)]/(2*b^(2/3))","A",1
2470,1,201,201,8.0731379,"\int \frac{-3 k-2 \left(1+k^2\right) x+k \left(1+k^2\right) x^2+4 k^2 x^3+k^3 x^4}{\left(\left(1-x^2\right) \left(1-k^2 x^2\right)\right)^{2/3} \left(-1+d-(2+d) k x-\left(d+k^2\right) x^2+d k x^3\right)} \, dx","IntegrateAlgebraic[(-3*k - 2*(1 + k^2)*x + k*(1 + k^2)*x^2 + 4*k^2*x^3 + k^3*x^4)/(((1 - x^2)*(1 - k^2*x^2))^(2/3)*(-1 + d - (2 + d)*k*x - (d + k^2)*x^2 + d*k*x^3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3}+\sqrt{3} k x}{1+k x+2 \sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{\sqrt[3]{d}}+\frac{\log \left(1+k x-\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{\sqrt[3]{d}}-\frac{\log \left(1+2 k x+k^2 x^2+\left(\sqrt[3]{d}+\sqrt[3]{d} k x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+d^{2/3} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3}+\sqrt{3} k x}{1+k x+2 \sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{\sqrt[3]{d}}+\frac{\log \left(1+k x-\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{\sqrt[3]{d}}-\frac{\log \left(1+2 k x+k^2 x^2+\left(\sqrt[3]{d}+\sqrt[3]{d} k x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+d^{2/3} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3] + Sqrt[3]*k*x)/(1 + k*x + 2*d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))])/d^(1/3) + Log[1 + k*x - d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3)]/d^(1/3) - Log[1 + 2*k*x + k^2*x^2 + (d^(1/3) + d^(1/3)*k*x)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3) + d^(2/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3)]/(2*d^(1/3))","A",1
2471,1,201,201,1.0296653,"\int \frac{-1+2 x^4}{\sqrt[4]{1+x^4} \left(-1+x^4+x^8\right)} \, dx","IntegrateAlgebraic[(-1 + 2*x^4)/((1 + x^4)^(1/4)*(-1 + x^4 + x^8)),x]","\frac{1}{2} \sqrt{\frac{1}{10} \left(11+5 \sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(-11+5 \sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)+\frac{1}{2} \sqrt{\frac{1}{10} \left(11+5 \sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(-11+5 \sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)","\frac{1}{2} \sqrt{\frac{1}{10} \left(11+5 \sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(-11+5 \sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)+\frac{1}{2} \sqrt{\frac{1}{10} \left(11+5 \sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(-11+5 \sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)",1,"(Sqrt[(11 + 5*Sqrt[5])/10]*ArcTan[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(1 + x^4)^(1/4)])/2 - (Sqrt[(-11 + 5*Sqrt[5])/10]*ArcTan[(Sqrt[1/2 + Sqrt[5]/2]*x)/(1 + x^4)^(1/4)])/2 + (Sqrt[(11 + 5*Sqrt[5])/10]*ArcTanh[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(1 + x^4)^(1/4)])/2 - (Sqrt[(-11 + 5*Sqrt[5])/10]*ArcTanh[(Sqrt[1/2 + Sqrt[5]/2]*x)/(1 + x^4)^(1/4)])/2","A",1
2472,1,203,203,0.7703004,"\int \frac{\left(-2 x+(1+k) x^2\right) \left(a-a (1+k) x+(1+a k) x^2\right)}{(-1+x) ((1-x) x (1-k x))^{2/3} (-1+k x) \left(b-b (1+k) x+(-1+b k) x^2\right)} \, dx","IntegrateAlgebraic[((-2*x + (1 + k)*x^2)*(a - a*(1 + k)*x + (1 + a*k)*x^2))/((-1 + x)*((1 - x)*x*(1 - k*x))^(2/3)*(-1 + k*x)*(b - b*(1 + k)*x + (-1 + b*k)*x^2)),x]","\frac{3 x^2}{2 \left(x-x^2-k x^2+k x^3\right)^{2/3}}+\frac{\left(\sqrt{3} a+\sqrt{3} b\right) \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{\sqrt[3]{b}}+\frac{(a+b) \log \left(x-\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{\sqrt[3]{b}}+\frac{(-a-b) \log \left(x^2+\sqrt[3]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 \sqrt[3]{b}}","\frac{3 x^2}{2 \left(x-x^2-k x^2+k x^3\right)^{2/3}}+\frac{\left(\sqrt{3} a+\sqrt{3} b\right) \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{\sqrt[3]{b}}+\frac{(a+b) \log \left(x-\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{\sqrt[3]{b}}+\frac{(-a-b) \log \left(x^2+\sqrt[3]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 \sqrt[3]{b}}",1,"(3*x^2)/(2*(x - x^2 - k*x^2 + k*x^3)^(2/3)) + ((Sqrt[3]*a + Sqrt[3]*b)*ArcTan[(Sqrt[3]*x)/(x + 2*b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/b^(1/3) + ((a + b)*Log[x - b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3)])/b^(1/3) + ((-a - b)*Log[x^2 + b^(1/3)*x*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + b^(2/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3)])/(2*b^(1/3))","A",1
2473,1,203,203,8.0664567,"\int \frac{-3 k+2 \left(1+k^2\right) x+k \left(1+k^2\right) x^2-4 k^2 x^3+k^3 x^4}{\left(\left(1-x^2\right) \left(1-k^2 x^2\right)\right)^{2/3} \left(1-d-(2+d) k x+\left(d+k^2\right) x^2+d k x^3\right)} \, dx","IntegrateAlgebraic[(-3*k + 2*(1 + k^2)*x + k*(1 + k^2)*x^2 - 4*k^2*x^3 + k^3*x^4)/(((1 - x^2)*(1 - k^2*x^2))^(2/3)*(1 - d - (2 + d)*k*x + (d + k^2)*x^2 + d*k*x^3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{-\sqrt{3}+\sqrt{3} k x}{-1+k x-2 \sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{\sqrt[3]{d}}+\frac{\log \left(-1+k x+\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{\sqrt[3]{d}}-\frac{\log \left(1-2 k x+k^2 x^2+\left(\sqrt[3]{d}-\sqrt[3]{d} k x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+d^{2/3} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{-\sqrt{3}+\sqrt{3} k x}{-1+k x-2 \sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{\sqrt[3]{d}}+\frac{\log \left(-1+k x+\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{\sqrt[3]{d}}-\frac{\log \left(1-2 k x+k^2 x^2+\left(\sqrt[3]{d}-\sqrt[3]{d} k x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+d^{2/3} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(-Sqrt[3] + Sqrt[3]*k*x)/(-1 + k*x - 2*d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))])/d^(1/3) + Log[-1 + k*x + d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3)]/d^(1/3) - Log[1 - 2*k*x + k^2*x^2 + (d^(1/3) - d^(1/3)*k*x)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3) + d^(2/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3)]/(2*d^(1/3))","A",1
2474,1,203,203,2.932171,"\int \frac{\sqrt[3]{-x+x^3} \left(-b+a x^6\right)}{-d+c x^6} \, dx","IntegrateAlgebraic[((-x + x^3)^(1/3)*(-b + a*x^6))/(-d + c*x^6),x]","\frac{a x \sqrt[3]{-x+x^3}}{2 c}+\frac{a \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)}{2 \sqrt{3} c}+\frac{a \log \left(-x+\sqrt[3]{-x+x^3}\right)}{6 c}-\frac{a \log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)}{12 c}+\frac{(b c-a d) \text{RootSum}\left[c-d+3 d \text{$\#$1}^3-3 d \text{$\#$1}^6+d \text{$\#$1}^9\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{-x+x^3}-x \text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^3}\&\right]}{6 c d}","\frac{a x \sqrt[3]{-x+x^3}}{2 c}+\frac{a \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)}{2 \sqrt{3} c}+\frac{a \log \left(-x+\sqrt[3]{-x+x^3}\right)}{6 c}-\frac{a \log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)}{12 c}+\frac{(b c-a d) \text{RootSum}\left[c-d+3 d \text{$\#$1}^3-3 d \text{$\#$1}^6+d \text{$\#$1}^9\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{-x+x^3}-x \text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^3}\&\right]}{6 c d}",1,"(a*x*(-x + x^3)^(1/3))/(2*c) + (a*ArcTan[(Sqrt[3]*x)/(x + 2*(-x + x^3)^(1/3))])/(2*Sqrt[3]*c) + (a*Log[-x + (-x + x^3)^(1/3)])/(6*c) - (a*Log[x^2 + x*(-x + x^3)^(1/3) + (-x + x^3)^(2/3)])/(12*c) + ((b*c - a*d)*RootSum[c - d + 3*d*#1^3 - 3*d*#1^6 + d*#1^9 & , (-(Log[x]*#1) + Log[(-x + x^3)^(1/3) - x*#1]*#1)/(-1 + #1^3) & ])/(6*c*d)","A",1
2475,1,203,203,0.0010295,"\int \frac{\sqrt[3]{-x+x^3} \left(-b+a x^6\right)}{-d+c x^6} \, dx","IntegrateAlgebraic[((-x + x^3)^(1/3)*(-b + a*x^6))/(-d + c*x^6),x]","\frac{a x \sqrt[3]{-x+x^3}}{2 c}+\frac{a \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)}{2 \sqrt{3} c}+\frac{a \log \left(-x+\sqrt[3]{-x+x^3}\right)}{6 c}-\frac{a \log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)}{12 c}+\frac{(b c-a d) \text{RootSum}\left[c-d+3 d \text{$\#$1}^3-3 d \text{$\#$1}^6+d \text{$\#$1}^9\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{-x+x^3}-x \text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^3}\&\right]}{6 c d}","\frac{a x \sqrt[3]{-x+x^3}}{2 c}+\frac{a \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)}{2 \sqrt{3} c}+\frac{a \log \left(-x+\sqrt[3]{-x+x^3}\right)}{6 c}-\frac{a \log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)}{12 c}+\frac{(b c-a d) \text{RootSum}\left[c-d+3 d \text{$\#$1}^3-3 d \text{$\#$1}^6+d \text{$\#$1}^9\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{-x+x^3}-x \text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^3}\&\right]}{6 c d}",1,"(a*x*(-x + x^3)^(1/3))/(2*c) + (a*ArcTan[(Sqrt[3]*x)/(x + 2*(-x + x^3)^(1/3))])/(2*Sqrt[3]*c) + (a*Log[-x + (-x + x^3)^(1/3)])/(6*c) - (a*Log[x^2 + x*(-x + x^3)^(1/3) + (-x + x^3)^(2/3)])/(12*c) + ((b*c - a*d)*RootSum[c - d + 3*d*#1^3 - 3*d*#1^6 + d*#1^9 & , (-(Log[x]*#1) + Log[(-x + x^3)^(1/3) - x*#1]*#1)/(-1 + #1^3) & ])/(6*c*d)","A",1
2476,1,203,203,0.5312157,"\int \frac{1}{1-x \sqrt{c+b x+a x^2}} \, dx","IntegrateAlgebraic[(1 - x*Sqrt[c + b*x + a*x^2])^(-1),x]","-2 \text{RootSum}\left[b^2-\sqrt{a} c^2-4 \sqrt{a} b \text{$\#$1}+b c \text{$\#$1}+4 a \text{$\#$1}^2-b \text{$\#$1}^3+\sqrt{a} \text{$\#$1}^4\&,\frac{\sqrt{a} c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)-b \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}+\sqrt{a} \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}^2}{-4 \sqrt{a} b+b c+8 a \text{$\#$1}-3 b \text{$\#$1}^2+4 \sqrt{a} \text{$\#$1}^3}\&\right]","-2 \text{RootSum}\left[b^2-\sqrt{a} c^2-4 \sqrt{a} b \text{$\#$1}+b c \text{$\#$1}+4 a \text{$\#$1}^2-b \text{$\#$1}^3+\sqrt{a} \text{$\#$1}^4\&,\frac{\sqrt{a} c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)-b \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}+\sqrt{a} \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}^2}{-4 \sqrt{a} b+b c+8 a \text{$\#$1}-3 b \text{$\#$1}^2+4 \sqrt{a} \text{$\#$1}^3}\&\right]",1,"-2*RootSum[b^2 - Sqrt[a]*c^2 - 4*Sqrt[a]*b*#1 + b*c*#1 + 4*a*#1^2 - b*#1^3 + Sqrt[a]*#1^4 & , (Sqrt[a]*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1] - b*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 + Sqrt[a]*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1^2)/(-4*Sqrt[a]*b + b*c + 8*a*#1 - 3*b*#1^2 + 4*Sqrt[a]*#1^3) & ]","A",1
2477,1,208,204,0.5111073,"\int \frac{-b+a x^2}{\left(-b+2 a x^2\right) \sqrt[3]{b^2 x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[(-b + a*x^2)/((-b + 2*a*x^2)*(b^2*x^2 + a^3*x^3)^(1/3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{2 a}-\frac{\log \left(a^2 x-a \sqrt[3]{b^2 x^2+a^3 x^3}\right)}{2 a}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{b^2 x^2+a^3 x^3}+\left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{4 a}-\frac{1}{4} \text{RootSum}\left[a^6-2 a b^3-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{2 a}-\frac{\log \left(-a x+\sqrt[3]{b^2 x^2+a^3 x^3}\right)}{2 a}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{b^2 x^2+a^3 x^3}+\left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{4 a}-\frac{1}{4} \text{RootSum}\left[a^6-2 a b^3-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*a*x)/(a*x + 2*(b^2*x^2 + a^3*x^3)^(1/3))])/(2*a) - Log[a^2*x - a*(b^2*x^2 + a^3*x^3)^(1/3)]/(2*a) + Log[a^2*x^2 + a*x*(b^2*x^2 + a^3*x^3)^(1/3) + (b^2*x^2 + a^3*x^3)^(2/3)]/(4*a) - RootSum[a^6 - 2*a*b^3 - 2*a^3*#1^3 + #1^6 & , (-Log[x] + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1])/#1 & ]/4","A",1
2478,1,208,204,0.0027676,"\int \frac{-b+a x^2}{\left(-b+2 a x^2\right) \sqrt[3]{b^2 x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[(-b + a*x^2)/((-b + 2*a*x^2)*(b^2*x^2 + a^3*x^3)^(1/3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{2 a}-\frac{\log \left(a^2 x-a \sqrt[3]{b^2 x^2+a^3 x^3}\right)}{2 a}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{b^2 x^2+a^3 x^3}+\left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{4 a}-\frac{1}{4} \text{RootSum}\left[a^6-2 a b^3-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{2 a}-\frac{\log \left(-a x+\sqrt[3]{b^2 x^2+a^3 x^3}\right)}{2 a}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{b^2 x^2+a^3 x^3}+\left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{4 a}-\frac{1}{4} \text{RootSum}\left[a^6-2 a b^3-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*a*x)/(a*x + 2*(b^2*x^2 + a^3*x^3)^(1/3))])/(2*a) - Log[a^2*x - a*(b^2*x^2 + a^3*x^3)^(1/3)]/(2*a) + Log[a^2*x^2 + a*x*(b^2*x^2 + a^3*x^3)^(1/3) + (b^2*x^2 + a^3*x^3)^(2/3)]/(4*a) - RootSum[a^6 - 2*a*b^3 - 2*a^3*#1^3 + #1^6 & , (-Log[x] + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1])/#1 & ]/4","A",1
2479,1,208,204,0.4827267,"\int \frac{b+a x^2}{\left(b+2 a x^2\right) \sqrt[3]{b^2 x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[(b + a*x^2)/((b + 2*a*x^2)*(b^2*x^2 + a^3*x^3)^(1/3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{2 a}-\frac{\log \left(a^2 x-a \sqrt[3]{b^2 x^2+a^3 x^3}\right)}{2 a}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{b^2 x^2+a^3 x^3}+\left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{4 a}-\frac{1}{4} \text{RootSum}\left[a^6+2 a b^3-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{2 a}-\frac{\log \left(-a x+\sqrt[3]{b^2 x^2+a^3 x^3}\right)}{2 a}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{b^2 x^2+a^3 x^3}+\left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{4 a}-\frac{1}{4} \text{RootSum}\left[a^6+2 a b^3-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*a*x)/(a*x + 2*(b^2*x^2 + a^3*x^3)^(1/3))])/(2*a) - Log[a^2*x - a*(b^2*x^2 + a^3*x^3)^(1/3)]/(2*a) + Log[a^2*x^2 + a*x*(b^2*x^2 + a^3*x^3)^(1/3) + (b^2*x^2 + a^3*x^3)^(2/3)]/(4*a) - RootSum[a^6 + 2*a*b^3 - 2*a^3*#1^3 + #1^6 & , (-Log[x] + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1])/#1 & ]/4","A",1
2480,1,208,204,0.0023351,"\int \frac{b+a x^2}{\left(b+2 a x^2\right) \sqrt[3]{b^2 x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[(b + a*x^2)/((b + 2*a*x^2)*(b^2*x^2 + a^3*x^3)^(1/3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{2 a}-\frac{\log \left(a^2 x-a \sqrt[3]{b^2 x^2+a^3 x^3}\right)}{2 a}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{b^2 x^2+a^3 x^3}+\left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{4 a}-\frac{1}{4} \text{RootSum}\left[a^6+2 a b^3-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{2 a}-\frac{\log \left(-a x+\sqrt[3]{b^2 x^2+a^3 x^3}\right)}{2 a}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{b^2 x^2+a^3 x^3}+\left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{4 a}-\frac{1}{4} \text{RootSum}\left[a^6+2 a b^3-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*a*x)/(a*x + 2*(b^2*x^2 + a^3*x^3)^(1/3))])/(2*a) - Log[a^2*x - a*(b^2*x^2 + a^3*x^3)^(1/3)]/(2*a) + Log[a^2*x^2 + a*x*(b^2*x^2 + a^3*x^3)^(1/3) + (b^2*x^2 + a^3*x^3)^(2/3)]/(4*a) - RootSum[a^6 + 2*a*b^3 - 2*a^3*#1^3 + #1^6 & , (-Log[x] + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1])/#1 & ]/4","A",1
2481,1,204,204,0.9107348,"\int \frac{x \left(4 a b-3 (a+b) x+2 x^2\right)}{\sqrt[3]{x^2 (-a+x) (-b+x)} \left(-a b+(a+b) x-x^2+d x^4\right)} \, dx","IntegrateAlgebraic[(x*(4*a*b - 3*(a + b)*x + 2*x^2))/((x^2*(-a + x)*(-b + x))^(1/3)*(-(a*b) + (a + b)*x - x^2 + d*x^4)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}{2 \sqrt[3]{d} x^2+\sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{\sqrt[3]{d}}+\frac{\log \left(-\sqrt[3]{d} x^2+\sqrt[3]{a b x^2+(-a-b) x^3+x^4}\right)}{\sqrt[3]{d}}-\frac{\log \left(d^{2/3} x^4+\sqrt[3]{d} x^2 \sqrt[3]{a b x^2+(-a-b) x^3+x^4}+\left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}{2 \sqrt[3]{d} x^2+\sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{\sqrt[3]{d}}+\frac{\log \left(-\sqrt[3]{d} x^2+\sqrt[3]{a b x^2+(-a-b) x^3+x^4}\right)}{\sqrt[3]{d}}-\frac{\log \left(d^{2/3} x^4+\sqrt[3]{d} x^2 \sqrt[3]{a b x^2+(-a-b) x^3+x^4}+\left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))/(2*d^(1/3)*x^2 + (a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))])/d^(1/3) + Log[-(d^(1/3)*x^2) + (a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3)]/d^(1/3) - Log[d^(2/3)*x^4 + d^(1/3)*x^2*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3) + (a*b*x^2 + (-a - b)*x^3 + x^4)^(2/3)]/(2*d^(1/3))","A",1
2482,1,203,204,0.4454586,"\int \frac{\sqrt[4]{b+a x^4} \left(2 b+3 a x^4\right)}{x^6 \left(b+a x^8\right)} \, dx","IntegrateAlgebraic[((b + a*x^4)^(1/4)*(2*b + 3*a*x^4))/(x^6*(b + a*x^8)),x]","\frac{\left(-2 b-17 a x^4\right) \sqrt[4]{b+a x^4}}{5 b x^5}-\frac{a \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{3 a^2 \log (x)+3 a b \log (x)-3 a^2 \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)-3 a b \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)-3 a \log (x) \text{$\#$1}^4+2 b \log (x) \text{$\#$1}^4+3 a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-2 b \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}^3+\text{$\#$1}^7}\&\right]}{8 b}","\frac{\left(-2 b-17 a x^4\right) \sqrt[4]{b+a x^4}}{5 b x^5}-\frac{a \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-3 a^2 \log (x)-3 a b \log (x)+3 a^2 \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)+3 a b \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)+3 a \log (x) \text{$\#$1}^4-2 b \log (x) \text{$\#$1}^4-3 a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+2 b \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}^3-\text{$\#$1}^7}\&\right]}{8 b}",1,"((-2*b - 17*a*x^4)*(b + a*x^4)^(1/4))/(5*b*x^5) - (a*RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (3*a^2*Log[x] + 3*a*b*Log[x] - 3*a^2*Log[(b + a*x^4)^(1/4) - x*#1] - 3*a*b*Log[(b + a*x^4)^(1/4) - x*#1] - 3*a*Log[x]*#1^4 + 2*b*Log[x]*#1^4 + 3*a*Log[(b + a*x^4)^(1/4) - x*#1]*#1^4 - 2*b*Log[(b + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3) + #1^7) & ])/(8*b)","A",1
2483,1,203,204,0.0016111,"\int \frac{\sqrt[4]{b+a x^4} \left(2 b+3 a x^4\right)}{x^6 \left(b+a x^8\right)} \, dx","IntegrateAlgebraic[((b + a*x^4)^(1/4)*(2*b + 3*a*x^4))/(x^6*(b + a*x^8)),x]","\frac{\left(-2 b-17 a x^4\right) \sqrt[4]{b+a x^4}}{5 b x^5}-\frac{a \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{3 a^2 \log (x)+3 a b \log (x)-3 a^2 \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)-3 a b \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)-3 a \log (x) \text{$\#$1}^4+2 b \log (x) \text{$\#$1}^4+3 a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-2 b \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}^3+\text{$\#$1}^7}\&\right]}{8 b}","\frac{\left(-2 b-17 a x^4\right) \sqrt[4]{b+a x^4}}{5 b x^5}-\frac{a \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-3 a^2 \log (x)-3 a b \log (x)+3 a^2 \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)+3 a b \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right)+3 a \log (x) \text{$\#$1}^4-2 b \log (x) \text{$\#$1}^4-3 a \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+2 b \log \left(\sqrt[4]{b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}^3-\text{$\#$1}^7}\&\right]}{8 b}",1,"((-2*b - 17*a*x^4)*(b + a*x^4)^(1/4))/(5*b*x^5) - (a*RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (3*a^2*Log[x] + 3*a*b*Log[x] - 3*a^2*Log[(b + a*x^4)^(1/4) - x*#1] - 3*a*b*Log[(b + a*x^4)^(1/4) - x*#1] - 3*a*Log[x]*#1^4 + 2*b*Log[x]*#1^4 + 3*a*Log[(b + a*x^4)^(1/4) - x*#1]*#1^4 - 2*b*Log[(b + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3) + #1^7) & ])/(8*b)","A",1
2484,1,212,204,0.2610903,"\int \frac{\sqrt{x+\sqrt{1+x}}}{1+x^2} \, dx","IntegrateAlgebraic[Sqrt[x + Sqrt[1 + x]]/(1 + x^2),x]","\frac{1}{2} \text{RootSum}\left[1-8 \text{$\#$1}+40 \text{$\#$1}^2-48 \text{$\#$1}^3+20 \text{$\#$1}^4+8 \text{$\#$1}^5-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right)+2 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}-2 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^5+\log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^6}{-1+10 \text{$\#$1}-18 \text{$\#$1}^2+10 \text{$\#$1}^3+5 \text{$\#$1}^4-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","\frac{1}{2} \text{RootSum}\left[1-8 \text{$\#$1}+40 \text{$\#$1}^2-48 \text{$\#$1}^3+20 \text{$\#$1}^4+8 \text{$\#$1}^5-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right)+2 \log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}-2 \log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^5+\log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^6}{-1+10 \text{$\#$1}-18 \text{$\#$1}^2+10 \text{$\#$1}^3+5 \text{$\#$1}^4-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]",1,"RootSum[1 - 8*#1 + 40*#1^2 - 48*#1^3 + 20*#1^4 + 8*#1^5 - 4*#1^6 + #1^8 & , (Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1] + 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1 - 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^5 + Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^6)/(-1 + 10*#1 - 18*#1^2 + 10*#1^3 + 5*#1^4 - 3*#1^5 + #1^7) & ]/2","A",1
2485,1,212,204,0.0009555,"\int \frac{\sqrt{x+\sqrt{1+x}}}{1+x^2} \, dx","IntegrateAlgebraic[Sqrt[x + Sqrt[1 + x]]/(1 + x^2),x]","\frac{1}{2} \text{RootSum}\left[1-8 \text{$\#$1}+40 \text{$\#$1}^2-48 \text{$\#$1}^3+20 \text{$\#$1}^4+8 \text{$\#$1}^5-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right)+2 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}-2 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^5+\log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^6}{-1+10 \text{$\#$1}-18 \text{$\#$1}^2+10 \text{$\#$1}^3+5 \text{$\#$1}^4-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","\frac{1}{2} \text{RootSum}\left[1-8 \text{$\#$1}+40 \text{$\#$1}^2-48 \text{$\#$1}^3+20 \text{$\#$1}^4+8 \text{$\#$1}^5-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right)+2 \log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}-2 \log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^5+\log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^6}{-1+10 \text{$\#$1}-18 \text{$\#$1}^2+10 \text{$\#$1}^3+5 \text{$\#$1}^4-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]",1,"RootSum[1 - 8*#1 + 40*#1^2 - 48*#1^3 + 20*#1^4 + 8*#1^5 - 4*#1^6 + #1^8 & , (Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1] + 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1 - 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^5 + Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^6)/(-1 + 10*#1 - 18*#1^2 + 10*#1^3 + 5*#1^4 - 3*#1^5 + #1^7) & ]/2","A",1
2486,1,212,204,0.3509377,"\int \frac{\sqrt{x+\sqrt{1+x}}}{x^2-\sqrt{1+x}} \, dx","IntegrateAlgebraic[Sqrt[x + Sqrt[1 + x]]/(x^2 - Sqrt[1 + x]),x]","2 \text{RootSum}\left[-1+6 \text{$\#$1}+3 \text{$\#$1}^2-2 \text{$\#$1}^3-8 \text{$\#$1}^4+16 \text{$\#$1}^5-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right)+2 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}-2 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^5+\log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^6}{3+3 \text{$\#$1}-3 \text{$\#$1}^2-16 \text{$\#$1}^3+40 \text{$\#$1}^4-12 \text{$\#$1}^5+4 \text{$\#$1}^7}\&\right]","2 \text{RootSum}\left[-1+6 \text{$\#$1}+3 \text{$\#$1}^2-2 \text{$\#$1}^3-8 \text{$\#$1}^4+16 \text{$\#$1}^5-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right)+2 \log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}-2 \log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^5+\log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^6}{3+3 \text{$\#$1}-3 \text{$\#$1}^2-16 \text{$\#$1}^3+40 \text{$\#$1}^4-12 \text{$\#$1}^5+4 \text{$\#$1}^7}\&\right]",1,"2*RootSum[-1 + 6*#1 + 3*#1^2 - 2*#1^3 - 8*#1^4 + 16*#1^5 - 4*#1^6 + #1^8 & , (Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1] + 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1 - 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^5 + Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^6)/(3 + 3*#1 - 3*#1^2 - 16*#1^3 + 40*#1^4 - 12*#1^5 + 4*#1^7) & ]","A",1
2487,1,212,204,0.0010717,"\int \frac{\sqrt{x+\sqrt{1+x}}}{x^2-\sqrt{1+x}} \, dx","IntegrateAlgebraic[Sqrt[x + Sqrt[1 + x]]/(x^2 - Sqrt[1 + x]),x]","2 \text{RootSum}\left[-1+6 \text{$\#$1}+3 \text{$\#$1}^2-2 \text{$\#$1}^3-8 \text{$\#$1}^4+16 \text{$\#$1}^5-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right)+2 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}-2 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^5+\log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^6}{3+3 \text{$\#$1}-3 \text{$\#$1}^2-16 \text{$\#$1}^3+40 \text{$\#$1}^4-12 \text{$\#$1}^5+4 \text{$\#$1}^7}\&\right]","2 \text{RootSum}\left[-1+6 \text{$\#$1}+3 \text{$\#$1}^2-2 \text{$\#$1}^3-8 \text{$\#$1}^4+16 \text{$\#$1}^5-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right)+2 \log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}-2 \log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^5+\log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^6}{3+3 \text{$\#$1}-3 \text{$\#$1}^2-16 \text{$\#$1}^3+40 \text{$\#$1}^4-12 \text{$\#$1}^5+4 \text{$\#$1}^7}\&\right]",1,"2*RootSum[-1 + 6*#1 + 3*#1^2 - 2*#1^3 - 8*#1^4 + 16*#1^5 - 4*#1^6 + #1^8 & , (Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1] + 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1 - 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^5 + Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^6)/(3 + 3*#1 - 3*#1^2 - 16*#1^3 + 40*#1^4 - 12*#1^5 + 4*#1^7) & ]","A",1
2488,1,205,205,0.6604287,"\int \frac{x^2}{\left(x^2 (-a+x)\right)^{2/3} \left(-a^2+2 a x+(-1+d) x^2\right)} \, dx","IntegrateAlgebraic[x^2/((x^2*(-a + x))^(2/3)*(-a^2 + 2*a*x + (-1 + d)*x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} x}{\sqrt[6]{d} x-2 \sqrt[3]{-a x^2+x^3}}\right)}{2 a d^{5/6}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} x}{\sqrt[6]{d} x+2 \sqrt[3]{-a x^2+x^3}}\right)}{2 a d^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{d} x}{\sqrt[3]{-a x^2+x^3}}\right)}{a d^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{d} x^2+\frac{\left(-a x^2+x^3\right)^{2/3}}{\sqrt[6]{d}}}{x \sqrt[3]{-a x^2+x^3}}\right)}{2 a d^{5/6}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} x}{\sqrt[6]{d} x-2 \sqrt[3]{-a x^2+x^3}}\right)}{2 a d^{5/6}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} x}{\sqrt[6]{d} x+2 \sqrt[3]{-a x^2+x^3}}\right)}{2 a d^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{d} x}{\sqrt[3]{-a x^2+x^3}}\right)}{a d^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{d} x^2+\frac{\left(-a x^2+x^3\right)^{2/3}}{\sqrt[6]{d}}}{x \sqrt[3]{-a x^2+x^3}}\right)}{2 a d^{5/6}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/6)*x)/(d^(1/6)*x - 2*(-(a*x^2) + x^3)^(1/3))])/(2*a*d^(5/6)) - (Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/6)*x)/(d^(1/6)*x + 2*(-(a*x^2) + x^3)^(1/3))])/(2*a*d^(5/6)) + ArcTanh[(d^(1/6)*x)/(-(a*x^2) + x^3)^(1/3)]/(a*d^(5/6)) + ArcTanh[(d^(1/6)*x^2 + (-(a*x^2) + x^3)^(2/3)/d^(1/6))/(x*(-(a*x^2) + x^3)^(1/3))]/(2*a*d^(5/6))","A",1
2489,1,205,205,0.4174036,"\int \frac{(-2+(1+k) x) \left(1-(1+k) x+(a+k) x^2\right)}{x ((1-x) x (1-k x))^{2/3} \left(1-(1+k) x+(-b+k) x^2\right)} \, dx","IntegrateAlgebraic[((-2 + (1 + k)*x)*(1 - (1 + k)*x + (a + k)*x^2))/(x*((1 - x)*x*(1 - k*x))^(2/3)*(1 - (1 + k)*x + (-b + k)*x^2)),x]","\frac{3 \sqrt[3]{x+(-1-k) x^2+k x^3}}{x}+\frac{\left(\sqrt{3} a+\sqrt{3} b\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} x}{\sqrt[3]{b} x+2 \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{b^{2/3}}+\frac{(a+b) \log \left(-\sqrt[3]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{b^{2/3}}+\frac{(-a-b) \log \left(b^{2/3} x^2+\sqrt[3]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 b^{2/3}}","\frac{3 \sqrt[3]{x+(-1-k) x^2+k x^3}}{x}+\frac{\left(\sqrt{3} a+\sqrt{3} b\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} x}{\sqrt[3]{b} x+2 \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{b^{2/3}}+\frac{(a+b) \log \left(-\sqrt[3]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{b^{2/3}}+\frac{(-a-b) \log \left(b^{2/3} x^2+\sqrt[3]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 b^{2/3}}",1,"(3*(x + (-1 - k)*x^2 + k*x^3)^(1/3))/x + ((Sqrt[3]*a + Sqrt[3]*b)*ArcTan[(Sqrt[3]*b^(1/3)*x)/(b^(1/3)*x + 2*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/b^(2/3) + ((a + b)*Log[-(b^(1/3)*x) + (x + (-1 - k)*x^2 + k*x^3)^(1/3)])/b^(2/3) + ((-a - b)*Log[b^(2/3)*x^2 + b^(1/3)*x*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + (x + (-1 - k)*x^2 + k*x^3)^(2/3)])/(2*b^(2/3))","A",1
2490,1,206,206,4.0927678,"\int \frac{-2 a^2 b x+a (3 a+2 b) x^2-4 a x^3+x^4}{\left(x^2 (-a+x) (-b+x)\right)^{2/3} \left(-a^2+2 a x-(1+b d) x^2+d x^3\right)} \, dx","IntegrateAlgebraic[(-2*a^2*b*x + a*(3*a + 2*b)*x^2 - 4*a*x^3 + x^4)/((x^2*(-a + x)*(-b + x))^(2/3)*(-a^2 + 2*a*x - (1 + b*d)*x^2 + d*x^3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a-\sqrt{3} x}{a-x-2 \sqrt[3]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{\sqrt[3]{d}}+\frac{\log \left(a-x+\sqrt[3]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}\right)}{\sqrt[3]{d}}-\frac{\log \left(a^2-2 a x+x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{a b x^2+(-a-b) x^3+x^4}+d^{2/3} \left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a-\sqrt{3} x}{a-x-2 \sqrt[3]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{\sqrt[3]{d}}+\frac{\log \left(a-x+\sqrt[3]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}\right)}{\sqrt[3]{d}}-\frac{\log \left(a^2-2 a x+x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{a b x^2+(-a-b) x^3+x^4}+d^{2/3} \left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*a - Sqrt[3]*x)/(a - x - 2*d^(1/3)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))])/d^(1/3) + Log[a - x + d^(1/3)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3)]/d^(1/3) - Log[a^2 - 2*a*x + x^2 + (-(a*d^(1/3)) + d^(1/3)*x)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3) + d^(2/3)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(2/3)]/(2*d^(1/3))","A",1
2491,1,206,206,0.2677722,"\int \frac{1}{\sqrt[3]{-8+12 x+54 x^2-135 x^3+81 x^4}} \, dx","IntegrateAlgebraic[(-8 + 12*x + 54*x^2 - 135*x^3 + 81*x^4)^(-1/3),x]","-\frac{\tan ^{-1}\left(\frac{-2 3^{5/6}+3\ 3^{5/6} x}{-2 \sqrt[3]{3}+3 \sqrt[3]{3} x+2 \sqrt[3]{-8+12 x+54 x^2-135 x^3+81 x^4}}\right)}{3^{5/6}}+\frac{\log \left(6-9 x+3^{2/3} \sqrt[3]{-8+12 x+54 x^2-135 x^3+81 x^4}\right)}{3 \sqrt[3]{3}}-\frac{\log \left(12-36 x+27 x^2+\left(-2 3^{2/3}+3\ 3^{2/3} x\right) \sqrt[3]{-8+12 x+54 x^2-135 x^3+81 x^4}+\sqrt[3]{3} \left(-8+12 x+54 x^2-135 x^3+81 x^4\right)^{2/3}\right)}{6 \sqrt[3]{3}}","-\frac{\tan ^{-1}\left(\frac{-2 3^{5/6}+3\ 3^{5/6} x}{-2 \sqrt[3]{3}+3 \sqrt[3]{3} x+2 \sqrt[3]{-8+12 x+54 x^2-135 x^3+81 x^4}}\right)}{3^{5/6}}+\frac{\log \left(6-9 x+3^{2/3} \sqrt[3]{-8+12 x+54 x^2-135 x^3+81 x^4}\right)}{3 \sqrt[3]{3}}-\frac{\log \left(12-36 x+27 x^2+\left(-2 3^{2/3}+3\ 3^{2/3} x\right) \sqrt[3]{-8+12 x+54 x^2-135 x^3+81 x^4}+\sqrt[3]{3} \left(-8+12 x+54 x^2-135 x^3+81 x^4\right)^{2/3}\right)}{6 \sqrt[3]{3}}",1,"-(ArcTan[(-2*3^(5/6) + 3*3^(5/6)*x)/(-2*3^(1/3) + 3*3^(1/3)*x + 2*(-8 + 12*x + 54*x^2 - 135*x^3 + 81*x^4)^(1/3))]/3^(5/6)) + Log[6 - 9*x + 3^(2/3)*(-8 + 12*x + 54*x^2 - 135*x^3 + 81*x^4)^(1/3)]/(3*3^(1/3)) - Log[12 - 36*x + 27*x^2 + (-2*3^(2/3) + 3*3^(2/3)*x)*(-8 + 12*x + 54*x^2 - 135*x^3 + 81*x^4)^(1/3) + 3^(1/3)*(-8 + 12*x + 54*x^2 - 135*x^3 + 81*x^4)^(2/3)]/(6*3^(1/3))","A",1
2492,1,206,206,6.4603091,"\int \frac{1+x^2}{\left(-1-x+x^2\right) \sqrt[3]{-1+x^6}} \, dx","IntegrateAlgebraic[(1 + x^2)/((-1 - x + x^2)*(-1 + x^6)^(1/3)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-1+x^6}}{-2^{2/3}+2^{2/3} x+2^{2/3} x^2+\sqrt[3]{-1+x^6}}\right)}{2^{2/3} \sqrt{3}}+\frac{\log \left(-2^{2/3}+2^{2/3} x+2^{2/3} x^2-2 \sqrt[3]{-1+x^6}\right)}{3\ 2^{2/3}}-\frac{\log \left(\sqrt[3]{2}-2 \sqrt[3]{2} x-\sqrt[3]{2} x^2+2 \sqrt[3]{2} x^3+\sqrt[3]{2} x^4+\left(-2^{2/3}+2^{2/3} x+2^{2/3} x^2\right) \sqrt[3]{-1+x^6}+2 \left(-1+x^6\right)^{2/3}\right)}{6\ 2^{2/3}}","\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-1+x^6}}{-2^{2/3}+2^{2/3} x+2^{2/3} x^2+\sqrt[3]{-1+x^6}}\right)}{2^{2/3} \sqrt{3}}+\frac{\log \left(-2^{2/3}+2^{2/3} x+2^{2/3} x^2-2 \sqrt[3]{-1+x^6}\right)}{3\ 2^{2/3}}-\frac{\log \left(\sqrt[3]{2}-2 \sqrt[3]{2} x-\sqrt[3]{2} x^2+2 \sqrt[3]{2} x^3+\sqrt[3]{2} x^4+\left(-2^{2/3}+2^{2/3} x+2^{2/3} x^2\right) \sqrt[3]{-1+x^6}+2 \left(-1+x^6\right)^{2/3}\right)}{6\ 2^{2/3}}",1,"ArcTan[(Sqrt[3]*(-1 + x^6)^(1/3))/(-2^(2/3) + 2^(2/3)*x + 2^(2/3)*x^2 + (-1 + x^6)^(1/3))]/(2^(2/3)*Sqrt[3]) + Log[-2^(2/3) + 2^(2/3)*x + 2^(2/3)*x^2 - 2*(-1 + x^6)^(1/3)]/(3*2^(2/3)) - Log[2^(1/3) - 2*2^(1/3)*x - 2^(1/3)*x^2 + 2*2^(1/3)*x^3 + 2^(1/3)*x^4 + (-2^(2/3) + 2^(2/3)*x + 2^(2/3)*x^2)*(-1 + x^6)^(1/3) + 2*(-1 + x^6)^(2/3)]/(6*2^(2/3))","A",1
2493,1,206,206,0.0010783,"\int \frac{1+x^2}{\left(-1-x+x^2\right) \sqrt[3]{-1+x^6}} \, dx","IntegrateAlgebraic[(1 + x^2)/((-1 - x + x^2)*(-1 + x^6)^(1/3)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-1+x^6}}{-2^{2/3}+2^{2/3} x+2^{2/3} x^2+\sqrt[3]{-1+x^6}}\right)}{2^{2/3} \sqrt{3}}+\frac{\log \left(-2^{2/3}+2^{2/3} x+2^{2/3} x^2-2 \sqrt[3]{-1+x^6}\right)}{3\ 2^{2/3}}-\frac{\log \left(\sqrt[3]{2}-2 \sqrt[3]{2} x-\sqrt[3]{2} x^2+2 \sqrt[3]{2} x^3+\sqrt[3]{2} x^4+\left(-2^{2/3}+2^{2/3} x+2^{2/3} x^2\right) \sqrt[3]{-1+x^6}+2 \left(-1+x^6\right)^{2/3}\right)}{6\ 2^{2/3}}","\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-1+x^6}}{-2^{2/3}+2^{2/3} x+2^{2/3} x^2+\sqrt[3]{-1+x^6}}\right)}{2^{2/3} \sqrt{3}}+\frac{\log \left(-2^{2/3}+2^{2/3} x+2^{2/3} x^2-2 \sqrt[3]{-1+x^6}\right)}{3\ 2^{2/3}}-\frac{\log \left(\sqrt[3]{2}-2 \sqrt[3]{2} x-\sqrt[3]{2} x^2+2 \sqrt[3]{2} x^3+\sqrt[3]{2} x^4+\left(-2^{2/3}+2^{2/3} x+2^{2/3} x^2\right) \sqrt[3]{-1+x^6}+2 \left(-1+x^6\right)^{2/3}\right)}{6\ 2^{2/3}}",1,"ArcTan[(Sqrt[3]*(-1 + x^6)^(1/3))/(-2^(2/3) + 2^(2/3)*x + 2^(2/3)*x^2 + (-1 + x^6)^(1/3))]/(2^(2/3)*Sqrt[3]) + Log[-2^(2/3) + 2^(2/3)*x + 2^(2/3)*x^2 - 2*(-1 + x^6)^(1/3)]/(3*2^(2/3)) - Log[2^(1/3) - 2*2^(1/3)*x - 2^(1/3)*x^2 + 2*2^(1/3)*x^3 + 2^(1/3)*x^4 + (-2^(2/3) + 2^(2/3)*x + 2^(2/3)*x^2)*(-1 + x^6)^(1/3) + 2*(-1 + x^6)^(2/3)]/(6*2^(2/3))","A",1
2494,1,206,206,5.4551801,"\int \frac{\left(2-2 x^4+3 x^5+4 x^6\right) \sqrt[3]{-x+2 x^3-x^5+x^6+x^7}}{\left(-1+x^2-x^4+x^5+x^6\right)^2} \, dx","IntegrateAlgebraic[((2 - 2*x^4 + 3*x^5 + 4*x^6)*(-x + 2*x^3 - x^5 + x^6 + x^7)^(1/3))/(-1 + x^2 - x^4 + x^5 + x^6)^2,x]","-\frac{x \sqrt[3]{-x+2 x^3-x^5+x^6+x^7}}{-1+x^2-x^4+x^5+x^6}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-x+2 x^3-x^5+x^6+x^7}}{2 x+\sqrt[3]{-x+2 x^3-x^5+x^6+x^7}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(-x+\sqrt[3]{-x+2 x^3-x^5+x^6+x^7}\right)-\frac{1}{6} \log \left(x^2+x \sqrt[3]{-x+2 x^3-x^5+x^6+x^7}+\left(-x+2 x^3-x^5+x^6+x^7\right)^{2/3}\right)","-\frac{x \sqrt[3]{-x+2 x^3-x^5+x^6+x^7}}{-1+x^2-x^4+x^5+x^6}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-x+2 x^3-x^5+x^6+x^7}}{2 x+\sqrt[3]{-x+2 x^3-x^5+x^6+x^7}}\right)}{\sqrt{3}}+\frac{1}{3} \log \left(-x+\sqrt[3]{-x+2 x^3-x^5+x^6+x^7}\right)-\frac{1}{6} \log \left(x^2+x \sqrt[3]{-x+2 x^3-x^5+x^6+x^7}+\left(-x+2 x^3-x^5+x^6+x^7\right)^{2/3}\right)",1,"-((x*(-x + 2*x^3 - x^5 + x^6 + x^7)^(1/3))/(-1 + x^2 - x^4 + x^5 + x^6)) - ArcTan[(Sqrt[3]*(-x + 2*x^3 - x^5 + x^6 + x^7)^(1/3))/(2*x + (-x + 2*x^3 - x^5 + x^6 + x^7)^(1/3))]/Sqrt[3] + Log[-x + (-x + 2*x^3 - x^5 + x^6 + x^7)^(1/3)]/3 - Log[x^2 + x*(-x + 2*x^3 - x^5 + x^6 + x^7)^(1/3) + (-x + 2*x^3 - x^5 + x^6 + x^7)^(2/3)]/6","A",1
2495,1,207,207,0.6570946,"\int \frac{-1+(-1+2 k) x}{\sqrt[3]{(1-x) x (1-k x)} \left(b-(1+2 b) x+(b+k) x^2\right)} \, dx","IntegrateAlgebraic[(-1 + (-1 + 2*k)*x)/(((1 - x)*x*(1 - k*x))^(1/3)*(b - (1 + 2*b)*x + (b + k)*x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{x+(-1-k) x^2+k x^3}}{2 \sqrt[3]{b}-2 \sqrt[3]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{\sqrt[3]{b}}+\frac{\log \left(-\sqrt[3]{b}+\sqrt[3]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{\sqrt[3]{b}}-\frac{\log \left(b^{2/3}-2 b^{2/3} x+b^{2/3} x^2+\left(\sqrt[3]{b}-\sqrt[3]{b} x\right) \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 \sqrt[3]{b}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{x+(-1-k) x^2+k x^3}}{2 \sqrt[3]{b}-2 \sqrt[3]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{\sqrt[3]{b}}+\frac{\log \left(-\sqrt[3]{b}+\sqrt[3]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{\sqrt[3]{b}}-\frac{\log \left(b^{2/3}-2 b^{2/3} x+b^{2/3} x^2+\left(\sqrt[3]{b}-\sqrt[3]{b} x\right) \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 \sqrt[3]{b}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(x + (-1 - k)*x^2 + k*x^3)^(1/3))/(2*b^(1/3) - 2*b^(1/3)*x + (x + (-1 - k)*x^2 + k*x^3)^(1/3))])/b^(1/3) + Log[-b^(1/3) + b^(1/3)*x + (x + (-1 - k)*x^2 + k*x^3)^(1/3)]/b^(1/3) - Log[b^(2/3) - 2*b^(2/3)*x + b^(2/3)*x^2 + (b^(1/3) - b^(1/3)*x)*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + (x + (-1 - k)*x^2 + k*x^3)^(2/3)]/(2*b^(1/3))","A",1
2496,1,207,207,0.6342985,"\int \frac{\sqrt{b^2+a^2 x^3} \left(2 b^2+c x^3+a^2 x^6\right)}{x^7 \left(-b^2+a^2 x^6\right)} \, dx","IntegrateAlgebraic[(Sqrt[b^2 + a^2*x^3]*(2*b^2 + c*x^3 + a^2*x^6))/(x^7*(-b^2 + a^2*x^6)),x]","\frac{\sqrt{b^2+a^2 x^3} \left(2 b^2+a^2 x^3+2 c x^3\right)}{6 b^2 x^6}-\frac{\sqrt{a-b} \left(3 a^2 b-a c\right) \tan ^{-1}\left(\frac{\sqrt{b^2+a^2 x^3}}{\sqrt{a-b} \sqrt{b}}\right)}{3 b^{5/2}}+\frac{\left(-a^4+12 a^2 b^2+2 a^2 c\right) \tanh ^{-1}\left(\frac{\sqrt{b^2+a^2 x^3}}{b}\right)}{6 b^3}-\frac{\sqrt{a+b} \left(3 a^2 b+a c\right) \tanh ^{-1}\left(\frac{\sqrt{b^2+a^2 x^3}}{\sqrt{b} \sqrt{a+b}}\right)}{3 b^{5/2}}","\frac{\sqrt{b^2+a^2 x^3} \left(2 b^2+a^2 x^3+2 c x^3\right)}{6 b^2 x^6}+\frac{\left(-a^4+12 a^2 b^2+2 a^2 c\right) \tanh ^{-1}\left(\frac{\sqrt{b^2+a^2 x^3}}{b}\right)}{6 b^3}-\frac{\sqrt{-a+b} \left(3 a^2 b-a c\right) \tanh ^{-1}\left(\frac{\sqrt{b^2+a^2 x^3}}{\sqrt{b} \sqrt{-a+b}}\right)}{3 b^{5/2}}-\frac{\sqrt{a+b} \left(3 a^2 b+a c\right) \tanh ^{-1}\left(\frac{\sqrt{b^2+a^2 x^3}}{\sqrt{b} \sqrt{a+b}}\right)}{3 b^{5/2}}",1,"(Sqrt[b^2 + a^2*x^3]*(2*b^2 + a^2*x^3 + 2*c*x^3))/(6*b^2*x^6) - (Sqrt[a - b]*(3*a^2*b - a*c)*ArcTan[Sqrt[b^2 + a^2*x^3]/(Sqrt[a - b]*Sqrt[b])])/(3*b^(5/2)) + ((-a^4 + 12*a^2*b^2 + 2*a^2*c)*ArcTanh[Sqrt[b^2 + a^2*x^3]/b])/(6*b^3) - (Sqrt[a + b]*(3*a^2*b + a*c)*ArcTanh[Sqrt[b^2 + a^2*x^3]/(Sqrt[b]*Sqrt[a + b])])/(3*b^(5/2))","A",1
2497,1,207,207,0.8373759,"\int \frac{x^7 (-4 a+3 x)}{\left(x^2 (-a+x)\right)^{2/3} \left(-a^2+2 a x-x^2+d x^8\right)} \, dx","IntegrateAlgebraic[(x^7*(-4*a + 3*x))/((x^2*(-a + x))^(2/3)*(-a^2 + 2*a*x - x^2 + d*x^8)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} x^2}{\sqrt[6]{d} x^2-2 \sqrt[3]{-a x^2+x^3}}\right)}{2 d^{5/6}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} x^2}{\sqrt[6]{d} x^2+2 \sqrt[3]{-a x^2+x^3}}\right)}{2 d^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{d} \left(-a x^2+x^3\right)^{2/3}}{a-x}\right)}{d^{5/6}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{d} x^4+\frac{\left(-a x^2+x^3\right)^{2/3}}{\sqrt[6]{d}}}{x^2 \sqrt[3]{-a x^2+x^3}}\right)}{2 d^{5/6}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} x^2}{\sqrt[6]{d} x^2-2 \sqrt[3]{-a x^2+x^3}}\right)}{2 d^{5/6}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} x^2}{\sqrt[6]{d} x^2+2 \sqrt[3]{-a x^2+x^3}}\right)}{2 d^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{d} \left(-a x^2+x^3\right)^{2/3}}{a-x}\right)}{d^{5/6}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{d} x^4+\frac{\left(-a x^2+x^3\right)^{2/3}}{\sqrt[6]{d}}}{x^2 \sqrt[3]{-a x^2+x^3}}\right)}{2 d^{5/6}}",1,"-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/6)*x^2)/(d^(1/6)*x^2 - 2*(-(a*x^2) + x^3)^(1/3))])/d^(5/6) + (Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/6)*x^2)/(d^(1/6)*x^2 + 2*(-(a*x^2) + x^3)^(1/3))])/(2*d^(5/6)) + ArcTanh[(d^(1/6)*(-(a*x^2) + x^3)^(2/3))/(a - x)]/d^(5/6) - ArcTanh[(d^(1/6)*x^4 + (-(a*x^2) + x^3)^(2/3)/d^(1/6))/(x^2*(-(a*x^2) + x^3)^(1/3))]/(2*d^(5/6))","A",1
2498,1,208,208,0.9359228,"\int \frac{-1+2 x+\left(-2 k+k^2\right) x^2}{((1-x) x (1-k x))^{2/3} \left(b-(1+2 b k) x+\left(1+b k^2\right) x^2\right)} \, dx","IntegrateAlgebraic[(-1 + 2*x + (-2*k + k^2)*x^2)/(((1 - x)*x*(1 - k*x))^(2/3)*(b - (1 + 2*b*k)*x + (1 + b*k^2)*x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}}{2 x-2 x^2+\sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}}\right)}{b^{2/3}}+\frac{\log \left(-x+x^2+\sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{b^{2/3}}-\frac{\log \left(x^2-2 x^3+x^4+\left(\sqrt[3]{b} x-\sqrt[3]{b} x^2\right) \left(x+(-1-k) x^2+k x^3\right)^{2/3}+b^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{4/3}\right)}{2 b^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}}{2 x-2 x^2+\sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}}\right)}{b^{2/3}}+\frac{\log \left(-x+x^2+\sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{b^{2/3}}-\frac{\log \left(x^2-2 x^3+x^4+\left(\sqrt[3]{b} x-\sqrt[3]{b} x^2\right) \left(x+(-1-k) x^2+k x^3\right)^{2/3}+b^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{4/3}\right)}{2 b^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3))/(2*x - 2*x^2 + b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3))])/b^(2/3) + Log[-x + x^2 + b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3)]/b^(2/3) - Log[x^2 - 2*x^3 + x^4 + (b^(1/3)*x - b^(1/3)*x^2)*(x + (-1 - k)*x^2 + k*x^3)^(2/3) + b^(2/3)*(x + (-1 - k)*x^2 + k*x^3)^(4/3)]/(2*b^(2/3))","A",1
2499,1,208,208,0.4953016,"\int \frac{1}{\sqrt[3]{-x^2+x^3} \left(-1+x^4\right)} \, dx","IntegrateAlgebraic[1/((-x^2 + x^3)^(1/3)*(-1 + x^4)),x]","-\frac{3 \left(-x^2+x^3\right)^{2/3}}{4 (-1+x) x}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-x^2+x^3}}\right)}{4 \sqrt[3]{2}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{-x^2+x^3}\right)}{4 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-x^2+x^3}+\sqrt[3]{2} \left(-x^2+x^3\right)^{2/3}\right)}{8 \sqrt[3]{2}}+\frac{1}{4} \text{RootSum}\left[2-2 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","-\frac{3 \left(-x^2+x^3\right)^{2/3}}{4 (-1+x) x}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-x^2+x^3}}\right)}{4 \sqrt[3]{2}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{-x^2+x^3}\right)}{4 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-x^2+x^3}+\sqrt[3]{2} \left(-x^2+x^3\right)^{2/3}\right)}{8 \sqrt[3]{2}}+\frac{1}{4} \text{RootSum}\left[2-2 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"(-3*(-x^2 + x^3)^(2/3))/(4*(-1 + x)*x) - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-x^2 + x^3)^(1/3))])/(4*2^(1/3)) + Log[-2*x + 2^(2/3)*(-x^2 + x^3)^(1/3)]/(4*2^(1/3)) - Log[2*x^2 + 2^(2/3)*x*(-x^2 + x^3)^(1/3) + 2^(1/3)*(-x^2 + x^3)^(2/3)]/(8*2^(1/3)) + RootSum[2 - 2*#1^3 + #1^6 & , (-Log[x] + Log[(-x^2 + x^3)^(1/3) - x*#1])/#1 & ]/4","A",1
2500,1,208,208,0.0009433,"\int \frac{1}{\sqrt[3]{-x^2+x^3} \left(-1+x^4\right)} \, dx","IntegrateAlgebraic[1/((-x^2 + x^3)^(1/3)*(-1 + x^4)),x]","-\frac{3 \left(-x^2+x^3\right)^{2/3}}{4 (-1+x) x}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-x^2+x^3}}\right)}{4 \sqrt[3]{2}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{-x^2+x^3}\right)}{4 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-x^2+x^3}+\sqrt[3]{2} \left(-x^2+x^3\right)^{2/3}\right)}{8 \sqrt[3]{2}}+\frac{1}{4} \text{RootSum}\left[2-2 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","-\frac{3 \left(-x^2+x^3\right)^{2/3}}{4 (-1+x) x}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-x^2+x^3}}\right)}{4 \sqrt[3]{2}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{-x^2+x^3}\right)}{4 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-x^2+x^3}+\sqrt[3]{2} \left(-x^2+x^3\right)^{2/3}\right)}{8 \sqrt[3]{2}}+\frac{1}{4} \text{RootSum}\left[2-2 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"(-3*(-x^2 + x^3)^(2/3))/(4*(-1 + x)*x) - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-x^2 + x^3)^(1/3))])/(4*2^(1/3)) + Log[-2*x + 2^(2/3)*(-x^2 + x^3)^(1/3)]/(4*2^(1/3)) - Log[2*x^2 + 2^(2/3)*x*(-x^2 + x^3)^(1/3) + 2^(1/3)*(-x^2 + x^3)^(2/3)]/(8*2^(1/3)) + RootSum[2 - 2*#1^3 + #1^6 & , (-Log[x] + Log[(-x^2 + x^3)^(1/3) - x*#1])/#1 & ]/4","A",1
2501,1,208,208,0.388992,"\int \frac{\sqrt[3]{-x^2+x^4}}{x \left(1+x^2\right)} \, dx","IntegrateAlgebraic[(-x^2 + x^4)^(1/3)/(x*(1 + x^2)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x^2}{-x^2+\sqrt[3]{2} \left(-x^2+x^4\right)^{2/3}}\right)}{2\ 2^{2/3}}+\frac{\log \left(-2 x^2+2^{2/3} \sqrt{3} x \sqrt[3]{-x^2+x^4}-\sqrt[3]{2} \left(-x^2+x^4\right)^{2/3}\right)}{4\ 2^{2/3}}-\frac{\log \left(2 x^2+\sqrt[3]{2} \left(-x^2+x^4\right)^{2/3}\right)}{2\ 2^{2/3}}+\frac{\log \left(2 x^2+2^{2/3} \sqrt{3} x \sqrt[3]{-x^2+x^4}+\sqrt[3]{2} \left(-x^2+x^4\right)^{2/3}\right)}{4\ 2^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x^2}{-x^2+\sqrt[3]{2} \left(-x^2+x^4\right)^{2/3}}\right)}{2\ 2^{2/3}}+\frac{\log \left(-2 x^2+2^{2/3} \sqrt{3} x \sqrt[3]{-x^2+x^4}-\sqrt[3]{2} \left(-x^2+x^4\right)^{2/3}\right)}{4\ 2^{2/3}}-\frac{\log \left(2 x^2+\sqrt[3]{2} \left(-x^2+x^4\right)^{2/3}\right)}{2\ 2^{2/3}}+\frac{\log \left(2 x^2+2^{2/3} \sqrt{3} x \sqrt[3]{-x^2+x^4}+\sqrt[3]{2} \left(-x^2+x^4\right)^{2/3}\right)}{4\ 2^{2/3}}",1,"-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*x^2)/(-x^2 + 2^(1/3)*(-x^2 + x^4)^(2/3))])/2^(2/3) + Log[-2*x^2 + 2^(2/3)*Sqrt[3]*x*(-x^2 + x^4)^(1/3) - 2^(1/3)*(-x^2 + x^4)^(2/3)]/(4*2^(2/3)) - Log[2*x^2 + 2^(1/3)*(-x^2 + x^4)^(2/3)]/(2*2^(2/3)) + Log[2*x^2 + 2^(2/3)*Sqrt[3]*x*(-x^2 + x^4)^(1/3) + 2^(1/3)*(-x^2 + x^4)^(2/3)]/(4*2^(2/3))","A",1
2502,1,208,208,0.9836795,"\int \frac{b+a x^6}{x^3 \left(-b+a x^3\right) \sqrt[4]{b x+a x^4}} \, dx","IntegrateAlgebraic[(b + a*x^6)/(x^3*(-b + a*x^3)*(b*x + a*x^4)^(1/4)),x]","\frac{4 \left(b x+a x^4\right)^{3/4}}{9 b x^3}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 \sqrt[4]{a}}-\frac{2^{3/4} (a+b) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 \sqrt[4]{a} b}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 \sqrt[4]{a}}-\frac{2^{3/4} (a+b) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 \sqrt[4]{a} b}","\frac{4 \left(b x+a x^4\right)^{3/4}}{9 b x^3}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 \sqrt[4]{a}}-\frac{2^{3/4} (a+b) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 \sqrt[4]{a} b}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 \sqrt[4]{a}}-\frac{2^{3/4} (a+b) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} \left(b x+a x^4\right)^{3/4}}{b+a x^3}\right)}{3 \sqrt[4]{a} b}",1,"(4*(b*x + a*x^4)^(3/4))/(9*b*x^3) + (2*ArcTan[(a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/(3*a^(1/4)) - (2^(3/4)*(a + b)*ArcTan[(2^(1/4)*a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/(3*a^(1/4)*b) + (2*ArcTanh[(a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/(3*a^(1/4)) - (2^(3/4)*(a + b)*ArcTanh[(2^(1/4)*a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/(3*a^(1/4)*b)","A",1
2503,1,338,208,0.4476124,"\int \frac{\sqrt{a x+\sqrt{-b+a x}}}{x^2 \sqrt{-b+a x}} \, dx","IntegrateAlgebraic[Sqrt[a*x + Sqrt[-b + a*x]]/(x^2*Sqrt[-b + a*x]),x]","\frac{\sqrt{-b+a x} \sqrt{a x+\sqrt{-b+a x}}}{b x}-a \text{RootSum}\left[b+b^2-4 b \text{$\#$1}+2 b \text{$\#$1}^2+\text{$\#$1}^4\&,\frac{-5 \log \left(-\sqrt{-b+a x}+\sqrt{a x+\sqrt{-b+a x}}-\text{$\#$1}\right)+2 \log \left(-\sqrt{-b+a x}+\sqrt{a x+\sqrt{-b+a x}}-\text{$\#$1}\right) \text{$\#$1}}{-b+b \text{$\#$1}+\text{$\#$1}^3}\&\right]+\frac{a \text{RootSum}\left[b+b^2-4 b \text{$\#$1}+2 b \text{$\#$1}^2+\text{$\#$1}^4\&,\frac{-19 b \log \left(-\sqrt{-b+a x}+\sqrt{a x+\sqrt{-b+a x}}-\text{$\#$1}\right)+4 b \log \left(-\sqrt{-b+a x}+\sqrt{a x+\sqrt{-b+a x}}-\text{$\#$1}\right) \text{$\#$1}+\log \left(-\sqrt{-b+a x}+\sqrt{a x+\sqrt{-b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2}{-b+b \text{$\#$1}+\text{$\#$1}^3}\&\right]}{4 b}","\frac{\sqrt{-b+a x} \sqrt{a x+\sqrt{-b+a x}}}{b x}-\frac{a \text{RootSum}\left[b+b^2-4 b \text{$\#$1}+2 b \text{$\#$1}^2+\text{$\#$1}^4\&,\frac{-b \log \left(\sqrt{-b+a x}-\sqrt{a x+\sqrt{-b+a x}}+\text{$\#$1}\right)+4 b \log \left(\sqrt{-b+a x}-\sqrt{a x+\sqrt{-b+a x}}+\text{$\#$1}\right) \text{$\#$1}-\log \left(\sqrt{-b+a x}-\sqrt{a x+\sqrt{-b+a x}}+\text{$\#$1}\right) \text{$\#$1}^2}{-b+b \text{$\#$1}+\text{$\#$1}^3}\&\right]}{4 b}",1,"(Sqrt[-b + a*x]*Sqrt[a*x + Sqrt[-b + a*x]])/(b*x) - a*RootSum[b + b^2 - 4*b*#1 + 2*b*#1^2 + #1^4 & , (-5*Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1] + 2*Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1)/(-b + b*#1 + #1^3) & ] + (a*RootSum[b + b^2 - 4*b*#1 + 2*b*#1^2 + #1^4 & , (-19*b*Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1] + 4*b*Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1 + Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1^2)/(-b + b*#1 + #1^3) & ])/(4*b)","A",1
2504,1,209,209,0.2816594,"\int \frac{3+x}{\sqrt[3]{-1+x^2} \left(5-x+2 x^2\right)} \, dx","IntegrateAlgebraic[(3 + x)/((-1 + x^2)^(1/3)*(5 - x + 2*x^2)),x]","\frac{\tan ^{-1}\left(\frac{3^{5/6} \sqrt[3]{-1+x^2}}{2 \sqrt[3]{2}-2 \sqrt[3]{2} x+\sqrt[3]{3} \sqrt[3]{-1+x^2}}\right)}{\sqrt[3]{2} \sqrt[6]{3}}+\frac{\log \left(-\sqrt[3]{2} 3^{2/3}+\sqrt[3]{2} 3^{2/3} x+3 \sqrt[3]{-1+x^2}\right)}{\sqrt[3]{2} 3^{2/3}}-\frac{\log \left(2^{2/3} \sqrt[3]{3}-2\ 2^{2/3} \sqrt[3]{3} x+2^{2/3} \sqrt[3]{3} x^2+\left(\sqrt[3]{2} 3^{2/3}-\sqrt[3]{2} 3^{2/3} x\right) \sqrt[3]{-1+x^2}+3 \left(-1+x^2\right)^{2/3}\right)}{2 \sqrt[3]{2} 3^{2/3}}","\frac{\tan ^{-1}\left(\frac{3^{5/6} \sqrt[3]{-1+x^2}}{2 \sqrt[3]{2}-2 \sqrt[3]{2} x+\sqrt[3]{3} \sqrt[3]{-1+x^2}}\right)}{\sqrt[3]{2} \sqrt[6]{3}}+\frac{\log \left(-\sqrt[3]{2} 3^{2/3}+\sqrt[3]{2} 3^{2/3} x+3 \sqrt[3]{-1+x^2}\right)}{\sqrt[3]{2} 3^{2/3}}-\frac{\log \left(2^{2/3} \sqrt[3]{3}-2\ 2^{2/3} \sqrt[3]{3} x+2^{2/3} \sqrt[3]{3} x^2+\left(\sqrt[3]{2} 3^{2/3}-\sqrt[3]{2} 3^{2/3} x\right) \sqrt[3]{-1+x^2}+3 \left(-1+x^2\right)^{2/3}\right)}{2 \sqrt[3]{2} 3^{2/3}}",1,"ArcTan[(3^(5/6)*(-1 + x^2)^(1/3))/(2*2^(1/3) - 2*2^(1/3)*x + 3^(1/3)*(-1 + x^2)^(1/3))]/(2^(1/3)*3^(1/6)) + Log[-(2^(1/3)*3^(2/3)) + 2^(1/3)*3^(2/3)*x + 3*(-1 + x^2)^(1/3)]/(2^(1/3)*3^(2/3)) - Log[2^(2/3)*3^(1/3) - 2*2^(2/3)*3^(1/3)*x + 2^(2/3)*3^(1/3)*x^2 + (2^(1/3)*3^(2/3) - 2^(1/3)*3^(2/3)*x)*(-1 + x^2)^(1/3) + 3*(-1 + x^2)^(2/3)]/(2*2^(1/3)*3^(2/3))","A",1
2505,1,209,209,0.3907091,"\int \frac{(-2+(1+k) x) \left(1-(1+k) x+(a+k) x^2\right)}{x^2 \sqrt[3]{(1-x) x (1-k x)} \left(1-(1+k) x+(-b+k) x^2\right)} \, dx","IntegrateAlgebraic[((-2 + (1 + k)*x)*(1 - (1 + k)*x + (a + k)*x^2))/(x^2*((1 - x)*x*(1 - k*x))^(1/3)*(1 - (1 + k)*x + (-b + k)*x^2)),x]","\frac{3 \left(x+(-1-k) x^2+k x^3\right)^{2/3}}{2 x^2}+\frac{\left(-\sqrt{3} a-\sqrt{3} b\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} x}{\sqrt[3]{b} x+2 \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{\sqrt[3]{b}}+\frac{(a+b) \log \left(-\sqrt[3]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{\sqrt[3]{b}}+\frac{(-a-b) \log \left(b^{2/3} x^2+\sqrt[3]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 \sqrt[3]{b}}","\frac{3 \left(x+(-1-k) x^2+k x^3\right)^{2/3}}{2 x^2}+\frac{\left(-\sqrt{3} a-\sqrt{3} b\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} x}{\sqrt[3]{b} x+2 \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{\sqrt[3]{b}}+\frac{(a+b) \log \left(-\sqrt[3]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{\sqrt[3]{b}}+\frac{(-a-b) \log \left(b^{2/3} x^2+\sqrt[3]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 \sqrt[3]{b}}",1,"(3*(x + (-1 - k)*x^2 + k*x^3)^(2/3))/(2*x^2) + ((-(Sqrt[3]*a) - Sqrt[3]*b)*ArcTan[(Sqrt[3]*b^(1/3)*x)/(b^(1/3)*x + 2*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/b^(1/3) + ((a + b)*Log[-(b^(1/3)*x) + (x + (-1 - k)*x^2 + k*x^3)^(1/3)])/b^(1/3) + ((-a - b)*Log[b^(2/3)*x^2 + b^(1/3)*x*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + (x + (-1 - k)*x^2 + k*x^3)^(2/3)])/(2*b^(1/3))","A",1
2506,1,209,209,0.4639981,"\int \frac{\left(-2+x^3\right) \sqrt[3]{x+x^3+x^4}}{\left(1+x^3\right) \left(1-x^2+x^3\right)} \, dx","IntegrateAlgebraic[((-2 + x^3)*(x + x^3 + x^4)^(1/3))/((1 + x^3)*(1 - x^2 + x^3)),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3+x^4}}\right)+\sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x+x^3+x^4}}\right)-\log \left(-x+\sqrt[3]{x+x^3+x^4}\right)+\sqrt[3]{2} \log \left(-2 x+2^{2/3} \sqrt[3]{x+x^3+x^4}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{x+x^3+x^4}+\left(x+x^3+x^4\right)^{2/3}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x+x^3+x^4}+\sqrt[3]{2} \left(x+x^3+x^4\right)^{2/3}\right)}{2^{2/3}}","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3+x^4}}\right)+\sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x+x^3+x^4}}\right)-\log \left(-x+\sqrt[3]{x+x^3+x^4}\right)+\sqrt[3]{2} \log \left(-2 x+2^{2/3} \sqrt[3]{x+x^3+x^4}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{x+x^3+x^4}+\left(x+x^3+x^4\right)^{2/3}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x+x^3+x^4}+\sqrt[3]{2} \left(x+x^3+x^4\right)^{2/3}\right)}{2^{2/3}}",1,"-(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^3 + x^4)^(1/3))]) + 2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(x + x^3 + x^4)^(1/3))] - Log[-x + (x + x^3 + x^4)^(1/3)] + 2^(1/3)*Log[-2*x + 2^(2/3)*(x + x^3 + x^4)^(1/3)] + Log[x^2 + x*(x + x^3 + x^4)^(1/3) + (x + x^3 + x^4)^(2/3)]/2 - Log[2*x^2 + 2^(2/3)*x*(x + x^3 + x^4)^(1/3) + 2^(1/3)*(x + x^3 + x^4)^(2/3)]/2^(2/3)","A",1
2507,1,209,209,0.4939973,"\int \frac{\sqrt[4]{b x^3+a x^4}}{-2 b+a x+2 x^2} \, dx","IntegrateAlgebraic[(b*x^3 + a*x^4)^(1/4)/(-2*b + a*x + 2*x^2),x]","-\sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)+\sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)+\frac{1}{2} \text{RootSum}\left[3 a^2-2 b-5 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{3 a^2 \log (x)-2 b \log (x)-3 a^2 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)+2 b \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)-2 a \log (x) \text{$\#$1}^4+2 a \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-5 a \text{$\#$1}^3+4 \text{$\#$1}^7}\&\right]","-\sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)+\sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)+\frac{1}{2} \text{RootSum}\left[3 a^2-2 b-5 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-3 a^2 \log (x)+2 b \log (x)+3 a^2 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)-2 b \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)+2 a \log (x) \text{$\#$1}^4-2 a \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{5 a \text{$\#$1}^3-4 \text{$\#$1}^7}\&\right]",1,"-(a^(1/4)*ArcTan[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)]) + a^(1/4)*ArcTanh[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)] + RootSum[3*a^2 - 2*b - 5*a*#1^4 + 2*#1^8 & , (3*a^2*Log[x] - 2*b*Log[x] - 3*a^2*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] + 2*b*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] - 2*a*Log[x]*#1^4 + 2*a*Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^4)/(-5*a*#1^3 + 4*#1^7) & ]/2","A",1
2508,1,209,209,0.0014927,"\int \frac{\sqrt[4]{b x^3+a x^4}}{-2 b+a x+2 x^2} \, dx","IntegrateAlgebraic[(b*x^3 + a*x^4)^(1/4)/(-2*b + a*x + 2*x^2),x]","-\sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)+\sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)+\frac{1}{2} \text{RootSum}\left[3 a^2-2 b-5 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{3 a^2 \log (x)-2 b \log (x)-3 a^2 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)+2 b \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)-2 a \log (x) \text{$\#$1}^4+2 a \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-5 a \text{$\#$1}^3+4 \text{$\#$1}^7}\&\right]","-\sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)+\sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)+\frac{1}{2} \text{RootSum}\left[3 a^2-2 b-5 a \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-3 a^2 \log (x)+2 b \log (x)+3 a^2 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)-2 b \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)+2 a \log (x) \text{$\#$1}^4-2 a \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{5 a \text{$\#$1}^3-4 \text{$\#$1}^7}\&\right]",1,"-(a^(1/4)*ArcTan[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)]) + a^(1/4)*ArcTanh[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)] + RootSum[3*a^2 - 2*b - 5*a*#1^4 + 2*#1^8 & , (3*a^2*Log[x] - 2*b*Log[x] - 3*a^2*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] + 2*b*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] - 2*a*Log[x]*#1^4 + 2*a*Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^4)/(-5*a*#1^3 + 4*#1^7) & ]/2","A",1
2509,1,209,209,0.4056821,"\int \frac{\left(1+x^3\right)^{2/3} \left(-1+2 x^6\right)}{x^6 \left(-1+2 x^3\right)} \, dx","IntegrateAlgebraic[((1 + x^3)^(2/3)*(-1 + 2*x^6))/(x^6*(-1 + 2*x^3)),x]","\frac{\left(-1-6 x^3\right) \left(1+x^3\right)^{2/3}}{5 x^5}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3}}\right)}{\sqrt{3}}+\sqrt[6]{3} \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2 \sqrt[3]{1+x^3}}\right)-\frac{1}{3} \log \left(-x+\sqrt[3]{1+x^3}\right)-\frac{\log \left(-3 x+3^{2/3} \sqrt[3]{1+x^3}\right)}{\sqrt[3]{3}}+\frac{1}{6} \log \left(x^2+x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)+\frac{\log \left(3 x^2+3^{2/3} x \sqrt[3]{1+x^3}+\sqrt[3]{3} \left(1+x^3\right)^{2/3}\right)}{2 \sqrt[3]{3}}","\frac{\left(-1-6 x^3\right) \left(1+x^3\right)^{2/3}}{5 x^5}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3}}\right)}{\sqrt{3}}+\sqrt[6]{3} \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2 \sqrt[3]{1+x^3}}\right)-\frac{1}{3} \log \left(-x+\sqrt[3]{1+x^3}\right)-\frac{\log \left(-3 x+3^{2/3} \sqrt[3]{1+x^3}\right)}{\sqrt[3]{3}}+\frac{1}{6} \log \left(x^2+x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)+\frac{\log \left(3 x^2+3^{2/3} x \sqrt[3]{1+x^3}+\sqrt[3]{3} \left(1+x^3\right)^{2/3}\right)}{2 \sqrt[3]{3}}",1,"((-1 - 6*x^3)*(1 + x^3)^(2/3))/(5*x^5) + ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^3)^(1/3))]/Sqrt[3] + 3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*(1 + x^3)^(1/3))] - Log[-x + (1 + x^3)^(1/3)]/3 - Log[-3*x + 3^(2/3)*(1 + x^3)^(1/3)]/3^(1/3) + Log[x^2 + x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/6 + Log[3*x^2 + 3^(2/3)*x*(1 + x^3)^(1/3) + 3^(1/3)*(1 + x^3)^(2/3)]/(2*3^(1/3))","A",1
2510,1,209,209,3.4981335,"\int \frac{-b+a x^4}{\left(b+a x^4\right) \sqrt[4]{b^2+c x^4+a^2 x^8}} \, dx","IntegrateAlgebraic[(-b + a*x^4)/((b + a*x^4)*(b^2 + c*x^4 + a^2*x^8)^(1/4)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{2 a b-c} x \sqrt[4]{b^2+c x^4+a^2 x^8}}{\sqrt{2 a b-c} x^2-\sqrt{b^2+c x^4+a^2 x^8}}\right)}{2 \sqrt{2} \sqrt[4]{2 a b-c}}-\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[4]{2 a b-c} x^2}{\sqrt{2}}+\frac{\sqrt{b^2+c x^4+a^2 x^8}}{\sqrt{2} \sqrt[4]{2 a b-c}}}{x \sqrt[4]{b^2+c x^4+a^2 x^8}}\right)}{2 \sqrt{2} \sqrt[4]{2 a b-c}}","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{2 a b-c} x \sqrt[4]{b^2+c x^4+a^2 x^8}}{\sqrt{2 a b-c} x^2-\sqrt{b^2+c x^4+a^2 x^8}}\right)}{2 \sqrt{2} \sqrt[4]{2 a b-c}}-\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[4]{2 a b-c} x^2}{\sqrt{2}}+\frac{\sqrt{b^2+c x^4+a^2 x^8}}{\sqrt{2} \sqrt[4]{2 a b-c}}}{x \sqrt[4]{b^2+c x^4+a^2 x^8}}\right)}{2 \sqrt{2} \sqrt[4]{2 a b-c}}",1,"ArcTan[(Sqrt[2]*(2*a*b - c)^(1/4)*x*(b^2 + c*x^4 + a^2*x^8)^(1/4))/(Sqrt[2*a*b - c]*x^2 - Sqrt[b^2 + c*x^4 + a^2*x^8])]/(2*Sqrt[2]*(2*a*b - c)^(1/4)) - ArcTanh[(((2*a*b - c)^(1/4)*x^2)/Sqrt[2] + Sqrt[b^2 + c*x^4 + a^2*x^8]/(Sqrt[2]*(2*a*b - c)^(1/4)))/(x*(b^2 + c*x^4 + a^2*x^8)^(1/4))]/(2*Sqrt[2]*(2*a*b - c)^(1/4))","A",1
2511,1,217,209,0.2722478,"\int \frac{\sqrt{x+\sqrt{1+x}}}{\sqrt{1+x} \left(1+x^2\right)} \, dx","IntegrateAlgebraic[Sqrt[x + Sqrt[1 + x]]/(Sqrt[1 + x]*(1 + x^2)),x]","-\frac{1}{2} \text{RootSum}\left[1-8 \text{$\#$1}+40 \text{$\#$1}^2-48 \text{$\#$1}^3+20 \text{$\#$1}^4+8 \text{$\#$1}^5-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right)+5 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2-5 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^4+2 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^5}{-1+10 \text{$\#$1}-18 \text{$\#$1}^2+10 \text{$\#$1}^3+5 \text{$\#$1}^4-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","-\frac{1}{2} \text{RootSum}\left[1-8 \text{$\#$1}+40 \text{$\#$1}^2-48 \text{$\#$1}^3+20 \text{$\#$1}^4+8 \text{$\#$1}^5-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right)+5 \log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^2-5 \log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^4+2 \log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^5}{-1+10 \text{$\#$1}-18 \text{$\#$1}^2+10 \text{$\#$1}^3+5 \text{$\#$1}^4-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]",1,"-1/2*RootSum[1 - 8*#1 + 40*#1^2 - 48*#1^3 + 20*#1^4 + 8*#1^5 - 4*#1^6 + #1^8 & , (-Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1] + 5*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^2 - 5*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^4 + 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^5)/(-1 + 10*#1 - 18*#1^2 + 10*#1^3 + 5*#1^4 - 3*#1^5 + #1^7) & ]","A",1
2512,1,217,209,0.0011307,"\int \frac{\sqrt{x+\sqrt{1+x}}}{\sqrt{1+x} \left(1+x^2\right)} \, dx","IntegrateAlgebraic[Sqrt[x + Sqrt[1 + x]]/(Sqrt[1 + x]*(1 + x^2)),x]","-\frac{1}{2} \text{RootSum}\left[1-8 \text{$\#$1}+40 \text{$\#$1}^2-48 \text{$\#$1}^3+20 \text{$\#$1}^4+8 \text{$\#$1}^5-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right)+5 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2-5 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^4+2 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^5}{-1+10 \text{$\#$1}-18 \text{$\#$1}^2+10 \text{$\#$1}^3+5 \text{$\#$1}^4-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","-\frac{1}{2} \text{RootSum}\left[1-8 \text{$\#$1}+40 \text{$\#$1}^2-48 \text{$\#$1}^3+20 \text{$\#$1}^4+8 \text{$\#$1}^5-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right)+5 \log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^2-5 \log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^4+2 \log \left(\sqrt{1+x}-\sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^5}{-1+10 \text{$\#$1}-18 \text{$\#$1}^2+10 \text{$\#$1}^3+5 \text{$\#$1}^4-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]",1,"-1/2*RootSum[1 - 8*#1 + 40*#1^2 - 48*#1^3 + 20*#1^4 + 8*#1^5 - 4*#1^6 + #1^8 & , (-Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1] + 5*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^2 - 5*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^4 + 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^5)/(-1 + 10*#1 - 18*#1^2 + 10*#1^3 + 5*#1^4 - 3*#1^5 + #1^7) & ]","A",1
2513,1,210,210,1.0638264,"\int \frac{(2+x)^2 \sqrt[3]{-19+66 x-30 x^2+9 x^3}}{(-3+2 x)^2 \left(-5+6 x-6 x^2+x^3\right)} \, dx","IntegrateAlgebraic[((2 + x)^2*(-19 + 66*x - 30*x^2 + 9*x^3)^(1/3))/((-3 + 2*x)^2*(-5 + 6*x - 6*x^2 + x^3)),x]","\frac{\sqrt[3]{-19+66 x-30 x^2+9 x^3}}{-3+2 x}+\frac{\sqrt[3]{2} \tan ^{-1}\left(\frac{-3 \sqrt{3}+2 \sqrt{3} x}{-3+2 x+2^{2/3} \sqrt[3]{-19+66 x-30 x^2+9 x^3}}\right)}{\sqrt{3}}+\frac{1}{3} \sqrt[3]{2} \log \left(6-4 x+2^{2/3} \sqrt[3]{-19+66 x-30 x^2+9 x^3}\right)-\frac{\log \left(18-24 x+8 x^2+\left(-3 2^{2/3}+2\ 2^{2/3} x\right) \sqrt[3]{-19+66 x-30 x^2+9 x^3}+\sqrt[3]{2} \left(-19+66 x-30 x^2+9 x^3\right)^{2/3}\right)}{3\ 2^{2/3}}","\frac{\sqrt[3]{-19+66 x-30 x^2+9 x^3}}{-3+2 x}+\frac{\sqrt[3]{2} \tan ^{-1}\left(\frac{-3 \sqrt{3}+2 \sqrt{3} x}{-3+2 x+2^{2/3} \sqrt[3]{-19+66 x-30 x^2+9 x^3}}\right)}{\sqrt{3}}+\frac{1}{3} \sqrt[3]{2} \log \left(6-4 x+2^{2/3} \sqrt[3]{-19+66 x-30 x^2+9 x^3}\right)-\frac{\log \left(18-24 x+8 x^2+\left(-3 2^{2/3}+2\ 2^{2/3} x\right) \sqrt[3]{-19+66 x-30 x^2+9 x^3}+\sqrt[3]{2} \left(-19+66 x-30 x^2+9 x^3\right)^{2/3}\right)}{3\ 2^{2/3}}",1,"(-19 + 66*x - 30*x^2 + 9*x^3)^(1/3)/(-3 + 2*x) + (2^(1/3)*ArcTan[(-3*Sqrt[3] + 2*Sqrt[3]*x)/(-3 + 2*x + 2^(2/3)*(-19 + 66*x - 30*x^2 + 9*x^3)^(1/3))])/Sqrt[3] + (2^(1/3)*Log[6 - 4*x + 2^(2/3)*(-19 + 66*x - 30*x^2 + 9*x^3)^(1/3)])/3 - Log[18 - 24*x + 8*x^2 + (-3*2^(2/3) + 2*2^(2/3)*x)*(-19 + 66*x - 30*x^2 + 9*x^3)^(1/3) + 2^(1/3)*(-19 + 66*x - 30*x^2 + 9*x^3)^(2/3)]/(3*2^(2/3))","A",1
2514,1,210,210,0.7098552,"\int \frac{1+x^3}{\left(-1+x^3\right) \sqrt[3]{x^2+x^4}} \, dx","IntegrateAlgebraic[(1 + x^3)/((-1 + x^3)*(x^2 + x^4)^(1/3)),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{x^2+x^4}}\right)}{\sqrt{3}}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x^2+x^4}}\right)}{\sqrt[3]{2} \sqrt{3}}-\frac{2}{3} \log \left(x+\sqrt[3]{x^2+x^4}\right)+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x^2+x^4}\right)}{3 \sqrt[3]{2}}+\frac{1}{3} \log \left(x^2-x \sqrt[3]{x^2+x^4}+\left(x^2+x^4\right)^{2/3}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x^2+x^4}+\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}\right)}{6 \sqrt[3]{2}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{x^2+x^4}}\right)}{\sqrt{3}}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x^2+x^4}}\right)}{\sqrt[3]{2} \sqrt{3}}-\frac{2}{3} \log \left(x+\sqrt[3]{x^2+x^4}\right)+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x^2+x^4}\right)}{3 \sqrt[3]{2}}+\frac{1}{3} \log \left(x^2-x \sqrt[3]{x^2+x^4}+\left(x^2+x^4\right)^{2/3}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x^2+x^4}+\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}\right)}{6 \sqrt[3]{2}}",1,"(-2*ArcTan[(Sqrt[3]*x)/(-x + 2*(x^2 + x^4)^(1/3))])/Sqrt[3] - ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(x^2 + x^4)^(1/3))]/(2^(1/3)*Sqrt[3]) - (2*Log[x + (x^2 + x^4)^(1/3)])/3 + Log[-2*x + 2^(2/3)*(x^2 + x^4)^(1/3)]/(3*2^(1/3)) + Log[x^2 - x*(x^2 + x^4)^(1/3) + (x^2 + x^4)^(2/3)]/3 - Log[2*x^2 + 2^(2/3)*x*(x^2 + x^4)^(1/3) + 2^(1/3)*(x^2 + x^4)^(2/3)]/(6*2^(1/3))","A",1
2515,1,210,210,0.8341405,"\int \frac{x \left(4 a b-3 (a+b) x+2 x^2\right)}{\sqrt[3]{x^2 (-a+x) (-b+x)} \left(-a b d+(a+b) d x-d x^2+x^4\right)} \, dx","IntegrateAlgebraic[(x*(4*a*b - 3*(a + b)*x + 2*x^2))/((x^2*(-a + x)*(-b + x))^(1/3)*(-(a*b*d) + (a + b)*d*x - d*x^2 + x^4)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}{2 x^2+\sqrt[3]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{d^{2/3}}+\frac{\log \left(x^2-\sqrt[3]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}\right)}{d^{2/3}}-\frac{\log \left(x^4+\sqrt[3]{d} x^2 \sqrt[3]{a b x^2+(-a-b) x^3+x^4}+d^{2/3} \left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}\right)}{2 d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}{2 x^2+\sqrt[3]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{d^{2/3}}+\frac{\log \left(x^2-\sqrt[3]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}\right)}{d^{2/3}}-\frac{\log \left(x^4+\sqrt[3]{d} x^2 \sqrt[3]{a b x^2+(-a-b) x^3+x^4}+d^{2/3} \left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}\right)}{2 d^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))/(2*x^2 + d^(1/3)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))])/d^(2/3) + Log[x^2 - d^(1/3)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3)]/d^(2/3) - Log[x^4 + d^(1/3)*x^2*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3) + d^(2/3)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(2/3)]/(2*d^(2/3))","A",1
2516,1,97,210,0.3375344,"\int \frac{\sqrt{b+a x^4}}{-b+a x^4} \, dx","IntegrateAlgebraic[Sqrt[b + a*x^4]/(-b + a*x^4),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x}{\sqrt{b+a x^4}}\right)}{2 \sqrt{2} \sqrt[4]{a} \sqrt[4]{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x}{\sqrt{b+a x^4}}\right)}{2 \sqrt{2} \sqrt[4]{a} \sqrt[4]{b}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x}{\sqrt{b}+\sqrt{a} x^2+\sqrt{b+a x^4}}\right)}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{6-4 \sqrt{2}} \sqrt[4]{a} \sqrt[4]{b} x}{\sqrt{b}+\sqrt{a} x^2+\sqrt{b+a x^4}}\right)}{2 \sqrt{2} \sqrt[4]{a} \sqrt[4]{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{6+4 \sqrt{2}} \sqrt[4]{a} \sqrt[4]{b} x}{\sqrt{b}+\sqrt{a} x^2+\sqrt{b+a x^4}}\right)}{2 \sqrt{2} \sqrt[4]{a} \sqrt[4]{b}}",1,"-1/2*ArcTan[(Sqrt[2]*a^(1/4)*b^(1/4)*x)/Sqrt[b + a*x^4]]/(Sqrt[2]*a^(1/4)*b^(1/4)) - ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*x)/Sqrt[b + a*x^4]]/(2*Sqrt[2]*a^(1/4)*b^(1/4))","A",1
2517,1,210,210,6.4592909,"\int \frac{1+x^2}{\left(-1+x+x^2\right) \sqrt[3]{-1+x^6}} \, dx","IntegrateAlgebraic[(1 + x^2)/((-1 + x + x^2)*(-1 + x^6)^(1/3)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-1+x^6}}{-2^{2/3}-2^{2/3} x+2^{2/3} x^2+\sqrt[3]{-1+x^6}}\right)}{2^{2/3} \sqrt{3}}-\frac{\log \left(-2^{2/3}-2^{2/3} x+2^{2/3} x^2-2 \sqrt[3]{-1+x^6}\right)}{3\ 2^{2/3}}+\frac{\log \left(\sqrt[3]{2}+2 \sqrt[3]{2} x-\sqrt[3]{2} x^2-2 \sqrt[3]{2} x^3+\sqrt[3]{2} x^4+\left(-2^{2/3}-2^{2/3} x+2^{2/3} x^2\right) \sqrt[3]{-1+x^6}+2 \left(-1+x^6\right)^{2/3}\right)}{6\ 2^{2/3}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-1+x^6}}{-2^{2/3}-2^{2/3} x+2^{2/3} x^2+\sqrt[3]{-1+x^6}}\right)}{2^{2/3} \sqrt{3}}-\frac{\log \left(-2^{2/3}-2^{2/3} x+2^{2/3} x^2-2 \sqrt[3]{-1+x^6}\right)}{3\ 2^{2/3}}+\frac{\log \left(\sqrt[3]{2}+2 \sqrt[3]{2} x-\sqrt[3]{2} x^2-2 \sqrt[3]{2} x^3+\sqrt[3]{2} x^4+\left(-2^{2/3}-2^{2/3} x+2^{2/3} x^2\right) \sqrt[3]{-1+x^6}+2 \left(-1+x^6\right)^{2/3}\right)}{6\ 2^{2/3}}",1,"-(ArcTan[(Sqrt[3]*(-1 + x^6)^(1/3))/(-2^(2/3) - 2^(2/3)*x + 2^(2/3)*x^2 + (-1 + x^6)^(1/3))]/(2^(2/3)*Sqrt[3])) - Log[-2^(2/3) - 2^(2/3)*x + 2^(2/3)*x^2 - 2*(-1 + x^6)^(1/3)]/(3*2^(2/3)) + Log[2^(1/3) + 2*2^(1/3)*x - 2^(1/3)*x^2 - 2*2^(1/3)*x^3 + 2^(1/3)*x^4 + (-2^(2/3) - 2^(2/3)*x + 2^(2/3)*x^2)*(-1 + x^6)^(1/3) + 2*(-1 + x^6)^(2/3)]/(6*2^(2/3))","A",1
2518,1,210,210,0.001007,"\int \frac{1+x^2}{\left(-1+x+x^2\right) \sqrt[3]{-1+x^6}} \, dx","IntegrateAlgebraic[(1 + x^2)/((-1 + x + x^2)*(-1 + x^6)^(1/3)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-1+x^6}}{-2^{2/3}-2^{2/3} x+2^{2/3} x^2+\sqrt[3]{-1+x^6}}\right)}{2^{2/3} \sqrt{3}}-\frac{\log \left(-2^{2/3}-2^{2/3} x+2^{2/3} x^2-2 \sqrt[3]{-1+x^6}\right)}{3\ 2^{2/3}}+\frac{\log \left(\sqrt[3]{2}+2 \sqrt[3]{2} x-\sqrt[3]{2} x^2-2 \sqrt[3]{2} x^3+\sqrt[3]{2} x^4+\left(-2^{2/3}-2^{2/3} x+2^{2/3} x^2\right) \sqrt[3]{-1+x^6}+2 \left(-1+x^6\right)^{2/3}\right)}{6\ 2^{2/3}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-1+x^6}}{-2^{2/3}-2^{2/3} x+2^{2/3} x^2+\sqrt[3]{-1+x^6}}\right)}{2^{2/3} \sqrt{3}}-\frac{\log \left(-2^{2/3}-2^{2/3} x+2^{2/3} x^2-2 \sqrt[3]{-1+x^6}\right)}{3\ 2^{2/3}}+\frac{\log \left(\sqrt[3]{2}+2 \sqrt[3]{2} x-\sqrt[3]{2} x^2-2 \sqrt[3]{2} x^3+\sqrt[3]{2} x^4+\left(-2^{2/3}-2^{2/3} x+2^{2/3} x^2\right) \sqrt[3]{-1+x^6}+2 \left(-1+x^6\right)^{2/3}\right)}{6\ 2^{2/3}}",1,"-(ArcTan[(Sqrt[3]*(-1 + x^6)^(1/3))/(-2^(2/3) - 2^(2/3)*x + 2^(2/3)*x^2 + (-1 + x^6)^(1/3))]/(2^(2/3)*Sqrt[3])) - Log[-2^(2/3) - 2^(2/3)*x + 2^(2/3)*x^2 - 2*(-1 + x^6)^(1/3)]/(3*2^(2/3)) + Log[2^(1/3) + 2*2^(1/3)*x - 2^(1/3)*x^2 - 2*2^(1/3)*x^3 + 2^(1/3)*x^4 + (-2^(2/3) - 2^(2/3)*x + 2^(2/3)*x^2)*(-1 + x^6)^(1/3) + 2*(-1 + x^6)^(2/3)]/(6*2^(2/3))","A",1
2519,1,210,210,0.5214511,"\int \frac{x^3 \sqrt[3]{-x^2+x^3}}{1+x^6} \, dx","IntegrateAlgebraic[(x^3*(-x^2 + x^3)^(1/3))/(1 + x^6),x]","\frac{1}{6} \text{RootSum}\left[2-2 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^3}\&\right]-\frac{1}{6} \text{RootSum}\left[1-2 \text{$\#$1}^3+5 \text{$\#$1}^6-4 \text{$\#$1}^9+\text{$\#$1}^{12}\&,\frac{\log (x) \text{$\#$1}-\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}+2 \log (x) \text{$\#$1}^4-2 \log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^4-\log (x) \text{$\#$1}^7+\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^7}{-1+5 \text{$\#$1}^3-6 \text{$\#$1}^6+2 \text{$\#$1}^9}\&\right]","\frac{1}{6} \text{RootSum}\left[2-2 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^3}\&\right]-\frac{1}{6} \text{RootSum}\left[1-2 \text{$\#$1}^3+5 \text{$\#$1}^6-4 \text{$\#$1}^9+\text{$\#$1}^{12}\&,\frac{\log (x) \text{$\#$1}-\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}+2 \log (x) \text{$\#$1}^4-2 \log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^4-\log (x) \text{$\#$1}^7+\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^7}{-1+5 \text{$\#$1}^3-6 \text{$\#$1}^6+2 \text{$\#$1}^9}\&\right]",1,"RootSum[2 - 2*#1^3 + #1^6 & , (-(Log[x]*#1) + Log[(-x^2 + x^3)^(1/3) - x*#1]*#1)/(-1 + #1^3) & ]/6 - RootSum[1 - 2*#1^3 + 5*#1^6 - 4*#1^9 + #1^12 & , (Log[x]*#1 - Log[(-x^2 + x^3)^(1/3) - x*#1]*#1 + 2*Log[x]*#1^4 - 2*Log[(-x^2 + x^3)^(1/3) - x*#1]*#1^4 - Log[x]*#1^7 + Log[(-x^2 + x^3)^(1/3) - x*#1]*#1^7)/(-1 + 5*#1^3 - 6*#1^6 + 2*#1^9) & ]/6","A",1
2520,1,211,210,10.877815,"\int \frac{b-2 c x^4+2 a x^8}{\sqrt[4]{-b+a x^4} \left(-2 b-c x^4+2 a x^8\right)} \, dx","IntegrateAlgebraic[(b - 2*c*x^4 + 2*a*x^8)/((-b + a*x^4)^(1/4)*(-2*b - c*x^4 + 2*a*x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2 \sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2 \sqrt[4]{a}}+\frac{1}{4} \text{RootSum}\left[2 a^2-2 a b+a c-4 a \text{$\#$1}^4-c \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{3 a \log (x)-c \log (x)-3 a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+c \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-3 \log (x) \text{$\#$1}^4+3 \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-4 a \text{$\#$1}-c \text{$\#$1}+4 \text{$\#$1}^5}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2 \sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2 \sqrt[4]{a}}+\frac{1}{4} \text{RootSum}\left[2 a^2-2 a b+a c-4 a \text{$\#$1}^4-c \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-3 a \log (x)+c \log (x)+3 a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-c \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+3 \log (x) \text{$\#$1}^4-3 \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{4 a \text{$\#$1}+c \text{$\#$1}-4 \text{$\#$1}^5}\&\right]",1,"ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(2*a^(1/4)) + ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(2*a^(1/4)) + RootSum[2*a^2 - 2*a*b + a*c - 4*a*#1^4 - c*#1^4 + 2*#1^8 & , (3*a*Log[x] - c*Log[x] - 3*a*Log[(-b + a*x^4)^(1/4) - x*#1] + c*Log[(-b + a*x^4)^(1/4) - x*#1] - 3*Log[x]*#1^4 + 3*Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4)/(-4*a*#1 - c*#1 + 4*#1^5) & ]/4","A",1
2521,1,211,210,0.0019814,"\int \frac{b-2 c x^4+2 a x^8}{\sqrt[4]{-b+a x^4} \left(-2 b-c x^4+2 a x^8\right)} \, dx","IntegrateAlgebraic[(b - 2*c*x^4 + 2*a*x^8)/((-b + a*x^4)^(1/4)*(-2*b - c*x^4 + 2*a*x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2 \sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2 \sqrt[4]{a}}+\frac{1}{4} \text{RootSum}\left[2 a^2-2 a b+a c-4 a \text{$\#$1}^4-c \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{3 a \log (x)-c \log (x)-3 a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+c \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-3 \log (x) \text{$\#$1}^4+3 \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-4 a \text{$\#$1}-c \text{$\#$1}+4 \text{$\#$1}^5}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2 \sqrt[4]{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2 \sqrt[4]{a}}+\frac{1}{4} \text{RootSum}\left[2 a^2-2 a b+a c-4 a \text{$\#$1}^4-c \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-3 a \log (x)+c \log (x)+3 a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-c \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+3 \log (x) \text{$\#$1}^4-3 \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{4 a \text{$\#$1}+c \text{$\#$1}-4 \text{$\#$1}^5}\&\right]",1,"ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(2*a^(1/4)) + ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(2*a^(1/4)) + RootSum[2*a^2 - 2*a*b + a*c - 4*a*#1^4 - c*#1^4 + 2*#1^8 & , (3*a*Log[x] - c*Log[x] - 3*a*Log[(-b + a*x^4)^(1/4) - x*#1] + c*Log[(-b + a*x^4)^(1/4) - x*#1] - 3*Log[x]*#1^4 + 3*Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4)/(-4*a*#1 - c*#1 + 4*#1^5) & ]/4","A",1
2522,1,211,211,0.5306907,"\int \frac{1}{x^3 \left(1+x^3\right) \sqrt[3]{-x^2+x^3}} \, dx","IntegrateAlgebraic[1/(x^3*(1 + x^3)*(-x^2 + x^3)^(1/3)),x]","\frac{3 \left(5+6 x+9 x^2\right) \left(-x^2+x^3\right)^{2/3}}{40 x^4}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-x^2+x^3}}\right)}{\sqrt[3]{2} \sqrt{3}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{-x^2+x^3}\right)}{3 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-x^2+x^3}+\sqrt[3]{2} \left(-x^2+x^3\right)^{2/3}\right)}{6 \sqrt[3]{2}}+\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{3 \left(5+6 x+9 x^2\right) \left(-x^2+x^3\right)^{2/3}}{40 x^4}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-x^2+x^3}}\right)}{\sqrt[3]{2} \sqrt{3}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{-x^2+x^3}\right)}{3 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-x^2+x^3}+\sqrt[3]{2} \left(-x^2+x^3\right)^{2/3}\right)}{6 \sqrt[3]{2}}+\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"(3*(5 + 6*x + 9*x^2)*(-x^2 + x^3)^(2/3))/(40*x^4) - ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-x^2 + x^3)^(1/3))]/(2^(1/3)*Sqrt[3]) + Log[-2*x + 2^(2/3)*(-x^2 + x^3)^(1/3)]/(3*2^(1/3)) - Log[2*x^2 + 2^(2/3)*x*(-x^2 + x^3)^(1/3) + 2^(1/3)*(-x^2 + x^3)^(2/3)]/(6*2^(1/3)) + RootSum[1 - #1^3 + #1^6 & , (-Log[x] + Log[(-x^2 + x^3)^(1/3) - x*#1])/#1 & ]/3","A",1
2523,1,211,211,0.4064253,"\int \frac{\left(-1+x^2\right) \sqrt[4]{x^3+x^4}}{1+x^2+x^4} \, dx","IntegrateAlgebraic[((-1 + x^2)*(x^3 + x^4)^(1/4))/(1 + x^2 + x^4),x]","-2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)+2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)-\frac{3}{2} \text{RootSum}\left[3-3 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 \text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]+\frac{1}{2} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{\log (x)-\log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]","-2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)+2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)-\frac{3}{2} \text{RootSum}\left[3-3 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 \text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]+\frac{1}{2} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{\log (x)-\log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]",1,"-2*ArcTan[x/(x^3 + x^4)^(1/4)] + 2*ArcTanh[x/(x^3 + x^4)^(1/4)] - (3*RootSum[3 - 3*#1^4 + #1^8 & , (-Log[x] + Log[(x^3 + x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(x^3 + x^4)^(1/4) - x*#1]*#1^4)/(-3*#1^3 + 2*#1^7) & ])/2 + RootSum[1 - #1^4 + #1^8 & , (Log[x] - Log[(x^3 + x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(x^3 + x^4)^(1/4) - x*#1]*#1^4)/(-#1^3 + 2*#1^7) & ]/2","A",1
2524,1,211,211,0.0010357,"\int \frac{\left(-1+x^2\right) \sqrt[4]{x^3+x^4}}{1+x^2+x^4} \, dx","IntegrateAlgebraic[((-1 + x^2)*(x^3 + x^4)^(1/4))/(1 + x^2 + x^4),x]","-2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)+2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)-\frac{3}{2} \text{RootSum}\left[3-3 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 \text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]+\frac{1}{2} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{\log (x)-\log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]","-2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)+2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)-\frac{3}{2} \text{RootSum}\left[3-3 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 \text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]+\frac{1}{2} \text{RootSum}\left[1-\text{$\#$1}^4+\text{$\#$1}^8\&,\frac{\log (x)-\log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]",1,"-2*ArcTan[x/(x^3 + x^4)^(1/4)] + 2*ArcTanh[x/(x^3 + x^4)^(1/4)] - (3*RootSum[3 - 3*#1^4 + #1^8 & , (-Log[x] + Log[(x^3 + x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(x^3 + x^4)^(1/4) - x*#1]*#1^4)/(-3*#1^3 + 2*#1^7) & ])/2 + RootSum[1 - #1^4 + #1^8 & , (Log[x] - Log[(x^3 + x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(x^3 + x^4)^(1/4) - x*#1]*#1^4)/(-#1^3 + 2*#1^7) & ]/2","A",1
2525,1,212,211,0.4805024,"\int \frac{\left(-b+a x^2\right) \sqrt[4]{-b x^2+a x^4}}{b-a x^2+x^4} \, dx","IntegrateAlgebraic[((-b + a*x^2)*(-(b*x^2) + a*x^4)^(1/4))/(b - a*x^2 + x^4),x]","-a^{5/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)+a^{5/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)-\frac{1}{2} \text{RootSum}\left[b-a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a b \log (x)+a b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)+a^2 \log (x) \text{$\#$1}^4-b \log (x) \text{$\#$1}^4-a^2 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]","-a^{5/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)+a^{5/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)-\frac{1}{2} \text{RootSum}\left[b-a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a b \log (x)-a b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)-a^2 \log (x) \text{$\#$1}^4+b \log (x) \text{$\#$1}^4+a^2 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}^3-2 \text{$\#$1}^7}\&\right]",1,"-(a^(5/4)*ArcTan[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)]) + a^(5/4)*ArcTanh[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)] - RootSum[b - a*#1^4 + #1^8 & , (-(a*b*Log[x]) + a*b*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1] + a^2*Log[x]*#1^4 - b*Log[x]*#1^4 - a^2*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4 + b*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3) + 2*#1^7) & ]/2","A",1
2526,1,212,211,0.0016469,"\int \frac{\left(-b+a x^2\right) \sqrt[4]{-b x^2+a x^4}}{b-a x^2+x^4} \, dx","IntegrateAlgebraic[((-b + a*x^2)*(-(b*x^2) + a*x^4)^(1/4))/(b - a*x^2 + x^4),x]","-a^{5/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)+a^{5/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)-\frac{1}{2} \text{RootSum}\left[b-a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a b \log (x)+a b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)+a^2 \log (x) \text{$\#$1}^4-b \log (x) \text{$\#$1}^4-a^2 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]","-a^{5/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)+a^{5/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)-\frac{1}{2} \text{RootSum}\left[b-a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a b \log (x)-a b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)-a^2 \log (x) \text{$\#$1}^4+b \log (x) \text{$\#$1}^4+a^2 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}^3-2 \text{$\#$1}^7}\&\right]",1,"-(a^(5/4)*ArcTan[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)]) + a^(5/4)*ArcTanh[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)] - RootSum[b - a*#1^4 + #1^8 & , (-(a*b*Log[x]) + a*b*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1] + a^2*Log[x]*#1^4 - b*Log[x]*#1^4 - a^2*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4 + b*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3) + 2*#1^7) & ]/2","A",1
2527,1,179,211,0.4418654,"\int \frac{\left(-b^2+a x^2\right)^2}{\left(b^2+a x^2\right)^2 \sqrt{b+\sqrt{b^2+a x^2}}} \, dx","IntegrateAlgebraic[(-b^2 + a*x^2)^2/((b^2 + a*x^2)^2*Sqrt[b + Sqrt[b^2 + a*x^2]]),x]","-\frac{b x}{\sqrt{b^2+a x^2} \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{2 x \left(2 b^2+a x^2\right)}{\left(b^2+a x^2\right) \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{\sqrt{a}}-\frac{\sqrt{2} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{\sqrt{a}}","-\frac{b x}{\sqrt{b^2+a x^2} \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{2 x \left(2 b^2+a x^2\right)}{\left(b^2+a x^2\right) \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{\sqrt{a}}-\frac{2 \sqrt{2} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2} \sqrt{b}}\right)}{\sqrt{a}}",1,"-((b*x)/(Sqrt[b^2 + a*x^2]*Sqrt[b + Sqrt[b^2 + a*x^2]])) + (2*x*(2*b^2 + a*x^2))/((b^2 + a*x^2)*Sqrt[b + Sqrt[b^2 + a*x^2]]) - (Sqrt[b]*ArcTan[(Sqrt[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/Sqrt[a] - (Sqrt[2]*Sqrt[b]*ArcTan[(Sqrt[a]*x)/(Sqrt[2]*Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/Sqrt[a]","A",1
2528,1,212,212,0.4754779,"\int \frac{(-2+(1+k) x) \left(a-a (1+k) x+(1+a k) x^2\right)}{(-1+x) \sqrt[3]{(1-x) x (1-k x)} (-1+k x) \left(b-b (1+k) x+(-1+b k) x^2\right)} \, dx","IntegrateAlgebraic[((-2 + (1 + k)*x)*(a - a*(1 + k)*x + (1 + a*k)*x^2))/((-1 + x)*((1 - x)*x*(1 - k*x))^(1/3)*(-1 + k*x)*(b - b*(1 + k)*x + (-1 + b*k)*x^2)),x]","\frac{3 \left(x-x^2-k x^2+k x^3\right)^{2/3}}{(-1+x) (-1+k x)}+\frac{\left(-\sqrt{3} a-\sqrt{3} b\right) \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{b^{2/3}}+\frac{(a+b) \log \left(x-\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{b^{2/3}}+\frac{(-a-b) \log \left(x^2+\sqrt[3]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 b^{2/3}}","\frac{3 \left(x-x^2-k x^2+k x^3\right)^{2/3}}{(-1+x) (-1+k x)}+\frac{\left(-\sqrt{3} a-\sqrt{3} b\right) \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{b^{2/3}}+\frac{(a+b) \log \left(x-\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{b^{2/3}}+\frac{(-a-b) \log \left(x^2+\sqrt[3]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 b^{2/3}}",1,"(3*(x - x^2 - k*x^2 + k*x^3)^(2/3))/((-1 + x)*(-1 + k*x)) + ((-(Sqrt[3]*a) - Sqrt[3]*b)*ArcTan[(Sqrt[3]*x)/(x + 2*b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/b^(2/3) + ((a + b)*Log[x - b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3)])/b^(2/3) + ((-a - b)*Log[x^2 + b^(1/3)*x*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + b^(2/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3)])/(2*b^(2/3))","A",1
2529,1,212,212,1.2390929,"\int \frac{\left(b+x^3\right)^3}{\sqrt[3]{a+x^3}} \, dx","IntegrateAlgebraic[(b + x^3)^3/(a + x^3)^(1/3),x]","\frac{1}{162} \left(a+x^3\right)^{2/3} \left(28 a^2 x-108 a b x+162 b^2 x-21 a x^4+81 b x^4+18 x^7\right)+\frac{1}{243} \left(-14 \sqrt{3} a^3+54 \sqrt{3} a^2 b-81 \sqrt{3} a b^2+81 \sqrt{3} b^3\right) \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{a+x^3}}\right)+\frac{1}{243} \left(14 a^3-54 a^2 b+81 a b^2-81 b^3\right) \log \left(-x+\sqrt[3]{a+x^3}\right)+\frac{1}{486} \left(-14 a^3+54 a^2 b-81 a b^2+81 b^3\right) \log \left(x^2+x \sqrt[3]{a+x^3}+\left(a+x^3\right)^{2/3}\right)","\frac{1}{162} \left(a+x^3\right)^{2/3} \left(28 a^2 x-108 a b x+162 b^2 x-21 a x^4+81 b x^4+18 x^7\right)+\frac{1}{243} \left(-14 \sqrt{3} a^3+54 \sqrt{3} a^2 b-81 \sqrt{3} a b^2+81 \sqrt{3} b^3\right) \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{a+x^3}}\right)+\frac{1}{243} \left(14 a^3-54 a^2 b+81 a b^2-81 b^3\right) \log \left(-x+\sqrt[3]{a+x^3}\right)+\frac{1}{486} \left(-14 a^3+54 a^2 b-81 a b^2+81 b^3\right) \log \left(x^2+x \sqrt[3]{a+x^3}+\left(a+x^3\right)^{2/3}\right)",1,"((a + x^3)^(2/3)*(28*a^2*x - 108*a*b*x + 162*b^2*x - 21*a*x^4 + 81*b*x^4 + 18*x^7))/162 + ((-14*Sqrt[3]*a^3 + 54*Sqrt[3]*a^2*b - 81*Sqrt[3]*a*b^2 + 81*Sqrt[3]*b^3)*ArcTan[(Sqrt[3]*x)/(x + 2*(a + x^3)^(1/3))])/243 + ((14*a^3 - 54*a^2*b + 81*a*b^2 - 81*b^3)*Log[-x + (a + x^3)^(1/3)])/243 + ((-14*a^3 + 54*a^2*b - 81*a*b^2 + 81*b^3)*Log[x^2 + x*(a + x^3)^(1/3) + (a + x^3)^(2/3)])/486","A",1
2530,1,212,212,0.4622724,"\int \frac{1-x^2+x^3}{\left(-1-x^2+x^3\right) \sqrt[3]{x^2+x^3}} \, dx","IntegrateAlgebraic[(1 - x^2 + x^3)/((-1 - x^2 + x^3)*(x^2 + x^3)^(1/3)),x]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{x^2+x^3}+\left(x^2+x^3\right)^{2/3}\right)+2 \text{RootSum}\left[-3+4 \text{$\#$1}^3-3 \text{$\#$1}^6+\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right)+2 \log (x) \text{$\#$1}^3-2 \log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3-\log (x) \text{$\#$1}^6+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^6}{4 \text{$\#$1}-6 \text{$\#$1}^4+3 \text{$\#$1}^7}\&\right]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{x^2+x^3}+\left(x^2+x^3\right)^{2/3}\right)+2 \text{RootSum}\left[-3+4 \text{$\#$1}^3-3 \text{$\#$1}^6+\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right)+2 \log (x) \text{$\#$1}^3-2 \log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3-\log (x) \text{$\#$1}^6+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^6}{4 \text{$\#$1}-6 \text{$\#$1}^4+3 \text{$\#$1}^7}\&\right]",1,"Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(x^2 + x^3)^(1/3))] - Log[-x + (x^2 + x^3)^(1/3)] + Log[x^2 + x*(x^2 + x^3)^(1/3) + (x^2 + x^3)^(2/3)]/2 + 2*RootSum[-3 + 4*#1^3 - 3*#1^6 + #1^9 & , (-Log[x] + Log[(x^2 + x^3)^(1/3) - x*#1] + 2*Log[x]*#1^3 - 2*Log[(x^2 + x^3)^(1/3) - x*#1]*#1^3 - Log[x]*#1^6 + Log[(x^2 + x^3)^(1/3) - x*#1]*#1^6)/(4*#1 - 6*#1^4 + 3*#1^7) & ]","A",1
2531,1,212,212,0.0015348,"\int \frac{1-x^2+x^3}{\left(-1-x^2+x^3\right) \sqrt[3]{x^2+x^3}} \, dx","IntegrateAlgebraic[(1 - x^2 + x^3)/((-1 - x^2 + x^3)*(x^2 + x^3)^(1/3)),x]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{x^2+x^3}+\left(x^2+x^3\right)^{2/3}\right)+2 \text{RootSum}\left[-3+4 \text{$\#$1}^3-3 \text{$\#$1}^6+\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right)+2 \log (x) \text{$\#$1}^3-2 \log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3-\log (x) \text{$\#$1}^6+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^6}{4 \text{$\#$1}-6 \text{$\#$1}^4+3 \text{$\#$1}^7}\&\right]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{x^2+x^3}+\left(x^2+x^3\right)^{2/3}\right)+2 \text{RootSum}\left[-3+4 \text{$\#$1}^3-3 \text{$\#$1}^6+\text{$\#$1}^9\&,\frac{-\log (x)+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right)+2 \log (x) \text{$\#$1}^3-2 \log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3-\log (x) \text{$\#$1}^6+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^6}{4 \text{$\#$1}-6 \text{$\#$1}^4+3 \text{$\#$1}^7}\&\right]",1,"Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(x^2 + x^3)^(1/3))] - Log[-x + (x^2 + x^3)^(1/3)] + Log[x^2 + x*(x^2 + x^3)^(1/3) + (x^2 + x^3)^(2/3)]/2 + 2*RootSum[-3 + 4*#1^3 - 3*#1^6 + #1^9 & , (-Log[x] + Log[(x^2 + x^3)^(1/3) - x*#1] + 2*Log[x]*#1^3 - 2*Log[(x^2 + x^3)^(1/3) - x*#1]*#1^3 - Log[x]*#1^6 + Log[(x^2 + x^3)^(1/3) - x*#1]*#1^6)/(4*#1 - 6*#1^4 + 3*#1^7) & ]","A",1
2532,1,212,212,5.6342244,"\int \frac{-3+\left(1-2 k^2\right) x+3 k^2 x^2+k^2 x^3}{\sqrt[3]{\left(1-x^2\right) \left(1-k^2 x^2\right)} \left(-1+d-(2+d) x-\left(1+d k^2\right) x^2+d k^2 x^3\right)} \, dx","IntegrateAlgebraic[(-3 + (1 - 2*k^2)*x + 3*k^2*x^2 + k^2*x^3)/(((1 - x^2)*(1 - k^2*x^2))^(1/3)*(-1 + d - (2 + d)*x - (1 + d*k^2)*x^2 + d*k^2*x^3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{2+2 x+\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{2/3}}+\frac{\log \left(1+x-\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{d^{2/3}}-\frac{\log \left(1+2 x+x^2+\left(\sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+d^{2/3} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{2+2 x+\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{2/3}}+\frac{\log \left(1+x-\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{d^{2/3}}-\frac{\log \left(1+2 x+x^2+\left(\sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+d^{2/3} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 d^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))/(2 + 2*x + d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))])/d^(2/3) + Log[1 + x - d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3)]/d^(2/3) - Log[1 + 2*x + x^2 + (d^(1/3) + d^(1/3)*x)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3) + d^(2/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3)]/(2*d^(2/3))","A",1
2533,1,97,212,0.486917,"\int \frac{x^2}{\left(-b+a x^4\right) \sqrt{b+a x^4}} \, dx","IntegrateAlgebraic[x^2/((-b + a*x^4)*Sqrt[b + a*x^4]),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x}{\sqrt{b+a x^4}}\right)}{4 \sqrt{2} a^{3/4} b^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x}{\sqrt{b+a x^4}}\right)}{4 \sqrt{2} a^{3/4} b^{3/4}}","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x}{\sqrt{b}+\sqrt{a} x^2+\sqrt{b+a x^4}}\right)}{2 \sqrt{2} a^{3/4} b^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{6-4 \sqrt{2}} \sqrt[4]{a} \sqrt[4]{b} x}{\sqrt{b}+\sqrt{a} x^2+\sqrt{b+a x^4}}\right)}{4 \sqrt{2} a^{3/4} b^{3/4}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{6+4 \sqrt{2}} \sqrt[4]{a} \sqrt[4]{b} x}{\sqrt{b}+\sqrt{a} x^2+\sqrt{b+a x^4}}\right)}{4 \sqrt{2} a^{3/4} b^{3/4}}",1,"ArcTan[(Sqrt[2]*a^(1/4)*b^(1/4)*x)/Sqrt[b + a*x^4]]/(4*Sqrt[2]*a^(3/4)*b^(3/4)) - ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*x)/Sqrt[b + a*x^4]]/(4*Sqrt[2]*a^(3/4)*b^(3/4))","A",1
2534,1,212,212,1.0036739,"\int \frac{(-1+x) (-1+k x) (-2+(1+k) x)}{\sqrt[3]{(1-x) x (1-k x)} \left(b-2 (b+b k) x+\left(b+4 b k+b k^2\right) x^2-2 b k (1+k) x^3+\left(-1+b k^2\right) x^4\right)} \, dx","IntegrateAlgebraic[((-1 + x)*(-1 + k*x)*(-2 + (1 + k)*x))/(((1 - x)*x*(1 - k*x))^(1/3)*(b - 2*(b + b*k)*x + (b + 4*b*k + b*k^2)*x^2 - 2*b*k*(1 + k)*x^3 + (-1 + b*k^2)*x^4)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x-2 \sqrt[6]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{2 b^{5/6}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[6]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{2 b^{5/6}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[6]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{b^{5/6}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[6]{b}}+\sqrt[6]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}}{x \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{2 b^{5/6}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x-2 \sqrt[6]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{2 b^{5/6}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[6]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{2 b^{5/6}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[6]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{b^{5/6}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[6]{b}}+\sqrt[6]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}}{x \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{2 b^{5/6}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x - 2*b^(1/6)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/(2*b^(5/6)) - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*b^(1/6)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/(2*b^(5/6)) - ArcTanh[x/(b^(1/6)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))]/b^(5/6) - ArcTanh[(x^2/b^(1/6) + b^(1/6)*(x + (-1 - k)*x^2 + k*x^3)^(2/3))/(x*(x + (-1 - k)*x^2 + k*x^3)^(1/3))]/(2*b^(5/6))","A",1
2535,1,190,212,0.3398889,"\int \frac{\left(-g+f x^2\right) \sqrt{c+\sqrt{b+a x}}}{e+d x^2} \, dx","IntegrateAlgebraic[((-g + f*x^2)*Sqrt[c + Sqrt[b + a*x]])/(e + d*x^2),x]","-\frac{4 \sqrt{c+\sqrt{b+a x}} \left(2 c^2 f-c f \sqrt{b+a x}-3 f (b+a x)\right)}{15 a d}-\frac{a (e f+d g) \text{RootSum}\left[b^2 d-2 b c^2 d+c^4 d+a^2 e+4 b c d \text{$\#$1}^2-4 c^3 d \text{$\#$1}^2-2 b d \text{$\#$1}^4+6 c^2 d \text{$\#$1}^4-4 c d \text{$\#$1}^6+d \text{$\#$1}^8\&,\frac{\log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}}{-b+c^2-2 c \text{$\#$1}^2+\text{$\#$1}^4}\&\right]}{2 d^2}","\frac{4 c f \sqrt{b+a x} \sqrt{c+\sqrt{b+a x}}}{15 a d}+\frac{4 f \left(3 b-2 c^2+3 a x\right) \sqrt{c+\sqrt{b+a x}}}{15 a d}-\frac{a (e f+d g) \text{RootSum}\left[b^2 d-2 b c^2 d+c^4 d+a^2 e+4 b c d \text{$\#$1}^2-4 c^3 d \text{$\#$1}^2-2 b d \text{$\#$1}^4+6 c^2 d \text{$\#$1}^4-4 c d \text{$\#$1}^6+d \text{$\#$1}^8\&,\frac{\log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}}{-b+c^2-2 c \text{$\#$1}^2+\text{$\#$1}^4}\&\right]}{2 d^2}",1,"(-4*Sqrt[c + Sqrt[b + a*x]]*(2*c^2*f - c*f*Sqrt[b + a*x] - 3*f*(b + a*x)))/(15*a*d) - (a*(e*f + d*g)*RootSum[b^2*d - 2*b*c^2*d + c^4*d + a^2*e + 4*b*c*d*#1^2 - 4*c^3*d*#1^2 - 2*b*d*#1^4 + 6*c^2*d*#1^4 - 4*c*d*#1^6 + d*#1^8 & , (Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1)/(-b + c^2 - 2*c*#1^2 + #1^4) & ])/(2*d^2)","A",1
2536,1,212,212,5.1368051,"\int \frac{\sqrt{b^2+a^2 x^2}}{d+c x^2+\sqrt{a x+\sqrt{b^2+a^2 x^2}}} \, dx","IntegrateAlgebraic[Sqrt[b^2 + a^2*x^2]/(d + c*x^2 + Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]),x]","\frac{a \log \left(a x+\sqrt{b^2+a^2 x^2}\right)}{c}-\frac{2 a \text{RootSum}\left[b^4 c-2 b^2 c \text{$\#$1}^4+4 a^2 d \text{$\#$1}^4+4 a^2 \text{$\#$1}^5+c \text{$\#$1}^8\&,\frac{b^2 c \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-a^2 d \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-a^2 \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}}{2 b^2 c-4 a^2 d-5 a^2 \text{$\#$1}-2 c \text{$\#$1}^4}\&\right]}{c}","\frac{a \log \left(a x+\sqrt{b^2+a^2 x^2}\right)}{c}-\frac{2 a \text{RootSum}\left[b^4 c-2 b^2 c \text{$\#$1}^4+4 a^2 d \text{$\#$1}^4+4 a^2 \text{$\#$1}^5+c \text{$\#$1}^8\&,\frac{b^2 c \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-a^2 d \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-a^2 \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}}{2 b^2 c-4 a^2 d-5 a^2 \text{$\#$1}-2 c \text{$\#$1}^4}\&\right]}{c}",1,"(a*Log[a*x + Sqrt[b^2 + a^2*x^2]])/c - (2*a*RootSum[b^4*c - 2*b^2*c*#1^4 + 4*a^2*d*#1^4 + 4*a^2*#1^5 + c*#1^8 & , (b^2*c*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1] - a^2*d*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1] - a^2*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]*#1)/(2*b^2*c - 4*a^2*d - 5*a^2*#1 - 2*c*#1^4) & ])/c","A",1
2537,1,212,212,0.0021329,"\int \frac{\sqrt{b^2+a^2 x^2}}{d+c x^2+\sqrt{a x+\sqrt{b^2+a^2 x^2}}} \, dx","IntegrateAlgebraic[Sqrt[b^2 + a^2*x^2]/(d + c*x^2 + Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]),x]","\frac{a \log \left(a x+\sqrt{b^2+a^2 x^2}\right)}{c}-\frac{2 a \text{RootSum}\left[b^4 c-2 b^2 c \text{$\#$1}^4+4 a^2 d \text{$\#$1}^4+4 a^2 \text{$\#$1}^5+c \text{$\#$1}^8\&,\frac{b^2 c \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-a^2 d \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-a^2 \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}}{2 b^2 c-4 a^2 d-5 a^2 \text{$\#$1}-2 c \text{$\#$1}^4}\&\right]}{c}","\frac{a \log \left(a x+\sqrt{b^2+a^2 x^2}\right)}{c}-\frac{2 a \text{RootSum}\left[b^4 c-2 b^2 c \text{$\#$1}^4+4 a^2 d \text{$\#$1}^4+4 a^2 \text{$\#$1}^5+c \text{$\#$1}^8\&,\frac{b^2 c \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-a^2 d \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-a^2 \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}}{2 b^2 c-4 a^2 d-5 a^2 \text{$\#$1}-2 c \text{$\#$1}^4}\&\right]}{c}",1,"(a*Log[a*x + Sqrt[b^2 + a^2*x^2]])/c - (2*a*RootSum[b^4*c - 2*b^2*c*#1^4 + 4*a^2*d*#1^4 + 4*a^2*#1^5 + c*#1^8 & , (b^2*c*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1] - a^2*d*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1] - a^2*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]*#1)/(2*b^2*c - 4*a^2*d - 5*a^2*#1 - 2*c*#1^4) & ])/c","A",1
2538,1,184,213,2.2042046,"\int \frac{\sqrt{1-\sqrt{1-\sqrt{1-\frac{1}{x}}}}}{x^2} \, dx","IntegrateAlgebraic[Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]]/x^2,x]","\frac{64}{315} \sqrt{1-\sqrt{1-\sqrt{\frac{-1+x}{x}}}} \sqrt{1-\sqrt{\frac{-1+x}{x}}}+\left(-\frac{8}{315} \sqrt{1-\sqrt{1-\sqrt{\frac{-1+x}{x}}}}+\frac{8}{63} \sqrt{1-\sqrt{1-\sqrt{\frac{-1+x}{x}}}} \sqrt{1-\sqrt{\frac{-1+x}{x}}}\right) \sqrt{\frac{-1+x}{x}}+\frac{8 \sqrt{1-\sqrt{1-\sqrt{\frac{-1+x}{x}}}} (-35+27 x)}{315 x}","-\frac{64}{315} \sqrt{1-\sqrt{1-\sqrt{\frac{-1+x}{x}}}}+\frac{64}{315} \sqrt{1-\sqrt{1-\sqrt{\frac{-1+x}{x}}}} \sqrt{1-\sqrt{\frac{-1+x}{x}}}+\left(-\frac{8}{315} \sqrt{1-\sqrt{1-\sqrt{\frac{-1+x}{x}}}}+\frac{8}{63} \sqrt{1-\sqrt{1-\sqrt{\frac{-1+x}{x}}}} \sqrt{1-\sqrt{\frac{-1+x}{x}}}\right) \sqrt{\frac{-1+x}{x}}+\frac{8 \sqrt{1-\sqrt{1-\sqrt{\frac{-1+x}{x}}}} (-1+x)}{9 x}",1,"(64*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]*Sqrt[1 - Sqrt[(-1 + x)/x]])/315 + ((-8*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]])/315 + (8*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]*Sqrt[1 - Sqrt[(-1 + x)/x]])/63)*Sqrt[(-1 + x)/x] + (8*Sqrt[1 - Sqrt[1 - Sqrt[(-1 + x)/x]]]*(-35 + 27*x))/(315*x)","A",1
2539,1,213,213,0.4448012,"\int \frac{\sqrt[3]{x^2+x^4}}{x \left(-1+x^2\right)} \, dx","IntegrateAlgebraic[(x^2 + x^4)^(1/3)/(x*(-1 + x^2)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}}\right)}{2\ 2^{2/3}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x^2+x^4}\right)}{2\ 2^{2/3}}+\frac{\log \left(2 x+2^{2/3} \sqrt[3]{x^2+x^4}\right)}{2\ 2^{2/3}}-\frac{\log \left(-2 x^2+2^{2/3} x \sqrt[3]{x^2+x^4}-\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}\right)}{4\ 2^{2/3}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x^2+x^4}+\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}\right)}{4\ 2^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}}\right)}{2\ 2^{2/3}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x^2+x^4}\right)}{2\ 2^{2/3}}+\frac{\log \left(2 x+2^{2/3} \sqrt[3]{x^2+x^4}\right)}{2\ 2^{2/3}}-\frac{\log \left(-2 x^2+2^{2/3} x \sqrt[3]{x^2+x^4}-\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}\right)}{4\ 2^{2/3}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x^2+x^4}+\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}\right)}{4\ 2^{2/3}}",1,"-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*x^2)/(x^2 + 2^(1/3)*(x^2 + x^4)^(2/3))])/2^(2/3) + Log[-2*x + 2^(2/3)*(x^2 + x^4)^(1/3)]/(2*2^(2/3)) + Log[2*x + 2^(2/3)*(x^2 + x^4)^(1/3)]/(2*2^(2/3)) - Log[-2*x^2 + 2^(2/3)*x*(x^2 + x^4)^(1/3) - 2^(1/3)*(x^2 + x^4)^(2/3)]/(4*2^(2/3)) - Log[2*x^2 + 2^(2/3)*x*(x^2 + x^4)^(1/3) + 2^(1/3)*(x^2 + x^4)^(2/3)]/(4*2^(2/3))","A",1
2540,1,213,213,0.5696613,"\int \frac{1}{(c+d x) \sqrt{b+a^2 x^2} \sqrt{a x-\sqrt{b+a^2 x^2}}} \, dx","IntegrateAlgebraic[1/((c + d*x)*Sqrt[b + a^2*x^2]*Sqrt[a*x - Sqrt[b + a^2*x^2]]),x]","-\frac{2 \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a x-\sqrt{b+a^2 x^2}}}{\sqrt{a c-\sqrt{a^2 c^2+b d^2}}}\right)}{\sqrt{a^2 c^2+b d^2} \sqrt{a c-\sqrt{a^2 c^2+b d^2}}}+\frac{2 \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a x-\sqrt{b+a^2 x^2}}}{\sqrt{a c+\sqrt{a^2 c^2+b d^2}}}\right)}{\sqrt{a^2 c^2+b d^2} \sqrt{a c+\sqrt{a^2 c^2+b d^2}}}","-\frac{2 \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a x-\sqrt{b+a^2 x^2}}}{\sqrt{a c-\sqrt{a^2 c^2+b d^2}}}\right)}{\sqrt{a^2 c^2+b d^2} \sqrt{a c-\sqrt{a^2 c^2+b d^2}}}+\frac{2 \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{a x-\sqrt{b+a^2 x^2}}}{\sqrt{a c+\sqrt{a^2 c^2+b d^2}}}\right)}{\sqrt{a^2 c^2+b d^2} \sqrt{a c+\sqrt{a^2 c^2+b d^2}}}",1,"(-2*Sqrt[d]*ArcTan[(Sqrt[d]*Sqrt[a*x - Sqrt[b + a^2*x^2]])/Sqrt[a*c - Sqrt[a^2*c^2 + b*d^2]]])/(Sqrt[a^2*c^2 + b*d^2]*Sqrt[a*c - Sqrt[a^2*c^2 + b*d^2]]) + (2*Sqrt[d]*ArcTan[(Sqrt[d]*Sqrt[a*x - Sqrt[b + a^2*x^2]])/Sqrt[a*c + Sqrt[a^2*c^2 + b*d^2]]])/(Sqrt[a^2*c^2 + b*d^2]*Sqrt[a*c + Sqrt[a^2*c^2 + b*d^2]])","A",1
2541,1,210,213,0.9378316,"\int \frac{\sqrt{1+x} \sqrt{1+\sqrt{1+x}}}{x^2 \sqrt{1+\sqrt{1+\sqrt{1+x}}}} \, dx","IntegrateAlgebraic[(Sqrt[1 + x]*Sqrt[1 + Sqrt[1 + x]])/(x^2*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]),x]","\frac{\sqrt{1+\sqrt{1+\sqrt{1+x}}}}{1-\sqrt{1+x}}+\frac{\left(3-\sqrt{1+x}\right) \sqrt{1+\sqrt{1+\sqrt{1+x}}}}{2 \left(-1+\sqrt{1+x}\right) \sqrt{1+\sqrt{1+x}}}-\frac{1}{4} \sqrt{17+25 \sqrt{2}} \tan ^{-1}\left(\sqrt{1+\sqrt{2}} \sqrt{1+\sqrt{1+\sqrt{1+x}}}\right)+\tanh ^{-1}\left(\sqrt{1+\sqrt{1+\sqrt{1+x}}}\right)-\frac{1}{4} \sqrt{-17+25 \sqrt{2}} \tanh ^{-1}\left(\sqrt{-1+\sqrt{2}} \sqrt{1+\sqrt{1+\sqrt{1+x}}}\right)","\frac{5 \sqrt{1+\sqrt{1+\sqrt{1+x}}}}{2 \left(-1+\sqrt{1+x}\right) \sqrt{1+\sqrt{1+x}}}-\frac{\left(1+\sqrt{1+\sqrt{1+x}}\right)^{5/2}}{2 \left(-1+\sqrt{1+x}\right) \sqrt{1+\sqrt{1+x}}}-\frac{1}{4} \sqrt{17+25 \sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{1+\sqrt{1+\sqrt{1+x}}}}{\sqrt{-1+\sqrt{2}}}\right)+\tanh ^{-1}\left(\sqrt{1+\sqrt{1+\sqrt{1+x}}}\right)-\frac{1}{4} \sqrt{-17+25 \sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{1+\sqrt{1+\sqrt{1+x}}}}{\sqrt{1+\sqrt{2}}}\right)",1,"Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]/(1 - Sqrt[1 + x]) + ((3 - Sqrt[1 + x])*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/(2*(-1 + Sqrt[1 + x])*Sqrt[1 + Sqrt[1 + x]]) - (Sqrt[17 + 25*Sqrt[2]]*ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]])/4 + ArcTanh[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]] - (Sqrt[-17 + 25*Sqrt[2]]*ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]])/4","A",1
2542,1,214,214,0.6142528,"\int \frac{-1+(2-k) x}{\sqrt[3]{(1-x) x (1-k x)} \left(b-(1+2 b k) x+\left(1+b k^2\right) x^2\right)} \, dx","IntegrateAlgebraic[(-1 + (2 - k)*x)/(((1 - x)*x*(1 - k*x))^(1/3)*(b - (1 + 2*b*k)*x + (1 + b*k^2)*x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{x+(-1-k) x^2+k x^3}}{2 \sqrt[3]{b}-2 \sqrt[3]{b} k x+\sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{\sqrt[3]{b}}+\frac{\log \left(-\sqrt[3]{b}+\sqrt[3]{b} k x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{\sqrt[3]{b}}-\frac{\log \left(b^{2/3}-2 b^{2/3} k x+b^{2/3} k^2 x^2+\left(\sqrt[3]{b}-\sqrt[3]{b} k x\right) \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 \sqrt[3]{b}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{x+(-1-k) x^2+k x^3}}{2 \sqrt[3]{b}-2 \sqrt[3]{b} k x+\sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{\sqrt[3]{b}}+\frac{\log \left(-\sqrt[3]{b}+\sqrt[3]{b} k x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{\sqrt[3]{b}}-\frac{\log \left(b^{2/3}-2 b^{2/3} k x+b^{2/3} k^2 x^2+\left(\sqrt[3]{b}-\sqrt[3]{b} k x\right) \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 \sqrt[3]{b}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(x + (-1 - k)*x^2 + k*x^3)^(1/3))/(2*b^(1/3) - 2*b^(1/3)*k*x + (x + (-1 - k)*x^2 + k*x^3)^(1/3))])/b^(1/3) + Log[-b^(1/3) + b^(1/3)*k*x + (x + (-1 - k)*x^2 + k*x^3)^(1/3)]/b^(1/3) - Log[b^(2/3) - 2*b^(2/3)*k*x + b^(2/3)*k^2*x^2 + (b^(1/3) - b^(1/3)*k*x)*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + (x + (-1 - k)*x^2 + k*x^3)^(2/3)]/(2*b^(1/3))","A",1
2543,1,214,214,0.5963019,"\int \frac{1}{x^6 \left(-1+x^3\right) \sqrt[3]{x^2+x^3}} \, dx","IntegrateAlgebraic[1/(x^6*(-1 + x^3)*(x^2 + x^3)^(1/3)),x]","\frac{3 \left(x^2+x^3\right)^{2/3} \left(3080-3300 x+3600 x^2+2495 x^3-2994 x^4+4491 x^5\right)}{52360 x^7}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x^2+x^3}}\right)}{\sqrt[3]{2} \sqrt{3}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x^2+x^3}\right)}{3 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x^2+x^3}+\sqrt[3]{2} \left(x^2+x^3\right)^{2/3}\right)}{6 \sqrt[3]{2}}+\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{3 \left(x^2+x^3\right)^{2/3} \left(3080-3300 x+3600 x^2+2495 x^3-2994 x^4+4491 x^5\right)}{52360 x^7}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x^2+x^3}}\right)}{\sqrt[3]{2} \sqrt{3}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x^2+x^3}\right)}{3 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x^2+x^3}+\sqrt[3]{2} \left(x^2+x^3\right)^{2/3}\right)}{6 \sqrt[3]{2}}+\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"(3*(x^2 + x^3)^(2/3)*(3080 - 3300*x + 3600*x^2 + 2495*x^3 - 2994*x^4 + 4491*x^5))/(52360*x^7) - ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(x^2 + x^3)^(1/3))]/(2^(1/3)*Sqrt[3]) + Log[-2*x + 2^(2/3)*(x^2 + x^3)^(1/3)]/(3*2^(1/3)) - Log[2*x^2 + 2^(2/3)*x*(x^2 + x^3)^(1/3) + 2^(1/3)*(x^2 + x^3)^(2/3)]/(6*2^(1/3)) + RootSum[1 - #1^3 + #1^6 & , (-Log[x] + Log[(x^2 + x^3)^(1/3) - x*#1])/#1 & ]/3","A",1
2544,1,214,214,0.0012571,"\int \frac{1}{x^6 \left(-1+x^3\right) \sqrt[3]{x^2+x^3}} \, dx","IntegrateAlgebraic[1/(x^6*(-1 + x^3)*(x^2 + x^3)^(1/3)),x]","\frac{3 \left(x^2+x^3\right)^{2/3} \left(3080-3300 x+3600 x^2+2495 x^3-2994 x^4+4491 x^5\right)}{52360 x^7}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x^2+x^3}}\right)}{\sqrt[3]{2} \sqrt{3}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x^2+x^3}\right)}{3 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x^2+x^3}+\sqrt[3]{2} \left(x^2+x^3\right)^{2/3}\right)}{6 \sqrt[3]{2}}+\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\frac{3 \left(x^2+x^3\right)^{2/3} \left(3080-3300 x+3600 x^2+2495 x^3-2994 x^4+4491 x^5\right)}{52360 x^7}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x^2+x^3}}\right)}{\sqrt[3]{2} \sqrt{3}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x^2+x^3}\right)}{3 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x^2+x^3}+\sqrt[3]{2} \left(x^2+x^3\right)^{2/3}\right)}{6 \sqrt[3]{2}}+\frac{1}{3} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"(3*(x^2 + x^3)^(2/3)*(3080 - 3300*x + 3600*x^2 + 2495*x^3 - 2994*x^4 + 4491*x^5))/(52360*x^7) - ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(x^2 + x^3)^(1/3))]/(2^(1/3)*Sqrt[3]) + Log[-2*x + 2^(2/3)*(x^2 + x^3)^(1/3)]/(3*2^(1/3)) - Log[2*x^2 + 2^(2/3)*x*(x^2 + x^3)^(1/3) + 2^(1/3)*(x^2 + x^3)^(2/3)]/(6*2^(1/3)) + RootSum[1 - #1^3 + #1^6 & , (-Log[x] + Log[(x^2 + x^3)^(1/3) - x*#1])/#1 & ]/3","A",1
2545,1,214,214,5.6297445,"\int \frac{3+\left(1-2 k^2\right) x-3 k^2 x^2+k^2 x^3}{\sqrt[3]{\left(1-x^2\right) \left(1-k^2 x^2\right)} \left(1-d-(2+d) x+\left(1+d k^2\right) x^2+d k^2 x^3\right)} \, dx","IntegrateAlgebraic[(3 + (1 - 2*k^2)*x - 3*k^2*x^2 + k^2*x^3)/(((1 - x^2)*(1 - k^2*x^2))^(1/3)*(1 - d - (2 + d)*x + (1 + d*k^2)*x^2 + d*k^2*x^3)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{2-2 x+\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{2/3}}-\frac{\log \left(-1+x+\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{d^{2/3}}+\frac{\log \left(1-2 x+x^2+\left(\sqrt[3]{d}-\sqrt[3]{d} x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+d^{2/3} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 d^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{2-2 x+\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{2/3}}-\frac{\log \left(-1+x+\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{d^{2/3}}+\frac{\log \left(1-2 x+x^2+\left(\sqrt[3]{d}-\sqrt[3]{d} x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+d^{2/3} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 d^{2/3}}",1,"-((Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))/(2 - 2*x + d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))])/d^(2/3)) - Log[-1 + x + d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3)]/d^(2/3) + Log[1 - 2*x + x^2 + (d^(1/3) - d^(1/3)*x)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3) + d^(2/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3)]/(2*d^(2/3))","A",1
2546,1,214,214,0.4677288,"\int \frac{\left(1+x^3\right)^{2/3} \left(8-4 x^3+x^6\right)}{x^6 \left(2+x^3\right)} \, dx","IntegrateAlgebraic[((1 + x^3)^(2/3)*(8 - 4*x^3 + x^6))/(x^6*(2 + x^3)),x]","\frac{2 \left(1+x^3\right)^{2/3} \left(-2+3 x^3\right)}{5 x^5}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3}}\right)}{\sqrt{3}}-\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2} \sqrt[3]{1+x^3}}\right)}{2^{2/3} \sqrt{3}}-\frac{1}{3} \log \left(-x+\sqrt[3]{1+x^3}\right)+\frac{5 \log \left(-x+\sqrt[3]{2} \sqrt[3]{1+x^3}\right)}{3\ 2^{2/3}}+\frac{1}{6} \log \left(x^2+x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)-\frac{5 \log \left(x^2+\sqrt[3]{2} x \sqrt[3]{1+x^3}+2^{2/3} \left(1+x^3\right)^{2/3}\right)}{6\ 2^{2/3}}","\frac{2 \left(1+x^3\right)^{2/3} \left(-2+3 x^3\right)}{5 x^5}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{1+x^3}}\right)}{\sqrt{3}}-\frac{5 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2} \sqrt[3]{1+x^3}}\right)}{2^{2/3} \sqrt{3}}-\frac{1}{3} \log \left(-x+\sqrt[3]{1+x^3}\right)+\frac{5 \log \left(-x+\sqrt[3]{2} \sqrt[3]{1+x^3}\right)}{3\ 2^{2/3}}+\frac{1}{6} \log \left(x^2+x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)-\frac{5 \log \left(x^2+\sqrt[3]{2} x \sqrt[3]{1+x^3}+2^{2/3} \left(1+x^3\right)^{2/3}\right)}{6\ 2^{2/3}}",1,"(2*(1 + x^3)^(2/3)*(-2 + 3*x^3))/(5*x^5) + ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^3)^(1/3))]/Sqrt[3] - (5*ArcTan[(Sqrt[3]*x)/(x + 2*2^(1/3)*(1 + x^3)^(1/3))])/(2^(2/3)*Sqrt[3]) - Log[-x + (1 + x^3)^(1/3)]/3 + (5*Log[-x + 2^(1/3)*(1 + x^3)^(1/3)])/(3*2^(2/3)) + Log[x^2 + x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/6 - (5*Log[x^2 + 2^(1/3)*x*(1 + x^3)^(1/3) + 2^(2/3)*(1 + x^3)^(2/3)])/(6*2^(2/3))","A",1
2547,1,214,214,0.4454876,"\int \frac{\left(-1+x^3\right)^{2/3} \left(8+2 x^3+x^6\right)}{x^6 \left(-2+x^3\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(8 + 2*x^3 + x^6))/(x^6*(-2 + x^3)),x]","\frac{\left(-1+x^3\right)^{2/3} \left(8+7 x^3\right)}{10 x^5}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{\sqrt{3}}-\frac{2 \sqrt[3]{2} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2} \sqrt[3]{-1+x^3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(-x+\sqrt[3]{-1+x^3}\right)+\frac{2}{3} \sqrt[3]{2} \log \left(-x+\sqrt[3]{2} \sqrt[3]{-1+x^3}\right)+\frac{1}{6} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)-\frac{1}{3} \sqrt[3]{2} \log \left(x^2+\sqrt[3]{2} x \sqrt[3]{-1+x^3}+2^{2/3} \left(-1+x^3\right)^{2/3}\right)","\frac{\left(-1+x^3\right)^{2/3} \left(8+7 x^3\right)}{10 x^5}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{\sqrt{3}}-\frac{2 \sqrt[3]{2} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2} \sqrt[3]{-1+x^3}}\right)}{\sqrt{3}}-\frac{1}{3} \log \left(-x+\sqrt[3]{-1+x^3}\right)+\frac{2}{3} \sqrt[3]{2} \log \left(-x+\sqrt[3]{2} \sqrt[3]{-1+x^3}\right)+\frac{1}{6} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)-\frac{1}{3} \sqrt[3]{2} \log \left(x^2+\sqrt[3]{2} x \sqrt[3]{-1+x^3}+2^{2/3} \left(-1+x^3\right)^{2/3}\right)",1,"((-1 + x^3)^(2/3)*(8 + 7*x^3))/(10*x^5) + ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^3)^(1/3))]/Sqrt[3] - (2*2^(1/3)*ArcTan[(Sqrt[3]*x)/(x + 2*2^(1/3)*(-1 + x^3)^(1/3))])/Sqrt[3] - Log[-x + (-1 + x^3)^(1/3)]/3 + (2*2^(1/3)*Log[-x + 2^(1/3)*(-1 + x^3)^(1/3)])/3 + Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)]/6 - (2^(1/3)*Log[x^2 + 2^(1/3)*x*(-1 + x^3)^(1/3) + 2^(2/3)*(-1 + x^3)^(2/3)])/3","A",1
2548,1,214,214,0.7074674,"\int \frac{\left(-2 q+p x^3\right) \sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left(c x^4+b x^2 \left(q+p x^3\right)+a \left(q+p x^3\right)^2\right)}{x^9} \, dx","IntegrateAlgebraic[((-2*q + p*x^3)*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]*(c*x^4 + b*x^2*(q + p*x^3) + a*(q + p*x^3)^2))/x^9,x]","\frac{\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left(3 a q^3+4 b q^2 x^2+9 a p q^2 x^3+6 c q x^4-3 a p q^2 x^4+8 b p q x^5-8 b p q x^6+9 a p^2 q x^6+6 c p x^7-3 a p^2 q x^7+4 b p^2 x^8+3 a p^3 x^9\right)}{12 x^8}+\left(2 c p q+a p^2 q^2\right) \log (x)+\frac{1}{2} \left(-2 c p q-a p^2 q^2\right) \log \left(q+p x^3+\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}\right)","\frac{\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left(3 a q^3+4 b q^2 x^2+9 a p q^2 x^3+6 c q x^4-3 a p q^2 x^4+8 b p q x^5-8 b p q x^6+9 a p^2 q x^6+6 c p x^7-3 a p^2 q x^7+4 b p^2 x^8+3 a p^3 x^9\right)}{12 x^8}+\left(2 c p q+a p^2 q^2\right) \log (x)+\frac{1}{2} \left(-2 c p q-a p^2 q^2\right) \log \left(q+p x^3+\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}\right)",1,"(Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]*(3*a*q^3 + 4*b*q^2*x^2 + 9*a*p*q^2*x^3 + 6*c*q*x^4 - 3*a*p*q^2*x^4 + 8*b*p*q*x^5 - 8*b*p*q*x^6 + 9*a*p^2*q*x^6 + 6*c*p*x^7 - 3*a*p^2*q*x^7 + 4*b*p^2*x^8 + 3*a*p^3*x^9))/(12*x^8) + (2*c*p*q + a*p^2*q^2)*Log[x] + ((-2*c*p*q - a*p^2*q^2)*Log[q + p*x^3 + Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]])/2","A",1
2549,1,215,215,0.5979961,"\int \frac{-b+a x^2}{\sqrt[3]{b^2 x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[(-b + a*x^2)/(b^2*x^2 + a^3*x^3)^(1/3),x]","\frac{\left(-4 b^2+3 a^3 x\right) \left(b^2 x^2+a^3 x^3\right)^{2/3}}{6 a^5 x}-\frac{\left(9 a^5 b-2 b^4\right) \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{3 \sqrt{3} a^6}+\frac{\left(9 a^5 b-2 b^4\right) \log \left(-a x+\sqrt[3]{b^2 x^2+a^3 x^3}\right)}{9 a^6}+\frac{\left(-9 a^5 b+2 b^4\right) \log \left(a^2 x^2+a x \sqrt[3]{b^2 x^2+a^3 x^3}+\left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{18 a^6}","\frac{\left(-4 b^2+3 a^3 x\right) \left(b^2 x^2+a^3 x^3\right)^{2/3}}{6 a^5 x}-\frac{\left(9 a^5 b-2 b^4\right) \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{3 \sqrt{3} a^6}+\frac{\left(9 a^5 b-2 b^4\right) \log \left(-a x+\sqrt[3]{b^2 x^2+a^3 x^3}\right)}{9 a^6}+\frac{\left(-9 a^5 b+2 b^4\right) \log \left(a^2 x^2+a x \sqrt[3]{b^2 x^2+a^3 x^3}+\left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{18 a^6}",1,"((-4*b^2 + 3*a^3*x)*(b^2*x^2 + a^3*x^3)^(2/3))/(6*a^5*x) - ((9*a^5*b - 2*b^4)*ArcTan[(Sqrt[3]*a*x)/(a*x + 2*(b^2*x^2 + a^3*x^3)^(1/3))])/(3*Sqrt[3]*a^6) + ((9*a^5*b - 2*b^4)*Log[-(a*x) + (b^2*x^2 + a^3*x^3)^(1/3)])/(9*a^6) + ((-9*a^5*b + 2*b^4)*Log[a^2*x^2 + a*x*(b^2*x^2 + a^3*x^3)^(1/3) + (b^2*x^2 + a^3*x^3)^(2/3)])/(18*a^6)","A",1
2550,1,221,215,0.6102062,"\int \frac{b+a x^2}{\sqrt[3]{b^2 x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[(b + a*x^2)/(b^2*x^2 + a^3*x^3)^(1/3),x]","\frac{\left(-4 b^2+3 a^3 x\right) \left(b^2 x^2+a^3 x^3\right)^{2/3}}{6 a^5 x}+\frac{\left(9 a^5 b+2 b^4\right) \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{3 \sqrt{3} a^6}+\frac{\left(-9 a^5 b-2 b^4\right) \log \left(a^7 x-a^6 \sqrt[3]{b^2 x^2+a^3 x^3}\right)}{9 a^6}+\frac{\left(9 a^5 b+2 b^4\right) \log \left(a^2 x^2+a x \sqrt[3]{b^2 x^2+a^3 x^3}+\left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{18 a^6}","\frac{\left(-4 b^2+3 a^3 x\right) \left(b^2 x^2+a^3 x^3\right)^{2/3}}{6 a^5 x}+\frac{\left(9 a^5 b+2 b^4\right) \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{3 \sqrt{3} a^6}+\frac{\left(-9 a^5 b-2 b^4\right) \log \left(-a x+\sqrt[3]{b^2 x^2+a^3 x^3}\right)}{9 a^6}+\frac{\left(9 a^5 b+2 b^4\right) \log \left(a^2 x^2+a x \sqrt[3]{b^2 x^2+a^3 x^3}+\left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{18 a^6}",1,"((-4*b^2 + 3*a^3*x)*(b^2*x^2 + a^3*x^3)^(2/3))/(6*a^5*x) + ((9*a^5*b + 2*b^4)*ArcTan[(Sqrt[3]*a*x)/(a*x + 2*(b^2*x^2 + a^3*x^3)^(1/3))])/(3*Sqrt[3]*a^6) + ((-9*a^5*b - 2*b^4)*Log[a^7*x - a^6*(b^2*x^2 + a^3*x^3)^(1/3)])/(9*a^6) + ((9*a^5*b + 2*b^4)*Log[a^2*x^2 + a*x*(b^2*x^2 + a^3*x^3)^(1/3) + (b^2*x^2 + a^3*x^3)^(2/3)])/(18*a^6)","A",1
2551,1,215,215,0.5442568,"\int \frac{-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3}{\sqrt[4]{x (-a+x)^2 (-b+x)^3} \left(b+\left(-1+a^2 d\right) x-2 a d x^2+d x^3\right)} \, dx","IntegrateAlgebraic[(-(a^2*b) + 4*a*b*x - (2*a + 3*b)*x^2 + 2*x^3)/((x*(-a + x)^2*(-b + x)^3)^(1/4)*(b + (-1 + a^2*d)*x - 2*a*d*x^2 + d*x^3)),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a^2 b^3 x+\left(3 a^2 b^2+2 a b^3\right) x^2+\left(-3 a^2 b-6 a b^2-b^3\right) x^3+\left(a^2+6 a b+3 b^2\right) x^4+(-2 a-3 b) x^5+x^6}}{b-x}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a^2 b^3 x+\left(3 a^2 b^2+2 a b^3\right) x^2+\left(-3 a^2 b-6 a b^2-b^3\right) x^3+\left(a^2+6 a b+3 b^2\right) x^4+(-2 a-3 b) x^5+x^6}}{b-x}\right)}{d^{3/4}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a^2 b^3 x+\left(3 a^2 b^2+2 a b^3\right) x^2+\left(-3 a^2 b-6 a b^2-b^3\right) x^3+\left(a^2+6 a b+3 b^2\right) x^4+(-2 a-3 b) x^5+x^6}}{b-x}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a^2 b^3 x+\left(3 a^2 b^2+2 a b^3\right) x^2+\left(-3 a^2 b-6 a b^2-b^3\right) x^3+\left(a^2+6 a b+3 b^2\right) x^4+(-2 a-3 b) x^5+x^6}}{b-x}\right)}{d^{3/4}}",1,"(-2*ArcTan[(d^(1/4)*(-(a^2*b^3*x) + (3*a^2*b^2 + 2*a*b^3)*x^2 + (-3*a^2*b - 6*a*b^2 - b^3)*x^3 + (a^2 + 6*a*b + 3*b^2)*x^4 + (-2*a - 3*b)*x^5 + x^6)^(1/4))/(b - x)])/d^(3/4) + (2*ArcTanh[(d^(1/4)*(-(a^2*b^3*x) + (3*a^2*b^2 + 2*a*b^3)*x^2 + (-3*a^2*b - 6*a*b^2 - b^3)*x^3 + (a^2 + 6*a*b + 3*b^2)*x^4 + (-2*a - 3*b)*x^5 + x^6)^(1/4))/(b - x)])/d^(3/4)","A",1
2552,1,214,215,1.8125036,"\int \frac{\left(b+a x^2\right) \sqrt[4]{b x^3+a x^4}}{x^2 \left(-b+a x^2\right)} \, dx","IntegrateAlgebraic[((b + a*x^2)*(b*x^3 + a*x^4)^(1/4))/(x^2*(-b + a*x^2)),x]","\frac{4 \sqrt[4]{b x^3+a x^4}}{x}-2 \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)+2 \sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)+a \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a \log (x)-b \log (x)-a \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)+b \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}^3+\text{$\#$1}^7}\&\right]","\frac{4 \sqrt[4]{b x^3+a x^4}}{x}-2 \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)+2 \sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)+a \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a \log (x)+b \log (x)+a \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)-b \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}^3-\text{$\#$1}^7}\&\right]",1,"(4*(b*x^3 + a*x^4)^(1/4))/x - 2*a^(1/4)*ArcTan[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)] + 2*a^(1/4)*ArcTanh[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)] + a*RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (a*Log[x] - b*Log[x] - a*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] + b*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] - Log[x]*#1^4 + Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3) + #1^7) & ]","A",1
2553,1,215,215,0.5948509,"\int \frac{a b^3-(6 a-b) b^2 x+9 a b x^2-(4 a+3 b) x^3+2 x^4}{\sqrt[4]{x (-a+x)^2 (-b+x)^3} \left(-a^2+\left(2 a-b^3 d\right) x+\left(-1+3 b^2 d\right) x^2-3 b d x^3+d x^4\right)} \, dx","IntegrateAlgebraic[(a*b^3 - (6*a - b)*b^2*x + 9*a*b*x^2 - (4*a + 3*b)*x^3 + 2*x^4)/((x*(-a + x)^2*(-b + x)^3)^(1/4)*(-a^2 + (2*a - b^3*d)*x + (-1 + 3*b^2*d)*x^2 - 3*b*d*x^3 + d*x^4)),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a^2 b^3 x+\left(3 a^2 b^2+2 a b^3\right) x^2+\left(-3 a^2 b-6 a b^2-b^3\right) x^3+\left(a^2+6 a b+3 b^2\right) x^4+(-2 a-3 b) x^5+x^6}}{a-x}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a^2 b^3 x+\left(3 a^2 b^2+2 a b^3\right) x^2+\left(-3 a^2 b-6 a b^2-b^3\right) x^3+\left(a^2+6 a b+3 b^2\right) x^4+(-2 a-3 b) x^5+x^6}}{a-x}\right)}{d^{3/4}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a^2 b^3 x+\left(3 a^2 b^2+2 a b^3\right) x^2+\left(-3 a^2 b-6 a b^2-b^3\right) x^3+\left(a^2+6 a b+3 b^2\right) x^4+(-2 a-3 b) x^5+x^6}}{a-x}\right)}{d^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{-a^2 b^3 x+\left(3 a^2 b^2+2 a b^3\right) x^2+\left(-3 a^2 b-6 a b^2-b^3\right) x^3+\left(a^2+6 a b+3 b^2\right) x^4+(-2 a-3 b) x^5+x^6}}{a-x}\right)}{d^{3/4}}",1,"(-2*ArcTan[(d^(1/4)*(-(a^2*b^3*x) + (3*a^2*b^2 + 2*a*b^3)*x^2 + (-3*a^2*b - 6*a*b^2 - b^3)*x^3 + (a^2 + 6*a*b + 3*b^2)*x^4 + (-2*a - 3*b)*x^5 + x^6)^(1/4))/(a - x)])/d^(3/4) + (2*ArcTanh[(d^(1/4)*(-(a^2*b^3*x) + (3*a^2*b^2 + 2*a*b^3)*x^2 + (-3*a^2*b - 6*a*b^2 - b^3)*x^3 + (a^2 + 6*a*b + 3*b^2)*x^4 + (-2*a - 3*b)*x^5 + x^6)^(1/4))/(a - x)])/d^(3/4)","A",1
2554,1,214,215,0.5295081,"\int \frac{\sqrt[4]{-b+a x^4} \left(b+c x^4+a x^8\right)}{x^6 \left(b+2 a x^8\right)} \, dx","IntegrateAlgebraic[((-b + a*x^4)^(1/4)*(b + c*x^4 + a*x^8))/(x^6*(b + 2*a*x^8)),x]","\frac{\sqrt[4]{-b+a x^4} \left(-b+a x^4-5 c x^4\right)}{5 b x^5}-\frac{a \text{RootSum}\left[a^2+2 a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a c \log (x)+2 b c \log (x)-a c \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-2 b c \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-b \log (x) \text{$\#$1}^4-c \log (x) \text{$\#$1}^4+b \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+c \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}^3+\text{$\#$1}^7}\&\right]}{8 b}","\frac{\sqrt[4]{-b+a x^4} \left(-b+a x^4-5 c x^4\right)}{5 b x^5}-\frac{a \text{RootSum}\left[a^2+2 a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a c \log (x)-2 b c \log (x)+a c \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+2 b c \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+b \log (x) \text{$\#$1}^4+c \log (x) \text{$\#$1}^4-b \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-c \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}^3-\text{$\#$1}^7}\&\right]}{8 b}",1,"((-b + a*x^4)^(1/4)*(-b + a*x^4 - 5*c*x^4))/(5*b*x^5) - (a*RootSum[a^2 + 2*a*b - 2*a*#1^4 + #1^8 & , (a*c*Log[x] + 2*b*c*Log[x] - a*c*Log[(-b + a*x^4)^(1/4) - x*#1] - 2*b*c*Log[(-b + a*x^4)^(1/4) - x*#1] - b*Log[x]*#1^4 - c*Log[x]*#1^4 + b*Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4 + c*Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3) + #1^7) & ])/(8*b)","A",1
2555,1,214,215,0.0015604,"\int \frac{\sqrt[4]{-b+a x^4} \left(b+c x^4+a x^8\right)}{x^6 \left(b+2 a x^8\right)} \, dx","IntegrateAlgebraic[((-b + a*x^4)^(1/4)*(b + c*x^4 + a*x^8))/(x^6*(b + 2*a*x^8)),x]","\frac{\sqrt[4]{-b+a x^4} \left(-b+a x^4-5 c x^4\right)}{5 b x^5}-\frac{a \text{RootSum}\left[a^2+2 a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a c \log (x)+2 b c \log (x)-a c \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-2 b c \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-b \log (x) \text{$\#$1}^4-c \log (x) \text{$\#$1}^4+b \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+c \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}^3+\text{$\#$1}^7}\&\right]}{8 b}","\frac{\sqrt[4]{-b+a x^4} \left(-b+a x^4-5 c x^4\right)}{5 b x^5}-\frac{a \text{RootSum}\left[a^2+2 a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a c \log (x)-2 b c \log (x)+a c \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+2 b c \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+b \log (x) \text{$\#$1}^4+c \log (x) \text{$\#$1}^4-b \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-c \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}^3-\text{$\#$1}^7}\&\right]}{8 b}",1,"((-b + a*x^4)^(1/4)*(-b + a*x^4 - 5*c*x^4))/(5*b*x^5) - (a*RootSum[a^2 + 2*a*b - 2*a*#1^4 + #1^8 & , (a*c*Log[x] + 2*b*c*Log[x] - a*c*Log[(-b + a*x^4)^(1/4) - x*#1] - 2*b*c*Log[(-b + a*x^4)^(1/4) - x*#1] - b*Log[x]*#1^4 - c*Log[x]*#1^4 + b*Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4 + c*Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3) + #1^7) & ])/(8*b)","A",1
2556,1,215,215,0.6646809,"\int \frac{1}{(d+c x) \sqrt{a x+\sqrt{b^2+a^2 x^2}}} \, dx","IntegrateAlgebraic[1/((d + c*x)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]),x]","\frac{2}{c \sqrt{a x+\sqrt{b^2+a^2 x^2}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{a d-\sqrt{b^2 c^2+a^2 d^2}}}\right)}{\sqrt{c} \sqrt{a d-\sqrt{b^2 c^2+a^2 d^2}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{a d+\sqrt{b^2 c^2+a^2 d^2}}}\right)}{\sqrt{c} \sqrt{a d+\sqrt{b^2 c^2+a^2 d^2}}}","\frac{2}{c \sqrt{a x+\sqrt{b^2+a^2 x^2}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{a d-\sqrt{b^2 c^2+a^2 d^2}}}\right)}{\sqrt{c} \sqrt{a d-\sqrt{b^2 c^2+a^2 d^2}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{a d+\sqrt{b^2 c^2+a^2 d^2}}}\right)}{\sqrt{c} \sqrt{a d+\sqrt{b^2 c^2+a^2 d^2}}}",1,"2/(c*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]) + (2*ArcTan[(Sqrt[c]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/Sqrt[a*d - Sqrt[b^2*c^2 + a^2*d^2]]])/(Sqrt[c]*Sqrt[a*d - Sqrt[b^2*c^2 + a^2*d^2]]) + (2*ArcTan[(Sqrt[c]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/Sqrt[a*d + Sqrt[b^2*c^2 + a^2*d^2]]])/(Sqrt[c]*Sqrt[a*d + Sqrt[b^2*c^2 + a^2*d^2]])","A",1
2557,1,215,215,0.644199,"\int \frac{\left(-q+2 p x^3\right) \sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left(b q x+c x^2+b p x^4+a \left(q+p x^3\right)^2\right)}{x^5} \, dx","IntegrateAlgebraic[((-q + 2*p*x^3)*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]*(b*q*x + c*x^2 + b*p*x^4 + a*(q + p*x^3)^2))/x^5,x]","\frac{\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left(3 a q^3+4 b q^2 x+6 c q x^2-3 a p q^2 x^2-8 b p q x^3+9 a p q^2 x^3+8 b p q x^4+6 c p x^5-3 a p^2 q x^5+9 a p^2 q x^6+4 b p^2 x^7+3 a p^3 x^9\right)}{12 x^4}+\frac{1}{2} \left(2 c p q+a p^2 q^2\right) \log (x)+\frac{1}{2} \left(-2 c p q-a p^2 q^2\right) \log \left(q+p x^3+\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right)","\frac{\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left(3 a q^3+4 b q^2 x+6 c q x^2-3 a p q^2 x^2-8 b p q x^3+9 a p q^2 x^3+8 b p q x^4+6 c p x^5-3 a p^2 q x^5+9 a p^2 q x^6+4 b p^2 x^7+3 a p^3 x^9\right)}{12 x^4}+\frac{1}{2} \left(2 c p q+a p^2 q^2\right) \log (x)+\frac{1}{2} \left(-2 c p q-a p^2 q^2\right) \log \left(q+p x^3+\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right)",1,"(Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]*(3*a*q^3 + 4*b*q^2*x + 6*c*q*x^2 - 3*a*p*q^2*x^2 - 8*b*p*q*x^3 + 9*a*p*q^2*x^3 + 8*b*p*q*x^4 + 6*c*p*x^5 - 3*a*p^2*q*x^5 + 9*a*p^2*q*x^6 + 4*b*p^2*x^7 + 3*a*p^3*x^9))/(12*x^4) + ((2*c*p*q + a*p^2*q^2)*Log[x])/2 + ((-2*c*p*q - a*p^2*q^2)*Log[q + p*x^3 + Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]])/2","A",1
2558,1,216,216,0.6290733,"\int \frac{-a (a-2 b)-2 b x+x^2}{((-a+x) (-b+x))^{2/3} \left(a^2+b d-(2 a+d) x+x^2\right)} \, dx","IntegrateAlgebraic[(-(a*(a - 2*b)) - 2*b*x + x^2)/(((-a + x)*(-b + x))^(2/3)*(a^2 + b*d - (2*a + d)*x + x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \left(a b+(-a-b) x+x^2\right)^{2/3}}{-2 b \sqrt[3]{d}+2 \sqrt[3]{d} x+\left(a b+(-a-b) x+x^2\right)^{2/3}}\right)}{\sqrt[3]{d}}+\frac{\log \left(b \sqrt{d}-\sqrt{d} x+\sqrt[6]{d} \left(a b+(-a-b) x+x^2\right)^{2/3}\right)}{\sqrt[3]{d}}-\frac{\log \left(b^2 d-2 b d x+d x^2+\left(-b d^{2/3}+d^{2/3} x\right) \left(a b+(-a-b) x+x^2\right)^{2/3}+\sqrt[3]{d} \left(a b+(-a-b) x+x^2\right)^{4/3}\right)}{2 \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \left(a b+(-a-b) x+x^2\right)^{2/3}}{-2 b \sqrt[3]{d}+2 \sqrt[3]{d} x+\left(a b+(-a-b) x+x^2\right)^{2/3}}\right)}{\sqrt[3]{d}}+\frac{\log \left(b \sqrt{d}-\sqrt{d} x+\sqrt[6]{d} \left(a b+(-a-b) x+x^2\right)^{2/3}\right)}{\sqrt[3]{d}}-\frac{\log \left(b^2 d-2 b d x+d x^2+\left(-b d^{2/3}+d^{2/3} x\right) \left(a b+(-a-b) x+x^2\right)^{2/3}+\sqrt[3]{d} \left(a b+(-a-b) x+x^2\right)^{4/3}\right)}{2 \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(a*b + (-a - b)*x + x^2)^(2/3))/(-2*b*d^(1/3) + 2*d^(1/3)*x + (a*b + (-a - b)*x + x^2)^(2/3))])/d^(1/3) + Log[b*Sqrt[d] - Sqrt[d]*x + d^(1/6)*(a*b + (-a - b)*x + x^2)^(2/3)]/d^(1/3) - Log[b^2*d - 2*b*d*x + d*x^2 + (-(b*d^(2/3)) + d^(2/3)*x)*(a*b + (-a - b)*x + x^2)^(2/3) + d^(1/3)*(a*b + (-a - b)*x + x^2)^(4/3)]/(2*d^(1/3))","A",1
2559,1,216,216,0.5438765,"\int \frac{a-2 b+x}{\sqrt[3]{(-a+x) (-b+x)} \left(b+a^2 d+(-1-2 a d) x+d x^2\right)} \, dx","IntegrateAlgebraic[(a - 2*b + x)/(((-a + x)*(-b + x))^(1/3)*(b + a^2*d + (-1 - 2*a*d)*x + d*x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a b+(-a-b) x+x^2}}{-2 a \sqrt[3]{d}+2 \sqrt[3]{d} x+\sqrt[3]{a b+(-a-b) x+x^2}}\right)}{\sqrt[3]{d}}+\frac{\log \left(a \sqrt[3]{d}-\sqrt[3]{d} x+\sqrt[3]{a b+(-a-b) x+x^2}\right)}{\sqrt[3]{d}}-\frac{\log \left(a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{a b+(-a-b) x+x^2}+\left(a b+(-a-b) x+x^2\right)^{2/3}\right)}{2 \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a b+(-a-b) x+x^2}}{-2 a \sqrt[3]{d}+2 \sqrt[3]{d} x+\sqrt[3]{a b+(-a-b) x+x^2}}\right)}{\sqrt[3]{d}}+\frac{\log \left(a \sqrt[3]{d}-\sqrt[3]{d} x+\sqrt[3]{a b+(-a-b) x+x^2}\right)}{\sqrt[3]{d}}-\frac{\log \left(a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{a b+(-a-b) x+x^2}+\left(a b+(-a-b) x+x^2\right)^{2/3}\right)}{2 \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(a*b + (-a - b)*x + x^2)^(1/3))/(-2*a*d^(1/3) + 2*d^(1/3)*x + (a*b + (-a - b)*x + x^2)^(1/3))])/d^(1/3) + Log[a*d^(1/3) - d^(1/3)*x + (a*b + (-a - b)*x + x^2)^(1/3)]/d^(1/3) - Log[a^2*d^(2/3) - 2*a*d^(2/3)*x + d^(2/3)*x^2 + (-(a*d^(1/3)) + d^(1/3)*x)*(a*b + (-a - b)*x + x^2)^(1/3) + (a*b + (-a - b)*x + x^2)^(2/3)]/(2*d^(1/3))","A",1
2560,1,216,216,0.5081298,"\int \frac{a-2 b+x}{\sqrt[3]{(-a+x) (-b+x)} \left(b+a^2 d-(1+2 a d) x+d x^2\right)} \, dx","IntegrateAlgebraic[(a - 2*b + x)/(((-a + x)*(-b + x))^(1/3)*(b + a^2*d - (1 + 2*a*d)*x + d*x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a b+(-a-b) x+x^2}}{-2 a \sqrt[3]{d}+2 \sqrt[3]{d} x+\sqrt[3]{a b+(-a-b) x+x^2}}\right)}{\sqrt[3]{d}}+\frac{\log \left(a \sqrt[3]{d}-\sqrt[3]{d} x+\sqrt[3]{a b+(-a-b) x+x^2}\right)}{\sqrt[3]{d}}-\frac{\log \left(a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{a b+(-a-b) x+x^2}+\left(a b+(-a-b) x+x^2\right)^{2/3}\right)}{2 \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a b+(-a-b) x+x^2}}{-2 a \sqrt[3]{d}+2 \sqrt[3]{d} x+\sqrt[3]{a b+(-a-b) x+x^2}}\right)}{\sqrt[3]{d}}+\frac{\log \left(a \sqrt[3]{d}-\sqrt[3]{d} x+\sqrt[3]{a b+(-a-b) x+x^2}\right)}{\sqrt[3]{d}}-\frac{\log \left(a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{a b+(-a-b) x+x^2}+\left(a b+(-a-b) x+x^2\right)^{2/3}\right)}{2 \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(a*b + (-a - b)*x + x^2)^(1/3))/(-2*a*d^(1/3) + 2*d^(1/3)*x + (a*b + (-a - b)*x + x^2)^(1/3))])/d^(1/3) + Log[a*d^(1/3) - d^(1/3)*x + (a*b + (-a - b)*x + x^2)^(1/3)]/d^(1/3) - Log[a^2*d^(2/3) - 2*a*d^(2/3)*x + d^(2/3)*x^2 + (-(a*d^(1/3)) + d^(1/3)*x)*(a*b + (-a - b)*x + x^2)^(1/3) + (a*b + (-a - b)*x + x^2)^(2/3)]/(2*d^(1/3))","A",1
2561,1,217,217,1.6848984,"\int \frac{-1+2 k x+(1-2 k) x^2}{((1-x) x (1-k x))^{2/3} \left(b-(1+2 b) x+(b+k) x^2\right)} \, dx","IntegrateAlgebraic[(-1 + 2*k*x + (1 - 2*k)*x^2)/(((1 - x)*x*(1 - k*x))^(2/3)*(b - (1 + 2*b)*x + (b + k)*x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}}{2 x-2 k x^2+\sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}}\right)}{b^{2/3}}+\frac{\log \left(-x+k x^2+\sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{b^{2/3}}-\frac{\log \left(x^2-2 k x^3+k^2 x^4+\left(\sqrt[3]{b} x-\sqrt[3]{b} k x^2\right) \left(x+(-1-k) x^2+k x^3\right)^{2/3}+b^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{4/3}\right)}{2 b^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}}{2 x-2 k x^2+\sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}}\right)}{b^{2/3}}+\frac{\log \left(-x+k x^2+\sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{b^{2/3}}-\frac{\log \left(x^2-2 k x^3+k^2 x^4+\left(\sqrt[3]{b} x-\sqrt[3]{b} k x^2\right) \left(x+(-1-k) x^2+k x^3\right)^{2/3}+b^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{4/3}\right)}{2 b^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3))/(2*x - 2*k*x^2 + b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3))])/b^(2/3) + Log[-x + k*x^2 + b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3)]/b^(2/3) - Log[x^2 - 2*k*x^3 + k^2*x^4 + (b^(1/3)*x - b^(1/3)*k*x^2)*(x + (-1 - k)*x^2 + k*x^3)^(2/3) + b^(2/3)*(x + (-1 - k)*x^2 + k*x^3)^(4/3)]/(2*b^(2/3))","A",1
2562,1,217,217,0.5143253,"\int \frac{-1+x}{(1+x) \sqrt[3]{-x^2+x^3}} \, dx","IntegrateAlgebraic[(-1 + x)/((1 + x)*(-x^2 + x^3)^(1/3)),x]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)-2^{2/3} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{-x^2+x^3}\right)+2^{2/3} \log \left(-2 x+2^{2/3} \sqrt[3]{-x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-x^2+x^3}+\sqrt[3]{2} \left(-x^2+x^3\right)^{2/3}\right)}{\sqrt[3]{2}}","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)-2^{2/3} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{-x^2+x^3}\right)+2^{2/3} \log \left(-2 x+2^{2/3} \sqrt[3]{-x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-x^2+x^3}+\sqrt[3]{2} \left(-x^2+x^3\right)^{2/3}\right)}{\sqrt[3]{2}}",1,"Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-x^2 + x^3)^(1/3))] - 2^(2/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-x^2 + x^3)^(1/3))] - Log[-x + (-x^2 + x^3)^(1/3)] + 2^(2/3)*Log[-2*x + 2^(2/3)*(-x^2 + x^3)^(1/3)] + Log[x^2 + x*(-x^2 + x^3)^(1/3) + (-x^2 + x^3)^(2/3)]/2 - Log[2*x^2 + 2^(2/3)*x*(-x^2 + x^3)^(1/3) + 2^(1/3)*(-x^2 + x^3)^(2/3)]/2^(1/3)","A",1
2563,1,217,217,0.7472757,"\int \frac{\left(-1+x+x^3+x^6\right)^{2/3} \left(3-2 x+3 x^6\right)}{\left(-1+x+x^6\right) \left(-1+x-x^3+x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x + x^3 + x^6)^(2/3)*(3 - 2*x + 3*x^6))/((-1 + x + x^6)*(-1 + x - x^3 + x^6)),x]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x+x^3+x^6}}\right)-2^{2/3} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-1+x+x^3+x^6}}\right)-\log \left(-x+\sqrt[3]{-1+x+x^3+x^6}\right)+2^{2/3} \log \left(-2 x+2^{2/3} \sqrt[3]{-1+x+x^3+x^6}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{-1+x+x^3+x^6}+\left(-1+x+x^3+x^6\right)^{2/3}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-1+x+x^3+x^6}+\sqrt[3]{2} \left(-1+x+x^3+x^6\right)^{2/3}\right)}{\sqrt[3]{2}}","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x+x^3+x^6}}\right)-2^{2/3} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-1+x+x^3+x^6}}\right)-\log \left(-x+\sqrt[3]{-1+x+x^3+x^6}\right)+2^{2/3} \log \left(-2 x+2^{2/3} \sqrt[3]{-1+x+x^3+x^6}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{-1+x+x^3+x^6}+\left(-1+x+x^3+x^6\right)^{2/3}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-1+x+x^3+x^6}+\sqrt[3]{2} \left(-1+x+x^3+x^6\right)^{2/3}\right)}{\sqrt[3]{2}}",1,"Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x + x^3 + x^6)^(1/3))] - 2^(2/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-1 + x + x^3 + x^6)^(1/3))] - Log[-x + (-1 + x + x^3 + x^6)^(1/3)] + 2^(2/3)*Log[-2*x + 2^(2/3)*(-1 + x + x^3 + x^6)^(1/3)] + Log[x^2 + x*(-1 + x + x^3 + x^6)^(1/3) + (-1 + x + x^3 + x^6)^(2/3)]/2 - Log[2*x^2 + 2^(2/3)*x*(-1 + x + x^3 + x^6)^(1/3) + 2^(1/3)*(-1 + x + x^3 + x^6)^(2/3)]/2^(1/3)","A",1
2564,1,217,217,0.7136862,"\int \frac{-1-2 x^4+2 x^8}{\sqrt[4]{-1+x^4} \left(-1-x^4+x^8\right)} \, dx","IntegrateAlgebraic[(-1 - 2*x^4 + 2*x^8)/((-1 + x^4)^(1/4)*(-1 - x^4 + x^8)),x]","\tan ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(-1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(-1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)","\tan ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(-1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(-1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-1+x^4}}\right)",1,"ArcTan[x/(-1 + x^4)^(1/4)] - (Sqrt[(-1 + Sqrt[5])/10]*ArcTan[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(-1 + x^4)^(1/4)])/2 - (Sqrt[(1 + Sqrt[5])/10]*ArcTan[(Sqrt[1/2 + Sqrt[5]/2]*x)/(-1 + x^4)^(1/4)])/2 + ArcTanh[x/(-1 + x^4)^(1/4)] - (Sqrt[(-1 + Sqrt[5])/10]*ArcTanh[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(-1 + x^4)^(1/4)])/2 - (Sqrt[(1 + Sqrt[5])/10]*ArcTanh[(Sqrt[1/2 + Sqrt[5]/2]*x)/(-1 + x^4)^(1/4)])/2","A",1
2565,1,217,217,0.660375,"\int \frac{-1+2 x^4+2 x^8}{\sqrt[4]{1+x^4} \left(-1+x^4+x^8\right)} \, dx","IntegrateAlgebraic[(-1 + 2*x^4 + 2*x^8)/((1 + x^4)^(1/4)*(-1 + x^4 + x^8)),x]","\tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(-1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(-1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)","\tan ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(-1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(-1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{1+x^4}}\right)",1,"ArcTan[x/(1 + x^4)^(1/4)] - (Sqrt[(-1 + Sqrt[5])/10]*ArcTan[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(1 + x^4)^(1/4)])/2 - (Sqrt[(1 + Sqrt[5])/10]*ArcTan[(Sqrt[1/2 + Sqrt[5]/2]*x)/(1 + x^4)^(1/4)])/2 + ArcTanh[x/(1 + x^4)^(1/4)] - (Sqrt[(-1 + Sqrt[5])/10]*ArcTanh[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(1 + x^4)^(1/4)])/2 - (Sqrt[(1 + Sqrt[5])/10]*ArcTanh[(Sqrt[1/2 + Sqrt[5]/2]*x)/(1 + x^4)^(1/4)])/2","A",1
2566,1,218,218,2.9513791,"\int \frac{-a b+(2 a-b) x}{\sqrt[3]{x (-a+x) (-b+x)} \left(-a^2+(2 a-b d) x+(-1+d) x^2\right)} \, dx","IntegrateAlgebraic[(-(a*b) + (2*a - b)*x)/((x*(-a + x)*(-b + x))^(1/3)*(-a^2 + (2*a - b*d)*x + (-1 + d)*x^2)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}}{-2 a+2 x+\sqrt[3]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}}\right)}{d^{2/3}}-\frac{\log \left(a-x+\sqrt[3]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}\right)}{d^{2/3}}+\frac{\log \left(a^2-2 a x+x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{a b x+(-a-b) x^2+x^3}+d^{2/3} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}\right)}{2 d^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}}{-2 a+2 x+\sqrt[3]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}}\right)}{d^{2/3}}-\frac{\log \left(a-x+\sqrt[3]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}\right)}{d^{2/3}}+\frac{\log \left(a^2-2 a x+x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{a b x+(-a-b) x^2+x^3}+d^{2/3} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}\right)}{2 d^{2/3}}",1,"-((Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(a*b*x + (-a - b)*x^2 + x^3)^(1/3))/(-2*a + 2*x + d^(1/3)*(a*b*x + (-a - b)*x^2 + x^3)^(1/3))])/d^(2/3)) - Log[a - x + d^(1/3)*(a*b*x + (-a - b)*x^2 + x^3)^(1/3)]/d^(2/3) + Log[a^2 - 2*a*x + x^2 + (-(a*d^(1/3)) + d^(1/3)*x)*(a*b*x + (-a - b)*x^2 + x^3)^(1/3) + d^(2/3)*(a*b*x + (-a - b)*x^2 + x^3)^(2/3)]/(2*d^(2/3))","A",1
2567,1,217,218,0.4778236,"\int \frac{\left(-4 b+a x^4\right) \sqrt[4]{-b+a x^4}}{x^6 \left(-8 b+a x^8\right)} \, dx","IntegrateAlgebraic[((-4*b + a*x^4)*(-b + a*x^4)^(1/4))/(x^6*(-8*b + a*x^8)),x]","\frac{\sqrt[4]{-b+a x^4} \left(-4 b+9 a x^4\right)}{40 b x^5}+\frac{a \text{RootSum}\left[8 a^2-a b-16 a \text{$\#$1}^4+8 \text{$\#$1}^8\&,\frac{8 a^2 \log (x)-a b \log (x)-8 a^2 \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+a b \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-8 a \log (x) \text{$\#$1}^4-4 b \log (x) \text{$\#$1}^4+8 a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+4 b \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}^3+\text{$\#$1}^7}\&\right]}{512 b}","\frac{\sqrt[4]{-b+a x^4} \left(-4 b+9 a x^4\right)}{40 b x^5}+\frac{a \text{RootSum}\left[8 a^2-a b-16 a \text{$\#$1}^4+8 \text{$\#$1}^8\&,\frac{-8 a^2 \log (x)+a b \log (x)+8 a^2 \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-a b \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+8 a \log (x) \text{$\#$1}^4+4 b \log (x) \text{$\#$1}^4-8 a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-4 b \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}^3-\text{$\#$1}^7}\&\right]}{512 b}",1,"((-b + a*x^4)^(1/4)*(-4*b + 9*a*x^4))/(40*b*x^5) + (a*RootSum[8*a^2 - a*b - 16*a*#1^4 + 8*#1^8 & , (8*a^2*Log[x] - a*b*Log[x] - 8*a^2*Log[(-b + a*x^4)^(1/4) - x*#1] + a*b*Log[(-b + a*x^4)^(1/4) - x*#1] - 8*a*Log[x]*#1^4 - 4*b*Log[x]*#1^4 + 8*a*Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4 + 4*b*Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3) + #1^7) & ])/(512*b)","A",1
2568,1,217,218,0.0016566,"\int \frac{\left(-4 b+a x^4\right) \sqrt[4]{-b+a x^4}}{x^6 \left(-8 b+a x^8\right)} \, dx","IntegrateAlgebraic[((-4*b + a*x^4)*(-b + a*x^4)^(1/4))/(x^6*(-8*b + a*x^8)),x]","\frac{\sqrt[4]{-b+a x^4} \left(-4 b+9 a x^4\right)}{40 b x^5}+\frac{a \text{RootSum}\left[8 a^2-a b-16 a \text{$\#$1}^4+8 \text{$\#$1}^8\&,\frac{8 a^2 \log (x)-a b \log (x)-8 a^2 \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+a b \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-8 a \log (x) \text{$\#$1}^4-4 b \log (x) \text{$\#$1}^4+8 a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+4 b \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}^3+\text{$\#$1}^7}\&\right]}{512 b}","\frac{\sqrt[4]{-b+a x^4} \left(-4 b+9 a x^4\right)}{40 b x^5}+\frac{a \text{RootSum}\left[8 a^2-a b-16 a \text{$\#$1}^4+8 \text{$\#$1}^8\&,\frac{-8 a^2 \log (x)+a b \log (x)+8 a^2 \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)-a b \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right)+8 a \log (x) \text{$\#$1}^4+4 b \log (x) \text{$\#$1}^4-8 a \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-4 b \log \left(\sqrt[4]{-b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}^3-\text{$\#$1}^7}\&\right]}{512 b}",1,"((-b + a*x^4)^(1/4)*(-4*b + 9*a*x^4))/(40*b*x^5) + (a*RootSum[8*a^2 - a*b - 16*a*#1^4 + 8*#1^8 & , (8*a^2*Log[x] - a*b*Log[x] - 8*a^2*Log[(-b + a*x^4)^(1/4) - x*#1] + a*b*Log[(-b + a*x^4)^(1/4) - x*#1] - 8*a*Log[x]*#1^4 - 4*b*Log[x]*#1^4 + 8*a*Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4 + 4*b*Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3) + #1^7) & ])/(512*b)","A",1
2569,1,219,219,0.4588244,"\int \frac{x}{(1+x) \sqrt[3]{-x^2+x^3}} \, dx","IntegrateAlgebraic[x/((1 + x)*(-x^2 + x^3)^(1/3)),x]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-x^2+x^3}}\right)}{\sqrt[3]{2}}-\log \left(-x+\sqrt[3]{-x^2+x^3}\right)+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{-x^2+x^3}\right)}{\sqrt[3]{2}}+\frac{1}{2} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-x^2+x^3}+\sqrt[3]{2} \left(-x^2+x^3\right)^{2/3}\right)}{2 \sqrt[3]{2}}","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-x^2+x^3}}\right)}{\sqrt[3]{2}}-\log \left(-x+\sqrt[3]{-x^2+x^3}\right)+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{-x^2+x^3}\right)}{\sqrt[3]{2}}+\frac{1}{2} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-x^2+x^3}+\sqrt[3]{2} \left(-x^2+x^3\right)^{2/3}\right)}{2 \sqrt[3]{2}}",1,"Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-x^2 + x^3)^(1/3))] - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-x^2 + x^3)^(1/3))])/2^(1/3) - Log[-x + (-x^2 + x^3)^(1/3)] + Log[-2*x + 2^(2/3)*(-x^2 + x^3)^(1/3)]/2^(1/3) + Log[x^2 + x*(-x^2 + x^3)^(1/3) + (-x^2 + x^3)^(2/3)]/2 - Log[2*x^2 + 2^(2/3)*x*(-x^2 + x^3)^(1/3) + 2^(1/3)*(-x^2 + x^3)^(2/3)]/(2*2^(1/3))","A",1
2570,1,219,219,0.4504044,"\int \frac{\sqrt[3]{-x^2+x^3}}{-1+x^2} \, dx","IntegrateAlgebraic[(-x^2 + x^3)^(1/3)/(-1 + x^2),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-x^2+x^3}}\right)}{2^{2/3}}-\log \left(-x+\sqrt[3]{-x^2+x^3}\right)+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{-x^2+x^3}\right)}{2^{2/3}}+\frac{1}{2} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-x^2+x^3}+\sqrt[3]{2} \left(-x^2+x^3\right)^{2/3}\right)}{2\ 2^{2/3}}","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-x^2+x^3}}\right)}{2^{2/3}}-\log \left(-x+\sqrt[3]{-x^2+x^3}\right)+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{-x^2+x^3}\right)}{2^{2/3}}+\frac{1}{2} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-x^2+x^3}+\sqrt[3]{2} \left(-x^2+x^3\right)^{2/3}\right)}{2\ 2^{2/3}}",1,"-(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-x^2 + x^3)^(1/3))]) + (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-x^2 + x^3)^(1/3))])/2^(2/3) - Log[-x + (-x^2 + x^3)^(1/3)] + Log[-2*x + 2^(2/3)*(-x^2 + x^3)^(1/3)]/2^(2/3) + Log[x^2 + x*(-x^2 + x^3)^(1/3) + (-x^2 + x^3)^(2/3)]/2 - Log[2*x^2 + 2^(2/3)*x*(-x^2 + x^3)^(1/3) + 2^(1/3)*(-x^2 + x^3)^(2/3)]/(2*2^(2/3))","A",1
2571,1,219,219,0.359489,"\int x^2 \sqrt{b+a^2 x^4} \sqrt{a x^2+\sqrt{b+a^2 x^4}} \, dx","IntegrateAlgebraic[x^2*Sqrt[b + a^2*x^4]*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]],x]","\frac{\sqrt{a} x \sqrt{b+a^2 x^4} \left(28 a b x^2+16 a^3 x^6\right) \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\sqrt{a} x \left(9 b^2+36 a^2 b x^4+16 a^4 x^8\right) \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{48 a^{3/2} b+96 a^{7/2} x^4+96 a^{5/2} x^2 \sqrt{b+a^2 x^4}}-\frac{3 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{b}}\right)}{16 \sqrt{2} a^{3/2}}","\frac{\sqrt{a} x \sqrt{b+a^2 x^4} \left(28 a b x^2+16 a^3 x^6\right) \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\sqrt{a} x \left(9 b^2+36 a^2 b x^4+16 a^4 x^8\right) \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{48 a^{3/2} b+96 a^{7/2} x^4+96 a^{5/2} x^2 \sqrt{b+a^2 x^4}}-\frac{3 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{b}}\right)}{16 \sqrt{2} a^{3/2}}",1,"(Sqrt[a]*x*Sqrt[b + a^2*x^4]*(28*a*b*x^2 + 16*a^3*x^6)*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + Sqrt[a]*x*(9*b^2 + 36*a^2*b*x^4 + 16*a^4*x^8)*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/(48*a^(3/2)*b + 96*a^(7/2)*x^4 + 96*a^(5/2)*x^2*Sqrt[b + a^2*x^4]) - (3*b^(3/2)*ArcTan[(Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[b]])/(16*Sqrt[2]*a^(3/2))","A",1
2572,1,220,220,7.1230283,"\int \frac{\left(-b x+a^2 x^2\right)^{3/2}}{\left(a x^2+x \sqrt{-b x+a^2 x^2}\right)^{3/2}} \, dx","IntegrateAlgebraic[(-(b*x) + a^2*x^2)^(3/2)/(a*x^2 + x*Sqrt[-(b*x) + a^2*x^2])^(3/2),x]","\frac{\sqrt{-b x+a^2 x^2} \left(115 b^2-88 a^2 b x+32 a^4 x^2\right) \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{40 b^2 x}+\sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)} \left(\frac{-145 a b^2+104 a^3 b x-32 a^5 x^2}{40 b^2}+\frac{9 \sqrt{b} \sqrt{-a x+\sqrt{-b x+a^2 x^2}} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{-a x+\sqrt{-b x+a^2 x^2}}}{\sqrt{b}}\right)}{8 \sqrt{2} \sqrt{a} x}\right)","\frac{\sqrt{-b x+a^2 x^2} \left(115 b^2-88 a^2 b x+32 a^4 x^2\right) \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{40 b^2 x}+\sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)} \left(\frac{-145 a b^2+104 a^3 b x-32 a^5 x^2}{40 b^2}+\frac{9 \sqrt{b} \sqrt{-a x+\sqrt{-b x+a^2 x^2}} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{-a x+\sqrt{-b x+a^2 x^2}}}{\sqrt{b}}\right)}{8 \sqrt{2} \sqrt{a} x}\right)",1,"(Sqrt[-(b*x) + a^2*x^2]*(115*b^2 - 88*a^2*b*x + 32*a^4*x^2)*Sqrt[x*(a*x + Sqrt[-(b*x) + a^2*x^2])])/(40*b^2*x) + Sqrt[x*(a*x + Sqrt[-(b*x) + a^2*x^2])]*((-145*a*b^2 + 104*a^3*b*x - 32*a^5*x^2)/(40*b^2) + (9*Sqrt[b]*Sqrt[-(a*x) + Sqrt[-(b*x) + a^2*x^2]]*ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[-(a*x) + Sqrt[-(b*x) + a^2*x^2]])/Sqrt[b]])/(8*Sqrt[2]*Sqrt[a]*x))","A",1
2573,1,221,221,5.6545798,"\int \frac{-3 k+\left(-2+k^2\right) x+3 k x^2+k^2 x^3}{\sqrt[3]{\left(1-x^2\right) \left(1-k^2 x^2\right)} \left(-1+d-(2+d) k x-\left(d+k^2\right) x^2+d k x^3\right)} \, dx","IntegrateAlgebraic[(-3*k + (-2 + k^2)*x + 3*k*x^2 + k^2*x^3)/(((1 - x^2)*(1 - k^2*x^2))^(1/3)*(-1 + d - (2 + d)*k*x - (d + k^2)*x^2 + d*k*x^3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{2+2 k x+\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{2/3}}+\frac{\log \left(1+k x-\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{d^{2/3}}-\frac{\log \left(1+2 k x+k^2 x^2+\left(\sqrt[3]{d}+\sqrt[3]{d} k x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+d^{2/3} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{2+2 k x+\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{2/3}}+\frac{\log \left(1+k x-\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{d^{2/3}}-\frac{\log \left(1+2 k x+k^2 x^2+\left(\sqrt[3]{d}+\sqrt[3]{d} k x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+d^{2/3} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 d^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))/(2 + 2*k*x + d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))])/d^(2/3) + Log[1 + k*x - d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3)]/d^(2/3) - Log[1 + 2*k*x + k^2*x^2 + (d^(1/3) + d^(1/3)*k*x)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3) + d^(2/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3)]/(2*d^(2/3))","A",1
2574,1,221,221,3.1373807,"\int \frac{a+b x}{x (-d+c x) \sqrt[4]{-x^3+x^4}} \, dx","IntegrateAlgebraic[(a + b*x)/(x*(-d + c*x)*(-x^3 + x^4)^(1/4)),x]","-\frac{4 a \left(-x^3+x^4\right)^{3/4}}{3 d x^3}+\frac{\sqrt{2} (a c+b d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c-d} \sqrt[4]{d} x \sqrt[4]{-x^3+x^4}}{\sqrt{c-d} x^2-\sqrt{d} \sqrt{-x^3+x^4}}\right)}{\sqrt[4]{c-d} d^{7/4}}-\frac{\sqrt{2} (a c+b d) \tanh ^{-1}\left(\frac{\sqrt{c-d} x^2+\sqrt{d} \sqrt{-x^3+x^4}}{\sqrt{2} \sqrt[4]{c-d} \sqrt[4]{d} x \sqrt[4]{-x^3+x^4}}\right)}{\sqrt[4]{c-d} d^{7/4}}","-\frac{4 a \left(-x^3+x^4\right)^{3/4}}{3 d x^3}+\frac{\sqrt{2} (a c+b d) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c-d} \sqrt[4]{d} x \sqrt[4]{-x^3+x^4}}{\sqrt{c-d} x^2-\sqrt{d} \sqrt{-x^3+x^4}}\right)}{\sqrt[4]{c-d} d^{7/4}}-\frac{\sqrt{2} (a c+b d) \tanh ^{-1}\left(\frac{\sqrt{c-d} x^2+\sqrt{d} \sqrt{-x^3+x^4}}{\sqrt{2} \sqrt[4]{c-d} \sqrt[4]{d} x \sqrt[4]{-x^3+x^4}}\right)}{\sqrt[4]{c-d} d^{7/4}}",1,"(-4*a*(-x^3 + x^4)^(3/4))/(3*d*x^3) + (Sqrt[2]*(a*c + b*d)*ArcTan[(Sqrt[2]*(c - d)^(1/4)*d^(1/4)*x*(-x^3 + x^4)^(1/4))/(Sqrt[c - d]*x^2 - Sqrt[d]*Sqrt[-x^3 + x^4])])/((c - d)^(1/4)*d^(7/4)) - (Sqrt[2]*(a*c + b*d)*ArcTanh[(Sqrt[c - d]*x^2 + Sqrt[d]*Sqrt[-x^3 + x^4])/(Sqrt[2]*(c - d)^(1/4)*d^(1/4)*x*(-x^3 + x^4)^(1/4))])/((c - d)^(1/4)*d^(7/4))","A",1
2575,1,221,221,1.0696582,"\int \frac{(-4+3 x) \sqrt[4]{\frac{-a+a x+b x^4}{-c+c x+d x^4}}}{(-1+x) x} \, dx","IntegrateAlgebraic[((-4 + 3*x)*((-a + a*x + b*x^4)/(-c + c*x + d*x^4))^(1/4))/((-1 + x)*x),x]","-\frac{2 \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt[4]{\frac{-a+a x+b x^4}{-c+c x+d x^4}}}{\sqrt[4]{a}}\right)}{\sqrt[4]{c}}+\frac{2 \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{\frac{-a+a x+b x^4}{-c+c x+d x^4}}}{\sqrt[4]{b}}\right)}{\sqrt[4]{d}}-\frac{2 \sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt[4]{\frac{-a+a x+b x^4}{-c+c x+d x^4}}}{\sqrt[4]{a}}\right)}{\sqrt[4]{c}}+\frac{2 \sqrt[4]{b} \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{\frac{-a+a x+b x^4}{-c+c x+d x^4}}}{\sqrt[4]{b}}\right)}{\sqrt[4]{d}}","-\frac{2 \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt[4]{\frac{-a+a x+b x^4}{-c+c x+d x^4}}}{\sqrt[4]{a}}\right)}{\sqrt[4]{c}}+\frac{2 \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{\frac{-a+a x+b x^4}{-c+c x+d x^4}}}{\sqrt[4]{b}}\right)}{\sqrt[4]{d}}-\frac{2 \sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt[4]{\frac{-a+a x+b x^4}{-c+c x+d x^4}}}{\sqrt[4]{a}}\right)}{\sqrt[4]{c}}+\frac{2 \sqrt[4]{b} \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{\frac{-a+a x+b x^4}{-c+c x+d x^4}}}{\sqrt[4]{b}}\right)}{\sqrt[4]{d}}",1,"(-2*a^(1/4)*ArcTan[(c^(1/4)*((-a + a*x + b*x^4)/(-c + c*x + d*x^4))^(1/4))/a^(1/4)])/c^(1/4) + (2*b^(1/4)*ArcTan[(d^(1/4)*((-a + a*x + b*x^4)/(-c + c*x + d*x^4))^(1/4))/b^(1/4)])/d^(1/4) - (2*a^(1/4)*ArcTanh[(c^(1/4)*((-a + a*x + b*x^4)/(-c + c*x + d*x^4))^(1/4))/a^(1/4)])/c^(1/4) + (2*b^(1/4)*ArcTanh[(d^(1/4)*((-a + a*x + b*x^4)/(-c + c*x + d*x^4))^(1/4))/b^(1/4)])/d^(1/4)","A",1
2576,1,221,221,0.5109649,"\int \frac{\left(1+x^3\right)^{2/3} \left(1-2 x^3+2 x^6\right)}{x^6 \left(-1-x^3+2 x^6\right)} \, dx","IntegrateAlgebraic[((1 + x^3)^(2/3)*(1 - 2*x^3 + 2*x^6))/(x^6*(-1 - x^3 + 2*x^6)),x]","\frac{\left(2-13 x^3\right) \left(1+x^3\right)^{2/3}}{10 x^5}-\frac{10 \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{1+x^3}}\right)}{3 \sqrt{3}}-\frac{2^{2/3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{1+x^3}}\right)}{3 \sqrt{3}}-\frac{10}{9} \log \left(x+\sqrt[3]{1+x^3}\right)+\frac{1}{9} 2^{2/3} \log \left(-2 x+2^{2/3} \sqrt[3]{1+x^3}\right)+\frac{5}{9} \log \left(x^2-x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{1+x^3}+\sqrt[3]{2} \left(1+x^3\right)^{2/3}\right)}{9 \sqrt[3]{2}}","\frac{\left(2-13 x^3\right) \left(1+x^3\right)^{2/3}}{10 x^5}-\frac{10 \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{1+x^3}}\right)}{3 \sqrt{3}}-\frac{2^{2/3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{1+x^3}}\right)}{3 \sqrt{3}}-\frac{10}{9} \log \left(x+\sqrt[3]{1+x^3}\right)+\frac{1}{9} 2^{2/3} \log \left(-2 x+2^{2/3} \sqrt[3]{1+x^3}\right)+\frac{5}{9} \log \left(x^2-x \sqrt[3]{1+x^3}+\left(1+x^3\right)^{2/3}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{1+x^3}+\sqrt[3]{2} \left(1+x^3\right)^{2/3}\right)}{9 \sqrt[3]{2}}",1,"((2 - 13*x^3)*(1 + x^3)^(2/3))/(10*x^5) - (10*ArcTan[(Sqrt[3]*x)/(-x + 2*(1 + x^3)^(1/3))])/(3*Sqrt[3]) - (2^(2/3)*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(1 + x^3)^(1/3))])/(3*Sqrt[3]) - (10*Log[x + (1 + x^3)^(1/3)])/9 + (2^(2/3)*Log[-2*x + 2^(2/3)*(1 + x^3)^(1/3)])/9 + (5*Log[x^2 - x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)])/9 - Log[2*x^2 + 2^(2/3)*x*(1 + x^3)^(1/3) + 2^(1/3)*(1 + x^3)^(2/3)]/(9*2^(1/3))","A",1
2577,1,222,222,0.6307038,"\int \frac{2}{(3+x) \left(2-8 x+8 x^2\right)^{2/3}} \, dx","IntegrateAlgebraic[2/((3 + x)*(2 - 8*x + 8*x^2)^(2/3)),x]","-\frac{3 \sqrt[3]{2} \sqrt[3]{1-4 x+4 x^2}}{7 (-1+2 x)}+\frac{1}{7} \sqrt[3]{\frac{2}{7}} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \left(1-4 x+4 x^2\right)^{2/3}}{2 \sqrt[3]{7}-4 \sqrt[3]{7} x+\left(1-4 x+4 x^2\right)^{2/3}}\right)-\frac{2}{21} \sqrt[3]{\frac{2}{7}} \log (-1+2 x)+\frac{1}{7} \sqrt[3]{\frac{2}{7}} \log \left(-\sqrt[3]{7}+2 \sqrt[3]{7} x+\left(1-4 x+4 x^2\right)^{2/3}\right)-\frac{\log \left(\sqrt[3]{7}-2 \sqrt[3]{7} x+7^{2/3} \sqrt[3]{1-4 x+4 x^2}+\left(1-4 x+4 x^2\right)^{2/3}\right)}{7\ 2^{2/3} \sqrt[3]{7}}","-\frac{3 \sqrt[3]{2} \sqrt[3]{1-4 x+4 x^2}}{7 (-1+2 x)}+\frac{1}{7} \sqrt[3]{\frac{2}{7}} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \left(1-4 x+4 x^2\right)^{2/3}}{2 \sqrt[3]{7}-4 \sqrt[3]{7} x+\left(1-4 x+4 x^2\right)^{2/3}}\right)-\frac{2}{21} \sqrt[3]{\frac{2}{7}} \log (-1+2 x)+\frac{1}{7} \sqrt[3]{\frac{2}{7}} \log \left(-\sqrt[3]{7}+2 \sqrt[3]{7} x+\left(1-4 x+4 x^2\right)^{2/3}\right)-\frac{\log \left(\sqrt[3]{7}-2 \sqrt[3]{7} x+7^{2/3} \sqrt[3]{1-4 x+4 x^2}+\left(1-4 x+4 x^2\right)^{2/3}\right)}{7\ 2^{2/3} \sqrt[3]{7}}",1,"(-3*2^(1/3)*(1 - 4*x + 4*x^2)^(1/3))/(7*(-1 + 2*x)) + ((2/7)^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*(1 - 4*x + 4*x^2)^(2/3))/(2*7^(1/3) - 4*7^(1/3)*x + (1 - 4*x + 4*x^2)^(2/3))])/7 - (2*(2/7)^(1/3)*Log[-1 + 2*x])/21 + ((2/7)^(1/3)*Log[-7^(1/3) + 2*7^(1/3)*x + (1 - 4*x + 4*x^2)^(2/3)])/7 - Log[7^(1/3) - 2*7^(1/3)*x + 7^(2/3)*(1 - 4*x + 4*x^2)^(1/3) + (1 - 4*x + 4*x^2)^(2/3)]/(7*2^(2/3)*7^(1/3))","A",1
2578,1,222,222,2.976077,"\int \frac{1-(-3+2 k) x-(4+k) x^2+3 k x^3}{\sqrt[3]{(1-x) x (1-k x)} \left(-1+(5+b) x-(10+b k) x^2+10 x^3-5 x^4+x^5\right)} \, dx","IntegrateAlgebraic[(1 - (-3 + 2*k)*x - (4 + k)*x^2 + 3*k*x^3)/(((1 - x)*x*(1 - k*x))^(1/3)*(-1 + (5 + b)*x - (10 + b*k)*x^2 + 10*x^3 - 5*x^4 + x^5)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}{2-4 x+2 x^2+\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{b^{2/3}}+\frac{\log \left(1-2 x+x^2-\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{b^{2/3}}-\frac{\log \left(1-4 x+6 x^2-4 x^3+x^4+\left(\sqrt[3]{b}-2 \sqrt[3]{b} x+\sqrt[3]{b} x^2\right) \sqrt[3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 b^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}{2-4 x+2 x^2+\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{b^{2/3}}+\frac{\log \left(1-2 x+x^2-\sqrt[3]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{b^{2/3}}-\frac{\log \left(1-4 x+6 x^2-4 x^3+x^4+\left(\sqrt[3]{b}-2 \sqrt[3]{b} x+\sqrt[3]{b} x^2\right) \sqrt[3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 b^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))/(2 - 4*x + 2*x^2 + b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/b^(2/3) + Log[1 - 2*x + x^2 - b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3)]/b^(2/3) - Log[1 - 4*x + 6*x^2 - 4*x^3 + x^4 + (b^(1/3) - 2*b^(1/3)*x + b^(1/3)*x^2)*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + b^(2/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3)]/(2*b^(2/3))","A",1
2579,1,223,223,0.5141593,"\int \frac{1}{x \sqrt[3]{(1+x) \left(q+2 q x+x^2\right)}} \, dx","IntegrateAlgebraic[1/(x*((1 + x)*(q + 2*q*x + x^2))^(1/3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{q+3 q x+(1+2 q) x^2+x^3}}{2 \sqrt[3]{q}+2 \sqrt[3]{q} x+\sqrt[3]{q+3 q x+(1+2 q) x^2+x^3}}\right)}{2 \sqrt[3]{q}}+\frac{\log \left(-\sqrt[3]{q}-\sqrt[3]{q} x+\sqrt[3]{q+3 q x+(1+2 q) x^2+x^3}\right)}{2 \sqrt[3]{q}}-\frac{\log \left(q^{2/3}+2 q^{2/3} x+q^{2/3} x^2+\left(\sqrt[3]{q}+\sqrt[3]{q} x\right) \sqrt[3]{q+3 q x+(1+2 q) x^2+x^3}+\left(q+3 q x+(1+2 q) x^2+x^3\right)^{2/3}\right)}{4 \sqrt[3]{q}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{q+3 q x+(1+2 q) x^2+x^3}}{2 \sqrt[3]{q}+2 \sqrt[3]{q} x+\sqrt[3]{q+3 q x+(1+2 q) x^2+x^3}}\right)}{2 \sqrt[3]{q}}+\frac{\log \left(-\sqrt[3]{q}-\sqrt[3]{q} x+\sqrt[3]{q+3 q x+(1+2 q) x^2+x^3}\right)}{2 \sqrt[3]{q}}-\frac{\log \left(q^{2/3}+2 q^{2/3} x+q^{2/3} x^2+\left(\sqrt[3]{q}+\sqrt[3]{q} x\right) \sqrt[3]{q+3 q x+(1+2 q) x^2+x^3}+\left(q+3 q x+(1+2 q) x^2+x^3\right)^{2/3}\right)}{4 \sqrt[3]{q}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(q + 3*q*x + (1 + 2*q)*x^2 + x^3)^(1/3))/(2*q^(1/3) + 2*q^(1/3)*x + (q + 3*q*x + (1 + 2*q)*x^2 + x^3)^(1/3))])/(2*q^(1/3)) + Log[-q^(1/3) - q^(1/3)*x + (q + 3*q*x + (1 + 2*q)*x^2 + x^3)^(1/3)]/(2*q^(1/3)) - Log[q^(2/3) + 2*q^(2/3)*x + q^(2/3)*x^2 + (q^(1/3) + q^(1/3)*x)*(q + 3*q*x + (1 + 2*q)*x^2 + x^3)^(1/3) + (q + 3*q*x + (1 + 2*q)*x^2 + x^3)^(2/3)]/(4*q^(1/3))","A",1
2580,1,223,223,0.9542694,"\int \frac{-1+2 x+\left(-2 k+k^2\right) x^2}{((1-x) x (1-k x))^{2/3} \left(1-(b+2 k) x+\left(b+k^2\right) x^2\right)} \, dx","IntegrateAlgebraic[(-1 + 2*x + (-2*k + k^2)*x^2)/(((1 - x)*x*(1 - k*x))^(2/3)*(1 - (b + 2*k)*x + (b + k^2)*x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \left(x+(-1-k) x^2+k x^3\right)^{2/3}}{2 \sqrt[3]{b} x-2 \sqrt[3]{b} x^2+\left(x+(-1-k) x^2+k x^3\right)^{2/3}}\right)}{\sqrt[3]{b}}+\frac{\log \left(-\sqrt{b} x+\sqrt{b} x^2+\sqrt[6]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{\sqrt[3]{b}}-\frac{\log \left(b x^2-2 b x^3+b x^4+\left(b^{2/3} x-b^{2/3} x^2\right) \left(x+(-1-k) x^2+k x^3\right)^{2/3}+\sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{4/3}\right)}{2 \sqrt[3]{b}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \left(x+(-1-k) x^2+k x^3\right)^{2/3}}{2 \sqrt[3]{b} x-2 \sqrt[3]{b} x^2+\left(x+(-1-k) x^2+k x^3\right)^{2/3}}\right)}{\sqrt[3]{b}}+\frac{\log \left(-\sqrt{b} x+\sqrt{b} x^2+\sqrt[6]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{\sqrt[3]{b}}-\frac{\log \left(b x^2-2 b x^3+b x^4+\left(b^{2/3} x-b^{2/3} x^2\right) \left(x+(-1-k) x^2+k x^3\right)^{2/3}+\sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{4/3}\right)}{2 \sqrt[3]{b}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(x + (-1 - k)*x^2 + k*x^3)^(2/3))/(2*b^(1/3)*x - 2*b^(1/3)*x^2 + (x + (-1 - k)*x^2 + k*x^3)^(2/3))])/b^(1/3) + Log[-(Sqrt[b]*x) + Sqrt[b]*x^2 + b^(1/6)*(x + (-1 - k)*x^2 + k*x^3)^(2/3)]/b^(1/3) - Log[b*x^2 - 2*b*x^3 + b*x^4 + (b^(2/3)*x - b^(2/3)*x^2)*(x + (-1 - k)*x^2 + k*x^3)^(2/3) + b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(4/3)]/(2*b^(1/3))","A",1
2581,1,223,223,5.6490245,"\int \frac{3 k+\left(-2+k^2\right) x-3 k x^2+k^2 x^3}{\sqrt[3]{\left(1-x^2\right) \left(1-k^2 x^2\right)} \left(1-d-(2+d) k x+\left(d+k^2\right) x^2+d k x^3\right)} \, dx","IntegrateAlgebraic[(3*k + (-2 + k^2)*x - 3*k*x^2 + k^2*x^3)/(((1 - x^2)*(1 - k^2*x^2))^(1/3)*(1 - d - (2 + d)*k*x + (d + k^2)*x^2 + d*k*x^3)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{2-2 k x+\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{2/3}}-\frac{\log \left(-1+k x+\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{d^{2/3}}+\frac{\log \left(1-2 k x+k^2 x^2+\left(\sqrt[3]{d}-\sqrt[3]{d} k x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+d^{2/3} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 d^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{2-2 k x+\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{2/3}}-\frac{\log \left(-1+k x+\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{d^{2/3}}+\frac{\log \left(1-2 k x+k^2 x^2+\left(\sqrt[3]{d}-\sqrt[3]{d} k x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+d^{2/3} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 d^{2/3}}",1,"-((Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))/(2 - 2*k*x + d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))])/d^(2/3)) - Log[-1 + k*x + d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3)]/d^(2/3) + Log[1 - 2*k*x + k^2*x^2 + (d^(1/3) - d^(1/3)*k*x)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3) + d^(2/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3)]/(2*d^(2/3))","A",1
2582,1,223,223,8.0940424,"\int \frac{3+2 \left(1+k^2\right) x-\left(1+k^2\right) x^2-4 k^2 x^3-k^2 x^4}{\left(\left(1-x^2\right) \left(1-k^2 x^2\right)\right)^{2/3} \left(1-d-(1+2 d) x-\left(d+k^2\right) x^2+k^2 x^3\right)} \, dx","IntegrateAlgebraic[(3 + 2*(1 + k^2)*x - (1 + k^2)*x^2 - 4*k^2*x^3 - k^2*x^4)/(((1 - x^2)*(1 - k^2*x^2))^(2/3)*(1 - d - (1 + 2*d)*x - (d + k^2)*x^2 + k^2*x^3)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d}+\sqrt{3} \sqrt[3]{d} x}{\sqrt[3]{d}+\sqrt[3]{d} x+2 \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{2/3}}-\frac{\log \left(-\sqrt[3]{d}-\sqrt[3]{d} x+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{d^{2/3}}+\frac{\log \left(d^{2/3}+2 d^{2/3} x+d^{2/3} x^2+\left(\sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 d^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d}+\sqrt{3} \sqrt[3]{d} x}{\sqrt[3]{d}+\sqrt[3]{d} x+2 \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{2/3}}-\frac{\log \left(-\sqrt[3]{d}-\sqrt[3]{d} x+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{d^{2/3}}+\frac{\log \left(d^{2/3}+2 d^{2/3} x+d^{2/3} x^2+\left(\sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 d^{2/3}}",1,"-((Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3) + Sqrt[3]*d^(1/3)*x)/(d^(1/3) + d^(1/3)*x + 2*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))])/d^(2/3)) - Log[-d^(1/3) - d^(1/3)*x + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3)]/d^(2/3) + Log[d^(2/3) + 2*d^(2/3)*x + d^(2/3)*x^2 + (d^(1/3) + d^(1/3)*x)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3) + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3)]/(2*d^(2/3))","A",1
2583,1,216,223,0.5618813,"\int \frac{\sqrt[4]{\frac{-1+x}{1+2 x}}-3 \left(\frac{-1+x}{1+2 x}\right)^{3/4}}{(-1+x) (1+x)^2 (-1+2 x)} \, dx","IntegrateAlgebraic[(((-1 + x)/(1 + 2*x))^(1/4) - 3*((-1 + x)/(1 + 2*x))^(3/4))/((-1 + x)*(1 + x)^2*(-1 + 2*x)),x]","\frac{-\sqrt[4]{\frac{-1+x}{1+2 x}}+3 \left(\frac{-1+x}{1+2 x}\right)^{3/4}}{2 \left(-2+\frac{-1+x}{1+2 x}\right)}-\frac{1}{72} \sqrt{24420+55819 \sqrt{2}} \tan ^{-1}\left(\frac{\sqrt[4]{\frac{-1+x}{1+2 x}}}{\sqrt[4]{2}}\right)-\frac{2}{9} \tan ^{-1}\left(\frac{-\frac{1}{2}+\sqrt{\frac{-1+x}{1+2 x}}}{\sqrt[4]{\frac{-1+x}{1+2 x}}}\right)+\frac{1}{72} \sqrt{-24420+55819 \sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{-1+x}{1+2 x}}}{\sqrt[4]{2}}\right)+\frac{10}{9} \tanh ^{-1}\left(\frac{2 \sqrt[4]{\frac{-1+x}{1+2 x}}}{1+2 \sqrt{\frac{-1+x}{1+2 x}}}\right)","\frac{(-1-2 x) \left(\frac{-1+x}{1+2 x}\right)^{3/4}}{2 (1+x)}+\frac{\sqrt[4]{\frac{-1+x}{1+2 x}} (1+2 x)}{6 (1+x)}-\frac{1}{72} \sqrt{24420+55819 \sqrt{2}} \tan ^{-1}\left(\frac{\sqrt[4]{\frac{-1+x}{1+2 x}}}{\sqrt[4]{2}}\right)+\frac{2}{9} \tan ^{-1}\left(\frac{2 \sqrt[4]{\frac{-1+x}{1+2 x}}}{-1+2 \sqrt{\frac{-1+x}{1+2 x}}}\right)+\frac{1}{72} \sqrt{-24420+55819 \sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{-1+x}{1+2 x}}}{\sqrt[4]{2}}\right)+\frac{10}{9} \tanh ^{-1}\left(\frac{2 \sqrt[4]{\frac{-1+x}{1+2 x}}}{1+2 \sqrt{\frac{-1+x}{1+2 x}}}\right)",1,"(-((-1 + x)/(1 + 2*x))^(1/4) + 3*((-1 + x)/(1 + 2*x))^(3/4))/(2*(-2 + (-1 + x)/(1 + 2*x))) - (Sqrt[24420 + 55819*Sqrt[2]]*ArcTan[((-1 + x)/(1 + 2*x))^(1/4)/2^(1/4)])/72 - (2*ArcTan[(-1/2 + Sqrt[(-1 + x)/(1 + 2*x)])/((-1 + x)/(1 + 2*x))^(1/4)])/9 + (Sqrt[-24420 + 55819*Sqrt[2]]*ArcTanh[((-1 + x)/(1 + 2*x))^(1/4)/2^(1/4)])/72 + (10*ArcTanh[(2*((-1 + x)/(1 + 2*x))^(1/4))/(1 + 2*Sqrt[(-1 + x)/(1 + 2*x)])])/9","A",1
2584,1,223,223,2.487997,"\int \frac{\left(-q+p x^2\right) \sqrt{q^2+p^2 x^4}}{b x^3+a \left(q+p x^2\right)^3} \, dx","IntegrateAlgebraic[((-q + p*x^2)*Sqrt[q^2 + p^2*x^4])/(b*x^3 + a*(q + p*x^2)^3),x]","\frac{1}{3} \text{RootSum}\left[8 a p^3 q^3+12 a p^2 q^2 \text{$\#$1}^2+8 b \text{$\#$1}^3+6 a p q \text{$\#$1}^4+a \text{$\#$1}^6\&,\frac{-4 p^2 q^2 \log (x)+4 p^2 q^2 \log \left(q+p x^2+\sqrt{q^2+p^2 x^4}-x \text{$\#$1}\right)+4 p q \log (x) \text{$\#$1}^2-4 p q \log \left(q+p x^2+\sqrt{q^2+p^2 x^4}-x \text{$\#$1}\right) \text{$\#$1}^2-\log (x) \text{$\#$1}^4+\log \left(q+p x^2+\sqrt{q^2+p^2 x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{4 a p^2 q^2 \text{$\#$1}+4 b \text{$\#$1}^2+4 a p q \text{$\#$1}^3+a \text{$\#$1}^5}\&\right]","\frac{1}{3} \text{RootSum}\left[8 a p^3 q^3+12 a p^2 q^2 \text{$\#$1}^2+8 b \text{$\#$1}^3+6 a p q \text{$\#$1}^4+a \text{$\#$1}^6\&,\frac{-4 p^2 q^2 \log (x)+4 p^2 q^2 \log \left(q+p x^2+\sqrt{q^2+p^2 x^4}-x \text{$\#$1}\right)+4 p q \log (x) \text{$\#$1}^2-4 p q \log \left(q+p x^2+\sqrt{q^2+p^2 x^4}-x \text{$\#$1}\right) \text{$\#$1}^2-\log (x) \text{$\#$1}^4+\log \left(q+p x^2+\sqrt{q^2+p^2 x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{4 a p^2 q^2 \text{$\#$1}+4 b \text{$\#$1}^2+4 a p q \text{$\#$1}^3+a \text{$\#$1}^5}\&\right]",1,"RootSum[8*a*p^3*q^3 + 12*a*p^2*q^2*#1^2 + 8*b*#1^3 + 6*a*p*q*#1^4 + a*#1^6 & , (-4*p^2*q^2*Log[x] + 4*p^2*q^2*Log[q + p*x^2 + Sqrt[q^2 + p^2*x^4] - x*#1] + 4*p*q*Log[x]*#1^2 - 4*p*q*Log[q + p*x^2 + Sqrt[q^2 + p^2*x^4] - x*#1]*#1^2 - Log[x]*#1^4 + Log[q + p*x^2 + Sqrt[q^2 + p^2*x^4] - x*#1]*#1^4)/(4*a*p^2*q^2*#1 + 4*b*#1^2 + 4*a*p*q*#1^3 + a*#1^5) & ]/3","A",1
2585,1,223,223,0.0019377,"\int \frac{\left(-q+p x^2\right) \sqrt{q^2+p^2 x^4}}{b x^3+a \left(q+p x^2\right)^3} \, dx","IntegrateAlgebraic[((-q + p*x^2)*Sqrt[q^2 + p^2*x^4])/(b*x^3 + a*(q + p*x^2)^3),x]","\frac{1}{3} \text{RootSum}\left[8 a p^3 q^3+12 a p^2 q^2 \text{$\#$1}^2+8 b \text{$\#$1}^3+6 a p q \text{$\#$1}^4+a \text{$\#$1}^6\&,\frac{-4 p^2 q^2 \log (x)+4 p^2 q^2 \log \left(q+p x^2+\sqrt{q^2+p^2 x^4}-x \text{$\#$1}\right)+4 p q \log (x) \text{$\#$1}^2-4 p q \log \left(q+p x^2+\sqrt{q^2+p^2 x^4}-x \text{$\#$1}\right) \text{$\#$1}^2-\log (x) \text{$\#$1}^4+\log \left(q+p x^2+\sqrt{q^2+p^2 x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{4 a p^2 q^2 \text{$\#$1}+4 b \text{$\#$1}^2+4 a p q \text{$\#$1}^3+a \text{$\#$1}^5}\&\right]","\frac{1}{3} \text{RootSum}\left[8 a p^3 q^3+12 a p^2 q^2 \text{$\#$1}^2+8 b \text{$\#$1}^3+6 a p q \text{$\#$1}^4+a \text{$\#$1}^6\&,\frac{-4 p^2 q^2 \log (x)+4 p^2 q^2 \log \left(q+p x^2+\sqrt{q^2+p^2 x^4}-x \text{$\#$1}\right)+4 p q \log (x) \text{$\#$1}^2-4 p q \log \left(q+p x^2+\sqrt{q^2+p^2 x^4}-x \text{$\#$1}\right) \text{$\#$1}^2-\log (x) \text{$\#$1}^4+\log \left(q+p x^2+\sqrt{q^2+p^2 x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{4 a p^2 q^2 \text{$\#$1}+4 b \text{$\#$1}^2+4 a p q \text{$\#$1}^3+a \text{$\#$1}^5}\&\right]",1,"RootSum[8*a*p^3*q^3 + 12*a*p^2*q^2*#1^2 + 8*b*#1^3 + 6*a*p*q*#1^4 + a*#1^6 & , (-4*p^2*q^2*Log[x] + 4*p^2*q^2*Log[q + p*x^2 + Sqrt[q^2 + p^2*x^4] - x*#1] + 4*p*q*Log[x]*#1^2 - 4*p*q*Log[q + p*x^2 + Sqrt[q^2 + p^2*x^4] - x*#1]*#1^2 - Log[x]*#1^4 + Log[q + p*x^2 + Sqrt[q^2 + p^2*x^4] - x*#1]*#1^4)/(4*a*p^2*q^2*#1 + 4*b*#1^2 + 4*a*p*q*#1^3 + a*#1^5) & ]/3","A",1
2586,1,294,223,0.9177856,"\int \frac{\left(-1+x^2\right)^2}{\left(1+x^2\right)^2 \sqrt{1+x^4} \sqrt{x^2+\sqrt{1+x^4}}} \, dx","IntegrateAlgebraic[(-1 + x^2)^2/((1 + x^2)^2*Sqrt[1 + x^4]*Sqrt[x^2 + Sqrt[1 + x^4]]),x]","\frac{x^2 \left(1+x^2\right)+x^2 \sqrt{1+x^4}}{x \left(1+x^2\right) \sqrt{x^2+\sqrt{1+x^4}}}+\sqrt{2} \tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^4}}{\sqrt{2}}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)-\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{-\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)-\sqrt{-1+\sqrt{2}} \tanh ^{-1}\left(\frac{-\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)","\frac{x^2 \left(1+x^2\right)+x^2 \sqrt{1+x^4}}{x \left(1+x^2\right) \sqrt{x^2+\sqrt{1+x^4}}}+\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-\sqrt{-1+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2 \left(-1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)",1,"(x^2*(1 + x^2) + x^2*Sqrt[1 + x^4])/(x*(1 + x^2)*Sqrt[x^2 + Sqrt[1 + x^4]]) + Sqrt[2]*ArcTan[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] - Sqrt[1 + Sqrt[2]]*ArcTan[(-Sqrt[1/2 + 1/Sqrt[2]] + Sqrt[1/2 + 1/Sqrt[2]]*x^2 + Sqrt[1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] - Sqrt[-1 + Sqrt[2]]*ArcTanh[(-Sqrt[-1/2 + 1/Sqrt[2]] + Sqrt[-1/2 + 1/Sqrt[2]]*x^2 + Sqrt[-1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])]","A",1
2587,1,224,224,8.0632256,"\int \frac{-3+2 \left(1+k^2\right) x+\left(1+k^2\right) x^2-4 k^2 x^3+k^2 x^4}{\left(\left(1-x^2\right) \left(1-k^2 x^2\right)\right)^{2/3} \left(-1+d+(-1-2 d) x+\left(d+k^2\right) x^2+k^2 x^3\right)} \, dx","IntegrateAlgebraic[(-3 + 2*(1 + k^2)*x + (1 + k^2)*x^2 - 4*k^2*x^3 + k^2*x^4)/(((1 - x^2)*(1 - k^2*x^2))^(2/3)*(-1 + d + (-1 - 2*d)*x + (d + k^2)*x^2 + k^2*x^3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{-\sqrt{3} \sqrt[3]{d}+\sqrt{3} \sqrt[3]{d} x}{-\sqrt[3]{d}+\sqrt[3]{d} x-2 \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{2/3}}+\frac{\log \left(-\sqrt[3]{d}+\sqrt[3]{d} x+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{d^{2/3}}-\frac{\log \left(d^{2/3}-2 d^{2/3} x+d^{2/3} x^2+\left(\sqrt[3]{d}-\sqrt[3]{d} x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{-\sqrt{3} \sqrt[3]{d}+\sqrt{3} \sqrt[3]{d} x}{-\sqrt[3]{d}+\sqrt[3]{d} x-2 \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{2/3}}+\frac{\log \left(-\sqrt[3]{d}+\sqrt[3]{d} x+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{d^{2/3}}-\frac{\log \left(d^{2/3}-2 d^{2/3} x+d^{2/3} x^2+\left(\sqrt[3]{d}-\sqrt[3]{d} x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 d^{2/3}}",1,"(Sqrt[3]*ArcTan[(-(Sqrt[3]*d^(1/3)) + Sqrt[3]*d^(1/3)*x)/(-d^(1/3) + d^(1/3)*x - 2*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))])/d^(2/3) + Log[-d^(1/3) + d^(1/3)*x + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3)]/d^(2/3) - Log[d^(2/3) - 2*d^(2/3)*x + d^(2/3)*x^2 + (d^(1/3) - d^(1/3)*x)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3) + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3)]/(2*d^(2/3))","A",1
2588,1,224,224,1.3445106,"\int \frac{x^3 (-2+(1+k) x)}{((1-x) x (1-k x))^{2/3} \left(1-(2+2 k) x+\left(1+4 k+k^2\right) x^2-\left(2 k+2 k^2\right) x^3+\left(-b+k^2\right) x^4\right)} \, dx","IntegrateAlgebraic[(x^3*(-2 + (1 + k)*x))/(((1 - x)*x*(1 - k*x))^(2/3)*(1 - (2 + 2*k)*x + (1 + 4*k + k^2)*x^2 - (2*k + 2*k^2)*x^3 + (-b + k^2)*x^4)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{b} x}{\sqrt[6]{b} x-2 \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{2 b^{5/6}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{b} x}{\sqrt[6]{b} x+2 \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{2 b^{5/6}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{b} x}{\sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{b^{5/6}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{b} x^2+\frac{\left(x+(-1-k) x^2+k x^3\right)^{2/3}}{\sqrt[6]{b}}}{x \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{2 b^{5/6}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{b} x}{\sqrt[6]{b} x-2 \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{2 b^{5/6}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{b} x}{\sqrt[6]{b} x+2 \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{2 b^{5/6}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{b} x}{\sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{b^{5/6}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{b} x^2+\frac{\left(x+(-1-k) x^2+k x^3\right)^{2/3}}{\sqrt[6]{b}}}{x \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{2 b^{5/6}}",1,"-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*b^(1/6)*x)/(b^(1/6)*x - 2*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/b^(5/6) + (Sqrt[3]*ArcTan[(Sqrt[3]*b^(1/6)*x)/(b^(1/6)*x + 2*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/(2*b^(5/6)) - ArcTanh[(b^(1/6)*x)/(x + (-1 - k)*x^2 + k*x^3)^(1/3)]/b^(5/6) - ArcTanh[(b^(1/6)*x^2 + (x + (-1 - k)*x^2 + k*x^3)^(2/3)/b^(1/6))/(x*(x + (-1 - k)*x^2 + k*x^3)^(1/3))]/(2*b^(5/6))","A",1
2589,1,295,224,1.1210822,"\int \frac{\left(1+x^2\right)^2}{\left(-1+x^2\right)^2 \sqrt{1+x^4} \sqrt{x^2+\sqrt{1+x^4}}} \, dx","IntegrateAlgebraic[(1 + x^2)^2/((-1 + x^2)^2*Sqrt[1 + x^4]*Sqrt[x^2 + Sqrt[1 + x^4]]),x]","\frac{-x^2 \left(-1+x^2\right)-x^2 \sqrt{1+x^4}}{x \left(-1+x^2\right) \sqrt{x^2+\sqrt{1+x^4}}}+\sqrt{2} \tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^4}}{\sqrt{2}}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)-\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{-\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)+\sqrt{-1+\sqrt{2}} \tanh ^{-1}\left(\frac{-\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)","\frac{-x^2 \left(-1+x^2\right)-x^2 \sqrt{1+x^4}}{x \left(-1+x^2\right) \sqrt{x^2+\sqrt{1+x^4}}}+\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2 \left(-1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)+\sqrt{-1+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)",1,"(-(x^2*(-1 + x^2)) - x^2*Sqrt[1 + x^4])/(x*(-1 + x^2)*Sqrt[x^2 + Sqrt[1 + x^4]]) + Sqrt[2]*ArcTan[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] - Sqrt[1 + Sqrt[2]]*ArcTan[(-Sqrt[-1/2 + 1/Sqrt[2]] + Sqrt[-1/2 + 1/Sqrt[2]]*x^2 + Sqrt[-1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] + Sqrt[-1 + Sqrt[2]]*ArcTanh[(-Sqrt[1/2 + 1/Sqrt[2]] + Sqrt[1/2 + 1/Sqrt[2]]*x^2 + Sqrt[1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])]","A",1
2590,1,295,224,1.080116,"\int \frac{\left(-1+x^2\right)^2 \sqrt{x^2+\sqrt{1+x^4}}}{\left(1+x^2\right)^2 \sqrt{1+x^4}} \, dx","IntegrateAlgebraic[((-1 + x^2)^2*Sqrt[x^2 + Sqrt[1 + x^4]])/((1 + x^2)^2*Sqrt[1 + x^4]),x]","\frac{-x^2 \left(-1+x^2\right)-x^2 \sqrt{1+x^4}}{x \left(1+x^2\right) \sqrt{x^2+\sqrt{1+x^4}}}+\sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\frac{-\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^4}}{\sqrt{2}}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)-\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{-\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)","\frac{-x^2 \left(-1+x^2\right)-x^2 \sqrt{1+x^4}}{x \left(1+x^2\right) \sqrt{x^2+\sqrt{1+x^4}}}+\sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2 \left(-1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)",1,"(-(x^2*(-1 + x^2)) - x^2*Sqrt[1 + x^4])/(x*(1 + x^2)*Sqrt[x^2 + Sqrt[1 + x^4]]) + Sqrt[-1 + Sqrt[2]]*ArcTan[(-Sqrt[1/2 + 1/Sqrt[2]] + Sqrt[1/2 + 1/Sqrt[2]]*x^2 + Sqrt[1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] + Sqrt[2]*ArcTanh[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] - Sqrt[1 + Sqrt[2]]*ArcTanh[(-Sqrt[-1/2 + 1/Sqrt[2]] + Sqrt[-1/2 + 1/Sqrt[2]]*x^2 + Sqrt[-1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])]","A",1
2591,1,178,224,0.2763106,"\int \frac{-1+x^2}{\left(1+x^2\right) \sqrt{1+\sqrt{1+\sqrt{1+x}}}} \, dx","IntegrateAlgebraic[(-1 + x^2)/((1 + x^2)*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]),x]","-\frac{16}{105} \left(-2+9 \sqrt{1+x}\right) \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\frac{8}{105} \sqrt{1+\sqrt{1+x}} \left(4+15 \sqrt{1+x}\right) \sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{RootSum}\left[2-8 \text{$\#$1}^4+8 \text{$\#$1}^6+14 \text{$\#$1}^8-32 \text{$\#$1}^{10}+24 \text{$\#$1}^{12}-8 \text{$\#$1}^{14}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{$\#$1}\right)}{-\text{$\#$1}+4 \text{$\#$1}^5-4 \text{$\#$1}^7+\text{$\#$1}^9}\&\right]","\frac{32}{105} \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\frac{32}{105} \sqrt{1+\sqrt{1+x}} \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\sqrt{1+x} \left(-\frac{48}{35} \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\frac{8}{7} \sqrt{1+\sqrt{1+x}} \sqrt{1+\sqrt{1+\sqrt{1+x}}}\right)-\text{RootSum}\left[2-8 \text{$\#$1}^4+8 \text{$\#$1}^6+14 \text{$\#$1}^8-32 \text{$\#$1}^{10}+24 \text{$\#$1}^{12}-8 \text{$\#$1}^{14}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{$\#$1}\right)}{-\text{$\#$1}+4 \text{$\#$1}^5-4 \text{$\#$1}^7+\text{$\#$1}^9}\&\right]",1,"(-16*(-2 + 9*Sqrt[1 + x])*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/105 + (8*Sqrt[1 + Sqrt[1 + x]]*(4 + 15*Sqrt[1 + x])*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/105 - RootSum[2 - 8*#1^4 + 8*#1^6 + 14*#1^8 - 32*#1^10 + 24*#1^12 - 8*#1^14 + #1^16 & , Log[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]] - #1]/(-#1 + 4*#1^5 - 4*#1^7 + #1^9) & ]","A",1
2592,1,178,224,0.0017121,"\int \frac{-1+x^2}{\left(1+x^2\right) \sqrt{1+\sqrt{1+\sqrt{1+x}}}} \, dx","IntegrateAlgebraic[(-1 + x^2)/((1 + x^2)*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]),x]","-\frac{16}{105} \left(-2+9 \sqrt{1+x}\right) \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\frac{8}{105} \sqrt{1+\sqrt{1+x}} \left(4+15 \sqrt{1+x}\right) \sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{RootSum}\left[2-8 \text{$\#$1}^4+8 \text{$\#$1}^6+14 \text{$\#$1}^8-32 \text{$\#$1}^{10}+24 \text{$\#$1}^{12}-8 \text{$\#$1}^{14}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{$\#$1}\right)}{-\text{$\#$1}+4 \text{$\#$1}^5-4 \text{$\#$1}^7+\text{$\#$1}^9}\&\right]","\frac{32}{105} \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\frac{32}{105} \sqrt{1+\sqrt{1+x}} \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\sqrt{1+x} \left(-\frac{48}{35} \sqrt{1+\sqrt{1+\sqrt{1+x}}}+\frac{8}{7} \sqrt{1+\sqrt{1+x}} \sqrt{1+\sqrt{1+\sqrt{1+x}}}\right)-\text{RootSum}\left[2-8 \text{$\#$1}^4+8 \text{$\#$1}^6+14 \text{$\#$1}^8-32 \text{$\#$1}^{10}+24 \text{$\#$1}^{12}-8 \text{$\#$1}^{14}+\text{$\#$1}^{16}\&,\frac{\log \left(\sqrt{1+\sqrt{1+\sqrt{1+x}}}-\text{$\#$1}\right)}{-\text{$\#$1}+4 \text{$\#$1}^5-4 \text{$\#$1}^7+\text{$\#$1}^9}\&\right]",1,"(-16*(-2 + 9*Sqrt[1 + x])*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/105 + (8*Sqrt[1 + Sqrt[1 + x]]*(4 + 15*Sqrt[1 + x])*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/105 - RootSum[2 - 8*#1^4 + 8*#1^6 + 14*#1^8 - 32*#1^10 + 24*#1^12 - 8*#1^14 + #1^16 & , Log[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]] - #1]/(-#1 + 4*#1^5 - 4*#1^7 + #1^9) & ]","A",1
2593,1,225,225,0.4476813,"\int \frac{(-2+x) \sqrt[3]{x-x^2+x^3}}{(-1+x) \left(-1+x+x^2\right)} \, dx","IntegrateAlgebraic[((-2 + x)*(x - x^2 + x^3)^(1/3))/((-1 + x)*(-1 + x + x^2)),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x-x^2+x^3}}\right)+\sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x-x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{x-x^2+x^3}\right)+\sqrt[3]{2} \log \left(-2 x+2^{2/3} \sqrt[3]{x-x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{x-x^2+x^3}+\left(x-x^2+x^3\right)^{2/3}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x-x^2+x^3}+\sqrt[3]{2} \left(x-x^2+x^3\right)^{2/3}\right)}{2^{2/3}}","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x-x^2+x^3}}\right)+\sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x-x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{x-x^2+x^3}\right)+\sqrt[3]{2} \log \left(-2 x+2^{2/3} \sqrt[3]{x-x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{x-x^2+x^3}+\left(x-x^2+x^3\right)^{2/3}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x-x^2+x^3}+\sqrt[3]{2} \left(x-x^2+x^3\right)^{2/3}\right)}{2^{2/3}}",1,"-(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(x - x^2 + x^3)^(1/3))]) + 2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(x - x^2 + x^3)^(1/3))] - Log[-x + (x - x^2 + x^3)^(1/3)] + 2^(1/3)*Log[-2*x + 2^(2/3)*(x - x^2 + x^3)^(1/3)] + Log[x^2 + x*(x - x^2 + x^3)^(1/3) + (x - x^2 + x^3)^(2/3)]/2 - Log[2*x^2 + 2^(2/3)*x*(x - x^2 + x^3)^(1/3) + 2^(1/3)*(x - x^2 + x^3)^(2/3)]/2^(2/3)","A",1
2594,1,225,225,0.6218085,"\int \frac{\left(1+x^2\right) \sqrt[3]{x^2+x^3}}{-1+x^2} \, dx","IntegrateAlgebraic[((1 + x^2)*(x^2 + x^3)^(1/3))/(-1 + x^2),x]","\frac{1}{6} (1+3 x) \sqrt[3]{x^2+x^3}-\frac{17 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^3}}\right)}{3 \sqrt{3}}+\sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x^2+x^3}}\right)-\frac{17}{9} \log \left(-x+\sqrt[3]{x^2+x^3}\right)+\sqrt[3]{2} \log \left(-2 x+2^{2/3} \sqrt[3]{x^2+x^3}\right)+\frac{17}{18} \log \left(x^2+x \sqrt[3]{x^2+x^3}+\left(x^2+x^3\right)^{2/3}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x^2+x^3}+\sqrt[3]{2} \left(x^2+x^3\right)^{2/3}\right)}{2^{2/3}}","\frac{1}{6} (1+3 x) \sqrt[3]{x^2+x^3}-\frac{17 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^3}}\right)}{3 \sqrt{3}}+\sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x^2+x^3}}\right)-\frac{17}{9} \log \left(-x+\sqrt[3]{x^2+x^3}\right)+\sqrt[3]{2} \log \left(-2 x+2^{2/3} \sqrt[3]{x^2+x^3}\right)+\frac{17}{18} \log \left(x^2+x \sqrt[3]{x^2+x^3}+\left(x^2+x^3\right)^{2/3}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x^2+x^3}+\sqrt[3]{2} \left(x^2+x^3\right)^{2/3}\right)}{2^{2/3}}",1,"((1 + 3*x)*(x^2 + x^3)^(1/3))/6 - (17*ArcTan[(Sqrt[3]*x)/(x + 2*(x^2 + x^3)^(1/3))])/(3*Sqrt[3]) + 2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(x^2 + x^3)^(1/3))] - (17*Log[-x + (x^2 + x^3)^(1/3)])/9 + 2^(1/3)*Log[-2*x + 2^(2/3)*(x^2 + x^3)^(1/3)] + (17*Log[x^2 + x*(x^2 + x^3)^(1/3) + (x^2 + x^3)^(2/3)])/18 - Log[2*x^2 + 2^(2/3)*x*(x^2 + x^3)^(1/3) + 2^(1/3)*(x^2 + x^3)^(2/3)]/2^(2/3)","A",1
2595,1,225,225,4.5579497,"\int \frac{-b+a x}{x \sqrt[3]{-b^3+a^3 x^3}} \, dx","IntegrateAlgebraic[(-b + a*x)/(x*(-b^3 + a^3*x^3)^(1/3)),x]","-\frac{\tan ^{-1}\left(\frac{\frac{x}{\sqrt{3}}+\frac{2 \sqrt[3]{-b^3+a^3 x^3}}{\sqrt{3} a}}{x}\right)}{\sqrt{3}}+\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-b^3+a^3 x^3}}{\sqrt{3} b}\right)}{\sqrt{3}}-\frac{2}{3} \tanh ^{-1}\left(\frac{-b-a x}{b-a x+2 \sqrt[3]{-b^3+a^3 x^3}}\right)-\frac{1}{6} \log \left(b^2-b \sqrt[3]{-b^3+a^3 x^3}+\left(-b^3+a^3 x^3\right)^{2/3}\right)+\frac{1}{6} \log \left(a^2 x^2+a x \sqrt[3]{-b^3+a^3 x^3}+\left(-b^3+a^3 x^3\right)^{2/3}\right)","-\frac{\tan ^{-1}\left(\frac{\frac{x}{\sqrt{3}}+\frac{2 \sqrt[3]{-b^3+a^3 x^3}}{\sqrt{3} a}}{x}\right)}{\sqrt{3}}+\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-b^3+a^3 x^3}}{\sqrt{3} b}\right)}{\sqrt{3}}-\frac{2}{3} \tanh ^{-1}\left(\frac{-b-a x}{b-a x+2 \sqrt[3]{-b^3+a^3 x^3}}\right)-\frac{1}{6} \log \left(b^2-b \sqrt[3]{-b^3+a^3 x^3}+\left(-b^3+a^3 x^3\right)^{2/3}\right)+\frac{1}{6} \log \left(a^2 x^2+a x \sqrt[3]{-b^3+a^3 x^3}+\left(-b^3+a^3 x^3\right)^{2/3}\right)",1,"-(ArcTan[(x/Sqrt[3] + (2*(-b^3 + a^3*x^3)^(1/3))/(Sqrt[3]*a))/x]/Sqrt[3]) + ArcTan[1/Sqrt[3] - (2*(-b^3 + a^3*x^3)^(1/3))/(Sqrt[3]*b)]/Sqrt[3] - (2*ArcTanh[(-b - a*x)/(b - a*x + 2*(-b^3 + a^3*x^3)^(1/3))])/3 - Log[b^2 - b*(-b^3 + a^3*x^3)^(1/3) + (-b^3 + a^3*x^3)^(2/3)]/6 + Log[a^2*x^2 + a*x*(-b^3 + a^3*x^3)^(1/3) + (-b^3 + a^3*x^3)^(2/3)]/6","A",1
2596,1,225,225,0.4945267,"\int \frac{\left(-2+x^3\right) \sqrt[3]{x+2 x^3+x^4}}{\left(1+x^3\right) \left(1+x^2+x^3\right)} \, dx","IntegrateAlgebraic[((-2 + x^3)*(x + 2*x^3 + x^4)^(1/3))/((1 + x^3)*(1 + x^2 + x^3)),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+2 x^3+x^4}}\right)+\sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x+2 x^3+x^4}}\right)-\log \left(-x+\sqrt[3]{x+2 x^3+x^4}\right)+\sqrt[3]{2} \log \left(-2 x+2^{2/3} \sqrt[3]{x+2 x^3+x^4}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{x+2 x^3+x^4}+\left(x+2 x^3+x^4\right)^{2/3}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x+2 x^3+x^4}+\sqrt[3]{2} \left(x+2 x^3+x^4\right)^{2/3}\right)}{2^{2/3}}","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+2 x^3+x^4}}\right)+\sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x+2 x^3+x^4}}\right)-\log \left(-x+\sqrt[3]{x+2 x^3+x^4}\right)+\sqrt[3]{2} \log \left(-2 x+2^{2/3} \sqrt[3]{x+2 x^3+x^4}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{x+2 x^3+x^4}+\left(x+2 x^3+x^4\right)^{2/3}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x+2 x^3+x^4}+\sqrt[3]{2} \left(x+2 x^3+x^4\right)^{2/3}\right)}{2^{2/3}}",1,"-(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(x + 2*x^3 + x^4)^(1/3))]) + 2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(x + 2*x^3 + x^4)^(1/3))] - Log[-x + (x + 2*x^3 + x^4)^(1/3)] + 2^(1/3)*Log[-2*x + 2^(2/3)*(x + 2*x^3 + x^4)^(1/3)] + Log[x^2 + x*(x + 2*x^3 + x^4)^(1/3) + (x + 2*x^3 + x^4)^(2/3)]/2 - Log[2*x^2 + 2^(2/3)*x*(x + 2*x^3 + x^4)^(1/3) + 2^(1/3)*(x + 2*x^3 + x^4)^(2/3)]/2^(2/3)","A",1
2597,1,225,225,0.7632724,"\int \frac{x^4}{\sqrt[4]{-1+x^4} \left(1-2 x^4+2 x^8\right)} \, dx","IntegrateAlgebraic[x^4/((-1 + x^4)^(1/4)*(1 - 2*x^4 + 2*x^8)),x]","-\frac{1}{8} \sqrt{2+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{-1+x^4}}{-x^2+\sqrt{-1+x^4}}\right)+\frac{1}{8} \sqrt{2-\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{-1+x^4}}{-x^2+\sqrt{-1+x^4}}\right)-\frac{1}{8} \sqrt{2+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{-1+x^4}}{x^2+\sqrt{-1+x^4}}\right)+\frac{1}{8} \sqrt{2-\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{-1+x^4}}{x^2+\sqrt{-1+x^4}}\right)","-\frac{1}{8} \sqrt{2+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{-1+x^4}}{-x^2+\sqrt{-1+x^4}}\right)+\frac{1}{8} \sqrt{2-\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{-1+x^4}}{-x^2+\sqrt{-1+x^4}}\right)-\frac{1}{8} \sqrt{2+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{-1+x^4}}{x^2+\sqrt{-1+x^4}}\right)+\frac{1}{8} \sqrt{2-\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{-1+x^4}}{x^2+\sqrt{-1+x^4}}\right)",1,"-1/8*(Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]]*x*(-1 + x^4)^(1/4))/(-x^2 + Sqrt[-1 + x^4])]) + (Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]]*x*(-1 + x^4)^(1/4))/(-x^2 + Sqrt[-1 + x^4])])/8 - (Sqrt[2 + Sqrt[2]]*ArcTanh[(Sqrt[2 - Sqrt[2]]*x*(-1 + x^4)^(1/4))/(x^2 + Sqrt[-1 + x^4])])/8 + (Sqrt[2 - Sqrt[2]]*ArcTanh[(Sqrt[2 + Sqrt[2]]*x*(-1 + x^4)^(1/4))/(x^2 + Sqrt[-1 + x^4])])/8","A",1
2598,1,225,225,0.6502511,"\int \frac{\left(-1+x^4\right)^{3/4}}{1-2 x^4+2 x^8} \, dx","IntegrateAlgebraic[(-1 + x^4)^(3/4)/(1 - 2*x^4 + 2*x^8),x]","-\frac{1}{8} \sqrt{2-\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{-1+x^4}}{-x^2+\sqrt{-1+x^4}}\right)-\frac{1}{8} \sqrt{2+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{-1+x^4}}{-x^2+\sqrt{-1+x^4}}\right)-\frac{1}{8} \sqrt{2-\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{-1+x^4}}{x^2+\sqrt{-1+x^4}}\right)-\frac{1}{8} \sqrt{2+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{-1+x^4}}{x^2+\sqrt{-1+x^4}}\right)","-\frac{1}{8} \sqrt{2-\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{-1+x^4}}{-x^2+\sqrt{-1+x^4}}\right)-\frac{1}{8} \sqrt{2+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{-1+x^4}}{-x^2+\sqrt{-1+x^4}}\right)-\frac{1}{8} \sqrt{2-\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{-1+x^4}}{x^2+\sqrt{-1+x^4}}\right)-\frac{1}{8} \sqrt{2+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{-1+x^4}}{x^2+\sqrt{-1+x^4}}\right)",1,"-1/8*(Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]]*x*(-1 + x^4)^(1/4))/(-x^2 + Sqrt[-1 + x^4])]) - (Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]]*x*(-1 + x^4)^(1/4))/(-x^2 + Sqrt[-1 + x^4])])/8 - (Sqrt[2 - Sqrt[2]]*ArcTanh[(Sqrt[2 - Sqrt[2]]*x*(-1 + x^4)^(1/4))/(x^2 + Sqrt[-1 + x^4])])/8 - (Sqrt[2 + Sqrt[2]]*ArcTanh[(Sqrt[2 + Sqrt[2]]*x*(-1 + x^4)^(1/4))/(x^2 + Sqrt[-1 + x^4])])/8","A",1
2599,1,225,225,0.7028388,"\int \frac{x^4}{\sqrt[4]{1+x^4} \left(1+2 x^4+2 x^8\right)} \, dx","IntegrateAlgebraic[x^4/((1 + x^4)^(1/4)*(1 + 2*x^4 + 2*x^8)),x]","\frac{1}{8} \sqrt{2+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{1+x^4}}{-x^2+\sqrt{1+x^4}}\right)-\frac{1}{8} \sqrt{2-\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{1+x^4}}{-x^2+\sqrt{1+x^4}}\right)+\frac{1}{8} \sqrt{2+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{1+x^4}}{x^2+\sqrt{1+x^4}}\right)-\frac{1}{8} \sqrt{2-\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{1+x^4}}{x^2+\sqrt{1+x^4}}\right)","\frac{1}{8} \sqrt{2+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{1+x^4}}{-x^2+\sqrt{1+x^4}}\right)-\frac{1}{8} \sqrt{2-\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{1+x^4}}{-x^2+\sqrt{1+x^4}}\right)+\frac{1}{8} \sqrt{2+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{1+x^4}}{x^2+\sqrt{1+x^4}}\right)-\frac{1}{8} \sqrt{2-\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{1+x^4}}{x^2+\sqrt{1+x^4}}\right)",1,"(Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]]*x*(1 + x^4)^(1/4))/(-x^2 + Sqrt[1 + x^4])])/8 - (Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]]*x*(1 + x^4)^(1/4))/(-x^2 + Sqrt[1 + x^4])])/8 + (Sqrt[2 + Sqrt[2]]*ArcTanh[(Sqrt[2 - Sqrt[2]]*x*(1 + x^4)^(1/4))/(x^2 + Sqrt[1 + x^4])])/8 - (Sqrt[2 - Sqrt[2]]*ArcTanh[(Sqrt[2 + Sqrt[2]]*x*(1 + x^4)^(1/4))/(x^2 + Sqrt[1 + x^4])])/8","A",1
2600,1,225,225,0.6272014,"\int \frac{\left(1+x^4\right)^{3/4}}{1+2 x^4+2 x^8} \, dx","IntegrateAlgebraic[(1 + x^4)^(3/4)/(1 + 2*x^4 + 2*x^8),x]","\frac{1}{8} \sqrt{2-\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{1+x^4}}{-x^2+\sqrt{1+x^4}}\right)+\frac{1}{8} \sqrt{2+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{1+x^4}}{-x^2+\sqrt{1+x^4}}\right)+\frac{1}{8} \sqrt{2-\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{1+x^4}}{x^2+\sqrt{1+x^4}}\right)+\frac{1}{8} \sqrt{2+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{1+x^4}}{x^2+\sqrt{1+x^4}}\right)","\frac{1}{8} \sqrt{2-\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{1+x^4}}{-x^2+\sqrt{1+x^4}}\right)+\frac{1}{8} \sqrt{2+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{1+x^4}}{-x^2+\sqrt{1+x^4}}\right)+\frac{1}{8} \sqrt{2-\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{1+x^4}}{x^2+\sqrt{1+x^4}}\right)+\frac{1}{8} \sqrt{2+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{1+x^4}}{x^2+\sqrt{1+x^4}}\right)",1,"(Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]]*x*(1 + x^4)^(1/4))/(-x^2 + Sqrt[1 + x^4])])/8 + (Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]]*x*(1 + x^4)^(1/4))/(-x^2 + Sqrt[1 + x^4])])/8 + (Sqrt[2 - Sqrt[2]]*ArcTanh[(Sqrt[2 - Sqrt[2]]*x*(1 + x^4)^(1/4))/(x^2 + Sqrt[1 + x^4])])/8 + (Sqrt[2 + Sqrt[2]]*ArcTanh[(Sqrt[2 + Sqrt[2]]*x*(1 + x^4)^(1/4))/(x^2 + Sqrt[1 + x^4])])/8","A",1
2601,1,225,225,6.2065039,"\int \frac{x^2 \sqrt{-b x+a^2 x^2}}{\left(a x^2+x \sqrt{-b x+a^2 x^2}\right)^{3/2}} \, dx","IntegrateAlgebraic[(x^2*Sqrt[-(b*x) + a^2*x^2])/(a*x^2 + x*Sqrt[-(b*x) + a^2*x^2])^(3/2),x]","\frac{\sqrt{-b x+a^2 x^2} \left(-15 b^2-104 a^2 b x+96 a^4 x^2\right) \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{120 a^2 b^2 x}+\sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)} \left(\frac{5 b^2+152 a^2 b x-96 a^4 x^2}{120 a b^2}+\frac{\sqrt{b} \sqrt{-a x+\sqrt{-b x+a^2 x^2}} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{-a x+\sqrt{-b x+a^2 x^2}}}{\sqrt{b}}\right)}{8 \sqrt{2} a^{5/2} x}\right)","\frac{\sqrt{-b x+a^2 x^2} \left(-15 b^2-104 a^2 b x+96 a^4 x^2\right) \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{120 a^2 b^2 x}+\sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)} \left(\frac{5 b^2+152 a^2 b x-96 a^4 x^2}{120 a b^2}+\frac{\sqrt{b} \sqrt{-a x+\sqrt{-b x+a^2 x^2}} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{-a x+\sqrt{-b x+a^2 x^2}}}{\sqrt{b}}\right)}{8 \sqrt{2} a^{5/2} x}\right)",1,"(Sqrt[-(b*x) + a^2*x^2]*(-15*b^2 - 104*a^2*b*x + 96*a^4*x^2)*Sqrt[x*(a*x + Sqrt[-(b*x) + a^2*x^2])])/(120*a^2*b^2*x) + Sqrt[x*(a*x + Sqrt[-(b*x) + a^2*x^2])]*((5*b^2 + 152*a^2*b*x - 96*a^4*x^2)/(120*a*b^2) + (Sqrt[b]*Sqrt[-(a*x) + Sqrt[-(b*x) + a^2*x^2]]*ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[-(a*x) + Sqrt[-(b*x) + a^2*x^2]])/Sqrt[b]])/(8*Sqrt[2]*a^(5/2)*x))","A",1
2602,1,225,225,1.4035562,"\int \frac{\sqrt{x^2+\sqrt{1+x^4}}}{(1+x) \sqrt{1+x^4}} \, dx","IntegrateAlgebraic[Sqrt[x^2 + Sqrt[1 + x^4]]/((1 + x)*Sqrt[1 + x^4]),x]","\sqrt{\frac{1}{2} \left(-1+\sqrt{2}\right)} \tan ^{-1}\left(\sqrt{1+\sqrt{2}} \sqrt{x^2+\sqrt{1+x^4}}\right)-\sqrt{\frac{1}{2} \left(-1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{-2+2 \sqrt{2}} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\sqrt{-1+\sqrt{2}} \sqrt{x^2+\sqrt{1+x^4}}\right)+\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2+2 \sqrt{2}} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)","\sqrt{\frac{1}{2} \left(-1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{x^2+\sqrt{1+x^4}}}{\sqrt{-1+\sqrt{2}}}\right)-\sqrt{\frac{1}{2} \left(-1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(-1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{x^2+\sqrt{1+x^4}}}{\sqrt{1+\sqrt{2}}}\right)+\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)",1,"Sqrt[(-1 + Sqrt[2])/2]*ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[x^2 + Sqrt[1 + x^4]]] - Sqrt[(-1 + Sqrt[2])/2]*ArcTan[(Sqrt[-2 + 2*Sqrt[2]]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2 + Sqrt[1 + x^4])] - Sqrt[(1 + Sqrt[2])/2]*ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[x^2 + Sqrt[1 + x^4]]] + Sqrt[(1 + Sqrt[2])/2]*ArcTanh[(Sqrt[2 + 2*Sqrt[2]]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2 + Sqrt[1 + x^4])]","A",1
2603,1,226,226,2.9505734,"\int \frac{-b+a x^2}{\left(-b+2 a x^3\right) \sqrt[3]{b^2 x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[(-b + a*x^2)/((-b + 2*a*x^3)*(b^2*x^2 + a^3*x^3)^(1/3)),x]","-\frac{1}{3} \text{RootSum}\left[a^9+2 a b^5-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-a^6 \log (x)+a b^3 \log (x)+a^6 \log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)-a b^3 \log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)+2 a^3 \log (x) \text{$\#$1}^3-2 a^3 \log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right) \text{$\#$1}^3-\log (x) \text{$\#$1}^6+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right) \text{$\#$1}^6}{a^6 \text{$\#$1}-2 a^3 \text{$\#$1}^4+\text{$\#$1}^7}\&\right]","-\frac{1}{3} \text{RootSum}\left[a^9+2 a b^5-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-a^6 \log (x)+a b^3 \log (x)+a^6 \log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)-a b^3 \log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)+2 a^3 \log (x) \text{$\#$1}^3-2 a^3 \log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right) \text{$\#$1}^3-\log (x) \text{$\#$1}^6+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right) \text{$\#$1}^6}{a^6 \text{$\#$1}-2 a^3 \text{$\#$1}^4+\text{$\#$1}^7}\&\right]",1,"-1/3*RootSum[a^9 + 2*a*b^5 - 3*a^6*#1^3 + 3*a^3*#1^6 - #1^9 & , (-(a^6*Log[x]) + a*b^3*Log[x] + a^6*Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1] - a*b^3*Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1] + 2*a^3*Log[x]*#1^3 - 2*a^3*Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1]*#1^3 - Log[x]*#1^6 + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1]*#1^6)/(a^6*#1 - 2*a^3*#1^4 + #1^7) & ]","A",1
2604,1,226,226,0.0023516,"\int \frac{-b+a x^2}{\left(-b+2 a x^3\right) \sqrt[3]{b^2 x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[(-b + a*x^2)/((-b + 2*a*x^3)*(b^2*x^2 + a^3*x^3)^(1/3)),x]","-\frac{1}{3} \text{RootSum}\left[a^9+2 a b^5-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-a^6 \log (x)+a b^3 \log (x)+a^6 \log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)-a b^3 \log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)+2 a^3 \log (x) \text{$\#$1}^3-2 a^3 \log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right) \text{$\#$1}^3-\log (x) \text{$\#$1}^6+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right) \text{$\#$1}^6}{a^6 \text{$\#$1}-2 a^3 \text{$\#$1}^4+\text{$\#$1}^7}\&\right]","-\frac{1}{3} \text{RootSum}\left[a^9+2 a b^5-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-a^6 \log (x)+a b^3 \log (x)+a^6 \log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)-a b^3 \log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)+2 a^3 \log (x) \text{$\#$1}^3-2 a^3 \log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right) \text{$\#$1}^3-\log (x) \text{$\#$1}^6+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right) \text{$\#$1}^6}{a^6 \text{$\#$1}-2 a^3 \text{$\#$1}^4+\text{$\#$1}^7}\&\right]",1,"-1/3*RootSum[a^9 + 2*a*b^5 - 3*a^6*#1^3 + 3*a^3*#1^6 - #1^9 & , (-(a^6*Log[x]) + a*b^3*Log[x] + a^6*Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1] - a*b^3*Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1] + 2*a^3*Log[x]*#1^3 - 2*a^3*Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1]*#1^3 - Log[x]*#1^6 + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1]*#1^6)/(a^6*#1 - 2*a^3*#1^4 + #1^7) & ]","A",1
2605,1,226,226,1.3678168,"\int \frac{2 a b x-3 a x^2+x^3}{\sqrt[3]{x^2 (-a+x) (-b+x)} \left(-a^2+2 a x-(1+b d) x^2+d x^3\right)} \, dx","IntegrateAlgebraic[(2*a*b*x - 3*a*x^2 + x^3)/((x^2*(-a + x)*(-b + x))^(1/3)*(-a^2 + 2*a*x - (1 + b*d)*x^2 + d*x^3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}{-2 a+2 x+\sqrt[3]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{d^{2/3}}+\frac{\log \left(a-x+\sqrt[3]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}\right)}{d^{2/3}}-\frac{\log \left(a^2-2 a x+x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{a b x^2+(-a-b) x^3+x^4}+d^{2/3} \left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}\right)}{2 d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}{-2 a+2 x+\sqrt[3]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{d^{2/3}}+\frac{\log \left(a-x+\sqrt[3]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}\right)}{d^{2/3}}-\frac{\log \left(a^2-2 a x+x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{a b x^2+(-a-b) x^3+x^4}+d^{2/3} \left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}\right)}{2 d^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))/(-2*a + 2*x + d^(1/3)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))])/d^(2/3) + Log[a - x + d^(1/3)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3)]/d^(2/3) - Log[a^2 - 2*a*x + x^2 + (-(a*d^(1/3)) + d^(1/3)*x)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3) + d^(2/3)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(2/3)]/(2*d^(2/3))","A",1
2606,1,226,226,3.7364306,"\int \frac{\left(-2+k^2\right) x+k^2 x^3}{\sqrt[3]{\left(1-x^2\right) \left(1-k^2 x^2\right)} \left(1-d+\left(-2+d k^2\right) x^2+x^4\right)} \, dx","IntegrateAlgebraic[((-2 + k^2)*x + k^2*x^3)/(((1 - x^2)*(1 - k^2*x^2))^(1/3)*(1 - d + (-2 + d*k^2)*x^2 + x^4)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{2-2 x^2+\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{2 d^{2/3}}+\frac{\log \left(-1+x^2+\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{2 d^{2/3}}-\frac{\log \left(1-2 x^2+x^4+\left(\sqrt[3]{d}-\sqrt[3]{d} x^2\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+d^{2/3} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{4 d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{2-2 x^2+\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{2 d^{2/3}}+\frac{\log \left(-1+x^2+\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{2 d^{2/3}}-\frac{\log \left(1-2 x^2+x^4+\left(\sqrt[3]{d}-\sqrt[3]{d} x^2\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+d^{2/3} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{4 d^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))/(2 - 2*x^2 + d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))])/(2*d^(2/3)) + Log[-1 + x^2 + d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3)]/(2*d^(2/3)) - Log[1 - 2*x^2 + x^4 + (d^(1/3) - d^(1/3)*x^2)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3) + d^(2/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3)]/(4*d^(2/3))","A",1
2607,1,226,226,0.9296912,"\int \frac{(1+2 x) \sqrt[4]{x^3+x^4}}{-1+x+x^2} \, dx","IntegrateAlgebraic[((1 + 2*x)*(x^3 + x^4)^(1/4))/(-1 + x + x^2),x]","2 \sqrt[4]{x^3+x^4}+\tan ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)-\sqrt{2 \left(1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{x^3+x^4}}\right)+\sqrt{2 \left(-1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{x^3+x^4}}\right)-\tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)+\sqrt{2 \left(1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{x^3+x^4}}\right)-\sqrt{2 \left(-1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{x^3+x^4}}\right)","2 \sqrt[4]{x^3+x^4}+\tan ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)-\sqrt{2 \left(1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{x^3+x^4}}\right)+\sqrt{2 \left(-1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{x^3+x^4}}\right)-\tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)+\sqrt{2 \left(1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{x^3+x^4}}\right)-\sqrt{2 \left(-1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{x^3+x^4}}\right)",1,"2*(x^3 + x^4)^(1/4) + ArcTan[x/(x^3 + x^4)^(1/4)] - Sqrt[2*(1 + Sqrt[5])]*ArcTan[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(x^3 + x^4)^(1/4)] + Sqrt[2*(-1 + Sqrt[5])]*ArcTan[(Sqrt[1/2 + Sqrt[5]/2]*x)/(x^3 + x^4)^(1/4)] - ArcTanh[x/(x^3 + x^4)^(1/4)] + Sqrt[2*(1 + Sqrt[5])]*ArcTanh[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(x^3 + x^4)^(1/4)] - Sqrt[2*(-1 + Sqrt[5])]*ArcTanh[(Sqrt[1/2 + Sqrt[5]/2]*x)/(x^3 + x^4)^(1/4)]","A",1
2608,1,226,226,5.5545655,"\int \frac{\left(-1+2 k^2\right) x-2 k^4 x^3+k^4 x^5}{\left(\left(1-x^2\right) \left(1-k^2 x^2\right)\right)^{2/3} \left(-1+d+\left(1-2 d k^2\right) x^2+d k^4 x^4\right)} \, dx","IntegrateAlgebraic[((-1 + 2*k^2)*x - 2*k^4*x^3 + k^4*x^5)/(((1 - x^2)*(1 - k^2*x^2))^(2/3)*(-1 + d + (1 - 2*d*k^2)*x^2 + d*k^4*x^4)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}}{2-2 x^2+\sqrt[3]{d} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}}\right)}{2 d^{2/3}}-\frac{\log \left(-1+x^2+\sqrt[3]{d} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 d^{2/3}}+\frac{\log \left(1-2 x^2+x^4+\left(\sqrt[3]{d}-\sqrt[3]{d} x^2\right) \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}+d^{2/3} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{4/3}\right)}{4 d^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}}{2-2 x^2+\sqrt[3]{d} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}}\right)}{2 d^{2/3}}-\frac{\log \left(-1+x^2+\sqrt[3]{d} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 d^{2/3}}+\frac{\log \left(1-2 x^2+x^4+\left(\sqrt[3]{d}-\sqrt[3]{d} x^2\right) \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}+d^{2/3} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{4/3}\right)}{4 d^{2/3}}",1,"-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3))/(2 - 2*x^2 + d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3))])/d^(2/3) - Log[-1 + x^2 + d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3)]/(2*d^(2/3)) + Log[1 - 2*x^2 + x^4 + (d^(1/3) - d^(1/3)*x^2)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3) + d^(2/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(4/3)]/(4*d^(2/3))","A",1
2609,1,227,227,1.2149007,"\int \frac{(-a+x) (-b+x) (-2 a b+(a+b) x)}{\sqrt[3]{x (-a+x) (-b+x)} \left(a^2 b^2 d-2 a b (a+b) d x+\left(a^2+4 a b+b^2\right) d x^2-2 (a+b) d x^3+(-1+d) x^4\right)} \, dx","IntegrateAlgebraic[((-a + x)*(-b + x)*(-2*a*b + (a + b)*x))/((x*(-a + x)*(-b + x))^(1/3)*(a^2*b^2*d - 2*a*b*(a + b)*d*x + (a^2 + 4*a*b + b^2)*d*x^2 - 2*(a + b)*d*x^3 + (-1 + d)*x^4)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x-2 \sqrt[6]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}}\right)}{2 d^{5/6}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[6]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}}\right)}{2 d^{5/6}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[6]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}}\right)}{d^{5/6}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[6]{d}}+\sqrt[6]{d} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}}{x \sqrt[3]{a b x+(-a-b) x^2+x^3}}\right)}{2 d^{5/6}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x-2 \sqrt[6]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}}\right)}{2 d^{5/6}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[6]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}}\right)}{2 d^{5/6}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt[6]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}}\right)}{d^{5/6}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[6]{d}}+\sqrt[6]{d} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}}{x \sqrt[3]{a b x+(-a-b) x^2+x^3}}\right)}{2 d^{5/6}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x - 2*d^(1/6)*(a*b*x + (-a - b)*x^2 + x^3)^(1/3))])/(2*d^(5/6)) - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*d^(1/6)*(a*b*x + (-a - b)*x^2 + x^3)^(1/3))])/(2*d^(5/6)) - ArcTanh[x/(d^(1/6)*(a*b*x + (-a - b)*x^2 + x^3)^(1/3))]/d^(5/6) - ArcTanh[(x^2/d^(1/6) + d^(1/6)*(a*b*x + (-a - b)*x^2 + x^3)^(2/3))/(x*(a*b*x + (-a - b)*x^2 + x^3)^(1/3))]/(2*d^(5/6))","A",1
2610,1,223,227,0.4884659,"\int \frac{\sqrt{b^2+a^2 x^3} \left(2 b^2+c x^3+a^2 x^6\right)}{x \left(b^2+a^2 x^6\right)} \, dx","IntegrateAlgebraic[(Sqrt[b^2 + a^2*x^3]*(2*b^2 + c*x^3 + a^2*x^6))/(x*(b^2 + a^2*x^6)),x]","\frac{2}{3} \sqrt{b^2+a^2 x^3}+\frac{\left(-(-1)^{3/4} a \sqrt{a-i b} b+\sqrt[4]{-1} \sqrt{a-i b} c\right) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{b^2+a^2 x^3}}{\sqrt{a-i b} \sqrt{b}}\right)}{3 a \sqrt{b}}+\frac{\left(-\sqrt[4]{-1} a \sqrt{a+i b} b+(-1)^{3/4} \sqrt{a+i b} c\right) \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{b^2+a^2 x^3}}{\sqrt{a+i b} \sqrt{b}}\right)}{3 a \sqrt{b}}-\frac{4}{3} b \tanh ^{-1}\left(\frac{\sqrt{b^2+a^2 x^3}}{b}\right)","\frac{2}{3} \sqrt{b^2+a^2 x^3}+\frac{\left(-(-1)^{3/4} a \sqrt{a-i b} b+\sqrt[4]{-1} \sqrt{a-i b} c\right) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{b^2+a^2 x^3}}{\sqrt{a-i b} \sqrt{b}}\right)}{3 a \sqrt{b}}-\frac{4}{3} b \tanh ^{-1}\left(\frac{\sqrt{b^2+a^2 x^3}}{b}\right)+\frac{\left(-(-1)^{3/4} a \sqrt{a+i b} b-\sqrt[4]{-1} \sqrt{a+i b} c\right) \tanh ^{-1}\left(\frac{(1+i) \sqrt{b^2+a^2 x^3}}{\sqrt{2} \sqrt{a+i b} \sqrt{b}}\right)}{3 a \sqrt{b}}",1,"(2*Sqrt[b^2 + a^2*x^3])/3 + ((-((-1)^(3/4)*a*Sqrt[a - I*b]*b) + (-1)^(1/4)*Sqrt[a - I*b]*c)*ArcTan[((-1)^(1/4)*Sqrt[b^2 + a^2*x^3])/(Sqrt[a - I*b]*Sqrt[b])])/(3*a*Sqrt[b]) + ((-((-1)^(1/4)*a*Sqrt[a + I*b]*b) + (-1)^(3/4)*Sqrt[a + I*b]*c)*ArcTan[((-1)^(3/4)*Sqrt[b^2 + a^2*x^3])/(Sqrt[a + I*b]*Sqrt[b])])/(3*a*Sqrt[b]) - (4*b*ArcTanh[Sqrt[b^2 + a^2*x^3]/b])/3","A",1
2611,1,227,227,5.7475842,"\int \frac{\sqrt{x+x^4} \left(b+a x^6\right)}{-d+c x^6} \, dx","IntegrateAlgebraic[(Sqrt[x + x^4]*(b + a*x^6))/(-d + c*x^6),x]","\frac{a x \sqrt{x+x^4}}{3 c}+\frac{\sqrt{-\left(\left(\sqrt{c}+\sqrt{d}\right) \sqrt{d}\right)} (b c+a d) \tan ^{-1}\left(\frac{\sqrt{-\sqrt{c} \sqrt{d}-d} x \sqrt{x+x^4}}{\sqrt{d} (1+x) \left(1-x+x^2\right)}\right)}{3 c^{3/2} d}-\frac{\sqrt{\left(\sqrt{c}-\sqrt{d}\right) \sqrt{d}} (b c+a d) \tan ^{-1}\left(\frac{\sqrt{\sqrt{c} \sqrt{d}-d} x \sqrt{x+x^4}}{\sqrt{d} (1+x) \left(1-x+x^2\right)}\right)}{3 c^{3/2} d}+\frac{a \tanh ^{-1}\left(\frac{x^2}{\sqrt{x+x^4}}\right)}{3 c}","\frac{a x \sqrt{x+x^4}}{3 c}+\frac{\sqrt{-\left(\left(\sqrt{c}+\sqrt{d}\right) \sqrt{d}\right)} (b c+a d) \tan ^{-1}\left(\frac{\sqrt{-\sqrt{c} \sqrt{d}-d} x \sqrt{x+x^4}}{\sqrt{d} (1+x) \left(1-x+x^2\right)}\right)}{3 c^{3/2} d}-\frac{\sqrt{\left(\sqrt{c}-\sqrt{d}\right) \sqrt{d}} (b c+a d) \tan ^{-1}\left(\frac{\sqrt{\sqrt{c} \sqrt{d}-d} x \sqrt{x+x^4}}{\sqrt{d} (1+x) \left(1-x+x^2\right)}\right)}{3 c^{3/2} d}+\frac{a \tanh ^{-1}\left(\frac{x^2}{\sqrt{x+x^4}}\right)}{3 c}",1,"(a*x*Sqrt[x + x^4])/(3*c) + (Sqrt[-((Sqrt[c] + Sqrt[d])*Sqrt[d])]*(b*c + a*d)*ArcTan[(Sqrt[-(Sqrt[c]*Sqrt[d]) - d]*x*Sqrt[x + x^4])/(Sqrt[d]*(1 + x)*(1 - x + x^2))])/(3*c^(3/2)*d) - (Sqrt[(Sqrt[c] - Sqrt[d])*Sqrt[d]]*(b*c + a*d)*ArcTan[(Sqrt[Sqrt[c]*Sqrt[d] - d]*x*Sqrt[x + x^4])/(Sqrt[d]*(1 + x)*(1 - x + x^2))])/(3*c^(3/2)*d) + (a*ArcTanh[x^2/Sqrt[x + x^4]])/(3*c)","A",1
2612,1,228,228,3.0555522,"\int \frac{x (-1+k x) \left(1-2 k x+(-1+2 k) x^2\right)}{((1-x) x (1-k x))^{2/3} \left(-1+4 x+(-6+b) x^2+(4-2 b k) x^3+\left(-1+b k^2\right) x^4\right)} \, dx","IntegrateAlgebraic[(x*(-1 + k*x)*(1 - 2*k*x + (-1 + 2*k)*x^2))/(((1 - x)*x*(1 - k*x))^(2/3)*(-1 + 4*x + (-6 + b)*x^2 + (4 - 2*b*k)*x^3 + (-1 + b*k^2)*x^4)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}}{2-4 x+2 x^2+\sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}}\right)}{2 b^{2/3}}-\frac{\log \left(1-2 x+x^2-\sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 b^{2/3}}+\frac{\log \left(1-4 x+6 x^2-4 x^3+x^4+\left(\sqrt[3]{b}-2 \sqrt[3]{b} x+\sqrt[3]{b} x^2\right) \left(x+(-1-k) x^2+k x^3\right)^{2/3}+b^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{4/3}\right)}{4 b^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}}{2-4 x+2 x^2+\sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}}\right)}{2 b^{2/3}}-\frac{\log \left(1-2 x+x^2-\sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 b^{2/3}}+\frac{\log \left(1-4 x+6 x^2-4 x^3+x^4+\left(\sqrt[3]{b}-2 \sqrt[3]{b} x+\sqrt[3]{b} x^2\right) \left(x+(-1-k) x^2+k x^3\right)^{2/3}+b^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{4/3}\right)}{4 b^{2/3}}",1,"-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3))/(2 - 4*x + 2*x^2 + b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3))])/b^(2/3) - Log[1 - 2*x + x^2 - b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3)]/(2*b^(2/3)) + Log[1 - 4*x + 6*x^2 - 4*x^3 + x^4 + (b^(1/3) - 2*b^(1/3)*x + b^(1/3)*x^2)*(x + (-1 - k)*x^2 + k*x^3)^(2/3) + b^(2/3)*(x + (-1 - k)*x^2 + k*x^3)^(4/3)]/(4*b^(2/3))","A",1
2613,1,228,228,20.9659131,"\int \frac{-3 a q+4 b p x^3+a p x^4}{\sqrt[3]{q+p x^4} \left(b^3 d+c q+3 a b^2 d x+3 a^2 b d x^2+a^3 d x^3+c p x^4\right)} \, dx","IntegrateAlgebraic[(-3*a*q + 4*b*p*x^3 + a*p*x^4)/((q + p*x^4)^(1/3)*(b^3*d + c*q + 3*a*b^2*d*x + 3*a^2*b*d*x^2 + a^3*d*x^3 + c*p*x^4)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{c} \sqrt[3]{q+p x^4}}{-2 b \sqrt[3]{d}-2 a \sqrt[3]{d} x+\sqrt[3]{c} \sqrt[3]{q+p x^4}}\right)}{c^{2/3} \sqrt[3]{d}}-\frac{\log \left(b \sqrt[3]{d}+a \sqrt[3]{d} x+\sqrt[3]{c} \sqrt[3]{q+p x^4}\right)}{c^{2/3} \sqrt[3]{d}}+\frac{\log \left(b^2 d^{2/3}+2 a b d^{2/3} x+a^2 d^{2/3} x^2+\left(-b \sqrt[3]{c} \sqrt[3]{d}-a \sqrt[3]{c} \sqrt[3]{d} x\right) \sqrt[3]{q+p x^4}+c^{2/3} \left(q+p x^4\right)^{2/3}\right)}{2 c^{2/3} \sqrt[3]{d}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{c} \sqrt[3]{q+p x^4}}{-2 b \sqrt[3]{d}-2 a \sqrt[3]{d} x+\sqrt[3]{c} \sqrt[3]{q+p x^4}}\right)}{c^{2/3} \sqrt[3]{d}}-\frac{\log \left(b \sqrt[3]{d}+a \sqrt[3]{d} x+\sqrt[3]{c} \sqrt[3]{q+p x^4}\right)}{c^{2/3} \sqrt[3]{d}}+\frac{\log \left(b^2 d^{2/3}+2 a b d^{2/3} x+a^2 d^{2/3} x^2+\left(-b \sqrt[3]{c} \sqrt[3]{d}-a \sqrt[3]{c} \sqrt[3]{d} x\right) \sqrt[3]{q+p x^4}+c^{2/3} \left(q+p x^4\right)^{2/3}\right)}{2 c^{2/3} \sqrt[3]{d}}",1,"-((Sqrt[3]*ArcTan[(Sqrt[3]*c^(1/3)*(q + p*x^4)^(1/3))/(-2*b*d^(1/3) - 2*a*d^(1/3)*x + c^(1/3)*(q + p*x^4)^(1/3))])/(c^(2/3)*d^(1/3))) - Log[b*d^(1/3) + a*d^(1/3)*x + c^(1/3)*(q + p*x^4)^(1/3)]/(c^(2/3)*d^(1/3)) + Log[b^2*d^(2/3) + 2*a*b*d^(2/3)*x + a^2*d^(2/3)*x^2 + (-(b*c^(1/3)*d^(1/3)) - a*c^(1/3)*d^(1/3)*x)*(q + p*x^4)^(1/3) + c^(2/3)*(q + p*x^4)^(2/3)]/(2*c^(2/3)*d^(1/3))","A",1
2614,1,239,228,0.621002,"\int \frac{x^2}{x^2-\sqrt{b+a x} \sqrt{c+\sqrt{b+a x}}} \, dx","IntegrateAlgebraic[x^2/(x^2 - Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]]),x]","\frac{b-c^2+a x}{a}-4 a \text{RootSum}\left[b^2-2 b c^2+c^4+a^2 c \text{$\#$1}+4 b c \text{$\#$1}^2-4 c^3 \text{$\#$1}^2-a^2 \text{$\#$1}^3-2 b \text{$\#$1}^4+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{c^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2-2 c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^6}{-a^2 c-8 b c \text{$\#$1}+8 c^3 \text{$\#$1}+3 a^2 \text{$\#$1}^2+8 b \text{$\#$1}^3-24 c^2 \text{$\#$1}^3+24 c \text{$\#$1}^5-8 \text{$\#$1}^7}\&\right]","x+4 a \text{RootSum}\left[b^2-2 b c^2+c^4+a^2 c \text{$\#$1}+4 b c \text{$\#$1}^2-4 c^3 \text{$\#$1}^2-a^2 \text{$\#$1}^3-2 b \text{$\#$1}^4+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-c^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2+2 c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^4-\log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^6}{-a^2 c-8 b c \text{$\#$1}+8 c^3 \text{$\#$1}+3 a^2 \text{$\#$1}^2+8 b \text{$\#$1}^3-24 c^2 \text{$\#$1}^3+24 c \text{$\#$1}^5-8 \text{$\#$1}^7}\&\right]",1,"(b - c^2 + a*x)/a - 4*a*RootSum[b^2 - 2*b*c^2 + c^4 + a^2*c*#1 + 4*b*c*#1^2 - 4*c^3*#1^2 - a^2*#1^3 - 2*b*#1^4 + 6*c^2*#1^4 - 4*c*#1^6 + #1^8 & , (c^2*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^2 - 2*c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^4 + Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^6)/(-(a^2*c) - 8*b*c*#1 + 8*c^3*#1 + 3*a^2*#1^2 + 8*b*#1^3 - 24*c^2*#1^3 + 24*c*#1^5 - 8*#1^7) & ]","A",1
2615,1,239,228,0.0022387,"\int \frac{x^2}{x^2-\sqrt{b+a x} \sqrt{c+\sqrt{b+a x}}} \, dx","IntegrateAlgebraic[x^2/(x^2 - Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]]),x]","\frac{b-c^2+a x}{a}-4 a \text{RootSum}\left[b^2-2 b c^2+c^4+a^2 c \text{$\#$1}+4 b c \text{$\#$1}^2-4 c^3 \text{$\#$1}^2-a^2 \text{$\#$1}^3-2 b \text{$\#$1}^4+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{c^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2-2 c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^6}{-a^2 c-8 b c \text{$\#$1}+8 c^3 \text{$\#$1}+3 a^2 \text{$\#$1}^2+8 b \text{$\#$1}^3-24 c^2 \text{$\#$1}^3+24 c \text{$\#$1}^5-8 \text{$\#$1}^7}\&\right]","x+4 a \text{RootSum}\left[b^2-2 b c^2+c^4+a^2 c \text{$\#$1}+4 b c \text{$\#$1}^2-4 c^3 \text{$\#$1}^2-a^2 \text{$\#$1}^3-2 b \text{$\#$1}^4+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-c^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2+2 c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^4-\log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^6}{-a^2 c-8 b c \text{$\#$1}+8 c^3 \text{$\#$1}+3 a^2 \text{$\#$1}^2+8 b \text{$\#$1}^3-24 c^2 \text{$\#$1}^3+24 c \text{$\#$1}^5-8 \text{$\#$1}^7}\&\right]",1,"(b - c^2 + a*x)/a - 4*a*RootSum[b^2 - 2*b*c^2 + c^4 + a^2*c*#1 + 4*b*c*#1^2 - 4*c^3*#1^2 - a^2*#1^3 - 2*b*#1^4 + 6*c^2*#1^4 - 4*c*#1^6 + #1^8 & , (c^2*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^2 - 2*c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^4 + Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^6)/(-(a^2*c) - 8*b*c*#1 + 8*c^3*#1 + 3*a^2*#1^2 + 8*b*#1^3 - 24*c^2*#1^3 + 24*c*#1^5 - 8*#1^7) & ]","A",1
2616,1,229,229,5.6203416,"\int \frac{3+\left(-1+2 k^2\right) x-3 k^2 x^2-k^2 x^3}{\sqrt[3]{\left(1-x^2\right) \left(1-k^2 x^2\right)} \left(1-d-(1+2 d) x-\left(d+k^2\right) x^2+k^2 x^3\right)} \, dx","IntegrateAlgebraic[(3 + (-1 + 2*k^2)*x - 3*k^2*x^2 - k^2*x^3)/(((1 - x^2)*(1 - k^2*x^2))^(1/3)*(1 - d - (1 + 2*d)*x - (d + k^2)*x^2 + k^2*x^3)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{2 \sqrt[3]{d}+2 \sqrt[3]{d} x+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{\sqrt[3]{d}}-\frac{\log \left(-\sqrt[3]{d}-\sqrt[3]{d} x+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{\sqrt[3]{d}}+\frac{\log \left(d^{2/3}+2 d^{2/3} x+d^{2/3} x^2+\left(\sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{2 \sqrt[3]{d}+2 \sqrt[3]{d} x+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{\sqrt[3]{d}}-\frac{\log \left(-\sqrt[3]{d}-\sqrt[3]{d} x+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{\sqrt[3]{d}}+\frac{\log \left(d^{2/3}+2 d^{2/3} x+d^{2/3} x^2+\left(\sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}",1,"-((Sqrt[3]*ArcTan[(Sqrt[3]*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))/(2*d^(1/3) + 2*d^(1/3)*x + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))])/d^(1/3)) - Log[-d^(1/3) - d^(1/3)*x + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3)]/d^(1/3) + Log[d^(2/3) + 2*d^(2/3)*x + d^(2/3)*x^2 + (d^(1/3) + d^(1/3)*x)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3) + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3)]/(2*d^(1/3))","A",1
2617,1,229,229,5.6209084,"\int \frac{3+\left(1-2 k^2\right) x-3 k^2 x^2+k^2 x^3}{\sqrt[3]{\left(1-x^2\right) \left(1-k^2 x^2\right)} \left(-1+d-(1+2 d) x+\left(d+k^2\right) x^2+k^2 x^3\right)} \, dx","IntegrateAlgebraic[(3 + (1 - 2*k^2)*x - 3*k^2*x^2 + k^2*x^3)/(((1 - x^2)*(1 - k^2*x^2))^(1/3)*(-1 + d - (1 + 2*d)*x + (d + k^2)*x^2 + k^2*x^3)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{2 \sqrt[3]{d}-2 \sqrt[3]{d} x+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{\sqrt[3]{d}}-\frac{\log \left(-\sqrt[3]{d}+\sqrt[3]{d} x+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{\sqrt[3]{d}}+\frac{\log \left(d^{2/3}-2 d^{2/3} x+d^{2/3} x^2+\left(\sqrt[3]{d}-\sqrt[3]{d} x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{2 \sqrt[3]{d}-2 \sqrt[3]{d} x+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{\sqrt[3]{d}}-\frac{\log \left(-\sqrt[3]{d}+\sqrt[3]{d} x+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{\sqrt[3]{d}}+\frac{\log \left(d^{2/3}-2 d^{2/3} x+d^{2/3} x^2+\left(\sqrt[3]{d}-\sqrt[3]{d} x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}",1,"-((Sqrt[3]*ArcTan[(Sqrt[3]*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))/(2*d^(1/3) - 2*d^(1/3)*x + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))])/d^(1/3)) - Log[-d^(1/3) + d^(1/3)*x + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3)]/d^(1/3) + Log[d^(2/3) - 2*d^(2/3)*x + d^(2/3)*x^2 + (d^(1/3) - d^(1/3)*x)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3) + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3)]/(2*d^(1/3))","A",1
2618,1,228,229,1.0005314,"\int \frac{\sqrt{-b^4+a^4 x^4} \left(b^4+a^4 x^4\right)}{b^8-c x^4+a^8 x^8} \, dx","IntegrateAlgebraic[(Sqrt[-b^4 + a^4*x^4]*(b^4 + a^4*x^4))/(b^8 - c*x^4 + a^8*x^8),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{2 a^4 b^4-c} x \sqrt{-b^4+a^4 x^4}}{b^4+\sqrt{2 a^4 b^4-c} x^2-a^4 x^4}\right)}{2 \sqrt{2} \sqrt[4]{2 a^4 b^4-c}}-\frac{\tanh ^{-1}\left(\frac{-\frac{b^4}{\sqrt{2} \sqrt[4]{2 a^4 b^4-c}}+\frac{\sqrt[4]{2 a^4 b^4-c} x^2}{\sqrt{2}}+\frac{a^4 x^4}{\sqrt{2} \sqrt[4]{2 a^4 b^4-c}}}{x \sqrt{-b^4+a^4 x^4}}\right)}{2 \sqrt{2} \sqrt[4]{2 a^4 b^4-c}}","-\frac{\tan ^{-1}\left(\frac{\frac{b^4}{\sqrt{2} \sqrt[4]{2 a^4 b^4-c}}+\frac{\sqrt[4]{2 a^4 b^4-c} x^2}{\sqrt{2}}-\frac{a^4 x^4}{\sqrt{2} \sqrt[4]{2 a^4 b^4-c}}}{x \sqrt{-b^4+a^4 x^4}}\right)}{2 \sqrt{2} \sqrt[4]{2 a^4 b^4-c}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{2 a^4 b^4-c} x \sqrt{-b^4+a^4 x^4}}{-b^4+\sqrt{2 a^4 b^4-c} x^2+a^4 x^4}\right)}{2 \sqrt{2} \sqrt[4]{2 a^4 b^4-c}}",1,"ArcTan[(Sqrt[2]*(2*a^4*b^4 - c)^(1/4)*x*Sqrt[-b^4 + a^4*x^4])/(b^4 + Sqrt[2*a^4*b^4 - c]*x^2 - a^4*x^4)]/(2*Sqrt[2]*(2*a^4*b^4 - c)^(1/4)) - ArcTanh[(-(b^4/(Sqrt[2]*(2*a^4*b^4 - c)^(1/4))) + ((2*a^4*b^4 - c)^(1/4)*x^2)/Sqrt[2] + (a^4*x^4)/(Sqrt[2]*(2*a^4*b^4 - c)^(1/4)))/(x*Sqrt[-b^4 + a^4*x^4])]/(2*Sqrt[2]*(2*a^4*b^4 - c)^(1/4))","A",1
2619,1,228,229,1.167094,"\int \frac{-b^8+a^8 x^8}{\sqrt{-b^4+a^4 x^4} \left(b^8-c x^4+a^8 x^8\right)} \, dx","IntegrateAlgebraic[(-b^8 + a^8*x^8)/(Sqrt[-b^4 + a^4*x^4]*(b^8 - c*x^4 + a^8*x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{2 a^4 b^4-c} x \sqrt{-b^4+a^4 x^4}}{b^4+\sqrt{2 a^4 b^4-c} x^2-a^4 x^4}\right)}{2 \sqrt{2} \sqrt[4]{2 a^4 b^4-c}}-\frac{\tanh ^{-1}\left(\frac{-\frac{b^4}{\sqrt{2} \sqrt[4]{2 a^4 b^4-c}}+\frac{\sqrt[4]{2 a^4 b^4-c} x^2}{\sqrt{2}}+\frac{a^4 x^4}{\sqrt{2} \sqrt[4]{2 a^4 b^4-c}}}{x \sqrt{-b^4+a^4 x^4}}\right)}{2 \sqrt{2} \sqrt[4]{2 a^4 b^4-c}}","-\frac{\tan ^{-1}\left(\frac{\frac{b^4}{\sqrt{2} \sqrt[4]{2 a^4 b^4-c}}+\frac{\sqrt[4]{2 a^4 b^4-c} x^2}{\sqrt{2}}-\frac{a^4 x^4}{\sqrt{2} \sqrt[4]{2 a^4 b^4-c}}}{x \sqrt{-b^4+a^4 x^4}}\right)}{2 \sqrt{2} \sqrt[4]{2 a^4 b^4-c}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{2 a^4 b^4-c} x \sqrt{-b^4+a^4 x^4}}{-b^4+\sqrt{2 a^4 b^4-c} x^2+a^4 x^4}\right)}{2 \sqrt{2} \sqrt[4]{2 a^4 b^4-c}}",1,"ArcTan[(Sqrt[2]*(2*a^4*b^4 - c)^(1/4)*x*Sqrt[-b^4 + a^4*x^4])/(b^4 + Sqrt[2*a^4*b^4 - c]*x^2 - a^4*x^4)]/(2*Sqrt[2]*(2*a^4*b^4 - c)^(1/4)) - ArcTanh[(-(b^4/(Sqrt[2]*(2*a^4*b^4 - c)^(1/4))) + ((2*a^4*b^4 - c)^(1/4)*x^2)/Sqrt[2] + (a^4*x^4)/(Sqrt[2]*(2*a^4*b^4 - c)^(1/4)))/(x*Sqrt[-b^4 + a^4*x^4])]/(2*Sqrt[2]*(2*a^4*b^4 - c)^(1/4))","A",1
2620,1,430,229,0.4819796,"\int \frac{\sqrt{a x+\sqrt{-b+a x}}}{x^2} \, dx","IntegrateAlgebraic[Sqrt[a*x + Sqrt[-b + a*x]]/x^2,x]","-\frac{\sqrt{a x+\sqrt{-b+a x}}}{x}-a \text{RootSum}\left[1-b+b^2+4 \text{$\#$1}+6 \text{$\#$1}^2+2 b \text{$\#$1}^2+4 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{b \log \left(-1-\sqrt{-b+a x}+\sqrt{a x+\sqrt{-b+a x}}-\text{$\#$1}\right)-\log \left(-1-\sqrt{-b+a x}+\sqrt{a x+\sqrt{-b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2}{1+3 \text{$\#$1}+b \text{$\#$1}+3 \text{$\#$1}^2+\text{$\#$1}^3}\&\right]+a \text{RootSum}\left[1-8 b+16 b^2+4 \text{$\#$1}-16 b \text{$\#$1}+6 \text{$\#$1}^2+8 b \text{$\#$1}^2+4 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{-\log \left(-1-2 \sqrt{-b+a x}+2 \sqrt{a x+\sqrt{-b+a x}}-\text{$\#$1}\right)+4 b \log \left(-1-2 \sqrt{-b+a x}+2 \sqrt{a x+\sqrt{-b+a x}}-\text{$\#$1}\right)+4 \log \left(-1-2 \sqrt{-b+a x}+2 \sqrt{a x+\sqrt{-b+a x}}-\text{$\#$1}\right) \text{$\#$1}-\log \left(-1-2 \sqrt{-b+a x}+2 \sqrt{a x+\sqrt{-b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2}{1-4 b+3 \text{$\#$1}+4 b \text{$\#$1}+3 \text{$\#$1}^2+\text{$\#$1}^3}\&\right]","-\frac{\sqrt{a x+\sqrt{-b+a x}}}{x}-\frac{1}{4} a \text{RootSum}\left[b+b^2-4 b \text{$\#$1}+2 b \text{$\#$1}^2+\text{$\#$1}^4\&,\frac{-\log \left(\sqrt{-b+a x}-\sqrt{a x+\sqrt{-b+a x}}+\text{$\#$1}\right)+2 b \log \left(\sqrt{-b+a x}-\sqrt{a x+\sqrt{-b+a x}}+\text{$\#$1}\right)+2 \log \left(\sqrt{-b+a x}-\sqrt{a x+\sqrt{-b+a x}}+\text{$\#$1}\right) \text{$\#$1}-2 \log \left(\sqrt{-b+a x}-\sqrt{a x+\sqrt{-b+a x}}+\text{$\#$1}\right) \text{$\#$1}^2}{-b+b \text{$\#$1}+\text{$\#$1}^3}\&\right]",1,"-(Sqrt[a*x + Sqrt[-b + a*x]]/x) - a*RootSum[1 - b + b^2 + 4*#1 + 6*#1^2 + 2*b*#1^2 + 4*#1^3 + #1^4 & , (b*Log[-1 - Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1] - Log[-1 - Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1^2)/(1 + 3*#1 + b*#1 + 3*#1^2 + #1^3) & ] + a*RootSum[1 - 8*b + 16*b^2 + 4*#1 - 16*b*#1 + 6*#1^2 + 8*b*#1^2 + 4*#1^3 + #1^4 & , (-Log[-1 - 2*Sqrt[-b + a*x] + 2*Sqrt[a*x + Sqrt[-b + a*x]] - #1] + 4*b*Log[-1 - 2*Sqrt[-b + a*x] + 2*Sqrt[a*x + Sqrt[-b + a*x]] - #1] + 4*Log[-1 - 2*Sqrt[-b + a*x] + 2*Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1 - Log[-1 - 2*Sqrt[-b + a*x] + 2*Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1^2)/(1 - 4*b + 3*#1 + 4*b*#1 + 3*#1^2 + #1^3) & ]","A",1
2621,1,230,230,0.8698232,"\int \frac{-1+2 k x+(1-2 k) x^2}{((1-x) x (1-k x))^{2/3} \left(1-(2+b) x+(1+b k) x^2\right)} \, dx","IntegrateAlgebraic[(-1 + 2*k*x + (1 - 2*k)*x^2)/(((1 - x)*x*(1 - k*x))^(2/3)*(1 - (2 + b)*x + (1 + b*k)*x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \left(x+(-1-k) x^2+k x^3\right)^{2/3}}{2 \sqrt[3]{b} x-2 \sqrt[3]{b} k x^2+\left(x+(-1-k) x^2+k x^3\right)^{2/3}}\right)}{\sqrt[3]{b}}+\frac{\log \left(-\sqrt{b} x+\sqrt{b} k x^2+\sqrt[6]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{\sqrt[3]{b}}-\frac{\log \left(b x^2-2 b k x^3+b k^2 x^4+\left(b^{2/3} x-b^{2/3} k x^2\right) \left(x+(-1-k) x^2+k x^3\right)^{2/3}+\sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{4/3}\right)}{2 \sqrt[3]{b}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \left(x+(-1-k) x^2+k x^3\right)^{2/3}}{2 \sqrt[3]{b} x-2 \sqrt[3]{b} k x^2+\left(x+(-1-k) x^2+k x^3\right)^{2/3}}\right)}{\sqrt[3]{b}}+\frac{\log \left(-\sqrt{b} x+\sqrt{b} k x^2+\sqrt[6]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{\sqrt[3]{b}}-\frac{\log \left(b x^2-2 b k x^3+b k^2 x^4+\left(b^{2/3} x-b^{2/3} k x^2\right) \left(x+(-1-k) x^2+k x^3\right)^{2/3}+\sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{4/3}\right)}{2 \sqrt[3]{b}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(x + (-1 - k)*x^2 + k*x^3)^(2/3))/(2*b^(1/3)*x - 2*b^(1/3)*k*x^2 + (x + (-1 - k)*x^2 + k*x^3)^(2/3))])/b^(1/3) + Log[-(Sqrt[b]*x) + Sqrt[b]*k*x^2 + b^(1/6)*(x + (-1 - k)*x^2 + k*x^3)^(2/3)]/b^(1/3) - Log[b*x^2 - 2*b*k*x^3 + b*k^2*x^4 + (b^(2/3)*x - b^(2/3)*k*x^2)*(x + (-1 - k)*x^2 + k*x^3)^(2/3) + b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(4/3)]/(2*b^(1/3))","A",1
2622,1,230,230,0.8035552,"\int \frac{x^3 \left(3+x^2\right)}{\left(1+x^2\right) \sqrt[3]{1+x^2-x^3} \left(1+x^2+x^3\right)} \, dx","IntegrateAlgebraic[(x^3*(3 + x^2))/((1 + x^2)*(1 + x^2 - x^3)^(1/3)*(1 + x^2 + x^3)),x]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{1+x^2-x^3}}\right)-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2^{2/3} \sqrt[3]{1+x^2-x^3}}\right)}{\sqrt[3]{2}}+\log \left(x+\sqrt[3]{1+x^2-x^3}\right)-\frac{\log \left(2 x+2^{2/3} \sqrt[3]{1+x^2-x^3}\right)}{\sqrt[3]{2}}-\frac{1}{2} \log \left(x^2-x \sqrt[3]{1+x^2-x^3}+\left(1+x^2-x^3\right)^{2/3}\right)+\frac{\log \left(-2 x^2+2^{2/3} x \sqrt[3]{1+x^2-x^3}-\sqrt[3]{2} \left(1+x^2-x^3\right)^{2/3}\right)}{2 \sqrt[3]{2}}","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{1+x^2-x^3}}\right)-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2^{2/3} \sqrt[3]{1+x^2-x^3}}\right)}{\sqrt[3]{2}}+\log \left(x+\sqrt[3]{1+x^2-x^3}\right)-\frac{\log \left(2 x+2^{2/3} \sqrt[3]{1+x^2-x^3}\right)}{\sqrt[3]{2}}-\frac{1}{2} \log \left(x^2-x \sqrt[3]{1+x^2-x^3}+\left(1+x^2-x^3\right)^{2/3}\right)+\frac{\log \left(-2 x^2+2^{2/3} x \sqrt[3]{1+x^2-x^3}-\sqrt[3]{2} \left(1+x^2-x^3\right)^{2/3}\right)}{2 \sqrt[3]{2}}",1,"Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(1 + x^2 - x^3)^(1/3))] - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2^(2/3)*(1 + x^2 - x^3)^(1/3))])/2^(1/3) + Log[x + (1 + x^2 - x^3)^(1/3)] - Log[2*x + 2^(2/3)*(1 + x^2 - x^3)^(1/3)]/2^(1/3) - Log[x^2 - x*(1 + x^2 - x^3)^(1/3) + (1 + x^2 - x^3)^(2/3)]/2 + Log[-2*x^2 + 2^(2/3)*x*(1 + x^2 - x^3)^(1/3) - 2^(1/3)*(1 + x^2 - x^3)^(2/3)]/(2*2^(1/3))","A",1
2623,1,230,230,0.6874162,"\int \frac{\left(1+x^2\right) \sqrt[4]{-x^3+x^4}}{-1+x+2 x^2} \, dx","IntegrateAlgebraic[((1 + x^2)*(-x^3 + x^4)^(1/4))/(-1 + x + 2*x^2),x]","\frac{1}{16} (-5+4 x) \sqrt[4]{-x^3+x^4}-\frac{57}{32} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)+\frac{4}{3} \sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)+\frac{5 \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-x^3+x^4}}{-x^2+\sqrt{-x^3+x^4}}\right)}{12 \sqrt{2}}+\frac{57}{32} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)-\frac{4}{3} \sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)-\frac{5 \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{-x^3+x^4}}{\sqrt{2}}}{x \sqrt[4]{-x^3+x^4}}\right)}{12 \sqrt{2}}","\frac{1}{16} (-5+4 x) \sqrt[4]{-x^3+x^4}-\frac{57}{32} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)+\frac{4}{3} \sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)+\frac{5 \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{-x^3+x^4}}{-x^2+\sqrt{-x^3+x^4}}\right)}{12 \sqrt{2}}+\frac{57}{32} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)-\frac{4}{3} \sqrt[4]{2} \tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^3+x^4}}\right)-\frac{5 \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{-x^3+x^4}}{\sqrt{2}}}{x \sqrt[4]{-x^3+x^4}}\right)}{12 \sqrt{2}}",1,"((-5 + 4*x)*(-x^3 + x^4)^(1/4))/16 - (57*ArcTan[x/(-x^3 + x^4)^(1/4)])/32 + (4*2^(1/4)*ArcTan[(2^(1/4)*x)/(-x^3 + x^4)^(1/4)])/3 + (5*ArcTan[(Sqrt[2]*x*(-x^3 + x^4)^(1/4))/(-x^2 + Sqrt[-x^3 + x^4])])/(12*Sqrt[2]) + (57*ArcTanh[x/(-x^3 + x^4)^(1/4)])/32 - (4*2^(1/4)*ArcTanh[(2^(1/4)*x)/(-x^3 + x^4)^(1/4)])/3 - (5*ArcTanh[(x^2/Sqrt[2] + Sqrt[-x^3 + x^4]/Sqrt[2])/(x*(-x^3 + x^4)^(1/4))])/(12*Sqrt[2])","A",1
2624,1,230,230,3.2491466,"\int \frac{\sqrt[4]{b x^2+a x^4} \left(b+a x^4+x^8\right)}{b+a x^4} \, dx","IntegrateAlgebraic[((b*x^2 + a*x^4)^(1/4)*(b + a*x^4 + x^8))/(b + a*x^4),x]","\frac{\sqrt[4]{b x^2+a x^4} \left(96 a^3 x-96 a b x-7 b^2 x+4 a b x^3+32 a^2 x^5\right)}{192 a^3}+\frac{\left(-32 a^3 b+32 a b^2-7 b^3\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{128 a^{15/4}}+\frac{\left(32 a^3 b-32 a b^2+7 b^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{128 a^{15/4}}-\frac{b^2 \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]}{4 a^2}","\frac{\sqrt[4]{b x^2+a x^4} \left(96 a^3 x-96 a b x-7 b^2 x+4 a b x^3+32 a^2 x^5\right)}{192 a^3}+\frac{\left(-32 a^3 b+32 a b^2-7 b^3\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{128 a^{15/4}}+\frac{\left(32 a^3 b-32 a b^2+7 b^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{128 a^{15/4}}-\frac{b^2 \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]}{4 a^2}",1,"((b*x^2 + a*x^4)^(1/4)*(96*a^3*x - 96*a*b*x - 7*b^2*x + 4*a*b*x^3 + 32*a^2*x^5))/(192*a^3) + ((-32*a^3*b + 32*a*b^2 - 7*b^3)*ArcTan[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/(128*a^(15/4)) + ((32*a^3*b - 32*a*b^2 + 7*b^3)*ArcTanh[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/(128*a^(15/4)) - (b^2*RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (-(Log[x]*#1) + Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1)/(-a + #1^4) & ])/(4*a^2)","A",1
2625,1,230,230,0.0018557,"\int \frac{\sqrt[4]{b x^2+a x^4} \left(b+a x^4+x^8\right)}{b+a x^4} \, dx","IntegrateAlgebraic[((b*x^2 + a*x^4)^(1/4)*(b + a*x^4 + x^8))/(b + a*x^4),x]","\frac{\sqrt[4]{b x^2+a x^4} \left(96 a^3 x-96 a b x-7 b^2 x+4 a b x^3+32 a^2 x^5\right)}{192 a^3}+\frac{\left(-32 a^3 b+32 a b^2-7 b^3\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{128 a^{15/4}}+\frac{\left(32 a^3 b-32 a b^2+7 b^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{128 a^{15/4}}-\frac{b^2 \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]}{4 a^2}","\frac{\sqrt[4]{b x^2+a x^4} \left(96 a^3 x-96 a b x-7 b^2 x+4 a b x^3+32 a^2 x^5\right)}{192 a^3}+\frac{\left(-32 a^3 b+32 a b^2-7 b^3\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{128 a^{15/4}}+\frac{\left(32 a^3 b-32 a b^2+7 b^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{128 a^{15/4}}-\frac{b^2 \text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]}{4 a^2}",1,"((b*x^2 + a*x^4)^(1/4)*(96*a^3*x - 96*a*b*x - 7*b^2*x + 4*a*b*x^3 + 32*a^2*x^5))/(192*a^3) + ((-32*a^3*b + 32*a*b^2 - 7*b^3)*ArcTan[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/(128*a^(15/4)) + ((32*a^3*b - 32*a*b^2 + 7*b^3)*ArcTanh[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/(128*a^(15/4)) - (b^2*RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (-(Log[x]*#1) + Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1)/(-a + #1^4) & ])/(4*a^2)","A",1
2626,0,0,231,3.8240962,"\int \frac{1}{\sqrt[3]{-1+x^2} \left(3+x^2\right)} \, dx","IntegrateAlgebraic[1/((-1 + x^2)^(1/3)*(3 + x^2)),x]","\int \frac{1}{\sqrt[3]{-1+x^2} \left(3+x^2\right)} \, dx","\frac{\tan ^{-1}\left(\frac{2^{2/3} x}{2^{2/3} \sqrt{3}+2 \sqrt{3} \sqrt[3]{-1+x^2}}\right)}{3\ 2^{2/3} \sqrt{3}}-\frac{\tanh ^{-1}\left(\frac{3\ 2^{2/3} x \sqrt[3]{-1+x^2}}{-3 \sqrt[3]{2}-\sqrt[3]{2} x^2+3\ 2^{2/3} \sqrt[3]{-1+x^2}-6 \left(-1+x^2\right)^{2/3}}\right)}{6\ 2^{2/3}}-\frac{i \tanh ^{-1}\left(\frac{2 i \sqrt[3]{2} \sqrt{3} x-i 2^{2/3} \sqrt{3} x \sqrt[3]{-1+x^2}}{-3 \sqrt[3]{2}+\sqrt[3]{2} x^2+3\ 2^{2/3} \sqrt[3]{-1+x^2}-6 \left(-1+x^2\right)^{2/3}}\right)}{6\ 2^{2/3} \sqrt{3}}",1,"Could not integrate","F",-1
2627,1,231,231,1.2274901,"\int \frac{1+x}{\left(1+3 x+x^2\right) \sqrt[3]{1-x^3}} \, dx","IntegrateAlgebraic[(1 + x)/((1 + 3*x + x^2)*(1 - x^3)^(1/3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{5 \sqrt{3} \sqrt[3]{1-x^3}}{2 \sqrt[3]{2} 5^{2/3}-2 \sqrt[3]{2} 5^{2/3} x+5 \sqrt[3]{1-x^3}}\right)}{\sqrt[3]{2} 5^{2/3}}+\frac{\log \left(-\sqrt[3]{2} 5^{2/3}+\sqrt[3]{2} 5^{2/3} x+5 \sqrt[3]{1-x^3}\right)}{\sqrt[3]{2} 5^{2/3}}-\frac{\log \left(2^{2/3} \sqrt[3]{5}-2\ 2^{2/3} \sqrt[3]{5} x+2^{2/3} \sqrt[3]{5} x^2+\left(\sqrt[3]{2} 5^{2/3}-\sqrt[3]{2} 5^{2/3} x\right) \sqrt[3]{1-x^3}+5 \left(1-x^3\right)^{2/3}\right)}{2 \sqrt[3]{2} 5^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{5 \sqrt{3} \sqrt[3]{1-x^3}}{2 \sqrt[3]{2} 5^{2/3}-2 \sqrt[3]{2} 5^{2/3} x+5 \sqrt[3]{1-x^3}}\right)}{\sqrt[3]{2} 5^{2/3}}+\frac{\log \left(-\sqrt[3]{2} 5^{2/3}+\sqrt[3]{2} 5^{2/3} x+5 \sqrt[3]{1-x^3}\right)}{\sqrt[3]{2} 5^{2/3}}-\frac{\log \left(2^{2/3} \sqrt[3]{5}-2\ 2^{2/3} \sqrt[3]{5} x+2^{2/3} \sqrt[3]{5} x^2+\left(\sqrt[3]{2} 5^{2/3}-\sqrt[3]{2} 5^{2/3} x\right) \sqrt[3]{1-x^3}+5 \left(1-x^3\right)^{2/3}\right)}{2 \sqrt[3]{2} 5^{2/3}}",1,"(Sqrt[3]*ArcTan[(5*Sqrt[3]*(1 - x^3)^(1/3))/(2*2^(1/3)*5^(2/3) - 2*2^(1/3)*5^(2/3)*x + 5*(1 - x^3)^(1/3))])/(2^(1/3)*5^(2/3)) + Log[-(2^(1/3)*5^(2/3)) + 2^(1/3)*5^(2/3)*x + 5*(1 - x^3)^(1/3)]/(2^(1/3)*5^(2/3)) - Log[2^(2/3)*5^(1/3) - 2*2^(2/3)*5^(1/3)*x + 2^(2/3)*5^(1/3)*x^2 + (2^(1/3)*5^(2/3) - 2^(1/3)*5^(2/3)*x)*(1 - x^3)^(1/3) + 5*(1 - x^3)^(2/3)]/(2*2^(1/3)*5^(2/3))","A",1
2628,1,231,231,8.1080305,"\int \frac{3 k+2 \left(1+k^2\right) x-k \left(1+k^2\right) x^2-4 k^2 x^3-k^3 x^4}{\left(\left(1-x^2\right) \left(1-k^2 x^2\right)\right)^{2/3} \left(1-d-(1+2 d) k x-\left(1+d k^2\right) x^2+k x^3\right)} \, dx","IntegrateAlgebraic[(3*k + 2*(1 + k^2)*x - k*(1 + k^2)*x^2 - 4*k^2*x^3 - k^3*x^4)/(((1 - x^2)*(1 - k^2*x^2))^(2/3)*(1 - d - (1 + 2*d)*k*x - (1 + d*k^2)*x^2 + k*x^3)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d}+\sqrt{3} \sqrt[3]{d} k x}{\sqrt[3]{d}+\sqrt[3]{d} k x+2 \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{2/3}}-\frac{\log \left(-\sqrt[3]{d}-\sqrt[3]{d} k x+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{d^{2/3}}+\frac{\log \left(d^{2/3}+2 d^{2/3} k x+d^{2/3} k^2 x^2+\left(\sqrt[3]{d}+\sqrt[3]{d} k x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 d^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d}+\sqrt{3} \sqrt[3]{d} k x}{\sqrt[3]{d}+\sqrt[3]{d} k x+2 \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{2/3}}-\frac{\log \left(-\sqrt[3]{d}-\sqrt[3]{d} k x+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{d^{2/3}}+\frac{\log \left(d^{2/3}+2 d^{2/3} k x+d^{2/3} k^2 x^2+\left(\sqrt[3]{d}+\sqrt[3]{d} k x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 d^{2/3}}",1,"-((Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3) + Sqrt[3]*d^(1/3)*k*x)/(d^(1/3) + d^(1/3)*k*x + 2*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))])/d^(2/3)) - Log[-d^(1/3) - d^(1/3)*k*x + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3)]/d^(2/3) + Log[d^(2/3) + 2*d^(2/3)*k*x + d^(2/3)*k^2*x^2 + (d^(1/3) + d^(1/3)*k*x)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3) + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3)]/(2*d^(2/3))","A",1
2629,1,261,231,5.7400616,"\int \frac{\sqrt{-x+x^4} \left(b+a x^6\right)}{-d+c x^6} \, dx","IntegrateAlgebraic[(Sqrt[-x + x^4]*(b + a*x^6))/(-d + c*x^6),x]","\frac{a x \sqrt{-x+x^4}}{3 c}-\frac{\sqrt{-\left(\left(\sqrt{c}+\sqrt{d}\right) \sqrt{d}\right)} (b c+a d) \tan ^{-1}\left(\frac{\sqrt{-\sqrt{c} \sqrt{d}-d} x \sqrt{-x+x^4}}{\sqrt{d} (-1+x) \left(1+x+x^2\right)}\right)}{3 c^{3/2} d}+\frac{\sqrt{\left(\sqrt{c}-\sqrt{d}\right) \sqrt{d}} (b c+a d) \tan ^{-1}\left(\frac{\sqrt{\sqrt{c} \sqrt{d}-d} x \sqrt{-x+x^4}}{\sqrt{d} (-1+x) \left(1+x+x^2\right)}\right)}{3 c^{3/2} d}+\frac{a \log \left(-x^2+\sqrt{-x+x^4}\right)}{6 c}-\frac{a \log \left(c x^2+c \sqrt{-x+x^4}\right)}{6 c}","\frac{a x \sqrt{-x+x^4}}{3 c}-\frac{\sqrt{-\left(\left(\sqrt{c}+\sqrt{d}\right) \sqrt{d}\right)} (b c+a d) \tan ^{-1}\left(\frac{\sqrt{-\sqrt{c} \sqrt{d}-d} x \sqrt{-x+x^4}}{\sqrt{d} (-1+x) \left(1+x+x^2\right)}\right)}{3 c^{3/2} d}+\frac{\sqrt{\left(\sqrt{c}-\sqrt{d}\right) \sqrt{d}} (b c+a d) \tan ^{-1}\left(\frac{\sqrt{\sqrt{c} \sqrt{d}-d} x \sqrt{-x+x^4}}{\sqrt{d} (-1+x) \left(1+x+x^2\right)}\right)}{3 c^{3/2} d}-\frac{a \tanh ^{-1}\left(\frac{x^2}{\sqrt{-x+x^4}}\right)}{3 c}",1,"(a*x*Sqrt[-x + x^4])/(3*c) - (Sqrt[-((Sqrt[c] + Sqrt[d])*Sqrt[d])]*(b*c + a*d)*ArcTan[(Sqrt[-(Sqrt[c]*Sqrt[d]) - d]*x*Sqrt[-x + x^4])/(Sqrt[d]*(-1 + x)*(1 + x + x^2))])/(3*c^(3/2)*d) + (Sqrt[(Sqrt[c] - Sqrt[d])*Sqrt[d]]*(b*c + a*d)*ArcTan[(Sqrt[Sqrt[c]*Sqrt[d] - d]*x*Sqrt[-x + x^4])/(Sqrt[d]*(-1 + x)*(1 + x + x^2))])/(3*c^(3/2)*d) + (a*Log[-x^2 + Sqrt[-x + x^4]])/(6*c) - (a*Log[c*x^2 + c*Sqrt[-x + x^4]])/(6*c)","A",1
2630,1,232,232,3.1675033,"\int \frac{a^2 b-2 a^2 x+(2 a-b) x^2}{(x (-a+x) (-b+x))^{2/3} \left(a^2 d+(b-2 a d) x+(-1+d) x^2\right)} \, dx","IntegrateAlgebraic[(a^2*b - 2*a^2*x + (2*a - b)*x^2)/((x*(-a + x)*(-b + x))^(2/3)*(a^2*d + (b - 2*a*d)*x + (-1 + d)*x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}}{-2 b x+2 x^2+\sqrt[3]{d} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}}\right)}{d^{2/3}}+\frac{\log \left(b x-x^2+\sqrt[3]{d} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}\right)}{d^{2/3}}-\frac{\log \left(b^2 x^2-2 b x^3+x^4+\left(-b \sqrt[3]{d} x+\sqrt[3]{d} x^2\right) \left(a b x+(-a-b) x^2+x^3\right)^{2/3}+d^{2/3} \left(a b x+(-a-b) x^2+x^3\right)^{4/3}\right)}{2 d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}}{-2 b x+2 x^2+\sqrt[3]{d} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}}\right)}{d^{2/3}}+\frac{\log \left(b x-x^2+\sqrt[3]{d} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}\right)}{d^{2/3}}-\frac{\log \left(b^2 x^2-2 b x^3+x^4+\left(-b \sqrt[3]{d} x+\sqrt[3]{d} x^2\right) \left(a b x+(-a-b) x^2+x^3\right)^{2/3}+d^{2/3} \left(a b x+(-a-b) x^2+x^3\right)^{4/3}\right)}{2 d^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(a*b*x + (-a - b)*x^2 + x^3)^(2/3))/(-2*b*x + 2*x^2 + d^(1/3)*(a*b*x + (-a - b)*x^2 + x^3)^(2/3))])/d^(2/3) + Log[b*x - x^2 + d^(1/3)*(a*b*x + (-a - b)*x^2 + x^3)^(2/3)]/d^(2/3) - Log[b^2*x^2 - 2*b*x^3 + x^4 + (-(b*d^(1/3)*x) + d^(1/3)*x^2)*(a*b*x + (-a - b)*x^2 + x^3)^(2/3) + d^(2/3)*(a*b*x + (-a - b)*x^2 + x^3)^(4/3)]/(2*d^(2/3))","A",1
2631,1,232,232,3.5711225,"\int \frac{2 a b^2 x-b (2 a+b) x^2+x^4}{\left(x (-a+x) (-b+x)^2\right)^{2/3} \left(-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right)} \, dx","IntegrateAlgebraic[(2*a*b^2*x - b*(2*a + b)*x^2 + x^4)/((x*(-a + x)*(-b + x)^2)^(2/3)*(-(a*b^2) + b*(2*a + b)*x - (a + 2*b + d)*x^2 + x^3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} x}{\sqrt[3]{d} x+2 \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)}{d^{2/3}}+\frac{\log \left(-\sqrt[3]{d} x+\sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}\right)}{d^{2/3}}-\frac{\log \left(d^{2/3} x^2+\sqrt[3]{d} x \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}+\left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{2/3}\right)}{2 d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} x}{\sqrt[3]{d} x+2 \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)}{d^{2/3}}+\frac{\log \left(-\sqrt[3]{d} x+\sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}\right)}{d^{2/3}}-\frac{\log \left(d^{2/3} x^2+\sqrt[3]{d} x \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}+\left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{2/3}\right)}{2 d^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*x)/(d^(1/3)*x + 2*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3))])/d^(2/3) + Log[-(d^(1/3)*x) + (-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3)]/d^(2/3) - Log[d^(2/3)*x^2 + d^(1/3)*x*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3) + (-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(2/3)]/(2*d^(2/3))","A",1
2632,1,232,232,8.1106004,"\int \frac{-3 k+2 \left(1+k^2\right) x+k \left(1+k^2\right) x^2-4 k^2 x^3+k^3 x^4}{\left(\left(1-x^2\right) \left(1-k^2 x^2\right)\right)^{2/3} \left(-1+d-(1+2 d) k x+\left(1+d k^2\right) x^2+k x^3\right)} \, dx","IntegrateAlgebraic[(-3*k + 2*(1 + k^2)*x + k*(1 + k^2)*x^2 - 4*k^2*x^3 + k^3*x^4)/(((1 - x^2)*(1 - k^2*x^2))^(2/3)*(-1 + d - (1 + 2*d)*k*x + (1 + d*k^2)*x^2 + k*x^3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{-\sqrt{3} \sqrt[3]{d}+\sqrt{3} \sqrt[3]{d} k x}{-\sqrt[3]{d}+\sqrt[3]{d} k x-2 \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{2/3}}+\frac{\log \left(-\sqrt[3]{d}+\sqrt[3]{d} k x+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{d^{2/3}}-\frac{\log \left(d^{2/3}-2 d^{2/3} k x+d^{2/3} k^2 x^2+\left(\sqrt[3]{d}-\sqrt[3]{d} k x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{-\sqrt{3} \sqrt[3]{d}+\sqrt{3} \sqrt[3]{d} k x}{-\sqrt[3]{d}+\sqrt[3]{d} k x-2 \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{d^{2/3}}+\frac{\log \left(-\sqrt[3]{d}+\sqrt[3]{d} k x+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{d^{2/3}}-\frac{\log \left(d^{2/3}-2 d^{2/3} k x+d^{2/3} k^2 x^2+\left(\sqrt[3]{d}-\sqrt[3]{d} k x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 d^{2/3}}",1,"(Sqrt[3]*ArcTan[(-(Sqrt[3]*d^(1/3)) + Sqrt[3]*d^(1/3)*k*x)/(-d^(1/3) + d^(1/3)*k*x - 2*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))])/d^(2/3) + Log[-d^(1/3) + d^(1/3)*k*x + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3)]/d^(2/3) - Log[d^(2/3) - 2*d^(2/3)*k*x + d^(2/3)*k^2*x^2 + (d^(1/3) - d^(1/3)*k*x)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3) + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3)]/(2*d^(2/3))","A",1
2633,1,232,232,0.4814148,"\int \frac{1-((-1+3 x) (3+4 x))^{3/2}}{1+((-1+3 x) (3+4 x))^{3/2}} \, dx","IntegrateAlgebraic[(1 - ((-1 + 3*x)*(3 + 4*x))^(3/2))/(1 + ((-1 + 3*x)*(3 + 4*x))^(3/2)),x]","-x-\frac{8 \tanh ^{-1}\left(\frac{-\frac{39}{\sqrt{217}}-\frac{52 x}{\sqrt{217}}+\frac{8 \sqrt{-3+5 x+12 x^2}}{\sqrt{217}}}{3+4 x}\right)}{3 \sqrt{217}}-\frac{4}{3} \text{RootSum}\left[144-156 \text{$\#$1}+145 \text{$\#$1}^2+13 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{12 \log (3+4 x)-12 \log \left(4 \sqrt{-3+5 x+12 x^2}-3 \text{$\#$1}-4 x \text{$\#$1}\right)+13 \log (3+4 x) \text{$\#$1}-13 \log \left(4 \sqrt{-3+5 x+12 x^2}-3 \text{$\#$1}-4 x \text{$\#$1}\right) \text{$\#$1}-\log (3+4 x) \text{$\#$1}^2+\log \left(4 \sqrt{-3+5 x+12 x^2}-3 \text{$\#$1}-4 x \text{$\#$1}\right) \text{$\#$1}^2}{-156+290 \text{$\#$1}+39 \text{$\#$1}^2+4 \text{$\#$1}^3}\&\right]","-x-\frac{8 \tanh ^{-1}\left(\frac{-\frac{39}{\sqrt{217}}-\frac{52 x}{\sqrt{217}}+\frac{8 \sqrt{-3+5 x+12 x^2}}{\sqrt{217}}}{3+4 x}\right)}{3 \sqrt{217}}-\frac{4}{3} \text{RootSum}\left[144-156 \text{$\#$1}+145 \text{$\#$1}^2+13 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{12 \log (3+4 x)-12 \log \left(4 \sqrt{-3+5 x+12 x^2}-3 \text{$\#$1}-4 x \text{$\#$1}\right)+13 \log (3+4 x) \text{$\#$1}-13 \log \left(4 \sqrt{-3+5 x+12 x^2}-3 \text{$\#$1}-4 x \text{$\#$1}\right) \text{$\#$1}-\log (3+4 x) \text{$\#$1}^2+\log \left(4 \sqrt{-3+5 x+12 x^2}-3 \text{$\#$1}-4 x \text{$\#$1}\right) \text{$\#$1}^2}{-156+290 \text{$\#$1}+39 \text{$\#$1}^2+4 \text{$\#$1}^3}\&\right]",1,"-x - (8*ArcTanh[(-39/Sqrt[217] - (52*x)/Sqrt[217] + (8*Sqrt[-3 + 5*x + 12*x^2])/Sqrt[217])/(3 + 4*x)])/(3*Sqrt[217]) - (4*RootSum[144 - 156*#1 + 145*#1^2 + 13*#1^3 + #1^4 & , (12*Log[3 + 4*x] - 12*Log[4*Sqrt[-3 + 5*x + 12*x^2] - 3*#1 - 4*x*#1] + 13*Log[3 + 4*x]*#1 - 13*Log[4*Sqrt[-3 + 5*x + 12*x^2] - 3*#1 - 4*x*#1]*#1 - Log[3 + 4*x]*#1^2 + Log[4*Sqrt[-3 + 5*x + 12*x^2] - 3*#1 - 4*x*#1]*#1^2)/(-156 + 290*#1 + 39*#1^2 + 4*#1^3) & ])/3","A",1
2634,1,232,232,7.3501798,"\int \frac{1}{\left(-b x+a^2 x^2\right)^{3/2} \left(a x^2+x \sqrt{-b x+a^2 x^2}\right)^{3/2}} \, dx","IntegrateAlgebraic[1/((-(b*x) + a^2*x^2)^(3/2)*(a*x^2 + x*Sqrt[-(b*x) + a^2*x^2])^(3/2)),x]","\frac{2 \sqrt{-b x+a^2 x^2} \left(210 b^3-200 a^2 b^2 x-456 a^4 b x^2+1601 a^6 x^3\right) \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{1155 b^5 x^4 \left(b-a^2 x\right)}+\sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)} \left(-\frac{4 \left(245 a b^2+461 a^3 b x+2533 a^5 x^2\right)}{1155 b^5 x^3}-\frac{6 a^{11/2} \sqrt{-a x+\sqrt{-b x+a^2 x^2}} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{-a x+\sqrt{-b x+a^2 x^2}}}{\sqrt{b}}\right)}{b^{11/2} x}\right)","\frac{2 \sqrt{-b x+a^2 x^2} \left(210 b^3-200 a^2 b^2 x-456 a^4 b x^2+1601 a^6 x^3\right) \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{1155 b^5 x^4 \left(b-a^2 x\right)}+\sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)} \left(-\frac{4 \left(245 a b^2+461 a^3 b x+2533 a^5 x^2\right)}{1155 b^5 x^3}-\frac{6 a^{11/2} \sqrt{-a x+\sqrt{-b x+a^2 x^2}} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{-a x+\sqrt{-b x+a^2 x^2}}}{\sqrt{b}}\right)}{b^{11/2} x}\right)",1,"(2*Sqrt[-(b*x) + a^2*x^2]*(210*b^3 - 200*a^2*b^2*x - 456*a^4*b*x^2 + 1601*a^6*x^3)*Sqrt[x*(a*x + Sqrt[-(b*x) + a^2*x^2])])/(1155*b^5*x^4*(b - a^2*x)) + Sqrt[x*(a*x + Sqrt[-(b*x) + a^2*x^2])]*((-4*(245*a*b^2 + 461*a^3*b*x + 2533*a^5*x^2))/(1155*b^5*x^3) - (6*a^(11/2)*Sqrt[-(a*x) + Sqrt[-(b*x) + a^2*x^2]]*ArcTan[(Sqrt[a]*Sqrt[-(a*x) + Sqrt[-(b*x) + a^2*x^2]])/Sqrt[b]])/(b^(11/2)*x))","A",1
2635,1,233,233,0.5419188,"\int \frac{1}{x \sqrt[3]{(-1+x) \left(q-2 q x+x^2\right)}} \, dx","IntegrateAlgebraic[1/(x*((-1 + x)*(q - 2*q*x + x^2))^(1/3)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-q+3 q x+(-1-2 q) x^2+x^3}}{-2 \sqrt[3]{q}+2 \sqrt[3]{q} x+\sqrt[3]{-q+3 q x+(-1-2 q) x^2+x^3}}\right)}{2 \sqrt[3]{q}}-\frac{\log \left(\sqrt[3]{q}-\sqrt[3]{q} x+\sqrt[3]{-q+3 q x+(-1-2 q) x^2+x^3}\right)}{2 \sqrt[3]{q}}+\frac{\log \left(q^{2/3}-2 q^{2/3} x+q^{2/3} x^2+\left(-\sqrt[3]{q}+\sqrt[3]{q} x\right) \sqrt[3]{-q+3 q x+(-1-2 q) x^2+x^3}+\left(-q+3 q x+(-1-2 q) x^2+x^3\right)^{2/3}\right)}{4 \sqrt[3]{q}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-q+3 q x+(-1-2 q) x^2+x^3}}{-2 \sqrt[3]{q}+2 \sqrt[3]{q} x+\sqrt[3]{-q+3 q x+(-1-2 q) x^2+x^3}}\right)}{2 \sqrt[3]{q}}-\frac{\log \left(\sqrt[3]{q}-\sqrt[3]{q} x+\sqrt[3]{-q+3 q x+(-1-2 q) x^2+x^3}\right)}{2 \sqrt[3]{q}}+\frac{\log \left(q^{2/3}-2 q^{2/3} x+q^{2/3} x^2+\left(-\sqrt[3]{q}+\sqrt[3]{q} x\right) \sqrt[3]{-q+3 q x+(-1-2 q) x^2+x^3}+\left(-q+3 q x+(-1-2 q) x^2+x^3\right)^{2/3}\right)}{4 \sqrt[3]{q}}",1,"-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*(-q + 3*q*x + (-1 - 2*q)*x^2 + x^3)^(1/3))/(-2*q^(1/3) + 2*q^(1/3)*x + (-q + 3*q*x + (-1 - 2*q)*x^2 + x^3)^(1/3))])/q^(1/3) - Log[q^(1/3) - q^(1/3)*x + (-q + 3*q*x + (-1 - 2*q)*x^2 + x^3)^(1/3)]/(2*q^(1/3)) + Log[q^(2/3) - 2*q^(2/3)*x + q^(2/3)*x^2 + (-q^(1/3) + q^(1/3)*x)*(-q + 3*q*x + (-1 - 2*q)*x^2 + x^3)^(1/3) + (-q + 3*q*x + (-1 - 2*q)*x^2 + x^3)^(2/3)]/(4*q^(1/3))","A",1
2636,1,233,233,2.9522795,"\int \frac{a b+(-2 a+b) x}{\sqrt[3]{x (-a+x) (-b+x)} \left(a^2 d+(b-2 a d) x+(-1+d) x^2\right)} \, dx","IntegrateAlgebraic[(a*b + (-2*a + b)*x)/((x*(-a + x)*(-b + x))^(1/3)*(a^2*d + (b - 2*a*d)*x + (-1 + d)*x^2)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a b x+(-a-b) x^2+x^3}}{-2 a \sqrt[3]{d}+2 \sqrt[3]{d} x+\sqrt[3]{a b x+(-a-b) x^2+x^3}}\right)}{\sqrt[3]{d}}-\frac{\log \left(a \sqrt[3]{d}-\sqrt[3]{d} x+\sqrt[3]{a b x+(-a-b) x^2+x^3}\right)}{\sqrt[3]{d}}+\frac{\log \left(a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{a b x+(-a-b) x^2+x^3}+\left(a b x+(-a-b) x^2+x^3\right)^{2/3}\right)}{2 \sqrt[3]{d}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a b x+(-a-b) x^2+x^3}}{-2 a \sqrt[3]{d}+2 \sqrt[3]{d} x+\sqrt[3]{a b x+(-a-b) x^2+x^3}}\right)}{\sqrt[3]{d}}-\frac{\log \left(a \sqrt[3]{d}-\sqrt[3]{d} x+\sqrt[3]{a b x+(-a-b) x^2+x^3}\right)}{\sqrt[3]{d}}+\frac{\log \left(a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{a b x+(-a-b) x^2+x^3}+\left(a b x+(-a-b) x^2+x^3\right)^{2/3}\right)}{2 \sqrt[3]{d}}",1,"-((Sqrt[3]*ArcTan[(Sqrt[3]*(a*b*x + (-a - b)*x^2 + x^3)^(1/3))/(-2*a*d^(1/3) + 2*d^(1/3)*x + (a*b*x + (-a - b)*x^2 + x^3)^(1/3))])/d^(1/3)) - Log[a*d^(1/3) - d^(1/3)*x + (a*b*x + (-a - b)*x^2 + x^3)^(1/3)]/d^(1/3) + Log[a^2*d^(2/3) - 2*a*d^(2/3)*x + d^(2/3)*x^2 + (-(a*d^(1/3)) + d^(1/3)*x)*(a*b*x + (-a - b)*x^2 + x^3)^(1/3) + (a*b*x + (-a - b)*x^2 + x^3)^(2/3)]/(2*d^(1/3))","A",1
2637,1,233,233,0.7433908,"\int \frac{1+x^4}{\left(-1-x^2+x^4\right) \sqrt[4]{-x^2+x^4}} \, dx","IntegrateAlgebraic[(1 + x^4)/((-1 - x^2 + x^4)*(-x^2 + x^4)^(1/4)),x]","\tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)-\sqrt{\frac{1}{2} \left(1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^2+x^4}}\right)-\sqrt{\frac{1}{2} \left(-1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^2+x^4}}\right)+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)-\sqrt{\frac{1}{2} \left(1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^2+x^4}}\right)-\sqrt{\frac{1}{2} \left(-1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^2+x^4}}\right)","\tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)-\sqrt{\frac{1}{2} \left(1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^2+x^4}}\right)-\sqrt{\frac{1}{2} \left(-1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^2+x^4}}\right)+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)-\sqrt{\frac{1}{2} \left(1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^2+x^4}}\right)-\sqrt{\frac{1}{2} \left(-1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^2+x^4}}\right)",1,"ArcTan[x/(-x^2 + x^4)^(1/4)] - Sqrt[(1 + Sqrt[5])/2]*ArcTan[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(-x^2 + x^4)^(1/4)] - Sqrt[(-1 + Sqrt[5])/2]*ArcTan[(Sqrt[1/2 + Sqrt[5]/2]*x)/(-x^2 + x^4)^(1/4)] + ArcTanh[x/(-x^2 + x^4)^(1/4)] - Sqrt[(1 + Sqrt[5])/2]*ArcTanh[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(-x^2 + x^4)^(1/4)] - Sqrt[(-1 + Sqrt[5])/2]*ArcTanh[(Sqrt[1/2 + Sqrt[5]/2]*x)/(-x^2 + x^4)^(1/4)]","A",1
2638,1,233,233,3.5140866,"\int \frac{x^3 \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}{\sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}} \, dx","IntegrateAlgebraic[(x^3*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])/Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]],x]","\frac{\sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}} \left(-255-136 a x^2+384 a^2 x^4\right) \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{1920 a^2}+\frac{\left(85 x+568 a x^3-384 a^2 x^5\right) \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{1920 a b}+\frac{17 \log \left(-a x-b \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}+\sqrt{2} \sqrt{a} \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}\right)}{128 \sqrt{2} a^{3/2} b}","\frac{\sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}} \left(-255-136 a x^2+384 a^2 x^4\right) \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{1920 a^2}+\frac{\left(85 x+568 a x^3-384 a^2 x^5\right) \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{1920 a b}+\frac{17 \log \left(-a x-b \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}+\sqrt{2} \sqrt{a} \sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}\right)}{128 \sqrt{2} a^{3/2} b}",1,"(Sqrt[-(a/b^2) + (a^2*x^2)/b^2]*(-255 - 136*a*x^2 + 384*a^2*x^4)*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]])/(1920*a^2) + ((85*x + 568*a*x^3 - 384*a^2*x^5)*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]])/(1920*a*b) + (17*Log[-(a*x) - b*Sqrt[-(a/b^2) + (a^2*x^2)/b^2] + Sqrt[2]*Sqrt[a]*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]]])/(128*Sqrt[2]*a^(3/2)*b)","A",1
2639,1,234,234,0.570948,"\int \frac{\sqrt[3]{-x+x^3} \left(-2+x^4\right)}{x^4 \left(1+x^2\right)} \, dx","IntegrateAlgebraic[((-x + x^3)^(1/3)*(-2 + x^4))/(x^4*(1 + x^2)),x]","-\frac{3 \left(-1+5 x^2\right) \sqrt[3]{-x+x^3}}{4 x^3}-\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-x+x^3}}\right)}{2^{2/3}}-\frac{1}{2} \log \left(-x+\sqrt[3]{-x+x^3}\right)-\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{-x+x^3}\right)}{2^{2/3}}+\frac{1}{4} \log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)+\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-x+x^3}+\sqrt[3]{2} \left(-x+x^3\right)^{2/3}\right)}{2\ 2^{2/3}}","-\frac{3 \left(-1+5 x^2\right) \sqrt[3]{-x+x^3}}{4 x^3}-\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-x+x^3}}\right)}{2^{2/3}}-\frac{1}{2} \log \left(-x+\sqrt[3]{-x+x^3}\right)-\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{-x+x^3}\right)}{2^{2/3}}+\frac{1}{4} \log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)+\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-x+x^3}+\sqrt[3]{2} \left(-x+x^3\right)^{2/3}\right)}{2\ 2^{2/3}}",1,"(-3*(-1 + 5*x^2)*(-x + x^3)^(1/3))/(4*x^3) - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-x + x^3)^(1/3))])/2 - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-x + x^3)^(1/3))])/2^(2/3) - Log[-x + (-x + x^3)^(1/3)]/2 - Log[-2*x + 2^(2/3)*(-x + x^3)^(1/3)]/2^(2/3) + Log[x^2 + x*(-x + x^3)^(1/3) + (-x + x^3)^(2/3)]/4 + Log[2*x^2 + 2^(2/3)*x*(-x + x^3)^(1/3) + 2^(1/3)*(-x + x^3)^(2/3)]/(2*2^(2/3))","A",1
2640,1,234,234,15.1876494,"\int \frac{2 \left(3 a q x-2 b p x^3+a p x^5\right)}{\sqrt[3]{q+p x^4} \left(b^3 d+c q+3 a b^2 d x^2+\left(3 a^2 b d+c p\right) x^4+a^3 d x^6\right)} \, dx","IntegrateAlgebraic[(2*(3*a*q*x - 2*b*p*x^3 + a*p*x^5))/((q + p*x^4)^(1/3)*(b^3*d + c*q + 3*a*b^2*d*x^2 + (3*a^2*b*d + c*p)*x^4 + a^3*d*x^6)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{c} \sqrt[3]{q+p x^4}}{-2 b \sqrt[3]{d}-2 a \sqrt[3]{d} x^2+\sqrt[3]{c} \sqrt[3]{q+p x^4}}\right)}{c^{2/3} \sqrt[3]{d}}+\frac{\log \left(b \sqrt[3]{d}+a \sqrt[3]{d} x^2+\sqrt[3]{c} \sqrt[3]{q+p x^4}\right)}{c^{2/3} \sqrt[3]{d}}-\frac{\log \left(b^2 d^{2/3}+2 a b d^{2/3} x^2+a^2 d^{2/3} x^4+\left(-b \sqrt[3]{c} \sqrt[3]{d}-a \sqrt[3]{c} \sqrt[3]{d} x^2\right) \sqrt[3]{q+p x^4}+c^{2/3} \left(q+p x^4\right)^{2/3}\right)}{2 c^{2/3} \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{c} \sqrt[3]{q+p x^4}}{-2 b \sqrt[3]{d}-2 a \sqrt[3]{d} x^2+\sqrt[3]{c} \sqrt[3]{q+p x^4}}\right)}{c^{2/3} \sqrt[3]{d}}+\frac{\log \left(b \sqrt[3]{d}+a \sqrt[3]{d} x^2+\sqrt[3]{c} \sqrt[3]{q+p x^4}\right)}{c^{2/3} \sqrt[3]{d}}-\frac{\log \left(b^2 d^{2/3}+2 a b d^{2/3} x^2+a^2 d^{2/3} x^4+\left(-b \sqrt[3]{c} \sqrt[3]{d}-a \sqrt[3]{c} \sqrt[3]{d} x^2\right) \sqrt[3]{q+p x^4}+c^{2/3} \left(q+p x^4\right)^{2/3}\right)}{2 c^{2/3} \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*c^(1/3)*(q + p*x^4)^(1/3))/(-2*b*d^(1/3) - 2*a*d^(1/3)*x^2 + c^(1/3)*(q + p*x^4)^(1/3))])/(c^(2/3)*d^(1/3)) + Log[b*d^(1/3) + a*d^(1/3)*x^2 + c^(1/3)*(q + p*x^4)^(1/3)]/(c^(2/3)*d^(1/3)) - Log[b^2*d^(2/3) + 2*a*b*d^(2/3)*x^2 + a^2*d^(2/3)*x^4 + (-(b*c^(1/3)*d^(1/3)) - a*c^(1/3)*d^(1/3)*x^2)*(q + p*x^4)^(1/3) + c^(2/3)*(q + p*x^4)^(2/3)]/(2*c^(2/3)*d^(1/3))","A",1
2641,1,235,235,4.3316941,"\int \frac{\left(b^2-2 b x+x^2\right) \left(-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3\right)}{\left(x (-a+x)^2 (-b+x)^3\right)^{3/4} \left(b d+\left(a^2-d\right) x-2 a x^2+x^3\right)} \, dx","IntegrateAlgebraic[((b^2 - 2*b*x + x^2)*(-(a^2*b) + 4*a*b*x - (2*a + 3*b)*x^2 + 2*x^3))/((x*(-a + x)^2*(-b + x)^3)^(3/4)*(b*d + (a^2 - d)*x - 2*a*x^2 + x^3)),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(-a^2 b^3 x+\left(3 a^2 b^2+2 a b^3\right) x^2+\left(-3 a^2 b-6 a b^2-b^3\right) x^3+\left(a^2+6 a b+3 b^2\right) x^4+(-2 a-3 b) x^5+x^6\right)^{3/4}}{(b-x)^2 x (-a+x)^2}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \left(-a^2 b^3 x+\left(3 a^2 b^2+2 a b^3\right) x^2+\left(-3 a^2 b-6 a b^2-b^3\right) x^3+\left(a^2+6 a b+3 b^2\right) x^4+(-2 a-3 b) x^5+x^6\right)^{3/4}}{(b-x)^2 x (-a+x)^2}\right)}{d^{3/4}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{d} \left(-a^2 b^3 x+\left(3 a^2 b^2+2 a b^3\right) x^2+\left(-3 a^2 b-6 a b^2-b^3\right) x^3+\left(a^2+6 a b+3 b^2\right) x^4+(-2 a-3 b) x^5+x^6\right)^{3/4}}{(b-x)^2 x (-a+x)^2}\right)}{d^{3/4}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{d} \left(-a^2 b^3 x+\left(3 a^2 b^2+2 a b^3\right) x^2+\left(-3 a^2 b-6 a b^2-b^3\right) x^3+\left(a^2+6 a b+3 b^2\right) x^4+(-2 a-3 b) x^5+x^6\right)^{3/4}}{(b-x)^2 x (-a+x)^2}\right)}{d^{3/4}}",1,"(2*ArcTan[(d^(1/4)*(-(a^2*b^3*x) + (3*a^2*b^2 + 2*a*b^3)*x^2 + (-3*a^2*b - 6*a*b^2 - b^3)*x^3 + (a^2 + 6*a*b + 3*b^2)*x^4 + (-2*a - 3*b)*x^5 + x^6)^(3/4))/((b - x)^2*x*(-a + x)^2)])/d^(3/4) - (2*ArcTanh[(d^(1/4)*(-(a^2*b^3*x) + (3*a^2*b^2 + 2*a*b^3)*x^2 + (-3*a^2*b - 6*a*b^2 - b^3)*x^3 + (a^2 + 6*a*b + 3*b^2)*x^4 + (-2*a - 3*b)*x^5 + x^6)^(3/4))/((b - x)^2*x*(-a + x)^2)])/d^(3/4)","A",1
2642,1,235,235,0.5392532,"\int \frac{(-2+(1+k) x) \left(1-2 (1+k) x+\left(1+4 k+k^2\right) x^2-2 \left(k+k^2\right) x^3+\left(a+k^2\right) x^4\right)}{x^3 ((1-x) x (1-k x))^{2/3} \left(1-(1+k) x+(-b+k) x^2\right)} \, dx","IntegrateAlgebraic[((-2 + (1 + k)*x)*(1 - 2*(1 + k)*x + (1 + 4*k + k^2)*x^2 - 2*(k + k^2)*x^3 + (a + k^2)*x^4))/(x^3*((1 - x)*x*(1 - k*x))^(2/3)*(1 - (1 + k)*x + (-b + k)*x^2)),x]","\frac{3 \left(1-x-k x+4 b x^2+k x^2\right) \sqrt[3]{x-x^2-k x^2+k x^3}}{4 x^3}+\frac{\left(\sqrt{3} a+\sqrt{3} b^2\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} x}{\sqrt[3]{b} x+2 \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{b^{2/3}}+\frac{\left(a+b^2\right) \log \left(-\sqrt[3]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{b^{2/3}}+\frac{\left(-a-b^2\right) \log \left(b^{2/3} x^2+\sqrt[3]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 b^{2/3}}","\frac{3 \left(1-x-k x+4 b x^2+k x^2\right) \sqrt[3]{x-x^2-k x^2+k x^3}}{4 x^3}+\frac{\left(\sqrt{3} a+\sqrt{3} b^2\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} x}{\sqrt[3]{b} x+2 \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{b^{2/3}}+\frac{\left(a+b^2\right) \log \left(-\sqrt[3]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{b^{2/3}}+\frac{\left(-a-b^2\right) \log \left(b^{2/3} x^2+\sqrt[3]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 b^{2/3}}",1,"(3*(1 - x - k*x + 4*b*x^2 + k*x^2)*(x - x^2 - k*x^2 + k*x^3)^(1/3))/(4*x^3) + ((Sqrt[3]*a + Sqrt[3]*b^2)*ArcTan[(Sqrt[3]*b^(1/3)*x)/(b^(1/3)*x + 2*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/b^(2/3) + ((a + b^2)*Log[-(b^(1/3)*x) + (x + (-1 - k)*x^2 + k*x^3)^(1/3)])/b^(2/3) + ((-a - b^2)*Log[b^(2/3)*x^2 + b^(1/3)*x*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + (x + (-1 - k)*x^2 + k*x^3)^(2/3)])/(2*b^(2/3))","A",1
2643,1,235,235,0.4546155,"\int \frac{\left(1+x^3\right) \sqrt[3]{-x^2+x^3}}{1+x^6} \, dx","IntegrateAlgebraic[((1 + x^3)*(-x^2 + x^3)^(1/3))/(1 + x^6),x]","\frac{1}{3} \text{RootSum}\left[2-2 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}^2}\&\right]-\frac{1}{6} \text{RootSum}\left[1-2 \text{$\#$1}^3+5 \text{$\#$1}^6-4 \text{$\#$1}^9+\text{$\#$1}^{12}\&,\frac{\log (x)-\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3+3 \log (x) \text{$\#$1}^6-3 \log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^6-2 \log (x) \text{$\#$1}^9+2 \log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^9}{-\text{$\#$1}^2+5 \text{$\#$1}^5-6 \text{$\#$1}^8+2 \text{$\#$1}^{11}}\&\right]","\frac{1}{3} \text{RootSum}\left[2-2 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}^2}\&\right]-\frac{1}{6} \text{RootSum}\left[1-2 \text{$\#$1}^3+5 \text{$\#$1}^6-4 \text{$\#$1}^9+\text{$\#$1}^{12}\&,\frac{\log (x)-\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^3+3 \log (x) \text{$\#$1}^6-3 \log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^6-2 \log (x) \text{$\#$1}^9+2 \log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^9}{-\text{$\#$1}^2+5 \text{$\#$1}^5-6 \text{$\#$1}^8+2 \text{$\#$1}^{11}}\&\right]",1,"RootSum[2 - 2*#1^3 + #1^6 & , (-Log[x] + Log[(-x^2 + x^3)^(1/3) - x*#1])/#1^2 & ]/3 - RootSum[1 - 2*#1^3 + 5*#1^6 - 4*#1^9 + #1^12 & , (Log[x] - Log[(-x^2 + x^3)^(1/3) - x*#1] + Log[x]*#1^3 - Log[(-x^2 + x^3)^(1/3) - x*#1]*#1^3 + 3*Log[x]*#1^6 - 3*Log[(-x^2 + x^3)^(1/3) - x*#1]*#1^6 - 2*Log[x]*#1^9 + 2*Log[(-x^2 + x^3)^(1/3) - x*#1]*#1^9)/(-#1^2 + 5*#1^5 - 6*#1^8 + 2*#1^11) & ]/6","A",1
2644,1,235,235,0.6353202,"\int \frac{\sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\left(d+c x^2\right) \sqrt{b+a^2 x^4}} \, dx","IntegrateAlgebraic[Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/((d + c*x^2)*Sqrt[b + a^2*x^4]),x]","-\frac{i \sqrt{a} \text{RootSum}\left[b^2 c+4 i a b d \text{$\#$1}+2 b c \text{$\#$1}^2-4 i a d \text{$\#$1}^3+c \text{$\#$1}^4\&,\frac{b \log \left(-i a x^2-i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)-\log \left(-i a x^2-i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2}{i a b d+b c \text{$\#$1}-3 i a d \text{$\#$1}^2+c \text{$\#$1}^3}\&\right]}{\sqrt{2}}","-\frac{i \sqrt{a} \text{RootSum}\left[b^2 c-4 i a b d \text{$\#$1}+2 b c \text{$\#$1}^2+4 i a d \text{$\#$1}^3+c \text{$\#$1}^4\&,\frac{b \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)-\log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2}{-i a b d+b c \text{$\#$1}+3 i a d \text{$\#$1}^2+c \text{$\#$1}^3}\&\right]}{\sqrt{2}}",1,"((-I)*Sqrt[a]*RootSum[b^2*c + (4*I)*a*b*d*#1 + 2*b*c*#1^2 - (4*I)*a*d*#1^3 + c*#1^4 & , (b*Log[(-I)*a*x^2 - I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1] - Log[(-I)*a*x^2 - I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1]*#1^2)/(I*a*b*d + b*c*#1 - (3*I)*a*d*#1^2 + c*#1^3) & ])/Sqrt[2]","A",1
2645,1,235,235,0.0025339,"\int \frac{\sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\left(d+c x^2\right) \sqrt{b+a^2 x^4}} \, dx","IntegrateAlgebraic[Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/((d + c*x^2)*Sqrt[b + a^2*x^4]),x]","-\frac{i \sqrt{a} \text{RootSum}\left[b^2 c+4 i a b d \text{$\#$1}+2 b c \text{$\#$1}^2-4 i a d \text{$\#$1}^3+c \text{$\#$1}^4\&,\frac{b \log \left(-i a x^2-i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)-\log \left(-i a x^2-i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2}{i a b d+b c \text{$\#$1}-3 i a d \text{$\#$1}^2+c \text{$\#$1}^3}\&\right]}{\sqrt{2}}","-\frac{i \sqrt{a} \text{RootSum}\left[b^2 c-4 i a b d \text{$\#$1}+2 b c \text{$\#$1}^2+4 i a d \text{$\#$1}^3+c \text{$\#$1}^4\&,\frac{b \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)-\log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2}{-i a b d+b c \text{$\#$1}+3 i a d \text{$\#$1}^2+c \text{$\#$1}^3}\&\right]}{\sqrt{2}}",1,"((-I)*Sqrt[a]*RootSum[b^2*c + (4*I)*a*b*d*#1 + 2*b*c*#1^2 - (4*I)*a*d*#1^3 + c*#1^4 & , (b*Log[(-I)*a*x^2 - I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1] - Log[(-I)*a*x^2 - I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1]*#1^2)/(I*a*b*d + b*c*#1 - (3*I)*a*d*#1^2 + c*#1^3) & ])/Sqrt[2]","A",1
2646,1,341,235,1.043265,"\int \frac{\sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{\left(b+a^2 x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]])/(b + a^2*x^2)^(3/2),x]","-\frac{\sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{a^2 x \sqrt{b+a^2 x^2}+a \left(b+a^2 x^2\right)}+\frac{2 \text{RootSum}\left[b+c^4-4 c^3 \text{$\#$1}^2+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-c^3+3 c^2 \text{$\#$1}^2-3 c \text{$\#$1}^4+\text{$\#$1}^6}\&\right]}{a}-\frac{\text{RootSum}\left[b+c^4-4 c^3 \text{$\#$1}^2+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-c \log \left(\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}-\text{$\#$1}\right)-5 \log \left(\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{c^3 \text{$\#$1}-3 c^2 \text{$\#$1}^3+3 c \text{$\#$1}^5-\text{$\#$1}^7}\&\right]}{4 a}","-\frac{\sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{a^2 x \sqrt{b+a^2 x^2}+a \left(b+a^2 x^2\right)}-\frac{\text{RootSum}\left[b+c^4-4 c^3 \text{$\#$1}^2+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-c \log \left(\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}-\text{$\#$1}\right)+3 \log \left(\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{c^3 \text{$\#$1}-3 c^2 \text{$\#$1}^3+3 c \text{$\#$1}^5-\text{$\#$1}^7}\&\right]}{4 a}",1,"-((Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]])/(a^2*x*Sqrt[b + a^2*x^2] + a*(b + a^2*x^2))) + (2*RootSum[b + c^4 - 4*c^3*#1^2 + 6*c^2*#1^4 - 4*c*#1^6 + #1^8 & , (Log[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] - #1]*#1)/(-c^3 + 3*c^2*#1^2 - 3*c*#1^4 + #1^6) & ])/a - RootSum[b + c^4 - 4*c^3*#1^2 + 6*c^2*#1^4 - 4*c*#1^6 + #1^8 & , (-(c*Log[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] - #1]) - 5*Log[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] - #1]*#1^2)/(c^3*#1 - 3*c^2*#1^3 + 3*c*#1^5 - #1^7) & ]/(4*a)","A",1
2647,1,341,235,0.0040748,"\int \frac{\sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{\left(b+a^2 x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]])/(b + a^2*x^2)^(3/2),x]","-\frac{\sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{a^2 x \sqrt{b+a^2 x^2}+a \left(b+a^2 x^2\right)}+\frac{2 \text{RootSum}\left[b+c^4-4 c^3 \text{$\#$1}^2+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-c^3+3 c^2 \text{$\#$1}^2-3 c \text{$\#$1}^4+\text{$\#$1}^6}\&\right]}{a}-\frac{\text{RootSum}\left[b+c^4-4 c^3 \text{$\#$1}^2+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-c \log \left(\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}-\text{$\#$1}\right)-5 \log \left(\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{c^3 \text{$\#$1}-3 c^2 \text{$\#$1}^3+3 c \text{$\#$1}^5-\text{$\#$1}^7}\&\right]}{4 a}","-\frac{\sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{a^2 x \sqrt{b+a^2 x^2}+a \left(b+a^2 x^2\right)}-\frac{\text{RootSum}\left[b+c^4-4 c^3 \text{$\#$1}^2+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-c \log \left(\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}-\text{$\#$1}\right)+3 \log \left(\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{c^3 \text{$\#$1}-3 c^2 \text{$\#$1}^3+3 c \text{$\#$1}^5-\text{$\#$1}^7}\&\right]}{4 a}",1,"-((Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]])/(a^2*x*Sqrt[b + a^2*x^2] + a*(b + a^2*x^2))) + (2*RootSum[b + c^4 - 4*c^3*#1^2 + 6*c^2*#1^4 - 4*c*#1^6 + #1^8 & , (Log[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] - #1]*#1)/(-c^3 + 3*c^2*#1^2 - 3*c*#1^4 + #1^6) & ])/a - RootSum[b + c^4 - 4*c^3*#1^2 + 6*c^2*#1^4 - 4*c*#1^6 + #1^8 & , (-(c*Log[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] - #1]) - 5*Log[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] - #1]*#1^2)/(c^3*#1 - 3*c^2*#1^3 + 3*c*#1^5 - #1^7) & ]/(4*a)","A",1
2648,1,236,236,3.6387127,"\int \frac{1+a k x+k x^2}{\left(-1+k x^2\right) \sqrt{\left(1-x^2\right) \left(1-k^2 x^2\right)}} \, dx","IntegrateAlgebraic[(1 + a*k*x + k*x^2)/((-1 + k*x^2)*Sqrt[(1 - x^2)*(1 - k^2*x^2)]),x]","\frac{\tan ^{-1}\left(\frac{\left(1-2 i \sqrt{k}-k\right) x}{1+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{\left(-1+\sqrt{k}\right) \left(1+\sqrt{k}\right)}+\frac{\tan ^{-1}\left(\frac{\left(1+2 i \sqrt{k}-k\right) x}{1+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{\left(-1+\sqrt{k}\right) \left(1+\sqrt{k}\right)}+\frac{a \sqrt{k} \tan ^{-1}\left(\frac{\left(2 \sqrt{k}-2 k^{3/2}\right) x^2}{1-2 k x^2+k^2 x^4+\left(1+k x^2\right) \sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{2 \left(-1+\sqrt{k}\right) \left(1+\sqrt{k}\right)}","\frac{\tan ^{-1}\left(\frac{\left(1-2 i \sqrt{k}-k\right) x}{1+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{\left(-1+\sqrt{k}\right) \left(1+\sqrt{k}\right)}+\frac{\tan ^{-1}\left(\frac{\left(1+2 i \sqrt{k}-k\right) x}{1+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{\left(-1+\sqrt{k}\right) \left(1+\sqrt{k}\right)}+\frac{a \sqrt{k} \tan ^{-1}\left(\frac{\left(2 \sqrt{k}-2 k^{3/2}\right) x^2}{1-2 k x^2+k^2 x^4+\left(1+k x^2\right) \sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{2 \left(-1+\sqrt{k}\right) \left(1+\sqrt{k}\right)}",1,"ArcTan[((1 - (2*I)*Sqrt[k] - k)*x)/(1 + k*x^2 + Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4])]/((-1 + Sqrt[k])*(1 + Sqrt[k])) + ArcTan[((1 + (2*I)*Sqrt[k] - k)*x)/(1 + k*x^2 + Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4])]/((-1 + Sqrt[k])*(1 + Sqrt[k])) + (a*Sqrt[k]*ArcTan[((2*Sqrt[k] - 2*k^(3/2))*x^2)/(1 - 2*k*x^2 + k^2*x^4 + (1 + k*x^2)*Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4])])/(2*(-1 + Sqrt[k])*(1 + Sqrt[k]))","A",1
2649,1,236,236,5.6543395,"\int \frac{3 k+\left(2-k^2\right) x-3 k x^2-k^2 x^3}{\sqrt[3]{\left(1-x^2\right) \left(1-k^2 x^2\right)} \left(1-d-(1+2 d) k x+\left(-1-d k^2\right) x^2+k x^3\right)} \, dx","IntegrateAlgebraic[(3*k + (2 - k^2)*x - 3*k*x^2 - k^2*x^3)/(((1 - x^2)*(1 - k^2*x^2))^(1/3)*(1 - d - (1 + 2*d)*k*x + (-1 - d*k^2)*x^2 + k*x^3)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{2 \sqrt[3]{d}+2 \sqrt[3]{d} k x+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{\sqrt[3]{d}}-\frac{\log \left(-\sqrt[3]{d}-\sqrt[3]{d} k x+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{\sqrt[3]{d}}+\frac{\log \left(d^{2/3}+2 d^{2/3} k x+d^{2/3} k^2 x^2+\left(\sqrt[3]{d}+\sqrt[3]{d} k x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{2 \sqrt[3]{d}+2 \sqrt[3]{d} k x+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{\sqrt[3]{d}}-\frac{\log \left(-\sqrt[3]{d}-\sqrt[3]{d} k x+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{\sqrt[3]{d}}+\frac{\log \left(d^{2/3}+2 d^{2/3} k x+d^{2/3} k^2 x^2+\left(\sqrt[3]{d}+\sqrt[3]{d} k x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}",1,"-((Sqrt[3]*ArcTan[(Sqrt[3]*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))/(2*d^(1/3) + 2*d^(1/3)*k*x + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))])/d^(1/3)) - Log[-d^(1/3) - d^(1/3)*k*x + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3)]/d^(1/3) + Log[d^(2/3) + 2*d^(2/3)*k*x + d^(2/3)*k^2*x^2 + (d^(1/3) + d^(1/3)*k*x)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3) + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3)]/(2*d^(1/3))","A",1
2650,1,236,236,5.6298294,"\int \frac{3 k+\left(-2+k^2\right) x-3 k x^2+k^2 x^3}{\sqrt[3]{\left(1-x^2\right) \left(1-k^2 x^2\right)} \left(-1+d-(1+2 d) k x+\left(1+d k^2\right) x^2+k x^3\right)} \, dx","IntegrateAlgebraic[(3*k + (-2 + k^2)*x - 3*k*x^2 + k^2*x^3)/(((1 - x^2)*(1 - k^2*x^2))^(1/3)*(-1 + d - (1 + 2*d)*k*x + (1 + d*k^2)*x^2 + k*x^3)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{2 \sqrt[3]{d}-2 \sqrt[3]{d} k x+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{\sqrt[3]{d}}-\frac{\log \left(-\sqrt[3]{d}+\sqrt[3]{d} k x+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{\sqrt[3]{d}}+\frac{\log \left(d^{2/3}-2 d^{2/3} k x+d^{2/3} k^2 x^2+\left(\sqrt[3]{d}-\sqrt[3]{d} k x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{2 \sqrt[3]{d}-2 \sqrt[3]{d} k x+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{\sqrt[3]{d}}-\frac{\log \left(-\sqrt[3]{d}+\sqrt[3]{d} k x+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{\sqrt[3]{d}}+\frac{\log \left(d^{2/3}-2 d^{2/3} k x+d^{2/3} k^2 x^2+\left(\sqrt[3]{d}-\sqrt[3]{d} k x\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}",1,"-((Sqrt[3]*ArcTan[(Sqrt[3]*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))/(2*d^(1/3) - 2*d^(1/3)*k*x + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))])/d^(1/3)) - Log[-d^(1/3) + d^(1/3)*k*x + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3)]/d^(1/3) + Log[d^(2/3) - 2*d^(2/3)*k*x + d^(2/3)*k^2*x^2 + (d^(1/3) - d^(1/3)*k*x)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3) + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3)]/(2*d^(1/3))","A",1
2651,1,235,236,0.6362329,"\int \frac{\sqrt[4]{-b x^3+a x^4}}{d+c x^2} \, dx","IntegrateAlgebraic[(-(b*x^3) + a*x^4)^(1/4)/(d + c*x^2),x]","-\frac{2 \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)}{c}+\frac{2 \sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)}{c}+\frac{\text{RootSum}\left[b^2 c+a^2 d-2 a d \text{$\#$1}^4+d \text{$\#$1}^8\&,\frac{b^2 c \log (x)+a^2 d \log (x)-b^2 c \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right)-a^2 d \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right)-a d \log (x) \text{$\#$1}^4+a d \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}^3+\text{$\#$1}^7}\&\right]}{2 c d}","-\frac{2 \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)}{c}+\frac{2 \sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)}{c}+\frac{\text{RootSum}\left[b^2 c+a^2 d-2 a d \text{$\#$1}^4+d \text{$\#$1}^8\&,\frac{-b^2 c \log (x)-a^2 d \log (x)+b^2 c \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right)+a^2 d \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right)+a d \log (x) \text{$\#$1}^4-a d \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}^3-\text{$\#$1}^7}\&\right]}{2 c d}",1,"(-2*a^(1/4)*ArcTan[(a^(1/4)*x)/(-(b*x^3) + a*x^4)^(1/4)])/c + (2*a^(1/4)*ArcTanh[(a^(1/4)*x)/(-(b*x^3) + a*x^4)^(1/4)])/c + RootSum[b^2*c + a^2*d - 2*a*d*#1^4 + d*#1^8 & , (b^2*c*Log[x] + a^2*d*Log[x] - b^2*c*Log[(-(b*x^3) + a*x^4)^(1/4) - x*#1] - a^2*d*Log[(-(b*x^3) + a*x^4)^(1/4) - x*#1] - a*d*Log[x]*#1^4 + a*d*Log[(-(b*x^3) + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3) + #1^7) & ]/(2*c*d)","A",1
2652,1,235,236,0.0016317,"\int \frac{\sqrt[4]{-b x^3+a x^4}}{d+c x^2} \, dx","IntegrateAlgebraic[(-(b*x^3) + a*x^4)^(1/4)/(d + c*x^2),x]","-\frac{2 \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)}{c}+\frac{2 \sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)}{c}+\frac{\text{RootSum}\left[b^2 c+a^2 d-2 a d \text{$\#$1}^4+d \text{$\#$1}^8\&,\frac{b^2 c \log (x)+a^2 d \log (x)-b^2 c \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right)-a^2 d \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right)-a d \log (x) \text{$\#$1}^4+a d \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}^3+\text{$\#$1}^7}\&\right]}{2 c d}","-\frac{2 \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)}{c}+\frac{2 \sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)}{c}+\frac{\text{RootSum}\left[b^2 c+a^2 d-2 a d \text{$\#$1}^4+d \text{$\#$1}^8\&,\frac{-b^2 c \log (x)-a^2 d \log (x)+b^2 c \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right)+a^2 d \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right)+a d \log (x) \text{$\#$1}^4-a d \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}^3-\text{$\#$1}^7}\&\right]}{2 c d}",1,"(-2*a^(1/4)*ArcTan[(a^(1/4)*x)/(-(b*x^3) + a*x^4)^(1/4)])/c + (2*a^(1/4)*ArcTanh[(a^(1/4)*x)/(-(b*x^3) + a*x^4)^(1/4)])/c + RootSum[b^2*c + a^2*d - 2*a*d*#1^4 + d*#1^8 & , (b^2*c*Log[x] + a^2*d*Log[x] - b^2*c*Log[(-(b*x^3) + a*x^4)^(1/4) - x*#1] - a^2*d*Log[(-(b*x^3) + a*x^4)^(1/4) - x*#1] - a*d*Log[x]*#1^4 + a*d*Log[(-(b*x^3) + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3) + #1^7) & ]/(2*c*d)","A",1
2653,1,236,236,0.4743463,"\int \frac{d+c x^4}{x \sqrt{a x+\sqrt{b^2+a^2 x^2}}} \, dx","IntegrateAlgebraic[(d + c*x^4)/(x*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]),x]","\frac{b^8 c}{72 a^4 \left(a x+\sqrt{b^2+a^2 x^2}\right)^{9/2}}-\frac{b^6 c}{20 a^4 \left(a x+\sqrt{b^2+a^2 x^2}\right)^{5/2}}+\frac{2 d}{\sqrt{a x+\sqrt{b^2+a^2 x^2}}}-\frac{b^2 c \left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}{12 a^4}+\frac{c \left(a x+\sqrt{b^2+a^2 x^2}\right)^{7/2}}{56 a^4}+\frac{2 d \tan ^{-1}\left(\frac{\sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{b}}\right)}{\sqrt{b}}-\frac{2 d \tanh ^{-1}\left(\frac{\sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{b}}\right)}{\sqrt{b}}","\frac{b^8 c}{72 a^4 \left(a x+\sqrt{b^2+a^2 x^2}\right)^{9/2}}-\frac{b^6 c}{20 a^4 \left(a x+\sqrt{b^2+a^2 x^2}\right)^{5/2}}+\frac{2 d}{\sqrt{a x+\sqrt{b^2+a^2 x^2}}}-\frac{b^2 c \left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}{12 a^4}+\frac{c \left(a x+\sqrt{b^2+a^2 x^2}\right)^{7/2}}{56 a^4}+\frac{2 d \tan ^{-1}\left(\frac{\sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{b}}\right)}{\sqrt{b}}-\frac{2 d \tanh ^{-1}\left(\frac{\sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{b}}\right)}{\sqrt{b}}",1,"(b^8*c)/(72*a^4*(a*x + Sqrt[b^2 + a^2*x^2])^(9/2)) - (b^6*c)/(20*a^4*(a*x + Sqrt[b^2 + a^2*x^2])^(5/2)) + (2*d)/Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - (b^2*c*(a*x + Sqrt[b^2 + a^2*x^2])^(3/2))/(12*a^4) + (c*(a*x + Sqrt[b^2 + a^2*x^2])^(7/2))/(56*a^4) + (2*d*ArcTan[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]/Sqrt[b]])/Sqrt[b] - (2*d*ArcTanh[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]/Sqrt[b]])/Sqrt[b]","A",1
2654,1,236,236,0.3492352,"\int \frac{\sqrt{1+x^2} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[(Sqrt[1 + x^2]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/Sqrt[x + Sqrt[1 + x^2]],x]","\frac{\left(-24993-2680 x-21570 x^2-4096 x^3+30720 x^4\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(1712+1814 x+4096 x^2+3072 x^3\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(-632-36930 x-4096 x^2+30720 x^3\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(278+4096 x+3072 x^2\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{26880 \left(x+\sqrt{1+x^2}\right)^{5/2}}-\frac{263}{256} \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)","\frac{\left(-24993-2680 x-21570 x^2-4096 x^3+30720 x^4\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(1712+1814 x+4096 x^2+3072 x^3\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(-632-36930 x-4096 x^2+30720 x^3\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(278+4096 x+3072 x^2\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{26880 \left(x+\sqrt{1+x^2}\right)^{5/2}}-\frac{263}{256} \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)",1,"((-24993 - 2680*x - 21570*x^2 - 4096*x^3 + 30720*x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (1712 + 1814*x + 4096*x^2 + 3072*x^3)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((-632 - 36930*x - 4096*x^2 + 30720*x^3)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (278 + 4096*x + 3072*x^2)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(26880*(x + Sqrt[1 + x^2])^(5/2)) - (263*ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]])/256","A",1
2655,1,236,236,0.327548,"\int \sqrt{1+x^2} \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[Sqrt[1 + x^2]*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]],x]","\frac{\left(31736+1545 x+112192 x^2-2560 x^3+40320 x^4\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-1542+40688 x+1536 x^2+2240 x^3\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(2825+92032 x-2560 x^2+40320 x^3\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(39568+1536 x+2240 x^2\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{55440 \left(x+\sqrt{1+x^2}\right)^{3/2}}-\frac{1}{16} \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)","\frac{\left(31736+1545 x+112192 x^2-2560 x^3+40320 x^4\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-1542+40688 x+1536 x^2+2240 x^3\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(2825+92032 x-2560 x^2+40320 x^3\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(39568+1536 x+2240 x^2\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{55440 \left(x+\sqrt{1+x^2}\right)^{3/2}}-\frac{1}{16} \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)",1,"((31736 + 1545*x + 112192*x^2 - 2560*x^3 + 40320*x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-1542 + 40688*x + 1536*x^2 + 2240*x^3)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((2825 + 92032*x - 2560*x^2 + 40320*x^3)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (39568 + 1536*x + 2240*x^2)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(55440*(x + Sqrt[1 + x^2])^(3/2)) - ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]]/16","A",1
2656,1,237,237,1.0809693,"\int \frac{\sqrt[4]{-x^3+x^4}}{x (-b+a x)} \, dx","IntegrateAlgebraic[(-x^3 + x^4)^(1/4)/(x*(-b + a*x)),x]","-\frac{2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)}{a}-\frac{\sqrt{2} \sqrt[4]{a-b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a-b} \sqrt[4]{b} x \sqrt[4]{-x^3+x^4}}{\sqrt{a-b} x^2-\sqrt{b} \sqrt{-x^3+x^4}}\right)}{a \sqrt[4]{b}}+\frac{2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)}{a}-\frac{\sqrt{2} \sqrt[4]{a-b} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{a-b} x^2}{\sqrt{2} \sqrt[4]{b}}+\frac{\sqrt[4]{b} \sqrt{-x^3+x^4}}{\sqrt{2} \sqrt[4]{a-b}}}{x \sqrt[4]{-x^3+x^4}}\right)}{a \sqrt[4]{b}}","-\frac{2 \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)}{a}-\frac{\sqrt{2} \sqrt[4]{a-b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a-b} \sqrt[4]{b} x \sqrt[4]{-x^3+x^4}}{\sqrt{a-b} x^2-\sqrt{b} \sqrt{-x^3+x^4}}\right)}{a \sqrt[4]{b}}+\frac{2 \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)}{a}-\frac{\sqrt{2} \sqrt[4]{a-b} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{a-b} x^2}{\sqrt{2} \sqrt[4]{b}}+\frac{\sqrt[4]{b} \sqrt{-x^3+x^4}}{\sqrt{2} \sqrt[4]{a-b}}}{x \sqrt[4]{-x^3+x^4}}\right)}{a \sqrt[4]{b}}",1,"(-2*ArcTan[x/(-x^3 + x^4)^(1/4)])/a - (Sqrt[2]*(a - b)^(1/4)*ArcTan[(Sqrt[2]*(a - b)^(1/4)*b^(1/4)*x*(-x^3 + x^4)^(1/4))/(Sqrt[a - b]*x^2 - Sqrt[b]*Sqrt[-x^3 + x^4])])/(a*b^(1/4)) + (2*ArcTanh[x/(-x^3 + x^4)^(1/4)])/a - (Sqrt[2]*(a - b)^(1/4)*ArcTanh[(((a - b)^(1/4)*x^2)/(Sqrt[2]*b^(1/4)) + (b^(1/4)*Sqrt[-x^3 + x^4])/(Sqrt[2]*(a - b)^(1/4)))/(x*(-x^3 + x^4)^(1/4))])/(a*b^(1/4))","A",1
2657,1,237,237,6.8887741,"\int \frac{x^3 (-2 a b+(a+b) x)}{(x (-a+x) (-b+x))^{2/3} \left(-a^2 b^2+2 a b (a+b) x-\left(a^2+4 a b+b^2\right) x^2+2 (a+b) x^3+(-1+d) x^4\right)} \, dx","IntegrateAlgebraic[(x^3*(-2*a*b + (a + b)*x))/((x*(-a + x)*(-b + x))^(2/3)*(-(a^2*b^2) + 2*a*b*(a + b)*x - (a^2 + 4*a*b + b^2)*x^2 + 2*(a + b)*x^3 + (-1 + d)*x^4)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} x}{\sqrt[6]{d} x-2 \sqrt[3]{a b x+(-a-b) x^2+x^3}}\right)}{2 d^{5/6}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} x}{\sqrt[6]{d} x+2 \sqrt[3]{a b x+(-a-b) x^2+x^3}}\right)}{2 d^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{d} x}{\sqrt[3]{a b x+(-a-b) x^2+x^3}}\right)}{d^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{d} x \sqrt[3]{a b x+(-a-b) x^2+x^3}}{\sqrt[3]{d} x^2+\left(a b x+(-a-b) x^2+x^3\right)^{2/3}}\right)}{2 d^{5/6}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} x}{\sqrt[6]{d} x-2 \sqrt[3]{a b x+(-a-b) x^2+x^3}}\right)}{2 d^{5/6}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} x}{\sqrt[6]{d} x+2 \sqrt[3]{a b x+(-a-b) x^2+x^3}}\right)}{2 d^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{d} x}{\sqrt[3]{a b x+(-a-b) x^2+x^3}}\right)}{d^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{d} x \sqrt[3]{a b x+(-a-b) x^2+x^3}}{\sqrt[3]{d} x^2+\left(a b x+(-a-b) x^2+x^3\right)^{2/3}}\right)}{2 d^{5/6}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/6)*x)/(d^(1/6)*x - 2*(a*b*x + (-a - b)*x^2 + x^3)^(1/3))])/(2*d^(5/6)) - (Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/6)*x)/(d^(1/6)*x + 2*(a*b*x + (-a - b)*x^2 + x^3)^(1/3))])/(2*d^(5/6)) + ArcTanh[(d^(1/6)*x)/(a*b*x + (-a - b)*x^2 + x^3)^(1/3)]/d^(5/6) + ArcTanh[(d^(1/6)*x*(a*b*x + (-a - b)*x^2 + x^3)^(1/3))/(d^(1/3)*x^2 + (a*b*x + (-a - b)*x^2 + x^3)^(2/3))]/(2*d^(5/6))","A",1
2658,1,237,237,0.6956372,"\int \frac{1+2 x^6}{\sqrt[3]{x+x^3} \left(-1+x^6\right)} \, dx","IntegrateAlgebraic[(1 + 2*x^6)/((x + x^3)^(1/3)*(-1 + x^6)),x]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x+x^3}}\right)}{2 \sqrt[3]{2}}-\log \left(-x+\sqrt[3]{x+x^3}\right)+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x+x^3}\right)}{2 \sqrt[3]{2}}+\frac{1}{2} \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x+x^3}+\sqrt[3]{2} \left(x+x^3\right)^{2/3}\right)}{4 \sqrt[3]{2}}+\frac{1}{2} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{x+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x+x^3}}\right)}{2 \sqrt[3]{2}}-\log \left(-x+\sqrt[3]{x+x^3}\right)+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x+x^3}\right)}{2 \sqrt[3]{2}}+\frac{1}{2} \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x+x^3}+\sqrt[3]{2} \left(x+x^3\right)^{2/3}\right)}{4 \sqrt[3]{2}}+\frac{1}{2} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{x+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^3)^(1/3))] - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(x + x^3)^(1/3))])/(2*2^(1/3)) - Log[-x + (x + x^3)^(1/3)] + Log[-2*x + 2^(2/3)*(x + x^3)^(1/3)]/(2*2^(1/3)) + Log[x^2 + x*(x + x^3)^(1/3) + (x + x^3)^(2/3)]/2 - Log[2*x^2 + 2^(2/3)*x*(x + x^3)^(1/3) + 2^(1/3)*(x + x^3)^(2/3)]/(4*2^(1/3)) + RootSum[1 - #1^3 + #1^6 & , (-Log[x] + Log[(x + x^3)^(1/3) - x*#1])/#1 & ]/2","A",1
2659,1,237,237,0.0012472,"\int \frac{1+2 x^6}{\sqrt[3]{x+x^3} \left(-1+x^6\right)} \, dx","IntegrateAlgebraic[(1 + 2*x^6)/((x + x^3)^(1/3)*(-1 + x^6)),x]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x+x^3}}\right)}{2 \sqrt[3]{2}}-\log \left(-x+\sqrt[3]{x+x^3}\right)+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x+x^3}\right)}{2 \sqrt[3]{2}}+\frac{1}{2} \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x+x^3}+\sqrt[3]{2} \left(x+x^3\right)^{2/3}\right)}{4 \sqrt[3]{2}}+\frac{1}{2} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{x+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x+x^3}}\right)}{2 \sqrt[3]{2}}-\log \left(-x+\sqrt[3]{x+x^3}\right)+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x+x^3}\right)}{2 \sqrt[3]{2}}+\frac{1}{2} \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x+x^3}+\sqrt[3]{2} \left(x+x^3\right)^{2/3}\right)}{4 \sqrt[3]{2}}+\frac{1}{2} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{x+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^3)^(1/3))] - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(x + x^3)^(1/3))])/(2*2^(1/3)) - Log[-x + (x + x^3)^(1/3)] + Log[-2*x + 2^(2/3)*(x + x^3)^(1/3)]/(2*2^(1/3)) + Log[x^2 + x*(x + x^3)^(1/3) + (x + x^3)^(2/3)]/2 - Log[2*x^2 + 2^(2/3)*x*(x + x^3)^(1/3) + 2^(1/3)*(x + x^3)^(2/3)]/(4*2^(1/3)) + RootSum[1 - #1^3 + #1^6 & , (-Log[x] + Log[(x + x^3)^(1/3) - x*#1])/#1 & ]/2","A",1
2660,1,238,238,0.5706768,"\int \frac{\left(-4 b+a x^3\right) \sqrt[3]{b+a x^3}}{x^5 \left(-2 b+a x^3\right)} \, dx","IntegrateAlgebraic[((-4*b + a*x^3)*(b + a*x^3)^(1/3))/(x^5*(-2*b + a*x^3)),x]","\frac{\left(-b-2 a x^3\right) \sqrt[3]{b+a x^3}}{2 b x^4}-\frac{a^{4/3} \tan ^{-1}\left(\frac{3^{5/6} \sqrt[3]{a} x}{\sqrt[3]{3} \sqrt[3]{a} x+2 \sqrt[3]{2} \sqrt[3]{b+a x^3}}\right)}{2 \sqrt[3]{2} \sqrt[6]{3} b}-\frac{a^{4/3} \log \left(-3 \sqrt[3]{a} x+\sqrt[3]{2} 3^{2/3} \sqrt[3]{b+a x^3}\right)}{2 \sqrt[3]{2} 3^{2/3} b}+\frac{a^{4/3} \log \left(3 a^{2/3} x^2+\sqrt[3]{2} 3^{2/3} \sqrt[3]{a} x \sqrt[3]{b+a x^3}+2^{2/3} \sqrt[3]{3} \left(b+a x^3\right)^{2/3}\right)}{4 \sqrt[3]{2} 3^{2/3} b}","\frac{\left(-b-2 a x^3\right) \sqrt[3]{b+a x^3}}{2 b x^4}-\frac{a^{4/3} \tan ^{-1}\left(\frac{3^{5/6} \sqrt[3]{a} x}{\sqrt[3]{3} \sqrt[3]{a} x+2 \sqrt[3]{2} \sqrt[3]{b+a x^3}}\right)}{2 \sqrt[3]{2} \sqrt[6]{3} b}-\frac{a^{4/3} \log \left(-3 \sqrt[3]{a} x+\sqrt[3]{2} 3^{2/3} \sqrt[3]{b+a x^3}\right)}{2 \sqrt[3]{2} 3^{2/3} b}+\frac{a^{4/3} \log \left(3 a^{2/3} x^2+\sqrt[3]{2} 3^{2/3} \sqrt[3]{a} x \sqrt[3]{b+a x^3}+2^{2/3} \sqrt[3]{3} \left(b+a x^3\right)^{2/3}\right)}{4 \sqrt[3]{2} 3^{2/3} b}",1,"((-b - 2*a*x^3)*(b + a*x^3)^(1/3))/(2*b*x^4) - (a^(4/3)*ArcTan[(3^(5/6)*a^(1/3)*x)/(3^(1/3)*a^(1/3)*x + 2*2^(1/3)*(b + a*x^3)^(1/3))])/(2*2^(1/3)*3^(1/6)*b) - (a^(4/3)*Log[-3*a^(1/3)*x + 2^(1/3)*3^(2/3)*(b + a*x^3)^(1/3)])/(2*2^(1/3)*3^(2/3)*b) + (a^(4/3)*Log[3*a^(2/3)*x^2 + 2^(1/3)*3^(2/3)*a^(1/3)*x*(b + a*x^3)^(1/3) + 2^(2/3)*3^(1/3)*(b + a*x^3)^(2/3)])/(4*2^(1/3)*3^(2/3)*b)","A",1
2661,1,238,238,4.143677,"\int \frac{-2 a^2 b x+a (3 a+2 b) x^2-4 a x^3+x^4}{\left(x^2 (-a+x) (-b+x)\right)^{2/3} \left(-a^2 d+2 a d x-(b+d) x^2+x^3\right)} \, dx","IntegrateAlgebraic[(-2*a^2*b*x + a*(3*a + 2*b)*x^2 - 4*a*x^3 + x^4)/((x^2*(-a + x)*(-b + x))^(2/3)*(-(a^2*d) + 2*a*d*x - (b + d)*x^2 + x^3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a \sqrt[3]{d}-\sqrt{3} \sqrt[3]{d} x}{a \sqrt[3]{d}-\sqrt[3]{d} x-2 \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{d^{2/3}}+\frac{\log \left(a \sqrt[3]{d}-\sqrt[3]{d} x+\sqrt[3]{a b x^2+(-a-b) x^3+x^4}\right)}{d^{2/3}}-\frac{\log \left(a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{a b x^2+(-a-b) x^3+x^4}+\left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}\right)}{2 d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a \sqrt[3]{d}-\sqrt{3} \sqrt[3]{d} x}{a \sqrt[3]{d}-\sqrt[3]{d} x-2 \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{d^{2/3}}+\frac{\log \left(a \sqrt[3]{d}-\sqrt[3]{d} x+\sqrt[3]{a b x^2+(-a-b) x^3+x^4}\right)}{d^{2/3}}-\frac{\log \left(a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{a b x^2+(-a-b) x^3+x^4}+\left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}\right)}{2 d^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*a*d^(1/3) - Sqrt[3]*d^(1/3)*x)/(a*d^(1/3) - d^(1/3)*x - 2*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))])/d^(2/3) + Log[a*d^(1/3) - d^(1/3)*x + (a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3)]/d^(2/3) - Log[a^2*d^(2/3) - 2*a*d^(2/3)*x + d^(2/3)*x^2 + (-(a*d^(1/3)) + d^(1/3)*x)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3) + (a*b*x^2 + (-a - b)*x^3 + x^4)^(2/3)]/(2*d^(2/3))","A",1
2662,1,238,238,0.5177716,"\int \frac{(-2+(1+k) x) \left(1-2 (1+k) x+\left(1+4 k+k^2\right) x^2-2 \left(k+k^2\right) x^3+\left(a+k^2\right) x^4\right)}{x^4 \sqrt[3]{(1-x) x (1-k x)} \left(1-(1+k) x+(-b+k) x^2\right)} \, dx","IntegrateAlgebraic[((-2 + (1 + k)*x)*(1 - 2*(1 + k)*x + (1 + 4*k + k^2)*x^2 - 2*(k + k^2)*x^3 + (a + k^2)*x^4))/(x^4*((1 - x)*x*(1 - k*x))^(1/3)*(1 - (1 + k)*x + (-b + k)*x^2)),x]","\frac{3 \left(2-2 x-2 k x+5 b x^2+2 k x^2\right) \left(x-x^2-k x^2+k x^3\right)^{2/3}}{10 x^4}+\frac{\left(-\sqrt{3} a-\sqrt{3} b^2\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} x}{\sqrt[3]{b} x+2 \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{\sqrt[3]{b}}+\frac{\left(a+b^2\right) \log \left(-\sqrt[3]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{\sqrt[3]{b}}+\frac{\left(-a-b^2\right) \log \left(b^{2/3} x^2+\sqrt[3]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 \sqrt[3]{b}}","\frac{3 \left(2-2 x-2 k x+5 b x^2+2 k x^2\right) \left(x-x^2-k x^2+k x^3\right)^{2/3}}{10 x^4}+\frac{\left(-\sqrt{3} a-\sqrt{3} b^2\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} x}{\sqrt[3]{b} x+2 \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{\sqrt[3]{b}}+\frac{\left(a+b^2\right) \log \left(-\sqrt[3]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{\sqrt[3]{b}}+\frac{\left(-a-b^2\right) \log \left(b^{2/3} x^2+\sqrt[3]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 \sqrt[3]{b}}",1,"(3*(2 - 2*x - 2*k*x + 5*b*x^2 + 2*k*x^2)*(x - x^2 - k*x^2 + k*x^3)^(2/3))/(10*x^4) + ((-(Sqrt[3]*a) - Sqrt[3]*b^2)*ArcTan[(Sqrt[3]*b^(1/3)*x)/(b^(1/3)*x + 2*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/b^(1/3) + ((a + b^2)*Log[-(b^(1/3)*x) + (x + (-1 - k)*x^2 + k*x^3)^(1/3)])/b^(1/3) + ((-a - b^2)*Log[b^(2/3)*x^2 + b^(1/3)*x*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + (x + (-1 - k)*x^2 + k*x^3)^(2/3)])/(2*b^(1/3))","A",1
2663,1,233,238,1.3255247,"\int \frac{x^6 \sqrt{x+x^4}}{b+a x^6} \, dx","IntegrateAlgebraic[(x^6*Sqrt[x + x^4])/(b + a*x^6),x]","\frac{x \sqrt{x+x^4}}{3 a}+\frac{\sqrt[4]{-1} \sqrt{\left(\sqrt{a}-i \sqrt{b}\right) \sqrt{b}} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{\sqrt{a} \sqrt{b}-i b} x \sqrt{x+x^4}}{\sqrt{b} (1+x) \left(1-x+x^2\right)}\right)}{3 a^{3/2}}+\frac{(-1)^{3/4} \sqrt{\left(\sqrt{a}+i \sqrt{b}\right) \sqrt{b}} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{\sqrt{a} \sqrt{b}+i b} x \sqrt{x+x^4}}{\sqrt{b} (1+x) \left(1-x+x^2\right)}\right)}{3 a^{3/2}}+\frac{\tanh ^{-1}\left(\frac{x^2}{\sqrt{x+x^4}}\right)}{3 a}","\frac{x \sqrt{x+x^4}}{3 a}+\frac{\sqrt[4]{-1} \sqrt{\left(\sqrt{a}-i \sqrt{b}\right) \sqrt{b}} \tan ^{-1}\left(\frac{(1+i) \sqrt{\sqrt{a} \sqrt{b}-i b} \sqrt{x+x^4}}{\sqrt{2} \left(\sqrt{a}-i \sqrt{b}\right) x^2}\right)}{3 a^{3/2}}+\frac{(-1)^{3/4} \sqrt{\left(\sqrt{a}+i \sqrt{b}\right) \sqrt{b}} \tan ^{-1}\left(\frac{(1+i) \sqrt{\sqrt{a} \sqrt{b}+i b} x \sqrt{x+x^4}}{\sqrt{2} \sqrt{b} (1+x) \left(1-x+x^2\right)}\right)}{3 a^{3/2}}+\frac{\tanh ^{-1}\left(\frac{x^2}{\sqrt{x+x^4}}\right)}{3 a}",1,"(x*Sqrt[x + x^4])/(3*a) + ((-1)^(1/4)*Sqrt[(Sqrt[a] - I*Sqrt[b])*Sqrt[b]]*ArcTan[((-1)^(3/4)*Sqrt[Sqrt[a]*Sqrt[b] - I*b]*x*Sqrt[x + x^4])/(Sqrt[b]*(1 + x)*(1 - x + x^2))])/(3*a^(3/2)) + ((-1)^(3/4)*Sqrt[(Sqrt[a] + I*Sqrt[b])*Sqrt[b]]*ArcTan[((-1)^(1/4)*Sqrt[Sqrt[a]*Sqrt[b] + I*b]*x*Sqrt[x + x^4])/(Sqrt[b]*(1 + x)*(1 - x + x^2))])/(3*a^(3/2)) + ArcTanh[x^2/Sqrt[x + x^4]]/(3*a)","A",1
2664,1,238,238,3.2191177,"\int \frac{\sqrt[4]{b x^2+a x^4} \left(-b-a x^4+x^8\right)}{-b+a x^4} \, dx","IntegrateAlgebraic[((b*x^2 + a*x^4)^(1/4)*(-b - a*x^4 + x^8))/(-b + a*x^4),x]","\frac{\sqrt[4]{b x^2+a x^4} \left(-96 a^3 x+96 a b x-7 b^2 x+4 a b x^3+32 a^2 x^5\right)}{192 a^3}+\frac{\left(32 a^3 b-32 a b^2-7 b^3\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{128 a^{15/4}}+\frac{\left(-32 a^3 b+32 a b^2+7 b^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{128 a^{15/4}}+\frac{\left(-2 a^2 b+b^2\right) \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]}{4 a^2}","\frac{\sqrt[4]{b x^2+a x^4} \left(-96 a^3 x+96 a b x-7 b^2 x+4 a b x^3+32 a^2 x^5\right)}{192 a^3}+\frac{\left(32 a^3 b-32 a b^2-7 b^3\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{128 a^{15/4}}+\frac{\left(-32 a^3 b+32 a b^2+7 b^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{128 a^{15/4}}+\frac{\left(-2 a^2 b+b^2\right) \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]}{4 a^2}",1,"((b*x^2 + a*x^4)^(1/4)*(-96*a^3*x + 96*a*b*x - 7*b^2*x + 4*a*b*x^3 + 32*a^2*x^5))/(192*a^3) + ((32*a^3*b - 32*a*b^2 - 7*b^3)*ArcTan[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/(128*a^(15/4)) + ((-32*a^3*b + 32*a*b^2 + 7*b^3)*ArcTanh[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/(128*a^(15/4)) + ((-2*a^2*b + b^2)*RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (-(Log[x]*#1) + Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1)/(-a + #1^4) & ])/(4*a^2)","A",1
2665,1,238,238,0.0017502,"\int \frac{\sqrt[4]{b x^2+a x^4} \left(-b-a x^4+x^8\right)}{-b+a x^4} \, dx","IntegrateAlgebraic[((b*x^2 + a*x^4)^(1/4)*(-b - a*x^4 + x^8))/(-b + a*x^4),x]","\frac{\sqrt[4]{b x^2+a x^4} \left(-96 a^3 x+96 a b x-7 b^2 x+4 a b x^3+32 a^2 x^5\right)}{192 a^3}+\frac{\left(32 a^3 b-32 a b^2-7 b^3\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{128 a^{15/4}}+\frac{\left(-32 a^3 b+32 a b^2+7 b^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{128 a^{15/4}}+\frac{\left(-2 a^2 b+b^2\right) \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]}{4 a^2}","\frac{\sqrt[4]{b x^2+a x^4} \left(-96 a^3 x+96 a b x-7 b^2 x+4 a b x^3+32 a^2 x^5\right)}{192 a^3}+\frac{\left(32 a^3 b-32 a b^2-7 b^3\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{128 a^{15/4}}+\frac{\left(-32 a^3 b+32 a b^2+7 b^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{128 a^{15/4}}+\frac{\left(-2 a^2 b+b^2\right) \text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}}{-a+\text{$\#$1}^4}\&\right]}{4 a^2}",1,"((b*x^2 + a*x^4)^(1/4)*(-96*a^3*x + 96*a*b*x - 7*b^2*x + 4*a*b*x^3 + 32*a^2*x^5))/(192*a^3) + ((32*a^3*b - 32*a*b^2 - 7*b^3)*ArcTan[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/(128*a^(15/4)) + ((-32*a^3*b + 32*a*b^2 + 7*b^3)*ArcTanh[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/(128*a^(15/4)) + ((-2*a^2*b + b^2)*RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (-(Log[x]*#1) + Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1)/(-a + #1^4) & ])/(4*a^2)","A",1
2666,1,238,238,5.3665022,"\int \frac{\left(-2+2 x^4+5 x^7\right) \sqrt[3]{x-x^3+x^5+x^8}}{\left(2+x^2+2 x^4+2 x^7\right)^2} \, dx","IntegrateAlgebraic[((-2 + 2*x^4 + 5*x^7)*(x - x^3 + x^5 + x^8)^(1/3))/(2 + x^2 + 2*x^4 + 2*x^7)^2,x]","-\frac{x \sqrt[3]{x-x^3+x^5+x^8}}{2 \left(2+x^2+2 x^4+2 x^7\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{x-x^3+x^5+x^8}}{2^{2/3} \sqrt[3]{3} x-\sqrt[3]{x-x^3+x^5+x^8}}\right)}{6 \sqrt[3]{2} \sqrt[6]{3}}+\frac{\log \left(2^{2/3} \sqrt[3]{3} x+2 \sqrt[3]{x-x^3+x^5+x^8}\right)}{6 \sqrt[3]{2} 3^{2/3}}-\frac{\log \left(\sqrt[3]{2} 3^{2/3} x^2-2^{2/3} \sqrt[3]{3} x \sqrt[3]{x-x^3+x^5+x^8}+2 \left(x-x^3+x^5+x^8\right)^{2/3}\right)}{12 \sqrt[3]{2} 3^{2/3}}","-\frac{x \sqrt[3]{x-x^3+x^5+x^8}}{2 \left(2+x^2+2 x^4+2 x^7\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{x-x^3+x^5+x^8}}{2^{2/3} \sqrt[3]{3} x-\sqrt[3]{x-x^3+x^5+x^8}}\right)}{6 \sqrt[3]{2} \sqrt[6]{3}}+\frac{\log \left(2^{2/3} \sqrt[3]{3} x+2 \sqrt[3]{x-x^3+x^5+x^8}\right)}{6 \sqrt[3]{2} 3^{2/3}}-\frac{\log \left(\sqrt[3]{2} 3^{2/3} x^2-2^{2/3} \sqrt[3]{3} x \sqrt[3]{x-x^3+x^5+x^8}+2 \left(x-x^3+x^5+x^8\right)^{2/3}\right)}{12 \sqrt[3]{2} 3^{2/3}}",1,"-1/2*(x*(x - x^3 + x^5 + x^8)^(1/3))/(2 + x^2 + 2*x^4 + 2*x^7) + ArcTan[(Sqrt[3]*(x - x^3 + x^5 + x^8)^(1/3))/(2^(2/3)*3^(1/3)*x - (x - x^3 + x^5 + x^8)^(1/3))]/(6*2^(1/3)*3^(1/6)) + Log[2^(2/3)*3^(1/3)*x + 2*(x - x^3 + x^5 + x^8)^(1/3)]/(6*2^(1/3)*3^(2/3)) - Log[2^(1/3)*3^(2/3)*x^2 - 2^(2/3)*3^(1/3)*x*(x - x^3 + x^5 + x^8)^(1/3) + 2*(x - x^3 + x^5 + x^8)^(2/3)]/(12*2^(1/3)*3^(2/3))","A",1
2667,1,239,239,1.2562517,"\int \frac{-b^5+a^5 x^5}{\sqrt{b^2 x+a^2 x^3} \left(b^5+a^5 x^5\right)} \, dx","IntegrateAlgebraic[(-b^5 + a^5*x^5)/(Sqrt[b^2*x + a^2*x^3]*(b^5 + a^5*x^5)),x]","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{5 \sqrt{a} \sqrt{b}}-\frac{2 \sqrt{2 \left(1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{5 \sqrt{a} \sqrt{b}}-\frac{2 \sqrt{2 \left(-1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{5 \sqrt{a} \sqrt{b}}","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{5 \sqrt{a} \sqrt{b}}-\frac{2 \sqrt{2 \left(1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{5 \sqrt{a} \sqrt{b}}-\frac{2 \sqrt{2 \left(-1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{5 \sqrt{a} \sqrt{b}}",1,"-1/5*(Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[b]*Sqrt[b^2*x + a^2*x^3])/(b^2 + a^2*x^2)])/(Sqrt[a]*Sqrt[b]) - (2*Sqrt[2*(1 + Sqrt[5])]*ArcTan[(Sqrt[-1/2 + Sqrt[5]/2]*Sqrt[a]*Sqrt[b]*Sqrt[b^2*x + a^2*x^3])/(b^2 + a^2*x^2)])/(5*Sqrt[a]*Sqrt[b]) - (2*Sqrt[2*(-1 + Sqrt[5])]*ArcTanh[(Sqrt[1/2 + Sqrt[5]/2]*Sqrt[a]*Sqrt[b]*Sqrt[b^2*x + a^2*x^3])/(b^2 + a^2*x^2)])/(5*Sqrt[a]*Sqrt[b])","A",1
2668,1,239,239,1.1991047,"\int \frac{b^5+a^5 x^5}{\sqrt{b^2 x+a^2 x^3} \left(-b^5+a^5 x^5\right)} \, dx","IntegrateAlgebraic[(b^5 + a^5*x^5)/(Sqrt[b^2*x + a^2*x^3]*(-b^5 + a^5*x^5)),x]","-\frac{2 \sqrt{2 \left(-1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{5 \sqrt{a} \sqrt{b}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{5 \sqrt{a} \sqrt{b}}-\frac{2 \sqrt{2 \left(1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{5 \sqrt{a} \sqrt{b}}","-\frac{2 \sqrt{2 \left(-1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{5 \sqrt{a} \sqrt{b}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{5 \sqrt{a} \sqrt{b}}-\frac{2 \sqrt{2 \left(1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{5 \sqrt{a} \sqrt{b}}",1,"(-2*Sqrt[2*(-1 + Sqrt[5])]*ArcTan[(Sqrt[1/2 + Sqrt[5]/2]*Sqrt[a]*Sqrt[b]*Sqrt[b^2*x + a^2*x^3])/(b^2 + a^2*x^2)])/(5*Sqrt[a]*Sqrt[b]) - (Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[b]*Sqrt[b^2*x + a^2*x^3])/(b^2 + a^2*x^2)])/(5*Sqrt[a]*Sqrt[b]) - (2*Sqrt[2*(1 + Sqrt[5])]*ArcTanh[(Sqrt[-1/2 + Sqrt[5]/2]*Sqrt[a]*Sqrt[b]*Sqrt[b^2*x + a^2*x^3])/(b^2 + a^2*x^2)])/(5*Sqrt[a]*Sqrt[b])","A",1
2669,1,239,239,1.0237952,"\int \frac{-2 b-a x^4+2 x^8}{x^4 \sqrt[4]{-b+a x^4} \left(-b+2 a x^4\right)} \, dx","IntegrateAlgebraic[(-2*b - a*x^4 + 2*x^8)/(x^4*(-b + a*x^4)^(1/4)*(-b + 2*a*x^4)),x]","\frac{2 \left(-b+a x^4\right)^{3/4}}{3 b x^3}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2 a^{5/4}}+\frac{\left(5 a^2-b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} x \sqrt[4]{-b+a x^4}}{-\sqrt{a} x^2+\sqrt{-b+a x^4}}\right)}{2 \sqrt{2} a^{5/4} b}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2 a^{5/4}}+\frac{\left(5 a^2-b\right) \tanh ^{-1}\left(\frac{\sqrt{a} x^2+\sqrt{-b+a x^4}}{\sqrt{2} \sqrt[4]{a} x \sqrt[4]{-b+a x^4}}\right)}{2 \sqrt{2} a^{5/4} b}","\frac{2 \left(-b+a x^4\right)^{3/4}}{3 b x^3}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2 a^{5/4}}+\frac{\left(5 a^2-b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} x \sqrt[4]{-b+a x^4}}{-\sqrt{a} x^2+\sqrt{-b+a x^4}}\right)}{2 \sqrt{2} a^{5/4} b}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b+a x^4}}\right)}{2 a^{5/4}}+\frac{\left(5 a^2-b\right) \tanh ^{-1}\left(\frac{\sqrt{a} x^2+\sqrt{-b+a x^4}}{\sqrt{2} \sqrt[4]{a} x \sqrt[4]{-b+a x^4}}\right)}{2 \sqrt{2} a^{5/4} b}",1,"(2*(-b + a*x^4)^(3/4))/(3*b*x^3) + ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(2*a^(5/4)) + ((5*a^2 - b)*ArcTan[(Sqrt[2]*a^(1/4)*x*(-b + a*x^4)^(1/4))/(-(Sqrt[a]*x^2) + Sqrt[-b + a*x^4])])/(2*Sqrt[2]*a^(5/4)*b) + ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(2*a^(5/4)) + ((5*a^2 - b)*ArcTanh[(Sqrt[a]*x^2 + Sqrt[-b + a*x^4])/(Sqrt[2]*a^(1/4)*x*(-b + a*x^4)^(1/4))])/(2*Sqrt[2]*a^(5/4)*b)","A",1
2670,1,278,239,2.2871652,"\int \frac{-1+x^{16}}{\sqrt{-1+x^4} \left(1+x^8+x^{16}\right)} \, dx","IntegrateAlgebraic[(-1 + x^16)/(Sqrt[-1 + x^4]*(1 + x^8 + x^16)),x]","\frac{\tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}-\frac{x^2}{\sqrt{2}}+\frac{x^4}{\sqrt{2}}}{x \sqrt{-1+x^4}}\right)}{4 \sqrt{2}}+\frac{\tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2} \sqrt[4]{3}}+\frac{\left(2 i-\sqrt{-1+4 i \sqrt{3}}\right) x^2}{\sqrt{2} \sqrt[4]{3}}+\frac{x^4}{\sqrt{2} \sqrt[4]{3}}}{x \sqrt{-1+x^4}}\right)}{4 \sqrt{2} \sqrt[4]{3}}-\frac{\tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{x^4}{\sqrt{2}}}{x \sqrt{-1+x^4}}\right)}{4 \sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{2} \sqrt[4]{3}}+\frac{\left(2 i+\sqrt{-1-4 i \sqrt{3}}\right) x^2}{\sqrt{2} \sqrt[4]{3}}+\frac{x^4}{\sqrt{2} \sqrt[4]{3}}}{x \sqrt{-1+x^4}}\right)}{4 \sqrt{2} \sqrt[4]{3}}","\frac{\tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}-\frac{x^2}{\sqrt{2}}+\frac{x^4}{\sqrt{2}}}{x \sqrt{-1+x^4}}\right)}{4 \sqrt{2}}+\frac{\tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2} \sqrt[4]{3}}-\frac{\sqrt[4]{3} x^2}{\sqrt{2}}+\frac{x^4}{\sqrt{2} \sqrt[4]{3}}}{x \sqrt{-1+x^4}}\right)}{4 \sqrt{2} \sqrt[4]{3}}-\frac{\tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{x^4}{\sqrt{2}}}{x \sqrt{-1+x^4}}\right)}{4 \sqrt{2}}-\frac{\tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{2} \sqrt[4]{3}}+\frac{\sqrt[4]{3} x^2}{\sqrt{2}}+\frac{x^4}{\sqrt{2} \sqrt[4]{3}}}{x \sqrt{-1+x^4}}\right)}{4 \sqrt{2} \sqrt[4]{3}}",1,"ArcTan[(-(1/Sqrt[2]) - x^2/Sqrt[2] + x^4/Sqrt[2])/(x*Sqrt[-1 + x^4])]/(4*Sqrt[2]) + ArcTan[(-(1/(Sqrt[2]*3^(1/4))) + ((2*I - Sqrt[-1 + (4*I)*Sqrt[3]])*x^2)/(Sqrt[2]*3^(1/4)) + x^4/(Sqrt[2]*3^(1/4)))/(x*Sqrt[-1 + x^4])]/(4*Sqrt[2]*3^(1/4)) - ArcTanh[(-(1/Sqrt[2]) + x^2/Sqrt[2] + x^4/Sqrt[2])/(x*Sqrt[-1 + x^4])]/(4*Sqrt[2]) - ArcTanh[(-(1/(Sqrt[2]*3^(1/4))) + ((2*I + Sqrt[-1 - (4*I)*Sqrt[3]])*x^2)/(Sqrt[2]*3^(1/4)) + x^4/(Sqrt[2]*3^(1/4)))/(x*Sqrt[-1 + x^4])]/(4*Sqrt[2]*3^(1/4))","C",1
2671,1,241,241,1.1561773,"\int \frac{2 a b x-3 a x^2+x^3}{\sqrt[3]{x^2 (-a+x) (-b+x)} \left(-a^2 d+2 a d x-(b+d) x^2+x^3\right)} \, dx","IntegrateAlgebraic[(2*a*b*x - 3*a*x^2 + x^3)/((x^2*(-a + x)*(-b + x))^(1/3)*(-(a^2*d) + 2*a*d*x - (b + d)*x^2 + x^3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}{-2 a \sqrt[3]{d}+2 \sqrt[3]{d} x+\sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{\sqrt[3]{d}}+\frac{\log \left(a \sqrt[3]{d}-\sqrt[3]{d} x+\sqrt[3]{a b x^2+(-a-b) x^3+x^4}\right)}{\sqrt[3]{d}}-\frac{\log \left(a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{a b x^2+(-a-b) x^3+x^4}+\left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}{-2 a \sqrt[3]{d}+2 \sqrt[3]{d} x+\sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{\sqrt[3]{d}}+\frac{\log \left(a \sqrt[3]{d}-\sqrt[3]{d} x+\sqrt[3]{a b x^2+(-a-b) x^3+x^4}\right)}{\sqrt[3]{d}}-\frac{\log \left(a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{a b x^2+(-a-b) x^3+x^4}+\left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))/(-2*a*d^(1/3) + 2*d^(1/3)*x + (a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))])/d^(1/3) + Log[a*d^(1/3) - d^(1/3)*x + (a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3)]/d^(1/3) - Log[a^2*d^(2/3) - 2*a*d^(2/3)*x + d^(2/3)*x^2 + (-(a*d^(1/3)) + d^(1/3)*x)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3) + (a*b*x^2 + (-a - b)*x^3 + x^4)^(2/3)]/(2*d^(1/3))","A",1
2672,1,240,241,0.8663751,"\int \frac{\left(d+c x^2\right) \sqrt[4]{b x^3+a x^4}}{x^2 \left(-d+c x^2\right)} \, dx","IntegrateAlgebraic[((d + c*x^2)*(b*x^3 + a*x^4)^(1/4))/(x^2*(-d + c*x^2)),x]","\frac{4 \sqrt[4]{b x^3+a x^4}}{x}-2 \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)+2 \sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)+\frac{\text{RootSum}\left[b^2 c-a^2 d+2 a d \text{$\#$1}^4-d \text{$\#$1}^8\&,\frac{-b^2 c \log (x)+a^2 d \log (x)+b^2 c \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)-a^2 d \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)-a d \log (x) \text{$\#$1}^4+a d \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}^3+\text{$\#$1}^7}\&\right]}{d}","\frac{4 \sqrt[4]{b x^3+a x^4}}{x}-2 \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)+2 \sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)+\frac{\text{RootSum}\left[b^2 c-a^2 d+2 a d \text{$\#$1}^4-d \text{$\#$1}^8\&,\frac{b^2 c \log (x)-a^2 d \log (x)-b^2 c \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)+a^2 d \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)+a d \log (x) \text{$\#$1}^4-a d \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}^3-\text{$\#$1}^7}\&\right]}{d}",1,"(4*(b*x^3 + a*x^4)^(1/4))/x - 2*a^(1/4)*ArcTan[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)] + 2*a^(1/4)*ArcTanh[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)] + RootSum[b^2*c - a^2*d + 2*a*d*#1^4 - d*#1^8 & , (-(b^2*c*Log[x]) + a^2*d*Log[x] + b^2*c*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] - a^2*d*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] - a*d*Log[x]*#1^4 + a*d*Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3) + #1^7) & ]/d","A",1
2673,1,240,241,0.0019935,"\int \frac{\left(d+c x^2\right) \sqrt[4]{b x^3+a x^4}}{x^2 \left(-d+c x^2\right)} \, dx","IntegrateAlgebraic[((d + c*x^2)*(b*x^3 + a*x^4)^(1/4))/(x^2*(-d + c*x^2)),x]","\frac{4 \sqrt[4]{b x^3+a x^4}}{x}-2 \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)+2 \sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)+\frac{\text{RootSum}\left[b^2 c-a^2 d+2 a d \text{$\#$1}^4-d \text{$\#$1}^8\&,\frac{-b^2 c \log (x)+a^2 d \log (x)+b^2 c \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)-a^2 d \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)-a d \log (x) \text{$\#$1}^4+a d \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}^3+\text{$\#$1}^7}\&\right]}{d}","\frac{4 \sqrt[4]{b x^3+a x^4}}{x}-2 \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)+2 \sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)+\frac{\text{RootSum}\left[b^2 c-a^2 d+2 a d \text{$\#$1}^4-d \text{$\#$1}^8\&,\frac{b^2 c \log (x)-a^2 d \log (x)-b^2 c \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)+a^2 d \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)+a d \log (x) \text{$\#$1}^4-a d \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}^3-\text{$\#$1}^7}\&\right]}{d}",1,"(4*(b*x^3 + a*x^4)^(1/4))/x - 2*a^(1/4)*ArcTan[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)] + 2*a^(1/4)*ArcTanh[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)] + RootSum[b^2*c - a^2*d + 2*a*d*#1^4 - d*#1^8 & , (-(b^2*c*Log[x]) + a^2*d*Log[x] + b^2*c*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] - a^2*d*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] - a*d*Log[x]*#1^4 + a*d*Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3) + #1^7) & ]/d","A",1
2674,1,241,241,0.6529724,"\int \frac{-1+x^4}{\left(1+x^4\right) \sqrt{a+b x+c x^2+b x^3+a x^4}} \, dx","IntegrateAlgebraic[(-1 + x^4)/((1 + x^4)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]),x]","-\frac{1}{2} \text{RootSum}\left[4 a^2-2 b^2-4 a c+c^2+8 \sqrt{a} b \text{$\#$1}-4 a \text{$\#$1}^2-2 c \text{$\#$1}^2+\text{$\#$1}^4\&,\frac{-2 a \log (x)+c \log (x)+2 a \log \left(\sqrt{a} \left(-1-x^2\right)+\sqrt{a+b x+c x^2+b x^3+a x^4}-x \text{$\#$1}\right)-c \log \left(\sqrt{a} \left(-1-x^2\right)+\sqrt{a+b x+c x^2+b x^3+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^2+\log \left(\sqrt{a} \left(-1-x^2\right)+\sqrt{a+b x+c x^2+b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^2}{-2 \sqrt{a} b+2 a \text{$\#$1}+c \text{$\#$1}-\text{$\#$1}^3}\&\right]","-\frac{1}{2} \text{RootSum}\left[4 a^2-2 b^2-4 a c+c^2+8 \sqrt{a} b \text{$\#$1}-4 a \text{$\#$1}^2-2 c \text{$\#$1}^2+\text{$\#$1}^4\&,\frac{-2 a \log (x)+c \log (x)+2 a \log \left(\sqrt{a} \left(-1-x^2\right)+\sqrt{a+b x+c x^2+b x^3+a x^4}-x \text{$\#$1}\right)-c \log \left(\sqrt{a} \left(-1-x^2\right)+\sqrt{a+b x+c x^2+b x^3+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^2+\log \left(\sqrt{a} \left(-1-x^2\right)+\sqrt{a+b x+c x^2+b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^2}{-2 \sqrt{a} b+2 a \text{$\#$1}+c \text{$\#$1}-\text{$\#$1}^3}\&\right]",1,"-1/2*RootSum[4*a^2 - 2*b^2 - 4*a*c + c^2 + 8*Sqrt[a]*b*#1 - 4*a*#1^2 - 2*c*#1^2 + #1^4 & , (-2*a*Log[x] + c*Log[x] + 2*a*Log[Sqrt[a]*(-1 - x^2) + Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4] - x*#1] - c*Log[Sqrt[a]*(-1 - x^2) + Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4] - x*#1] - Log[x]*#1^2 + Log[Sqrt[a]*(-1 - x^2) + Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4] - x*#1]*#1^2)/(-2*Sqrt[a]*b + 2*a*#1 + c*#1 - #1^3) & ]","A",1
2675,1,241,241,5.554876,"\int \frac{\left(-2+k^2\right) x+k^2 x^3}{\sqrt[3]{\left(1-x^2\right) \left(1-k^2 x^2\right)} \left(-1+d+\left(-2 d+k^2\right) x^2+d x^4\right)} \, dx","IntegrateAlgebraic[((-2 + k^2)*x + k^2*x^3)/(((1 - x^2)*(1 - k^2*x^2))^(1/3)*(-1 + d + (-2*d + k^2)*x^2 + d*x^4)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{2 \sqrt[3]{d}-2 \sqrt[3]{d} x^2+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{2 \sqrt[3]{d}}+\frac{\log \left(-\sqrt[3]{d}+\sqrt[3]{d} x^2+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{2 \sqrt[3]{d}}-\frac{\log \left(d^{2/3}-2 d^{2/3} x^2+d^{2/3} x^4+\left(\sqrt[3]{d}-\sqrt[3]{d} x^2\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{4 \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{2 \sqrt[3]{d}-2 \sqrt[3]{d} x^2+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{2 \sqrt[3]{d}}+\frac{\log \left(-\sqrt[3]{d}+\sqrt[3]{d} x^2+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{2 \sqrt[3]{d}}-\frac{\log \left(d^{2/3}-2 d^{2/3} x^2+d^{2/3} x^4+\left(\sqrt[3]{d}-\sqrt[3]{d} x^2\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{4 \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))/(2*d^(1/3) - 2*d^(1/3)*x^2 + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))])/(2*d^(1/3)) + Log[-d^(1/3) + d^(1/3)*x^2 + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3)]/(2*d^(1/3)) - Log[d^(2/3) - 2*d^(2/3)*x^2 + d^(2/3)*x^4 + (d^(1/3) - d^(1/3)*x^2)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3) + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3)]/(4*d^(1/3))","A",1
2676,1,241,241,5.5560064,"\int \frac{\left(-1+2 k^2\right) x-2 k^4 x^3+k^4 x^5}{\left(\left(1-x^2\right) \left(1-k^2 x^2\right)\right)^{2/3} \left(1-d+\left(d-2 k^2\right) x^2+k^4 x^4\right)} \, dx","IntegrateAlgebraic[((-1 + 2*k^2)*x - 2*k^4*x^3 + k^4*x^5)/(((1 - x^2)*(1 - k^2*x^2))^(2/3)*(1 - d + (d - 2*k^2)*x^2 + k^4*x^4)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}}{2 \sqrt[3]{d}-2 \sqrt[3]{d} x^2+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}}\right)}{2 \sqrt[3]{d}}-\frac{\log \left(-\sqrt[3]{d}+\sqrt[3]{d} x^2+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}+\frac{\log \left(d^{2/3}-2 d^{2/3} x^2+d^{2/3} x^4+\left(\sqrt[3]{d}-\sqrt[3]{d} x^2\right) \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{4/3}\right)}{4 \sqrt[3]{d}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}}{2 \sqrt[3]{d}-2 \sqrt[3]{d} x^2+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}}\right)}{2 \sqrt[3]{d}}-\frac{\log \left(-\sqrt[3]{d}+\sqrt[3]{d} x^2+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}+\frac{\log \left(d^{2/3}-2 d^{2/3} x^2+d^{2/3} x^4+\left(\sqrt[3]{d}-\sqrt[3]{d} x^2\right) \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{4/3}\right)}{4 \sqrt[3]{d}}",1,"-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3))/(2*d^(1/3) - 2*d^(1/3)*x^2 + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3))])/d^(1/3) - Log[-d^(1/3) + d^(1/3)*x^2 + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3)]/(2*d^(1/3)) + Log[d^(2/3) - 2*d^(2/3)*x^2 + d^(2/3)*x^4 + (d^(1/3) - d^(1/3)*x^2)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3) + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(4/3)]/(4*d^(1/3))","A",1
2677,1,241,241,16.1353572,"\int \frac{\left(-2+x^6\right) \left(1-x^4+x^6\right)}{\sqrt[4]{1+x^6} \left(1+2 x^6+x^8+x^{12}\right)} \, dx","IntegrateAlgebraic[((-2 + x^6)*(1 - x^4 + x^6))/((1 + x^6)^(1/4)*(1 + 2*x^6 + x^8 + x^12)),x]","\frac{1}{2} \sqrt{\frac{1}{2} \left(2-\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{1+x^6}}{-x^2+\sqrt{1+x^6}}\right)-\frac{1}{2} \sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{1+x^6}}{-x^2+\sqrt{1+x^6}}\right)+\frac{1}{2} \sqrt{\frac{1}{2} \left(2-\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{1+x^6}}{x^2+\sqrt{1+x^6}}\right)-\frac{1}{2} \sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{1+x^6}}{x^2+\sqrt{1+x^6}}\right)","\frac{1}{2} \sqrt{\frac{1}{2} \left(2-\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{1+x^6}}{-x^2+\sqrt{1+x^6}}\right)-\frac{1}{2} \sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{1+x^6}}{-x^2+\sqrt{1+x^6}}\right)+\frac{1}{2} \sqrt{\frac{1}{2} \left(2-\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{1+x^6}}{x^2+\sqrt{1+x^6}}\right)-\frac{1}{2} \sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{1+x^6}}{x^2+\sqrt{1+x^6}}\right)",1,"(Sqrt[(2 - Sqrt[2])/2]*ArcTan[(Sqrt[2 - Sqrt[2]]*x*(1 + x^6)^(1/4))/(-x^2 + Sqrt[1 + x^6])])/2 - (Sqrt[(2 + Sqrt[2])/2]*ArcTan[(Sqrt[2 + Sqrt[2]]*x*(1 + x^6)^(1/4))/(-x^2 + Sqrt[1 + x^6])])/2 + (Sqrt[(2 - Sqrt[2])/2]*ArcTanh[(Sqrt[2 - Sqrt[2]]*x*(1 + x^6)^(1/4))/(x^2 + Sqrt[1 + x^6])])/2 - (Sqrt[(2 + Sqrt[2])/2]*ArcTanh[(Sqrt[2 + Sqrt[2]]*x*(1 + x^6)^(1/4))/(x^2 + Sqrt[1 + x^6])])/2","A",1
2678,1,242,242,1.2017245,"\int \frac{\left(-q+2 p x^3\right) \left(a q+b x+a p x^3\right) \sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^3 \left(c q+d x+c p x^3\right)} \, dx","IntegrateAlgebraic[((-q + 2*p*x^3)*(a*q + b*x + a*p*x^3)*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6])/(x^3*(c*q + d*x + c*p*x^3)),x]","\frac{\left(a c q+2 b c x-2 a d x+a c p x^3\right) \sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{2 c^2 x^2}-\frac{2 (-b c+a d) \sqrt{-d^2+2 c^2 p q} \tan ^{-1}\left(\frac{\sqrt{-d^2+2 c^2 p q} x}{c q+d x+c p x^3+c \sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}}\right)}{c^3}+\frac{\left(b c d-a d^2+a c^2 p q\right) \log (x)}{c^3}+\frac{\left(-b c d+a d^2-a c^2 p q\right) \log \left(q+p x^3+\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right)}{c^3}","\frac{\left(a c q+2 b c x-2 a d x+a c p x^3\right) \sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{2 c^2 x^2}-\frac{2 (-b c+a d) \sqrt{-d^2+2 c^2 p q} \tan ^{-1}\left(\frac{\sqrt{-d^2+2 c^2 p q} x}{c q+d x+c p x^3+c \sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}}\right)}{c^3}+\frac{\left(b c d-a d^2+a c^2 p q\right) \log (x)}{c^3}+\frac{\left(-b c d+a d^2-a c^2 p q\right) \log \left(q+p x^3+\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right)}{c^3}",1,"((a*c*q + 2*b*c*x - 2*a*d*x + a*c*p*x^3)*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6])/(2*c^2*x^2) - (2*(-(b*c) + a*d)*Sqrt[-d^2 + 2*c^2*p*q]*ArcTan[(Sqrt[-d^2 + 2*c^2*p*q]*x)/(c*q + d*x + c*p*x^3 + c*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6])])/c^3 + ((b*c*d - a*d^2 + a*c^2*p*q)*Log[x])/c^3 + ((-(b*c*d) + a*d^2 - a*c^2*p*q)*Log[q + p*x^3 + Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]])/c^3","A",1
2679,1,326,242,1.0494998,"\int \frac{-1+x^2}{\left(1+x^2\right) \sqrt{x^2+\sqrt{1+x^4}}} \, dx","IntegrateAlgebraic[(-1 + x^2)/((1 + x^2)*Sqrt[x^2 + Sqrt[1 + x^4]]),x]","\frac{x}{2 \sqrt{x^2+\sqrt{1+x^4}}}+2 \sqrt{2} \tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^4}}{\sqrt{2}}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)-2 \sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{-\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)+\frac{\tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^4}}{\sqrt{2}}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)}{\sqrt{2}}-2 \sqrt{-1+\sqrt{2}} \tanh ^{-1}\left(\frac{-\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)","\frac{x}{2 \sqrt{x^2+\sqrt{1+x^4}}}+2 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-2 \sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)}{\sqrt{2}}-2 \sqrt{-1+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2 \left(-1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)",1,"x/(2*Sqrt[x^2 + Sqrt[1 + x^4]]) + 2*Sqrt[2]*ArcTan[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] - 2*Sqrt[1 + Sqrt[2]]*ArcTan[(-Sqrt[1/2 + 1/Sqrt[2]] + Sqrt[1/2 + 1/Sqrt[2]]*x^2 + Sqrt[1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] + ArcTanh[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])]/Sqrt[2] - 2*Sqrt[-1 + Sqrt[2]]*ArcTanh[(-Sqrt[-1/2 + 1/Sqrt[2]] + Sqrt[-1/2 + 1/Sqrt[2]]*x^2 + Sqrt[-1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])]","A",1
2680,1,326,242,1.1021169,"\int \frac{\left(-1+x^2\right) \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2} \, dx","IntegrateAlgebraic[((-1 + x^2)*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2),x]","\frac{1}{2} x \sqrt{x^2+\sqrt{1+x^4}}+\frac{\tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^4}}{\sqrt{2}}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)}{\sqrt{2}}-2 \sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\frac{-\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)-2 \sqrt{2} \tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^4}}{\sqrt{2}}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)+2 \sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{-\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)","\frac{1}{2} x \sqrt{x^2+\sqrt{1+x^4}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)}{\sqrt{2}}-2 \sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)+2 \sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2 \left(-1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)",1,"(x*Sqrt[x^2 + Sqrt[1 + x^4]])/2 + ArcTan[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])]/Sqrt[2] - 2*Sqrt[-1 + Sqrt[2]]*ArcTan[(-Sqrt[1/2 + 1/Sqrt[2]] + Sqrt[1/2 + 1/Sqrt[2]]*x^2 + Sqrt[1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] - 2*Sqrt[2]*ArcTanh[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] + 2*Sqrt[1 + Sqrt[2]]*ArcTanh[(-Sqrt[-1/2 + 1/Sqrt[2]] + Sqrt[-1/2 + 1/Sqrt[2]]*x^2 + Sqrt[-1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])]","A",1
2681,1,242,242,0.3143159,"\int \sqrt{1+x^2} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[Sqrt[1 + x^2]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]],x]","\frac{\left(78032-1254 x+193024 x^2-3072 x^3+35840 x^4\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(184+345 x+2048 x^2+2560 x^3\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(282+175104 x-3072 x^2+35840 x^3\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-935+2048 x+2560 x^2\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{80640 x \sqrt{1+x^2}+40320 \left(1+2 x^2\right)}-\frac{251}{128} \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)","\frac{\left(78032-1254 x+193024 x^2-3072 x^3+35840 x^4\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(184+345 x+2048 x^2+2560 x^3\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(282+175104 x-3072 x^2+35840 x^3\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-935+2048 x+2560 x^2\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{80640 x \sqrt{1+x^2}+40320 \left(1+2 x^2\right)}-\frac{251}{128} \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)",1,"((78032 - 1254*x + 193024*x^2 - 3072*x^3 + 35840*x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (184 + 345*x + 2048*x^2 + 2560*x^3)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((282 + 175104*x - 3072*x^2 + 35840*x^3)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-935 + 2048*x + 2560*x^2)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(80640*x*Sqrt[1 + x^2] + 40320*(1 + 2*x^2)) - (251*ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]])/128","A",1
2682,1,243,243,0.6043698,"\int \frac{x (-a+x)}{\left(x^2 (-a+x)\right)^{2/3} \left(a^2 d-2 a d x+(-1+d) x^2\right)} \, dx","IntegrateAlgebraic[(x*(-a + x))/((x^2*(-a + x))^(2/3)*(a^2*d - 2*a*d*x + (-1 + d)*x^2)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{d} \left(-a x^2+x^3\right)^{2/3}}\right)}{2 a d^{2/3}}+\frac{\log \left(x-\sqrt[6]{d} \sqrt[3]{-a x^2+x^3}\right)}{2 a d^{2/3}}+\frac{\log \left(x+\sqrt[6]{d} \sqrt[3]{-a x^2+x^3}\right)}{2 a d^{2/3}}-\frac{\log \left(x^2-\sqrt[6]{d} x \sqrt[3]{-a x^2+x^3}+\sqrt[3]{d} \left(-a x^2+x^3\right)^{2/3}\right)}{4 a d^{2/3}}-\frac{\log \left(x^2+\sqrt[6]{d} x \sqrt[3]{-a x^2+x^3}+\sqrt[3]{d} \left(-a x^2+x^3\right)^{2/3}\right)}{4 a d^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{d} \left(-a x^2+x^3\right)^{2/3}}\right)}{2 a d^{2/3}}+\frac{\log \left(x-\sqrt[6]{d} \sqrt[3]{-a x^2+x^3}\right)}{2 a d^{2/3}}+\frac{\log \left(x+\sqrt[6]{d} \sqrt[3]{-a x^2+x^3}\right)}{2 a d^{2/3}}-\frac{\log \left(x^2-\sqrt[6]{d} x \sqrt[3]{-a x^2+x^3}+\sqrt[3]{d} \left(-a x^2+x^3\right)^{2/3}\right)}{4 a d^{2/3}}-\frac{\log \left(x^2+\sqrt[6]{d} x \sqrt[3]{-a x^2+x^3}+\sqrt[3]{d} \left(-a x^2+x^3\right)^{2/3}\right)}{4 a d^{2/3}}",1,"-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*d^(1/3)*(-(a*x^2) + x^3)^(2/3))])/(a*d^(2/3)) + Log[x - d^(1/6)*(-(a*x^2) + x^3)^(1/3)]/(2*a*d^(2/3)) + Log[x + d^(1/6)*(-(a*x^2) + x^3)^(1/3)]/(2*a*d^(2/3)) - Log[x^2 - d^(1/6)*x*(-(a*x^2) + x^3)^(1/3) + d^(1/3)*(-(a*x^2) + x^3)^(2/3)]/(4*a*d^(2/3)) - Log[x^2 + d^(1/6)*x*(-(a*x^2) + x^3)^(1/3) + d^(1/3)*(-(a*x^2) + x^3)^(2/3)]/(4*a*d^(2/3))","A",1
2683,1,243,243,0.5611846,"\int \frac{x}{\sqrt[3]{x^2 (-a+x)} \left(a^2 d-2 a d x+(-1+d) x^2\right)} \, dx","IntegrateAlgebraic[x/((x^2*(-a + x))^(1/3)*(a^2*d - 2*a*d*x + (-1 + d)*x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{d} \left(-a x^2+x^3\right)^{2/3}}\right)}{2 a \sqrt[3]{d}}+\frac{\log \left(x-\sqrt[6]{d} \sqrt[3]{-a x^2+x^3}\right)}{2 a \sqrt[3]{d}}+\frac{\log \left(x+\sqrt[6]{d} \sqrt[3]{-a x^2+x^3}\right)}{2 a \sqrt[3]{d}}-\frac{\log \left(x^2-\sqrt[6]{d} x \sqrt[3]{-a x^2+x^3}+\sqrt[3]{d} \left(-a x^2+x^3\right)^{2/3}\right)}{4 a \sqrt[3]{d}}-\frac{\log \left(x^2+\sqrt[6]{d} x \sqrt[3]{-a x^2+x^3}+\sqrt[3]{d} \left(-a x^2+x^3\right)^{2/3}\right)}{4 a \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{d} \left(-a x^2+x^3\right)^{2/3}}\right)}{2 a \sqrt[3]{d}}+\frac{\log \left(x-\sqrt[6]{d} \sqrt[3]{-a x^2+x^3}\right)}{2 a \sqrt[3]{d}}+\frac{\log \left(x+\sqrt[6]{d} \sqrt[3]{-a x^2+x^3}\right)}{2 a \sqrt[3]{d}}-\frac{\log \left(x^2-\sqrt[6]{d} x \sqrt[3]{-a x^2+x^3}+\sqrt[3]{d} \left(-a x^2+x^3\right)^{2/3}\right)}{4 a \sqrt[3]{d}}-\frac{\log \left(x^2+\sqrt[6]{d} x \sqrt[3]{-a x^2+x^3}+\sqrt[3]{d} \left(-a x^2+x^3\right)^{2/3}\right)}{4 a \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*d^(1/3)*(-(a*x^2) + x^3)^(2/3))])/(2*a*d^(1/3)) + Log[x - d^(1/6)*(-(a*x^2) + x^3)^(1/3)]/(2*a*d^(1/3)) + Log[x + d^(1/6)*(-(a*x^2) + x^3)^(1/3)]/(2*a*d^(1/3)) - Log[x^2 - d^(1/6)*x*(-(a*x^2) + x^3)^(1/3) + d^(1/3)*(-(a*x^2) + x^3)^(2/3)]/(4*a*d^(1/3)) - Log[x^2 + d^(1/6)*x*(-(a*x^2) + x^3)^(1/3) + d^(1/3)*(-(a*x^2) + x^3)^(2/3)]/(4*a*d^(1/3))","A",1
2684,1,243,243,0.6462619,"\int \frac{-a x+x^2}{\left(x^2 (-a+x)\right)^{2/3} \left(a^2 d-2 a d x+(-1+d) x^2\right)} \, dx","IntegrateAlgebraic[(-(a*x) + x^2)/((x^2*(-a + x))^(2/3)*(a^2*d - 2*a*d*x + (-1 + d)*x^2)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{d} \left(-a x^2+x^3\right)^{2/3}}\right)}{2 a d^{2/3}}+\frac{\log \left(x-\sqrt[6]{d} \sqrt[3]{-a x^2+x^3}\right)}{2 a d^{2/3}}+\frac{\log \left(x+\sqrt[6]{d} \sqrt[3]{-a x^2+x^3}\right)}{2 a d^{2/3}}-\frac{\log \left(x^2-\sqrt[6]{d} x \sqrt[3]{-a x^2+x^3}+\sqrt[3]{d} \left(-a x^2+x^3\right)^{2/3}\right)}{4 a d^{2/3}}-\frac{\log \left(x^2+\sqrt[6]{d} x \sqrt[3]{-a x^2+x^3}+\sqrt[3]{d} \left(-a x^2+x^3\right)^{2/3}\right)}{4 a d^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{d} \left(-a x^2+x^3\right)^{2/3}}\right)}{2 a d^{2/3}}+\frac{\log \left(x-\sqrt[6]{d} \sqrt[3]{-a x^2+x^3}\right)}{2 a d^{2/3}}+\frac{\log \left(x+\sqrt[6]{d} \sqrt[3]{-a x^2+x^3}\right)}{2 a d^{2/3}}-\frac{\log \left(x^2-\sqrt[6]{d} x \sqrt[3]{-a x^2+x^3}+\sqrt[3]{d} \left(-a x^2+x^3\right)^{2/3}\right)}{4 a d^{2/3}}-\frac{\log \left(x^2+\sqrt[6]{d} x \sqrt[3]{-a x^2+x^3}+\sqrt[3]{d} \left(-a x^2+x^3\right)^{2/3}\right)}{4 a d^{2/3}}",1,"-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*d^(1/3)*(-(a*x^2) + x^3)^(2/3))])/(a*d^(2/3)) + Log[x - d^(1/6)*(-(a*x^2) + x^3)^(1/3)]/(2*a*d^(2/3)) + Log[x + d^(1/6)*(-(a*x^2) + x^3)^(1/3)]/(2*a*d^(2/3)) - Log[x^2 - d^(1/6)*x*(-(a*x^2) + x^3)^(1/3) + d^(1/3)*(-(a*x^2) + x^3)^(2/3)]/(4*a*d^(2/3)) - Log[x^2 + d^(1/6)*x*(-(a*x^2) + x^3)^(1/3) + d^(1/3)*(-(a*x^2) + x^3)^(2/3)]/(4*a*d^(2/3))","A",1
2685,1,243,243,5.4288975,"\int \frac{-1+x}{(1+x) \sqrt[3]{-1+x^3}} \, dx","IntegrateAlgebraic[(-1 + x)/((1 + x)*(-1 + x^3)^(1/3)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{\sqrt{3}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{-\frac{\sqrt[3]{2}}{\sqrt{3}}+\frac{\sqrt[3]{2} x}{\sqrt{3}}+\frac{\sqrt[3]{-1+x^3}}{\sqrt{3}}}{\sqrt[3]{-1+x^3}}\right)}{\sqrt[3]{2}}+\frac{\log \left(-\sqrt[3]{2}+\sqrt[3]{2} x-2 \sqrt[3]{-1+x^3}\right)}{\sqrt[3]{2}}-\frac{1}{3} \log \left(-x+\sqrt[3]{-1+x^3}\right)+\frac{1}{6} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)-\frac{\log \left(2^{2/3}-2\ 2^{2/3} x+2^{2/3} x^2+\left(-2 \sqrt[3]{2}+2 \sqrt[3]{2} x\right) \sqrt[3]{-1+x^3}+4 \left(-1+x^3\right)^{2/3}\right)}{2 \sqrt[3]{2}}","\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-1+x^3}}\right)}{\sqrt{3}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{-\frac{\sqrt[3]{2}}{\sqrt{3}}+\frac{\sqrt[3]{2} x}{\sqrt{3}}+\frac{\sqrt[3]{-1+x^3}}{\sqrt{3}}}{\sqrt[3]{-1+x^3}}\right)}{\sqrt[3]{2}}+\frac{\log \left(-\sqrt[3]{2}+\sqrt[3]{2} x-2 \sqrt[3]{-1+x^3}\right)}{\sqrt[3]{2}}-\frac{1}{3} \log \left(-x+\sqrt[3]{-1+x^3}\right)+\frac{1}{6} \log \left(x^2+x \sqrt[3]{-1+x^3}+\left(-1+x^3\right)^{2/3}\right)-\frac{\log \left(2^{2/3}-2\ 2^{2/3} x+2^{2/3} x^2+\left(-2 \sqrt[3]{2}+2 \sqrt[3]{2} x\right) \sqrt[3]{-1+x^3}+4 \left(-1+x^3\right)^{2/3}\right)}{2 \sqrt[3]{2}}",1,"ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^3)^(1/3))]/Sqrt[3] - (Sqrt[3]*ArcTan[(-(2^(1/3)/Sqrt[3]) + (2^(1/3)*x)/Sqrt[3] + (-1 + x^3)^(1/3)/Sqrt[3])/(-1 + x^3)^(1/3)])/2^(1/3) + Log[-2^(1/3) + 2^(1/3)*x - 2*(-1 + x^3)^(1/3)]/2^(1/3) - Log[-x + (-1 + x^3)^(1/3)]/3 + Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)]/6 - Log[2^(2/3) - 2*2^(2/3)*x + 2^(2/3)*x^2 + (-2*2^(1/3) + 2*2^(1/3)*x)*(-1 + x^3)^(1/3) + 4*(-1 + x^3)^(2/3)]/(2*2^(1/3))","A",1
2686,1,243,243,0.6881049,"\int \frac{\left(1+x^2\right) \sqrt[3]{-x^2+x^3}}{-1+x^2} \, dx","IntegrateAlgebraic[((1 + x^2)*(-x^2 + x^3)^(1/3))/(-1 + x^2),x]","\frac{1}{6} (-1+3 x) \sqrt[3]{-x^2+x^3}-\frac{17 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)}{3 \sqrt{3}}+\sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-x^2+x^3}}\right)-\frac{17}{9} \log \left(-x+\sqrt[3]{-x^2+x^3}\right)+\sqrt[3]{2} \log \left(-2 x+2^{2/3} \sqrt[3]{-x^2+x^3}\right)+\frac{17}{18} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-x^2+x^3}+\sqrt[3]{2} \left(-x^2+x^3\right)^{2/3}\right)}{2^{2/3}}","\frac{1}{6} (-1+3 x) \sqrt[3]{-x^2+x^3}-\frac{17 \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x^2+x^3}}\right)}{3 \sqrt{3}}+\sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-x^2+x^3}}\right)-\frac{17}{9} \log \left(-x+\sqrt[3]{-x^2+x^3}\right)+\sqrt[3]{2} \log \left(-2 x+2^{2/3} \sqrt[3]{-x^2+x^3}\right)+\frac{17}{18} \log \left(x^2+x \sqrt[3]{-x^2+x^3}+\left(-x^2+x^3\right)^{2/3}\right)-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-x^2+x^3}+\sqrt[3]{2} \left(-x^2+x^3\right)^{2/3}\right)}{2^{2/3}}",1,"((-1 + 3*x)*(-x^2 + x^3)^(1/3))/6 - (17*ArcTan[(Sqrt[3]*x)/(x + 2*(-x^2 + x^3)^(1/3))])/(3*Sqrt[3]) + 2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-x^2 + x^3)^(1/3))] - (17*Log[-x + (-x^2 + x^3)^(1/3)])/9 + 2^(1/3)*Log[-2*x + 2^(2/3)*(-x^2 + x^3)^(1/3)] + (17*Log[x^2 + x*(-x^2 + x^3)^(1/3) + (-x^2 + x^3)^(2/3)])/18 - Log[2*x^2 + 2^(2/3)*x*(-x^2 + x^3)^(1/3) + 2^(1/3)*(-x^2 + x^3)^(2/3)]/2^(2/3)","A",1
2687,1,243,243,3.1471701,"\int \frac{b+a x^4}{\left(-b-a x^2+x^4\right) \sqrt[4]{-b x^2+a x^4}} \, dx","IntegrateAlgebraic[(b + a*x^4)/((-b - a*x^2 + x^4)*(-(b*x^2) + a*x^4)^(1/4)),x]","a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)+a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)+\frac{1}{2} \text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a \log (x)+2 a^2 \log (x)-a \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)-2 a^2 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4-a \log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+a \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]","a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)+a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)+\frac{1}{2} \text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a \log (x)-2 a^2 \log (x)+a \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)+2 a^2 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4+a \log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-a \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{3 a \text{$\#$1}-2 \text{$\#$1}^5}\&\right]",1,"a^(3/4)*ArcTan[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)] + a^(3/4)*ArcTanh[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)] + RootSum[2*a^2 - b - 3*a*#1^4 + #1^8 & , (a*Log[x] + 2*a^2*Log[x] - a*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1] - 2*a^2*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1] - Log[x]*#1^4 - a*Log[x]*#1^4 + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4 + a*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1 + 2*#1^5) & ]/2","A",1
2688,1,243,243,0.0021559,"\int \frac{b+a x^4}{\left(-b-a x^2+x^4\right) \sqrt[4]{-b x^2+a x^4}} \, dx","IntegrateAlgebraic[(b + a*x^4)/((-b - a*x^2 + x^4)*(-(b*x^2) + a*x^4)^(1/4)),x]","a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)+a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)+\frac{1}{2} \text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a \log (x)+2 a^2 \log (x)-a \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)-2 a^2 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^4-a \log (x) \text{$\#$1}^4+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+a \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 a \text{$\#$1}+2 \text{$\#$1}^5}\&\right]","a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)+a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)+\frac{1}{2} \text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a \log (x)-2 a^2 \log (x)+a \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)+2 a^2 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4+a \log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-a \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{3 a \text{$\#$1}-2 \text{$\#$1}^5}\&\right]",1,"a^(3/4)*ArcTan[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)] + a^(3/4)*ArcTanh[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)] + RootSum[2*a^2 - b - 3*a*#1^4 + #1^8 & , (a*Log[x] + 2*a^2*Log[x] - a*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1] - 2*a^2*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1] - Log[x]*#1^4 - a*Log[x]*#1^4 + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4 + a*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1 + 2*#1^5) & ]/2","A",1
2689,1,243,243,3.8849829,"\int \frac{\left(1-2 k^2\right) x+k^2 x^3}{\sqrt[3]{\left(1-x^2\right) \left(1-k^2 x^2\right)} \left(1-d+\left(d-2 k^2\right) x^2+k^4 x^4\right)} \, dx","IntegrateAlgebraic[((1 - 2*k^2)*x + k^2*x^3)/(((1 - x^2)*(1 - k^2*x^2))^(1/3)*(1 - d + (d - 2*k^2)*x^2 + k^4*x^4)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{2-2 k^2 x^2+\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{2 d^{2/3}}+\frac{\log \left(-1+k^2 x^2+\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{2 d^{2/3}}-\frac{\log \left(1-2 k^2 x^2+k^4 x^4+\left(\sqrt[3]{d}-\sqrt[3]{d} k^2 x^2\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+d^{2/3} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{4 d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{2-2 k^2 x^2+\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{2 d^{2/3}}+\frac{\log \left(-1+k^2 x^2+\sqrt[3]{d} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{2 d^{2/3}}-\frac{\log \left(1-2 k^2 x^2+k^4 x^4+\left(\sqrt[3]{d}-\sqrt[3]{d} k^2 x^2\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+d^{2/3} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{4 d^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))/(2 - 2*k^2*x^2 + d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))])/(2*d^(2/3)) + Log[-1 + k^2*x^2 + d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3)]/(2*d^(2/3)) - Log[1 - 2*k^2*x^2 + k^4*x^4 + (d^(1/3) - d^(1/3)*k^2*x^2)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3) + d^(2/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3)]/(4*d^(2/3))","A",1
2690,1,243,243,5.5757621,"\int \frac{\left(2-k^2\right) x-2 x^3+k^2 x^5}{\left(\left(1-x^2\right) \left(1-k^2 x^2\right)\right)^{2/3} \left(-1+d+\left(-2 d+k^2\right) x^2+d x^4\right)} \, dx","IntegrateAlgebraic[((2 - k^2)*x - 2*x^3 + k^2*x^5)/(((1 - x^2)*(1 - k^2*x^2))^(2/3)*(-1 + d + (-2*d + k^2)*x^2 + d*x^4)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}}{2-2 k^2 x^2+\sqrt[3]{d} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}}\right)}{2 d^{2/3}}-\frac{\log \left(-1+k^2 x^2+\sqrt[3]{d} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 d^{2/3}}+\frac{\log \left(1-2 k^2 x^2+k^4 x^4+\left(\sqrt[3]{d}-\sqrt[3]{d} k^2 x^2\right) \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}+d^{2/3} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{4/3}\right)}{4 d^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}}{2-2 k^2 x^2+\sqrt[3]{d} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}}\right)}{2 d^{2/3}}-\frac{\log \left(-1+k^2 x^2+\sqrt[3]{d} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 d^{2/3}}+\frac{\log \left(1-2 k^2 x^2+k^4 x^4+\left(\sqrt[3]{d}-\sqrt[3]{d} k^2 x^2\right) \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}+d^{2/3} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{4/3}\right)}{4 d^{2/3}}",1,"-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3))/(2 - 2*k^2*x^2 + d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3))])/d^(2/3) - Log[-1 + k^2*x^2 + d^(1/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3)]/(2*d^(2/3)) + Log[1 - 2*k^2*x^2 + k^4*x^4 + (d^(1/3) - d^(1/3)*k^2*x^2)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3) + d^(2/3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(4/3)]/(4*d^(2/3))","A",1
2691,1,211,243,0.3775203,"\int \frac{1}{x^4 \sqrt{b+\sqrt{b^2+a x^2}}} \, dx","IntegrateAlgebraic[1/(x^4*Sqrt[b + Sqrt[b^2 + a*x^2]]),x]","\frac{a^{3/2} \left(-56 b^3-8 b^2 \sqrt{b^2+a x^2}\right)+a^{5/2} \left(5 b x^2+15 x^2 \sqrt{b^2+a x^2}\right)}{\frac{384 a^{3/2} b^4 x^3 \sqrt{b^2+a x^2}}{\left(b+\sqrt{b^2+a x^2}\right)^{3/2}}+\frac{192 a^{3/2} b^3 x^3 \left(2 b^2+a x^2\right)}{\left(b+\sqrt{b^2+a x^2}\right)^{3/2}}}+\frac{5 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{64 \sqrt{2} b^{7/2}}","\frac{a^{3/2} \left(-56 b^3-8 b^2 \sqrt{b^2+a x^2}\right)+a^{5/2} \left(5 b x^2+15 x^2 \sqrt{b^2+a x^2}\right)}{\frac{384 a^{3/2} b^4 x^3 \sqrt{b^2+a x^2}}{\left(b+\sqrt{b^2+a x^2}\right)^{3/2}}+\frac{192 a^{3/2} b^3 x^3 \left(2 b^2+a x^2\right)}{\left(b+\sqrt{b^2+a x^2}\right)^{3/2}}}+\frac{5 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2} \sqrt{b}}\right)}{32 \sqrt{2} b^{7/2}}",1,"(a^(3/2)*(-56*b^3 - 8*b^2*Sqrt[b^2 + a*x^2]) + a^(5/2)*(5*b*x^2 + 15*x^2*Sqrt[b^2 + a*x^2]))/((384*a^(3/2)*b^4*x^3*Sqrt[b^2 + a*x^2])/(b + Sqrt[b^2 + a*x^2])^(3/2) + (192*a^(3/2)*b^3*x^3*(2*b^2 + a*x^2))/(b + Sqrt[b^2 + a*x^2])^(3/2)) + (5*a^(3/2)*ArcTan[(Sqrt[a]*x)/(Sqrt[2]*Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/(64*Sqrt[2]*b^(7/2))","A",1
2692,1,244,244,0.6708895,"\int \frac{-a (a-5 b)-(3 a+5 b) x+4 x^2}{\sqrt[3]{(-a+x) (-b+x)} \left(-a^5+b d-\left(-5 a^4+d\right) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right)} \, dx","IntegrateAlgebraic[(-(a*(a - 5*b)) - (3*a + 5*b)*x + 4*x^2)/(((-a + x)*(-b + x))^(1/3)*(-a^5 + b*d - (-5*a^4 + d)*x - 10*a^3*x^2 + 10*a^2*x^3 - 5*a*x^4 + x^5)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{a b+(-a-b) x+x^2}}{2 a^2-4 a x+2 x^2+\sqrt[3]{d} \sqrt[3]{a b+(-a-b) x+x^2}}\right)}{d^{2/3}}+\frac{\log \left(a^2-2 a x+x^2-\sqrt[3]{d} \sqrt[3]{a b+(-a-b) x+x^2}\right)}{d^{2/3}}-\frac{\log \left(a^4-4 a^3 x+6 a^2 x^2-4 a x^3+x^4+d^{2/3} \left(a b+(-a-b) x+x^2\right)^{2/3}+\sqrt[3]{a b+(-a-b) x+x^2} \left(a^2 \sqrt[3]{d}-2 a \sqrt[3]{d} x+\sqrt[3]{d} x^2\right)\right)}{2 d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{a b+(-a-b) x+x^2}}{2 a^2-4 a x+2 x^2+\sqrt[3]{d} \sqrt[3]{a b+(-a-b) x+x^2}}\right)}{d^{2/3}}+\frac{\log \left(a^2-2 a x+x^2-\sqrt[3]{d} \sqrt[3]{a b+(-a-b) x+x^2}\right)}{d^{2/3}}-\frac{\log \left(a^4-4 a^3 x+6 a^2 x^2-4 a x^3+x^4+d^{2/3} \left(a b+(-a-b) x+x^2\right)^{2/3}+\sqrt[3]{a b+(-a-b) x+x^2} \left(a^2 \sqrt[3]{d}-2 a \sqrt[3]{d} x+\sqrt[3]{d} x^2\right)\right)}{2 d^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(a*b + (-a - b)*x + x^2)^(1/3))/(2*a^2 - 4*a*x + 2*x^2 + d^(1/3)*(a*b + (-a - b)*x + x^2)^(1/3))])/d^(2/3) + Log[a^2 - 2*a*x + x^2 - d^(1/3)*(a*b + (-a - b)*x + x^2)^(1/3)]/d^(2/3) - Log[a^4 - 4*a^3*x + 6*a^2*x^2 - 4*a*x^3 + x^4 + d^(2/3)*(a*b + (-a - b)*x + x^2)^(2/3) + (a*b + (-a - b)*x + x^2)^(1/3)*(a^2*d^(1/3) - 2*a*d^(1/3)*x + d^(1/3)*x^2)]/(2*d^(2/3))","A",1
2693,1,266,244,1.3199928,"\int \frac{x^6 \sqrt{-x+x^4}}{b+a x^6} \, dx","IntegrateAlgebraic[(x^6*Sqrt[-x + x^4])/(b + a*x^6),x]","\frac{x \sqrt{-x+x^4}}{3 a}-\frac{\sqrt[4]{-1} \sqrt{\left(\sqrt{a}-i \sqrt{b}\right) \sqrt{b}} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{\sqrt{a} \sqrt{b}-i b} x \sqrt{-x+x^4}}{\sqrt{b} (-1+x) \left(1+x+x^2\right)}\right)}{3 a^{3/2}}-\frac{(-1)^{3/4} \sqrt{\left(\sqrt{a}+i \sqrt{b}\right) \sqrt{b}} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{\sqrt{a} \sqrt{b}+i b} x \sqrt{-x+x^4}}{\sqrt{b} (-1+x) \left(1+x+x^2\right)}\right)}{3 a^{3/2}}+\frac{\log \left(-x^2+\sqrt{-x+x^4}\right)}{6 a}-\frac{\log \left(a x^2+a \sqrt{-x+x^4}\right)}{6 a}","\frac{x \sqrt{-x+x^4}}{3 a}-\frac{\sqrt[4]{-1} \sqrt{\left(\sqrt{a}-i \sqrt{b}\right) \sqrt{b}} \tan ^{-1}\left(\frac{(1+i) \sqrt{\sqrt{a} \sqrt{b}-i b} \sqrt{-x+x^4}}{\sqrt{2} \left(\sqrt{a}-i \sqrt{b}\right) x^2}\right)}{3 a^{3/2}}-\frac{(-1)^{3/4} \sqrt{\left(\sqrt{a}+i \sqrt{b}\right) \sqrt{b}} \tan ^{-1}\left(\frac{(1+i) \sqrt{\sqrt{a} \sqrt{b}+i b} x \sqrt{-x+x^4}}{\sqrt{2} \sqrt{b} (-1+x) \left(1+x+x^2\right)}\right)}{3 a^{3/2}}-\frac{\tanh ^{-1}\left(\frac{x^2}{\sqrt{-x+x^4}}\right)}{3 a}",1,"(x*Sqrt[-x + x^4])/(3*a) - ((-1)^(1/4)*Sqrt[(Sqrt[a] - I*Sqrt[b])*Sqrt[b]]*ArcTan[((-1)^(3/4)*Sqrt[Sqrt[a]*Sqrt[b] - I*b]*x*Sqrt[-x + x^4])/(Sqrt[b]*(-1 + x)*(1 + x + x^2))])/(3*a^(3/2)) - ((-1)^(3/4)*Sqrt[(Sqrt[a] + I*Sqrt[b])*Sqrt[b]]*ArcTan[((-1)^(1/4)*Sqrt[Sqrt[a]*Sqrt[b] + I*b]*x*Sqrt[-x + x^4])/(Sqrt[b]*(-1 + x)*(1 + x + x^2))])/(3*a^(3/2)) + Log[-x^2 + Sqrt[-x + x^4]]/(6*a) - Log[a*x^2 + a*Sqrt[-x + x^4]]/(6*a)","A",1
2694,1,244,244,0.956728,"\int \frac{b^6+a^6 x^6}{\sqrt{-b^2 x+a^2 x^3} \left(-b^6+a^6 x^6\right)} \, dx","IntegrateAlgebraic[(b^6 + a^6*x^6)/(Sqrt[-(b^2*x) + a^2*x^3]*(-b^6 + a^6*x^6)),x]","\frac{2 \sqrt{-b^2 x+a^2 x^3}}{3 \left(b^2-a^2 x^2\right)}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} 3^{3/4} \sqrt{a} \sqrt{b} \sqrt{-b^2 x+a^2 x^3}}{-\sqrt{3} b^2-3 a b x+\sqrt{3} a^2 x^2}\right)}{3 \sqrt[4]{3} \sqrt{a} \sqrt{b}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{-\frac{b^{3/2}}{\sqrt{2} \sqrt[4]{3} \sqrt{a}}+\frac{\sqrt[4]{3} \sqrt{a} \sqrt{b} x}{\sqrt{2}}+\frac{a^{3/2} x^2}{\sqrt{2} \sqrt[4]{3} \sqrt{b}}}{\sqrt{-b^2 x+a^2 x^3}}\right)}{3 \sqrt[4]{3} \sqrt{a} \sqrt{b}}","\frac{2 \sqrt{-b^2 x+a^2 x^3}}{3 \left(b^2-a^2 x^2\right)}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} 3^{3/4} \sqrt{a} \sqrt{b} \sqrt{-b^2 x+a^2 x^3}}{-\sqrt{3} b^2-3 a b x+\sqrt{3} a^2 x^2}\right)}{3 \sqrt[4]{3} \sqrt{a} \sqrt{b}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{-\frac{b^{3/2}}{\sqrt{2} \sqrt[4]{3} \sqrt{a}}+\frac{\sqrt[4]{3} \sqrt{a} \sqrt{b} x}{\sqrt{2}}+\frac{a^{3/2} x^2}{\sqrt{2} \sqrt[4]{3} \sqrt{b}}}{\sqrt{-b^2 x+a^2 x^3}}\right)}{3 \sqrt[4]{3} \sqrt{a} \sqrt{b}}",1,"(2*Sqrt[-(b^2*x) + a^2*x^3])/(3*(b^2 - a^2*x^2)) - (Sqrt[2]*ArcTan[(Sqrt[2]*3^(3/4)*Sqrt[a]*Sqrt[b]*Sqrt[-(b^2*x) + a^2*x^3])/(-(Sqrt[3]*b^2) - 3*a*b*x + Sqrt[3]*a^2*x^2)])/(3*3^(1/4)*Sqrt[a]*Sqrt[b]) - (Sqrt[2]*ArcTanh[(-(b^(3/2)/(Sqrt[2]*3^(1/4)*Sqrt[a])) + (3^(1/4)*Sqrt[a]*Sqrt[b]*x)/Sqrt[2] + (a^(3/2)*x^2)/(Sqrt[2]*3^(1/4)*Sqrt[b]))/Sqrt[-(b^2*x) + a^2*x^3]])/(3*3^(1/4)*Sqrt[a]*Sqrt[b])","A",1
2695,1,245,245,3.1864709,"\int \frac{a^2 b-2 a^2 x+(2 a-b) x^2}{(x (-a+x) (-b+x))^{2/3} \left(a^2+(-2 a+b d) x+(1-d) x^2\right)} \, dx","IntegrateAlgebraic[(a^2*b - 2*a^2*x + (2*a - b)*x^2)/((x*(-a + x)*(-b + x))^(2/3)*(a^2 + (-2*a + b*d)*x + (1 - d)*x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}}{-2 b \sqrt[3]{d} x+2 \sqrt[3]{d} x^2+\left(a b x+(-a-b) x^2+x^3\right)^{2/3}}\right)}{\sqrt[3]{d}}+\frac{\log \left(b \sqrt{d} x-\sqrt{d} x^2+\sqrt[6]{d} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}\right)}{\sqrt[3]{d}}-\frac{\log \left(b^2 d x^2-2 b d x^3+d x^4+\left(-b d^{2/3} x+d^{2/3} x^2\right) \left(a b x+(-a-b) x^2+x^3\right)^{2/3}+\sqrt[3]{d} \left(a b x+(-a-b) x^2+x^3\right)^{4/3}\right)}{2 \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}}{-2 b \sqrt[3]{d} x+2 \sqrt[3]{d} x^2+\left(a b x+(-a-b) x^2+x^3\right)^{2/3}}\right)}{\sqrt[3]{d}}+\frac{\log \left(b \sqrt{d} x-\sqrt{d} x^2+\sqrt[6]{d} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}\right)}{\sqrt[3]{d}}-\frac{\log \left(b^2 d x^2-2 b d x^3+d x^4+\left(-b d^{2/3} x+d^{2/3} x^2\right) \left(a b x+(-a-b) x^2+x^3\right)^{2/3}+\sqrt[3]{d} \left(a b x+(-a-b) x^2+x^3\right)^{4/3}\right)}{2 \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(a*b*x + (-a - b)*x^2 + x^3)^(2/3))/(-2*b*d^(1/3)*x + 2*d^(1/3)*x^2 + (a*b*x + (-a - b)*x^2 + x^3)^(2/3))])/d^(1/3) + Log[b*Sqrt[d]*x - Sqrt[d]*x^2 + d^(1/6)*(a*b*x + (-a - b)*x^2 + x^3)^(2/3)]/d^(1/3) - Log[b^2*d*x^2 - 2*b*d*x^3 + d*x^4 + (-(b*d^(2/3)*x) + d^(2/3)*x^2)*(a*b*x + (-a - b)*x^2 + x^3)^(2/3) + d^(1/3)*(a*b*x + (-a - b)*x^2 + x^3)^(4/3)]/(2*d^(1/3))","A",1
2696,1,245,245,0.5677669,"\int \frac{b^6+a^6 x^6}{\sqrt{b^2 x+a^2 x^3} \left(-b^6+a^6 x^6\right)} \, dx","IntegrateAlgebraic[(b^6 + a^6*x^6)/(Sqrt[b^2*x + a^2*x^3]*(-b^6 + a^6*x^6)),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{3 \sqrt{a} \sqrt{b}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{3 \sqrt{2} \sqrt{a} \sqrt{b}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{3 \sqrt{a} \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{3 \sqrt{2} \sqrt{a} \sqrt{b}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{3 \sqrt{a} \sqrt{b}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{3 \sqrt{2} \sqrt{a} \sqrt{b}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{3 \sqrt{a} \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right)}{3 \sqrt{2} \sqrt{a} \sqrt{b}}",1,"(-2*ArcTan[(Sqrt[a]*Sqrt[b]*Sqrt[b^2*x + a^2*x^3])/(b^2 + a^2*x^2)])/(3*Sqrt[a]*Sqrt[b]) - ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[b]*Sqrt[b^2*x + a^2*x^3])/(b^2 + a^2*x^2)]/(3*Sqrt[2]*Sqrt[a]*Sqrt[b]) - (2*ArcTanh[(Sqrt[a]*Sqrt[b]*Sqrt[b^2*x + a^2*x^3])/(b^2 + a^2*x^2)])/(3*Sqrt[a]*Sqrt[b]) - ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[b]*Sqrt[b^2*x + a^2*x^3])/(b^2 + a^2*x^2)]/(3*Sqrt[2]*Sqrt[a]*Sqrt[b])","A",1
2697,1,245,245,0.6811934,"\int \frac{\left(2+5 x^7\right) \sqrt[3]{-x-x^3+x^8}}{\left(-1+x^7\right) \left(-1+x^2+x^7\right)} \, dx","IntegrateAlgebraic[((2 + 5*x^7)*(-x - x^3 + x^8)^(1/3))/((-1 + x^7)*(-1 + x^2 + x^7)),x]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-x-x^3+x^8}}\right)-\sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2^{2/3} \sqrt[3]{-x-x^3+x^8}}\right)-\log \left(x+\sqrt[3]{-x-x^3+x^8}\right)+\sqrt[3]{2} \log \left(2 x+2^{2/3} \sqrt[3]{-x-x^3+x^8}\right)+\frac{1}{2} \log \left(x^2-x \sqrt[3]{-x-x^3+x^8}+\left(-x-x^3+x^8\right)^{2/3}\right)-\frac{\log \left(-2 x^2+2^{2/3} x \sqrt[3]{-x-x^3+x^8}-\sqrt[3]{2} \left(-x-x^3+x^8\right)^{2/3}\right)}{2^{2/3}}","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{-x-x^3+x^8}}\right)-\sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2^{2/3} \sqrt[3]{-x-x^3+x^8}}\right)-\log \left(x+\sqrt[3]{-x-x^3+x^8}\right)+\sqrt[3]{2} \log \left(2 x+2^{2/3} \sqrt[3]{-x-x^3+x^8}\right)+\frac{1}{2} \log \left(x^2-x \sqrt[3]{-x-x^3+x^8}+\left(-x-x^3+x^8\right)^{2/3}\right)-\frac{\log \left(-2 x^2+2^{2/3} x \sqrt[3]{-x-x^3+x^8}-\sqrt[3]{2} \left(-x-x^3+x^8\right)^{2/3}\right)}{2^{2/3}}",1,"Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(-x - x^3 + x^8)^(1/3))] - 2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2^(2/3)*(-x - x^3 + x^8)^(1/3))] - Log[x + (-x - x^3 + x^8)^(1/3)] + 2^(1/3)*Log[2*x + 2^(2/3)*(-x - x^3 + x^8)^(1/3)] + Log[x^2 - x*(-x - x^3 + x^8)^(1/3) + (-x - x^3 + x^8)^(2/3)]/2 - Log[-2*x^2 + 2^(2/3)*x*(-x - x^3 + x^8)^(1/3) - 2^(1/3)*(-x - x^3 + x^8)^(2/3)]/2^(2/3)","A",1
2698,1,225,245,4.0020884,"\int \frac{x^6 \left(-4+x^3\right)}{\left(-1+x^3\right)^{3/4} \left(1-2 x^3+x^6+x^8\right)} \, dx","IntegrateAlgebraic[(x^6*(-4 + x^3))/((-1 + x^3)^(3/4)*(1 - 2*x^3 + x^6 + x^8)),x]","\frac{1}{2} \sqrt{2-\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{-1+x^3}}{-x^2+\sqrt{-1+x^3}}\right)+\frac{1}{2} \sqrt{2+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{-1+x^3}}{-x^2+\sqrt{-1+x^3}}\right)-\frac{1}{2} \sqrt{2-\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{-1+x^3}}{x^2+\sqrt{-1+x^3}}\right)-\frac{1}{2} \sqrt{2+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{-1+x^3}}{x^2+\sqrt{-1+x^3}}\right)","\frac{1}{2} \sqrt{2+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{-1+x^3}}{-x^2+\sqrt{-1+x^3}}\right)-\frac{1}{2} \sqrt{2-\sqrt{2}} \tan ^{-1}\left(\frac{\left(\sqrt{\frac{2}{2-\sqrt{2}}}-\frac{2}{\sqrt{2-\sqrt{2}}}\right) x \sqrt[4]{-1+x^3}}{-x^2+\sqrt{-1+x^3}}\right)-\frac{1}{2} \sqrt{2-\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{-1+x^3}}{x^2+\sqrt{-1+x^3}}\right)-\frac{1}{2} \sqrt{2+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{-1+x^3}}{x^2+\sqrt{-1+x^3}}\right)",1,"(Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]]*x*(-1 + x^3)^(1/4))/(-x^2 + Sqrt[-1 + x^3])])/2 + (Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]]*x*(-1 + x^3)^(1/4))/(-x^2 + Sqrt[-1 + x^3])])/2 - (Sqrt[2 - Sqrt[2]]*ArcTanh[(Sqrt[2 - Sqrt[2]]*x*(-1 + x^3)^(1/4))/(x^2 + Sqrt[-1 + x^3])])/2 - (Sqrt[2 + Sqrt[2]]*ArcTanh[(Sqrt[2 + Sqrt[2]]*x*(-1 + x^3)^(1/4))/(x^2 + Sqrt[-1 + x^3])])/2","A",1
2699,1,225,245,13.3156728,"\int \frac{x^6 \left(4+x^5\right)}{\left(-1+x^5\right)^{3/4} \left(1-2 x^5+x^8+x^{10}\right)} \, dx","IntegrateAlgebraic[(x^6*(4 + x^5))/((-1 + x^5)^(3/4)*(1 - 2*x^5 + x^8 + x^10)),x]","-\frac{1}{2} \sqrt{2-\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{-1+x^5}}{-x^2+\sqrt{-1+x^5}}\right)-\frac{1}{2} \sqrt{2+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{-1+x^5}}{-x^2+\sqrt{-1+x^5}}\right)+\frac{1}{2} \sqrt{2-\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{-1+x^5}}{x^2+\sqrt{-1+x^5}}\right)+\frac{1}{2} \sqrt{2+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{-1+x^5}}{x^2+\sqrt{-1+x^5}}\right)","-\frac{1}{2} \sqrt{2+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{-1+x^5}}{-x^2+\sqrt{-1+x^5}}\right)+\frac{1}{2} \sqrt{2-\sqrt{2}} \tan ^{-1}\left(\frac{\left(\sqrt{\frac{2}{2-\sqrt{2}}}-\frac{2}{\sqrt{2-\sqrt{2}}}\right) x \sqrt[4]{-1+x^5}}{-x^2+\sqrt{-1+x^5}}\right)+\frac{1}{2} \sqrt{2-\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{-1+x^5}}{x^2+\sqrt{-1+x^5}}\right)+\frac{1}{2} \sqrt{2+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{-1+x^5}}{x^2+\sqrt{-1+x^5}}\right)",1,"-1/2*(Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]]*x*(-1 + x^5)^(1/4))/(-x^2 + Sqrt[-1 + x^5])]) - (Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]]*x*(-1 + x^5)^(1/4))/(-x^2 + Sqrt[-1 + x^5])])/2 + (Sqrt[2 - Sqrt[2]]*ArcTanh[(Sqrt[2 - Sqrt[2]]*x*(-1 + x^5)^(1/4))/(x^2 + Sqrt[-1 + x^5])])/2 + (Sqrt[2 + Sqrt[2]]*ArcTanh[(Sqrt[2 + Sqrt[2]]*x*(-1 + x^5)^(1/4))/(x^2 + Sqrt[-1 + x^5])])/2","A",1
2700,1,223,245,0.5991942,"\int \frac{\sqrt{b+a x} \left(-g+f x^2\right)}{\left(e+d x^2\right) \sqrt{c+\sqrt{b+a x}}} \, dx","IntegrateAlgebraic[(Sqrt[b + a*x]*(-g + f*x^2))/((e + d*x^2)*Sqrt[c + Sqrt[b + a*x]]),x]","\frac{4 \sqrt{c+\sqrt{b+a x}} \left(8 c^2 f-4 c f \sqrt{b+a x}+3 f (b+a x)\right)}{15 a d}-\frac{a (e f+d g) \text{RootSum}\left[b^2 d-2 b c^2 d+c^4 d+a^2 e+4 b c d \text{$\#$1}^2-4 c^3 d \text{$\#$1}^2-2 b d \text{$\#$1}^4+6 c^2 d \text{$\#$1}^4-4 c d \text{$\#$1}^6+d \text{$\#$1}^8\&,\frac{-c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right)+\log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2}{-b \text{$\#$1}+c^2 \text{$\#$1}-2 c \text{$\#$1}^3+\text{$\#$1}^5}\&\right]}{2 d^2}","-\frac{16 c f \sqrt{b+a x} \sqrt{c+\sqrt{b+a x}}}{15 a d}+\frac{4 f \left(3 b+8 c^2+3 a x\right) \sqrt{c+\sqrt{b+a x}}}{15 a d}-\frac{a (e f+d g) \text{RootSum}\left[b^2 d-2 b c^2 d+c^4 d+a^2 e+4 b c d \text{$\#$1}^2-4 c^3 d \text{$\#$1}^2-2 b d \text{$\#$1}^4+6 c^2 d \text{$\#$1}^4-4 c d \text{$\#$1}^6+d \text{$\#$1}^8\&,\frac{-c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right)+\log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2}{-b \text{$\#$1}+c^2 \text{$\#$1}-2 c \text{$\#$1}^3+\text{$\#$1}^5}\&\right]}{2 d^2}",1,"(4*Sqrt[c + Sqrt[b + a*x]]*(8*c^2*f - 4*c*f*Sqrt[b + a*x] + 3*f*(b + a*x)))/(15*a*d) - (a*(e*f + d*g)*RootSum[b^2*d - 2*b*c^2*d + c^4*d + a^2*e + 4*b*c*d*#1^2 - 4*c^3*d*#1^2 - 2*b*d*#1^4 + 6*c^2*d*#1^4 - 4*c*d*#1^6 + d*#1^8 & , (-(c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]) + Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^2)/(-(b*#1) + c^2*#1 - 2*c*#1^3 + #1^5) & ])/(2*d^2)","A",1
2701,1,329,245,1.313968,"\int \frac{\sqrt{1+x^4} \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2} \, dx","IntegrateAlgebraic[(Sqrt[1 + x^4]*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2),x]","\frac{1}{2} x \sqrt{x^2+\sqrt{1+x^4}}-\frac{\tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^4}}{\sqrt{2}}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)}{\sqrt{2}}+\sqrt{2 \left(-1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{-\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^4}}{\sqrt{2}}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)+\sqrt{2 \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{-\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)","\frac{1}{2} x \sqrt{x^2+\sqrt{1+x^4}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)}{\sqrt{2}}+\sqrt{2 \left(-1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)+\sqrt{2 \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2 \left(-1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)",1,"(x*Sqrt[x^2 + Sqrt[1 + x^4]])/2 - ArcTan[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])]/Sqrt[2] + Sqrt[2*(-1 + Sqrt[2])]*ArcTan[(-Sqrt[1/2 + 1/Sqrt[2]] + Sqrt[1/2 + 1/Sqrt[2]]*x^2 + Sqrt[1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] - Sqrt[2]*ArcTanh[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] + Sqrt[2*(1 + Sqrt[2])]*ArcTanh[(-Sqrt[-1/2 + 1/Sqrt[2]] + Sqrt[-1/2 + 1/Sqrt[2]]*x^2 + Sqrt[-1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])]","A",1
2702,1,245,245,0.1723201,"\int \frac{x}{x^2-\sqrt{1+x} \sqrt{1+\sqrt{1+x}}} \, dx","IntegrateAlgebraic[x/(x^2 - Sqrt[1 + x]*Sqrt[1 + Sqrt[1 + x]]),x]","\frac{2}{55} \left(15+\sqrt{5}\right) \log \left(1+\sqrt{5}-2 \sqrt{1+\sqrt{1+x}}\right)-\frac{2}{55} \left(-15+\sqrt{5}\right) \log \left(-1+\sqrt{5}+2 \sqrt{1+\sqrt{1+x}}\right)+\frac{4}{11} \text{RootSum}\left[-1+\text{$\#$1}-\text{$\#$1}^2-2 \text{$\#$1}^3+\text{$\#$1}^4+\text{$\#$1}^5\&,\frac{\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)-5 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}-12 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2+6 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^3+8 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^4}{1-2 \text{$\#$1}-6 \text{$\#$1}^2+4 \text{$\#$1}^3+5 \text{$\#$1}^4}\&\right]","\frac{2}{55} \left(15+\sqrt{5}\right) \log \left(1+\sqrt{5}-2 \sqrt{1+\sqrt{1+x}}\right)-\frac{2}{55} \left(-15+\sqrt{5}\right) \log \left(-1+\sqrt{5}+2 \sqrt{1+\sqrt{1+x}}\right)+\frac{4}{11} \text{RootSum}\left[-1+\text{$\#$1}-\text{$\#$1}^2-2 \text{$\#$1}^3+\text{$\#$1}^4+\text{$\#$1}^5\&,\frac{\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)-5 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}-12 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2+6 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^3+8 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^4}{1-2 \text{$\#$1}-6 \text{$\#$1}^2+4 \text{$\#$1}^3+5 \text{$\#$1}^4}\&\right]",1,"(2*(15 + Sqrt[5])*Log[1 + Sqrt[5] - 2*Sqrt[1 + Sqrt[1 + x]]])/55 - (2*(-15 + Sqrt[5])*Log[-1 + Sqrt[5] + 2*Sqrt[1 + Sqrt[1 + x]]])/55 + (4*RootSum[-1 + #1 - #1^2 - 2*#1^3 + #1^4 + #1^5 & , (Log[Sqrt[1 + Sqrt[1 + x]] - #1] - 5*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1 - 12*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^2 + 6*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^3 + 8*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^4)/(1 - 2*#1 - 6*#1^2 + 4*#1^3 + 5*#1^4) & ])/11","A",1
2703,1,245,245,0.0012636,"\int \frac{x}{x^2-\sqrt{1+x} \sqrt{1+\sqrt{1+x}}} \, dx","IntegrateAlgebraic[x/(x^2 - Sqrt[1 + x]*Sqrt[1 + Sqrt[1 + x]]),x]","\frac{2}{55} \left(15+\sqrt{5}\right) \log \left(1+\sqrt{5}-2 \sqrt{1+\sqrt{1+x}}\right)-\frac{2}{55} \left(-15+\sqrt{5}\right) \log \left(-1+\sqrt{5}+2 \sqrt{1+\sqrt{1+x}}\right)+\frac{4}{11} \text{RootSum}\left[-1+\text{$\#$1}-\text{$\#$1}^2-2 \text{$\#$1}^3+\text{$\#$1}^4+\text{$\#$1}^5\&,\frac{\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)-5 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}-12 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2+6 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^3+8 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^4}{1-2 \text{$\#$1}-6 \text{$\#$1}^2+4 \text{$\#$1}^3+5 \text{$\#$1}^4}\&\right]","\frac{2}{55} \left(15+\sqrt{5}\right) \log \left(1+\sqrt{5}-2 \sqrt{1+\sqrt{1+x}}\right)-\frac{2}{55} \left(-15+\sqrt{5}\right) \log \left(-1+\sqrt{5}+2 \sqrt{1+\sqrt{1+x}}\right)+\frac{4}{11} \text{RootSum}\left[-1+\text{$\#$1}-\text{$\#$1}^2-2 \text{$\#$1}^3+\text{$\#$1}^4+\text{$\#$1}^5\&,\frac{\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)-5 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}-12 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2+6 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^3+8 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^4}{1-2 \text{$\#$1}-6 \text{$\#$1}^2+4 \text{$\#$1}^3+5 \text{$\#$1}^4}\&\right]",1,"(2*(15 + Sqrt[5])*Log[1 + Sqrt[5] - 2*Sqrt[1 + Sqrt[1 + x]]])/55 - (2*(-15 + Sqrt[5])*Log[-1 + Sqrt[5] + 2*Sqrt[1 + Sqrt[1 + x]]])/55 + (4*RootSum[-1 + #1 - #1^2 - 2*#1^3 + #1^4 + #1^5 & , (Log[Sqrt[1 + Sqrt[1 + x]] - #1] - 5*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1 - 12*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^2 + 6*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^3 + 8*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^4)/(1 - 2*#1 - 6*#1^2 + 4*#1^3 + 5*#1^4) & ])/11","A",1
2704,1,328,246,0.4785183,"\int \frac{\sqrt{b+a x} \sqrt{1+\sqrt{b+a x}}}{x^2 \sqrt{1+\sqrt{1+\sqrt{b+a x}}}} \, dx","IntegrateAlgebraic[(Sqrt[b + a*x]*Sqrt[1 + Sqrt[b + a*x]])/(x^2*Sqrt[1 + Sqrt[1 + Sqrt[b + a*x]]]),x]","\frac{\sqrt{1+\sqrt{b+a x}} \sqrt{1+\sqrt{1+\sqrt{b+a x}}}}{x}-\frac{\left(a+a \sqrt{b+a x}\right) \sqrt{1+\sqrt{1+\sqrt{b+a x}}}}{a x}+a \text{RootSum}\left[b-4 \text{$\#$1}^4+4 \text{$\#$1}^6-\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{1+\sqrt{1+\sqrt{b+a x}}}-\text{$\#$1}\right)+\log \left(\sqrt{1+\sqrt{1+\sqrt{b+a x}}}-\text{$\#$1}\right) \text{$\#$1}^2}{-2 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]-\frac{1}{8} a \text{RootSum}\left[b-4 \text{$\#$1}^4+4 \text{$\#$1}^6-\text{$\#$1}^8\&,\frac{6 \log \left(\sqrt{1+\sqrt{1+\sqrt{b+a x}}}-\text{$\#$1}\right)-7 \log \left(\sqrt{1+\sqrt{1+\sqrt{b+a x}}}-\text{$\#$1}\right) \text{$\#$1}^2+3 \log \left(\sqrt{1+\sqrt{1+\sqrt{b+a x}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","-\frac{\sqrt{1+\sqrt{1+\sqrt{b+a x}}}}{x}-\frac{\sqrt{b+a x} \sqrt{1+\sqrt{1+\sqrt{b+a x}}}}{x}+\frac{\sqrt{1+\sqrt{b+a x}} \sqrt{1+\sqrt{1+\sqrt{b+a x}}}}{x}+\frac{1}{8} a \text{RootSum}\left[b-4 \text{$\#$1}^4+4 \text{$\#$1}^6-\text{$\#$1}^8\&,\frac{2 \log \left(\sqrt{1+\sqrt{1+\sqrt{b+a x}}}-\text{$\#$1}\right)-9 \log \left(\sqrt{1+\sqrt{1+\sqrt{b+a x}}}-\text{$\#$1}\right) \text{$\#$1}^2+5 \log \left(\sqrt{1+\sqrt{1+\sqrt{b+a x}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]",1,"(Sqrt[1 + Sqrt[b + a*x]]*Sqrt[1 + Sqrt[1 + Sqrt[b + a*x]]])/x - ((a + a*Sqrt[b + a*x])*Sqrt[1 + Sqrt[1 + Sqrt[b + a*x]]])/(a*x) + a*RootSum[b - 4*#1^4 + 4*#1^6 - #1^8 & , (-Log[Sqrt[1 + Sqrt[1 + Sqrt[b + a*x]]] - #1] + Log[Sqrt[1 + Sqrt[1 + Sqrt[b + a*x]]] - #1]*#1^2)/(-2*#1^3 + #1^5) & ] - (a*RootSum[b - 4*#1^4 + 4*#1^6 - #1^8 & , (6*Log[Sqrt[1 + Sqrt[1 + Sqrt[b + a*x]]] - #1] - 7*Log[Sqrt[1 + Sqrt[1 + Sqrt[b + a*x]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[1 + Sqrt[b + a*x]]] - #1]*#1^4)/(2*#1^3 - 3*#1^5 + #1^7) & ])/8","A",1
2705,1,328,246,0.0024865,"\int \frac{\sqrt{b+a x} \sqrt{1+\sqrt{b+a x}}}{x^2 \sqrt{1+\sqrt{1+\sqrt{b+a x}}}} \, dx","IntegrateAlgebraic[(Sqrt[b + a*x]*Sqrt[1 + Sqrt[b + a*x]])/(x^2*Sqrt[1 + Sqrt[1 + Sqrt[b + a*x]]]),x]","\frac{\sqrt{1+\sqrt{b+a x}} \sqrt{1+\sqrt{1+\sqrt{b+a x}}}}{x}-\frac{\left(a+a \sqrt{b+a x}\right) \sqrt{1+\sqrt{1+\sqrt{b+a x}}}}{a x}+a \text{RootSum}\left[b-4 \text{$\#$1}^4+4 \text{$\#$1}^6-\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{1+\sqrt{1+\sqrt{b+a x}}}-\text{$\#$1}\right)+\log \left(\sqrt{1+\sqrt{1+\sqrt{b+a x}}}-\text{$\#$1}\right) \text{$\#$1}^2}{-2 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]-\frac{1}{8} a \text{RootSum}\left[b-4 \text{$\#$1}^4+4 \text{$\#$1}^6-\text{$\#$1}^8\&,\frac{6 \log \left(\sqrt{1+\sqrt{1+\sqrt{b+a x}}}-\text{$\#$1}\right)-7 \log \left(\sqrt{1+\sqrt{1+\sqrt{b+a x}}}-\text{$\#$1}\right) \text{$\#$1}^2+3 \log \left(\sqrt{1+\sqrt{1+\sqrt{b+a x}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","-\frac{\sqrt{1+\sqrt{1+\sqrt{b+a x}}}}{x}-\frac{\sqrt{b+a x} \sqrt{1+\sqrt{1+\sqrt{b+a x}}}}{x}+\frac{\sqrt{1+\sqrt{b+a x}} \sqrt{1+\sqrt{1+\sqrt{b+a x}}}}{x}+\frac{1}{8} a \text{RootSum}\left[b-4 \text{$\#$1}^4+4 \text{$\#$1}^6-\text{$\#$1}^8\&,\frac{2 \log \left(\sqrt{1+\sqrt{1+\sqrt{b+a x}}}-\text{$\#$1}\right)-9 \log \left(\sqrt{1+\sqrt{1+\sqrt{b+a x}}}-\text{$\#$1}\right) \text{$\#$1}^2+5 \log \left(\sqrt{1+\sqrt{1+\sqrt{b+a x}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]",1,"(Sqrt[1 + Sqrt[b + a*x]]*Sqrt[1 + Sqrt[1 + Sqrt[b + a*x]]])/x - ((a + a*Sqrt[b + a*x])*Sqrt[1 + Sqrt[1 + Sqrt[b + a*x]]])/(a*x) + a*RootSum[b - 4*#1^4 + 4*#1^6 - #1^8 & , (-Log[Sqrt[1 + Sqrt[1 + Sqrt[b + a*x]]] - #1] + Log[Sqrt[1 + Sqrt[1 + Sqrt[b + a*x]]] - #1]*#1^2)/(-2*#1^3 + #1^5) & ] - (a*RootSum[b - 4*#1^4 + 4*#1^6 - #1^8 & , (6*Log[Sqrt[1 + Sqrt[1 + Sqrt[b + a*x]]] - #1] - 7*Log[Sqrt[1 + Sqrt[1 + Sqrt[b + a*x]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[1 + Sqrt[b + a*x]]] - #1]*#1^4)/(2*#1^3 - 3*#1^5 + #1^7) & ])/8","A",1
2706,1,248,247,0.66241,"\int \frac{x^4 \sqrt[4]{b x^2+a x^4}}{b+a x^2+x^4} \, dx","IntegrateAlgebraic[(x^4*(b*x^2 + a*x^4)^(1/4))/(b + a*x^2 + x^4),x]","\frac{1}{2} x \sqrt[4]{b x^2+a x^4}+\frac{\left(4 a^2-b\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{4 a^{3/4}}+\frac{\left(-4 a^2+b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{4 a^{3/4}}+\frac{1}{2} \text{RootSum}\left[b-a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a b \log (x)+a b \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)+a^2 \log (x) \text{$\#$1}^4-b \log (x) \text{$\#$1}^4-a^2 \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+b \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]","\frac{1}{2} x \sqrt[4]{b x^2+a x^4}+\frac{\left(4 a^2-b\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{4 a^{3/4}}+\frac{\left(-4 a^2+b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{4 a^{3/4}}+\frac{1}{2} \text{RootSum}\left[b-a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a b \log (x)-a b \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)-a^2 \log (x) \text{$\#$1}^4+b \log (x) \text{$\#$1}^4+a^2 \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-b \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}^3-2 \text{$\#$1}^7}\&\right]",1,"(x*(b*x^2 + a*x^4)^(1/4))/2 + ((4*a^2 - b)*ArcTan[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/(4*a^(3/4)) + ((-4*a^2 + b)*ArcTanh[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/(4*a^(3/4)) + RootSum[b - a*#1^4 + #1^8 & , (-(a*b*Log[x]) + a*b*Log[(b*x^2 + a*x^4)^(1/4) - x*#1] + a^2*Log[x]*#1^4 - b*Log[x]*#1^4 - a^2*Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1^4 + b*Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3) + 2*#1^7) & ]/2","A",1
2707,1,248,247,0.0015589,"\int \frac{x^4 \sqrt[4]{b x^2+a x^4}}{b+a x^2+x^4} \, dx","IntegrateAlgebraic[(x^4*(b*x^2 + a*x^4)^(1/4))/(b + a*x^2 + x^4),x]","\frac{1}{2} x \sqrt[4]{b x^2+a x^4}+\frac{\left(4 a^2-b\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{4 a^{3/4}}+\frac{\left(-4 a^2+b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{4 a^{3/4}}+\frac{1}{2} \text{RootSum}\left[b-a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a b \log (x)+a b \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)+a^2 \log (x) \text{$\#$1}^4-b \log (x) \text{$\#$1}^4-a^2 \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+b \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]","\frac{1}{2} x \sqrt[4]{b x^2+a x^4}+\frac{\left(4 a^2-b\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{4 a^{3/4}}+\frac{\left(-4 a^2+b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{4 a^{3/4}}+\frac{1}{2} \text{RootSum}\left[b-a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a b \log (x)-a b \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)-a^2 \log (x) \text{$\#$1}^4+b \log (x) \text{$\#$1}^4+a^2 \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-b \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}^3-2 \text{$\#$1}^7}\&\right]",1,"(x*(b*x^2 + a*x^4)^(1/4))/2 + ((4*a^2 - b)*ArcTan[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/(4*a^(3/4)) + ((-4*a^2 + b)*ArcTanh[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/(4*a^(3/4)) + RootSum[b - a*#1^4 + #1^8 & , (-(a*b*Log[x]) + a*b*Log[(b*x^2 + a*x^4)^(1/4) - x*#1] + a^2*Log[x]*#1^4 - b*Log[x]*#1^4 - a^2*Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1^4 + b*Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3) + 2*#1^7) & ]/2","A",1
2708,1,247,247,23.1070698,"\int \frac{(b+a x) \left(-3 a q+4 b p x^3+a p x^4\right)}{\left(q+p x^4\right)^{2/3} \left(b^3 c+d q+3 a b^2 c x+3 a^2 b c x^2+a^3 c x^3+d p x^4\right)} \, dx","IntegrateAlgebraic[((b + a*x)*(-3*a*q + 4*b*p*x^3 + a*p*x^4))/((q + p*x^4)^(2/3)*(b^3*c + d*q + 3*a*b^2*c*x + 3*a^2*b*c*x^2 + a^3*c*x^3 + d*p*x^4)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} b \sqrt[3]{c}+\sqrt{3} a \sqrt[3]{c} x}{b \sqrt[3]{c}+a \sqrt[3]{c} x-2 \sqrt[3]{d} \sqrt[3]{q+p x^4}}\right)}{c^{2/3} \sqrt[3]{d}}+\frac{\log \left(a b \sqrt[3]{c}+a^2 \sqrt[3]{c} x+a \sqrt[3]{d} \sqrt[3]{q+p x^4}\right)}{c^{2/3} \sqrt[3]{d}}-\frac{\log \left(a^2 b^2 c^{2/3}+2 a^3 b c^{2/3} x+a^4 c^{2/3} x^2+\left(-a^2 b \sqrt[3]{c} \sqrt[3]{d}-a^3 \sqrt[3]{c} \sqrt[3]{d} x\right) \sqrt[3]{q+p x^4}+a^2 d^{2/3} \left(q+p x^4\right)^{2/3}\right)}{2 c^{2/3} \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} b \sqrt[3]{c}+\sqrt{3} a \sqrt[3]{c} x}{b \sqrt[3]{c}+a \sqrt[3]{c} x-2 \sqrt[3]{d} \sqrt[3]{q+p x^4}}\right)}{c^{2/3} \sqrt[3]{d}}+\frac{\log \left(a b \sqrt[3]{c}+a^2 \sqrt[3]{c} x+a \sqrt[3]{d} \sqrt[3]{q+p x^4}\right)}{c^{2/3} \sqrt[3]{d}}-\frac{\log \left(a^2 b^2 c^{2/3}+2 a^3 b c^{2/3} x+a^4 c^{2/3} x^2+\left(-a^2 b \sqrt[3]{c} \sqrt[3]{d}-a^3 \sqrt[3]{c} \sqrt[3]{d} x\right) \sqrt[3]{q+p x^4}+a^2 d^{2/3} \left(q+p x^4\right)^{2/3}\right)}{2 c^{2/3} \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*b*c^(1/3) + Sqrt[3]*a*c^(1/3)*x)/(b*c^(1/3) + a*c^(1/3)*x - 2*d^(1/3)*(q + p*x^4)^(1/3))])/(c^(2/3)*d^(1/3)) + Log[a*b*c^(1/3) + a^2*c^(1/3)*x + a*d^(1/3)*(q + p*x^4)^(1/3)]/(c^(2/3)*d^(1/3)) - Log[a^2*b^2*c^(2/3) + 2*a^3*b*c^(2/3)*x + a^4*c^(2/3)*x^2 + (-(a^2*b*c^(1/3)*d^(1/3)) - a^3*c^(1/3)*d^(1/3)*x)*(q + p*x^4)^(1/3) + a^2*d^(2/3)*(q + p*x^4)^(2/3)]/(2*c^(2/3)*d^(1/3))","A",1
2709,1,247,247,0.8229197,"\int \frac{\sqrt{b^2+a^2 x^3} \left(2 b^2+c x^3+a^2 x^6\right)}{x^7 \left(b^2+a^2 x^6\right)} \, dx","IntegrateAlgebraic[(Sqrt[b^2 + a^2*x^3]*(2*b^2 + c*x^3 + a^2*x^6))/(x^7*(b^2 + a^2*x^6)),x]","\frac{\sqrt{b^2+a^2 x^3} \left(-2 b^2-a^2 x^3-2 c x^3\right)}{6 b^2 x^6}+\frac{\sqrt{a-i b} \left((-1)^{3/4} a^2 b-\sqrt[4]{-1} a c\right) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{b^2+a^2 x^3}}{\sqrt{a-i b} \sqrt{b}}\right)}{3 b^{5/2}}-\frac{\sqrt{a+i b} \left(-\sqrt[4]{-1} a^2 b+(-1)^{3/4} a c\right) \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{b^2+a^2 x^3}}{\sqrt{a+i b} \sqrt{b}}\right)}{3 b^{5/2}}+\frac{\left(a^4+4 a^2 b^2-2 a^2 c\right) \tanh ^{-1}\left(\frac{\sqrt{b^2+a^2 x^3}}{b}\right)}{6 b^3}","\frac{\sqrt{b^2+a^2 x^3} \left(-2 b^2-a^2 x^3-2 c x^3\right)}{6 b^2 x^6}+\frac{\sqrt{a-i b} \left((-1)^{3/4} a^2 b-\sqrt[4]{-1} a c\right) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{b^2+a^2 x^3}}{\sqrt{a-i b} \sqrt{b}}\right)}{3 b^{5/2}}-\frac{\sqrt{a+i b} \left(-\sqrt[4]{-1} a^2 b+(-1)^{3/4} a c\right) \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{b^2+a^2 x^3}}{\sqrt{a+i b} \sqrt{b}}\right)}{3 b^{5/2}}+\frac{\left(a^4+4 a^2 b^2-2 a^2 c\right) \tanh ^{-1}\left(\frac{\sqrt{b^2+a^2 x^3}}{b}\right)}{6 b^3}",1,"(Sqrt[b^2 + a^2*x^3]*(-2*b^2 - a^2*x^3 - 2*c*x^3))/(6*b^2*x^6) + (Sqrt[a - I*b]*((-1)^(3/4)*a^2*b - (-1)^(1/4)*a*c)*ArcTan[((-1)^(1/4)*Sqrt[b^2 + a^2*x^3])/(Sqrt[a - I*b]*Sqrt[b])])/(3*b^(5/2)) - (Sqrt[a + I*b]*(-((-1)^(1/4)*a^2*b) + (-1)^(3/4)*a*c)*ArcTan[((-1)^(3/4)*Sqrt[b^2 + a^2*x^3])/(Sqrt[a + I*b]*Sqrt[b])])/(3*b^(5/2)) + ((a^4 + 4*a^2*b^2 - 2*a^2*c)*ArcTanh[Sqrt[b^2 + a^2*x^3]/b])/(6*b^3)","A",1
2710,1,37,248,0.9836897,"\int \frac{\left(-2+x^4\right) \sqrt{2+x^4}}{\left(2+x^2+x^4\right) \left(2+2 x^2+x^4\right)} \, dx","IntegrateAlgebraic[((-2 + x^4)*Sqrt[2 + x^4])/((2 + x^2 + x^4)*(2 + 2*x^2 + x^4)),x]","\tan ^{-1}\left(\frac{x}{\sqrt{2+x^4}}\right)-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{2+x^4}}\right)","2 \sqrt[4]{2} \text{RootSum}\left[1-4 \text{$\#$1}^2+12 \sqrt{2} \text{$\#$1}^2+70 \text{$\#$1}^4-24 \sqrt{2} \text{$\#$1}^4-4 \text{$\#$1}^6+12 \sqrt{2} \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(-\sqrt{2}+2^{3/4} x-x^2\right) \text{$\#$1}+\log \left(\sqrt{2+x^4}+\sqrt{2} \text{$\#$1}-2^{3/4} x \text{$\#$1}+x^2 \text{$\#$1}\right) \text{$\#$1}-\log \left(-\sqrt{2}+2^{3/4} x-x^2\right) \text{$\#$1}^3+\log \left(\sqrt{2+x^4}+\sqrt{2} \text{$\#$1}-2^{3/4} x \text{$\#$1}+x^2 \text{$\#$1}\right) \text{$\#$1}^3}{-1+3 \sqrt{2}+35 \text{$\#$1}^2-12 \sqrt{2} \text{$\#$1}^2-3 \text{$\#$1}^4+9 \sqrt{2} \text{$\#$1}^4+\text{$\#$1}^6}\&\right]",1,"ArcTan[x/Sqrt[2 + x^4]] - Sqrt[2]*ArcTan[(Sqrt[2]*x)/Sqrt[2 + x^4]]","A",1
2711,1,253,248,1.7797947,"\int \frac{x^2 \left(-2+x^8\right) \sqrt[4]{2-2 x^4+x^8}}{\left(2+x^8\right) \left(4-x^4+2 x^8\right)} \, dx","IntegrateAlgebraic[(x^2*(-2 + x^8)*(2 - 2*x^4 + x^8)^(1/4))/((2 + x^8)*(4 - x^4 + 2*x^8)),x]","\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{2-2 x^4+x^8}}{\sqrt{2} x^2-\sqrt{2-2 x^4+x^8}}\right)}{2 \sqrt[4]{2}}-\frac{\sqrt[4]{3} \tan ^{-1}\left(\frac{-\frac{\sqrt[4]{3} x^2}{2^{3/4}}+\frac{\sqrt{2-2 x^4+x^8}}{\sqrt[4]{6}}}{x \sqrt[4]{2-2 x^4+x^8}}\right)}{2\ 2^{3/4}}+\frac{\tanh ^{-1}\left(\frac{2 \sqrt[4]{2} x \sqrt[4]{2-2 x^4+x^8}}{2 x^2+\sqrt{2} \sqrt{2-2 x^4+x^8}}\right)}{2 \sqrt[4]{2}}-\frac{\sqrt[4]{3} \tanh ^{-1}\left(\frac{6^{3/4} x \sqrt[4]{2-2 x^4+x^8}}{3 x^2+\sqrt{6} \sqrt{2-2 x^4+x^8}}\right)}{2\ 2^{3/4}}","\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{2-2 x^4+x^8}}{\sqrt{2} x^2-\sqrt{2-2 x^4+x^8}}\right)}{2 \sqrt[4]{2}}+\frac{\sqrt[4]{3} \tan ^{-1}\left(\frac{6^{3/4} x \sqrt[4]{2-2 x^4+x^8}}{-3 x^2+\sqrt{6} \sqrt{2-2 x^4+x^8}}\right)}{2\ 2^{3/4}}+\frac{\tanh ^{-1}\left(\frac{2 \sqrt[4]{2} x \sqrt[4]{2-2 x^4+x^8}}{2 x^2+\sqrt{2} \sqrt{2-2 x^4+x^8}}\right)}{2 \sqrt[4]{2}}-\frac{\sqrt[4]{3} \tanh ^{-1}\left(\frac{6^{3/4} x \sqrt[4]{2-2 x^4+x^8}}{3 x^2+\sqrt{6} \sqrt{2-2 x^4+x^8}}\right)}{2\ 2^{3/4}}",1,"ArcTan[(2^(3/4)*x*(2 - 2*x^4 + x^8)^(1/4))/(Sqrt[2]*x^2 - Sqrt[2 - 2*x^4 + x^8])]/(2*2^(1/4)) - (3^(1/4)*ArcTan[(-((3^(1/4)*x^2)/2^(3/4)) + Sqrt[2 - 2*x^4 + x^8]/6^(1/4))/(x*(2 - 2*x^4 + x^8)^(1/4))])/(2*2^(3/4)) + ArcTanh[(2*2^(1/4)*x*(2 - 2*x^4 + x^8)^(1/4))/(2*x^2 + Sqrt[2]*Sqrt[2 - 2*x^4 + x^8])]/(2*2^(1/4)) - (3^(1/4)*ArcTanh[(6^(3/4)*x*(2 - 2*x^4 + x^8)^(1/4))/(3*x^2 + Sqrt[6]*Sqrt[2 - 2*x^4 + x^8])])/(2*2^(3/4))","A",1
2712,1,242,248,0.3469559,"\int \frac{-1+x^2}{\left(1+x^2\right) \sqrt{x+\sqrt{1+x}}} \, dx","IntegrateAlgebraic[(-1 + x^2)/((1 + x^2)*Sqrt[x + Sqrt[1 + x]]),x]","2 \sqrt{x+\sqrt{1+x}}+\log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}\right)+32 \text{RootSum}\left[625-1000 \text{$\#$1}+300 \text{$\#$1}^2-120 \text{$\#$1}^3+470 \text{$\#$1}^4-24 \text{$\#$1}^5+12 \text{$\#$1}^6-8 \text{$\#$1}^7+\text{$\#$1}^8\&,\frac{5 \log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^2-2 \log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^3+\log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^4}{-125+75 \text{$\#$1}-45 \text{$\#$1}^2+235 \text{$\#$1}^3-15 \text{$\#$1}^4+9 \text{$\#$1}^5-7 \text{$\#$1}^6+\text{$\#$1}^7}\&\right]","2 \sqrt{x+\sqrt{1+x}}+\log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}\right)+32 \text{RootSum}\left[625-1000 \text{$\#$1}+300 \text{$\#$1}^2-120 \text{$\#$1}^3+470 \text{$\#$1}^4-24 \text{$\#$1}^5+12 \text{$\#$1}^6-8 \text{$\#$1}^7+\text{$\#$1}^8\&,\frac{5 \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2-2 \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^3+\log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^4}{-125+75 \text{$\#$1}-45 \text{$\#$1}^2+235 \text{$\#$1}^3-15 \text{$\#$1}^4+9 \text{$\#$1}^5-7 \text{$\#$1}^6+\text{$\#$1}^7}\&\right]",1,"2*Sqrt[x + Sqrt[1 + x]] + Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]] + 32*RootSum[625 - 1000*#1 + 300*#1^2 - 120*#1^3 + 470*#1^4 - 24*#1^5 + 12*#1^6 - 8*#1^7 + #1^8 & , (5*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]] + #1]*#1^2 - 2*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]] + #1]*#1^3 + Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]] + #1]*#1^4)/(-125 + 75*#1 - 45*#1^2 + 235*#1^3 - 15*#1^4 + 9*#1^5 - 7*#1^6 + #1^7) & ]","A",1
2713,1,242,248,0.0019661,"\int \frac{-1+x^2}{\left(1+x^2\right) \sqrt{x+\sqrt{1+x}}} \, dx","IntegrateAlgebraic[(-1 + x^2)/((1 + x^2)*Sqrt[x + Sqrt[1 + x]]),x]","2 \sqrt{x+\sqrt{1+x}}+\log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}\right)+32 \text{RootSum}\left[625-1000 \text{$\#$1}+300 \text{$\#$1}^2-120 \text{$\#$1}^3+470 \text{$\#$1}^4-24 \text{$\#$1}^5+12 \text{$\#$1}^6-8 \text{$\#$1}^7+\text{$\#$1}^8\&,\frac{5 \log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^2-2 \log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^3+\log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^4}{-125+75 \text{$\#$1}-45 \text{$\#$1}^2+235 \text{$\#$1}^3-15 \text{$\#$1}^4+9 \text{$\#$1}^5-7 \text{$\#$1}^6+\text{$\#$1}^7}\&\right]","2 \sqrt{x+\sqrt{1+x}}+\log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}\right)+32 \text{RootSum}\left[625-1000 \text{$\#$1}+300 \text{$\#$1}^2-120 \text{$\#$1}^3+470 \text{$\#$1}^4-24 \text{$\#$1}^5+12 \text{$\#$1}^6-8 \text{$\#$1}^7+\text{$\#$1}^8\&,\frac{5 \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2-2 \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^3+\log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^4}{-125+75 \text{$\#$1}-45 \text{$\#$1}^2+235 \text{$\#$1}^3-15 \text{$\#$1}^4+9 \text{$\#$1}^5-7 \text{$\#$1}^6+\text{$\#$1}^7}\&\right]",1,"2*Sqrt[x + Sqrt[1 + x]] + Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]] + 32*RootSum[625 - 1000*#1 + 300*#1^2 - 120*#1^3 + 470*#1^4 - 24*#1^5 + 12*#1^6 - 8*#1^7 + #1^8 & , (5*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]] + #1]*#1^2 - 2*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]] + #1]*#1^3 + Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]] + #1]*#1^4)/(-125 + 75*#1 - 45*#1^2 + 235*#1^3 - 15*#1^4 + 9*#1^5 - 7*#1^6 + #1^7) & ]","A",1
2714,1,248,248,2.4484778,"\int \frac{1}{\sqrt{-b+a^2 x^2} \sqrt[3]{a x^2+x \sqrt{-b+a^2 x^2}}} \, dx","IntegrateAlgebraic[1/(Sqrt[-b + a^2*x^2]*(a*x^2 + x*Sqrt[-b + a^2*x^2])^(1/3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{2} \sqrt[3]{a} \sqrt[3]{a x^2+x \sqrt{-b+a^2 x^2}}}{\sqrt{3} \sqrt[3]{b}}\right)}{2^{2/3} a^{2/3} \sqrt[3]{b}}+\frac{\log \left(-1+\frac{\sqrt[3]{2} \sqrt[3]{a} \sqrt[3]{a x^2+x \sqrt{-b+a^2 x^2}}}{\sqrt[3]{b}}\right)}{2^{2/3} a^{2/3} \sqrt[3]{b}}-\frac{\log \left(1+\frac{\sqrt[3]{2} \sqrt[3]{a} \sqrt[3]{a x^2+x \sqrt{-b+a^2 x^2}}}{\sqrt[3]{b}}+\frac{2^{2/3} a^{2/3} \left(a x^2+x \sqrt{-b+a^2 x^2}\right)^{2/3}}{b^{2/3}}\right)}{2\ 2^{2/3} a^{2/3} \sqrt[3]{b}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{2} \sqrt[3]{a} \sqrt[3]{a x^2+x \sqrt{-b+a^2 x^2}}}{\sqrt{3} \sqrt[3]{b}}\right)}{2^{2/3} a^{2/3} \sqrt[3]{b}}+\frac{\log \left(-1+\frac{\sqrt[3]{2} \sqrt[3]{a} \sqrt[3]{a x^2+x \sqrt{-b+a^2 x^2}}}{\sqrt[3]{b}}\right)}{2^{2/3} a^{2/3} \sqrt[3]{b}}-\frac{\log \left(1+\frac{\sqrt[3]{2} \sqrt[3]{a} \sqrt[3]{a x^2+x \sqrt{-b+a^2 x^2}}}{\sqrt[3]{b}}+\frac{2^{2/3} a^{2/3} \left(a x^2+x \sqrt{-b+a^2 x^2}\right)^{2/3}}{b^{2/3}}\right)}{2\ 2^{2/3} a^{2/3} \sqrt[3]{b}}",1,"(Sqrt[3]*ArcTan[1/Sqrt[3] + (2*2^(1/3)*a^(1/3)*(a*x^2 + x*Sqrt[-b + a^2*x^2])^(1/3))/(Sqrt[3]*b^(1/3))])/(2^(2/3)*a^(2/3)*b^(1/3)) + Log[-1 + (2^(1/3)*a^(1/3)*(a*x^2 + x*Sqrt[-b + a^2*x^2])^(1/3))/b^(1/3)]/(2^(2/3)*a^(2/3)*b^(1/3)) - Log[1 + (2^(1/3)*a^(1/3)*(a*x^2 + x*Sqrt[-b + a^2*x^2])^(1/3))/b^(1/3) + (2^(2/3)*a^(2/3)*(a*x^2 + x*Sqrt[-b + a^2*x^2])^(2/3))/b^(2/3)]/(2*2^(2/3)*a^(2/3)*b^(1/3))","A",1
2715,1,232,248,0.5591159,"\int \frac{\sqrt{-b+a x}}{1+\sqrt{a x+\sqrt{-b+a x}}} \, dx","IntegrateAlgebraic[Sqrt[-b + a*x]/(1 + Sqrt[a*x + Sqrt[-b + a*x]]),x]","-\frac{2 \sqrt{-b+a x}}{a}+\frac{\sqrt{a x+\sqrt{-b+a x}} \left(-3+2 \sqrt{-b+a x}\right)}{2 a}-\frac{4 (-3+2 b) \tan ^{-1}\left(\frac{1-2 \sqrt{-b+a x}+2 \sqrt{a x+\sqrt{-b+a x}}}{\sqrt{-5+4 b}}\right)}{a \sqrt{-5+4 b}}+\frac{(-19+4 b) \log \left(a \left(-1-2 \sqrt{-b+a x}\right)+2 a \sqrt{a x+\sqrt{-b+a x}}\right)}{4 a}+\frac{2 \log \left(1-2 b-2 (-b+a x)+\sqrt{a x+\sqrt{-b+a x}} \left(-1+2 \sqrt{-b+a x}\right)\right)}{a}","-\frac{3 \sqrt{a x+\sqrt{-b+a x}}}{2 a}+\sqrt{-b+a x} \left(-\frac{2}{a}+\frac{\sqrt{a x+\sqrt{-b+a x}}}{a}\right)-\frac{4 (-3+2 b) \tan ^{-1}\left(\frac{1-2 \sqrt{-b+a x}+2 \sqrt{a x+\sqrt{-b+a x}}}{\sqrt{-5+4 b}}\right)}{a \sqrt{-5+4 b}}+\frac{(-19+4 b) \log \left(1+2 \sqrt{-b+a x}-2 \sqrt{a x+\sqrt{-b+a x}}\right)}{4 a}+\frac{2 \log \left(1-2 a x-\sqrt{a x+\sqrt{-b+a x}}+2 \sqrt{-b+a x} \sqrt{a x+\sqrt{-b+a x}}\right)}{a}",1,"(-2*Sqrt[-b + a*x])/a + (Sqrt[a*x + Sqrt[-b + a*x]]*(-3 + 2*Sqrt[-b + a*x]))/(2*a) - (4*(-3 + 2*b)*ArcTan[(1 - 2*Sqrt[-b + a*x] + 2*Sqrt[a*x + Sqrt[-b + a*x]])/Sqrt[-5 + 4*b]])/(a*Sqrt[-5 + 4*b]) + ((-19 + 4*b)*Log[a*(-1 - 2*Sqrt[-b + a*x]) + 2*a*Sqrt[a*x + Sqrt[-b + a*x]]])/(4*a) + (2*Log[1 - 2*b - 2*(-b + a*x) + Sqrt[a*x + Sqrt[-b + a*x]]*(-1 + 2*Sqrt[-b + a*x])])/a","A",1
2716,1,91,249,0.3624524,"\int x \sqrt{1+\sqrt{2}+\sqrt{2} x+x^2} \, dx","IntegrateAlgebraic[x*Sqrt[1 + Sqrt[2] + Sqrt[2]*x + x^2],x]","\frac{1}{12} \sqrt{1+\sqrt{2}+\sqrt{2} x+x^2} \left(1+4 \sqrt{2}+\sqrt{2} x+4 x^2\right)+\frac{1}{8} \left(4+\sqrt{2}\right) \log \left(\sqrt{2}+2 x-2 \sqrt{1+\sqrt{2}+\sqrt{2} x+x^2}\right)","\frac{234639860780381988229271893709089+599643393101900122105957184226864 x+1129650644023239063399012556878492 x^2+1387443210853276135686753647625176 x^3+1378491402159939739304774333547654 x^4+1027228435068267876872457367335328 x^5+628530823497779895944692654279120 x^6+287852425478772976908575001303368 x^7+105092569428910116088968496479929 x^8+24945598143999361560228474400848 x^9+4150400378441530192773231110852 x^{10}+\sqrt{2} \left(165915436694475540805635728313976+424011909556064190889798701120701 x+798783630760582994655573305295524 x^2+981070502905588448116241178725596 x^3+974740618274645593869948214329596 x^4+726360192264417334196307996315058 x^5+444438407480045173855448430747052 x^6+203542402037035701060244675374692 x^7+74311668495500397843180232944316 x^8+17639201608376503319286232221121 x^9+2934776252235219007085702870688 x^{10}\right)}{12 \sqrt{2} \left(1+\sqrt{2}+\sqrt{2} x+x^2\right)^{3/2} \left(4276282378397179351700892708442+5009976441455984103472318426114 x+5009976441455984103472318426114 x^2+2075200189220765096386615555426 x^3+733694063058804751771425717672 x^4\right)+12 \left(1+\sqrt{2}+\sqrt{2} x+x^2\right)^{3/2} \left(6047576536066366651665626203827+7085176630676749199858933981540 x+7085176630676749199858933981540 x^2+2934776252235219007085702870688 x^3+1037600094610382548193307777713 x^4\right)}+\left(\frac{1}{2}+\frac{1}{4 \sqrt{2}}\right) \log \left(\sqrt{2}+2 x-2 \sqrt{1+\sqrt{2}+\sqrt{2} x+x^2}\right)",1,"(Sqrt[1 + Sqrt[2] + Sqrt[2]*x + x^2]*(1 + 4*Sqrt[2] + Sqrt[2]*x + 4*x^2))/12 + ((4 + Sqrt[2])*Log[Sqrt[2] + 2*x - 2*Sqrt[1 + Sqrt[2] + Sqrt[2]*x + x^2]])/8","A",1
2717,1,247,249,0.6380472,"\int \frac{-a x+x^2}{\left(x^2 (-a+x)\right)^{2/3} \left(a^2-2 a x+(1-d) x^2\right)} \, dx","IntegrateAlgebraic[(-(a*x) + x^2)/((x^2*(-a + x))^(2/3)*(a^2 - 2*a*x + (1 - d)*x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\frac{x^2}{\sqrt{3}}+\frac{2 \left(-a x^2+x^3\right)^{2/3}}{\sqrt{3} \sqrt[3]{d}}}{x^2}\right)}{2 a \sqrt[3]{d}}+\frac{\log \left(-\sqrt[6]{d} x+\sqrt[3]{-a x^2+x^3}\right)}{2 a \sqrt[3]{d}}+\frac{\log \left(\sqrt[6]{d} x+\sqrt[3]{-a x^2+x^3}\right)}{2 a \sqrt[3]{d}}-\frac{\log \left(\sqrt[3]{d} x^2-\sqrt[6]{d} x \sqrt[3]{-a x^2+x^3}+\left(-a x^2+x^3\right)^{2/3}\right)}{4 a \sqrt[3]{d}}-\frac{\log \left(\sqrt[3]{d} x^2+\sqrt[6]{d} x \sqrt[3]{-a x^2+x^3}+\left(-a x^2+x^3\right)^{2/3}\right)}{4 a \sqrt[3]{d}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} x^2}{\sqrt[3]{d} x^2+2 \left(-a x^2+x^3\right)^{2/3}}\right)}{2 a \sqrt[3]{d}}+\frac{\log \left(-\sqrt[6]{d} x+\sqrt[3]{-a x^2+x^3}\right)}{2 a \sqrt[3]{d}}+\frac{\log \left(\sqrt[6]{d} x+\sqrt[3]{-a x^2+x^3}\right)}{2 a \sqrt[3]{d}}-\frac{\log \left(\sqrt[3]{d} x^2-\sqrt[6]{d} x \sqrt[3]{-a x^2+x^3}+\left(-a x^2+x^3\right)^{2/3}\right)}{4 a \sqrt[3]{d}}-\frac{\log \left(\sqrt[3]{d} x^2+\sqrt[6]{d} x \sqrt[3]{-a x^2+x^3}+\left(-a x^2+x^3\right)^{2/3}\right)}{4 a \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(x^2/Sqrt[3] + (2*(-(a*x^2) + x^3)^(2/3))/(Sqrt[3]*d^(1/3)))/x^2])/(2*a*d^(1/3)) + Log[-(d^(1/6)*x) + (-(a*x^2) + x^3)^(1/3)]/(2*a*d^(1/3)) + Log[d^(1/6)*x + (-(a*x^2) + x^3)^(1/3)]/(2*a*d^(1/3)) - Log[d^(1/3)*x^2 - d^(1/6)*x*(-(a*x^2) + x^3)^(1/3) + (-(a*x^2) + x^3)^(2/3)]/(4*a*d^(1/3)) - Log[d^(1/3)*x^2 + d^(1/6)*x*(-(a*x^2) + x^3)^(1/3) + (-(a*x^2) + x^3)^(2/3)]/(4*a*d^(1/3))","A",1
2718,1,247,249,0.5212115,"\int \frac{x}{\sqrt[3]{x^2 (-a+x)} \left(-a^2+2 a x+(-1+d) x^2\right)} \, dx","IntegrateAlgebraic[x/((x^2*(-a + x))^(1/3)*(-a^2 + 2*a*x + (-1 + d)*x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\frac{x^2}{\sqrt{3}}+\frac{2 \left(-a x^2+x^3\right)^{2/3}}{\sqrt{3} \sqrt[3]{d}}}{x^2}\right)}{2 a d^{2/3}}-\frac{\log \left(-\sqrt[6]{d} x+\sqrt[3]{-a x^2+x^3}\right)}{2 a d^{2/3}}-\frac{\log \left(\sqrt[6]{d} x+\sqrt[3]{-a x^2+x^3}\right)}{2 a d^{2/3}}+\frac{\log \left(\sqrt[3]{d} x^2-\sqrt[6]{d} x \sqrt[3]{-a x^2+x^3}+\left(-a x^2+x^3\right)^{2/3}\right)}{4 a d^{2/3}}+\frac{\log \left(\sqrt[3]{d} x^2+\sqrt[6]{d} x \sqrt[3]{-a x^2+x^3}+\left(-a x^2+x^3\right)^{2/3}\right)}{4 a d^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} x^2}{\sqrt[3]{d} x^2+2 \left(-a x^2+x^3\right)^{2/3}}\right)}{2 a d^{2/3}}-\frac{\log \left(-\sqrt[6]{d} x+\sqrt[3]{-a x^2+x^3}\right)}{2 a d^{2/3}}-\frac{\log \left(\sqrt[6]{d} x+\sqrt[3]{-a x^2+x^3}\right)}{2 a d^{2/3}}+\frac{\log \left(\sqrt[3]{d} x^2-\sqrt[6]{d} x \sqrt[3]{-a x^2+x^3}+\left(-a x^2+x^3\right)^{2/3}\right)}{4 a d^{2/3}}+\frac{\log \left(\sqrt[3]{d} x^2+\sqrt[6]{d} x \sqrt[3]{-a x^2+x^3}+\left(-a x^2+x^3\right)^{2/3}\right)}{4 a d^{2/3}}",1,"(Sqrt[3]*ArcTan[(x^2/Sqrt[3] + (2*(-(a*x^2) + x^3)^(2/3))/(Sqrt[3]*d^(1/3)))/x^2])/(2*a*d^(2/3)) - Log[-(d^(1/6)*x) + (-(a*x^2) + x^3)^(1/3)]/(2*a*d^(2/3)) - Log[d^(1/6)*x + (-(a*x^2) + x^3)^(1/3)]/(2*a*d^(2/3)) + Log[d^(1/3)*x^2 - d^(1/6)*x*(-(a*x^2) + x^3)^(1/3) + (-(a*x^2) + x^3)^(2/3)]/(4*a*d^(2/3)) + Log[d^(1/3)*x^2 + d^(1/6)*x*(-(a*x^2) + x^3)^(1/3) + (-(a*x^2) + x^3)^(2/3)]/(4*a*d^(2/3))","A",1
2719,1,249,249,1.4184119,"\int \frac{-a b x^2+x^4}{\left(x^2 (-a+x) (-b+x)\right)^{2/3} \left(a^2 b^2-2 a b (a+b) x+\left(a^2+4 a b+b^2-d\right) x^2-2 (a+b) x^3+x^4\right)} \, dx","IntegrateAlgebraic[(-(a*b*x^2) + x^4)/((x^2*(-a + x)*(-b + x))^(2/3)*(a^2*b^2 - 2*a*b*(a + b)*x + (a^2 + 4*a*b + b^2 - d)*x^2 - 2*(a + b)*x^3 + x^4)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} x}{\sqrt[6]{d} x-2 \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{2 d^{5/6}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} x}{\sqrt[6]{d} x+2 \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{2 d^{5/6}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{d} x}{\sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{d^{5/6}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{d} x^2+\frac{\left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}}{\sqrt[6]{d}}}{x \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{2 d^{5/6}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} x}{\sqrt[6]{d} x-2 \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{2 d^{5/6}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} x}{\sqrt[6]{d} x+2 \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{2 d^{5/6}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{d} x}{\sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{d^{5/6}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{d} x^2+\frac{\left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}}{\sqrt[6]{d}}}{x \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{2 d^{5/6}}",1,"-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/6)*x)/(d^(1/6)*x - 2*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))])/d^(5/6) + (Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/6)*x)/(d^(1/6)*x + 2*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))])/(2*d^(5/6)) - ArcTanh[(d^(1/6)*x)/(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3)]/d^(5/6) - ArcTanh[(d^(1/6)*x^2 + (a*b*x^2 + (-a - b)*x^3 + x^4)^(2/3)/d^(1/6))/(x*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))]/(2*d^(5/6))","A",1
2720,1,237,249,0.2939322,"\int \frac{\sqrt[4]{\frac{1+4 x^2+a x^2+6 x^4+4 a x^4+4 x^6+6 a x^6+x^8+4 a x^8+a x^{10}}{x^2}}}{x} \, dx","IntegrateAlgebraic[((1 + 4*x^2 + a*x^2 + 6*x^4 + 4*a*x^4 + 4*x^6 + 6*a*x^6 + x^8 + 4*a*x^8 + a*x^10)/x^2)^(1/4)/x,x]","\frac{\left(-4+x^2\right) \sqrt[4]{\frac{1+4 x^2+a x^2+6 x^4+4 a x^4+4 x^6+6 a x^6+x^8+4 a x^8+a x^{10}}{x^2}}}{2 \left(1+x^2\right)}+\frac{(-1-4 a) \tan ^{-1}\left(\frac{\sqrt[4]{a} \left(1+x^2\right)}{\sqrt[4]{\frac{1+4 x^2+a x^2+6 x^4+4 a x^4+4 x^6+6 a x^6+x^8+4 a x^8+a x^{10}}{x^2}}}\right)}{4 a^{3/4}}+\frac{(1+4 a) \tanh ^{-1}\left(\frac{\sqrt[4]{a} \left(1+x^2\right)}{\sqrt[4]{\frac{1+4 x^2+a x^2+6 x^4+4 a x^4+4 x^6+6 a x^6+x^8+4 a x^8+a x^{10}}{x^2}}}\right)}{4 a^{3/4}}","\frac{\sqrt[4]{\frac{1+4 x^2+a x^2+6 x^4+4 a x^4+4 x^6+6 a x^6+x^8+4 a x^8+a x^{10}}{x^2}} \left(-2 \sqrt[4]{1+a x^2}+\frac{1}{2} x^2 \sqrt[4]{1+a x^2}+\frac{\sqrt{x} \tan ^{-1}\left(\frac{\sqrt[4]{1+a x^2}}{\sqrt[4]{a} \sqrt{x}}\right)}{4 a^{3/4}}+\sqrt[4]{a} \sqrt{x} \tan ^{-1}\left(\frac{\sqrt[4]{1+a x^2}}{\sqrt[4]{a} \sqrt{x}}\right)+\frac{\sqrt{x} \tanh ^{-1}\left(\frac{\sqrt[4]{1+a x^2}}{\sqrt[4]{a} \sqrt{x}}\right)}{4 a^{3/4}}+\sqrt[4]{a} \sqrt{x} \tanh ^{-1}\left(\frac{\sqrt[4]{1+a x^2}}{\sqrt[4]{a} \sqrt{x}}\right)\right)}{\left(1+x^2\right) \sqrt[4]{1+a x^2}}",1,"((-4 + x^2)*((1 + 4*x^2 + a*x^2 + 6*x^4 + 4*a*x^4 + 4*x^6 + 6*a*x^6 + x^8 + 4*a*x^8 + a*x^10)/x^2)^(1/4))/(2*(1 + x^2)) + ((-1 - 4*a)*ArcTan[(a^(1/4)*(1 + x^2))/((1 + 4*x^2 + a*x^2 + 6*x^4 + 4*a*x^4 + 4*x^6 + 6*a*x^6 + x^8 + 4*a*x^8 + a*x^10)/x^2)^(1/4)])/(4*a^(3/4)) + ((1 + 4*a)*ArcTanh[(a^(1/4)*(1 + x^2))/((1 + 4*x^2 + a*x^2 + 6*x^4 + 4*a*x^4 + 4*x^6 + 6*a*x^6 + x^8 + 4*a*x^8 + a*x^10)/x^2)^(1/4)])/(4*a^(3/4))","A",1
2721,1,250,250,5.4178145,"\int \frac{\sqrt[3]{2-8 x+8 x^2}}{3+x} \, dx","IntegrateAlgebraic[(2 - 8*x + 8*x^2)^(1/3)/(3 + x),x]","-\frac{\sqrt[3]{(1-2 x)^2} \left(7+7^{2/3} \sqrt[3]{-1+2 x}\right) \left(7-7^{2/3} \sqrt[3]{-1+2 x}+\sqrt[3]{7} (-1+2 x)^{2/3}\right)^2 \left(\frac{3 (-1+2 x)^{2/3}}{2^{2/3}}+\sqrt[3]{2} \sqrt{3} 7^{2/3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+2 x}}{\sqrt{3} \sqrt[3]{7}}\right)+\sqrt[3]{2} 7^{2/3} \log \left(7+7^{2/3} \sqrt[3]{-1+2 x}\right)-\left(\frac{7}{2}\right)^{2/3} \log \left(-7+7^{2/3} \sqrt[3]{-1+2 x}-\sqrt[3]{7} (-1+2 x)^{2/3}\right)\right)}{14 (3+x) \sqrt[3]{-1+2 x} \left(\sqrt[3]{7}-2 \sqrt[3]{7} x-7 \sqrt[3]{-1+2 x}+(-7+14 x)^{2/3}\right)}","-\frac{\sqrt[3]{(1-2 x)^2} \left(7+7^{2/3} \sqrt[3]{-1+2 x}\right) \left(7-7^{2/3} \sqrt[3]{-1+2 x}+\sqrt[3]{7} (-1+2 x)^{2/3}\right)^2 \left(\frac{3 (-1+2 x)^{2/3}}{2^{2/3}}+\sqrt[3]{2} \sqrt{3} 7^{2/3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1+2 x}}{\sqrt{3} \sqrt[3]{7}}\right)+\sqrt[3]{2} 7^{2/3} \log \left(7+7^{2/3} \sqrt[3]{-1+2 x}\right)-\left(\frac{7}{2}\right)^{2/3} \log \left(-7+7^{2/3} \sqrt[3]{-1+2 x}-\sqrt[3]{7} (-1+2 x)^{2/3}\right)\right)}{14 (3+x) \sqrt[3]{-1+2 x} \left(\sqrt[3]{7}-2 \sqrt[3]{7} x-7 \sqrt[3]{-1+2 x}+(-7+14 x)^{2/3}\right)}",1,"-1/14*(((1 - 2*x)^2)^(1/3)*(7 + 7^(2/3)*(-1 + 2*x)^(1/3))*(7 - 7^(2/3)*(-1 + 2*x)^(1/3) + 7^(1/3)*(-1 + 2*x)^(2/3))^2*((3*(-1 + 2*x)^(2/3))/2^(2/3) + 2^(1/3)*Sqrt[3]*7^(2/3)*ArcTan[1/Sqrt[3] - (2*(-1 + 2*x)^(1/3))/(Sqrt[3]*7^(1/3))] + 2^(1/3)*7^(2/3)*Log[7 + 7^(2/3)*(-1 + 2*x)^(1/3)] - (7/2)^(2/3)*Log[-7 + 7^(2/3)*(-1 + 2*x)^(1/3) - 7^(1/3)*(-1 + 2*x)^(2/3)]))/((3 + x)*(-1 + 2*x)^(1/3)*(7^(1/3) - 2*7^(1/3)*x - 7*(-1 + 2*x)^(1/3) + (-7 + 14*x)^(2/3)))","A",1
2722,1,250,250,17.8044546,"\int \frac{(b+a x) \left(-a q+b p x^2\right)}{\left(q+p x^3\right)^{2/3} \left(b^3 c+d q+3 a b^2 c x+3 a^2 b c x^2+\left(a^3 c+d p\right) x^3\right)} \, dx","IntegrateAlgebraic[((b + a*x)*(-(a*q) + b*p*x^2))/((q + p*x^3)^(2/3)*(b^3*c + d*q + 3*a*b^2*c*x + 3*a^2*b*c*x^2 + (a^3*c + d*p)*x^3)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{3} b \sqrt[3]{c}+\sqrt{3} a \sqrt[3]{c} x}{b \sqrt[3]{c}+a \sqrt[3]{c} x-2 \sqrt[3]{d} \sqrt[3]{q+p x^3}}\right)}{\sqrt{3} c^{2/3} \sqrt[3]{d}}+\frac{\log \left(b^2 \sqrt[3]{c}+a b \sqrt[3]{c} x+b \sqrt[3]{d} \sqrt[3]{q+p x^3}\right)}{3 c^{2/3} \sqrt[3]{d}}-\frac{\log \left(b^4 c^{2/3}+2 a b^3 c^{2/3} x+a^2 b^2 c^{2/3} x^2+\left(-b^3 \sqrt[3]{c} \sqrt[3]{d}-a b^2 \sqrt[3]{c} \sqrt[3]{d} x\right) \sqrt[3]{q+p x^3}+b^2 d^{2/3} \left(q+p x^3\right)^{2/3}\right)}{6 c^{2/3} \sqrt[3]{d}}","\frac{\tan ^{-1}\left(\frac{\sqrt{3} b \sqrt[3]{c}+\sqrt{3} a \sqrt[3]{c} x}{b \sqrt[3]{c}+a \sqrt[3]{c} x-2 \sqrt[3]{d} \sqrt[3]{q+p x^3}}\right)}{\sqrt{3} c^{2/3} \sqrt[3]{d}}+\frac{\log \left(b^2 \sqrt[3]{c}+a b \sqrt[3]{c} x+b \sqrt[3]{d} \sqrt[3]{q+p x^3}\right)}{3 c^{2/3} \sqrt[3]{d}}-\frac{\log \left(b^4 c^{2/3}+2 a b^3 c^{2/3} x+a^2 b^2 c^{2/3} x^2+\left(-b^3 \sqrt[3]{c} \sqrt[3]{d}-a b^2 \sqrt[3]{c} \sqrt[3]{d} x\right) \sqrt[3]{q+p x^3}+b^2 d^{2/3} \left(q+p x^3\right)^{2/3}\right)}{6 c^{2/3} \sqrt[3]{d}}",1,"ArcTan[(Sqrt[3]*b*c^(1/3) + Sqrt[3]*a*c^(1/3)*x)/(b*c^(1/3) + a*c^(1/3)*x - 2*d^(1/3)*(q + p*x^3)^(1/3))]/(Sqrt[3]*c^(2/3)*d^(1/3)) + Log[b^2*c^(1/3) + a*b*c^(1/3)*x + b*d^(1/3)*(q + p*x^3)^(1/3)]/(3*c^(2/3)*d^(1/3)) - Log[b^4*c^(2/3) + 2*a*b^3*c^(2/3)*x + a^2*b^2*c^(2/3)*x^2 + (-(b^3*c^(1/3)*d^(1/3)) - a*b^2*c^(1/3)*d^(1/3)*x)*(q + p*x^3)^(1/3) + b^2*d^(2/3)*(q + p*x^3)^(2/3)]/(6*c^(2/3)*d^(1/3))","A",1
2723,1,250,250,1.0540271,"\int \frac{(-b+x) \left(-a (a-2 b)-2 b x+x^2\right)}{((-a+x) (-b+x))^{2/3} \left(a^4-b^2 d-2 \left(2 a^3-b d\right) x+\left(6 a^2-d\right) x^2-4 a x^3+x^4\right)} \, dx","IntegrateAlgebraic[((-b + x)*(-(a*(a - 2*b)) - 2*b*x + x^2))/(((-a + x)*(-b + x))^(2/3)*(a^4 - b^2*d - 2*(2*a^3 - b*d)*x + (6*a^2 - d)*x^2 - 4*a*x^3 + x^4)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \left(a b+(-a-b) x+x^2\right)^{2/3}}{2 a^2-4 a x+2 x^2+\sqrt[3]{d} \left(a b+(-a-b) x+x^2\right)^{2/3}}\right)}{2 d^{2/3}}+\frac{\log \left(a^2-2 a x+x^2-\sqrt[3]{d} \left(a b+(-a-b) x+x^2\right)^{2/3}\right)}{2 d^{2/3}}-\frac{\log \left(a^4-4 a^3 x+6 a^2 x^2-4 a x^3+x^4+d^{2/3} \left(a b+(-a-b) x+x^2\right)^{4/3}+\left(a b+(-a-b) x+x^2\right)^{2/3} \left(a^2 \sqrt[3]{d}-2 a \sqrt[3]{d} x+\sqrt[3]{d} x^2\right)\right)}{4 d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \left(a b+(-a-b) x+x^2\right)^{2/3}}{2 a^2-4 a x+2 x^2+\sqrt[3]{d} \left(a b+(-a-b) x+x^2\right)^{2/3}}\right)}{2 d^{2/3}}+\frac{\log \left(a^2-2 a x+x^2-\sqrt[3]{d} \left(a b+(-a-b) x+x^2\right)^{2/3}\right)}{2 d^{2/3}}-\frac{\log \left(a^4-4 a^3 x+6 a^2 x^2-4 a x^3+x^4+d^{2/3} \left(a b+(-a-b) x+x^2\right)^{4/3}+\left(a b+(-a-b) x+x^2\right)^{2/3} \left(a^2 \sqrt[3]{d}-2 a \sqrt[3]{d} x+\sqrt[3]{d} x^2\right)\right)}{4 d^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(a*b + (-a - b)*x + x^2)^(2/3))/(2*a^2 - 4*a*x + 2*x^2 + d^(1/3)*(a*b + (-a - b)*x + x^2)^(2/3))])/(2*d^(2/3)) + Log[a^2 - 2*a*x + x^2 - d^(1/3)*(a*b + (-a - b)*x + x^2)^(2/3)]/(2*d^(2/3)) - Log[a^4 - 4*a^3*x + 6*a^2*x^2 - 4*a*x^3 + x^4 + d^(2/3)*(a*b + (-a - b)*x + x^2)^(4/3) + (a*b + (-a - b)*x + x^2)^(2/3)*(a^2*d^(1/3) - 2*a*d^(1/3)*x + d^(1/3)*x^2)]/(4*d^(2/3))","A",1
2724,1,250,250,0.5448529,"\int \frac{\sqrt{1+x^5} \left(2+x^5\right)}{x^6 \left(-1-x^5+a x^{10}\right)} \, dx","IntegrateAlgebraic[(Sqrt[1 + x^5]*(2 + x^5))/(x^6*(-1 - x^5 + a*x^10)),x]","\frac{2 \sqrt{1+x^5}}{5 x^5}+\frac{\left(\sqrt{2} \sqrt{a}+4 \sqrt{2} a^{3/2}+\sqrt{2} \sqrt{a} \sqrt{1+4 a}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{1+x^5}}{\sqrt{-1-2 a-\sqrt{1+4 a}}}\right)}{5 \sqrt{1+4 a} \sqrt{-1-2 a-\sqrt{1+4 a}}}+\frac{\left(-\sqrt{2} \sqrt{a}-4 \sqrt{2} a^{3/2}+\sqrt{2} \sqrt{a} \sqrt{1+4 a}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{1+x^5}}{\sqrt{-1-2 a+\sqrt{1+4 a}}}\right)}{5 \sqrt{1+4 a} \sqrt{-1-2 a+\sqrt{1+4 a}}}","\frac{2 \sqrt{1+x^5}}{5 x^5}+\frac{\left(\sqrt{2} \sqrt{a}+4 \sqrt{2} a^{3/2}+\sqrt{2} \sqrt{a} \sqrt{1+4 a}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{1+x^5}}{\sqrt{-1-2 a-\sqrt{1+4 a}}}\right)}{5 \sqrt{1+4 a} \sqrt{-1-2 a-\sqrt{1+4 a}}}+\frac{\left(-\sqrt{2} \sqrt{a}-4 \sqrt{2} a^{3/2}+\sqrt{2} \sqrt{a} \sqrt{1+4 a}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{1+x^5}}{\sqrt{-1-2 a+\sqrt{1+4 a}}}\right)}{5 \sqrt{1+4 a} \sqrt{-1-2 a+\sqrt{1+4 a}}}",1,"(2*Sqrt[1 + x^5])/(5*x^5) + ((Sqrt[2]*Sqrt[a] + 4*Sqrt[2]*a^(3/2) + Sqrt[2]*Sqrt[a]*Sqrt[1 + 4*a])*ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[1 + x^5])/Sqrt[-1 - 2*a - Sqrt[1 + 4*a]]])/(5*Sqrt[1 + 4*a]*Sqrt[-1 - 2*a - Sqrt[1 + 4*a]]) + ((-(Sqrt[2]*Sqrt[a]) - 4*Sqrt[2]*a^(3/2) + Sqrt[2]*Sqrt[a]*Sqrt[1 + 4*a])*ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[1 + x^5])/Sqrt[-1 - 2*a + Sqrt[1 + 4*a]]])/(5*Sqrt[1 + 4*a]*Sqrt[-1 - 2*a + Sqrt[1 + 4*a]])","A",1
2725,1,250,250,0.3467978,"\int \frac{\left(-2+x^5\right) \sqrt{-1+x^5}}{x^6 \left(1-x^5+a x^{10}\right)} \, dx","IntegrateAlgebraic[((-2 + x^5)*Sqrt[-1 + x^5])/(x^6*(1 - x^5 + a*x^10)),x]","\frac{2 \sqrt{-1+x^5}}{5 x^5}+\frac{\left(\sqrt{2} \sqrt{a}+\sqrt{2} \sqrt{1-4 a} \sqrt{a}-4 \sqrt{2} a^{3/2}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{-1+x^5}}{\sqrt{-1-\sqrt{1-4 a}+2 a}}\right)}{5 \sqrt{1-4 a} \sqrt{-1-\sqrt{1-4 a}+2 a}}+\frac{\left(-\sqrt{2} \sqrt{a}+\sqrt{2} \sqrt{1-4 a} \sqrt{a}+4 \sqrt{2} a^{3/2}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{-1+x^5}}{\sqrt{-1+\sqrt{1-4 a}+2 a}}\right)}{5 \sqrt{1-4 a} \sqrt{-1+\sqrt{1-4 a}+2 a}}","\frac{2 \sqrt{-1+x^5}}{5 x^5}+\frac{\left(\sqrt{2} \sqrt{a}+\sqrt{2} \sqrt{1-4 a} \sqrt{a}-4 \sqrt{2} a^{3/2}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{-1+x^5}}{\sqrt{-1-\sqrt{1-4 a}+2 a}}\right)}{5 \sqrt{1-4 a} \sqrt{-1-\sqrt{1-4 a}+2 a}}+\frac{\left(-\sqrt{2} \sqrt{a}+\sqrt{2} \sqrt{1-4 a} \sqrt{a}+4 \sqrt{2} a^{3/2}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{-1+x^5}}{\sqrt{-1+\sqrt{1-4 a}+2 a}}\right)}{5 \sqrt{1-4 a} \sqrt{-1+\sqrt{1-4 a}+2 a}}",1,"(2*Sqrt[-1 + x^5])/(5*x^5) + ((Sqrt[2]*Sqrt[a] + Sqrt[2]*Sqrt[1 - 4*a]*Sqrt[a] - 4*Sqrt[2]*a^(3/2))*ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[-1 + x^5])/Sqrt[-1 - Sqrt[1 - 4*a] + 2*a]])/(5*Sqrt[1 - 4*a]*Sqrt[-1 - Sqrt[1 - 4*a] + 2*a]) + ((-(Sqrt[2]*Sqrt[a]) + Sqrt[2]*Sqrt[1 - 4*a]*Sqrt[a] + 4*Sqrt[2]*a^(3/2))*ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[-1 + x^5])/Sqrt[-1 + Sqrt[1 - 4*a] + 2*a]])/(5*Sqrt[1 - 4*a]*Sqrt[-1 + Sqrt[1 - 4*a] + 2*a])","A",1
2726,1,250,250,0.1678061,"\int \frac{1}{x^2-\sqrt{1+x} \sqrt{1+\sqrt{1+x}}} \, dx","IntegrateAlgebraic[(x^2 - Sqrt[1 + x]*Sqrt[1 + Sqrt[1 + x]])^(-1),x]","\frac{2}{55} \left(-5+7 \sqrt{5}\right) \log \left(1+\sqrt{5}-2 \sqrt{1+\sqrt{1+x}}\right)-\frac{2}{55} \left(5+7 \sqrt{5}\right) \log \left(-1+\sqrt{5}+2 \sqrt{1+\sqrt{1+x}}\right)+\frac{4}{11} \text{RootSum}\left[-1+\text{$\#$1}-\text{$\#$1}^2-2 \text{$\#$1}^3+\text{$\#$1}^4+\text{$\#$1}^5\&,\frac{7 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}-7 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2-2 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^3+\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^4}{1-2 \text{$\#$1}-6 \text{$\#$1}^2+4 \text{$\#$1}^3+5 \text{$\#$1}^4}\&\right]","\frac{2}{55} \left(-5+7 \sqrt{5}\right) \log \left(1+\sqrt{5}-2 \sqrt{1+\sqrt{1+x}}\right)-\frac{2}{55} \left(5+7 \sqrt{5}\right) \log \left(-1+\sqrt{5}+2 \sqrt{1+\sqrt{1+x}}\right)+\frac{4}{11} \text{RootSum}\left[-1+\text{$\#$1}-\text{$\#$1}^2-2 \text{$\#$1}^3+\text{$\#$1}^4+\text{$\#$1}^5\&,\frac{7 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}-7 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2-2 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^3+\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^4}{1-2 \text{$\#$1}-6 \text{$\#$1}^2+4 \text{$\#$1}^3+5 \text{$\#$1}^4}\&\right]",1,"(2*(-5 + 7*Sqrt[5])*Log[1 + Sqrt[5] - 2*Sqrt[1 + Sqrt[1 + x]]])/55 - (2*(5 + 7*Sqrt[5])*Log[-1 + Sqrt[5] + 2*Sqrt[1 + Sqrt[1 + x]]])/55 + (4*RootSum[-1 + #1 - #1^2 - 2*#1^3 + #1^4 + #1^5 & , (7*Log[Sqrt[1 + Sqrt[1 + x]] - #1] - 2*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1 - 7*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^2 - 2*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^3 + Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^4)/(1 - 2*#1 - 6*#1^2 + 4*#1^3 + 5*#1^4) & ])/11","A",1
2727,1,250,250,0.0014979,"\int \frac{1}{x^2-\sqrt{1+x} \sqrt{1+\sqrt{1+x}}} \, dx","IntegrateAlgebraic[(x^2 - Sqrt[1 + x]*Sqrt[1 + Sqrt[1 + x]])^(-1),x]","\frac{2}{55} \left(-5+7 \sqrt{5}\right) \log \left(1+\sqrt{5}-2 \sqrt{1+\sqrt{1+x}}\right)-\frac{2}{55} \left(5+7 \sqrt{5}\right) \log \left(-1+\sqrt{5}+2 \sqrt{1+\sqrt{1+x}}\right)+\frac{4}{11} \text{RootSum}\left[-1+\text{$\#$1}-\text{$\#$1}^2-2 \text{$\#$1}^3+\text{$\#$1}^4+\text{$\#$1}^5\&,\frac{7 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}-7 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2-2 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^3+\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^4}{1-2 \text{$\#$1}-6 \text{$\#$1}^2+4 \text{$\#$1}^3+5 \text{$\#$1}^4}\&\right]","\frac{2}{55} \left(-5+7 \sqrt{5}\right) \log \left(1+\sqrt{5}-2 \sqrt{1+\sqrt{1+x}}\right)-\frac{2}{55} \left(5+7 \sqrt{5}\right) \log \left(-1+\sqrt{5}+2 \sqrt{1+\sqrt{1+x}}\right)+\frac{4}{11} \text{RootSum}\left[-1+\text{$\#$1}-\text{$\#$1}^2-2 \text{$\#$1}^3+\text{$\#$1}^4+\text{$\#$1}^5\&,\frac{7 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}-7 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2-2 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^3+\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^4}{1-2 \text{$\#$1}-6 \text{$\#$1}^2+4 \text{$\#$1}^3+5 \text{$\#$1}^4}\&\right]",1,"(2*(-5 + 7*Sqrt[5])*Log[1 + Sqrt[5] - 2*Sqrt[1 + Sqrt[1 + x]]])/55 - (2*(5 + 7*Sqrt[5])*Log[-1 + Sqrt[5] + 2*Sqrt[1 + Sqrt[1 + x]]])/55 + (4*RootSum[-1 + #1 - #1^2 - 2*#1^3 + #1^4 + #1^5 & , (7*Log[Sqrt[1 + Sqrt[1 + x]] - #1] - 2*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1 - 7*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^2 - 2*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^3 + Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^4)/(1 - 2*#1 - 6*#1^2 + 4*#1^3 + 5*#1^4) & ])/11","A",1
2728,1,250,250,1.3155537,"\int \frac{x \left(x^2 c_3-c_4\right)}{\sqrt{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \left(x+3 x^2 c_3+3 c_4\right) \left(-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right)} \, dx","IntegrateAlgebraic[(x*(x^2*C[3] - C[4]))/(Sqrt[(x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4])]*(x + 3*x^2*C[3] + 3*C[4])*(-x^2 + x^4*C[3]^2 + 2*x^2*C[3]*C[4] + C[4]^2)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{1-c_0} \sqrt{-1+c_1} \sqrt{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{-1+c_0}\right) \sqrt{-1+c_1}}{2 \sqrt{1-c_0}}-\frac{\tan ^{-1}\left(\frac{\sqrt{-1-c_0} \sqrt{1+c_1} \sqrt{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{1+c_0}\right) \sqrt{1+c_1}}{4 \sqrt{-1-c_0}}+\frac{3 \tan ^{-1}\left(\frac{\sqrt{1-3 c_0} \sqrt{-1+3 c_1} \sqrt{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{-1+3 c_0}\right) \sqrt{-1+3 c_1}}{4 \sqrt{1-3 c_0}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{1-c_0} \sqrt{-1+c_1} \sqrt{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{-1+c_0}\right) \sqrt{-1+c_1}}{2 \sqrt{1-c_0}}-\frac{\tan ^{-1}\left(\frac{\sqrt{-1-c_0} \sqrt{1+c_1} \sqrt{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{1+c_0}\right) \sqrt{1+c_1}}{4 \sqrt{-1-c_0}}+\frac{3 \tan ^{-1}\left(\frac{\sqrt{1-3 c_0} \sqrt{-1+3 c_1} \sqrt{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{-1+3 c_0}\right) \sqrt{-1+3 c_1}}{4 \sqrt{1-3 c_0}}",1,"-1/2*(ArcTan[(Sqrt[1 - C[0]]*Sqrt[-1 + C[1]]*Sqrt[(x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4])])/(-1 + C[0])]*Sqrt[-1 + C[1]])/Sqrt[1 - C[0]] - (ArcTan[(Sqrt[-1 - C[0]]*Sqrt[1 + C[1]]*Sqrt[(x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4])])/(1 + C[0])]*Sqrt[1 + C[1]])/(4*Sqrt[-1 - C[0]]) + (3*ArcTan[(Sqrt[1 - 3*C[0]]*Sqrt[-1 + 3*C[1]]*Sqrt[(x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4])])/(-1 + 3*C[0])]*Sqrt[-1 + 3*C[1]])/(4*Sqrt[1 - 3*C[0]])","A",1
2729,1,233,251,1.1060639,"\int \frac{1}{\left(-b+a x^4\right)^2 \sqrt[4]{-b x^2+a x^4}} \, dx","IntegrateAlgebraic[1/((-b + a*x^4)^2*(-(b*x^2) + a*x^4)^(1/4)),x]","\frac{\left(-b-a x^2\right) \left(-b x^2+a x^4\right)^{3/4}}{4 b^2 (-a+b) x \left(b-a x^4\right)}-\frac{\text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{4 b^2}+\frac{\text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 a \log (x)+2 a \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}+\text{$\#$1}^5}\&\right]}{32 (a-b) b}","\frac{\left(-b-a x^2\right) \left(-b x^2+a x^4\right)^{3/4}}{4 b^2 (-a+b) x \left(b-a x^4\right)}-\frac{\text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-8 a^2 \log (x)+6 a b \log (x)+8 a^2 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)-6 a b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)+8 a \log (x) \text{$\#$1}^4-7 b \log (x) \text{$\#$1}^4-8 a \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+7 b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}-\text{$\#$1}^5}\&\right]}{32 (a-b) b^2}",1,"((-b - a*x^2)*(-(b*x^2) + a*x^4)^(3/4))/(4*b^2*(-a + b)*x*(b - a*x^4)) - RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (-Log[x] + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1])/#1 & ]/(4*b^2) + RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (-2*a*Log[x] + 2*a*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1) + #1^5) & ]/(32*(a - b)*b)","A",1
2730,1,233,251,0.0019976,"\int \frac{1}{\left(-b+a x^4\right)^2 \sqrt[4]{-b x^2+a x^4}} \, dx","IntegrateAlgebraic[1/((-b + a*x^4)^2*(-(b*x^2) + a*x^4)^(1/4)),x]","\frac{\left(-b-a x^2\right) \left(-b x^2+a x^4\right)^{3/4}}{4 b^2 (-a+b) x \left(b-a x^4\right)}-\frac{\text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{4 b^2}+\frac{\text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 a \log (x)+2 a \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^4-\log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}+\text{$\#$1}^5}\&\right]}{32 (a-b) b}","\frac{\left(-b-a x^2\right) \left(-b x^2+a x^4\right)^{3/4}}{4 b^2 (-a+b) x \left(b-a x^4\right)}-\frac{\text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-8 a^2 \log (x)+6 a b \log (x)+8 a^2 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)-6 a b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)+8 a \log (x) \text{$\#$1}^4-7 b \log (x) \text{$\#$1}^4-8 a \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+7 b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}-\text{$\#$1}^5}\&\right]}{32 (a-b) b^2}",1,"((-b - a*x^2)*(-(b*x^2) + a*x^4)^(3/4))/(4*b^2*(-a + b)*x*(b - a*x^4)) - RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (-Log[x] + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1])/#1 & ]/(4*b^2) + RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (-2*a*Log[x] + 2*a*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1) + #1^5) & ]/(32*(a - b)*b)","A",1
2731,1,251,251,1.2162019,"\int \frac{\left(-2 q+p x^3\right) \left(a q+b x^2+a p x^3\right) \sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^5 \left(c q+d x^2+c p x^3\right)} \, dx","IntegrateAlgebraic[((-2*q + p*x^3)*(a*q + b*x^2 + a*p*x^3)*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6])/(x^5*(c*q + d*x^2 + c*p*x^3)),x]","\frac{\left(a c q+2 b c x^2-2 a d x^2+a c p x^3\right) \sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{2 c^2 x^4}-\frac{2 (-b c+a d) \sqrt{-d^2+2 c^2 p q} \tan ^{-1}\left(\frac{\sqrt{-d^2+2 c^2 p q} x^2}{c q+d x^2+c p x^3+c \sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}}\right)}{c^3}+\frac{2 \left(b c d-a d^2+a c^2 p q\right) \log (x)}{c^3}+\frac{\left(-b c d+a d^2-a c^2 p q\right) \log \left(q+p x^3+\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}\right)}{c^3}","\frac{\left(a c q+2 b c x^2-2 a d x^2+a c p x^3\right) \sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{2 c^2 x^4}-\frac{2 (-b c+a d) \sqrt{-d^2+2 c^2 p q} \tan ^{-1}\left(\frac{\sqrt{-d^2+2 c^2 p q} x^2}{c q+d x^2+c p x^3+c \sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}}\right)}{c^3}+\frac{2 \left(b c d-a d^2+a c^2 p q\right) \log (x)}{c^3}+\frac{\left(-b c d+a d^2-a c^2 p q\right) \log \left(q+p x^3+\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}\right)}{c^3}",1,"((a*c*q + 2*b*c*x^2 - 2*a*d*x^2 + a*c*p*x^3)*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6])/(2*c^2*x^4) - (2*(-(b*c) + a*d)*Sqrt[-d^2 + 2*c^2*p*q]*ArcTan[(Sqrt[-d^2 + 2*c^2*p*q]*x^2)/(c*q + d*x^2 + c*p*x^3 + c*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6])])/c^3 + (2*(b*c*d - a*d^2 + a*c^2*p*q)*Log[x])/c^3 + ((-(b*c*d) + a*d^2 - a*c^2*p*q)*Log[q + p*x^3 + Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]])/c^3","A",1
2732,1,252,252,3.1888272,"\int -\frac{(-1+x) x \left(-1+2 x+\left(-2 k+k^2\right) x^2\right)}{((1-x) x (1-k x))^{2/3} \left(1-4 k x+\left(-b+6 k^2\right) x^2+\left(2 b-4 k^3\right) x^3+\left(-b+k^4\right) x^4\right)} \, dx","IntegrateAlgebraic[-(((-1 + x)*x*(-1 + 2*x + (-2*k + k^2)*x^2))/(((1 - x)*x*(1 - k*x))^(2/3)*(1 - 4*k*x + (-b + 6*k^2)*x^2 + (2*b - 4*k^3)*x^3 + (-b + k^4)*x^4))),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}}{2-4 k x+2 k^2 x^2+\sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}}\right)}{2 b^{2/3}}+\frac{\log \left(1-2 k x+k^2 x^2-\sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 b^{2/3}}-\frac{\log \left(1-4 k x+6 k^2 x^2-4 k^3 x^3+k^4 x^4+\left(\sqrt[3]{b}-2 \sqrt[3]{b} k x+\sqrt[3]{b} k^2 x^2\right) \left(x+(-1-k) x^2+k x^3\right)^{2/3}+b^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{4/3}\right)}{4 b^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}}{2-4 k x+2 k^2 x^2+\sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}}\right)}{2 b^{2/3}}+\frac{\log \left(1-2 k x+k^2 x^2-\sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 b^{2/3}}-\frac{\log \left(1-4 k x+6 k^2 x^2-4 k^3 x^3+k^4 x^4+\left(\sqrt[3]{b}-2 \sqrt[3]{b} k x+\sqrt[3]{b} k^2 x^2\right) \left(x+(-1-k) x^2+k x^3\right)^{2/3}+b^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{4/3}\right)}{4 b^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3))/(2 - 4*k*x + 2*k^2*x^2 + b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3))])/(2*b^(2/3)) + Log[1 - 2*k*x + k^2*x^2 - b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3)]/(2*b^(2/3)) - Log[1 - 4*k*x + 6*k^2*x^2 - 4*k^3*x^3 + k^4*x^4 + (b^(1/3) - 2*b^(1/3)*k*x + b^(1/3)*k^2*x^2)*(x + (-1 - k)*x^2 + k*x^3)^(2/3) + b^(2/3)*(x + (-1 - k)*x^2 + k*x^3)^(4/3)]/(4*b^(2/3))","A",1
2733,1,252,252,0.5113135,"\int \frac{\left(-2 q+p x^3\right) \sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left(b x^8+a \left(q+p x^3\right)^4\right)}{x^{13}} \, dx","IntegrateAlgebraic[((-2*q + p*x^3)*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]*(b*x^8 + a*(q + p*x^3)^4))/x^13,x]","\frac{\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left(2 a q^5+10 a p q^4 x^3-a p q^4 x^4+20 a p^2 q^3 x^6-3 a p^2 q^3 x^7+6 b q x^8-3 a p^2 q^3 x^8+20 a p^3 q^2 x^9-3 a p^3 q^2 x^{10}+6 b p x^{11}-3 a p^3 q^2 x^{11}+10 a p^4 q x^{12}-a p^4 q x^{13}+2 a p^5 x^{15}\right)}{12 x^{12}}+\left(2 b p q+a p^3 q^3\right) \log (x)+\frac{1}{2} \left(-2 b p q-a p^3 q^3\right) \log \left(q+p x^3+\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}\right)","\frac{\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left(2 a q^5+10 a p q^4 x^3-a p q^4 x^4+20 a p^2 q^3 x^6-3 a p^2 q^3 x^7+6 b q x^8-3 a p^2 q^3 x^8+20 a p^3 q^2 x^9-3 a p^3 q^2 x^{10}+6 b p x^{11}-3 a p^3 q^2 x^{11}+10 a p^4 q x^{12}-a p^4 q x^{13}+2 a p^5 x^{15}\right)}{12 x^{12}}+\left(2 b p q+a p^3 q^3\right) \log (x)+\frac{1}{2} \left(-2 b p q-a p^3 q^3\right) \log \left(q+p x^3+\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}\right)",1,"(Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]*(2*a*q^5 + 10*a*p*q^4*x^3 - a*p*q^4*x^4 + 20*a*p^2*q^3*x^6 - 3*a*p^2*q^3*x^7 + 6*b*q*x^8 - 3*a*p^2*q^3*x^8 + 20*a*p^3*q^2*x^9 - 3*a*p^3*q^2*x^10 + 6*b*p*x^11 - 3*a*p^3*q^2*x^11 + 10*a*p^4*q*x^12 - a*p^4*q*x^13 + 2*a*p^5*x^15))/(12*x^12) + (2*b*p*q + a*p^3*q^3)*Log[x] + ((-2*b*p*q - a*p^3*q^3)*Log[q + p*x^3 + Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]])/2","A",1
2734,1,253,252,0.1699512,"\int \frac{x^2}{x^2-\sqrt{1+x} \sqrt{1+\sqrt{1+x}}} \, dx","IntegrateAlgebraic[x^2/(x^2 - Sqrt[1 + x]*Sqrt[1 + Sqrt[1 + x]]),x]","1+x+\frac{4}{55} \left(5+4 \sqrt{5}\right) \log \left(1+\sqrt{5}-2 \sqrt{1+\sqrt{1+x}}\right)-\frac{4}{55} \left(-5+4 \sqrt{5}\right) \log \left(-1+\sqrt{5}+2 \sqrt{1+\sqrt{1+x}}\right)-\frac{4}{11} \text{RootSum}\left[-1+\text{$\#$1}-\text{$\#$1}^2-2 \text{$\#$1}^3+\text{$\#$1}^4+\text{$\#$1}^5\&,\frac{3 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)+7 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}-3 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2-4 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^3+2 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^4}{1-2 \text{$\#$1}-6 \text{$\#$1}^2+4 \text{$\#$1}^3+5 \text{$\#$1}^4}\&\right]","x+\frac{4}{55} \left(5+4 \sqrt{5}\right) \log \left(1+\sqrt{5}-2 \sqrt{1+\sqrt{1+x}}\right)-\frac{4}{55} \left(-5+4 \sqrt{5}\right) \log \left(-1+\sqrt{5}+2 \sqrt{1+\sqrt{1+x}}\right)-\frac{4}{11} \text{RootSum}\left[-1+\text{$\#$1}-\text{$\#$1}^2-2 \text{$\#$1}^3+\text{$\#$1}^4+\text{$\#$1}^5\&,\frac{3 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)+7 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}-3 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2-4 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^3+2 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^4}{1-2 \text{$\#$1}-6 \text{$\#$1}^2+4 \text{$\#$1}^3+5 \text{$\#$1}^4}\&\right]",1,"1 + x + (4*(5 + 4*Sqrt[5])*Log[1 + Sqrt[5] - 2*Sqrt[1 + Sqrt[1 + x]]])/55 - (4*(-5 + 4*Sqrt[5])*Log[-1 + Sqrt[5] + 2*Sqrt[1 + Sqrt[1 + x]]])/55 - (4*RootSum[-1 + #1 - #1^2 - 2*#1^3 + #1^4 + #1^5 & , (3*Log[Sqrt[1 + Sqrt[1 + x]] - #1] + 7*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1 - 3*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^2 - 4*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^3 + 2*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^4)/(1 - 2*#1 - 6*#1^2 + 4*#1^3 + 5*#1^4) & ])/11","A",1
2735,1,253,252,0.001349,"\int \frac{x^2}{x^2-\sqrt{1+x} \sqrt{1+\sqrt{1+x}}} \, dx","IntegrateAlgebraic[x^2/(x^2 - Sqrt[1 + x]*Sqrt[1 + Sqrt[1 + x]]),x]","1+x+\frac{4}{55} \left(5+4 \sqrt{5}\right) \log \left(1+\sqrt{5}-2 \sqrt{1+\sqrt{1+x}}\right)-\frac{4}{55} \left(-5+4 \sqrt{5}\right) \log \left(-1+\sqrt{5}+2 \sqrt{1+\sqrt{1+x}}\right)-\frac{4}{11} \text{RootSum}\left[-1+\text{$\#$1}-\text{$\#$1}^2-2 \text{$\#$1}^3+\text{$\#$1}^4+\text{$\#$1}^5\&,\frac{3 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)+7 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}-3 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2-4 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^3+2 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^4}{1-2 \text{$\#$1}-6 \text{$\#$1}^2+4 \text{$\#$1}^3+5 \text{$\#$1}^4}\&\right]","x+\frac{4}{55} \left(5+4 \sqrt{5}\right) \log \left(1+\sqrt{5}-2 \sqrt{1+\sqrt{1+x}}\right)-\frac{4}{55} \left(-5+4 \sqrt{5}\right) \log \left(-1+\sqrt{5}+2 \sqrt{1+\sqrt{1+x}}\right)-\frac{4}{11} \text{RootSum}\left[-1+\text{$\#$1}-\text{$\#$1}^2-2 \text{$\#$1}^3+\text{$\#$1}^4+\text{$\#$1}^5\&,\frac{3 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)+7 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}-3 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2-4 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^3+2 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^4}{1-2 \text{$\#$1}-6 \text{$\#$1}^2+4 \text{$\#$1}^3+5 \text{$\#$1}^4}\&\right]",1,"1 + x + (4*(5 + 4*Sqrt[5])*Log[1 + Sqrt[5] - 2*Sqrt[1 + Sqrt[1 + x]]])/55 - (4*(-5 + 4*Sqrt[5])*Log[-1 + Sqrt[5] + 2*Sqrt[1 + Sqrt[1 + x]]])/55 - (4*RootSum[-1 + #1 - #1^2 - 2*#1^3 + #1^4 + #1^5 & , (3*Log[Sqrt[1 + Sqrt[1 + x]] - #1] + 7*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1 - 3*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^2 - 4*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^3 + 2*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^4)/(1 - 2*#1 - 6*#1^2 + 4*#1^3 + 5*#1^4) & ])/11","A",1
2736,1,411,253,1.1866066,"\int \frac{1}{x \sqrt[5]{1-4 x+6 x^2-4 x^3+x^4}} \, dx","IntegrateAlgebraic[1/(x*(1 - 4*x + 6*x^2 - 4*x^3 + x^4)^(1/5)),x]","-\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{10+2 \sqrt{5}} \sqrt[5]{1-4 x+6 x^2-4 x^3+x^4}}{-4+4 x+\left(-1+\sqrt{5}\right) \sqrt[5]{1-4 x+6 x^2-4 x^3+x^4}}\right)+\sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{10-2 \sqrt{5}} \sqrt[5]{1-4 x+6 x^2-4 x^3+x^4}}{4-4 x+\left(1+\sqrt{5}\right) \sqrt[5]{1-4 x+6 x^2-4 x^3+x^4}}\right)-\frac{4}{5} \log (-1+x)+\frac{1}{5} \log \left(1-4 x+6 x^2-4 x^3+x^4\right)+\log \left(-1+x+\sqrt[5]{1-4 x+6 x^2-4 x^3+x^4}\right)+\frac{1}{4} \left(-1-\sqrt{5}\right) \log \left(2-4 x+2 x^2+\left(1+\sqrt{5} (1-x)-x\right) \sqrt[5]{1-4 x+6 x^2-4 x^3+x^4}+2 \left(1-4 x+6 x^2-4 x^3+x^4\right)^{2/5}\right)+\frac{1}{4} \left(-1+\sqrt{5}\right) \log \left(2-4 x+2 x^2+\left(1+\sqrt{5} (-1+x)-x\right) \sqrt[5]{1-4 x+6 x^2-4 x^3+x^4}+2 \left(1-4 x+6 x^2-4 x^3+x^4\right)^{2/5}\right)","\frac{(-1+x)^{4/5} \left(-\sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)} \tan ^{-1}\left(\sqrt{\frac{5}{10-2 \sqrt{5}}}+\frac{1}{\sqrt{10-2 \sqrt{5}}}-\frac{4 \sqrt[5]{-1+x}}{\sqrt{10-2 \sqrt{5}}}\right)+\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} \tan ^{-1}\left(\sqrt{\frac{5}{10+2 \sqrt{5}}}-\frac{1}{\sqrt{10+2 \sqrt{5}}}+\frac{4 \sqrt[5]{-1+x}}{\sqrt{10+2 \sqrt{5}}}\right)+\log \left(1+\sqrt[5]{-1+x}\right)+\frac{1}{4} \left(-1-\sqrt{5}\right) \log \left(2+\left(-1-\sqrt{5}\right) \sqrt[5]{-1+x}+2 (-1+x)^{2/5}\right)+\frac{1}{4} \left(-1+\sqrt{5}\right) \log \left(2+\left(-1+\sqrt{5}\right) \sqrt[5]{-1+x}+2 (-1+x)^{2/5}\right)\right)}{\sqrt[5]{(-1+x)^4}}",1,"-(Sqrt[(5 + Sqrt[5])/2]*ArcTan[(Sqrt[10 + 2*Sqrt[5]]*(1 - 4*x + 6*x^2 - 4*x^3 + x^4)^(1/5))/(-4 + 4*x + (-1 + Sqrt[5])*(1 - 4*x + 6*x^2 - 4*x^3 + x^4)^(1/5))]) + Sqrt[(5 - Sqrt[5])/2]*ArcTan[(Sqrt[10 - 2*Sqrt[5]]*(1 - 4*x + 6*x^2 - 4*x^3 + x^4)^(1/5))/(4 - 4*x + (1 + Sqrt[5])*(1 - 4*x + 6*x^2 - 4*x^3 + x^4)^(1/5))] - (4*Log[-1 + x])/5 + Log[1 - 4*x + 6*x^2 - 4*x^3 + x^4]/5 + Log[-1 + x + (1 - 4*x + 6*x^2 - 4*x^3 + x^4)^(1/5)] + ((-1 - Sqrt[5])*Log[2 - 4*x + 2*x^2 + (1 + Sqrt[5]*(1 - x) - x)*(1 - 4*x + 6*x^2 - 4*x^3 + x^4)^(1/5) + 2*(1 - 4*x + 6*x^2 - 4*x^3 + x^4)^(2/5)])/4 + ((-1 + Sqrt[5])*Log[2 - 4*x + 2*x^2 + (1 + Sqrt[5]*(-1 + x) - x)*(1 - 4*x + 6*x^2 - 4*x^3 + x^4)^(1/5) + 2*(1 - 4*x + 6*x^2 - 4*x^3 + x^4)^(2/5)])/4","A",1
2737,1,254,254,0.6929421,"\int \frac{1-x+x^2}{\left(-1+x^2\right) \sqrt[3]{x^2+x^4}} \, dx","IntegrateAlgebraic[(1 - x + x^2)/((-1 + x^2)*(x^2 + x^4)^(1/3)),x]","-\frac{3 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2^{2/3} \sqrt[3]{x^2+x^4}}\right)}{4 \sqrt[3]{2}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x^2+x^4}}\right)}{4 \sqrt[3]{2}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x^2+x^4}\right)}{4 \sqrt[3]{2}}-\frac{3 \log \left(2 x+2^{2/3} \sqrt[3]{x^2+x^4}\right)}{4 \sqrt[3]{2}}+\frac{3 \log \left(-2 x^2+2^{2/3} x \sqrt[3]{x^2+x^4}-\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}\right)}{8 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x^2+x^4}+\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}\right)}{8 \sqrt[3]{2}}","-\frac{3 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2^{2/3} \sqrt[3]{x^2+x^4}}\right)}{4 \sqrt[3]{2}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x^2+x^4}}\right)}{4 \sqrt[3]{2}}+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x^2+x^4}\right)}{4 \sqrt[3]{2}}-\frac{3 \log \left(2 x+2^{2/3} \sqrt[3]{x^2+x^4}\right)}{4 \sqrt[3]{2}}+\frac{3 \log \left(-2 x^2+2^{2/3} x \sqrt[3]{x^2+x^4}-\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}\right)}{8 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x^2+x^4}+\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}\right)}{8 \sqrt[3]{2}}",1,"(-3*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2^(2/3)*(x^2 + x^4)^(1/3))])/(4*2^(1/3)) - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(x^2 + x^4)^(1/3))])/(4*2^(1/3)) + Log[-2*x + 2^(2/3)*(x^2 + x^4)^(1/3)]/(4*2^(1/3)) - (3*Log[2*x + 2^(2/3)*(x^2 + x^4)^(1/3)])/(4*2^(1/3)) + (3*Log[-2*x^2 + 2^(2/3)*x*(x^2 + x^4)^(1/3) - 2^(1/3)*(x^2 + x^4)^(2/3)])/(8*2^(1/3)) - Log[2*x^2 + 2^(2/3)*x*(x^2 + x^4)^(1/3) + 2^(1/3)*(x^2 + x^4)^(2/3)]/(8*2^(1/3))","A",1
2738,1,254,254,0.6248869,"\int \frac{1+x+x^2}{\left(-1+x^2\right) \sqrt[3]{x^2+x^4}} \, dx","IntegrateAlgebraic[(1 + x + x^2)/((-1 + x^2)*(x^2 + x^4)^(1/3)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2^{2/3} \sqrt[3]{x^2+x^4}}\right)}{4 \sqrt[3]{2}}-\frac{3 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x^2+x^4}}\right)}{4 \sqrt[3]{2}}+\frac{3 \log \left(-2 x+2^{2/3} \sqrt[3]{x^2+x^4}\right)}{4 \sqrt[3]{2}}-\frac{\log \left(2 x+2^{2/3} \sqrt[3]{x^2+x^4}\right)}{4 \sqrt[3]{2}}+\frac{\log \left(-2 x^2+2^{2/3} x \sqrt[3]{x^2+x^4}-\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}\right)}{8 \sqrt[3]{2}}-\frac{3 \log \left(2 x^2+2^{2/3} x \sqrt[3]{x^2+x^4}+\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}\right)}{8 \sqrt[3]{2}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2^{2/3} \sqrt[3]{x^2+x^4}}\right)}{4 \sqrt[3]{2}}-\frac{3 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x^2+x^4}}\right)}{4 \sqrt[3]{2}}+\frac{3 \log \left(-2 x+2^{2/3} \sqrt[3]{x^2+x^4}\right)}{4 \sqrt[3]{2}}-\frac{\log \left(2 x+2^{2/3} \sqrt[3]{x^2+x^4}\right)}{4 \sqrt[3]{2}}+\frac{\log \left(-2 x^2+2^{2/3} x \sqrt[3]{x^2+x^4}-\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}\right)}{8 \sqrt[3]{2}}-\frac{3 \log \left(2 x^2+2^{2/3} x \sqrt[3]{x^2+x^4}+\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}\right)}{8 \sqrt[3]{2}}",1,"-1/4*(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2^(2/3)*(x^2 + x^4)^(1/3))])/2^(1/3) - (3*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(x^2 + x^4)^(1/3))])/(4*2^(1/3)) + (3*Log[-2*x + 2^(2/3)*(x^2 + x^4)^(1/3)])/(4*2^(1/3)) - Log[2*x + 2^(2/3)*(x^2 + x^4)^(1/3)]/(4*2^(1/3)) + Log[-2*x^2 + 2^(2/3)*x*(x^2 + x^4)^(1/3) - 2^(1/3)*(x^2 + x^4)^(2/3)]/(8*2^(1/3)) - (3*Log[2*x^2 + 2^(2/3)*x*(x^2 + x^4)^(1/3) + 2^(1/3)*(x^2 + x^4)^(2/3)])/(8*2^(1/3))","A",1
2739,1,232,254,0.3090992,"\int \frac{\left(-1+x^2\right) \sqrt{1-x-x^2+x^3+x^4}}{x^2 \left(1+x^2\right)} \, dx","IntegrateAlgebraic[((-1 + x^2)*Sqrt[1 - x - x^2 + x^3 + x^4])/(x^2*(1 + x^2)),x]","\frac{\sqrt{1-x-x^2+x^3+x^4}}{x}+\frac{\log (x)}{2}-\frac{1}{2} \log \left(2-x-2 x^2+2 \sqrt{1-x-x^2+x^3+x^4}\right)-\frac{1}{2} \text{RootSum}\left[5-16 \text{$\#$1}+14 \text{$\#$1}^2+\text{$\#$1}^4\&,\frac{-\log (x)+\log \left(1-x^2+\sqrt{1-x-x^2+x^3+x^4}-x \text{$\#$1}\right)+8 \log (x) \text{$\#$1}-8 \log \left(1-x^2+\sqrt{1-x-x^2+x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}-3 \log (x) \text{$\#$1}^2+3 \log \left(1-x^2+\sqrt{1-x-x^2+x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^2}{-4+7 \text{$\#$1}+\text{$\#$1}^3}\&\right]","\frac{\sqrt{1-x-x^2+x^3+x^4}}{x}+\frac{\log (x)}{2}-\frac{1}{2} \log \left(2-x-2 x^2+2 \sqrt{1-x-x^2+x^3+x^4}\right)+\text{RootSum}\left[9+12 \text{$\#$1}+62 \text{$\#$1}^2-4 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{9 \log (x)-9 \log \left(2-x-2 x^2+2 \sqrt{1-x-x^2+x^3+x^4}+x \text{$\#$1}\right)-10 \log (x) \text{$\#$1}+10 \log \left(2-x-2 x^2+2 \sqrt{1-x-x^2+x^3+x^4}+x \text{$\#$1}\right) \text{$\#$1}-3 \log (x) \text{$\#$1}^2+3 \log \left(2-x-2 x^2+2 \sqrt{1-x-x^2+x^3+x^4}+x \text{$\#$1}\right) \text{$\#$1}^2}{3+31 \text{$\#$1}-3 \text{$\#$1}^2+\text{$\#$1}^3}\&\right]",1,"Sqrt[1 - x - x^2 + x^3 + x^4]/x + Log[x]/2 - Log[2 - x - 2*x^2 + 2*Sqrt[1 - x - x^2 + x^3 + x^4]]/2 - RootSum[5 - 16*#1 + 14*#1^2 + #1^4 & , (-Log[x] + Log[1 - x^2 + Sqrt[1 - x - x^2 + x^3 + x^4] - x*#1] + 8*Log[x]*#1 - 8*Log[1 - x^2 + Sqrt[1 - x - x^2 + x^3 + x^4] - x*#1]*#1 - 3*Log[x]*#1^2 + 3*Log[1 - x^2 + Sqrt[1 - x - x^2 + x^3 + x^4] - x*#1]*#1^2)/(-4 + 7*#1 + #1^3) & ]/2","A",1
2740,1,232,254,0.0013069,"\int \frac{\left(-1+x^2\right) \sqrt{1-x-x^2+x^3+x^4}}{x^2 \left(1+x^2\right)} \, dx","IntegrateAlgebraic[((-1 + x^2)*Sqrt[1 - x - x^2 + x^3 + x^4])/(x^2*(1 + x^2)),x]","\frac{\sqrt{1-x-x^2+x^3+x^4}}{x}+\frac{\log (x)}{2}-\frac{1}{2} \log \left(2-x-2 x^2+2 \sqrt{1-x-x^2+x^3+x^4}\right)-\frac{1}{2} \text{RootSum}\left[5-16 \text{$\#$1}+14 \text{$\#$1}^2+\text{$\#$1}^4\&,\frac{-\log (x)+\log \left(1-x^2+\sqrt{1-x-x^2+x^3+x^4}-x \text{$\#$1}\right)+8 \log (x) \text{$\#$1}-8 \log \left(1-x^2+\sqrt{1-x-x^2+x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}-3 \log (x) \text{$\#$1}^2+3 \log \left(1-x^2+\sqrt{1-x-x^2+x^3+x^4}-x \text{$\#$1}\right) \text{$\#$1}^2}{-4+7 \text{$\#$1}+\text{$\#$1}^3}\&\right]","\frac{\sqrt{1-x-x^2+x^3+x^4}}{x}+\frac{\log (x)}{2}-\frac{1}{2} \log \left(2-x-2 x^2+2 \sqrt{1-x-x^2+x^3+x^4}\right)+\text{RootSum}\left[9+12 \text{$\#$1}+62 \text{$\#$1}^2-4 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{9 \log (x)-9 \log \left(2-x-2 x^2+2 \sqrt{1-x-x^2+x^3+x^4}+x \text{$\#$1}\right)-10 \log (x) \text{$\#$1}+10 \log \left(2-x-2 x^2+2 \sqrt{1-x-x^2+x^3+x^4}+x \text{$\#$1}\right) \text{$\#$1}-3 \log (x) \text{$\#$1}^2+3 \log \left(2-x-2 x^2+2 \sqrt{1-x-x^2+x^3+x^4}+x \text{$\#$1}\right) \text{$\#$1}^2}{3+31 \text{$\#$1}-3 \text{$\#$1}^2+\text{$\#$1}^3}\&\right]",1,"Sqrt[1 - x - x^2 + x^3 + x^4]/x + Log[x]/2 - Log[2 - x - 2*x^2 + 2*Sqrt[1 - x - x^2 + x^3 + x^4]]/2 - RootSum[5 - 16*#1 + 14*#1^2 + #1^4 & , (-Log[x] + Log[1 - x^2 + Sqrt[1 - x - x^2 + x^3 + x^4] - x*#1] + 8*Log[x]*#1 - 8*Log[1 - x^2 + Sqrt[1 - x - x^2 + x^3 + x^4] - x*#1]*#1 - 3*Log[x]*#1^2 + 3*Log[1 - x^2 + Sqrt[1 - x - x^2 + x^3 + x^4] - x*#1]*#1^2)/(-4 + 7*#1 + #1^3) & ]/2","A",1
2741,1,254,254,0.6249321,"\int \frac{\left(-1+x^3\right)^{2/3} \left(4+x^3\right)}{x^6 \left(-2-x^3+x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(4 + x^3))/(x^6*(-2 - x^3 + x^6)),x]","\frac{\left(8-13 x^3\right) \left(-1+x^3\right)^{2/3}}{20 x^5}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2} \sqrt[3]{-1+x^3}}\right)}{2\ 2^{2/3} \sqrt{3}}+\frac{2^{2/3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-1+x^3}}\right)}{\sqrt{3}}+\frac{\log \left(-x+\sqrt[3]{2} \sqrt[3]{-1+x^3}\right)}{6\ 2^{2/3}}-\frac{1}{3} 2^{2/3} \log \left(-2 x+2^{2/3} \sqrt[3]{-1+x^3}\right)+\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-1+x^3}+\sqrt[3]{2} \left(-1+x^3\right)^{2/3}\right)}{3 \sqrt[3]{2}}-\frac{\log \left(x^2+\sqrt[3]{2} x \sqrt[3]{-1+x^3}+2^{2/3} \left(-1+x^3\right)^{2/3}\right)}{12\ 2^{2/3}}","\frac{\left(8-13 x^3\right) \left(-1+x^3\right)^{2/3}}{20 x^5}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2} \sqrt[3]{-1+x^3}}\right)}{2\ 2^{2/3} \sqrt{3}}+\frac{2^{2/3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-1+x^3}}\right)}{\sqrt{3}}+\frac{\log \left(-x+\sqrt[3]{2} \sqrt[3]{-1+x^3}\right)}{6\ 2^{2/3}}-\frac{1}{3} 2^{2/3} \log \left(-2 x+2^{2/3} \sqrt[3]{-1+x^3}\right)+\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-1+x^3}+\sqrt[3]{2} \left(-1+x^3\right)^{2/3}\right)}{3 \sqrt[3]{2}}-\frac{\log \left(x^2+\sqrt[3]{2} x \sqrt[3]{-1+x^3}+2^{2/3} \left(-1+x^3\right)^{2/3}\right)}{12\ 2^{2/3}}",1,"((8 - 13*x^3)*(-1 + x^3)^(2/3))/(20*x^5) - ArcTan[(Sqrt[3]*x)/(x + 2*2^(1/3)*(-1 + x^3)^(1/3))]/(2*2^(2/3)*Sqrt[3]) + (2^(2/3)*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-1 + x^3)^(1/3))])/Sqrt[3] + Log[-x + 2^(1/3)*(-1 + x^3)^(1/3)]/(6*2^(2/3)) - (2^(2/3)*Log[-2*x + 2^(2/3)*(-1 + x^3)^(1/3)])/3 + Log[2*x^2 + 2^(2/3)*x*(-1 + x^3)^(1/3) + 2^(1/3)*(-1 + x^3)^(2/3)]/(3*2^(1/3)) - Log[x^2 + 2^(1/3)*x*(-1 + x^3)^(1/3) + 2^(2/3)*(-1 + x^3)^(2/3)]/(12*2^(2/3))","A",1
2742,1,254,254,0.9020774,"\int \frac{\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{-b+a^2 x^2} \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}} \, dx","IntegrateAlgebraic[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)/(Sqrt[-b + a^2*x^2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4)),x]","-\frac{4 \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{a \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}-\frac{4 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[3]{c}}\right)}{\sqrt{3} a c^{2/3}}+\frac{4 \log \left(-\sqrt[3]{c}+\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{3 a c^{2/3}}-\frac{2 \log \left(c^{2/3}+\sqrt[3]{c} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}\right)}{3 a c^{2/3}}","-\frac{4 \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{a \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}-\frac{4 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[3]{c}}\right)}{\sqrt{3} a c^{2/3}}+\frac{4 \log \left(-\sqrt[3]{c}+\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{3 a c^{2/3}}-\frac{2 \log \left(c^{2/3}+\sqrt[3]{c} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}\right)}{3 a c^{2/3}}",1,"(-4*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3))/(a*(a*x + Sqrt[-b + a^2*x^2])^(1/4)) - (4*ArcTan[1/Sqrt[3] + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3))/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*a*c^(2/3)) + (4*Log[-c^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)])/(3*a*c^(2/3)) - (2*Log[c^(2/3) + c^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3)])/(3*a*c^(2/3))","A",1
2743,1,255,255,0.9463947,"\int \frac{\left(x+x^2\right) \sqrt[4]{x^3+x^4}}{-1+x+x^2} \, dx","IntegrateAlgebraic[((x + x^2)*(x^3 + x^4)^(1/4))/(-1 + x + x^2),x]","\frac{1}{8} (1+4 x) \sqrt[4]{x^3+x^4}-\frac{29}{16} \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)+\sqrt{\frac{1}{5} \left(2+2 \sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{x^3+x^4}}\right)+\sqrt{\frac{1}{5} \left(-2+2 \sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{x^3+x^4}}\right)+\frac{29}{16} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)-\sqrt{\frac{1}{5} \left(2+2 \sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{x^3+x^4}}\right)-\sqrt{\frac{1}{5} \left(-2+2 \sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{x^3+x^4}}\right)","\frac{1}{8} (1+4 x) \sqrt[4]{x^3+x^4}-\frac{29}{16} \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)+\sqrt{\frac{1}{5} \left(2+2 \sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{x^3+x^4}}\right)+\sqrt{\frac{1}{5} \left(-2+2 \sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{x^3+x^4}}\right)+\frac{29}{16} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^4}}\right)-\sqrt{\frac{1}{5} \left(2+2 \sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{x^3+x^4}}\right)-\sqrt{\frac{1}{5} \left(-2+2 \sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{x^3+x^4}}\right)",1,"((1 + 4*x)*(x^3 + x^4)^(1/4))/8 - (29*ArcTan[x/(x^3 + x^4)^(1/4)])/16 + Sqrt[(2 + 2*Sqrt[5])/5]*ArcTan[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(x^3 + x^4)^(1/4)] + Sqrt[(-2 + 2*Sqrt[5])/5]*ArcTan[(Sqrt[1/2 + Sqrt[5]/2]*x)/(x^3 + x^4)^(1/4)] + (29*ArcTanh[x/(x^3 + x^4)^(1/4)])/16 - Sqrt[(2 + 2*Sqrt[5])/5]*ArcTanh[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(x^3 + x^4)^(1/4)] - Sqrt[(-2 + 2*Sqrt[5])/5]*ArcTanh[(Sqrt[1/2 + Sqrt[5]/2]*x)/(x^3 + x^4)^(1/4)]","A",1
2744,1,255,255,0.4586419,"\int \frac{\left(b+x^4\right)^2}{\sqrt[4]{a b^4+4 a b^3 x^4+b^4 x^4+6 a b^2 x^8+4 b^3 x^8+4 a b x^{12}+6 b^2 x^{12}+a x^{16}+4 b x^{16}+x^{20}}} \, dx","IntegrateAlgebraic[(b + x^4)^2/(a*b^4 + 4*a*b^3*x^4 + b^4*x^4 + 6*a*b^2*x^8 + 4*b^3*x^8 + 4*a*b*x^12 + 6*b^2*x^12 + a*x^16 + 4*b*x^16 + x^20)^(1/4),x]","\frac{x \left(a b^4+\left(4 a b^3+b^4\right) x^4+\left(6 a b^2+4 b^3\right) x^8+\left(4 a b+6 b^2\right) x^{12}+(a+4 b) x^{16}+x^{20}\right)^{3/4}}{4 \left(b+x^4\right)^3}+\frac{1}{8} (a-4 b) \tan ^{-1}\left(\frac{\sqrt[4]{a b^4+\left(4 a b^3+b^4\right) x^4+\left(6 a b^2+4 b^3\right) x^8+\left(4 a b+6 b^2\right) x^{12}+(a+4 b) x^{16}+x^{20}}}{x \left(b+x^4\right)}\right)+\frac{1}{8} (-a+4 b) \tanh ^{-1}\left(\frac{\sqrt[4]{a b^4+\left(4 a b^3+b^4\right) x^4+\left(6 a b^2+4 b^3\right) x^8+\left(4 a b+6 b^2\right) x^{12}+(a+4 b) x^{16}+x^{20}}}{x \left(b+x^4\right)}\right)","\frac{x \left(a b^4+\left(4 a b^3+b^4\right) x^4+\left(6 a b^2+4 b^3\right) x^8+\left(4 a b+6 b^2\right) x^{12}+(a+4 b) x^{16}+x^{20}\right)^{3/4}}{4 \left(b+x^4\right)^3}+\frac{1}{8} (a-4 b) \tan ^{-1}\left(\frac{\sqrt[4]{a b^4+\left(4 a b^3+b^4\right) x^4+\left(6 a b^2+4 b^3\right) x^8+\left(4 a b+6 b^2\right) x^{12}+(a+4 b) x^{16}+x^{20}}}{x \left(b+x^4\right)}\right)+\frac{1}{8} (-a+4 b) \tanh ^{-1}\left(\frac{\sqrt[4]{a b^4+\left(4 a b^3+b^4\right) x^4+\left(6 a b^2+4 b^3\right) x^8+\left(4 a b+6 b^2\right) x^{12}+(a+4 b) x^{16}+x^{20}}}{x \left(b+x^4\right)}\right)",1,"(x*(a*b^4 + (4*a*b^3 + b^4)*x^4 + (6*a*b^2 + 4*b^3)*x^8 + (4*a*b + 6*b^2)*x^12 + (a + 4*b)*x^16 + x^20)^(3/4))/(4*(b + x^4)^3) + ((a - 4*b)*ArcTan[(a*b^4 + (4*a*b^3 + b^4)*x^4 + (6*a*b^2 + 4*b^3)*x^8 + (4*a*b + 6*b^2)*x^12 + (a + 4*b)*x^16 + x^20)^(1/4)/(x*(b + x^4))])/8 + ((-a + 4*b)*ArcTanh[(a*b^4 + (4*a*b^3 + b^4)*x^4 + (6*a*b^2 + 4*b^3)*x^8 + (4*a*b + 6*b^2)*x^12 + (a + 4*b)*x^16 + x^20)^(1/4)/(x*(b + x^4))])/8","A",1
2745,1,255,255,0.4375508,"\int \frac{\left(-q+2 p x^3\right) \sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left(b x^4+a \left(q+p x^3\right)^4\right)}{x^7} \, dx","IntegrateAlgebraic[((-q + 2*p*x^3)*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]*(b*x^4 + a*(q + p*x^3)^4))/x^7,x]","\frac{\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left(2 a q^5-a p q^4 x^2+10 a p q^4 x^3+6 b q x^4-3 a p^2 q^3 x^4-3 a p^2 q^3 x^5+20 a p^2 q^3 x^6+6 b p x^7-3 a p^3 q^2 x^7-3 a p^3 q^2 x^8+20 a p^3 q^2 x^9-a p^4 q x^{11}+10 a p^4 q x^{12}+2 a p^5 x^{15}\right)}{12 x^6}+\frac{1}{2} \left(2 b p q+a p^3 q^3\right) \log (x)+\frac{1}{2} \left(-2 b p q-a p^3 q^3\right) \log \left(q+p x^3+\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right)","\frac{\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left(2 a q^5-a p q^4 x^2+10 a p q^4 x^3+6 b q x^4-3 a p^2 q^3 x^4-3 a p^2 q^3 x^5+20 a p^2 q^3 x^6+6 b p x^7-3 a p^3 q^2 x^7-3 a p^3 q^2 x^8+20 a p^3 q^2 x^9-a p^4 q x^{11}+10 a p^4 q x^{12}+2 a p^5 x^{15}\right)}{12 x^6}+\frac{1}{2} \left(2 b p q+a p^3 q^3\right) \log (x)+\frac{1}{2} \left(-2 b p q-a p^3 q^3\right) \log \left(q+p x^3+\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right)",1,"(Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]*(2*a*q^5 - a*p*q^4*x^2 + 10*a*p*q^4*x^3 + 6*b*q*x^4 - 3*a*p^2*q^3*x^4 - 3*a*p^2*q^3*x^5 + 20*a*p^2*q^3*x^6 + 6*b*p*x^7 - 3*a*p^3*q^2*x^7 - 3*a*p^3*q^2*x^8 + 20*a*p^3*q^2*x^9 - a*p^4*q*x^11 + 10*a*p^4*q*x^12 + 2*a*p^5*x^15))/(12*x^6) + ((2*b*p*q + a*p^3*q^3)*Log[x])/2 + ((-2*b*p*q - a*p^3*q^3)*Log[q + p*x^3 + Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]])/2","A",1
2746,1,255,255,0.3370697,"\int \frac{1-\sqrt{x-\sqrt{1+x^2}}}{x^4-2 x^2 \sqrt{1+x^2}} \, dx","IntegrateAlgebraic[(1 - Sqrt[x - Sqrt[1 + x^2]])/(x^4 - 2*x^2*Sqrt[1 + x^2]),x]","-\frac{1}{\left(-1-x+\sqrt{1+x^2}\right) \left(1+\sqrt{x-\sqrt{1+x^2}}\right)}+\frac{1}{2} \tan ^{-1}\left(\sqrt{x-\sqrt{1+x^2}}\right)+\frac{1}{2} \tanh ^{-1}\left(\sqrt{x-\sqrt{1+x^2}}\right)-\frac{1}{2} \text{RootSum}\left[8-32 \text{$\#$1}+80 \text{$\#$1}^2-128 \text{$\#$1}^3+128 \text{$\#$1}^4-80 \text{$\#$1}^5+32 \text{$\#$1}^6-8 \text{$\#$1}^7+\text{$\#$1}^8\&,\frac{\log \left(-1+\sqrt{x-\sqrt{1+x^2}}+\text{$\#$1}\right) \text{$\#$1}-2 \log \left(-1+\sqrt{x-\sqrt{1+x^2}}+\text{$\#$1}\right) \text{$\#$1}^2+\log \left(-1+\sqrt{x-\sqrt{1+x^2}}+\text{$\#$1}\right) \text{$\#$1}^3}{4-16 \text{$\#$1}+32 \text{$\#$1}^2-32 \text{$\#$1}^3+18 \text{$\#$1}^4-6 \text{$\#$1}^5+\text{$\#$1}^6}\&\right]","-\frac{1}{\left(-1-x+\sqrt{1+x^2}\right) \left(1+\sqrt{x-\sqrt{1+x^2}}\right)}+\frac{1}{2} \tan ^{-1}\left(\sqrt{x-\sqrt{1+x^2}}\right)+\frac{1}{2} \tanh ^{-1}\left(\sqrt{x-\sqrt{1+x^2}}\right)-\frac{1}{2} \text{RootSum}\left[8-32 \text{$\#$1}+80 \text{$\#$1}^2-128 \text{$\#$1}^3+128 \text{$\#$1}^4-80 \text{$\#$1}^5+32 \text{$\#$1}^6-8 \text{$\#$1}^7+\text{$\#$1}^8\&,\frac{\log \left(-1+\sqrt{x-\sqrt{1+x^2}}+\text{$\#$1}\right) \text{$\#$1}-2 \log \left(-1+\sqrt{x-\sqrt{1+x^2}}+\text{$\#$1}\right) \text{$\#$1}^2+\log \left(-1+\sqrt{x-\sqrt{1+x^2}}+\text{$\#$1}\right) \text{$\#$1}^3}{4-16 \text{$\#$1}+32 \text{$\#$1}^2-32 \text{$\#$1}^3+18 \text{$\#$1}^4-6 \text{$\#$1}^5+\text{$\#$1}^6}\&\right]",1,"-(1/((-1 - x + Sqrt[1 + x^2])*(1 + Sqrt[x - Sqrt[1 + x^2]]))) + ArcTan[Sqrt[x - Sqrt[1 + x^2]]]/2 + ArcTanh[Sqrt[x - Sqrt[1 + x^2]]]/2 - RootSum[8 - 32*#1 + 80*#1^2 - 128*#1^3 + 128*#1^4 - 80*#1^5 + 32*#1^6 - 8*#1^7 + #1^8 & , (Log[-1 + Sqrt[x - Sqrt[1 + x^2]] + #1]*#1 - 2*Log[-1 + Sqrt[x - Sqrt[1 + x^2]] + #1]*#1^2 + Log[-1 + Sqrt[x - Sqrt[1 + x^2]] + #1]*#1^3)/(4 - 16*#1 + 32*#1^2 - 32*#1^3 + 18*#1^4 - 6*#1^5 + #1^6) & ]/2","A",1
2747,1,255,255,1.2623561,"\int \frac{1}{\sqrt{-b+a^2 x^2} \sqrt[6]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}} \, dx","IntegrateAlgebraic[1/(Sqrt[-b + a^2*x^2]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6)),x]","-\frac{4 \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[6]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[6]{c}}\right)}{a \sqrt[6]{c}}+\frac{4 \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[6]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[6]{c}}\right)}{a \sqrt[6]{c}}-\frac{8 \tanh ^{-1}\left(\frac{\sqrt[6]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[6]{c}}\right)}{a \sqrt[6]{c}}-\frac{4 \tanh ^{-1}\left(\frac{\sqrt[6]{c}+\frac{\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[6]{c}}}{\sqrt[6]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}\right)}{a \sqrt[6]{c}}","-\frac{4 \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[6]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[6]{c}}\right)}{a \sqrt[6]{c}}+\frac{4 \sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[6]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[6]{c}}\right)}{a \sqrt[6]{c}}-\frac{8 \tanh ^{-1}\left(\frac{\sqrt[6]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[6]{c}}\right)}{a \sqrt[6]{c}}-\frac{4 \tanh ^{-1}\left(\frac{\sqrt[6]{c}+\frac{\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[6]{c}}}{\sqrt[6]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}\right)}{a \sqrt[6]{c}}",1,"(-4*Sqrt[3]*ArcTan[1/Sqrt[3] - (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6))/(Sqrt[3]*c^(1/6))])/(a*c^(1/6)) + (4*Sqrt[3]*ArcTan[1/Sqrt[3] + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6))/(Sqrt[3]*c^(1/6))])/(a*c^(1/6)) - (8*ArcTanh[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6)/c^(1/6)])/(a*c^(1/6)) - (4*ArcTanh[(c^(1/6) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)/c^(1/6))/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6)])/(a*c^(1/6))","A",1
2748,1,256,256,4.7944316,"\int \frac{\left(1-2 k^2\right) x+k^2 x^3}{\sqrt[3]{\left(1-x^2\right) \left(1-k^2 x^2\right)} \left(-1+d+\left(1-2 d k^2\right) x^2+d k^4 x^4\right)} \, dx","IntegrateAlgebraic[((1 - 2*k^2)*x + k^2*x^3)/(((1 - x^2)*(1 - k^2*x^2))^(1/3)*(-1 + d + (1 - 2*d*k^2)*x^2 + d*k^4*x^4)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{2 \sqrt[3]{d}-2 \sqrt[3]{d} k^2 x^2+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{2 \sqrt[3]{d}}+\frac{\log \left(-\sqrt[3]{d}+\sqrt[3]{d} k^2 x^2+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{2 \sqrt[3]{d}}-\frac{\log \left(d^{2/3}-2 d^{2/3} k^2 x^2+d^{2/3} k^4 x^4+\left(\sqrt[3]{d}-\sqrt[3]{d} k^2 x^2\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{4 \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}{2 \sqrt[3]{d}-2 \sqrt[3]{d} k^2 x^2+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{2 \sqrt[3]{d}}+\frac{\log \left(-\sqrt[3]{d}+\sqrt[3]{d} k^2 x^2+\sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}\right)}{2 \sqrt[3]{d}}-\frac{\log \left(d^{2/3}-2 d^{2/3} k^2 x^2+d^{2/3} k^4 x^4+\left(\sqrt[3]{d}-\sqrt[3]{d} k^2 x^2\right) \sqrt[3]{1+\left(-1-k^2\right) x^2+k^2 x^4}+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{4 \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))/(2*d^(1/3) - 2*d^(1/3)*k^2*x^2 + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3))])/(2*d^(1/3)) + Log[-d^(1/3) + d^(1/3)*k^2*x^2 + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3)]/(2*d^(1/3)) - Log[d^(2/3) - 2*d^(2/3)*k^2*x^2 + d^(2/3)*k^4*x^4 + (d^(1/3) - d^(1/3)*k^2*x^2)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3) + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3)]/(4*d^(1/3))","A",1
2749,1,256,256,4.7045393,"\int \frac{\left(2-k^2\right) x-2 x^3+k^2 x^5}{\left(\left(1-x^2\right) \left(1-k^2 x^2\right)\right)^{2/3} \left(1-d+\left(-2+d k^2\right) x^2+x^4\right)} \, dx","IntegrateAlgebraic[((2 - k^2)*x - 2*x^3 + k^2*x^5)/(((1 - x^2)*(1 - k^2*x^2))^(2/3)*(1 - d + (-2 + d*k^2)*x^2 + x^4)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}}{2 \sqrt[3]{d}-2 \sqrt[3]{d} k^2 x^2+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}}\right)}{2 \sqrt[3]{d}}-\frac{\log \left(-\sqrt[3]{d}+\sqrt[3]{d} k^2 x^2+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}+\frac{\log \left(d^{2/3}-2 d^{2/3} k^2 x^2+d^{2/3} k^4 x^4+\left(\sqrt[3]{d}-\sqrt[3]{d} k^2 x^2\right) \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{4/3}\right)}{4 \sqrt[3]{d}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}}{2 \sqrt[3]{d}-2 \sqrt[3]{d} k^2 x^2+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}}\right)}{2 \sqrt[3]{d}}-\frac{\log \left(-\sqrt[3]{d}+\sqrt[3]{d} k^2 x^2+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}+\frac{\log \left(d^{2/3}-2 d^{2/3} k^2 x^2+d^{2/3} k^4 x^4+\left(\sqrt[3]{d}-\sqrt[3]{d} k^2 x^2\right) \left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{2/3}+\left(1+\left(-1-k^2\right) x^2+k^2 x^4\right)^{4/3}\right)}{4 \sqrt[3]{d}}",1,"-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3))/(2*d^(1/3) - 2*d^(1/3)*k^2*x^2 + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3))])/d^(1/3) - Log[-d^(1/3) + d^(1/3)*k^2*x^2 + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3)]/(2*d^(1/3)) + Log[d^(2/3) - 2*d^(2/3)*k^2*x^2 + d^(2/3)*k^4*x^4 + (d^(1/3) - d^(1/3)*k^2*x^2)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(2/3) + (1 + (-1 - k^2)*x^2 + k^2*x^4)^(4/3)]/(4*d^(1/3))","A",1
2750,1,257,257,0.8706167,"\int \frac{\left(-1+x^4\right) \sqrt[4]{-x^2+x^4}}{-1-x^2+x^4} \, dx","IntegrateAlgebraic[((-1 + x^4)*(-x^2 + x^4)^(1/4))/(-1 - x^2 + x^4),x]","\frac{1}{2} x \sqrt[4]{-x^2+x^4}-\frac{3}{4} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)+\sqrt{\frac{1}{10} \left(1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^2+x^4}}\right)+\sqrt{\frac{1}{10} \left(-1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^2+x^4}}\right)+\frac{3}{4} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)-\sqrt{\frac{1}{10} \left(1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^2+x^4}}\right)-\sqrt{\frac{1}{10} \left(-1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^2+x^4}}\right)","\frac{1}{2} x \sqrt[4]{-x^2+x^4}-\frac{3}{4} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)+\sqrt{\frac{1}{10} \left(1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^2+x^4}}\right)+\sqrt{\frac{1}{10} \left(-1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^2+x^4}}\right)+\frac{3}{4} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)-\sqrt{\frac{1}{10} \left(1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^2+x^4}}\right)-\sqrt{\frac{1}{10} \left(-1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^2+x^4}}\right)",1,"(x*(-x^2 + x^4)^(1/4))/2 - (3*ArcTan[x/(-x^2 + x^4)^(1/4)])/4 + Sqrt[(1 + Sqrt[5])/10]*ArcTan[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(-x^2 + x^4)^(1/4)] + Sqrt[(-1 + Sqrt[5])/10]*ArcTan[(Sqrt[1/2 + Sqrt[5]/2]*x)/(-x^2 + x^4)^(1/4)] + (3*ArcTanh[x/(-x^2 + x^4)^(1/4)])/4 - Sqrt[(1 + Sqrt[5])/10]*ArcTanh[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(-x^2 + x^4)^(1/4)] - Sqrt[(-1 + Sqrt[5])/10]*ArcTanh[(Sqrt[1/2 + Sqrt[5]/2]*x)/(-x^2 + x^4)^(1/4)]","A",1
2751,1,257,257,3.0106628,"\int \frac{1+(3-2 k) x-(4+k) x^2+3 k x^3}{\sqrt[3]{(1-x) x (1-k x)} \left(-b+(1+5 b) x-(10 b+k) x^2+10 b x^3-5 b x^4+b x^5\right)} \, dx","IntegrateAlgebraic[(1 + (3 - 2*k)*x - (4 + k)*x^2 + 3*k*x^3)/(((1 - x)*x*(1 - k*x))^(1/3)*(-b + (1 + 5*b)*x - (10*b + k)*x^2 + 10*b*x^3 - 5*b*x^4 + b*x^5)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{x+(-1-k) x^2+k x^3}}{2 \sqrt[3]{b}-4 \sqrt[3]{b} x+2 \sqrt[3]{b} x^2+\sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{\sqrt[3]{b}}+\frac{\log \left(-\sqrt[3]{b}+2 \sqrt[3]{b} x-\sqrt[3]{b} x^2+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{\sqrt[3]{b}}-\frac{\log \left(b^{2/3}-4 b^{2/3} x+6 b^{2/3} x^2-4 b^{2/3} x^3+b^{2/3} x^4+\left(\sqrt[3]{b}-2 \sqrt[3]{b} x+\sqrt[3]{b} x^2\right) \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 \sqrt[3]{b}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{x+(-1-k) x^2+k x^3}}{2 \sqrt[3]{b}-4 \sqrt[3]{b} x+2 \sqrt[3]{b} x^2+\sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{\sqrt[3]{b}}+\frac{\log \left(-\sqrt[3]{b}+2 \sqrt[3]{b} x-\sqrt[3]{b} x^2+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{\sqrt[3]{b}}-\frac{\log \left(b^{2/3}-4 b^{2/3} x+6 b^{2/3} x^2-4 b^{2/3} x^3+b^{2/3} x^4+\left(\sqrt[3]{b}-2 \sqrt[3]{b} x+\sqrt[3]{b} x^2\right) \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 \sqrt[3]{b}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(x + (-1 - k)*x^2 + k*x^3)^(1/3))/(2*b^(1/3) - 4*b^(1/3)*x + 2*b^(1/3)*x^2 + (x + (-1 - k)*x^2 + k*x^3)^(1/3))])/b^(1/3) + Log[-b^(1/3) + 2*b^(1/3)*x - b^(1/3)*x^2 + (x + (-1 - k)*x^2 + k*x^3)^(1/3)]/b^(1/3) - Log[b^(2/3) - 4*b^(2/3)*x + 6*b^(2/3)*x^2 - 4*b^(2/3)*x^3 + b^(2/3)*x^4 + (b^(1/3) - 2*b^(1/3)*x + b^(1/3)*x^2)*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + (x + (-1 - k)*x^2 + k*x^3)^(2/3)]/(2*b^(1/3))","A",1
2752,1,257,257,0.6137326,"\int \frac{x^3}{\sqrt[3]{-x^2+x^3} \left(-1+x^6\right)} \, dx","IntegrateAlgebraic[x^3/((-x^2 + x^3)^(1/3)*(-1 + x^6)),x]","-\frac{\left(-x^2+x^3\right)^{2/3}}{2 (-1+x) x}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-x^2+x^3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}-\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{-x^2+x^3}\right)}{6 \sqrt[3]{2}}+\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-x^2+x^3}+\sqrt[3]{2} \left(-x^2+x^3\right)^{2/3}\right)}{12 \sqrt[3]{2}}+\frac{1}{6} \text{RootSum}\left[3-3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]-\frac{1}{6} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","-\frac{\left(-x^2+x^3\right)^{2/3}}{2 (-1+x) x}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{-x^2+x^3}}\right)}{2 \sqrt[3]{2} \sqrt{3}}-\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{-x^2+x^3}\right)}{6 \sqrt[3]{2}}+\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{-x^2+x^3}+\sqrt[3]{2} \left(-x^2+x^3\right)^{2/3}\right)}{12 \sqrt[3]{2}}+\frac{1}{6} \text{RootSum}\left[3-3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]-\frac{1}{6} \text{RootSum}\left[1-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-\log (x)+\log \left(\sqrt[3]{-x^2+x^3}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]",1,"-1/2*(-x^2 + x^3)^(2/3)/((-1 + x)*x) + ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-x^2 + x^3)^(1/3))]/(2*2^(1/3)*Sqrt[3]) - Log[-2*x + 2^(2/3)*(-x^2 + x^3)^(1/3)]/(6*2^(1/3)) + Log[2*x^2 + 2^(2/3)*x*(-x^2 + x^3)^(1/3) + 2^(1/3)*(-x^2 + x^3)^(2/3)]/(12*2^(1/3)) + RootSum[3 - 3*#1^3 + #1^6 & , (-Log[x] + Log[(-x^2 + x^3)^(1/3) - x*#1])/#1 & ]/6 - RootSum[1 - #1^3 + #1^6 & , (-Log[x] + Log[(-x^2 + x^3)^(1/3) - x*#1])/#1 & ]/6","A",1
2753,1,257,257,0.2881852,"\int \frac{\sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{1-x^2} \, dx","IntegrateAlgebraic[(Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^2),x]","-\frac{1}{2} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}-3 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]-\frac{1}{2} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^3+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^5}{-2+4 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]","-\frac{1}{2} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}-3 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]-\frac{1}{2} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^3+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^5}{-2+4 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]",1,"-1/2*RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1 - 3*#1^3 + #1^5) & ] - RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^3 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^5)/(-2 + 4*#1^2 - 3*#1^4 + #1^6) & ]/2","A",1
2754,1,257,257,0.0021287,"\int \frac{\sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{1-x^2} \, dx","IntegrateAlgebraic[(Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^2),x]","-\frac{1}{2} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}-3 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]-\frac{1}{2} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^3+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^5}{-2+4 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]","-\frac{1}{2} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}-3 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]-\frac{1}{2} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^3+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^5}{-2+4 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]",1,"-1/2*RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1 - 3*#1^3 + #1^5) & ] - RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^3 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^5)/(-2 + 4*#1^2 - 3*#1^4 + #1^6) & ]/2","A",1
2755,1,257,257,0.5491955,"\int \frac{1}{\sqrt{-b+a^2 x^2} \sqrt[4]{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}} \, dx","IntegrateAlgebraic[1/(Sqrt[-b + a^2*x^2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)),x]","-\frac{4 \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}}{a c \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}-\frac{4 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[3]{c}}\right)}{\sqrt{3} a c^{4/3}}-\frac{4 \log \left(-\sqrt[3]{c}+\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{3 a c^{4/3}}+\frac{2 \log \left(c^{2/3}+\sqrt[3]{c} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}\right)}{3 a c^{4/3}}","-\frac{4 \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}}{a c \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}-\frac{4 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[3]{c}}\right)}{\sqrt{3} a c^{4/3}}-\frac{4 \log \left(-\sqrt[3]{c}+\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{3 a c^{4/3}}+\frac{2 \log \left(c^{2/3}+\sqrt[3]{c} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}\right)}{3 a c^{4/3}}",1,"(-4*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3))/(a*c*(a*x + Sqrt[-b + a^2*x^2])^(1/4)) - (4*ArcTan[1/Sqrt[3] + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3))/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*a*c^(4/3)) - (4*Log[-c^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)])/(3*a*c^(4/3)) + (2*Log[c^(2/3) + c^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3)])/(3*a*c^(4/3))","A",1
2756,1,258,258,3.425017,"\int \frac{(-b+x) \left(a b-2 b x+x^2\right)}{\left(x (-a+x) (-b+x)^2\right)^{2/3} \left(b-(1+a d) x+d x^2\right)} \, dx","IntegrateAlgebraic[((-b + x)*(a*b - 2*b*x + x^2))/((x*(-a + x)*(-b + x)^2)^(2/3)*(b - (1 + a*d)*x + d*x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} b-\sqrt{3} x}{b-x-2 \sqrt[3]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)}{\sqrt[3]{d}}+\frac{\log \left(b-x+\sqrt[3]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}\right)}{\sqrt[3]{d}}-\frac{\log \left(b^2-2 b x+x^2+\left(-b \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}+d^{2/3} \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} b-\sqrt{3} x}{b-x-2 \sqrt[3]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)}{\sqrt[3]{d}}+\frac{\log \left(b-x+\sqrt[3]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}\right)}{\sqrt[3]{d}}-\frac{\log \left(b^2-2 b x+x^2+\left(-b \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}+d^{2/3} \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*b - Sqrt[3]*x)/(b - x - 2*d^(1/3)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3))])/d^(1/3) + Log[b - x + d^(1/3)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3)]/d^(1/3) - Log[b^2 - 2*b*x + x^2 + (-(b*d^(1/3)) + d^(1/3)*x)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3) + d^(2/3)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(2/3)]/(2*d^(1/3))","A",1
2757,1,258,258,5.2670432,"\int \frac{(a-5 b+4 x) \left(-a^3+3 a^2 x-3 a x^2+x^3\right)}{((-a+x) (-b+x))^{2/3} \left(-a^5+b d-\left(-5 a^4+d\right) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right)} \, dx","IntegrateAlgebraic[((a - 5*b + 4*x)*(-a^3 + 3*a^2*x - 3*a*x^2 + x^3))/(((-a + x)*(-b + x))^(2/3)*(-a^5 + b*d - (-5*a^4 + d)*x - 10*a^3*x^2 + 10*a^2*x^3 - 5*a*x^4 + x^5)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a^2-2 \sqrt{3} a x+\sqrt{3} x^2}{a^2-2 a x+x^2+2 \sqrt[3]{d} \sqrt[3]{a b+(-a-b) x+x^2}}\right)}{\sqrt[3]{d}}+\frac{\log \left(a^3-2 a^2 x+a x^2-a \sqrt[3]{d} \sqrt[3]{a b+(-a-b) x+x^2}\right)}{\sqrt[3]{d}}-\frac{\log \left(a^6-4 a^5 x+6 a^4 x^2-4 a^3 x^3+a^2 x^4+a^2 d^{2/3} \left(a b+(-a-b) x+x^2\right)^{2/3}+\sqrt[3]{a b+(-a-b) x+x^2} \left(a^4 \sqrt[3]{d}-2 a^3 \sqrt[3]{d} x+a^2 \sqrt[3]{d} x^2\right)\right)}{2 \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a^2-2 \sqrt{3} a x+\sqrt{3} x^2}{a^2-2 a x+x^2+2 \sqrt[3]{d} \sqrt[3]{a b+(-a-b) x+x^2}}\right)}{\sqrt[3]{d}}+\frac{\log \left(a^3-2 a^2 x+a x^2-a \sqrt[3]{d} \sqrt[3]{a b+(-a-b) x+x^2}\right)}{\sqrt[3]{d}}-\frac{\log \left(a^6-4 a^5 x+6 a^4 x^2-4 a^3 x^3+a^2 x^4+a^2 d^{2/3} \left(a b+(-a-b) x+x^2\right)^{2/3}+\sqrt[3]{a b+(-a-b) x+x^2} \left(a^4 \sqrt[3]{d}-2 a^3 \sqrt[3]{d} x+a^2 \sqrt[3]{d} x^2\right)\right)}{2 \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*a^2 - 2*Sqrt[3]*a*x + Sqrt[3]*x^2)/(a^2 - 2*a*x + x^2 + 2*d^(1/3)*(a*b + (-a - b)*x + x^2)^(1/3))])/d^(1/3) + Log[a^3 - 2*a^2*x + a*x^2 - a*d^(1/3)*(a*b + (-a - b)*x + x^2)^(1/3)]/d^(1/3) - Log[a^6 - 4*a^5*x + 6*a^4*x^2 - 4*a^3*x^3 + a^2*x^4 + a^2*d^(2/3)*(a*b + (-a - b)*x + x^2)^(2/3) + (a*b + (-a - b)*x + x^2)^(1/3)*(a^4*d^(1/3) - 2*a^3*d^(1/3)*x + a^2*d^(1/3)*x^2)]/(2*d^(1/3))","A",1
2758,1,297,258,0.1389432,"\int \frac{1-x^3 \sqrt[3]{5+4 x}-x^3 (5+4 x)^{2/3}}{1-x \sqrt{5+4 x}} \, dx","IntegrateAlgebraic[(1 - x^3*(5 + 4*x)^(1/3) - x^3*(5 + 4*x)^(2/3))/(1 - x*Sqrt[5 + 4*x]),x]","\frac{3 \left(51731680 \sqrt[6]{5+4 x}-32332300 \sqrt[3]{5+4 x}-16166150 (5+4 x)^{2/3}+16166150 (5+4 x)^{5/6}+11547250 (5+4 x)^{7/6}+1616615 (5+4 x)^{4/3}+1293292 (5+4 x)^{5/3}-2939300 (5+4 x)^{11/6}-2487100 (5+4 x)^{13/6}+190190 (5+4 x)^{17/6}+170170 (5+4 x)^{19/6}\right)}{103463360}-3 \log \left(1+\sqrt[6]{5+4 x}\right)+\frac{1}{2} \text{RootSum}\left[-4-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-4 \log \left(\sqrt[6]{5+4 x}-\text{$\#$1}\right)+3 \log \left(\sqrt[6]{5+4 x}-\text{$\#$1}\right) \text{$\#$1}-8 \log \left(\sqrt[6]{5+4 x}-\text{$\#$1}\right) \text{$\#$1}^2+2 \log \left(\sqrt[6]{5+4 x}-\text{$\#$1}\right) \text{$\#$1}^3-\log \left(\sqrt[6]{5+4 x}-\text{$\#$1}\right) \text{$\#$1}^4+2 \log \left(\sqrt[6]{5+4 x}-\text{$\#$1}\right) \text{$\#$1}^5}{-\text{$\#$1}^2+2 \text{$\#$1}^5}\&\right]","\frac{3}{64} (-15+4 x) \sqrt[3]{5+4 x}+\frac{3}{160} (5+4 x)^{2/3} (-15+8 x)+\frac{3}{748} (5+4 x)^{5/6} \left(45-30 x+22 x^2\right)+\frac{3 \sqrt[6]{5+4 x} \left(4583-150 x+70 x^2+728 x^3\right)}{6916}-3 \log \left(1+\sqrt[6]{5+4 x}\right)+\frac{1}{2} \text{RootSum}\left[-4-\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-4 \log \left(\sqrt[6]{5+4 x}-\text{$\#$1}\right)+3 \log \left(\sqrt[6]{5+4 x}-\text{$\#$1}\right) \text{$\#$1}-8 \log \left(\sqrt[6]{5+4 x}-\text{$\#$1}\right) \text{$\#$1}^2+2 \log \left(\sqrt[6]{5+4 x}-\text{$\#$1}\right) \text{$\#$1}^3-\log \left(\sqrt[6]{5+4 x}-\text{$\#$1}\right) \text{$\#$1}^4+2 \log \left(\sqrt[6]{5+4 x}-\text{$\#$1}\right) \text{$\#$1}^5}{-\text{$\#$1}^2+2 \text{$\#$1}^5}\&\right]",1,"(3*(51731680*(5 + 4*x)^(1/6) - 32332300*(5 + 4*x)^(1/3) - 16166150*(5 + 4*x)^(2/3) + 16166150*(5 + 4*x)^(5/6) + 11547250*(5 + 4*x)^(7/6) + 1616615*(5 + 4*x)^(4/3) + 1293292*(5 + 4*x)^(5/3) - 2939300*(5 + 4*x)^(11/6) - 2487100*(5 + 4*x)^(13/6) + 190190*(5 + 4*x)^(17/6) + 170170*(5 + 4*x)^(19/6)))/103463360 - 3*Log[1 + (5 + 4*x)^(1/6)] + RootSum[-4 - #1^3 + #1^6 & , (-4*Log[(5 + 4*x)^(1/6) - #1] + 3*Log[(5 + 4*x)^(1/6) - #1]*#1 - 8*Log[(5 + 4*x)^(1/6) - #1]*#1^2 + 2*Log[(5 + 4*x)^(1/6) - #1]*#1^3 - Log[(5 + 4*x)^(1/6) - #1]*#1^4 + 2*Log[(5 + 4*x)^(1/6) - #1]*#1^5)/(-#1^2 + 2*#1^5) & ]/2","A",1
2759,1,259,259,1.223759,"\int \frac{x^3 \left(x^3 c_3-2 c_4\right) \sqrt{\frac{x^2 c_0+x^3 c_3+c_4}{x^2 c_1+x^3 c_3+c_4}}}{\left(x^2+2 x^3 c_3+2 c_4\right) \left(-x^4+x^6 c_3{}^2+2 x^3 c_3 c_4+c_4{}^2\right)} \, dx","IntegrateAlgebraic[(x^3*(x^3*C[3] - 2*C[4])*Sqrt[(x^2*C[0] + x^3*C[3] + C[4])/(x^2*C[1] + x^3*C[3] + C[4])])/((x^2 + 2*x^3*C[3] + 2*C[4])*(-x^4 + x^6*C[3]^2 + 2*x^3*C[3]*C[4] + C[4]^2)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{1-c_0} \sqrt{-1+c_1} \sqrt{\frac{x^2 c_0+x^3 c_3+c_4}{x^2 c_1+x^3 c_3+c_4}}}{-1+c_0}\right) \sqrt{1-c_0}}{\sqrt{-1+c_1}}+\frac{\tan ^{-1}\left(\frac{\sqrt{-1-c_0} \sqrt{1+c_1} \sqrt{\frac{x^2 c_0+x^3 c_3+c_4}{x^2 c_1+x^3 c_3+c_4}}}{1+c_0}\right) \sqrt{-1-c_0}}{3 \sqrt{1+c_1}}-\frac{4 \tan ^{-1}\left(\frac{\sqrt{1-2 c_0} \sqrt{-1+2 c_1} \sqrt{\frac{x^2 c_0+x^3 c_3+c_4}{x^2 c_1+x^3 c_3+c_4}}}{-1+2 c_0}\right) \sqrt{1-2 c_0}}{3 \sqrt{-1+2 c_1}}","\frac{\tan ^{-1}\left(\frac{\sqrt{1-c_0} \sqrt{-1+c_1} \sqrt{\frac{x^2 c_0+x^3 c_3+c_4}{x^2 c_1+x^3 c_3+c_4}}}{-1+c_0}\right) \sqrt{1-c_0}}{\sqrt{-1+c_1}}+\frac{\tan ^{-1}\left(\frac{\sqrt{-1-c_0} \sqrt{1+c_1} \sqrt{\frac{x^2 c_0+x^3 c_3+c_4}{x^2 c_1+x^3 c_3+c_4}}}{1+c_0}\right) \sqrt{-1-c_0}}{3 \sqrt{1+c_1}}-\frac{4 \tan ^{-1}\left(\frac{\sqrt{1-2 c_0} \sqrt{-1+2 c_1} \sqrt{\frac{x^2 c_0+x^3 c_3+c_4}{x^2 c_1+x^3 c_3+c_4}}}{-1+2 c_0}\right) \sqrt{1-2 c_0}}{3 \sqrt{-1+2 c_1}}",1,"(ArcTan[(Sqrt[1 - C[0]]*Sqrt[-1 + C[1]]*Sqrt[(x^2*C[0] + x^3*C[3] + C[4])/(x^2*C[1] + x^3*C[3] + C[4])])/(-1 + C[0])]*Sqrt[1 - C[0]])/Sqrt[-1 + C[1]] + (ArcTan[(Sqrt[-1 - C[0]]*Sqrt[1 + C[1]]*Sqrt[(x^2*C[0] + x^3*C[3] + C[4])/(x^2*C[1] + x^3*C[3] + C[4])])/(1 + C[0])]*Sqrt[-1 - C[0]])/(3*Sqrt[1 + C[1]]) - (4*ArcTan[(Sqrt[1 - 2*C[0]]*Sqrt[-1 + 2*C[1]]*Sqrt[(x^2*C[0] + x^3*C[3] + C[4])/(x^2*C[1] + x^3*C[3] + C[4])])/(-1 + 2*C[0])]*Sqrt[1 - 2*C[0]])/(3*Sqrt[-1 + 2*C[1]])","A",1
2760,1,259,259,1.943748,"\int \frac{x^5 \left(2 x^5 c_3-3 c_4\right) \sqrt{\frac{x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}}}{\left(x^3+2 x^5 c_3+2 c_4\right) \left(-x^6+x^{10} c_3{}^2+2 x^5 c_3 c_4+c_4{}^2\right)} \, dx","IntegrateAlgebraic[(x^5*(2*x^5*C[3] - 3*C[4])*Sqrt[(x^3*C[0] + x^5*C[3] + C[4])/(x^3*C[1] + x^5*C[3] + C[4])])/((x^3 + 2*x^5*C[3] + 2*C[4])*(-x^6 + x^10*C[3]^2 + 2*x^5*C[3]*C[4] + C[4]^2)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{1-c_0} \sqrt{-1+c_1} \sqrt{\frac{x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}}}{-1+c_0}\right) \sqrt{1-c_0}}{\sqrt{-1+c_1}}+\frac{\tan ^{-1}\left(\frac{\sqrt{-1-c_0} \sqrt{1+c_1} \sqrt{\frac{x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}}}{1+c_0}\right) \sqrt{-1-c_0}}{3 \sqrt{1+c_1}}-\frac{4 \tan ^{-1}\left(\frac{\sqrt{1-2 c_0} \sqrt{-1+2 c_1} \sqrt{\frac{x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}}}{-1+2 c_0}\right) \sqrt{1-2 c_0}}{3 \sqrt{-1+2 c_1}}","\frac{\tan ^{-1}\left(\frac{\sqrt{1-c_0} \sqrt{-1+c_1} \sqrt{\frac{x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}}}{-1+c_0}\right) \sqrt{1-c_0}}{\sqrt{-1+c_1}}+\frac{\tan ^{-1}\left(\frac{\sqrt{-1-c_0} \sqrt{1+c_1} \sqrt{\frac{x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}}}{1+c_0}\right) \sqrt{-1-c_0}}{3 \sqrt{1+c_1}}-\frac{4 \tan ^{-1}\left(\frac{\sqrt{1-2 c_0} \sqrt{-1+2 c_1} \sqrt{\frac{x^3 c_0+x^5 c_3+c_4}{x^3 c_1+x^5 c_3+c_4}}}{-1+2 c_0}\right) \sqrt{1-2 c_0}}{3 \sqrt{-1+2 c_1}}",1,"(ArcTan[(Sqrt[1 - C[0]]*Sqrt[-1 + C[1]]*Sqrt[(x^3*C[0] + x^5*C[3] + C[4])/(x^3*C[1] + x^5*C[3] + C[4])])/(-1 + C[0])]*Sqrt[1 - C[0]])/Sqrt[-1 + C[1]] + (ArcTan[(Sqrt[-1 - C[0]]*Sqrt[1 + C[1]]*Sqrt[(x^3*C[0] + x^5*C[3] + C[4])/(x^3*C[1] + x^5*C[3] + C[4])])/(1 + C[0])]*Sqrt[-1 - C[0]])/(3*Sqrt[1 + C[1]]) - (4*ArcTan[(Sqrt[1 - 2*C[0]]*Sqrt[-1 + 2*C[1]]*Sqrt[(x^3*C[0] + x^5*C[3] + C[4])/(x^3*C[1] + x^5*C[3] + C[4])])/(-1 + 2*C[0])]*Sqrt[1 - 2*C[0]])/(3*Sqrt[-1 + 2*C[1]])","A",1
2761,1,274,260,0.3992205,"\int \frac{\left(-1+x^2\right)^2 \left(x+x^3\right)}{\sqrt{1+x^4} \left(1-2 x^2+4 x^4-2 x^6+x^8\right)} \, dx","IntegrateAlgebraic[((-1 + x^2)^2*(x + x^3))/(Sqrt[1 + x^4]*(1 - 2*x^2 + 4*x^4 - 2*x^6 + x^8)),x]","\frac{1}{4} \text{RootSum}\left[1-4 \text{$\#$1}+12 \text{$\#$1}^2-4 \text{$\#$1}^3-10 \text{$\#$1}^4+4 \text{$\#$1}^5+12 \text{$\#$1}^6+4 \text{$\#$1}^7+\text{$\#$1}^8\&,\frac{-\log \left(-x^2+\sqrt{1+x^4}-\text{$\#$1}\right)+2 \log \left(-x^2+\sqrt{1+x^4}-\text{$\#$1}\right) \text{$\#$1}+7 \log \left(-x^2+\sqrt{1+x^4}-\text{$\#$1}\right) \text{$\#$1}^2-12 \log \left(-x^2+\sqrt{1+x^4}-\text{$\#$1}\right) \text{$\#$1}^3-7 \log \left(-x^2+\sqrt{1+x^4}-\text{$\#$1}\right) \text{$\#$1}^4+2 \log \left(-x^2+\sqrt{1+x^4}-\text{$\#$1}\right) \text{$\#$1}^5+\log \left(-x^2+\sqrt{1+x^4}-\text{$\#$1}\right) \text{$\#$1}^6}{-1+6 \text{$\#$1}-3 \text{$\#$1}^2-10 \text{$\#$1}^3+5 \text{$\#$1}^4+18 \text{$\#$1}^5+7 \text{$\#$1}^6+2 \text{$\#$1}^7}\&\right]","\frac{1}{4} \text{RootSum}\left[1+4 \text{$\#$1}+12 \text{$\#$1}^2+4 \text{$\#$1}^3-10 \text{$\#$1}^4-4 \text{$\#$1}^5+12 \text{$\#$1}^6-4 \text{$\#$1}^7+\text{$\#$1}^8\&,\frac{-\log \left(x^2+\sqrt{1+x^4}-\text{$\#$1}\right)-2 \log \left(x^2+\sqrt{1+x^4}-\text{$\#$1}\right) \text{$\#$1}+7 \log \left(x^2+\sqrt{1+x^4}-\text{$\#$1}\right) \text{$\#$1}^2+12 \log \left(x^2+\sqrt{1+x^4}-\text{$\#$1}\right) \text{$\#$1}^3-7 \log \left(x^2+\sqrt{1+x^4}-\text{$\#$1}\right) \text{$\#$1}^4-2 \log \left(x^2+\sqrt{1+x^4}-\text{$\#$1}\right) \text{$\#$1}^5+\log \left(x^2+\sqrt{1+x^4}-\text{$\#$1}\right) \text{$\#$1}^6}{1+6 \text{$\#$1}+3 \text{$\#$1}^2-10 \text{$\#$1}^3-5 \text{$\#$1}^4+18 \text{$\#$1}^5-7 \text{$\#$1}^6+2 \text{$\#$1}^7}\&\right]",1,"RootSum[1 - 4*#1 + 12*#1^2 - 4*#1^3 - 10*#1^4 + 4*#1^5 + 12*#1^6 + 4*#1^7 + #1^8 & , (-Log[-x^2 + Sqrt[1 + x^4] - #1] + 2*Log[-x^2 + Sqrt[1 + x^4] - #1]*#1 + 7*Log[-x^2 + Sqrt[1 + x^4] - #1]*#1^2 - 12*Log[-x^2 + Sqrt[1 + x^4] - #1]*#1^3 - 7*Log[-x^2 + Sqrt[1 + x^4] - #1]*#1^4 + 2*Log[-x^2 + Sqrt[1 + x^4] - #1]*#1^5 + Log[-x^2 + Sqrt[1 + x^4] - #1]*#1^6)/(-1 + 6*#1 - 3*#1^2 - 10*#1^3 + 5*#1^4 + 18*#1^5 + 7*#1^6 + 2*#1^7) & ]/4","A",1
2762,1,261,261,1.474673,"\int \frac{\left(-1+x+2 x^2\right) \sqrt[4]{-x^3+x^4}}{-1-x+x^2} \, dx","IntegrateAlgebraic[((-1 + x + 2*x^2)*(-x^3 + x^4)^(1/4))/(-1 - x + x^2),x]","\frac{1}{4} (11+4 x) \sqrt[4]{-x^3+x^4}-\frac{49}{8} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)+\sqrt{2 \left(11+5 \sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^3+x^4}}\right)-\sqrt{2 \left(-11+5 \sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^3+x^4}}\right)+\frac{49}{8} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)-\sqrt{2 \left(11+5 \sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^3+x^4}}\right)+\sqrt{2 \left(-11+5 \sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^3+x^4}}\right)","\frac{1}{4} (11+4 x) \sqrt[4]{-x^3+x^4}-\frac{49}{8} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)+\sqrt{2 \left(11+5 \sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^3+x^4}}\right)-\sqrt{2 \left(-11+5 \sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^3+x^4}}\right)+\frac{49}{8} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)-\sqrt{2 \left(11+5 \sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^3+x^4}}\right)+\sqrt{2 \left(-11+5 \sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^3+x^4}}\right)",1,"((11 + 4*x)*(-x^3 + x^4)^(1/4))/4 - (49*ArcTan[x/(-x^3 + x^4)^(1/4)])/8 + Sqrt[2*(11 + 5*Sqrt[5])]*ArcTan[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(-x^3 + x^4)^(1/4)] - Sqrt[2*(-11 + 5*Sqrt[5])]*ArcTan[(Sqrt[1/2 + Sqrt[5]/2]*x)/(-x^3 + x^4)^(1/4)] + (49*ArcTanh[x/(-x^3 + x^4)^(1/4)])/8 - Sqrt[2*(11 + 5*Sqrt[5])]*ArcTanh[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(-x^3 + x^4)^(1/4)] + Sqrt[2*(-11 + 5*Sqrt[5])]*ArcTanh[(Sqrt[1/2 + Sqrt[5]/2]*x)/(-x^3 + x^4)^(1/4)]","A",1
2763,1,241,261,16.3989174,"\int \frac{\left(2+x^6\right) \left(-1+x^4+x^6\right)}{\sqrt[4]{-1+x^6} \left(1-2 x^6+x^8+x^{12}\right)} \, dx","IntegrateAlgebraic[((2 + x^6)*(-1 + x^4 + x^6))/((-1 + x^6)^(1/4)*(1 - 2*x^6 + x^8 + x^12)),x]","-\frac{1}{2} \sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{-1+x^6}}{-x^2+\sqrt{-1+x^6}}\right)-\frac{1}{2} \sqrt{\frac{1}{2} \left(2-\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{-1+x^6}}{-x^2+\sqrt{-1+x^6}}\right)-\frac{1}{2} \sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{-1+x^6}}{x^2+\sqrt{-1+x^6}}\right)-\frac{1}{2} \sqrt{\frac{1}{2} \left(2-\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{-1+x^6}}{x^2+\sqrt{-1+x^6}}\right)","-\frac{1}{2} \sqrt{\frac{1}{2} \left(2-\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{-1+x^6}}{-x^2+\sqrt{-1+x^6}}\right)+\frac{1}{2} \sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\left(\sqrt{\frac{2}{2-\sqrt{2}}}-\frac{2}{\sqrt{2-\sqrt{2}}}\right) x \sqrt[4]{-1+x^6}}{-x^2+\sqrt{-1+x^6}}\right)-\frac{1}{2} \sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{-1+x^6}}{x^2+\sqrt{-1+x^6}}\right)-\frac{1}{2} \sqrt{\frac{1}{2} \left(2-\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{-1+x^6}}{x^2+\sqrt{-1+x^6}}\right)",1,"-1/2*(Sqrt[(2 + Sqrt[2])/2]*ArcTan[(Sqrt[2 - Sqrt[2]]*x*(-1 + x^6)^(1/4))/(-x^2 + Sqrt[-1 + x^6])]) - (Sqrt[(2 - Sqrt[2])/2]*ArcTan[(Sqrt[2 + Sqrt[2]]*x*(-1 + x^6)^(1/4))/(-x^2 + Sqrt[-1 + x^6])])/2 - (Sqrt[(2 + Sqrt[2])/2]*ArcTanh[(Sqrt[2 - Sqrt[2]]*x*(-1 + x^6)^(1/4))/(x^2 + Sqrt[-1 + x^6])])/2 - (Sqrt[(2 - Sqrt[2])/2]*ArcTanh[(Sqrt[2 + Sqrt[2]]*x*(-1 + x^6)^(1/4))/(x^2 + Sqrt[-1 + x^6])])/2","A",1
2764,1,261,261,1.0385894,"\int \frac{1}{(d+c x)^2 \sqrt{a x+\sqrt{b^2+a^2 x^2}}} \, dx","IntegrateAlgebraic[1/((d + c*x)^2*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]),x]","-\frac{1}{c (d+c x) \sqrt{a x+\sqrt{b^2+a^2 x^2}}}-\frac{a \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{a d-\sqrt{b^2 c^2+a^2 d^2}}}\right)}{\sqrt{c} \sqrt{b^2 c^2+a^2 d^2} \sqrt{a d-\sqrt{b^2 c^2+a^2 d^2}}}+\frac{a \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{a d+\sqrt{b^2 c^2+a^2 d^2}}}\right)}{\sqrt{c} \sqrt{b^2 c^2+a^2 d^2} \sqrt{a d+\sqrt{b^2 c^2+a^2 d^2}}}","-\frac{1}{c (d+c x) \sqrt{a x+\sqrt{b^2+a^2 x^2}}}-\frac{a \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{a d-\sqrt{b^2 c^2+a^2 d^2}}}\right)}{\sqrt{c} \sqrt{b^2 c^2+a^2 d^2} \sqrt{a d-\sqrt{b^2 c^2+a^2 d^2}}}+\frac{a \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{a d+\sqrt{b^2 c^2+a^2 d^2}}}\right)}{\sqrt{c} \sqrt{b^2 c^2+a^2 d^2} \sqrt{a d+\sqrt{b^2 c^2+a^2 d^2}}}",1,"-(1/(c*(d + c*x)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])) - (a*ArcTan[(Sqrt[c]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/Sqrt[a*d - Sqrt[b^2*c^2 + a^2*d^2]]])/(Sqrt[c]*Sqrt[b^2*c^2 + a^2*d^2]*Sqrt[a*d - Sqrt[b^2*c^2 + a^2*d^2]]) + (a*ArcTan[(Sqrt[c]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/Sqrt[a*d + Sqrt[b^2*c^2 + a^2*d^2]]])/(Sqrt[c]*Sqrt[b^2*c^2 + a^2*d^2]*Sqrt[a*d + Sqrt[b^2*c^2 + a^2*d^2]])","A",1
2765,1,341,261,2.2178034,"\int \frac{\left(1+x^2\right)^2}{\left(-1+x^2\right)^2 \sqrt{x^2+\sqrt{1+x^4}}} \, dx","IntegrateAlgebraic[(1 + x^2)^2/((-1 + x^2)^2*Sqrt[x^2 + Sqrt[1 + x^4]]),x]","\frac{x \left(-5+x^2\right)}{2 \left(-1+x^2\right) \sqrt{x^2+\sqrt{1+x^4}}}-4 \sqrt{2} \tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^4}}{\sqrt{2}}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)+\sqrt{14+10 \sqrt{2}} \tan ^{-1}\left(\frac{-\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)+\frac{\tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^4}}{\sqrt{2}}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)}{\sqrt{2}}-\sqrt{-14+10 \sqrt{2}} \tanh ^{-1}\left(\frac{-\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)","\frac{x \left(-5+x^2\right)}{2 \left(-1+x^2\right) \sqrt{x^2+\sqrt{1+x^4}}}-4 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)+\sqrt{2 \left(7+5 \sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(-1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)}{\sqrt{2}}-\sqrt{2 \left(-7+5 \sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)",1,"(x*(-5 + x^2))/(2*(-1 + x^2)*Sqrt[x^2 + Sqrt[1 + x^4]]) - 4*Sqrt[2]*ArcTan[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] + Sqrt[14 + 10*Sqrt[2]]*ArcTan[(-Sqrt[-1/2 + 1/Sqrt[2]] + Sqrt[-1/2 + 1/Sqrt[2]]*x^2 + Sqrt[-1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] + ArcTanh[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])]/Sqrt[2] - Sqrt[-14 + 10*Sqrt[2]]*ArcTanh[(-Sqrt[1/2 + 1/Sqrt[2]] + Sqrt[1/2 + 1/Sqrt[2]]*x^2 + Sqrt[1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])]","A",1
2766,1,262,262,7.1670078,"\int \frac{a (a b+a c-2 b c)-2 \left(a^2-b c\right) x+(2 a-b-c) x^2}{((-a+x) (-b+x) (-c+x))^{2/3} \left(a^2-b c d+(-2 a+b d+c d) x+(1-d) x^2\right)} \, dx","IntegrateAlgebraic[(a*(a*b + a*c - 2*b*c) - 2*(a^2 - b*c)*x + (2*a - b - c)*x^2)/(((-a + x)*(-b + x)*(-c + x))^(2/3)*(a^2 - b*c*d + (-2*a + b*d + c*d)*x + (1 - d)*x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a-\sqrt{3} x}{a-x-2 \sqrt[3]{d} \sqrt[3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}\right)}{\sqrt[3]{d}}+\frac{\log \left(a-x+\sqrt[3]{d} \sqrt[3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}\right)}{\sqrt[3]{d}}-\frac{\log \left(a^2-2 a x+x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}+d^{2/3} \left(-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3\right)^{2/3}\right)}{2 \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a-\sqrt{3} x}{a-x-2 \sqrt[3]{d} \sqrt[3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}\right)}{\sqrt[3]{d}}+\frac{\log \left(a-x+\sqrt[3]{d} \sqrt[3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}\right)}{\sqrt[3]{d}}-\frac{\log \left(a^2-2 a x+x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}+d^{2/3} \left(-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3\right)^{2/3}\right)}{2 \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*a - Sqrt[3]*x)/(a - x - 2*d^(1/3)*(-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3)^(1/3))])/d^(1/3) + Log[a - x + d^(1/3)*(-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3)^(1/3)]/d^(1/3) - Log[a^2 - 2*a*x + x^2 + (-(a*d^(1/3)) + d^(1/3)*x)*(-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3)^(1/3) + d^(2/3)*(-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3)^(2/3)]/(2*d^(1/3))","A",1
2767,1,262,262,2.3207293,"\int \frac{1+x^4}{\left(-1+x^4\right) \sqrt{a+b x+c x^2+b x^3+a x^4}} \, dx","IntegrateAlgebraic[(1 + x^4)/((-1 + x^4)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{2 a-c} x}{\sqrt{a}+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{\sqrt{2 a-c}}-\frac{\sqrt{-2 a-2 b-c} \tan ^{-1}\left(\frac{\sqrt{-2 a-2 b-c} x}{\sqrt{a}-2 \sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{2 (2 a+2 b+c)}-\frac{\sqrt{-2 a+2 b-c} \tan ^{-1}\left(\frac{\sqrt{-2 a+2 b-c} x}{\sqrt{a}+2 \sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{2 (2 a-2 b+c)}","\frac{\tan ^{-1}\left(\frac{\sqrt{2 a-c} x}{\sqrt{a}+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{\sqrt{2 a-c}}-\frac{\sqrt{-2 a-2 b-c} \tan ^{-1}\left(\frac{\sqrt{-2 a-2 b-c} x}{\sqrt{a}-2 \sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{2 (2 a+2 b+c)}-\frac{\sqrt{-2 a+2 b-c} \tan ^{-1}\left(\frac{\sqrt{-2 a+2 b-c} x}{\sqrt{a}+2 \sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{2 (2 a-2 b+c)}",1,"ArcTan[(Sqrt[2*a - c]*x)/(Sqrt[a] + Sqrt[a]*x^2 - Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])]/Sqrt[2*a - c] - (Sqrt[-2*a - 2*b - c]*ArcTan[(Sqrt[-2*a - 2*b - c]*x)/(Sqrt[a] - 2*Sqrt[a]*x + Sqrt[a]*x^2 - Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])])/(2*(2*a + 2*b + c)) - (Sqrt[-2*a + 2*b - c]*ArcTan[(Sqrt[-2*a + 2*b - c]*x)/(Sqrt[a] + 2*Sqrt[a]*x + Sqrt[a]*x^2 - Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])])/(2*(2*a - 2*b + c))","A",1
2768,1,342,262,1.945325,"\int \frac{\left(-1+x^2\right)^2}{\left(1+x^2\right)^2 \sqrt{x^2+\sqrt{1+x^4}}} \, dx","IntegrateAlgebraic[(-1 + x^2)^2/((1 + x^2)^2*Sqrt[x^2 + Sqrt[1 + x^4]]),x]","\frac{x \left(5+x^2\right)}{2 \left(1+x^2\right) \sqrt{x^2+\sqrt{1+x^4}}}+4 \sqrt{2} \tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^4}}{\sqrt{2}}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)-\sqrt{14+10 \sqrt{2}} \tan ^{-1}\left(\frac{-\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)+\frac{\tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^4}}{\sqrt{2}}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)}{\sqrt{2}}-\sqrt{-14+10 \sqrt{2}} \tanh ^{-1}\left(\frac{-\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)","\frac{x \left(5+x^2\right)}{2 \left(1+x^2\right) \sqrt{x^2+\sqrt{1+x^4}}}+4 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-\sqrt{2 \left(7+5 \sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)}{\sqrt{2}}-\sqrt{2 \left(-7+5 \sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2 \left(-1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)",1,"(x*(5 + x^2))/(2*(1 + x^2)*Sqrt[x^2 + Sqrt[1 + x^4]]) + 4*Sqrt[2]*ArcTan[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] - Sqrt[14 + 10*Sqrt[2]]*ArcTan[(-Sqrt[1/2 + 1/Sqrt[2]] + Sqrt[1/2 + 1/Sqrt[2]]*x^2 + Sqrt[1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] + ArcTanh[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])]/Sqrt[2] - Sqrt[-14 + 10*Sqrt[2]]*ArcTanh[(-Sqrt[-1/2 + 1/Sqrt[2]] + Sqrt[-1/2 + 1/Sqrt[2]]*x^2 + Sqrt[-1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])]","A",1
2769,1,263,263,0.6779018,"\int \frac{2 a b^2-b (2 a+b) x+x^3}{\sqrt[3]{x (-a+x) (-b+x)^2} \left(-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right)} \, dx","IntegrateAlgebraic[(2*a*b^2 - b*(2*a + b)*x + x^3)/((x*(-a + x)*(-b + x)^2)^(1/3)*(-(a*b^2) + b*(2*a + b)*x - (a + 2*b + d)*x^2 + x^3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{2 \sqrt[3]{d} x+\sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)}{\sqrt[3]{d}}+\frac{\log \left(-\sqrt[3]{d} x+\sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}\right)}{\sqrt[3]{d}}-\frac{\log \left(d^{2/3} x^2+\sqrt[3]{d} x \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}+\left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{2 \sqrt[3]{d} x+\sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)}{\sqrt[3]{d}}+\frac{\log \left(-\sqrt[3]{d} x+\sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}\right)}{\sqrt[3]{d}}-\frac{\log \left(d^{2/3} x^2+\sqrt[3]{d} x \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}+\left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3))/(2*d^(1/3)*x + (-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3))])/d^(1/3) + Log[-(d^(1/3)*x) + (-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3)]/d^(1/3) - Log[d^(2/3)*x^2 + d^(1/3)*x*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3) + (-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(2/3)]/(2*d^(1/3))","A",1
2770,1,263,263,4.6207752,"\int \frac{-d+c x}{x \sqrt[3]{-b+a x^3}} \, dx","IntegrateAlgebraic[(-d + c*x)/(x*(-b + a*x^3)^(1/3)),x]","-\frac{c \tan ^{-1}\left(\frac{\frac{x}{\sqrt{3}}+\frac{2 \sqrt[3]{-b+a x^3}}{\sqrt{3} \sqrt[3]{a}}}{x}\right)}{\sqrt{3} \sqrt[3]{a}}+\frac{d \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-b+a x^3}}{\sqrt{3} \sqrt[3]{b}}\right)}{\sqrt{3} \sqrt[3]{b}}+\frac{d \log \left(\sqrt[3]{b}+\sqrt[3]{-b+a x^3}\right)}{3 \sqrt[3]{b}}-\frac{c \log \left(-\sqrt[3]{a} x+\sqrt[3]{-b+a x^3}\right)}{3 \sqrt[3]{a}}-\frac{d \log \left(b^{2/3}-\sqrt[3]{b} \sqrt[3]{-b+a x^3}+\left(-b+a x^3\right)^{2/3}\right)}{6 \sqrt[3]{b}}+\frac{c \log \left(a^{2/3} x^2+\sqrt[3]{a} x \sqrt[3]{-b+a x^3}+\left(-b+a x^3\right)^{2/3}\right)}{6 \sqrt[3]{a}}","-\frac{c \tan ^{-1}\left(\frac{\frac{x}{\sqrt{3}}+\frac{2 \sqrt[3]{-b+a x^3}}{\sqrt{3} \sqrt[3]{a}}}{x}\right)}{\sqrt{3} \sqrt[3]{a}}+\frac{d \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-b+a x^3}}{\sqrt{3} \sqrt[3]{b}}\right)}{\sqrt{3} \sqrt[3]{b}}+\frac{d \log \left(\sqrt[3]{b}+\sqrt[3]{-b+a x^3}\right)}{3 \sqrt[3]{b}}-\frac{c \log \left(-\sqrt[3]{a} x+\sqrt[3]{-b+a x^3}\right)}{3 \sqrt[3]{a}}-\frac{d \log \left(b^{2/3}-\sqrt[3]{b} \sqrt[3]{-b+a x^3}+\left(-b+a x^3\right)^{2/3}\right)}{6 \sqrt[3]{b}}+\frac{c \log \left(a^{2/3} x^2+\sqrt[3]{a} x \sqrt[3]{-b+a x^3}+\left(-b+a x^3\right)^{2/3}\right)}{6 \sqrt[3]{a}}",1,"-((c*ArcTan[(x/Sqrt[3] + (2*(-b + a*x^3)^(1/3))/(Sqrt[3]*a^(1/3)))/x])/(Sqrt[3]*a^(1/3))) + (d*ArcTan[1/Sqrt[3] - (2*(-b + a*x^3)^(1/3))/(Sqrt[3]*b^(1/3))])/(Sqrt[3]*b^(1/3)) + (d*Log[b^(1/3) + (-b + a*x^3)^(1/3)])/(3*b^(1/3)) - (c*Log[-(a^(1/3)*x) + (-b + a*x^3)^(1/3)])/(3*a^(1/3)) - (d*Log[b^(2/3) - b^(1/3)*(-b + a*x^3)^(1/3) + (-b + a*x^3)^(2/3)])/(6*b^(1/3)) + (c*Log[a^(2/3)*x^2 + a^(1/3)*x*(-b + a*x^3)^(1/3) + (-b + a*x^3)^(2/3)])/(6*a^(1/3))","A",1
2771,1,227,263,1.5384571,"\int \frac{-b^4+c^2 x^2+a^4 x^4}{\sqrt{-b^4+a^4 x^4} \left(b^4+a^4 x^4\right)} \, dx","IntegrateAlgebraic[(-b^4 + c^2*x^2 + a^4*x^4)/(Sqrt[-b^4 + a^4*x^4]*(b^4 + a^4*x^4)),x]","\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 a^2 b^2-i c^2\right) \tan ^{-1}\left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{-b^4+a^4 x^4}}{a b x}\right)}{a^3 b^3}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 a^2 b^2+i c^2\right) \log \left(i b^2-(1-i) a b x+a^2 x^2+\sqrt{-b^4+a^4 x^4}\right)}{a^3 b^3}-\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 a^2 b^2+i c^2\right) \log \left(-a^3 b^5+(1+i) a^4 b^4 x+i a^5 b^3 x^2+i a^3 b^3 \sqrt{-b^4+a^4 x^4}\right)}{a^3 b^3}","-\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(2 a^2 b^2+i c^2\right) \tan ^{-1}\left(\frac{(1+i) a b x}{i b^2+a^2 x^2+\sqrt{-b^4+a^4 x^4}}\right)}{a^3 b^3}+\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \left(2 a^2 b^2-i c^2\right) \tanh ^{-1}\left(\frac{(1+i) b^2+(1-i) a^2 x^2+(1-i) \sqrt{-b^4+a^4 x^4}}{2 \sqrt{3-2 \sqrt{2}} a b x}\right)}{a^3 b^3}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 i a^2 b^2+c^2\right) \tanh ^{-1}\left(\frac{(1+i) b^2+(1-i) a^2 x^2+(1-i) \sqrt{-b^4+a^4 x^4}}{2 \sqrt{3+2 \sqrt{2}} a b x}\right)}{a^3 b^3}",1,"((1/8 + I/8)*(2*a^2*b^2 - I*c^2)*ArcTan[((1/2 + I/2)*Sqrt[-b^4 + a^4*x^4])/(a*b*x)])/(a^3*b^3) + ((1/8 + I/8)*(2*a^2*b^2 + I*c^2)*Log[I*b^2 - (1 - I)*a*b*x + a^2*x^2 + Sqrt[-b^4 + a^4*x^4]])/(a^3*b^3) - ((1/8 + I/8)*(2*a^2*b^2 + I*c^2)*Log[-(a^3*b^5) + (1 + I)*a^4*b^4*x + I*a^5*b^3*x^2 + I*a^3*b^3*Sqrt[-b^4 + a^4*x^4]])/(a^3*b^3)","A",1
2772,1,265,265,2.1676726,"\int \frac{x^2}{\left(1-x^4\right) \sqrt{a+b x+c x^2+b x^3+a x^4}} \, dx","IntegrateAlgebraic[x^2/((1 - x^4)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{2 a-c} x}{\sqrt{a}+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{2 \sqrt{2 a-c}}+\frac{\sqrt{-2 a-2 b-c} \tan ^{-1}\left(\frac{\sqrt{-2 a-2 b-c} x}{\sqrt{a}-2 \sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{4 (2 a+2 b+c)}+\frac{\sqrt{-2 a+2 b-c} \tan ^{-1}\left(\frac{\sqrt{-2 a+2 b-c} x}{\sqrt{a}+2 \sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{4 (2 a-2 b+c)}","\frac{\tan ^{-1}\left(\frac{\sqrt{2 a-c} x}{\sqrt{a}+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{2 \sqrt{2 a-c}}+\frac{\sqrt{-2 a-2 b-c} \tan ^{-1}\left(\frac{\sqrt{-2 a-2 b-c} x}{\sqrt{a}-2 \sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{4 (2 a+2 b+c)}+\frac{\sqrt{-2 a+2 b-c} \tan ^{-1}\left(\frac{\sqrt{-2 a+2 b-c} x}{\sqrt{a}+2 \sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{4 (2 a-2 b+c)}",1,"ArcTan[(Sqrt[2*a - c]*x)/(Sqrt[a] + Sqrt[a]*x^2 - Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])]/(2*Sqrt[2*a - c]) + (Sqrt[-2*a - 2*b - c]*ArcTan[(Sqrt[-2*a - 2*b - c]*x)/(Sqrt[a] - 2*Sqrt[a]*x + Sqrt[a]*x^2 - Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])])/(4*(2*a + 2*b + c)) + (Sqrt[-2*a + 2*b - c]*ArcTan[(Sqrt[-2*a + 2*b - c]*x)/(Sqrt[a] + 2*Sqrt[a]*x + Sqrt[a]*x^2 - Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])])/(4*(2*a - 2*b + c))","A",1
2773,1,265,265,3.384972,"\int \frac{x (-b+x) \left(-a^2 b+2 a^2 x+(-2 a+b) x^2\right)}{(x (-a+x) (-b+x))^{2/3} \left(-a^4+4 a^3 x+\left(-6 a^2+b^2 d\right) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right)} \, dx","IntegrateAlgebraic[(x*(-b + x)*(-(a^2*b) + 2*a^2*x + (-2*a + b)*x^2))/((x*(-a + x)*(-b + x))^(2/3)*(-a^4 + 4*a^3*x + (-6*a^2 + b^2*d)*x^2 + 2*(2*a - b*d)*x^3 + (-1 + d)*x^4)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}}{2 a^2-4 a x+2 x^2+\sqrt[3]{d} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}}\right)}{2 d^{2/3}}+\frac{\log \left(a^2-2 a x+x^2-\sqrt[3]{d} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}\right)}{2 d^{2/3}}-\frac{\log \left(a^4-4 a^3 x+6 a^2 x^2-4 a x^3+x^4+\left(a^2 \sqrt[3]{d}-2 a \sqrt[3]{d} x+\sqrt[3]{d} x^2\right) \left(a b x+(-a-b) x^2+x^3\right)^{2/3}+d^{2/3} \left(a b x+(-a-b) x^2+x^3\right)^{4/3}\right)}{4 d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}}{2 a^2-4 a x+2 x^2+\sqrt[3]{d} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}}\right)}{2 d^{2/3}}+\frac{\log \left(a^2-2 a x+x^2-\sqrt[3]{d} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}\right)}{2 d^{2/3}}-\frac{\log \left(a^4-4 a^3 x+6 a^2 x^2-4 a x^3+x^4+\left(a^2 \sqrt[3]{d}-2 a \sqrt[3]{d} x+\sqrt[3]{d} x^2\right) \left(a b x+(-a-b) x^2+x^3\right)^{2/3}+d^{2/3} \left(a b x+(-a-b) x^2+x^3\right)^{4/3}\right)}{4 d^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(a*b*x + (-a - b)*x^2 + x^3)^(2/3))/(2*a^2 - 4*a*x + 2*x^2 + d^(1/3)*(a*b*x + (-a - b)*x^2 + x^3)^(2/3))])/(2*d^(2/3)) + Log[a^2 - 2*a*x + x^2 - d^(1/3)*(a*b*x + (-a - b)*x^2 + x^3)^(2/3)]/(2*d^(2/3)) - Log[a^4 - 4*a^3*x + 6*a^2*x^2 - 4*a*x^3 + x^4 + (a^2*d^(1/3) - 2*a*d^(1/3)*x + d^(1/3)*x^2)*(a*b*x + (-a - b)*x^2 + x^3)^(2/3) + d^(2/3)*(a*b*x + (-a - b)*x^2 + x^3)^(4/3)]/(4*d^(2/3))","A",1
2774,1,265,265,0.735586,"\int \frac{\left(-1+x^3\right)^{2/3} \left(8-8 x^3+x^6\right)}{x^6 \left(-4+x^3\right) \left(-2+x^3\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(8 - 8*x^3 + x^6))/(x^6*(-4 + x^3)*(-2 + x^3)),x]","\frac{\left(-1+x^3\right)^{2/3} \left(-8+13 x^3\right)}{40 x^5}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2} \sqrt[3]{-1+x^3}}\right)}{2\ 2^{2/3} \sqrt{3}}+\frac{\sqrt[6]{3} \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2\ 2^{2/3} \sqrt[3]{-1+x^3}}\right)}{8 \sqrt[3]{2}}+\frac{\log \left(-x+\sqrt[3]{2} \sqrt[3]{-1+x^3}\right)}{6\ 2^{2/3}}-\frac{\log \left(-3 x+6^{2/3} \sqrt[3]{-1+x^3}\right)}{8 \sqrt[3]{6}}-\frac{\log \left(x^2+\sqrt[3]{2} x \sqrt[3]{-1+x^3}+2^{2/3} \left(-1+x^3\right)^{2/3}\right)}{12\ 2^{2/3}}+\frac{\log \left(3 x^2+6^{2/3} x \sqrt[3]{-1+x^3}+2 \sqrt[3]{6} \left(-1+x^3\right)^{2/3}\right)}{16 \sqrt[3]{6}}","\frac{\left(-1+x^3\right)^{2/3} \left(-8+13 x^3\right)}{40 x^5}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2} \sqrt[3]{-1+x^3}}\right)}{2\ 2^{2/3} \sqrt{3}}+\frac{\sqrt[6]{3} \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2\ 2^{2/3} \sqrt[3]{-1+x^3}}\right)}{8 \sqrt[3]{2}}+\frac{\log \left(-x+\sqrt[3]{2} \sqrt[3]{-1+x^3}\right)}{6\ 2^{2/3}}-\frac{\log \left(-3 x+6^{2/3} \sqrt[3]{-1+x^3}\right)}{8 \sqrt[3]{6}}-\frac{\log \left(x^2+\sqrt[3]{2} x \sqrt[3]{-1+x^3}+2^{2/3} \left(-1+x^3\right)^{2/3}\right)}{12\ 2^{2/3}}+\frac{\log \left(3 x^2+6^{2/3} x \sqrt[3]{-1+x^3}+2 \sqrt[3]{6} \left(-1+x^3\right)^{2/3}\right)}{16 \sqrt[3]{6}}",1,"((-1 + x^3)^(2/3)*(-8 + 13*x^3))/(40*x^5) - ArcTan[(Sqrt[3]*x)/(x + 2*2^(1/3)*(-1 + x^3)^(1/3))]/(2*2^(2/3)*Sqrt[3]) + (3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*2^(2/3)*(-1 + x^3)^(1/3))])/(8*2^(1/3)) + Log[-x + 2^(1/3)*(-1 + x^3)^(1/3)]/(6*2^(2/3)) - Log[-3*x + 6^(2/3)*(-1 + x^3)^(1/3)]/(8*6^(1/3)) - Log[x^2 + 2^(1/3)*x*(-1 + x^3)^(1/3) + 2^(2/3)*(-1 + x^3)^(2/3)]/(12*2^(2/3)) + Log[3*x^2 + 6^(2/3)*x*(-1 + x^3)^(1/3) + 2*6^(1/3)*(-1 + x^3)^(2/3)]/(16*6^(1/3))","A",1
2775,1,288,265,0.9588584,"\int \frac{\left(c+\left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4}\right)^{4/3}}{\sqrt{-b+a^2 x^2}} \, dx","IntegrateAlgebraic[(c + (a*x + Sqrt[-b + a^2*x^2])^(3/4))^(4/3)/Sqrt[-b + a^2*x^2],x]","\frac{5 c \sqrt[3]{c+\left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4}}}{a}+\frac{\left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4} \sqrt[3]{c+\left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4}}}{a}-\frac{4 c^{4/3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{c+\left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4}}}{\sqrt{3} \sqrt[3]{c}}\right)}{\sqrt{3} a}+\frac{4 c^{4/3} \log \left(-\sqrt[3]{c}+\sqrt[3]{c+\left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4}}\right)}{3 a}-\frac{2 c^{4/3} \log \left(c^{2/3}+\sqrt[3]{c} \sqrt[3]{c+\left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4}}+\left(c+\left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4}\right)^{2/3}\right)}{3 a}","\frac{4 c \sqrt[3]{c+\left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4}}}{a}+\frac{\left(c+\left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4}\right)^{4/3}}{a}-\frac{4 c^{4/3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{c+\left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4}}}{\sqrt{3} \sqrt[3]{c}}\right)}{\sqrt{3} a}+\frac{4 c^{4/3} \log \left(-\sqrt[3]{c}+\sqrt[3]{c+\left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4}}\right)}{3 a}-\frac{2 c^{4/3} \log \left(c^{2/3}+\sqrt[3]{c} \sqrt[3]{c+\left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4}}+\left(c+\left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4}\right)^{2/3}\right)}{3 a}",1,"(5*c*(c + (a*x + Sqrt[-b + a^2*x^2])^(3/4))^(1/3))/a + ((a*x + Sqrt[-b + a^2*x^2])^(3/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(3/4))^(1/3))/a - (4*c^(4/3)*ArcTan[1/Sqrt[3] + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(3/4))^(1/3))/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*a) + (4*c^(4/3)*Log[-c^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(3/4))^(1/3)])/(3*a) - (2*c^(4/3)*Log[c^(2/3) + c^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(3/4))^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(3/4))^(2/3)])/(3*a)","A",1
2776,1,266,266,1.387757,"\int \frac{1}{(-b+a x) \sqrt[3]{b^3+a^3 x^3}} \, dx","IntegrateAlgebraic[1/((-b + a*x)*(b^3 + a^3*x^3)^(1/3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b^3+a^3 x^3}}{\sqrt[3]{2} b+\sqrt[3]{2} a x+\sqrt[3]{b^3+a^3 x^3}}\right)}{2 \sqrt[3]{2} a b}+\frac{\log \left(\sqrt[3]{2} \sqrt{a} b^{3/2}+\sqrt[3]{2} a^{3/2} \sqrt{b} x-2 \sqrt{a} \sqrt{b} \sqrt[3]{b^3+a^3 x^3}\right)}{2 \sqrt[3]{2} a b}-\frac{\log \left(2^{2/3} a b^3+2\ 2^{2/3} a^2 b^2 x+2^{2/3} a^3 b x^2+\left(2 \sqrt[3]{2} a b^2+2 \sqrt[3]{2} a^2 b x\right) \sqrt[3]{b^3+a^3 x^3}+4 a b \left(b^3+a^3 x^3\right)^{2/3}\right)}{4 \sqrt[3]{2} a b}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b^3+a^3 x^3}}{\sqrt[3]{2} b+\sqrt[3]{2} a x+\sqrt[3]{b^3+a^3 x^3}}\right)}{2 \sqrt[3]{2} a b}+\frac{\log \left(\sqrt[3]{2} \sqrt{a} b^{3/2}+\sqrt[3]{2} a^{3/2} \sqrt{b} x-2 \sqrt{a} \sqrt{b} \sqrt[3]{b^3+a^3 x^3}\right)}{2 \sqrt[3]{2} a b}-\frac{\log \left(2^{2/3} a b^3+2\ 2^{2/3} a^2 b^2 x+2^{2/3} a^3 b x^2+\left(2 \sqrt[3]{2} a b^2+2 \sqrt[3]{2} a^2 b x\right) \sqrt[3]{b^3+a^3 x^3}+4 a b \left(b^3+a^3 x^3\right)^{2/3}\right)}{4 \sqrt[3]{2} a b}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(b^3 + a^3*x^3)^(1/3))/(2^(1/3)*b + 2^(1/3)*a*x + (b^3 + a^3*x^3)^(1/3))])/(2*2^(1/3)*a*b) + Log[2^(1/3)*Sqrt[a]*b^(3/2) + 2^(1/3)*a^(3/2)*Sqrt[b]*x - 2*Sqrt[a]*Sqrt[b]*(b^3 + a^3*x^3)^(1/3)]/(2*2^(1/3)*a*b) - Log[2^(2/3)*a*b^3 + 2*2^(2/3)*a^2*b^2*x + 2^(2/3)*a^3*b*x^2 + (2*2^(1/3)*a*b^2 + 2*2^(1/3)*a^2*b*x)*(b^3 + a^3*x^3)^(1/3) + 4*a*b*(b^3 + a^3*x^3)^(2/3)]/(4*2^(1/3)*a*b)","A",1
2777,1,266,266,3.7884179,"\int \frac{\sqrt[3]{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{\sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}} \, dx","IntegrateAlgebraic[(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])^(1/3)/Sqrt[-(a/b^2) + (a^2*x^2)/b^2],x]","\frac{3 b \sqrt[3]{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{2 a}-\frac{\sqrt{3} b \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{2} \sqrt[3]{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{\sqrt{3}}\right)}{2 \sqrt[3]{2} a}+\frac{b \log \left(-1+\sqrt[3]{2} \sqrt[3]{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}\right)}{2 \sqrt[3]{2} a}-\frac{b \log \left(1+\sqrt[3]{2} \sqrt[3]{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}+2^{2/3} \left(a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}\right)^{2/3}\right)}{4 \sqrt[3]{2} a}","\frac{3 b \sqrt[3]{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{2 a}-\frac{\sqrt{3} b \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{2} \sqrt[3]{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{\sqrt{3}}\right)}{2 \sqrt[3]{2} a}+\frac{b \log \left(-1+\sqrt[3]{2} \sqrt[3]{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}\right)}{2 \sqrt[3]{2} a}-\frac{b \log \left(1+\sqrt[3]{2} \sqrt[3]{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}+2^{2/3} \left(a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}\right)^{2/3}\right)}{4 \sqrt[3]{2} a}",1,"(3*b*(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])^(1/3))/(2*a) - (Sqrt[3]*b*ArcTan[1/Sqrt[3] + (2*2^(1/3)*(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])^(1/3))/Sqrt[3]])/(2*2^(1/3)*a) + (b*Log[-1 + 2^(1/3)*(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])^(1/3)])/(2*2^(1/3)*a) - (b*Log[1 + 2^(1/3)*(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])^(1/3) + 2^(2/3)*(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])^(2/3)])/(4*2^(1/3)*a)","A",1
2778,1,181,266,0.2494335,"\int \frac{\sqrt{1+x^2} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{1-x^2} \, dx","IntegrateAlgebraic[(Sqrt[1 + x^2]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^2),x]","-4 \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+4 \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)-\text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{-2 \text{$\#$1}+\text{$\#$1}^3}\&\right]+\text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{2-2 \text{$\#$1}^2+\text{$\#$1}^4}\&\right]","-4 \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+4 \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)-\text{RootSum}\left[-1-8 \text{$\#$1}^2-8 \text{$\#$1}^3+14 \text{$\#$1}^4+32 \text{$\#$1}^5+24 \text{$\#$1}^6+8 \text{$\#$1}^7+\text{$\#$1}^8\&,\frac{\log \left(-1+\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{-1+\text{$\#$1}+3 \text{$\#$1}^2+\text{$\#$1}^3}\&\right]+\text{RootSum}\left[-1+8 \text{$\#$1}^2+8 \text{$\#$1}^3+18 \text{$\#$1}^4+32 \text{$\#$1}^5+24 \text{$\#$1}^6+8 \text{$\#$1}^7+\text{$\#$1}^8\&,\frac{\log \left(-1+\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+\log \left(-1+\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{1+4 \text{$\#$1}^2+4 \text{$\#$1}^3+\text{$\#$1}^4}\&\right]",1,"-4*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + 4*ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/(-2*#1 + #1^3) & ] + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(2 - 2*#1^2 + #1^4) & ]","A",1
2779,1,181,266,0.0018063,"\int \frac{\sqrt{1+x^2} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{1-x^2} \, dx","IntegrateAlgebraic[(Sqrt[1 + x^2]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^2),x]","-4 \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+4 \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)-\text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{-2 \text{$\#$1}+\text{$\#$1}^3}\&\right]+\text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{2-2 \text{$\#$1}^2+\text{$\#$1}^4}\&\right]","-4 \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+4 \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)-\text{RootSum}\left[-1-8 \text{$\#$1}^2-8 \text{$\#$1}^3+14 \text{$\#$1}^4+32 \text{$\#$1}^5+24 \text{$\#$1}^6+8 \text{$\#$1}^7+\text{$\#$1}^8\&,\frac{\log \left(-1+\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{-1+\text{$\#$1}+3 \text{$\#$1}^2+\text{$\#$1}^3}\&\right]+\text{RootSum}\left[-1+8 \text{$\#$1}^2+8 \text{$\#$1}^3+18 \text{$\#$1}^4+32 \text{$\#$1}^5+24 \text{$\#$1}^6+8 \text{$\#$1}^7+\text{$\#$1}^8\&,\frac{\log \left(-1+\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+\log \left(-1+\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{1+4 \text{$\#$1}^2+4 \text{$\#$1}^3+\text{$\#$1}^4}\&\right]",1,"-4*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + 4*ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/(-2*#1 + #1^3) & ] + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(2 - 2*#1^2 + #1^4) & ]","A",1
2780,1,267,267,5.8104249,"\int \frac{-b+x}{\left((-a+x) (-b+x)^2\right)^{2/3} (b-a d+(-1+d) x)} \, dx","IntegrateAlgebraic[(-b + x)/(((-a + x)*(-b + x)^2)^(2/3)*(b - a*d + (-1 + d)*x)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} b-\sqrt{3} x}{b-x-2 \sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}\right)}{(a-b) \sqrt[3]{d}}+\frac{\log \left(b-x+\sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}\right)}{(a-b) \sqrt[3]{d}}-\frac{\log \left(b^2-2 b x+x^2+\left(-b \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}+d^{2/3} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}\right)}{2 (a-b) \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} b-\sqrt{3} x}{b-x-2 \sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}\right)}{(a-b) \sqrt[3]{d}}+\frac{\log \left(b-x+\sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}\right)}{(a-b) \sqrt[3]{d}}-\frac{\log \left(b^2-2 b x+x^2+\left(-b \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}+d^{2/3} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}\right)}{2 (a-b) \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*b - Sqrt[3]*x)/(b - x - 2*d^(1/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3))])/((a - b)*d^(1/3)) + Log[b - x + d^(1/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3)]/((a - b)*d^(1/3)) - Log[b^2 - 2*b*x + x^2 + (-(b*d^(1/3)) + d^(1/3)*x)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3) + d^(2/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(2/3)]/(2*(a - b)*d^(1/3))","A",1
2781,1,269,267,0.5744283,"\int \frac{-a q x+b p x^3}{\sqrt{q+p x^4} \left(b^2 c+d q+2 a b c x^2+\left(a^2 c+d p\right) x^4\right)} \, dx","IntegrateAlgebraic[(-(a*q*x) + b*p*x^3)/(Sqrt[q + p*x^4]*(b^2*c + d*q + 2*a*b*c*x^2 + (a^2*c + d*p)*x^4)),x]","-\frac{1}{4} \text{RootSum}\left[a^2 c q^2+d p q^2+4 a b c \sqrt{p} q \text{$\#$1}+4 b^2 c p \text{$\#$1}^2-2 a^2 c q \text{$\#$1}^2+2 d p q \text{$\#$1}^2-4 a b c \sqrt{p} \text{$\#$1}^3+a^2 c \text{$\#$1}^4+d p \text{$\#$1}^4\&,\frac{b p q \log \left(-\sqrt{p} x^2+\sqrt{q+p x^4}-\text{$\#$1}\right)-2 a \sqrt{p} q \log \left(-\sqrt{p} x^2+\sqrt{q+p x^4}-\text{$\#$1}\right) \text{$\#$1}-b p \log \left(-\sqrt{p} x^2+\sqrt{q+p x^4}-\text{$\#$1}\right) \text{$\#$1}^2}{a b c \sqrt{p} q+2 b^2 c p \text{$\#$1}-a^2 c q \text{$\#$1}+d p q \text{$\#$1}-3 a b c \sqrt{p} \text{$\#$1}^2+a^2 c \text{$\#$1}^3+d p \text{$\#$1}^3}\&\right]","-\frac{1}{4} \text{RootSum}\left[a^2 c q^2+d p q^2-4 a b c \sqrt{p} q \text{$\#$1}+4 b^2 c p \text{$\#$1}^2-2 a^2 c q \text{$\#$1}^2+2 d p q \text{$\#$1}^2+4 a b c \sqrt{p} \text{$\#$1}^3+a^2 c \text{$\#$1}^4+d p \text{$\#$1}^4\&,\frac{b p q \log \left(\sqrt{p} x^2+\sqrt{q+p x^4}-\text{$\#$1}\right)+2 a \sqrt{p} q \log \left(\sqrt{p} x^2+\sqrt{q+p x^4}-\text{$\#$1}\right) \text{$\#$1}-b p \log \left(\sqrt{p} x^2+\sqrt{q+p x^4}-\text{$\#$1}\right) \text{$\#$1}^2}{-a b c \sqrt{p} q+2 b^2 c p \text{$\#$1}-a^2 c q \text{$\#$1}+d p q \text{$\#$1}+3 a b c \sqrt{p} \text{$\#$1}^2+a^2 c \text{$\#$1}^3+d p \text{$\#$1}^3}\&\right]",1,"-1/4*RootSum[a^2*c*q^2 + d*p*q^2 + 4*a*b*c*Sqrt[p]*q*#1 + 4*b^2*c*p*#1^2 - 2*a^2*c*q*#1^2 + 2*d*p*q*#1^2 - 4*a*b*c*Sqrt[p]*#1^3 + a^2*c*#1^4 + d*p*#1^4 & , (b*p*q*Log[-(Sqrt[p]*x^2) + Sqrt[q + p*x^4] - #1] - 2*a*Sqrt[p]*q*Log[-(Sqrt[p]*x^2) + Sqrt[q + p*x^4] - #1]*#1 - b*p*Log[-(Sqrt[p]*x^2) + Sqrt[q + p*x^4] - #1]*#1^2)/(a*b*c*Sqrt[p]*q + 2*b^2*c*p*#1 - a^2*c*q*#1 + d*p*q*#1 - 3*a*b*c*Sqrt[p]*#1^2 + a^2*c*#1^3 + d*p*#1^3) & ]","A",1
2782,1,267,267,21.7393374,"\int \frac{(b+a x)^2 \left(-2 a q+3 b p x^2+a p x^3\right)}{\sqrt{q+p x^3} \left(b^4 c+d q^2+4 a b^3 c x+6 a^2 b^2 c x^2+\left(4 a^3 b c+2 d p q\right) x^3+a^4 c x^4+d p^2 x^6\right)} \, dx","IntegrateAlgebraic[((b + a*x)^2*(-2*a*q + 3*b*p*x^2 + a*p*x^3))/(Sqrt[q + p*x^3]*(b^4*c + d*q^2 + 4*a*b^3*c*x + 6*a^2*b^2*c*x^2 + (4*a^3*b*c + 2*d*p*q)*x^3 + a^4*c*x^4 + d*p^2*x^6)),x]","\frac{\tan ^{-1}\left(\frac{b \sqrt[4]{c}+a \sqrt[4]{c} x}{b \sqrt[4]{c}+a \sqrt[4]{c} x-\sqrt{2} \sqrt[4]{d} \sqrt{q+p x^3}}\right)}{\sqrt{2} c^{3/4} \sqrt[4]{d}}-\frac{\tan ^{-1}\left(\frac{b \sqrt[4]{c}+a \sqrt[4]{c} x}{b \sqrt[4]{c}+a \sqrt[4]{c} x+\sqrt{2} \sqrt[4]{d} \sqrt{q+p x^3}}\right)}{\sqrt{2} c^{3/4} \sqrt[4]{d}}+\frac{\tanh ^{-1}\left(\frac{\left(\sqrt{2} b \sqrt[4]{c} \sqrt[4]{d}+\sqrt{2} a \sqrt[4]{c} \sqrt[4]{d} x\right) \sqrt{q+p x^3}}{b^2 \sqrt{c}+\sqrt{d} q+2 a b \sqrt{c} x+a^2 \sqrt{c} x^2+\sqrt{d} p x^3}\right)}{\sqrt{2} c^{3/4} \sqrt[4]{d}}","\frac{\tan ^{-1}\left(\frac{b \sqrt[4]{c}+a \sqrt[4]{c} x}{b \sqrt[4]{c}+a \sqrt[4]{c} x-\sqrt{2} \sqrt[4]{d} \sqrt{q+p x^3}}\right)}{\sqrt{2} c^{3/4} \sqrt[4]{d}}-\frac{\tan ^{-1}\left(\frac{b \sqrt[4]{c}+a \sqrt[4]{c} x}{b \sqrt[4]{c}+a \sqrt[4]{c} x+\sqrt{2} \sqrt[4]{d} \sqrt{q+p x^3}}\right)}{\sqrt{2} c^{3/4} \sqrt[4]{d}}+\frac{\tanh ^{-1}\left(\frac{\left(\sqrt{2} b \sqrt[4]{c} \sqrt[4]{d}+\sqrt{2} a \sqrt[4]{c} \sqrt[4]{d} x\right) \sqrt{q+p x^3}}{b^2 \sqrt{c}+\sqrt{d} q+2 a b \sqrt{c} x+a^2 \sqrt{c} x^2+\sqrt{d} p x^3}\right)}{\sqrt{2} c^{3/4} \sqrt[4]{d}}",1,"ArcTan[(b*c^(1/4) + a*c^(1/4)*x)/(b*c^(1/4) + a*c^(1/4)*x - Sqrt[2]*d^(1/4)*Sqrt[q + p*x^3])]/(Sqrt[2]*c^(3/4)*d^(1/4)) - ArcTan[(b*c^(1/4) + a*c^(1/4)*x)/(b*c^(1/4) + a*c^(1/4)*x + Sqrt[2]*d^(1/4)*Sqrt[q + p*x^3])]/(Sqrt[2]*c^(3/4)*d^(1/4)) + ArcTanh[((Sqrt[2]*b*c^(1/4)*d^(1/4) + Sqrt[2]*a*c^(1/4)*d^(1/4)*x)*Sqrt[q + p*x^3])/(b^2*Sqrt[c] + Sqrt[d]*q + 2*a*b*Sqrt[c]*x + a^2*Sqrt[c]*x^2 + Sqrt[d]*p*x^3)]/(Sqrt[2]*c^(3/4)*d^(1/4))","A",1
2783,1,420,267,0.6159794,"\int \frac{\left(1+x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1-x^2\right)^2 \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[((1 + x^2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/((1 - x^2)^2*Sqrt[x + Sqrt[1 + x^2]]),x]","-\frac{x \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(-1+x^2\right) \sqrt{x+\sqrt{1+x^2}}}+\frac{1}{2} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{2 \text{$\#$1}-3 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]-\frac{1}{8} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+5 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{1}{2} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-2+4 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]-\frac{1}{8} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+5 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","-\frac{x \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(-1+x^2\right) \sqrt{x+\sqrt{1+x^2}}}-\frac{1}{8} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-\frac{1}{8} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]",1,"-((x*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/((-1 + x^2)*Sqrt[x + Sqrt[1 + x^2]])) + RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/(2*#1 - 3*#1^3 + #1^5) & ]/2 - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 5*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2)/(2*#1^3 - 3*#1^5 + #1^7) & ]/8 + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(-2 + 4*#1^2 - 3*#1^4 + #1^6) & ]/2 - RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 5*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ]/8","A",1
2784,1,420,267,0.0024244,"\int \frac{\left(1+x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1-x^2\right)^2 \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[((1 + x^2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/((1 - x^2)^2*Sqrt[x + Sqrt[1 + x^2]]),x]","-\frac{x \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(-1+x^2\right) \sqrt{x+\sqrt{1+x^2}}}+\frac{1}{2} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{2 \text{$\#$1}-3 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]-\frac{1}{8} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+5 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{1}{2} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-2+4 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]-\frac{1}{8} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+5 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","-\frac{x \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(-1+x^2\right) \sqrt{x+\sqrt{1+x^2}}}-\frac{1}{8} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-\frac{1}{8} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]",1,"-((x*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/((-1 + x^2)*Sqrt[x + Sqrt[1 + x^2]])) + RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/(2*#1 - 3*#1^3 + #1^5) & ]/2 - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 5*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2)/(2*#1^3 - 3*#1^5 + #1^7) & ]/8 + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(-2 + 4*#1^2 - 3*#1^4 + #1^6) & ]/2 - RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 5*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ]/8","A",1
2785,1,268,268,1.0556618,"\int \frac{-1+k^4 x^4}{\sqrt{(1-x) x \left(1-k^2 x\right)} \left(a+b x^2+a k^4 x^4\right)} \, dx","IntegrateAlgebraic[(-1 + k^4*x^4)/(Sqrt[(1 - x)*x*(1 - k^2*x)]*(a + b*x^2 + a*k^4*x^4)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{a+a k^2-\sqrt{a} \sqrt{-b+2 a k^2}} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{\sqrt{a} (-1+x) \left(-1+k^2 x\right)}\right)}{\sqrt{a} \sqrt{a+a k^2-\sqrt{a} \sqrt{-b+2 a k^2}}}-\frac{\sqrt{\sqrt{a} \left(\sqrt{a}+\sqrt{a} k^2+\sqrt{-b+2 a k^2}\right)} \tan ^{-1}\left(\frac{\sqrt{a+a k^2+\sqrt{a} \sqrt{-b+2 a k^2}} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{\sqrt{a} (-1+x) \left(-1+k^2 x\right)}\right)}{a \left(\sqrt{a}+\sqrt{a} k^2+\sqrt{-b+2 a k^2}\right)}","-\frac{\tan ^{-1}\left(\frac{\sqrt{a+a k^2-\sqrt{a} \sqrt{-b+2 a k^2}} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{\sqrt{a} (-1+x) \left(-1+k^2 x\right)}\right)}{\sqrt{a} \sqrt{a+a k^2-\sqrt{a} \sqrt{-b+2 a k^2}}}-\frac{\sqrt{\sqrt{a} \left(\sqrt{a}+\sqrt{a} k^2+\sqrt{-b+2 a k^2}\right)} \tan ^{-1}\left(\frac{\sqrt{a+a k^2+\sqrt{a} \sqrt{-b+2 a k^2}} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{\sqrt{a} (-1+x) \left(-1+k^2 x\right)}\right)}{a \left(\sqrt{a}+\sqrt{a} k^2+\sqrt{-b+2 a k^2}\right)}",1,"-(ArcTan[(Sqrt[a + a*k^2 - Sqrt[a]*Sqrt[-b + 2*a*k^2]]*Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3])/(Sqrt[a]*(-1 + x)*(-1 + k^2*x))]/(Sqrt[a]*Sqrt[a + a*k^2 - Sqrt[a]*Sqrt[-b + 2*a*k^2]])) - (Sqrt[Sqrt[a]*(Sqrt[a] + Sqrt[a]*k^2 + Sqrt[-b + 2*a*k^2])]*ArcTan[(Sqrt[a + a*k^2 + Sqrt[a]*Sqrt[-b + 2*a*k^2]]*Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3])/(Sqrt[a]*(-1 + x)*(-1 + k^2*x))])/(a*(Sqrt[a] + Sqrt[a]*k^2 + Sqrt[-b + 2*a*k^2]))","A",1
2786,-1,0,268,180.000678,"\int \frac{x^5 (-4 a+3 x)}{\sqrt[3]{x^2 (-a+x)} \left(-a^2+2 a x-x^2+d x^8\right)} \, dx","IntegrateAlgebraic[(x^5*(-4*a + 3*x))/((x^2*(-a + x))^(1/3)*(-a^2 + 2*a*x - x^2 + d*x^8)),x]","\text{\$Aborted}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} x^4}{\sqrt[3]{d} x^4+2 \left(-a x^2+x^3\right)^{2/3}}\right)}{2 d^{2/3}}+\frac{\log \left(-a \sqrt[6]{d} x^2+a \sqrt[3]{-a x^2+x^3}\right)}{2 d^{2/3}}+\frac{\log \left(a \sqrt[6]{d} x^2+a \sqrt[3]{-a x^2+x^3}\right)}{2 d^{2/3}}-\frac{\log \left(a^2 \sqrt[3]{d} x^4-a^2 \sqrt[6]{d} x^2 \sqrt[3]{-a x^2+x^3}+a^2 \left(-a x^2+x^3\right)^{2/3}\right)}{4 d^{2/3}}-\frac{\log \left(a^2 \sqrt[3]{d} x^4+a^2 \sqrt[6]{d} x^2 \sqrt[3]{-a x^2+x^3}+a^2 \left(-a x^2+x^3\right)^{2/3}\right)}{4 d^{2/3}}",1,"$Aborted","F",-1
2787,1,34,269,0.0603222,"\int \frac{\sqrt{1-x}}{8 (1+x)^{7/2}} \, dx","IntegrateAlgebraic[Sqrt[1 - x]/(8*(1 + x)^(7/2)),x]","-\frac{(1-x)^{3/2} \left(5+\frac{3 (1-x)}{1+x}\right)}{240 (1+x)^{3/2}}","\frac{\left(\frac{1}{960}-\frac{\sqrt{1-x}}{960}\right) \left(-1+\sqrt{1+x}\right)^9 \left(15+\frac{15 \left(-1+\sqrt{1-x}\right)^8}{\left(-1+\sqrt{1+x}\right)^8}-\frac{30 \left(-1+\sqrt{1-x}\right)^7}{\left(-1+\sqrt{1+x}\right)^7}+\frac{40 \left(-1+\sqrt{1-x}\right)^6}{\left(-1+\sqrt{1+x}\right)^6}+\frac{50 \left(-1+\sqrt{1-x}\right)^5}{\left(-1+\sqrt{1+x}\right)^5}-\frac{46 \left(-1+\sqrt{1-x}\right)^4}{\left(-1+\sqrt{1+x}\right)^4}-\frac{50 \left(-1+\sqrt{1-x}\right)^3}{\left(-1+\sqrt{1+x}\right)^3}+\frac{40 \left(-1+\sqrt{1-x}\right)^2}{\left(-1+\sqrt{1+x}\right)^2}+\frac{-30+30 \sqrt{1-x}}{-1+\sqrt{1+x}}\right)}{\left(1+x-2 \sqrt{1+x}+\sqrt{1-x} \sqrt{1+x}\right)^5}",1,"-1/240*((1 - x)^(3/2)*(5 + (3*(1 - x))/(1 + x)))/(1 + x)^(3/2)","A",1
2788,1,269,269,0.7072937,"\int \frac{2 a b^2-b (2 a+b) x+x^3}{\sqrt[3]{x (-a+x) (-b+x)^2} \left(-a b^2 d+b (2 a+b) d x-(1+a d+2 b d) x^2+d x^3\right)} \, dx","IntegrateAlgebraic[(2*a*b^2 - b*(2*a + b)*x + x^3)/((x*(-a + x)*(-b + x)^2)^(1/3)*(-(a*b^2*d) + b*(2*a + b)*d*x - (1 + a*d + 2*b*d)*x^2 + d*x^3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{2 x+\sqrt[3]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)}{d^{2/3}}+\frac{\log \left(x-\sqrt[3]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}\right)}{d^{2/3}}-\frac{\log \left(x^2+\sqrt[3]{d} x \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}+d^{2/3} \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{2/3}\right)}{2 d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{2 x+\sqrt[3]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)}{d^{2/3}}+\frac{\log \left(x-\sqrt[3]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}\right)}{d^{2/3}}-\frac{\log \left(x^2+\sqrt[3]{d} x \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}+d^{2/3} \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{2/3}\right)}{2 d^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3))/(2*x + d^(1/3)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3))])/d^(2/3) + Log[x - d^(1/3)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3)]/d^(2/3) - Log[x^2 + d^(1/3)*x*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3) + d^(2/3)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(2/3)]/(2*d^(2/3))","A",1
2789,1,269,269,0.9640459,"\int \frac{\left(-x+x^2\right) \sqrt[4]{-x^3+x^4}}{-1-x+x^2} \, dx","IntegrateAlgebraic[((-x + x^2)*(-x^3 + x^4)^(1/4))/(-1 - x + x^2),x]","\frac{1}{8} (-1+4 x) \sqrt[4]{-x^3+x^4}-\frac{29}{16} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)+\sqrt{\frac{1}{5} \left(2+2 \sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^3+x^4}}\right)+\sqrt{\frac{1}{5} \left(-2+2 \sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^3+x^4}}\right)+\frac{29}{16} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)-\sqrt{\frac{1}{5} \left(2+2 \sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^3+x^4}}\right)-\sqrt{\frac{1}{5} \left(-2+2 \sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^3+x^4}}\right)","\frac{1}{8} (-1+4 x) \sqrt[4]{-x^3+x^4}-\frac{29}{16} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)+\sqrt{\frac{1}{5} \left(2+2 \sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^3+x^4}}\right)+\sqrt{\frac{1}{5} \left(-2+2 \sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^3+x^4}}\right)+\frac{29}{16} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^3+x^4}}\right)-\sqrt{\frac{1}{5} \left(2+2 \sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^3+x^4}}\right)-\sqrt{\frac{1}{5} \left(-2+2 \sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} x}{\sqrt[4]{-x^3+x^4}}\right)",1,"((-1 + 4*x)*(-x^3 + x^4)^(1/4))/8 - (29*ArcTan[x/(-x^3 + x^4)^(1/4)])/16 + Sqrt[(2 + 2*Sqrt[5])/5]*ArcTan[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(-x^3 + x^4)^(1/4)] + Sqrt[(-2 + 2*Sqrt[5])/5]*ArcTan[(Sqrt[1/2 + Sqrt[5]/2]*x)/(-x^3 + x^4)^(1/4)] + (29*ArcTanh[x/(-x^3 + x^4)^(1/4)])/16 - Sqrt[(2 + 2*Sqrt[5])/5]*ArcTanh[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(-x^3 + x^4)^(1/4)] - Sqrt[(-2 + 2*Sqrt[5])/5]*ArcTanh[(Sqrt[1/2 + Sqrt[5]/2]*x)/(-x^3 + x^4)^(1/4)]","A",1
2790,1,246,269,0.5638497,"\int \frac{\sqrt{b+a x} \sqrt{c+\sqrt{b+a x}}}{x \sqrt{d+\sqrt{c+\sqrt{b+a x}}}} \, dx","IntegrateAlgebraic[(Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]])/(x*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]]),x]","-\frac{32}{15} d \sqrt{c+\sqrt{b+a x}} \sqrt{d+\sqrt{c+\sqrt{b+a x}}}+\frac{8}{15} \left(3 c+8 d^2+3 \sqrt{b+a x}\right) \sqrt{d+\sqrt{c+\sqrt{b+a x}}}-b \text{RootSum}\left[b-c^2+2 c d^2-d^4-4 c d \text{$\#$1}^2+4 d^3 \text{$\#$1}^2+2 c \text{$\#$1}^4-6 d^2 \text{$\#$1}^4+4 d \text{$\#$1}^6-\text{$\#$1}^8\&,\frac{d \log \left(\sqrt{d+\sqrt{c+\sqrt{b+a x}}}-\text{$\#$1}\right)-\log \left(\sqrt{d+\sqrt{c+\sqrt{b+a x}}}-\text{$\#$1}\right) \text{$\#$1}^2}{-c \text{$\#$1}+d^2 \text{$\#$1}-2 d \text{$\#$1}^3+\text{$\#$1}^5}\&\right]","\frac{8}{15} \left(3 c+8 d^2\right) \sqrt{d+\sqrt{c+\sqrt{b+a x}}}+\frac{8}{5} \sqrt{b+a x} \sqrt{d+\sqrt{c+\sqrt{b+a x}}}-\frac{32}{15} d \sqrt{c+\sqrt{b+a x}} \sqrt{d+\sqrt{c+\sqrt{b+a x}}}-b \text{RootSum}\left[b-c^2+2 c d^2-d^4-4 c d \text{$\#$1}^2+4 d^3 \text{$\#$1}^2+2 c \text{$\#$1}^4-6 d^2 \text{$\#$1}^4+4 d \text{$\#$1}^6-\text{$\#$1}^8\&,\frac{d \log \left(\sqrt{d+\sqrt{c+\sqrt{b+a x}}}-\text{$\#$1}\right)-\log \left(\sqrt{d+\sqrt{c+\sqrt{b+a x}}}-\text{$\#$1}\right) \text{$\#$1}^2}{-c \text{$\#$1}+d^2 \text{$\#$1}-2 d \text{$\#$1}^3+\text{$\#$1}^5}\&\right]",1,"(-32*d*Sqrt[c + Sqrt[b + a*x]]*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]])/15 + (8*(3*c + 8*d^2 + 3*Sqrt[b + a*x])*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]])/15 - b*RootSum[b - c^2 + 2*c*d^2 - d^4 - 4*c*d*#1^2 + 4*d^3*#1^2 + 2*c*#1^4 - 6*d^2*#1^4 + 4*d*#1^6 - #1^8 & , (d*Log[Sqrt[d + Sqrt[c + Sqrt[b + a*x]]] - #1] - Log[Sqrt[d + Sqrt[c + Sqrt[b + a*x]]] - #1]*#1^2)/(-(c*#1) + d^2*#1 - 2*d*#1^3 + #1^5) & ]","A",1
2791,1,246,269,0.0026605,"\int \frac{\sqrt{b+a x} \sqrt{c+\sqrt{b+a x}}}{x \sqrt{d+\sqrt{c+\sqrt{b+a x}}}} \, dx","IntegrateAlgebraic[(Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]])/(x*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]]),x]","-\frac{32}{15} d \sqrt{c+\sqrt{b+a x}} \sqrt{d+\sqrt{c+\sqrt{b+a x}}}+\frac{8}{15} \left(3 c+8 d^2+3 \sqrt{b+a x}\right) \sqrt{d+\sqrt{c+\sqrt{b+a x}}}-b \text{RootSum}\left[b-c^2+2 c d^2-d^4-4 c d \text{$\#$1}^2+4 d^3 \text{$\#$1}^2+2 c \text{$\#$1}^4-6 d^2 \text{$\#$1}^4+4 d \text{$\#$1}^6-\text{$\#$1}^8\&,\frac{d \log \left(\sqrt{d+\sqrt{c+\sqrt{b+a x}}}-\text{$\#$1}\right)-\log \left(\sqrt{d+\sqrt{c+\sqrt{b+a x}}}-\text{$\#$1}\right) \text{$\#$1}^2}{-c \text{$\#$1}+d^2 \text{$\#$1}-2 d \text{$\#$1}^3+\text{$\#$1}^5}\&\right]","\frac{8}{15} \left(3 c+8 d^2\right) \sqrt{d+\sqrt{c+\sqrt{b+a x}}}+\frac{8}{5} \sqrt{b+a x} \sqrt{d+\sqrt{c+\sqrt{b+a x}}}-\frac{32}{15} d \sqrt{c+\sqrt{b+a x}} \sqrt{d+\sqrt{c+\sqrt{b+a x}}}-b \text{RootSum}\left[b-c^2+2 c d^2-d^4-4 c d \text{$\#$1}^2+4 d^3 \text{$\#$1}^2+2 c \text{$\#$1}^4-6 d^2 \text{$\#$1}^4+4 d \text{$\#$1}^6-\text{$\#$1}^8\&,\frac{d \log \left(\sqrt{d+\sqrt{c+\sqrt{b+a x}}}-\text{$\#$1}\right)-\log \left(\sqrt{d+\sqrt{c+\sqrt{b+a x}}}-\text{$\#$1}\right) \text{$\#$1}^2}{-c \text{$\#$1}+d^2 \text{$\#$1}-2 d \text{$\#$1}^3+\text{$\#$1}^5}\&\right]",1,"(-32*d*Sqrt[c + Sqrt[b + a*x]]*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]])/15 + (8*(3*c + 8*d^2 + 3*Sqrt[b + a*x])*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]])/15 - b*RootSum[b - c^2 + 2*c*d^2 - d^4 - 4*c*d*#1^2 + 4*d^3*#1^2 + 2*c*#1^4 - 6*d^2*#1^4 + 4*d*#1^6 - #1^8 & , (d*Log[Sqrt[d + Sqrt[c + Sqrt[b + a*x]]] - #1] - Log[Sqrt[d + Sqrt[c + Sqrt[b + a*x]]] - #1]*#1^2)/(-(c*#1) + d^2*#1 - 2*d*#1^3 + #1^5) & ]","A",1
2792,1,269,269,0.4095804,"\int \frac{1}{\sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}} \, dx","IntegrateAlgebraic[1/Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]],x]","\frac{(6 b-16 a x) \sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+(-9 b+8 a x) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\sqrt{-b+a^2 x^2} \left(-16 \sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+8 \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{12 a^2 x+12 a \sqrt{-b+a^2 x^2}}+\frac{3 b \tanh ^{-1}\left(\sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{4 a}","\frac{(6 b-16 a x) \sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+(-9 b+8 a x) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\sqrt{-b+a^2 x^2} \left(-16 \sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+8 \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{12 a^2 x+12 a \sqrt{-b+a^2 x^2}}+\frac{3 b \tanh ^{-1}\left(\sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{4 a}",1,"((6*b - 16*a*x)*Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]] + (-9*b + 8*a*x)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]] + Sqrt[-b + a^2*x^2]*(-16*Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]] + 8*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]))/(12*a^2*x + 12*a*Sqrt[-b + a^2*x^2]) + (3*b*ArcTanh[Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]])/(4*a)","A",1
2793,1,270,270,0.5517835,"\int \frac{(-1+x) (-1+k x) \left(-2 x+(1+k) x^2\right)}{((1-x) x (1-k x))^{2/3} \left(b-2 b (1+k) x+\left(b+4 b k+b k^2\right) x^2-2 b k (1+k) x^3+\left(-1+b k^2\right) x^4\right)} \, dx","IntegrateAlgebraic[((-1 + x)*(-1 + k*x)*(-2*x + (1 + k)*x^2))/(((1 - x)*x*(1 - k*x))^(2/3)*(b - 2*b*(1 + k)*x + (b + 4*b*k + b*k^2)*x^2 - 2*b*k*(1 + k)*x^3 + (-1 + b*k^2)*x^4)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}}\right)}{2 b^{2/3}}+\frac{\log \left(x-\sqrt[6]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{2 b^{2/3}}+\frac{\log \left(x+\sqrt[6]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{2 b^{2/3}}-\frac{\log \left(x^2-\sqrt[6]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+\sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{4 b^{2/3}}-\frac{\log \left(x^2+\sqrt[6]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+\sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{4 b^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}}\right)}{2 b^{2/3}}+\frac{\log \left(x-\sqrt[6]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{2 b^{2/3}}+\frac{\log \left(x+\sqrt[6]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{2 b^{2/3}}-\frac{\log \left(x^2-\sqrt[6]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+\sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{4 b^{2/3}}-\frac{\log \left(x^2+\sqrt[6]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+\sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{4 b^{2/3}}",1,"-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3))])/b^(2/3) + Log[x - b^(1/6)*(x + (-1 - k)*x^2 + k*x^3)^(1/3)]/(2*b^(2/3)) + Log[x + b^(1/6)*(x + (-1 - k)*x^2 + k*x^3)^(1/3)]/(2*b^(2/3)) - Log[x^2 - b^(1/6)*x*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3)]/(4*b^(2/3)) - Log[x^2 + b^(1/6)*x*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3)]/(4*b^(2/3))","A",1
2794,1,270,270,6.6447693,"\int \frac{\left(-1+2 (-2+k) x+3 k x^2\right) \left(-1+3 x-3 x^2+x^3\right)}{((1-x) x (1-k x))^{2/3} \left(-b+(1+5 b) x-(10 b+k) x^2+10 b x^3-5 b x^4+b x^5\right)} \, dx","IntegrateAlgebraic[((-1 + 2*(-2 + k)*x + 3*k*x^2)*(-1 + 3*x - 3*x^2 + x^3))/(((1 - x)*x*(1 - k*x))^(2/3)*(-b + (1 + 5*b)*x - (10*b + k)*x^2 + 10*b*x^3 - 5*b*x^4 + b*x^5)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b}-2 \sqrt{3} \sqrt[3]{b} x+\sqrt{3} \sqrt[3]{b} x^2}{\sqrt[3]{b}-2 \sqrt[3]{b} x+\sqrt[3]{b} x^2+2 \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{b^{2/3}}+\frac{\log \left(-\sqrt[3]{b}+2 \sqrt[3]{b} x-\sqrt[3]{b} x^2+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{b^{2/3}}-\frac{\log \left(b^{2/3}-4 b^{2/3} x+6 b^{2/3} x^2-4 b^{2/3} x^3+b^{2/3} x^4+\left(\sqrt[3]{b}-2 \sqrt[3]{b} x+\sqrt[3]{b} x^2\right) \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 b^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b}-2 \sqrt{3} \sqrt[3]{b} x+\sqrt{3} \sqrt[3]{b} x^2}{\sqrt[3]{b}-2 \sqrt[3]{b} x+\sqrt[3]{b} x^2+2 \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{b^{2/3}}+\frac{\log \left(-\sqrt[3]{b}+2 \sqrt[3]{b} x-\sqrt[3]{b} x^2+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{b^{2/3}}-\frac{\log \left(b^{2/3}-4 b^{2/3} x+6 b^{2/3} x^2-4 b^{2/3} x^3+b^{2/3} x^4+\left(\sqrt[3]{b}-2 \sqrt[3]{b} x+\sqrt[3]{b} x^2\right) \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 b^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*b^(1/3) - 2*Sqrt[3]*b^(1/3)*x + Sqrt[3]*b^(1/3)*x^2)/(b^(1/3) - 2*b^(1/3)*x + b^(1/3)*x^2 + 2*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/b^(2/3) + Log[-b^(1/3) + 2*b^(1/3)*x - b^(1/3)*x^2 + (x + (-1 - k)*x^2 + k*x^3)^(1/3)]/b^(2/3) - Log[b^(2/3) - 4*b^(2/3)*x + 6*b^(2/3)*x^2 - 4*b^(2/3)*x^3 + b^(2/3)*x^4 + (b^(1/3) - 2*b^(1/3)*x + b^(1/3)*x^2)*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + (x + (-1 - k)*x^2 + k*x^3)^(2/3)]/(2*b^(2/3))","A",1
2795,1,271,270,0.6303103,"\int \frac{\sqrt[4]{2 b+a x^4} \left(-4 b+a x^8\right)}{x^6 \left(-4 b+c x^4+a x^8\right)} \, dx","IntegrateAlgebraic[((2*b + a*x^4)^(1/4)*(-4*b + a*x^8))/(x^6*(-4*b + c*x^4 + a*x^8)),x]","\frac{\sqrt[4]{2 b+a x^4} \left(-4 b-2 a x^4-5 c x^4\right)}{20 b x^5}-\frac{c \text{RootSum}\left[2 a^2-2 a b+a c-4 a \text{$\#$1}^4-c \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{2 a^2 \log (x)-2 a b \log (x)+a c \log (x)-2 a^2 \log \left(\sqrt[4]{2 b+a x^4}-x \text{$\#$1}\right)+2 a b \log \left(\sqrt[4]{2 b+a x^4}-x \text{$\#$1}\right)-a c \log \left(\sqrt[4]{2 b+a x^4}-x \text{$\#$1}\right)-2 a \log (x) \text{$\#$1}^4-c \log (x) \text{$\#$1}^4+2 a \log \left(\sqrt[4]{2 b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+c \log \left(\sqrt[4]{2 b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-4 a \text{$\#$1}^3-c \text{$\#$1}^3+4 \text{$\#$1}^7}\&\right]}{16 b}","\frac{\sqrt[4]{2 b+a x^4} \left(-4 b-2 a x^4-5 c x^4\right)}{20 b x^5}-\frac{c \text{RootSum}\left[2 a^2-2 a b+a c-4 a \text{$\#$1}^4-c \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-2 a^2 \log (x)+2 a b \log (x)-a c \log (x)+2 a^2 \log \left(\sqrt[4]{2 b+a x^4}-x \text{$\#$1}\right)-2 a b \log \left(\sqrt[4]{2 b+a x^4}-x \text{$\#$1}\right)+a c \log \left(\sqrt[4]{2 b+a x^4}-x \text{$\#$1}\right)+2 a \log (x) \text{$\#$1}^4+c \log (x) \text{$\#$1}^4-2 a \log \left(\sqrt[4]{2 b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-c \log \left(\sqrt[4]{2 b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{4 a \text{$\#$1}^3+c \text{$\#$1}^3-4 \text{$\#$1}^7}\&\right]}{16 b}",1,"((2*b + a*x^4)^(1/4)*(-4*b - 2*a*x^4 - 5*c*x^4))/(20*b*x^5) - (c*RootSum[2*a^2 - 2*a*b + a*c - 4*a*#1^4 - c*#1^4 + 2*#1^8 & , (2*a^2*Log[x] - 2*a*b*Log[x] + a*c*Log[x] - 2*a^2*Log[(2*b + a*x^4)^(1/4) - x*#1] + 2*a*b*Log[(2*b + a*x^4)^(1/4) - x*#1] - a*c*Log[(2*b + a*x^4)^(1/4) - x*#1] - 2*a*Log[x]*#1^4 - c*Log[x]*#1^4 + 2*a*Log[(2*b + a*x^4)^(1/4) - x*#1]*#1^4 + c*Log[(2*b + a*x^4)^(1/4) - x*#1]*#1^4)/(-4*a*#1^3 - c*#1^3 + 4*#1^7) & ])/(16*b)","A",1
2796,1,271,270,0.0017137,"\int \frac{\sqrt[4]{2 b+a x^4} \left(-4 b+a x^8\right)}{x^6 \left(-4 b+c x^4+a x^8\right)} \, dx","IntegrateAlgebraic[((2*b + a*x^4)^(1/4)*(-4*b + a*x^8))/(x^6*(-4*b + c*x^4 + a*x^8)),x]","\frac{\sqrt[4]{2 b+a x^4} \left(-4 b-2 a x^4-5 c x^4\right)}{20 b x^5}-\frac{c \text{RootSum}\left[2 a^2-2 a b+a c-4 a \text{$\#$1}^4-c \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{2 a^2 \log (x)-2 a b \log (x)+a c \log (x)-2 a^2 \log \left(\sqrt[4]{2 b+a x^4}-x \text{$\#$1}\right)+2 a b \log \left(\sqrt[4]{2 b+a x^4}-x \text{$\#$1}\right)-a c \log \left(\sqrt[4]{2 b+a x^4}-x \text{$\#$1}\right)-2 a \log (x) \text{$\#$1}^4-c \log (x) \text{$\#$1}^4+2 a \log \left(\sqrt[4]{2 b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+c \log \left(\sqrt[4]{2 b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-4 a \text{$\#$1}^3-c \text{$\#$1}^3+4 \text{$\#$1}^7}\&\right]}{16 b}","\frac{\sqrt[4]{2 b+a x^4} \left(-4 b-2 a x^4-5 c x^4\right)}{20 b x^5}-\frac{c \text{RootSum}\left[2 a^2-2 a b+a c-4 a \text{$\#$1}^4-c \text{$\#$1}^4+2 \text{$\#$1}^8\&,\frac{-2 a^2 \log (x)+2 a b \log (x)-a c \log (x)+2 a^2 \log \left(\sqrt[4]{2 b+a x^4}-x \text{$\#$1}\right)-2 a b \log \left(\sqrt[4]{2 b+a x^4}-x \text{$\#$1}\right)+a c \log \left(\sqrt[4]{2 b+a x^4}-x \text{$\#$1}\right)+2 a \log (x) \text{$\#$1}^4+c \log (x) \text{$\#$1}^4-2 a \log \left(\sqrt[4]{2 b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-c \log \left(\sqrt[4]{2 b+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{4 a \text{$\#$1}^3+c \text{$\#$1}^3-4 \text{$\#$1}^7}\&\right]}{16 b}",1,"((2*b + a*x^4)^(1/4)*(-4*b - 2*a*x^4 - 5*c*x^4))/(20*b*x^5) - (c*RootSum[2*a^2 - 2*a*b + a*c - 4*a*#1^4 - c*#1^4 + 2*#1^8 & , (2*a^2*Log[x] - 2*a*b*Log[x] + a*c*Log[x] - 2*a^2*Log[(2*b + a*x^4)^(1/4) - x*#1] + 2*a*b*Log[(2*b + a*x^4)^(1/4) - x*#1] - a*c*Log[(2*b + a*x^4)^(1/4) - x*#1] - 2*a*Log[x]*#1^4 - c*Log[x]*#1^4 + 2*a*Log[(2*b + a*x^4)^(1/4) - x*#1]*#1^4 + c*Log[(2*b + a*x^4)^(1/4) - x*#1]*#1^4)/(-4*a*#1^3 - c*#1^3 + 4*#1^7) & ])/(16*b)","A",1
2797,1,598,270,23.7438005,"\int \frac{-b^{12}+a^{12} x^{12}}{\sqrt{-b^4+a^4 x^4} \left(b^{12}+a^{12} x^{12}\right)} \, dx","IntegrateAlgebraic[(-b^12 + a^12*x^12)/(Sqrt[-b^4 + a^4*x^4]*(b^12 + a^12*x^12)),x]","-\frac{\left(\frac{1}{6}-\frac{i}{6}\right) \tan ^{-1}\left(\frac{(1+i) a b x}{i b^2+a^2 x^2+\sqrt{-b^4+a^4 x^4}}\right)}{a b}+\frac{(-1)^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} a b x}{i b^2+a^2 x^2+\sqrt{-b^4+a^4 x^4}}\right)}{3 a b}-\frac{i \left(-\sqrt{2} \sqrt[4]{-7-4 \sqrt{3}}+\sqrt{6} \sqrt[4]{-7-4 \sqrt{3}}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{-7-4 \sqrt{3}} \left(b^2-i a^2 x^2-i \sqrt{-b^4+a^4 x^4}\right)}{\sqrt{2} a b x}\right)}{6 a b}+\frac{i \left(\sqrt{2} \sqrt[4]{-7+4 \sqrt{3}}+\sqrt{6} \sqrt[4]{-7+4 \sqrt{3}}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{-7+4 \sqrt{3}} \left(b^2-i a^2 x^2-i \sqrt{-b^4+a^4 x^4}\right)}{\sqrt{2} a b x}\right)}{6 a b}-\frac{\sqrt[4]{-1} \tanh ^{-1}\left(\frac{\sqrt{2} a b x}{i b^2+a^2 x^2+\sqrt{-b^4+a^4 x^4}}\right)}{3 a b}+\frac{\left(\frac{1}{24}-\frac{i}{24}\right) \log \left(i b^4-(1+i) a b^3 x-(1-i) a^3 b x^3-i a^4 x^4+\left(b^2-(1-i) a b x-i a^2 x^2\right) \sqrt{-b^4+a^4 x^4}\right)}{a b}-\frac{\left(\frac{1}{24}-\frac{i}{24}\right) \log \left(-a b^5-(1-i) a^2 b^4 x+(1+i) a^4 b^2 x^3+a^5 b x^4+\left(i a b^3+(1+i) a^2 b^2 x+a^3 b x^2\right) \sqrt{-b^4+a^4 x^4}\right)}{a b}","\frac{\tan ^{-1}\left(\frac{\sqrt{2} a b x \sqrt{-b^4+a^4 x^4}}{b^4+a^2 b^2 x^2-a^4 x^4}\right)}{3 \sqrt{2} a b}-\frac{\tan ^{-1}\left(\frac{\frac{b^3}{2 a}+a b x^2-\frac{a^3 x^4}{2 b}}{x \sqrt{-b^4+a^4 x^4}}\right)}{12 a b}+\frac{\tanh ^{-1}\left(\frac{\frac{b^3}{2 a}-a b x^2-\frac{a^3 x^4}{2 b}}{x \sqrt{-b^4+a^4 x^4}}\right)}{12 a b}+\frac{\tanh ^{-1}\left(\frac{\frac{b^3}{\sqrt{2} a}-\frac{a b x^2}{\sqrt{2}}-\frac{a^3 x^4}{\sqrt{2} b}}{x \sqrt{-b^4+a^4 x^4}}\right)}{3 \sqrt{2} a b}",1,"((-1/6 + I/6)*ArcTan[((1 + I)*a*b*x)/(I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4])])/(a*b) + ((-1)^(3/4)*ArcTan[(Sqrt[2]*a*b*x)/(I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4])])/(3*a*b) - ((I/6)*(-(Sqrt[2]*(-7 - 4*Sqrt[3])^(1/4)) + Sqrt[6]*(-7 - 4*Sqrt[3])^(1/4))*ArcTanh[((-7 - 4*Sqrt[3])^(1/4)*(b^2 - I*a^2*x^2 - I*Sqrt[-b^4 + a^4*x^4]))/(Sqrt[2]*a*b*x)])/(a*b) + ((I/6)*(Sqrt[2]*(-7 + 4*Sqrt[3])^(1/4) + Sqrt[6]*(-7 + 4*Sqrt[3])^(1/4))*ArcTanh[((-7 + 4*Sqrt[3])^(1/4)*(b^2 - I*a^2*x^2 - I*Sqrt[-b^4 + a^4*x^4]))/(Sqrt[2]*a*b*x)])/(a*b) - ((-1)^(1/4)*ArcTanh[(Sqrt[2]*a*b*x)/(I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4])])/(3*a*b) + ((1/24 - I/24)*Log[I*b^4 - (1 + I)*a*b^3*x - (1 - I)*a^3*b*x^3 - I*a^4*x^4 + (b^2 - (1 - I)*a*b*x - I*a^2*x^2)*Sqrt[-b^4 + a^4*x^4]])/(a*b) - ((1/24 - I/24)*Log[-(a*b^5) - (1 - I)*a^2*b^4*x + (1 + I)*a^4*b^2*x^3 + a^5*b*x^4 + (I*a*b^3 + (1 + I)*a^2*b^2*x + a^3*b*x^2)*Sqrt[-b^4 + a^4*x^4]])/(a*b)","C",0
2798,1,270,270,1.0950099,"\int \frac{1}{1-(1+x) \sqrt{c+b x+a x^2}} \, dx","IntegrateAlgebraic[(1 - (1 + x)*Sqrt[c + b*x + a*x^2])^(-1),x]","2 \text{RootSum}\left[-b^2-\sqrt{a} b c+\sqrt{a} c^2+4 \sqrt{a} b \text{$\#$1}+b^2 \text{$\#$1}+2 a c \text{$\#$1}-b c \text{$\#$1}-4 a \text{$\#$1}^2-3 \sqrt{a} b \text{$\#$1}^2+2 a \text{$\#$1}^3+b \text{$\#$1}^3-\sqrt{a} \text{$\#$1}^4\&,\frac{\sqrt{a} c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)-b \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}+\sqrt{a} \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}^2}{4 \sqrt{a} b+b^2+2 a c-b c-8 a \text{$\#$1}-6 \sqrt{a} b \text{$\#$1}+6 a \text{$\#$1}^2+3 b \text{$\#$1}^2-4 \sqrt{a} \text{$\#$1}^3}\&\right]","2 \text{RootSum}\left[-b^2-\sqrt{a} b c+\sqrt{a} c^2+4 \sqrt{a} b \text{$\#$1}+b^2 \text{$\#$1}+2 a c \text{$\#$1}-b c \text{$\#$1}-4 a \text{$\#$1}^2-3 \sqrt{a} b \text{$\#$1}^2+2 a \text{$\#$1}^3+b \text{$\#$1}^3-\sqrt{a} \text{$\#$1}^4\&,\frac{\sqrt{a} c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)-b \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}+\sqrt{a} \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}^2}{4 \sqrt{a} b+b^2+2 a c-b c-8 a \text{$\#$1}-6 \sqrt{a} b \text{$\#$1}+6 a \text{$\#$1}^2+3 b \text{$\#$1}^2-4 \sqrt{a} \text{$\#$1}^3}\&\right]",1,"2*RootSum[-b^2 - Sqrt[a]*b*c + Sqrt[a]*c^2 + 4*Sqrt[a]*b*#1 + b^2*#1 + 2*a*c*#1 - b*c*#1 - 4*a*#1^2 - 3*Sqrt[a]*b*#1^2 + 2*a*#1^3 + b*#1^3 - Sqrt[a]*#1^4 & , (Sqrt[a]*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1] - b*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 + Sqrt[a]*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1^2)/(4*Sqrt[a]*b + b^2 + 2*a*c - b*c - 8*a*#1 - 6*Sqrt[a]*b*#1 + 6*a*#1^2 + 3*b*#1^2 - 4*Sqrt[a]*#1^3) & ]","A",1
2799,1,271,271,0.6124827,"\int \frac{\left(1-x^4\right) \sqrt[4]{1-x-4 x^2+4 x^3+6 x^4-6 x^5-4 x^6+4 x^7+x^8-x^9}}{1+x^4} \, dx","IntegrateAlgebraic[((1 - x^4)*(1 - x - 4*x^2 + 4*x^3 + 6*x^4 - 6*x^5 - 4*x^6 + 4*x^7 + x^8 - x^9)^(1/4))/(1 + x^4),x]","-\frac{4 \left(-17+8 x+9 x^2\right) \sqrt[4]{1-x-4 x^2+4 x^3+6 x^4-6 x^5-4 x^6+4 x^7+x^8-x^9}}{117 (1+x)}-\frac{1}{2} \text{RootSum}\left[2-4 \text{$\#$1}^4+6 \text{$\#$1}^8-4 \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{2 \log \left(-1+x^2\right) \text{$\#$1}^5-2 \log \left(\sqrt[4]{1-x-4 x^2+4 x^3+6 x^4-6 x^5-4 x^6+4 x^7+x^8-x^9}+\text{$\#$1}-x^2 \text{$\#$1}\right) \text{$\#$1}^5-\log \left(-1+x^2\right) \text{$\#$1}^9+\log \left(\sqrt[4]{1-x-4 x^2+4 x^3+6 x^4-6 x^5-4 x^6+4 x^7+x^8-x^9}+\text{$\#$1}-x^2 \text{$\#$1}\right) \text{$\#$1}^9}{-1+3 \text{$\#$1}^4-3 \text{$\#$1}^8+\text{$\#$1}^{12}}\&\right]","-\frac{4 \left(-17+8 x+9 x^2\right) \sqrt[4]{1-x-4 x^2+4 x^3+6 x^4-6 x^5-4 x^6+4 x^7+x^8-x^9}}{117 (1+x)}-\frac{1}{2} \text{RootSum}\left[2-4 \text{$\#$1}^4+6 \text{$\#$1}^8-4 \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{2 \log \left(-1+x^2\right) \text{$\#$1}^5-2 \log \left(\sqrt[4]{1-x-4 x^2+4 x^3+6 x^4-6 x^5-4 x^6+4 x^7+x^8-x^9}+\text{$\#$1}-x^2 \text{$\#$1}\right) \text{$\#$1}^5-\log \left(-1+x^2\right) \text{$\#$1}^9+\log \left(\sqrt[4]{1-x-4 x^2+4 x^3+6 x^4-6 x^5-4 x^6+4 x^7+x^8-x^9}+\text{$\#$1}-x^2 \text{$\#$1}\right) \text{$\#$1}^9}{-1+3 \text{$\#$1}^4-3 \text{$\#$1}^8+\text{$\#$1}^{12}}\&\right]",1,"(-4*(-17 + 8*x + 9*x^2)*(1 - x - 4*x^2 + 4*x^3 + 6*x^4 - 6*x^5 - 4*x^6 + 4*x^7 + x^8 - x^9)^(1/4))/(117*(1 + x)) - RootSum[2 - 4*#1^4 + 6*#1^8 - 4*#1^12 + #1^16 & , (2*Log[-1 + x^2]*#1^5 - 2*Log[(1 - x - 4*x^2 + 4*x^3 + 6*x^4 - 6*x^5 - 4*x^6 + 4*x^7 + x^8 - x^9)^(1/4) + #1 - x^2*#1]*#1^5 - Log[-1 + x^2]*#1^9 + Log[(1 - x - 4*x^2 + 4*x^3 + 6*x^4 - 6*x^5 - 4*x^6 + 4*x^7 + x^8 - x^9)^(1/4) + #1 - x^2*#1]*#1^9)/(-1 + 3*#1^4 - 3*#1^8 + #1^12) & ]/2","A",1
2800,1,272,272,0.7101051,"\int \frac{-a (a-5 b)-(3 a+5 b) x+4 x^2}{\sqrt[3]{(-a+x) (-b+x)} \left(b-a^5 d-\left(1-5 a^4 d\right) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right)} \, dx","IntegrateAlgebraic[(-(a*(a - 5*b)) - (3*a + 5*b)*x + 4*x^2)/(((-a + x)*(-b + x))^(1/3)*(b - a^5*d - (1 - 5*a^4*d)*x - 10*a^3*d*x^2 + 10*a^2*d*x^3 - 5*a*d*x^4 + d*x^5)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a b+(-a-b) x+x^2}}{2 a^2 \sqrt[3]{d}-4 a \sqrt[3]{d} x+2 \sqrt[3]{d} x^2+\sqrt[3]{a b+(-a-b) x+x^2}}\right)}{\sqrt[3]{d}}+\frac{\log \left(a^2 \sqrt{d}-2 a \sqrt{d} x+\sqrt{d} x^2-\sqrt[6]{d} \sqrt[3]{a b+(-a-b) x+x^2}\right)}{\sqrt[3]{d}}-\frac{\log \left(a^4 d-4 a^3 d x+6 a^2 d x^2-4 a d x^3+d x^4+\sqrt[3]{d} \left(a b+(-a-b) x+x^2\right)^{2/3}+\sqrt[3]{a b+(-a-b) x+x^2} \left(a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2\right)\right)}{2 \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a b+(-a-b) x+x^2}}{2 a^2 \sqrt[3]{d}-4 a \sqrt[3]{d} x+2 \sqrt[3]{d} x^2+\sqrt[3]{a b+(-a-b) x+x^2}}\right)}{\sqrt[3]{d}}+\frac{\log \left(a^2 \sqrt{d}-2 a \sqrt{d} x+\sqrt{d} x^2-\sqrt[6]{d} \sqrt[3]{a b+(-a-b) x+x^2}\right)}{\sqrt[3]{d}}-\frac{\log \left(a^4 d-4 a^3 d x+6 a^2 d x^2-4 a d x^3+d x^4+\sqrt[3]{d} \left(a b+(-a-b) x+x^2\right)^{2/3}+\sqrt[3]{a b+(-a-b) x+x^2} \left(a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2\right)\right)}{2 \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(a*b + (-a - b)*x + x^2)^(1/3))/(2*a^2*d^(1/3) - 4*a*d^(1/3)*x + 2*d^(1/3)*x^2 + (a*b + (-a - b)*x + x^2)^(1/3))])/d^(1/3) + Log[a^2*Sqrt[d] - 2*a*Sqrt[d]*x + Sqrt[d]*x^2 - d^(1/6)*(a*b + (-a - b)*x + x^2)^(1/3)]/d^(1/3) - Log[a^4*d - 4*a^3*d*x + 6*a^2*d*x^2 - 4*a*d*x^3 + d*x^4 + d^(1/3)*(a*b + (-a - b)*x + x^2)^(2/3) + (a*b + (-a - b)*x + x^2)^(1/3)*(a^2*d^(2/3) - 2*a*d^(2/3)*x + d^(2/3)*x^2)]/(2*d^(1/3))","A",1
2801,1,167,272,0.5319164,"\int \frac{1+x^8}{\sqrt[4]{-x^2+x^4} \left(-1+x^8\right)} \, dx","IntegrateAlgebraic[(1 + x^8)/((-x^2 + x^4)^(1/4)*(-1 + x^8)),x]","-\frac{\left(-x^2+x^4\right)^{3/4}}{x \left(-1+x^2\right)}+\tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{2 \sqrt[4]{2}}+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{2 \sqrt[4]{2}}+\frac{1}{4} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","-\frac{\left(-x^2+x^4\right)^{3/4}}{x \left(-1+x^2\right)}+\tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{2 \sqrt[4]{2}}+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{2 \sqrt[4]{2}}+\frac{5}{8} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}^3+\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}^3}{-1+\text{$\#$1}^4}\&\right]+\frac{1}{8} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 \log (x)-2 \log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right)+3 \log (x) \text{$\#$1}^4-3 \log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}+\text{$\#$1}^5}\&\right]",1,"-((-x^2 + x^4)^(3/4)/(x*(-1 + x^2))) + ArcTan[x/(-x^2 + x^4)^(1/4)] - ArcTan[(2^(1/4)*x)/(-x^2 + x^4)^(1/4)]/(2*2^(1/4)) + ArcTanh[x/(-x^2 + x^4)^(1/4)] - ArcTanh[(2^(1/4)*x)/(-x^2 + x^4)^(1/4)]/(2*2^(1/4)) + RootSum[2 - 2*#1^4 + #1^8 & , (-Log[x] + Log[(-x^2 + x^4)^(1/4) - x*#1])/#1 & ]/4","A",1
2802,1,167,272,0.0018701,"\int \frac{1+x^8}{\sqrt[4]{-x^2+x^4} \left(-1+x^8\right)} \, dx","IntegrateAlgebraic[(1 + x^8)/((-x^2 + x^4)^(1/4)*(-1 + x^8)),x]","-\frac{\left(-x^2+x^4\right)^{3/4}}{x \left(-1+x^2\right)}+\tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{2 \sqrt[4]{2}}+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{2 \sqrt[4]{2}}+\frac{1}{4} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x)+\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right)}{\text{$\#$1}}\&\right]","-\frac{\left(-x^2+x^4\right)^{3/4}}{x \left(-1+x^2\right)}+\tan ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{2 \sqrt[4]{2}}+\tanh ^{-1}\left(\frac{x}{\sqrt[4]{-x^2+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{-x^2+x^4}}\right)}{2 \sqrt[4]{2}}+\frac{5}{8} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-\log (x) \text{$\#$1}^3+\log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}^3}{-1+\text{$\#$1}^4}\&\right]+\frac{1}{8} \text{RootSum}\left[2-2 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 \log (x)-2 \log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right)+3 \log (x) \text{$\#$1}^4-3 \log \left(\sqrt[4]{-x^2+x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}+\text{$\#$1}^5}\&\right]",1,"-((-x^2 + x^4)^(3/4)/(x*(-1 + x^2))) + ArcTan[x/(-x^2 + x^4)^(1/4)] - ArcTan[(2^(1/4)*x)/(-x^2 + x^4)^(1/4)]/(2*2^(1/4)) + ArcTanh[x/(-x^2 + x^4)^(1/4)] - ArcTanh[(2^(1/4)*x)/(-x^2 + x^4)^(1/4)]/(2*2^(1/4)) + RootSum[2 - 2*#1^4 + #1^8 & , (-Log[x] + Log[(-x^2 + x^4)^(1/4) - x*#1])/#1 & ]/4","A",1
2803,1,273,273,6.3710304,"\int \frac{a b-(a+b) x+x^2}{\left((-a+x) (-b+x)^2\right)^{2/3} \left(a^2-b^2 d-2 (a-b d) x+(1-d) x^2\right)} \, dx","IntegrateAlgebraic[(a*b - (a + b)*x + x^2)/(((-a + x)*(-b + x)^2)^(2/3)*(a^2 - b^2*d - 2*(a - b*d)*x + (1 - d)*x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a-\sqrt{3} x}{a-x-2 \sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}\right)}{2 (a-b) \sqrt[3]{d}}+\frac{\log \left(a-x+\sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}\right)}{2 (a-b) \sqrt[3]{d}}-\frac{\log \left(a^2-2 a x+x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}+d^{2/3} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}\right)}{4 (a-b) \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a-\sqrt{3} x}{a-x-2 \sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}\right)}{2 (a-b) \sqrt[3]{d}}+\frac{\log \left(a-x+\sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}\right)}{2 (a-b) \sqrt[3]{d}}-\frac{\log \left(a^2-2 a x+x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}+d^{2/3} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}\right)}{4 (a-b) \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*a - Sqrt[3]*x)/(a - x - 2*d^(1/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3))])/(2*(a - b)*d^(1/3)) + Log[a - x + d^(1/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3)]/(2*(a - b)*d^(1/3)) - Log[a^2 - 2*a*x + x^2 + (-(a*d^(1/3)) + d^(1/3)*x)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3) + d^(2/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(2/3)]/(4*(a - b)*d^(1/3))","A",1
2804,1,389,273,1.5991981,"\int \frac{\sqrt{x^2+\sqrt{1+x^4}}}{\left(-1+x^4\right) \sqrt{1+x^4}} \, dx","IntegrateAlgebraic[Sqrt[x^2 + Sqrt[1 + x^4]]/((-1 + x^4)*Sqrt[1 + x^4]),x]","\frac{1}{2} \sqrt{\frac{1}{2} \left(-1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{-\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)-\frac{1}{2} \sqrt{\frac{1}{2} \left(-1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{-\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)-\frac{1}{2} \sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{-\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)-\frac{1}{2} \sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{-\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)","\frac{1}{2} \sqrt{\frac{1}{2} \left(-1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(-1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{2} \left(-1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2 \left(-1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)",1,"(Sqrt[(-1 + Sqrt[2])/2]*ArcTan[(-Sqrt[-1/2 + 1/Sqrt[2]] + Sqrt[-1/2 + 1/Sqrt[2]]*x^2 + Sqrt[-1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])])/2 - (Sqrt[(-1 + Sqrt[2])/2]*ArcTan[(-Sqrt[1/2 + 1/Sqrt[2]] + Sqrt[1/2 + 1/Sqrt[2]]*x^2 + Sqrt[1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])])/2 - (Sqrt[(1 + Sqrt[2])/2]*ArcTanh[(-Sqrt[-1/2 + 1/Sqrt[2]] + Sqrt[-1/2 + 1/Sqrt[2]]*x^2 + Sqrt[-1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])])/2 - (Sqrt[(1 + Sqrt[2])/2]*ArcTanh[(-Sqrt[1/2 + 1/Sqrt[2]] + Sqrt[1/2 + 1/Sqrt[2]]*x^2 + Sqrt[1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])])/2","A",1
2805,1,269,274,0.8741663,"\int \frac{-1+x^6}{\sqrt[3]{-x^2+x^4} \left(1+x^6\right)} \, dx","IntegrateAlgebraic[(-1 + x^6)/((-x^2 + x^4)^(1/3)*(1 + x^6)),x]","-\frac{2}{3} \tan ^{-1}\left(\frac{x}{\sqrt[3]{-x^2+x^4}}\right)-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{2} x}{\sqrt[3]{-x^2+x^4}}\right)}{3 \sqrt[3]{2}}-\frac{1}{3} i \left(-i+\sqrt{3}\right) \tan ^{-1}\left(\frac{\left(1-i \sqrt{3}\right) x}{2 \sqrt[3]{-x^2+x^4}}\right)+\frac{1}{3} i \left(i+\sqrt{3}\right) \tan ^{-1}\left(\frac{\left(1+i \sqrt{3}\right) x}{2 \sqrt[3]{-x^2+x^4}}\right)-\frac{\tan ^{-1}\left(\frac{2^{2/3} x \sqrt[3]{-x^2+x^4}}{-2 x^2+\sqrt[3]{2} \left(-x^2+x^4\right)^{2/3}}\right)}{6 \sqrt[3]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[3]{2} x^2}{\sqrt{3}}+\frac{\left(-x^2+x^4\right)^{2/3}}{\sqrt[3]{2} \sqrt{3}}}{x \sqrt[3]{-x^2+x^4}}\right)}{2 \sqrt[3]{2} \sqrt{3}}","-\frac{2}{3} \tan ^{-1}\left(\frac{x}{\sqrt[3]{-x^2+x^4}}\right)-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{2} x}{\sqrt[3]{-x^2+x^4}}\right)}{3 \sqrt[3]{2}}-\frac{1}{3} \tan ^{-1}\left(\frac{x \sqrt[3]{-x^2+x^4}}{-x^2+\left(-x^2+x^4\right)^{2/3}}\right)-\frac{\tan ^{-1}\left(\frac{2^{2/3} x \sqrt[3]{-x^2+x^4}}{-2 x^2+\sqrt[3]{2} \left(-x^2+x^4\right)^{2/3}}\right)}{6 \sqrt[3]{2}}-\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{3}}+\frac{\left(-x^2+x^4\right)^{2/3}}{\sqrt{3}}}{x \sqrt[3]{-x^2+x^4}}\right)}{\sqrt{3}}-\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[3]{2} x^2}{\sqrt{3}}+\frac{\left(-x^2+x^4\right)^{2/3}}{\sqrt[3]{2} \sqrt{3}}}{x \sqrt[3]{-x^2+x^4}}\right)}{2 \sqrt[3]{2} \sqrt{3}}",1,"(-2*ArcTan[x/(-x^2 + x^4)^(1/3)])/3 - ArcTan[(2^(1/3)*x)/(-x^2 + x^4)^(1/3)]/(3*2^(1/3)) - (I/3)*(-I + Sqrt[3])*ArcTan[((1 - I*Sqrt[3])*x)/(2*(-x^2 + x^4)^(1/3))] + (I/3)*(I + Sqrt[3])*ArcTan[((1 + I*Sqrt[3])*x)/(2*(-x^2 + x^4)^(1/3))] - ArcTan[(2^(2/3)*x*(-x^2 + x^4)^(1/3))/(-2*x^2 + 2^(1/3)*(-x^2 + x^4)^(2/3))]/(6*2^(1/3)) - ArcTanh[((2^(1/3)*x^2)/Sqrt[3] + (-x^2 + x^4)^(2/3)/(2^(1/3)*Sqrt[3]))/(x*(-x^2 + x^4)^(1/3))]/(2*2^(1/3)*Sqrt[3])","C",1
2806,1,274,274,1.3691783,"\int \frac{\sqrt{x^2+\sqrt{1+x^4}}}{1+x} \, dx","IntegrateAlgebraic[Sqrt[x^2 + Sqrt[1 + x^4]]/(1 + x),x]","\sqrt{x^2+\sqrt{1+x^4}}-\sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\sqrt{1+\sqrt{2}} \sqrt{x^2+\sqrt{1+x^4}}\right)+\sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{-2+2 \sqrt{2}} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\sqrt{-1+\sqrt{2}} \sqrt{x^2+\sqrt{1+x^4}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)+\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2+2 \sqrt{2}} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)","\sqrt{x^2+\sqrt{1+x^4}}-\sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{x^2+\sqrt{1+x^4}}}{\sqrt{-1+\sqrt{2}}}\right)+\sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2 \left(-1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{x^2+\sqrt{1+x^4}}}{\sqrt{1+\sqrt{2}}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)+\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)",1,"Sqrt[x^2 + Sqrt[1 + x^4]] - Sqrt[-1 + Sqrt[2]]*ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[x^2 + Sqrt[1 + x^4]]] + Sqrt[-1 + Sqrt[2]]*ArcTan[(Sqrt[-2 + 2*Sqrt[2]]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2 + Sqrt[1 + x^4])] - Sqrt[1 + Sqrt[2]]*ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[x^2 + Sqrt[1 + x^4]]] - Sqrt[2]*ArcTanh[(Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2 + Sqrt[1 + x^4])] + Sqrt[1 + Sqrt[2]]*ArcTanh[(Sqrt[2 + 2*Sqrt[2]]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2 + Sqrt[1 + x^4])]","A",1
2807,1,318,275,0.8516436,"\int \frac{b+a x^2}{\left(b-a x^2\right) \sqrt[4]{b x^3+a x^5}} \, dx","IntegrateAlgebraic[(b + a*x^2)/((b - a*x^2)*(b*x^3 + a*x^5)^(1/4)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[8]{a} \sqrt[8]{b} x}{\sqrt[4]{b x^3+a x^5}}\right)}{\sqrt[4]{2} \sqrt[8]{a} \sqrt[8]{b}}-\frac{\tan ^{-1}\left(\frac{\sqrt[8]{a} \sqrt[8]{b} x}{\sqrt[8]{a} \sqrt[8]{b} x-\sqrt[4]{2} \sqrt[4]{b x^3+a x^5}}\right)}{2^{3/4} \sqrt[8]{a} \sqrt[8]{b}}+\frac{\tan ^{-1}\left(\frac{\sqrt[8]{a} \sqrt[8]{b} x}{\sqrt[8]{a} \sqrt[8]{b} x+\sqrt[4]{2} \sqrt[4]{b x^3+a x^5}}\right)}{2^{3/4} \sqrt[8]{a} \sqrt[8]{b}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[8]{a} \sqrt[8]{b} x}{\sqrt[4]{b x^3+a x^5}}\right)}{\sqrt[4]{2} \sqrt[8]{a} \sqrt[8]{b}}+\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[8]{a} \sqrt[8]{b} x^2}{\sqrt[4]{2}}+\frac{\sqrt{b x^3+a x^5}}{2^{3/4} \sqrt[8]{a} \sqrt[8]{b}}}{x \sqrt[4]{b x^3+a x^5}}\right)}{2^{3/4} \sqrt[8]{a} \sqrt[8]{b}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[8]{a} \sqrt[8]{b} x}{\sqrt[4]{b x^3+a x^5}}\right)}{\sqrt[4]{2} \sqrt[8]{a} \sqrt[8]{b}}-\frac{\tan ^{-1}\left(\frac{2^{3/4} \sqrt[8]{a} \sqrt[8]{b} x \sqrt[4]{b x^3+a x^5}}{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x^2-\sqrt{b x^3+a x^5}}\right)}{2^{3/4} \sqrt[8]{a} \sqrt[8]{b}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[8]{a} \sqrt[8]{b} x}{\sqrt[4]{b x^3+a x^5}}\right)}{\sqrt[4]{2} \sqrt[8]{a} \sqrt[8]{b}}+\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[8]{a} \sqrt[8]{b} x^2}{\sqrt[4]{2}}+\frac{\sqrt{b x^3+a x^5}}{2^{3/4} \sqrt[8]{a} \sqrt[8]{b}}}{x \sqrt[4]{b x^3+a x^5}}\right)}{2^{3/4} \sqrt[8]{a} \sqrt[8]{b}}",1,"ArcTan[(2^(1/4)*a^(1/8)*b^(1/8)*x)/(b*x^3 + a*x^5)^(1/4)]/(2^(1/4)*a^(1/8)*b^(1/8)) - ArcTan[(a^(1/8)*b^(1/8)*x)/(a^(1/8)*b^(1/8)*x - 2^(1/4)*(b*x^3 + a*x^5)^(1/4))]/(2^(3/4)*a^(1/8)*b^(1/8)) + ArcTan[(a^(1/8)*b^(1/8)*x)/(a^(1/8)*b^(1/8)*x + 2^(1/4)*(b*x^3 + a*x^5)^(1/4))]/(2^(3/4)*a^(1/8)*b^(1/8)) + ArcTanh[(2^(1/4)*a^(1/8)*b^(1/8)*x)/(b*x^3 + a*x^5)^(1/4)]/(2^(1/4)*a^(1/8)*b^(1/8)) + ArcTanh[((a^(1/8)*b^(1/8)*x^2)/2^(1/4) + Sqrt[b*x^3 + a*x^5]/(2^(3/4)*a^(1/8)*b^(1/8)))/(x*(b*x^3 + a*x^5)^(1/4))]/(2^(3/4)*a^(1/8)*b^(1/8))","A",1
2808,1,257,275,2.3289564,"\int \frac{\left(4+x^3\right) \left(1+x^3+x^4\right)}{\sqrt[4]{1+x^3} \left(1+2 x^3+x^6+x^8\right)} \, dx","IntegrateAlgebraic[((4 + x^3)*(1 + x^3 + x^4))/((1 + x^3)^(1/4)*(1 + 2*x^3 + x^6 + x^8)),x]","-\sqrt{\frac{1}{2} \left(2-\sqrt{2}\right)} \tan ^{-1}\left(\frac{-\sqrt{1-\frac{1}{\sqrt{2}}} x^2+\sqrt{1-\frac{1}{\sqrt{2}}} \sqrt{1+x^3}}{x \sqrt[4]{1+x^3}}\right)-\sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} \tan ^{-1}\left(\frac{-\sqrt{1+\frac{1}{\sqrt{2}}} x^2+\sqrt{1+\frac{1}{\sqrt{2}}} \sqrt{1+x^3}}{x \sqrt[4]{1+x^3}}\right)+\sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{1+x^3}}{x^2+\sqrt{1+x^3}}\right)+\sqrt{\frac{1}{2} \left(2-\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{1+x^3}}{x^2+\sqrt{1+x^3}}\right)","-\sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\frac{\sqrt{2-\sqrt{2}} x^2}{-2+\sqrt{2}}-\frac{\sqrt{2-\sqrt{2}} \sqrt{1+x^3}}{-2+\sqrt{2}}}{x \sqrt[4]{1+x^3}}\right)-\sqrt{\frac{1}{2} \left(2-\sqrt{2}\right)} \tan ^{-1}\left(\frac{-\frac{x^2}{\sqrt{2+\sqrt{2}}}+\frac{\sqrt{1+x^3}}{\sqrt{2+\sqrt{2}}}}{x \sqrt[4]{1+x^3}}\right)+\sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{1+x^3}}{x^2+\sqrt{1+x^3}}\right)+\sqrt{\frac{1}{2} \left(2-\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{1+x^3}}{x^2+\sqrt{1+x^3}}\right)",1,"-(Sqrt[(2 - Sqrt[2])/2]*ArcTan[(-(Sqrt[1 - 1/Sqrt[2]]*x^2) + Sqrt[1 - 1/Sqrt[2]]*Sqrt[1 + x^3])/(x*(1 + x^3)^(1/4))]) - Sqrt[(2 + Sqrt[2])/2]*ArcTan[(-(Sqrt[1 + 1/Sqrt[2]]*x^2) + Sqrt[1 + 1/Sqrt[2]]*Sqrt[1 + x^3])/(x*(1 + x^3)^(1/4))] + Sqrt[(2 + Sqrt[2])/2]*ArcTanh[(Sqrt[2 - Sqrt[2]]*x*(1 + x^3)^(1/4))/(x^2 + Sqrt[1 + x^3])] + Sqrt[(2 - Sqrt[2])/2]*ArcTanh[(Sqrt[2 + Sqrt[2]]*x*(1 + x^3)^(1/4))/(x^2 + Sqrt[1 + x^3])]","A",1
2809,1,277,275,0.1908645,"\int \frac{x^2 \sqrt{1+x}}{x^2-\sqrt{1+x} \sqrt{1+\sqrt{1+x}}} \, dx","IntegrateAlgebraic[(x^2*Sqrt[1 + x])/(x^2 - Sqrt[1 + x]*Sqrt[1 + Sqrt[1 + x]]),x]","4 \sqrt{1+\sqrt{1+x}}+\frac{2}{3} \left(-2+(1+x)^{3/2}\right)+\frac{2}{55} \left(25+9 \sqrt{5}\right) \log \left(1+\sqrt{5}-2 \sqrt{1+\sqrt{1+x}}\right)-\frac{2}{55} \left(-25+9 \sqrt{5}\right) \log \left(-1+\sqrt{5}+2 \sqrt{1+\sqrt{1+x}}\right)-\frac{4}{11} \text{RootSum}\left[-1+\text{$\#$1}-\text{$\#$1}^2-2 \text{$\#$1}^3+\text{$\#$1}^4+\text{$\#$1}^5\&,\frac{-9 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)+\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}-2 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^3+5 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^4}{1-2 \text{$\#$1}-6 \text{$\#$1}^2+4 \text{$\#$1}^3+5 \text{$\#$1}^4}\&\right]","\frac{2}{3} (1+x)^{3/2}+4 \sqrt{1+\sqrt{1+x}}+\frac{2}{55} \left(25+9 \sqrt{5}\right) \log \left(1+\sqrt{5}-2 \sqrt{1+\sqrt{1+x}}\right)-\frac{2}{55} \left(-25+9 \sqrt{5}\right) \log \left(-1+\sqrt{5}+2 \sqrt{1+\sqrt{1+x}}\right)-\frac{4}{11} \text{RootSum}\left[-1+\text{$\#$1}-\text{$\#$1}^2-2 \text{$\#$1}^3+\text{$\#$1}^4+\text{$\#$1}^5\&,\frac{-9 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)+\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}-2 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^3+5 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^4}{1-2 \text{$\#$1}-6 \text{$\#$1}^2+4 \text{$\#$1}^3+5 \text{$\#$1}^4}\&\right]",1,"4*Sqrt[1 + Sqrt[1 + x]] + (2*(-2 + (1 + x)^(3/2)))/3 + (2*(25 + 9*Sqrt[5])*Log[1 + Sqrt[5] - 2*Sqrt[1 + Sqrt[1 + x]]])/55 - (2*(-25 + 9*Sqrt[5])*Log[-1 + Sqrt[5] + 2*Sqrt[1 + Sqrt[1 + x]]])/55 - (4*RootSum[-1 + #1 - #1^2 - 2*#1^3 + #1^4 + #1^5 & , (-9*Log[Sqrt[1 + Sqrt[1 + x]] - #1] + Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1 - 2*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^3 + 5*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^4)/(1 - 2*#1 - 6*#1^2 + 4*#1^3 + 5*#1^4) & ])/11","A",1
2810,1,277,275,0.0012027,"\int \frac{x^2 \sqrt{1+x}}{x^2-\sqrt{1+x} \sqrt{1+\sqrt{1+x}}} \, dx","IntegrateAlgebraic[(x^2*Sqrt[1 + x])/(x^2 - Sqrt[1 + x]*Sqrt[1 + Sqrt[1 + x]]),x]","4 \sqrt{1+\sqrt{1+x}}+\frac{2}{3} \left(-2+(1+x)^{3/2}\right)+\frac{2}{55} \left(25+9 \sqrt{5}\right) \log \left(1+\sqrt{5}-2 \sqrt{1+\sqrt{1+x}}\right)-\frac{2}{55} \left(-25+9 \sqrt{5}\right) \log \left(-1+\sqrt{5}+2 \sqrt{1+\sqrt{1+x}}\right)-\frac{4}{11} \text{RootSum}\left[-1+\text{$\#$1}-\text{$\#$1}^2-2 \text{$\#$1}^3+\text{$\#$1}^4+\text{$\#$1}^5\&,\frac{-9 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)+\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}-2 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^3+5 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^4}{1-2 \text{$\#$1}-6 \text{$\#$1}^2+4 \text{$\#$1}^3+5 \text{$\#$1}^4}\&\right]","\frac{2}{3} (1+x)^{3/2}+4 \sqrt{1+\sqrt{1+x}}+\frac{2}{55} \left(25+9 \sqrt{5}\right) \log \left(1+\sqrt{5}-2 \sqrt{1+\sqrt{1+x}}\right)-\frac{2}{55} \left(-25+9 \sqrt{5}\right) \log \left(-1+\sqrt{5}+2 \sqrt{1+\sqrt{1+x}}\right)-\frac{4}{11} \text{RootSum}\left[-1+\text{$\#$1}-\text{$\#$1}^2-2 \text{$\#$1}^3+\text{$\#$1}^4+\text{$\#$1}^5\&,\frac{-9 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right)+\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}-2 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^3+5 \log \left(\sqrt{1+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^4}{1-2 \text{$\#$1}-6 \text{$\#$1}^2+4 \text{$\#$1}^3+5 \text{$\#$1}^4}\&\right]",1,"4*Sqrt[1 + Sqrt[1 + x]] + (2*(-2 + (1 + x)^(3/2)))/3 + (2*(25 + 9*Sqrt[5])*Log[1 + Sqrt[5] - 2*Sqrt[1 + Sqrt[1 + x]]])/55 - (2*(-25 + 9*Sqrt[5])*Log[-1 + Sqrt[5] + 2*Sqrt[1 + Sqrt[1 + x]]])/55 - (4*RootSum[-1 + #1 - #1^2 - 2*#1^3 + #1^4 + #1^5 & , (-9*Log[Sqrt[1 + Sqrt[1 + x]] - #1] + Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1 - 2*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^3 + 5*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^4)/(1 - 2*#1 - 6*#1^2 + 4*#1^3 + 5*#1^4) & ])/11","A",1
2811,1,276,276,1.3844515,"\int \frac{\left(-b-a x+x^4\right) \sqrt[4]{b x^3+a x^4}}{-b+a x} \, dx","IntegrateAlgebraic[((-b - a*x + x^4)*(b*x^3 + a*x^4)^(1/4))/(-b + a*x),x]","\frac{\sqrt[4]{b x^3+a x^4} \left(-65280 a^4 b+32705 b^4-15360 a^5 x+16420 a b^3 x+10400 a^2 b^2 x^2+8064 a^3 b x^3+6144 a^4 x^4\right)}{30720 a^5}+\frac{\left(19712 a^4 b^2-9843 b^5\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{4096 a^{23/4}}-\frac{2 \sqrt[4]{2} \left(2 a^4 b^2-b^5\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{a^{23/4}}+\frac{\left(-19712 a^4 b^2+9843 b^5\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{4096 a^{23/4}}+\frac{2 \sqrt[4]{2} \left(2 a^4 b^2-b^5\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{a^{23/4}}","\frac{\sqrt[4]{b x^3+a x^4} \left(-65280 a^4 b+32705 b^4-15360 a^5 x+16420 a b^3 x+10400 a^2 b^2 x^2+8064 a^3 b x^3+6144 a^4 x^4\right)}{30720 a^5}+\frac{\left(19712 a^4 b^2-9843 b^5\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{4096 a^{23/4}}-\frac{2 \sqrt[4]{2} \left(2 a^4 b^2-b^5\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{a^{23/4}}+\frac{\left(-19712 a^4 b^2+9843 b^5\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{4096 a^{23/4}}+\frac{2 \sqrt[4]{2} \left(2 a^4 b^2-b^5\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{a^{23/4}}",1,"((b*x^3 + a*x^4)^(1/4)*(-65280*a^4*b + 32705*b^4 - 15360*a^5*x + 16420*a*b^3*x + 10400*a^2*b^2*x^2 + 8064*a^3*b*x^3 + 6144*a^4*x^4))/(30720*a^5) + ((19712*a^4*b^2 - 9843*b^5)*ArcTan[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(4096*a^(23/4)) - (2*2^(1/4)*(2*a^4*b^2 - b^5)*ArcTan[(2^(1/4)*a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/a^(23/4) + ((-19712*a^4*b^2 + 9843*b^5)*ArcTanh[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(4096*a^(23/4)) + (2*2^(1/4)*(2*a^4*b^2 - b^5)*ArcTanh[(2^(1/4)*a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/a^(23/4)","A",1
2812,1,276,276,3.1531087,"\int \frac{(-4 a+b+3 x) \left(-b^3+3 b^2 x-3 b x^2+x^3\right)}{\left((-a+x) (-b+x)^2\right)^{2/3} \left(a+b^4 d-\left(1+4 b^3 d\right) x+6 b^2 d x^2-4 b d x^3+d x^4\right)} \, dx","IntegrateAlgebraic[((-4*a + b + 3*x)*(-b^3 + 3*b^2*x - 3*b*x^2 + x^3))/(((-a + x)*(-b + x)^2)^(2/3)*(a + b^4*d - (1 + 4*b^3*d)*x + 6*b^2*d*x^2 - 4*b*d*x^3 + d*x^4)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}}{-2 a+2 x+\sqrt[3]{d} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}}\right)}{d^{2/3}}+\frac{\log \left(a-x+\sqrt[3]{d} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}\right)}{d^{2/3}}-\frac{\log \left(a^2-2 a x+x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}+d^{2/3} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{4/3}\right)}{2 d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}}{-2 a+2 x+\sqrt[3]{d} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}}\right)}{d^{2/3}}+\frac{\log \left(a-x+\sqrt[3]{d} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}\right)}{d^{2/3}}-\frac{\log \left(a^2-2 a x+x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}+d^{2/3} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{4/3}\right)}{2 d^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(2/3))/(-2*a + 2*x + d^(1/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(2/3))])/d^(2/3) + Log[a - x + d^(1/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(2/3)]/d^(2/3) - Log[a^2 - 2*a*x + x^2 + (-(a*d^(1/3)) + d^(1/3)*x)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(2/3) + d^(2/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(4/3)]/(2*d^(2/3))","A",1
2813,1,274,276,0.6262948,"\int \frac{x^2 (-2+(1+k) x)}{\sqrt[3]{(1-x) x (1-k x)} \left(1-(2+2 k) x+\left(1+4 k+k^2\right) x^2-\left(2 k+2 k^2\right) x^3+\left(-b+k^2\right) x^4\right)} \, dx","IntegrateAlgebraic[(x^2*(-2 + (1 + k)*x))/(((1 - x)*x*(1 - k*x))^(1/3)*(1 - (2 + 2*k)*x + (1 + 4*k + k^2)*x^2 - (2*k + 2*k^2)*x^3 + (-b + k^2)*x^4)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\frac{x^2}{\sqrt{3}}+\frac{2 \left(x+(-1-k) x^2+k x^3\right)^{2/3}}{\sqrt{3} \sqrt[3]{b}}}{x^2}\right)}{2 b^{2/3}}+\frac{\log \left(-\sqrt[6]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{2 b^{2/3}}+\frac{\log \left(\sqrt[6]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{2 b^{2/3}}-\frac{\log \left(\sqrt[3]{b} x^2-\sqrt[6]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{4 b^{2/3}}-\frac{\log \left(\sqrt[3]{b} x^2+\sqrt[6]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{4 b^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} x^2}{\sqrt[3]{b} x^2+2 \left(x+(-1-k) x^2+k x^3\right)^{2/3}}\right)}{2 b^{2/3}}+\frac{\log \left(-\sqrt[6]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{2 b^{2/3}}+\frac{\log \left(\sqrt[6]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{2 b^{2/3}}-\frac{\log \left(\sqrt[3]{b} x^2-\sqrt[6]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{4 b^{2/3}}-\frac{\log \left(\sqrt[3]{b} x^2+\sqrt[6]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{4 b^{2/3}}",1,"-1/2*(Sqrt[3]*ArcTan[(x^2/Sqrt[3] + (2*(x + (-1 - k)*x^2 + k*x^3)^(2/3))/(Sqrt[3]*b^(1/3)))/x^2])/b^(2/3) + Log[-(b^(1/6)*x) + (x + (-1 - k)*x^2 + k*x^3)^(1/3)]/(2*b^(2/3)) + Log[b^(1/6)*x + (x + (-1 - k)*x^2 + k*x^3)^(1/3)]/(2*b^(2/3)) - Log[b^(1/3)*x^2 - b^(1/6)*x*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + (x + (-1 - k)*x^2 + k*x^3)^(2/3)]/(4*b^(2/3)) - Log[b^(1/3)*x^2 + b^(1/6)*x*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + (x + (-1 - k)*x^2 + k*x^3)^(2/3)]/(4*b^(2/3))","A",1
2814,1,276,276,0.4507961,"\int \frac{1}{c+\sqrt{a x+\sqrt{b^2+a^2 x^2}}} \, dx","IntegrateAlgebraic[(c + Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])^(-1),x]","\frac{2 b^2 c^3+2 a b^2 c x+4 a^2 c^3 x^2+\left(-b^2 c^2+3 a b^2 x+2 a c^4 x\right) \sqrt{a x+\sqrt{b^2+a^2 x^2}}+\sqrt{b^2+a^2 x^2} \left(2 b^2 c+4 a c^3 x+\left(3 b^2+2 c^4\right) \sqrt{a x+\sqrt{b^2+a^2 x^2}}\right)}{2 a c^3 \left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}+\frac{b^2 \log \left(a x+\sqrt{b^2+a^2 x^2}\right)}{2 a c^3}-\frac{b^2 \log \left(c+\sqrt{a x+\sqrt{b^2+a^2 x^2}}\right)}{a c^3}-\frac{c \log \left(c+\sqrt{a x+\sqrt{b^2+a^2 x^2}}\right)}{a}","\frac{2 b^2 c^3+2 a b^2 c x+4 a^2 c^3 x^2+\left(-b^2 c^2+3 a b^2 x+2 a c^4 x\right) \sqrt{a x+\sqrt{b^2+a^2 x^2}}+\sqrt{b^2+a^2 x^2} \left(2 b^2 c+4 a c^3 x+\left(3 b^2+2 c^4\right) \sqrt{a x+\sqrt{b^2+a^2 x^2}}\right)}{2 a c^3 \left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}+\frac{b^2 \log \left(a x+\sqrt{b^2+a^2 x^2}\right)}{2 a c^3}-\frac{b^2 \log \left(c+\sqrt{a x+\sqrt{b^2+a^2 x^2}}\right)}{a c^3}-\frac{c \log \left(c+\sqrt{a x+\sqrt{b^2+a^2 x^2}}\right)}{a}",1,"(2*b^2*c^3 + 2*a*b^2*c*x + 4*a^2*c^3*x^2 + (-(b^2*c^2) + 3*a*b^2*x + 2*a*c^4*x)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] + Sqrt[b^2 + a^2*x^2]*(2*b^2*c + 4*a*c^3*x + (3*b^2 + 2*c^4)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]))/(2*a*c^3*(a*x + Sqrt[b^2 + a^2*x^2])^(3/2)) + (b^2*Log[a*x + Sqrt[b^2 + a^2*x^2]])/(2*a*c^3) - (b^2*Log[c + Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]])/(a*c^3) - (c*Log[c + Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]])/a","A",1
2815,1,218,278,16.2460138,"\int \frac{(-7+x) \sqrt[3]{1+x-5 x^2+3 x^3}}{(-5+x) (-1+x)^3} \, dx","IntegrateAlgebraic[((-7 + x)*(1 + x - 5*x^2 + 3*x^3)^(1/3))/((-5 + x)*(-1 + x)^3),x]","\frac{\sqrt[3]{-1+x} (1+3 x)^{2/3} \sqrt[3]{(-1+x)^2 (1+3 x)} \left(-\frac{3 \sqrt[3]{1+3 x} \left(4+\frac{3 (1+3 x)}{-1+x}\right)}{32 \sqrt[3]{-1+x}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{\sqrt[3]{2} \sqrt[3]{1+3 x}}{\sqrt{3} \sqrt[3]{-1+x}}\right)}{4 \sqrt[3]{2}}-\frac{\log \left(-2+\frac{\sqrt[3]{2} \sqrt[3]{1+3 x}}{\sqrt[3]{-1+x}}\right)}{4 \sqrt[3]{2}}+\frac{\log \left(4+\frac{2 \sqrt[3]{2} \sqrt[3]{1+3 x}}{\sqrt[3]{-1+x}}+\frac{2^{2/3} (1+3 x)^{2/3}}{(-1+x)^{2/3}}\right)}{8 \sqrt[3]{2}}\right)}{-1-2 x+3 x^2}","\frac{(-1+x)^{4/3} (1+3 x)^{2/3} \left(-\frac{21 \sqrt[3]{3} \left(-16 \sqrt[3]{1+3 x}+7 (1+3 x)^{4/3}\right)}{64 (-3+3 x)^{4/3}}+\frac{3 \sqrt[3]{3} \left(-80 \sqrt[3]{1+3 x}+23 (1+3 x)^{4/3}\right)}{64 (-3+3 x)^{4/3}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{3^{5/6} \sqrt[3]{1+3 x}}{2\ 2^{2/3} \sqrt[3]{-3+3 x}+\sqrt[3]{3} \sqrt[3]{1+3 x}}\right)}{4 \sqrt[3]{2}}-\frac{\log \left(6^{2/3} \sqrt[3]{-3+3 x}-3 \sqrt[3]{1+3 x}\right)}{4 \sqrt[3]{2}}+\frac{\log \left(2 \sqrt[3]{6} (-3+3 x)^{2/3}+6^{2/3} \sqrt[3]{-3+3 x} \sqrt[3]{1+3 x}+3 (1+3 x)^{2/3}\right)}{8 \sqrt[3]{2}}\right)}{\left((-1+x)^2 (1+3 x)\right)^{2/3}}",1,"((-1 + x)^(1/3)*(1 + 3*x)^(2/3)*((-1 + x)^2*(1 + 3*x))^(1/3)*((-3*(1 + 3*x)^(1/3)*(4 + (3*(1 + 3*x))/(-1 + x)))/(32*(-1 + x)^(1/3)) + (Sqrt[3]*ArcTan[1/Sqrt[3] + (2^(1/3)*(1 + 3*x)^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))])/(4*2^(1/3)) - Log[-2 + (2^(1/3)*(1 + 3*x)^(1/3))/(-1 + x)^(1/3)]/(4*2^(1/3)) + Log[4 + (2*2^(1/3)*(1 + 3*x)^(1/3))/(-1 + x)^(1/3) + (2^(2/3)*(1 + 3*x)^(2/3))/(-1 + x)^(2/3)]/(8*2^(1/3))))/(-1 - 2*x + 3*x^2)","A",1
2816,1,278,278,1.104155,"\int \frac{b^2+a^2 x^2}{\left(-b^2+a^2 x^2\right)^3 \sqrt[3]{-b x^2+a x^3}} \, dx","IntegrateAlgebraic[(b^2 + a^2*x^2)/((-b^2 + a^2*x^2)^3*(-(b*x^2) + a*x^3)^(1/3)),x]","-\frac{\left(-b x^2+a x^3\right)^{2/3} \left(-1190 b^4+91 a b^3 x+1503 a^2 b^2 x^2-67 a^3 b x^3-625 a^4 x^4\right)}{672 b^4 x (b-a x)^3 (b+a x)^2}-\frac{59 \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a} x}{\sqrt[3]{a} x+2^{2/3} \sqrt[3]{-b x^2+a x^3}}\right)}{48 \sqrt[3]{2} \sqrt{3} \sqrt[3]{a} b^4}+\frac{59 \log \left(-2 \sqrt[3]{a} x+2^{2/3} \sqrt[3]{-b x^2+a x^3}\right)}{144 \sqrt[3]{2} \sqrt[3]{a} b^4}-\frac{59 \log \left(2 a^{2/3} x^2+2^{2/3} \sqrt[3]{a} x \sqrt[3]{-b x^2+a x^3}+\sqrt[3]{2} \left(-b x^2+a x^3\right)^{2/3}\right)}{288 \sqrt[3]{2} \sqrt[3]{a} b^4}","-\frac{\left(-b x^2+a x^3\right)^{2/3} \left(-1190 b^4+91 a b^3 x+1503 a^2 b^2 x^2-67 a^3 b x^3-625 a^4 x^4\right)}{672 b^4 x (b-a x)^3 (b+a x)^2}-\frac{59 \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a} x}{\sqrt[3]{a} x+2^{2/3} \sqrt[3]{-b x^2+a x^3}}\right)}{48 \sqrt[3]{2} \sqrt{3} \sqrt[3]{a} b^4}+\frac{59 \log \left(-2 \sqrt[3]{a} x+2^{2/3} \sqrt[3]{-b x^2+a x^3}\right)}{144 \sqrt[3]{2} \sqrt[3]{a} b^4}-\frac{59 \log \left(2 a^{2/3} x^2+2^{2/3} \sqrt[3]{a} x \sqrt[3]{-b x^2+a x^3}+\sqrt[3]{2} \left(-b x^2+a x^3\right)^{2/3}\right)}{288 \sqrt[3]{2} \sqrt[3]{a} b^4}",1,"-1/672*((-(b*x^2) + a*x^3)^(2/3)*(-1190*b^4 + 91*a*b^3*x + 1503*a^2*b^2*x^2 - 67*a^3*b*x^3 - 625*a^4*x^4))/(b^4*x*(b - a*x)^3*(b + a*x)^2) - (59*ArcTan[(Sqrt[3]*a^(1/3)*x)/(a^(1/3)*x + 2^(2/3)*(-(b*x^2) + a*x^3)^(1/3))])/(48*2^(1/3)*Sqrt[3]*a^(1/3)*b^4) + (59*Log[-2*a^(1/3)*x + 2^(2/3)*(-(b*x^2) + a*x^3)^(1/3)])/(144*2^(1/3)*a^(1/3)*b^4) - (59*Log[2*a^(2/3)*x^2 + 2^(2/3)*a^(1/3)*x*(-(b*x^2) + a*x^3)^(1/3) + 2^(1/3)*(-(b*x^2) + a*x^3)^(2/3)])/(288*2^(1/3)*a^(1/3)*b^4)","A",1
2817,1,278,278,0.6767286,"\int \frac{d+c x}{(-d+c x) \sqrt{a x+\sqrt{b^2+a^2 x^2}}} \, dx","IntegrateAlgebraic[(d + c*x)/((-d + c*x)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]),x]","\frac{2 (6 a d+3 a c x) \sqrt{b^2+a^2 x^2}+2 \left(b^2 c+6 a^2 d x+3 a^2 c x^2\right)}{3 a c \left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}+\frac{4 d \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{-a d-\sqrt{b^2 c^2+a^2 d^2}}}\right)}{\sqrt{c} \sqrt{-a d-\sqrt{b^2 c^2+a^2 d^2}}}+\frac{4 d \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{-a d+\sqrt{b^2 c^2+a^2 d^2}}}\right)}{\sqrt{c} \sqrt{-a d+\sqrt{b^2 c^2+a^2 d^2}}}","\frac{2 (6 a d+3 a c x) \sqrt{b^2+a^2 x^2}+2 \left(b^2 c+6 a^2 d x+3 a^2 c x^2\right)}{3 a c \left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}+\frac{4 d \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{-a d-\sqrt{b^2 c^2+a^2 d^2}}}\right)}{\sqrt{c} \sqrt{-a d-\sqrt{b^2 c^2+a^2 d^2}}}+\frac{4 d \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{-a d+\sqrt{b^2 c^2+a^2 d^2}}}\right)}{\sqrt{c} \sqrt{-a d+\sqrt{b^2 c^2+a^2 d^2}}}",1,"(2*(6*a*d + 3*a*c*x)*Sqrt[b^2 + a^2*x^2] + 2*(b^2*c + 6*a^2*d*x + 3*a^2*c*x^2))/(3*a*c*(a*x + Sqrt[b^2 + a^2*x^2])^(3/2)) + (4*d*ArcTan[(Sqrt[c]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/Sqrt[-(a*d) - Sqrt[b^2*c^2 + a^2*d^2]]])/(Sqrt[c]*Sqrt[-(a*d) - Sqrt[b^2*c^2 + a^2*d^2]]) + (4*d*ArcTan[(Sqrt[c]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/Sqrt[-(a*d) + Sqrt[b^2*c^2 + a^2*d^2]]])/(Sqrt[c]*Sqrt[-(a*d) + Sqrt[b^2*c^2 + a^2*d^2]])","A",1
2818,1,280,279,1.4406555,"\int \frac{\sqrt[4]{-b x^3+a x^4}}{-d-2 c x+x^2} \, dx","IntegrateAlgebraic[(-(b*x^3) + a*x^4)^(1/4)/(-d - 2*c*x + x^2),x]","-2 \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)+2 \sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)+\frac{1}{2} \text{RootSum}\left[b^2-2 a b c-a^2 d+2 b c \text{$\#$1}^4+2 a d \text{$\#$1}^4-d \text{$\#$1}^8\&,\frac{-b^2 \log (x)+2 a b c \log (x)+a^2 d \log (x)+b^2 \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right)-2 a b c \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right)-a^2 d \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right)-a d \log (x) \text{$\#$1}^4+a d \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-b c \text{$\#$1}^3-a d \text{$\#$1}^3+d \text{$\#$1}^7}\&\right]","-2 \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)+2 \sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)+\frac{1}{2} \text{RootSum}\left[b^2-2 a b c-a^2 d+2 b c \text{$\#$1}^4+2 a d \text{$\#$1}^4-d \text{$\#$1}^8\&,\frac{b^2 \log (x)-2 a b c \log (x)-a^2 d \log (x)-b^2 \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right)+2 a b c \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right)+a^2 d \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right)+a d \log (x) \text{$\#$1}^4-a d \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{b c \text{$\#$1}^3+a d \text{$\#$1}^3-d \text{$\#$1}^7}\&\right]",1,"-2*a^(1/4)*ArcTan[(a^(1/4)*x)/(-(b*x^3) + a*x^4)^(1/4)] + 2*a^(1/4)*ArcTanh[(a^(1/4)*x)/(-(b*x^3) + a*x^4)^(1/4)] + RootSum[b^2 - 2*a*b*c - a^2*d + 2*b*c*#1^4 + 2*a*d*#1^4 - d*#1^8 & , (-(b^2*Log[x]) + 2*a*b*c*Log[x] + a^2*d*Log[x] + b^2*Log[(-(b*x^3) + a*x^4)^(1/4) - x*#1] - 2*a*b*c*Log[(-(b*x^3) + a*x^4)^(1/4) - x*#1] - a^2*d*Log[(-(b*x^3) + a*x^4)^(1/4) - x*#1] - a*d*Log[x]*#1^4 + a*d*Log[(-(b*x^3) + a*x^4)^(1/4) - x*#1]*#1^4)/(-(b*c*#1^3) - a*d*#1^3 + d*#1^7) & ]/2","A",1
2819,1,280,279,0.0017905,"\int \frac{\sqrt[4]{-b x^3+a x^4}}{-d-2 c x+x^2} \, dx","IntegrateAlgebraic[(-(b*x^3) + a*x^4)^(1/4)/(-d - 2*c*x + x^2),x]","-2 \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)+2 \sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)+\frac{1}{2} \text{RootSum}\left[b^2-2 a b c-a^2 d+2 b c \text{$\#$1}^4+2 a d \text{$\#$1}^4-d \text{$\#$1}^8\&,\frac{-b^2 \log (x)+2 a b c \log (x)+a^2 d \log (x)+b^2 \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right)-2 a b c \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right)-a^2 d \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right)-a d \log (x) \text{$\#$1}^4+a d \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-b c \text{$\#$1}^3-a d \text{$\#$1}^3+d \text{$\#$1}^7}\&\right]","-2 \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)+2 \sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)+\frac{1}{2} \text{RootSum}\left[b^2-2 a b c-a^2 d+2 b c \text{$\#$1}^4+2 a d \text{$\#$1}^4-d \text{$\#$1}^8\&,\frac{b^2 \log (x)-2 a b c \log (x)-a^2 d \log (x)-b^2 \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right)+2 a b c \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right)+a^2 d \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right)+a d \log (x) \text{$\#$1}^4-a d \log \left(\sqrt[4]{-b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{b c \text{$\#$1}^3+a d \text{$\#$1}^3-d \text{$\#$1}^7}\&\right]",1,"-2*a^(1/4)*ArcTan[(a^(1/4)*x)/(-(b*x^3) + a*x^4)^(1/4)] + 2*a^(1/4)*ArcTanh[(a^(1/4)*x)/(-(b*x^3) + a*x^4)^(1/4)] + RootSum[b^2 - 2*a*b*c - a^2*d + 2*b*c*#1^4 + 2*a*d*#1^4 - d*#1^8 & , (-(b^2*Log[x]) + 2*a*b*c*Log[x] + a^2*d*Log[x] + b^2*Log[(-(b*x^3) + a*x^4)^(1/4) - x*#1] - 2*a*b*c*Log[(-(b*x^3) + a*x^4)^(1/4) - x*#1] - a^2*d*Log[(-(b*x^3) + a*x^4)^(1/4) - x*#1] - a*d*Log[x]*#1^4 + a*d*Log[(-(b*x^3) + a*x^4)^(1/4) - x*#1]*#1^4)/(-(b*c*#1^3) - a*d*#1^3 + d*#1^7) & ]/2","A",1
2820,1,279,279,3.4672875,"\int \frac{x (-1+k x) (-1+(-1+2 k) x)}{\sqrt[3]{(1-x) x (1-k x)} \left(-1+4 x+(-6+b) x^2+(4-2 b k) x^3+\left(-1+b k^2\right) x^4\right)} \, dx","IntegrateAlgebraic[(x*(-1 + k*x)*(-1 + (-1 + 2*k)*x))/(((1 - x)*x*(1 - k*x))^(1/3)*(-1 + 4*x + (-6 + b)*x^2 + (4 - 2*b*k)*x^3 + (-1 + b*k^2)*x^4)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}{2-2 x+\sqrt[6]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{2 b^{5/6}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}{-2+2 x+\sqrt[6]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{2 b^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}{-1+x}\right)}{b^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\left(-\sqrt[6]{b}+\sqrt[6]{b} x\right) \sqrt[3]{x+(-1-k) x^2+k x^3}}{1-2 x+x^2+\sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}}\right)}{2 b^{5/6}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}{2-2 x+\sqrt[6]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{2 b^{5/6}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}{-2+2 x+\sqrt[6]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{2 b^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}{-1+x}\right)}{b^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\left(-\sqrt[6]{b}+\sqrt[6]{b} x\right) \sqrt[3]{x+(-1-k) x^2+k x^3}}{1-2 x+x^2+\sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}}\right)}{2 b^{5/6}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*b^(1/6)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))/(2 - 2*x + b^(1/6)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/(2*b^(5/6)) - (Sqrt[3]*ArcTan[(Sqrt[3]*b^(1/6)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))/(-2 + 2*x + b^(1/6)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/(2*b^(5/6)) + ArcTanh[(b^(1/6)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))/(-1 + x)]/b^(5/6) + ArcTanh[((-b^(1/6) + b^(1/6)*x)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))/(1 - 2*x + x^2 + b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3))]/(2*b^(5/6))","A",1
2821,1,279,279,2.7425356,"\int \frac{1+(-2+3 k) x-\left(k+4 k^2\right) x^2+3 k^2 x^3}{\sqrt[3]{(1-x) x (1-k x)} \left(-b+(1+5 b k) x-\left(1+10 b k^2\right) x^2+10 b k^3 x^3-5 b k^4 x^4+b k^5 x^5\right)} \, dx","IntegrateAlgebraic[(1 + (-2 + 3*k)*x - (k + 4*k^2)*x^2 + 3*k^2*x^3)/(((1 - x)*x*(1 - k*x))^(1/3)*(-b + (1 + 5*b*k)*x - (1 + 10*b*k^2)*x^2 + 10*b*k^3*x^3 - 5*b*k^4*x^4 + b*k^5*x^5)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{x+(-1-k) x^2+k x^3}}{2 \sqrt[3]{b}-4 \sqrt[3]{b} k x+2 \sqrt[3]{b} k^2 x^2+\sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{\sqrt[3]{b}}+\frac{\log \left(-\sqrt[3]{b}+2 \sqrt[3]{b} k x-\sqrt[3]{b} k^2 x^2+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{\sqrt[3]{b}}-\frac{\log \left(b^{2/3}-4 b^{2/3} k x+6 b^{2/3} k^2 x^2-4 b^{2/3} k^3 x^3+b^{2/3} k^4 x^4+\left(\sqrt[3]{b}-2 \sqrt[3]{b} k x+\sqrt[3]{b} k^2 x^2\right) \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 \sqrt[3]{b}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{x+(-1-k) x^2+k x^3}}{2 \sqrt[3]{b}-4 \sqrt[3]{b} k x+2 \sqrt[3]{b} k^2 x^2+\sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{\sqrt[3]{b}}+\frac{\log \left(-\sqrt[3]{b}+2 \sqrt[3]{b} k x-\sqrt[3]{b} k^2 x^2+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{\sqrt[3]{b}}-\frac{\log \left(b^{2/3}-4 b^{2/3} k x+6 b^{2/3} k^2 x^2-4 b^{2/3} k^3 x^3+b^{2/3} k^4 x^4+\left(\sqrt[3]{b}-2 \sqrt[3]{b} k x+\sqrt[3]{b} k^2 x^2\right) \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 \sqrt[3]{b}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(x + (-1 - k)*x^2 + k*x^3)^(1/3))/(2*b^(1/3) - 4*b^(1/3)*k*x + 2*b^(1/3)*k^2*x^2 + (x + (-1 - k)*x^2 + k*x^3)^(1/3))])/b^(1/3) + Log[-b^(1/3) + 2*b^(1/3)*k*x - b^(1/3)*k^2*x^2 + (x + (-1 - k)*x^2 + k*x^3)^(1/3)]/b^(1/3) - Log[b^(2/3) - 4*b^(2/3)*k*x + 6*b^(2/3)*k^2*x^2 - 4*b^(2/3)*k^3*x^3 + b^(2/3)*k^4*x^4 + (b^(1/3) - 2*b^(1/3)*k*x + b^(1/3)*k^2*x^2)*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + (x + (-1 - k)*x^2 + k*x^3)^(2/3)]/(2*b^(1/3))","A",1
2822,1,282,282,0.3277369,"\int \frac{(d+c x)^2}{\sqrt{a x+\sqrt{b^2+a^2 x^2}}} \, dx","IntegrateAlgebraic[(d + c*x)^2/Sqrt[a*x + Sqrt[b^2 + a^2*x^2]],x]","\frac{2 \sqrt{b^2+a^2 x^2} \left(4 b^4 c d-4 b^4 c^2 x+25 a^2 b^2 d^2 x+30 a^2 b^2 c d x^2-3 a^2 b^2 c^2 x^3+60 a^4 d^2 x^3+40 a^4 c d x^4+12 a^4 c^2 x^5\right)}{15 a^2 \left(a x+\sqrt{b^2+a^2 x^2}\right)^{7/2}}+\frac{2 \left(-8 b^6 c^2+35 a^2 b^4 d^2+98 a^2 b^4 c d x-49 a^2 b^4 c^2 x^2+385 a^4 b^2 d^2 x^2+350 a^4 b^2 c d x^3+21 a^4 b^2 c^2 x^4+420 a^6 d^2 x^4+280 a^6 c d x^5+84 a^6 c^2 x^6\right)}{105 a^3 \left(a x+\sqrt{b^2+a^2 x^2}\right)^{7/2}}","\frac{2 \sqrt{b^2+a^2 x^2} \left(4 b^4 c d-4 b^4 c^2 x+25 a^2 b^2 d^2 x+30 a^2 b^2 c d x^2-3 a^2 b^2 c^2 x^3+60 a^4 d^2 x^3+40 a^4 c d x^4+12 a^4 c^2 x^5\right)}{15 a^2 \left(a x+\sqrt{b^2+a^2 x^2}\right)^{7/2}}+\frac{2 \left(-8 b^6 c^2+35 a^2 b^4 d^2+98 a^2 b^4 c d x-49 a^2 b^4 c^2 x^2+385 a^4 b^2 d^2 x^2+350 a^4 b^2 c d x^3+21 a^4 b^2 c^2 x^4+420 a^6 d^2 x^4+280 a^6 c d x^5+84 a^6 c^2 x^6\right)}{105 a^3 \left(a x+\sqrt{b^2+a^2 x^2}\right)^{7/2}}",1,"(2*Sqrt[b^2 + a^2*x^2]*(4*b^4*c*d - 4*b^4*c^2*x + 25*a^2*b^2*d^2*x + 30*a^2*b^2*c*d*x^2 - 3*a^2*b^2*c^2*x^3 + 60*a^4*d^2*x^3 + 40*a^4*c*d*x^4 + 12*a^4*c^2*x^5))/(15*a^2*(a*x + Sqrt[b^2 + a^2*x^2])^(7/2)) + (2*(-8*b^6*c^2 + 35*a^2*b^4*d^2 + 98*a^2*b^4*c*d*x - 49*a^2*b^4*c^2*x^2 + 385*a^4*b^2*d^2*x^2 + 350*a^4*b^2*c*d*x^3 + 21*a^4*b^2*c^2*x^4 + 420*a^6*d^2*x^4 + 280*a^6*c*d*x^5 + 84*a^6*c^2*x^6))/(105*a^3*(a*x + Sqrt[b^2 + a^2*x^2])^(7/2))","A",1
2823,1,288,282,0.3164476,"\int \frac{1-\sqrt[3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}}{x^2+\sqrt[3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}} \, dx","IntegrateAlgebraic[(1 - (1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(1/3))/(x^2 + (1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(1/3)),x]","\frac{1}{3} (1-3 x)+3 \text{RootSum}\left[-3-28 \text{$\#$1}+6 \text{$\#$1}^3+3 \text{$\#$1}^4-3 \text{$\#$1}^6-3 \text{$\#$1}^7+\text{$\#$1}^{10}\&,\frac{-10 \log \left(1+x^3\right) \text{$\#$1}^2+10 \log \left(\sqrt[3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}-\text{$\#$1}-x^3 \text{$\#$1}\right) \text{$\#$1}^2+2 \log \left(1+x^3\right) \text{$\#$1}^5-2 \log \left(\sqrt[3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}-\text{$\#$1}-x^3 \text{$\#$1}\right) \text{$\#$1}^5-\log \left(1+x^3\right) \text{$\#$1}^8+\log \left(\sqrt[3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}-\text{$\#$1}-x^3 \text{$\#$1}\right) \text{$\#$1}^8}{-28+18 \text{$\#$1}^2+12 \text{$\#$1}^3-18 \text{$\#$1}^5-21 \text{$\#$1}^6+10 \text{$\#$1}^9}\&\right]","-x+3 \text{RootSum}\left[-3-28 \text{$\#$1}+6 \text{$\#$1}^3+3 \text{$\#$1}^4-3 \text{$\#$1}^6-3 \text{$\#$1}^7+\text{$\#$1}^{10}\&,\frac{-10 \log \left(1+x^3\right) \text{$\#$1}^2+10 \log \left(\sqrt[3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}-\text{$\#$1}-x^3 \text{$\#$1}\right) \text{$\#$1}^2+2 \log \left(1+x^3\right) \text{$\#$1}^5-2 \log \left(\sqrt[3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}-\text{$\#$1}-x^3 \text{$\#$1}\right) \text{$\#$1}^5-\log \left(1+x^3\right) \text{$\#$1}^8+\log \left(\sqrt[3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}-\text{$\#$1}-x^3 \text{$\#$1}\right) \text{$\#$1}^8}{-28+18 \text{$\#$1}^2+12 \text{$\#$1}^3-18 \text{$\#$1}^5-21 \text{$\#$1}^6+10 \text{$\#$1}^9}\&\right]",1,"(1 - 3*x)/3 + 3*RootSum[-3 - 28*#1 + 6*#1^3 + 3*#1^4 - 3*#1^6 - 3*#1^7 + #1^10 & , (-10*Log[1 + x^3]*#1^2 + 10*Log[(1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(1/3) - #1 - x^3*#1]*#1^2 + 2*Log[1 + x^3]*#1^5 - 2*Log[(1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(1/3) - #1 - x^3*#1]*#1^5 - Log[1 + x^3]*#1^8 + Log[(1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(1/3) - #1 - x^3*#1]*#1^8)/(-28 + 18*#1^2 + 12*#1^3 - 18*#1^5 - 21*#1^6 + 10*#1^9) & ]","A",1
2824,1,285,285,2.923784,"\int \frac{1-k x}{(1+(-2+k) x) ((1-x) x (1-k x))^{2/3}} \, dx","IntegrateAlgebraic[(1 - k*x)/((1 + (-2 + k)*x)*((1 - x)*x*(1 - k*x))^(2/3)),x]","\frac{(-1)^{2/3} \sqrt{3} \tan ^{-1}\left(\frac{-\sqrt{3}+\sqrt{3} k x}{-1+k x-2 \sqrt[3]{-1} 2^{2/3} \sqrt[3]{-1+k} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{2^{2/3} \sqrt[3]{-1+k}}+\frac{(-1)^{2/3} \log \left(-1+k x+\sqrt[3]{-1} 2^{2/3} \sqrt[3]{-1+k} \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{2^{2/3} \sqrt[3]{-1+k}}-\frac{(-1)^{2/3} \log \left(1-2 k x+k^2 x^2+\left(\sqrt[3]{-1} 2^{2/3} \sqrt[3]{-1+k}-\sqrt[3]{-1} 2^{2/3} \sqrt[3]{-1+k} k x\right) \sqrt[3]{x+(-1-k) x^2+k x^3}+2 (-1)^{2/3} \sqrt[3]{2} (-1+k)^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2\ 2^{2/3} \sqrt[3]{-1+k}}","\frac{(-1)^{2/3} \sqrt{3} \tan ^{-1}\left(\frac{-\sqrt{3}+\sqrt{3} k x}{-1+k x-2 \sqrt[3]{-1} 2^{2/3} \sqrt[3]{-1+k} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{2^{2/3} \sqrt[3]{-1+k}}+\frac{(-1)^{2/3} \log \left(-1+k x+\sqrt[3]{-1} 2^{2/3} \sqrt[3]{-1+k} \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{2^{2/3} \sqrt[3]{-1+k}}-\frac{(-1)^{2/3} \log \left(1-2 k x+k^2 x^2+\left(\sqrt[3]{-1} 2^{2/3} \sqrt[3]{-1+k}-\sqrt[3]{-1} 2^{2/3} \sqrt[3]{-1+k} k x\right) \sqrt[3]{x+(-1-k) x^2+k x^3}+2 (-1)^{2/3} \sqrt[3]{2} (-1+k)^{2/3} \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2\ 2^{2/3} \sqrt[3]{-1+k}}",1,"((-1)^(2/3)*Sqrt[3]*ArcTan[(-Sqrt[3] + Sqrt[3]*k*x)/(-1 + k*x - 2*(-1)^(1/3)*2^(2/3)*(-1 + k)^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/(2^(2/3)*(-1 + k)^(1/3)) + ((-1)^(2/3)*Log[-1 + k*x + (-1)^(1/3)*2^(2/3)*(-1 + k)^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(1/3)])/(2^(2/3)*(-1 + k)^(1/3)) - ((-1)^(2/3)*Log[1 - 2*k*x + k^2*x^2 + ((-1)^(1/3)*2^(2/3)*(-1 + k)^(1/3) - (-1)^(1/3)*2^(2/3)*(-1 + k)^(1/3)*k*x)*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + 2*(-1)^(2/3)*2^(1/3)*(-1 + k)^(2/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3)])/(2*2^(2/3)*(-1 + k)^(1/3))","A",1
2825,1,285,285,0.3612646,"\int \frac{1+x^2+x^3}{\left(-1+x^2+x^3\right) \sqrt[3]{x^2+x^3}} \, dx","IntegrateAlgebraic[(1 + x^2 + x^3)/((-1 + x^2 + x^3)*(x^2 + x^3)^(1/3)),x]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{x^2+x^3}+\left(x^2+x^3\right)^{2/3}\right)+2 \text{RootSum}\left[-1-\text{$\#$1}+\text{$\#$1}^3\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}}{-1+3 \text{$\#$1}^2}\&\right]+2 \text{RootSum}\left[1-\text{$\#$1}+\text{$\#$1}^2-2 \text{$\#$1}^3+\text{$\#$1}^4+\text{$\#$1}^6\&,\frac{2 \log (x) \text{$\#$1}-2 \log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2-2 \log (x) \text{$\#$1}^4+2 \log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^4}{-1+2 \text{$\#$1}-6 \text{$\#$1}^2+4 \text{$\#$1}^3+6 \text{$\#$1}^5}\&\right]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{x^2+x^3}+\left(x^2+x^3\right)^{2/3}\right)+2 \text{RootSum}\left[-1-\text{$\#$1}+\text{$\#$1}^3\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}}{-1+3 \text{$\#$1}^2}\&\right]+2 \text{RootSum}\left[1-\text{$\#$1}+\text{$\#$1}^2-2 \text{$\#$1}^3+\text{$\#$1}^4+\text{$\#$1}^6\&,\frac{2 \log (x) \text{$\#$1}-2 \log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2-2 \log (x) \text{$\#$1}^4+2 \log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^4}{-1+2 \text{$\#$1}-6 \text{$\#$1}^2+4 \text{$\#$1}^3+6 \text{$\#$1}^5}\&\right]",1,"Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(x^2 + x^3)^(1/3))] - Log[-x + (x^2 + x^3)^(1/3)] + Log[x^2 + x*(x^2 + x^3)^(1/3) + (x^2 + x^3)^(2/3)]/2 + 2*RootSum[-1 - #1 + #1^3 & , (-(Log[x]*#1) + Log[(x^2 + x^3)^(1/3) - x*#1]*#1)/(-1 + 3*#1^2) & ] + 2*RootSum[1 - #1 + #1^2 - 2*#1^3 + #1^4 + #1^6 & , (2*Log[x]*#1 - 2*Log[(x^2 + x^3)^(1/3) - x*#1]*#1 - Log[x]*#1^2 + Log[(x^2 + x^3)^(1/3) - x*#1]*#1^2 - 2*Log[x]*#1^4 + 2*Log[(x^2 + x^3)^(1/3) - x*#1]*#1^4)/(-1 + 2*#1 - 6*#1^2 + 4*#1^3 + 6*#1^5) & ]","A",1
2826,1,285,285,0.0013467,"\int \frac{1+x^2+x^3}{\left(-1+x^2+x^3\right) \sqrt[3]{x^2+x^3}} \, dx","IntegrateAlgebraic[(1 + x^2 + x^3)/((-1 + x^2 + x^3)*(x^2 + x^3)^(1/3)),x]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{x^2+x^3}+\left(x^2+x^3\right)^{2/3}\right)+2 \text{RootSum}\left[-1-\text{$\#$1}+\text{$\#$1}^3\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}}{-1+3 \text{$\#$1}^2}\&\right]+2 \text{RootSum}\left[1-\text{$\#$1}+\text{$\#$1}^2-2 \text{$\#$1}^3+\text{$\#$1}^4+\text{$\#$1}^6\&,\frac{2 \log (x) \text{$\#$1}-2 \log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2-2 \log (x) \text{$\#$1}^4+2 \log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^4}{-1+2 \text{$\#$1}-6 \text{$\#$1}^2+4 \text{$\#$1}^3+6 \text{$\#$1}^5}\&\right]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^3}}\right)-\log \left(-x+\sqrt[3]{x^2+x^3}\right)+\frac{1}{2} \log \left(x^2+x \sqrt[3]{x^2+x^3}+\left(x^2+x^3\right)^{2/3}\right)+2 \text{RootSum}\left[-1-\text{$\#$1}+\text{$\#$1}^3\&,\frac{-\log (x) \text{$\#$1}+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}}{-1+3 \text{$\#$1}^2}\&\right]+2 \text{RootSum}\left[1-\text{$\#$1}+\text{$\#$1}^2-2 \text{$\#$1}^3+\text{$\#$1}^4+\text{$\#$1}^6\&,\frac{2 \log (x) \text{$\#$1}-2 \log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}-\log (x) \text{$\#$1}^2+\log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^2-2 \log (x) \text{$\#$1}^4+2 \log \left(\sqrt[3]{x^2+x^3}-x \text{$\#$1}\right) \text{$\#$1}^4}{-1+2 \text{$\#$1}-6 \text{$\#$1}^2+4 \text{$\#$1}^3+6 \text{$\#$1}^5}\&\right]",1,"Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(x^2 + x^3)^(1/3))] - Log[-x + (x^2 + x^3)^(1/3)] + Log[x^2 + x*(x^2 + x^3)^(1/3) + (x^2 + x^3)^(2/3)]/2 + 2*RootSum[-1 - #1 + #1^3 & , (-(Log[x]*#1) + Log[(x^2 + x^3)^(1/3) - x*#1]*#1)/(-1 + 3*#1^2) & ] + 2*RootSum[1 - #1 + #1^2 - 2*#1^3 + #1^4 + #1^6 & , (2*Log[x]*#1 - 2*Log[(x^2 + x^3)^(1/3) - x*#1]*#1 - Log[x]*#1^2 + Log[(x^2 + x^3)^(1/3) - x*#1]*#1^2 - 2*Log[x]*#1^4 + 2*Log[(x^2 + x^3)^(1/3) - x*#1]*#1^4)/(-1 + 2*#1 - 6*#1^2 + 4*#1^3 + 6*#1^5) & ]","A",1
2827,1,166,286,0.2308709,"\int \frac{\sqrt{a x+\sqrt{-b+a x}}}{x \sqrt{-b+a x}} \, dx","IntegrateAlgebraic[Sqrt[a*x + Sqrt[-b + a*x]]/(x*Sqrt[-b + a*x]),x]","-2 \log \left(-1-2 \sqrt{-b+a x}+2 \sqrt{a x+\sqrt{-b+a x}}\right)-\text{RootSum}\left[b+b^2-4 b \text{$\#$1}+2 b \text{$\#$1}^2+\text{$\#$1}^4\&,\frac{b \log \left(-\sqrt{-b+a x}+\sqrt{a x+\sqrt{-b+a x}}-\text{$\#$1}\right)-\log \left(-\sqrt{-b+a x}+\sqrt{a x+\sqrt{-b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2}{-b+b \text{$\#$1}+\text{$\#$1}^3}\&\right]","-2 \log \left(1+2 \sqrt{-b+a x}-2 \sqrt{a x+\sqrt{-b+a x}}\right)-2 \text{RootSum}\left[1-8 b+16 b^2+4 \text{$\#$1}-16 b \text{$\#$1}+6 \text{$\#$1}^2+8 b \text{$\#$1}^2+4 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{-\log \left(1+2 \sqrt{-b+a x}-2 \sqrt{a x+\sqrt{-b+a x}}+\text{$\#$1}\right)+4 b \log \left(1+2 \sqrt{-b+a x}-2 \sqrt{a x+\sqrt{-b+a x}}+\text{$\#$1}\right)-2 \log \left(1+2 \sqrt{-b+a x}-2 \sqrt{a x+\sqrt{-b+a x}}+\text{$\#$1}\right) \text{$\#$1}-\log \left(1+2 \sqrt{-b+a x}-2 \sqrt{a x+\sqrt{-b+a x}}+\text{$\#$1}\right) \text{$\#$1}^2}{1-4 b+3 \text{$\#$1}+4 b \text{$\#$1}+3 \text{$\#$1}^2+\text{$\#$1}^3}\&\right]",1,"-2*Log[-1 - 2*Sqrt[-b + a*x] + 2*Sqrt[a*x + Sqrt[-b + a*x]]] - RootSum[b + b^2 - 4*b*#1 + 2*b*#1^2 + #1^4 & , (b*Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1] - Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1^2)/(-b + b*#1 + #1^3) & ]","A",1
2828,1,286,286,0.5024759,"\int \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}} \, dx","IntegrateAlgebraic[Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]],x]","\frac{\left(-75 b-8 a c^2 x+60 a^2 x^2\right) \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+\left(16 c^3+6 a c x\right) \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+\sqrt{b+a^2 x^2} \left(\left(-8 c^2+60 a x\right) \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+6 c \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}\right)}{105 a \sqrt{a x+\sqrt{b+a^2 x^2}}}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{\sqrt{c}}\right)}{a \sqrt{c}}","\frac{\left(-75 b-8 a c^2 x+60 a^2 x^2\right) \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+\left(16 c^3+6 a c x\right) \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+\sqrt{b+a^2 x^2} \left(\left(-8 c^2+60 a x\right) \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+6 c \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}\right)}{105 a \sqrt{a x+\sqrt{b+a^2 x^2}}}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{\sqrt{c}}\right)}{a \sqrt{c}}",1,"((-75*b - 8*a*c^2*x + 60*a^2*x^2)*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] + (16*c^3 + 6*a*c*x)*Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] + Sqrt[b + a^2*x^2]*((-8*c^2 + 60*a*x)*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] + 6*c*Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]))/(105*a*Sqrt[a*x + Sqrt[b + a^2*x^2]]) - (b*ArcTanh[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/Sqrt[c]])/(a*Sqrt[c])","A",1
2829,1,287,287,0.9081336,"\int x^4 \sqrt{b+a^2 x^4} \sqrt{a x^2+\sqrt{b+a^2 x^4}} \, dx","IntegrateAlgebraic[x^4*Sqrt[b + a^2*x^4]*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]],x]","\frac{\sqrt{a} x \sqrt{b+a^2 x^4} \left(104 a b^2 x^2+264 a^3 b x^6+192 a^5 x^{10}\right) \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\sqrt{a} x \left(39 b^3+212 a^2 b^2 x^4+360 a^4 b x^8+192 a^6 x^{12}\right) \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{1152 a^{7/2} b x^2+1536 a^{11/2} x^6+384 a^{5/2} b \sqrt{b+a^2 x^4}+1536 a^{9/2} x^4 \sqrt{b+a^2 x^4}}-\frac{13 b^2 \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{128 \sqrt{2} a^{5/2}}","\frac{\sqrt{a} x \sqrt{b+a^2 x^4} \left(104 a b^2 x^2+264 a^3 b x^6+192 a^5 x^{10}\right) \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\sqrt{a} x \left(39 b^3+212 a^2 b^2 x^4+360 a^4 b x^8+192 a^6 x^{12}\right) \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{1152 a^{7/2} b x^2+1536 a^{11/2} x^6+384 a^{5/2} b \sqrt{b+a^2 x^4}+1536 a^{9/2} x^4 \sqrt{b+a^2 x^4}}-\frac{13 b^2 \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{128 \sqrt{2} a^{5/2}}",1,"(Sqrt[a]*x*Sqrt[b + a^2*x^4]*(104*a*b^2*x^2 + 264*a^3*b*x^6 + 192*a^5*x^10)*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + Sqrt[a]*x*(39*b^3 + 212*a^2*b^2*x^4 + 360*a^4*b*x^8 + 192*a^6*x^12)*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/(1152*a^(7/2)*b*x^2 + 1536*a^(11/2)*x^6 + 384*a^(5/2)*b*Sqrt[b + a^2*x^4] + 1536*a^(9/2)*x^4*Sqrt[b + a^2*x^4]) - (13*b^2*Log[I*a*x^2 + I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]])/(128*Sqrt[2]*a^(5/2))","A",1
2830,1,287,287,1.0006815,"\int \frac{\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{\left(b+a^2 x^2\right)^{3/2} \sqrt{a x+\sqrt{b+a^2 x^2}}} \, dx","IntegrateAlgebraic[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/((b + a^2*x^2)^(3/2)*Sqrt[a*x + Sqrt[b + a^2*x^2]]),x]","\frac{a x \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+\sqrt{b+a^2 x^2} \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{a^2 b x \sqrt{b+a^2 x^2}+a b \left(b+a^2 x^2\right)}+\frac{\text{RootSum}\left[b+c^4-4 c^3 \text{$\#$1}^2+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-c \log \left(\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}-\text{$\#$1}\right)-\log \left(\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{c \text{$\#$1}-\text{$\#$1}^3}\&\right]}{4 a b}","\frac{a x \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+\sqrt{b+a^2 x^2} \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{a^2 b x \sqrt{b+a^2 x^2}+a b \left(b+a^2 x^2\right)}+\frac{\text{RootSum}\left[b+c^4-4 c^3 \text{$\#$1}^2+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-c \log \left(\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}-\text{$\#$1}\right)-\log \left(\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{c \text{$\#$1}-\text{$\#$1}^3}\&\right]}{4 a b}",1,"(a*x*Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] + Sqrt[b + a^2*x^2]*Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]])/(a^2*b*x*Sqrt[b + a^2*x^2] + a*b*(b + a^2*x^2)) + RootSum[b + c^4 - 4*c^3*#1^2 + 6*c^2*#1^4 - 4*c*#1^6 + #1^8 & , (-(c*Log[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] - #1]) - Log[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] - #1]*#1^2)/(c*#1 - #1^3) & ]/(4*a*b)","A",1
2831,1,287,287,0.0034661,"\int \frac{\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{\left(b+a^2 x^2\right)^{3/2} \sqrt{a x+\sqrt{b+a^2 x^2}}} \, dx","IntegrateAlgebraic[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/((b + a^2*x^2)^(3/2)*Sqrt[a*x + Sqrt[b + a^2*x^2]]),x]","\frac{a x \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+\sqrt{b+a^2 x^2} \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{a^2 b x \sqrt{b+a^2 x^2}+a b \left(b+a^2 x^2\right)}+\frac{\text{RootSum}\left[b+c^4-4 c^3 \text{$\#$1}^2+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-c \log \left(\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}-\text{$\#$1}\right)-\log \left(\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{c \text{$\#$1}-\text{$\#$1}^3}\&\right]}{4 a b}","\frac{a x \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+\sqrt{b+a^2 x^2} \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{a^2 b x \sqrt{b+a^2 x^2}+a b \left(b+a^2 x^2\right)}+\frac{\text{RootSum}\left[b+c^4-4 c^3 \text{$\#$1}^2+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-c \log \left(\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}-\text{$\#$1}\right)-\log \left(\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{c \text{$\#$1}-\text{$\#$1}^3}\&\right]}{4 a b}",1,"(a*x*Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] + Sqrt[b + a^2*x^2]*Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]])/(a^2*b*x*Sqrt[b + a^2*x^2] + a*b*(b + a^2*x^2)) + RootSum[b + c^4 - 4*c^3*#1^2 + 6*c^2*#1^4 - 4*c*#1^6 + #1^8 & , (-(c*Log[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] - #1]) - Log[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] - #1]*#1^2)/(c*#1 - #1^3) & ]/(4*a*b)","A",1
2832,1,291,288,0.7057943,"\int \frac{x^2}{\left(b+a x^2\right) \sqrt[3]{x+x^3}} \, dx","IntegrateAlgebraic[x^2/((b + a*x^2)*(x + x^3)^(1/3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)}{2 a}+\frac{\sqrt{3} \sqrt[3]{b} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a-b} x}{\sqrt[3]{a-b} x-2 \sqrt[3]{b} \sqrt[3]{x+x^3}}\right)}{2 a \sqrt[3]{a-b}}-\frac{\log \left(-a x+a \sqrt[3]{x+x^3}\right)}{2 a}-\frac{\sqrt[3]{b} \log \left(\sqrt[3]{a-b} x+\sqrt[3]{b} \sqrt[3]{x+x^3}\right)}{2 a \sqrt[3]{a-b}}+\frac{\log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)}{4 a}+\frac{\sqrt[3]{b} \log \left((a-b)^{2/3} x^2-\sqrt[3]{a-b} \sqrt[3]{b} x \sqrt[3]{x+x^3}+b^{2/3} \left(x+x^3\right)^{2/3}\right)}{4 a \sqrt[3]{a-b}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)}{2 a}+\frac{\sqrt{3} \sqrt[3]{b} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a-b} x}{\sqrt[3]{a-b} x-2 \sqrt[3]{b} \sqrt[3]{x+x^3}}\right)}{2 a \sqrt[3]{a-b}}-\frac{\log \left(-x+\sqrt[3]{x+x^3}\right)}{2 a}-\frac{\sqrt[3]{b} \log \left(\sqrt[3]{a-b} x+\sqrt[3]{b} \sqrt[3]{x+x^3}\right)}{2 a \sqrt[3]{a-b}}+\frac{\log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)}{4 a}+\frac{\sqrt[3]{b} \log \left((a-b)^{2/3} x^2-\sqrt[3]{a-b} \sqrt[3]{b} x \sqrt[3]{x+x^3}+b^{2/3} \left(x+x^3\right)^{2/3}\right)}{4 a \sqrt[3]{a-b}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^3)^(1/3))])/(2*a) + (Sqrt[3]*b^(1/3)*ArcTan[(Sqrt[3]*(a - b)^(1/3)*x)/((a - b)^(1/3)*x - 2*b^(1/3)*(x + x^3)^(1/3))])/(2*a*(a - b)^(1/3)) - Log[-(a*x) + a*(x + x^3)^(1/3)]/(2*a) - (b^(1/3)*Log[(a - b)^(1/3)*x + b^(1/3)*(x + x^3)^(1/3)])/(2*a*(a - b)^(1/3)) + Log[x^2 + x*(x + x^3)^(1/3) + (x + x^3)^(2/3)]/(4*a) + (b^(1/3)*Log[(a - b)^(2/3)*x^2 - (a - b)^(1/3)*b^(1/3)*x*(x + x^3)^(1/3) + b^(2/3)*(x + x^3)^(2/3)])/(4*a*(a - b)^(1/3))","A",1
2833,1,288,288,0.7987575,"\int \frac{1+x^6}{\sqrt[3]{x^2+x^4} \left(-1+x^6\right)} \, dx","IntegrateAlgebraic[(1 + x^6)/((x^2 + x^4)^(1/3)*(-1 + x^6)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{x^2+x^4}}\right)}{\sqrt{3}}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^4}}\right)}{\sqrt{3}}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2^{2/3} \sqrt[3]{x^2+x^4}}\right)}{2 \sqrt[3]{2} \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x^2+x^4}}\right)}{2 \sqrt[3]{2} \sqrt{3}}-\frac{2}{3} \tanh ^{-1}\left(\frac{x}{\sqrt[3]{x^2+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt[3]{2} x}{\sqrt[3]{x^2+x^4}}\right)}{3 \sqrt[3]{2}}-\frac{1}{3} \tanh ^{-1}\left(\frac{x^2+\left(x^2+x^4\right)^{2/3}}{x \sqrt[3]{x^2+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt[3]{2} x^2+\frac{\left(x^2+x^4\right)^{2/3}}{\sqrt[3]{2}}}{x \sqrt[3]{x^2+x^4}}\right)}{6 \sqrt[3]{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{x^2+x^4}}\right)}{\sqrt{3}}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^4}}\right)}{\sqrt{3}}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2^{2/3} \sqrt[3]{x^2+x^4}}\right)}{2 \sqrt[3]{2} \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x^2+x^4}}\right)}{2 \sqrt[3]{2} \sqrt{3}}-\frac{2}{3} \tanh ^{-1}\left(\frac{x}{\sqrt[3]{x^2+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt[3]{2} x}{\sqrt[3]{x^2+x^4}}\right)}{3 \sqrt[3]{2}}-\frac{1}{3} \tanh ^{-1}\left(\frac{x^2+\left(x^2+x^4\right)^{2/3}}{x \sqrt[3]{x^2+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt[3]{2} x^2+\frac{\left(x^2+x^4\right)^{2/3}}{\sqrt[3]{2}}}{x \sqrt[3]{x^2+x^4}}\right)}{6 \sqrt[3]{2}}",1,"-(ArcTan[(Sqrt[3]*x)/(-x + 2*(x^2 + x^4)^(1/3))]/Sqrt[3]) - ArcTan[(Sqrt[3]*x)/(x + 2*(x^2 + x^4)^(1/3))]/Sqrt[3] - ArcTan[(Sqrt[3]*x)/(-x + 2^(2/3)*(x^2 + x^4)^(1/3))]/(2*2^(1/3)*Sqrt[3]) - ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(x^2 + x^4)^(1/3))]/(2*2^(1/3)*Sqrt[3]) - (2*ArcTanh[x/(x^2 + x^4)^(1/3)])/3 - ArcTanh[(2^(1/3)*x)/(x^2 + x^4)^(1/3)]/(3*2^(1/3)) - ArcTanh[(x^2 + (x^2 + x^4)^(2/3))/(x*(x^2 + x^4)^(1/3))]/3 - ArcTanh[(2^(1/3)*x^2 + (x^2 + x^4)^(2/3)/2^(1/3))/(x*(x^2 + x^4)^(1/3))]/(6*2^(1/3))","A",1
2834,1,288,288,0.3491148,"\int \frac{\sqrt{-2+3 x^4} \left(-1+3 x^4\right)}{x^3 \left(1-3 x^4+3 x^8\right)} \, dx","IntegrateAlgebraic[(Sqrt[-2 + 3*x^4]*(-1 + 3*x^4))/(x^3*(1 - 3*x^4 + 3*x^8)),x]","\frac{\sqrt{-2+3 x^4}}{2 x^2}+\frac{1}{2} \sqrt{3} \text{RootSum}\left[4+4 \text{$\#$1}-2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{-2 \log \left(\sqrt{3} x^2+\sqrt{-2+3 x^4}-\text{$\#$1}\right)-\log \left(\sqrt{3} x^2+\sqrt{-2+3 x^4}-\text{$\#$1}\right) \text{$\#$1}+\log \left(\sqrt{3} x^2+\sqrt{-2+3 x^4}-\text{$\#$1}\right) \text{$\#$1}^2}{2-3 \text{$\#$1}^2+2 \text{$\#$1}^3}\&\right]-\frac{1}{2} \sqrt{3} \text{RootSum}\left[4-4 \text{$\#$1}+2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{-2 \log \left(\sqrt{3} x^2+\sqrt{-2+3 x^4}-\text{$\#$1}\right)+\log \left(\sqrt{3} x^2+\sqrt{-2+3 x^4}-\text{$\#$1}\right) \text{$\#$1}+\log \left(\sqrt{3} x^2+\sqrt{-2+3 x^4}-\text{$\#$1}\right) \text{$\#$1}^2}{-2+3 \text{$\#$1}^2+2 \text{$\#$1}^3}\&\right]","\frac{\sqrt{-2+3 x^4}}{2 x^2}+\frac{1}{2} \sqrt{3} \text{RootSum}\left[4+4 \text{$\#$1}-2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{-2 \log \left(\sqrt{3} x^2+\sqrt{-2+3 x^4}-\text{$\#$1}\right)-\log \left(\sqrt{3} x^2+\sqrt{-2+3 x^4}-\text{$\#$1}\right) \text{$\#$1}+\log \left(\sqrt{3} x^2+\sqrt{-2+3 x^4}-\text{$\#$1}\right) \text{$\#$1}^2}{2-3 \text{$\#$1}^2+2 \text{$\#$1}^3}\&\right]-\frac{1}{2} \sqrt{3} \text{RootSum}\left[4-4 \text{$\#$1}+2 \text{$\#$1}^3+\text{$\#$1}^4\&,\frac{-2 \log \left(\sqrt{3} x^2+\sqrt{-2+3 x^4}-\text{$\#$1}\right)+\log \left(\sqrt{3} x^2+\sqrt{-2+3 x^4}-\text{$\#$1}\right) \text{$\#$1}+\log \left(\sqrt{3} x^2+\sqrt{-2+3 x^4}-\text{$\#$1}\right) \text{$\#$1}^2}{-2+3 \text{$\#$1}^2+2 \text{$\#$1}^3}\&\right]",1,"Sqrt[-2 + 3*x^4]/(2*x^2) + (Sqrt[3]*RootSum[4 + 4*#1 - 2*#1^3 + #1^4 & , (-2*Log[Sqrt[3]*x^2 + Sqrt[-2 + 3*x^4] - #1] - Log[Sqrt[3]*x^2 + Sqrt[-2 + 3*x^4] - #1]*#1 + Log[Sqrt[3]*x^2 + Sqrt[-2 + 3*x^4] - #1]*#1^2)/(2 - 3*#1^2 + 2*#1^3) & ])/2 - (Sqrt[3]*RootSum[4 - 4*#1 + 2*#1^3 + #1^4 & , (-2*Log[Sqrt[3]*x^2 + Sqrt[-2 + 3*x^4] - #1] + Log[Sqrt[3]*x^2 + Sqrt[-2 + 3*x^4] - #1]*#1 + Log[Sqrt[3]*x^2 + Sqrt[-2 + 3*x^4] - #1]*#1^2)/(-2 + 3*#1^2 + 2*#1^3) & ])/2","A",1
2835,1,288,288,8.1597206,"\int \frac{1}{\left(-b x+a^2 x^2\right)^{5/2} \left(a x^2+x \sqrt{-b x+a^2 x^2}\right)^{3/2}} \, dx","IntegrateAlgebraic[1/((-(b*x) + a^2*x^2)^(5/2)*(a*x^2 + x*Sqrt[-(b*x) + a^2*x^2])^(3/2)),x]","\frac{\sqrt{-b x+a^2 x^2} \left(-4368 b^5+2688 a^2 b^4 x+5248 a^4 b^3 x^2+12416 a^6 b^2 x^3-148243 a^8 b x^4+121339 a^{10} x^5\right) \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{16380 b^7 x^5 \left(b-a^2 x\right)^2}+\sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)} \left(-\frac{-4872 a b^4-9352 a^3 b^3 x-24840 a^5 b^2 x^2-229768 a^7 b x^3+283847 a^9 x^4}{8190 b^7 x^4 \left(b-a^2 x\right)}+\frac{109 a^{15/2} \sqrt{-a x+\sqrt{-b x+a^2 x^2}} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{-a x+\sqrt{-b x+a^2 x^2}}}{\sqrt{b}}\right)}{4 b^{15/2} x}\right)","\frac{\sqrt{-b x+a^2 x^2} \left(-4368 b^5+2688 a^2 b^4 x+5248 a^4 b^3 x^2+12416 a^6 b^2 x^3-148243 a^8 b x^4+121339 a^{10} x^5\right) \sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)}}{16380 b^7 x^5 \left(b-a^2 x\right)^2}+\sqrt{x \left(a x+\sqrt{-b x+a^2 x^2}\right)} \left(-\frac{-4872 a b^4-9352 a^3 b^3 x-24840 a^5 b^2 x^2-229768 a^7 b x^3+283847 a^9 x^4}{8190 b^7 x^4 \left(b-a^2 x\right)}+\frac{109 a^{15/2} \sqrt{-a x+\sqrt{-b x+a^2 x^2}} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{-a x+\sqrt{-b x+a^2 x^2}}}{\sqrt{b}}\right)}{4 b^{15/2} x}\right)",1,"(Sqrt[-(b*x) + a^2*x^2]*(-4368*b^5 + 2688*a^2*b^4*x + 5248*a^4*b^3*x^2 + 12416*a^6*b^2*x^3 - 148243*a^8*b*x^4 + 121339*a^10*x^5)*Sqrt[x*(a*x + Sqrt[-(b*x) + a^2*x^2])])/(16380*b^7*x^5*(b - a^2*x)^2) + Sqrt[x*(a*x + Sqrt[-(b*x) + a^2*x^2])]*(-1/8190*(-4872*a*b^4 - 9352*a^3*b^3*x - 24840*a^5*b^2*x^2 - 229768*a^7*b*x^3 + 283847*a^9*x^4)/(b^7*x^4*(b - a^2*x)) + (109*a^(15/2)*Sqrt[-(a*x) + Sqrt[-(b*x) + a^2*x^2]]*ArcTan[(Sqrt[a]*Sqrt[-(a*x) + Sqrt[-(b*x) + a^2*x^2]])/Sqrt[b]])/(4*b^(15/2)*x))","A",1
2836,1,289,289,1.0824911,"\int \frac{(a-2 b+x) (-b+x)}{\sqrt[3]{(-a+x) (-b+x)} \left(a^4-b^2 d-2 \left(2 a^3-b d\right) x+\left(6 a^2-d\right) x^2-4 a x^3+x^4\right)} \, dx","IntegrateAlgebraic[((a - 2*b + x)*(-b + x))/(((-a + x)*(-b + x))^(1/3)*(a^4 - b^2*d - 2*(2*a^3 - b*d)*x + (6*a^2 - d)*x^2 - 4*a*x^3 + x^4)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{a b+(-a-b) x+x^2}}{2 a-2 x+\sqrt[6]{d} \sqrt[3]{a b+(-a-b) x+x^2}}\right)}{2 d^{5/6}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{a b+(-a-b) x+x^2}}{-2 a+2 x+\sqrt[6]{d} \sqrt[3]{a b+(-a-b) x+x^2}}\right)}{2 d^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{d} \sqrt[3]{a b+(-a-b) x+x^2}}{a-x}\right)}{d^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\left(a \sqrt[6]{d}-\sqrt[6]{d} x\right) \sqrt[3]{a b+(-a-b) x+x^2}}{a^2-2 a x+x^2+\sqrt[3]{d} \left(a b+(-a-b) x+x^2\right)^{2/3}}\right)}{2 d^{5/6}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{a b+(-a-b) x+x^2}}{2 a-2 x+\sqrt[6]{d} \sqrt[3]{a b+(-a-b) x+x^2}}\right)}{2 d^{5/6}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{a b+(-a-b) x+x^2}}{-2 a+2 x+\sqrt[6]{d} \sqrt[3]{a b+(-a-b) x+x^2}}\right)}{2 d^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{d} \sqrt[3]{a b+(-a-b) x+x^2}}{a-x}\right)}{d^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\left(a \sqrt[6]{d}-\sqrt[6]{d} x\right) \sqrt[3]{a b+(-a-b) x+x^2}}{a^2-2 a x+x^2+\sqrt[3]{d} \left(a b+(-a-b) x+x^2\right)^{2/3}}\right)}{2 d^{5/6}}",1,"-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/6)*(a*b + (-a - b)*x + x^2)^(1/3))/(2*a - 2*x + d^(1/6)*(a*b + (-a - b)*x + x^2)^(1/3))])/d^(5/6) + (Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/6)*(a*b + (-a - b)*x + x^2)^(1/3))/(-2*a + 2*x + d^(1/6)*(a*b + (-a - b)*x + x^2)^(1/3))])/(2*d^(5/6)) + ArcTanh[(d^(1/6)*(a*b + (-a - b)*x + x^2)^(1/3))/(a - x)]/d^(5/6) + ArcTanh[((a*d^(1/6) - d^(1/6)*x)*(a*b + (-a - b)*x + x^2)^(1/3))/(a^2 - 2*a*x + x^2 + d^(1/3)*(a*b + (-a - b)*x + x^2)^(2/3))]/(2*d^(5/6))","A",1
2837,1,289,289,3.7911017,"\int \frac{(-1+x) x (1+(-2+k) x)}{\sqrt[3]{(1-x) x (1-k x)} \left(1-4 k x+\left(-b+6 k^2\right) x^2+\left(2 b-4 k^3\right) x^3+\left(-b+k^4\right) x^4\right)} \, dx","IntegrateAlgebraic[((-1 + x)*x*(1 + (-2 + k)*x))/(((1 - x)*x*(1 - k*x))^(1/3)*(1 - 4*k*x + (-b + 6*k^2)*x^2 + (2*b - 4*k^3)*x^3 + (-b + k^4)*x^4)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}{2-2 k x+\sqrt[6]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{2 b^{5/6}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}{-2+2 k x+\sqrt[6]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{2 b^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}{-1+k x}\right)}{b^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\left(-\sqrt[6]{b}+\sqrt[6]{b} k x\right) \sqrt[3]{x+(-1-k) x^2+k x^3}}{1-2 k x+k^2 x^2+\sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}}\right)}{2 b^{5/6}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}{2-2 k x+\sqrt[6]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{2 b^{5/6}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}{-2+2 k x+\sqrt[6]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{2 b^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{b} \sqrt[3]{x+(-1-k) x^2+k x^3}}{-1+k x}\right)}{b^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\left(-\sqrt[6]{b}+\sqrt[6]{b} k x\right) \sqrt[3]{x+(-1-k) x^2+k x^3}}{1-2 k x+k^2 x^2+\sqrt[3]{b} \left(x+(-1-k) x^2+k x^3\right)^{2/3}}\right)}{2 b^{5/6}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*b^(1/6)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))/(2 - 2*k*x + b^(1/6)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/(2*b^(5/6)) - (Sqrt[3]*ArcTan[(Sqrt[3]*b^(1/6)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))/(-2 + 2*k*x + b^(1/6)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/(2*b^(5/6)) + ArcTanh[(b^(1/6)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))/(-1 + k*x)]/b^(5/6) + ArcTanh[((-b^(1/6) + b^(1/6)*k*x)*(x + (-1 - k)*x^2 + k*x^3)^(1/3))/(1 - 2*k*x + k^2*x^2 + b^(1/3)*(x + (-1 - k)*x^2 + k*x^3)^(2/3))]/(2*b^(5/6))","A",1
2838,1,289,289,0.6683222,"\int \frac{\left(-1+x^3\right)^{2/3} \left(4+x^6\right)}{x^6 \left(-4+x^6\right)} \, dx","IntegrateAlgebraic[((-1 + x^3)^(2/3)*(4 + x^6))/(x^6*(-4 + x^6)),x]","\frac{\left(1-x^3\right) \left(-1+x^3\right)^{2/3}}{5 x^5}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2} \sqrt[3]{-1+x^3}}\right)}{2\ 2^{2/3} \sqrt{3}}+\frac{\sqrt[6]{3} \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2 \sqrt[3]{2} \sqrt[3]{-1+x^3}}\right)}{2\ 2^{2/3}}+\frac{\log \left(-x+\sqrt[3]{2} \sqrt[3]{-1+x^3}\right)}{6\ 2^{2/3}}-\frac{\log \left(-3 x+\sqrt[3]{2} 3^{2/3} \sqrt[3]{-1+x^3}\right)}{2\ 2^{2/3} \sqrt[3]{3}}-\frac{\log \left(x^2+\sqrt[3]{2} x \sqrt[3]{-1+x^3}+2^{2/3} \left(-1+x^3\right)^{2/3}\right)}{12\ 2^{2/3}}+\frac{\log \left(3 x^2+\sqrt[3]{2} 3^{2/3} x \sqrt[3]{-1+x^3}+2^{2/3} \sqrt[3]{3} \left(-1+x^3\right)^{2/3}\right)}{4\ 2^{2/3} \sqrt[3]{3}}","\frac{\left(1-x^3\right) \left(-1+x^3\right)^{2/3}}{5 x^5}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{2} \sqrt[3]{-1+x^3}}\right)}{2\ 2^{2/3} \sqrt{3}}+\frac{\sqrt[6]{3} \tan ^{-1}\left(\frac{3^{5/6} x}{\sqrt[3]{3} x+2 \sqrt[3]{2} \sqrt[3]{-1+x^3}}\right)}{2\ 2^{2/3}}+\frac{\log \left(-x+\sqrt[3]{2} \sqrt[3]{-1+x^3}\right)}{6\ 2^{2/3}}-\frac{\log \left(-3 x+\sqrt[3]{2} 3^{2/3} \sqrt[3]{-1+x^3}\right)}{2\ 2^{2/3} \sqrt[3]{3}}-\frac{\log \left(x^2+\sqrt[3]{2} x \sqrt[3]{-1+x^3}+2^{2/3} \left(-1+x^3\right)^{2/3}\right)}{12\ 2^{2/3}}+\frac{\log \left(3 x^2+\sqrt[3]{2} 3^{2/3} x \sqrt[3]{-1+x^3}+2^{2/3} \sqrt[3]{3} \left(-1+x^3\right)^{2/3}\right)}{4\ 2^{2/3} \sqrt[3]{3}}",1,"((1 - x^3)*(-1 + x^3)^(2/3))/(5*x^5) - ArcTan[(Sqrt[3]*x)/(x + 2*2^(1/3)*(-1 + x^3)^(1/3))]/(2*2^(2/3)*Sqrt[3]) + (3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*2^(1/3)*(-1 + x^3)^(1/3))])/(2*2^(2/3)) + Log[-x + 2^(1/3)*(-1 + x^3)^(1/3)]/(6*2^(2/3)) - Log[-3*x + 2^(1/3)*3^(2/3)*(-1 + x^3)^(1/3)]/(2*2^(2/3)*3^(1/3)) - Log[x^2 + 2^(1/3)*x*(-1 + x^3)^(1/3) + 2^(2/3)*(-1 + x^3)^(2/3)]/(12*2^(2/3)) + Log[3*x^2 + 2^(1/3)*3^(2/3)*x*(-1 + x^3)^(1/3) + 2^(2/3)*3^(1/3)*(-1 + x^3)^(2/3)]/(4*2^(2/3)*3^(1/3))","A",1
2839,1,289,289,1.2210843,"\int \frac{\sqrt[6]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{-b+a^2 x^2}} \, dx","IntegrateAlgebraic[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6)/Sqrt[-b + a^2*x^2],x]","\frac{24 \sqrt[6]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{a}+\frac{4 \sqrt{3} \sqrt[6]{c} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[6]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[6]{c}}\right)}{a}-\frac{4 \sqrt{3} \sqrt[6]{c} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[6]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[6]{c}}\right)}{a}-\frac{8 \sqrt[6]{c} \tanh ^{-1}\left(\frac{\sqrt[6]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[6]{c}}\right)}{a}-\frac{4 \sqrt[6]{c} \tanh ^{-1}\left(\frac{\sqrt[6]{c}+\frac{\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[6]{c}}}{\sqrt[6]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}\right)}{a}","\frac{24 \sqrt[6]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{a}+\frac{4 \sqrt{3} \sqrt[6]{c} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[6]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[6]{c}}\right)}{a}-\frac{4 \sqrt{3} \sqrt[6]{c} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[6]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[6]{c}}\right)}{a}-\frac{8 \sqrt[6]{c} \tanh ^{-1}\left(\frac{\sqrt[6]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[6]{c}}\right)}{a}-\frac{4 \sqrt[6]{c} \tanh ^{-1}\left(\frac{\sqrt[6]{c}+\frac{\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[6]{c}}}{\sqrt[6]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}\right)}{a}",1,"(24*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6))/a + (4*Sqrt[3]*c^(1/6)*ArcTan[1/Sqrt[3] - (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6))/(Sqrt[3]*c^(1/6))])/a - (4*Sqrt[3]*c^(1/6)*ArcTan[1/Sqrt[3] + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6))/(Sqrt[3]*c^(1/6))])/a - (8*c^(1/6)*ArcTanh[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6)/c^(1/6)])/a - (4*c^(1/6)*ArcTanh[(c^(1/6) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)/c^(1/6))/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6)])/a","A",1
2840,1,290,290,3.3583133,"\int \frac{(-b+x) \left(a b-2 b x+x^2\right)}{\left(x (-a+x) (-b+x)^2\right)^{2/3} \left(b d-(a+d) x+x^2\right)} \, dx","IntegrateAlgebraic[((-b + x)*(a*b - 2*b*x + x^2))/((x*(-a + x)*(-b + x)^2)^(2/3)*(b*d - (a + d)*x + x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} b \sqrt[3]{d}-\sqrt{3} \sqrt[3]{d} x}{b \sqrt[3]{d}-\sqrt[3]{d} x-2 \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)}{d^{2/3}}+\frac{\log \left(b \sqrt[3]{d}-\sqrt[3]{d} x+\sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}\right)}{d^{2/3}}-\frac{\log \left(b^2 d^{2/3}-2 b d^{2/3} x+d^{2/3} x^2+\left(-b \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}+\left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{2/3}\right)}{2 d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} b \sqrt[3]{d}-\sqrt{3} \sqrt[3]{d} x}{b \sqrt[3]{d}-\sqrt[3]{d} x-2 \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)}{d^{2/3}}+\frac{\log \left(b \sqrt[3]{d}-\sqrt[3]{d} x+\sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}\right)}{d^{2/3}}-\frac{\log \left(b^2 d^{2/3}-2 b d^{2/3} x+d^{2/3} x^2+\left(-b \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}+\left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{2/3}\right)}{2 d^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*b*d^(1/3) - Sqrt[3]*d^(1/3)*x)/(b*d^(1/3) - d^(1/3)*x - 2*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3))])/d^(2/3) + Log[b*d^(1/3) - d^(1/3)*x + (-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3)]/d^(2/3) - Log[b^2*d^(2/3) - 2*b*d^(2/3)*x + d^(2/3)*x^2 + (-(b*d^(1/3)) + d^(1/3)*x)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3) + (-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(2/3)]/(2*d^(2/3))","A",1
2841,1,290,290,4.6392002,"\int \frac{-d+c x^4}{x \sqrt[3]{-b+a x^3}} \, dx","IntegrateAlgebraic[(-d + c*x^4)/(x*(-b + a*x^3)^(1/3)),x]","\frac{c x \left(-b+a x^3\right)^{2/3}}{3 a}-\frac{b c \tan ^{-1}\left(\frac{\frac{x}{\sqrt{3}}+\frac{2 \sqrt[3]{-b+a x^3}}{\sqrt{3} \sqrt[3]{a}}}{x}\right)}{3 \sqrt{3} a^{4/3}}+\frac{d \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-b+a x^3}}{\sqrt{3} \sqrt[3]{b}}\right)}{\sqrt{3} \sqrt[3]{b}}+\frac{d \log \left(\sqrt[3]{b}+\sqrt[3]{-b+a x^3}\right)}{3 \sqrt[3]{b}}-\frac{b c \log \left(-\sqrt[3]{a} x+\sqrt[3]{-b+a x^3}\right)}{9 a^{4/3}}-\frac{d \log \left(b^{2/3}-\sqrt[3]{b} \sqrt[3]{-b+a x^3}+\left(-b+a x^3\right)^{2/3}\right)}{6 \sqrt[3]{b}}+\frac{b c \log \left(a^{2/3} x^2+\sqrt[3]{a} x \sqrt[3]{-b+a x^3}+\left(-b+a x^3\right)^{2/3}\right)}{18 a^{4/3}}","\frac{c x \left(-b+a x^3\right)^{2/3}}{3 a}-\frac{b c \tan ^{-1}\left(\frac{\frac{x}{\sqrt{3}}+\frac{2 \sqrt[3]{-b+a x^3}}{\sqrt{3} \sqrt[3]{a}}}{x}\right)}{3 \sqrt{3} a^{4/3}}+\frac{d \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-b+a x^3}}{\sqrt{3} \sqrt[3]{b}}\right)}{\sqrt{3} \sqrt[3]{b}}+\frac{d \log \left(\sqrt[3]{b}+\sqrt[3]{-b+a x^3}\right)}{3 \sqrt[3]{b}}-\frac{b c \log \left(-\sqrt[3]{a} x+\sqrt[3]{-b+a x^3}\right)}{9 a^{4/3}}-\frac{d \log \left(b^{2/3}-\sqrt[3]{b} \sqrt[3]{-b+a x^3}+\left(-b+a x^3\right)^{2/3}\right)}{6 \sqrt[3]{b}}+\frac{b c \log \left(a^{2/3} x^2+\sqrt[3]{a} x \sqrt[3]{-b+a x^3}+\left(-b+a x^3\right)^{2/3}\right)}{18 a^{4/3}}",1,"(c*x*(-b + a*x^3)^(2/3))/(3*a) - (b*c*ArcTan[(x/Sqrt[3] + (2*(-b + a*x^3)^(1/3))/(Sqrt[3]*a^(1/3)))/x])/(3*Sqrt[3]*a^(4/3)) + (d*ArcTan[1/Sqrt[3] - (2*(-b + a*x^3)^(1/3))/(Sqrt[3]*b^(1/3))])/(Sqrt[3]*b^(1/3)) + (d*Log[b^(1/3) + (-b + a*x^3)^(1/3)])/(3*b^(1/3)) - (b*c*Log[-(a^(1/3)*x) + (-b + a*x^3)^(1/3)])/(9*a^(4/3)) - (d*Log[b^(2/3) - b^(1/3)*(-b + a*x^3)^(1/3) + (-b + a*x^3)^(2/3)])/(6*b^(1/3)) + (b*c*Log[a^(2/3)*x^2 + a^(1/3)*x*(-b + a*x^3)^(1/3) + (-b + a*x^3)^(2/3)])/(18*a^(4/3))","A",1
2842,1,291,291,0.5649594,"\int \frac{a b-2 b x+x^2}{\sqrt[3]{x (-a+x) (-b+x)^2} \left(b-(1+a d) x+d x^2\right)} \, dx","IntegrateAlgebraic[(a*b - 2*b*x + x^2)/((x*(-a + x)*(-b + x)^2)^(1/3)*(b - (1 + a*d)*x + d*x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{-2 b+2 x+\sqrt[3]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)}{d^{2/3}}+\frac{\log \left(b-x+\sqrt[3]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}\right)}{d^{2/3}}-\frac{\log \left(b^2-2 b x+x^2+\left(-b \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}+d^{2/3} \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{2/3}\right)}{2 d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{-2 b+2 x+\sqrt[3]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)}{d^{2/3}}+\frac{\log \left(b-x+\sqrt[3]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}\right)}{d^{2/3}}-\frac{\log \left(b^2-2 b x+x^2+\left(-b \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}+d^{2/3} \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{2/3}\right)}{2 d^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3))/(-2*b + 2*x + d^(1/3)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3))])/d^(2/3) + Log[b - x + d^(1/3)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3)]/d^(2/3) - Log[b^2 - 2*b*x + x^2 + (-(b*d^(1/3)) + d^(1/3)*x)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3) + d^(2/3)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(2/3)]/(2*d^(2/3))","A",1
2843,1,291,291,0.6935049,"\int \frac{-a b^2+(4 a-b) b x-3 a x^2+x^3}{\sqrt[3]{x (-a+x) (-b+x)^2} \left(-a^2+\left(2 a+b^2 d\right) x-(1+2 b d) x^2+d x^3\right)} \, dx","IntegrateAlgebraic[(-(a*b^2) + (4*a - b)*b*x - 3*a*x^2 + x^3)/((x*(-a + x)*(-b + x)^2)^(1/3)*(-a^2 + (2*a + b^2*d)*x - (1 + 2*b*d)*x^2 + d*x^3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{-2 a+2 x+\sqrt[3]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)}{d^{2/3}}+\frac{\log \left(a-x+\sqrt[3]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}\right)}{d^{2/3}}-\frac{\log \left(a^2-2 a x+x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}+d^{2/3} \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{2/3}\right)}{2 d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{-2 a+2 x+\sqrt[3]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)}{d^{2/3}}+\frac{\log \left(a-x+\sqrt[3]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}\right)}{d^{2/3}}-\frac{\log \left(a^2-2 a x+x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}+d^{2/3} \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{2/3}\right)}{2 d^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3))/(-2*a + 2*x + d^(1/3)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3))])/d^(2/3) + Log[a - x + d^(1/3)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3)]/d^(2/3) - Log[a^2 - 2*a*x + x^2 + (-(a*d^(1/3)) + d^(1/3)*x)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3) + d^(2/3)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(2/3)]/(2*d^(2/3))","A",1
2844,1,291,291,3.1082241,"\int \frac{(-4 a+b+3 x) \left(-b^3+3 b^2 x-3 b x^2+x^3\right)}{\left((-a+x) (-b+x)^2\right)^{2/3} \left(b^4+a d-\left(4 b^3+d\right) x+6 b^2 x^2-4 b x^3+x^4\right)} \, dx","IntegrateAlgebraic[((-4*a + b + 3*x)*(-b^3 + 3*b^2*x - 3*b*x^2 + x^3))/(((-a + x)*(-b + x)^2)^(2/3)*(b^4 + a*d - (4*b^3 + d)*x + 6*b^2*x^2 - 4*b*x^3 + x^4)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}}{-2 a \sqrt[3]{d}+2 \sqrt[3]{d} x+\left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}}\right)}{\sqrt[3]{d}}+\frac{\log \left(a \sqrt{d}-\sqrt{d} x+\sqrt[6]{d} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}\right)}{\sqrt[3]{d}}-\frac{\log \left(a^2 d-2 a d x+d x^2+\left(-a d^{2/3}+d^{2/3} x\right) \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}+\sqrt[3]{d} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{4/3}\right)}{2 \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}}{-2 a \sqrt[3]{d}+2 \sqrt[3]{d} x+\left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}}\right)}{\sqrt[3]{d}}+\frac{\log \left(a \sqrt{d}-\sqrt{d} x+\sqrt[6]{d} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}\right)}{\sqrt[3]{d}}-\frac{\log \left(a^2 d-2 a d x+d x^2+\left(-a d^{2/3}+d^{2/3} x\right) \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}+\sqrt[3]{d} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{4/3}\right)}{2 \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(2/3))/(-2*a*d^(1/3) + 2*d^(1/3)*x + (-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(2/3))])/d^(1/3) + Log[a*Sqrt[d] - Sqrt[d]*x + d^(1/6)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(2/3)]/d^(1/3) - Log[a^2*d - 2*a*d*x + d*x^2 + (-(a*d^(2/3)) + d^(2/3)*x)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(2/3) + d^(1/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(4/3)]/(2*d^(1/3))","A",1
2845,1,293,291,1.3845348,"\int \frac{\sqrt[4]{-b x^3+a x^4}}{x (-d+c x)} \, dx","IntegrateAlgebraic[(-(b*x^3) + a*x^4)^(1/4)/(x*(-d + c*x)),x]","-\frac{2 \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)}{c}-\frac{\sqrt{2} \sqrt[4]{b c-a d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{d} \sqrt[4]{b c-a d} x \sqrt[4]{-b x^3+a x^4}}{\sqrt{b c-a d} x^2-\sqrt{d} \sqrt{-b x^3+a x^4}}\right)}{c \sqrt[4]{d}}+\frac{2 \sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)}{c}-\frac{\sqrt{2} \sqrt[4]{b c-a d} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b c-a d} x^2}{\sqrt{2} \sqrt[4]{d}}+\frac{\sqrt[4]{d} \sqrt{-b x^3+a x^4}}{\sqrt{2} \sqrt[4]{b c-a d}}}{x \sqrt[4]{-b x^3+a x^4}}\right)}{c \sqrt[4]{d}}","-\frac{2 \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)}{c}-\frac{(1-i) \sqrt[4]{-b c+a d} \tan ^{-1}\left(\frac{(1+i) \sqrt[4]{d} \sqrt[4]{-b c+a d} x \sqrt[4]{-b x^3+a x^4}}{\sqrt{-b c+a d} x^2-i \sqrt{d} \sqrt{-b x^3+a x^4}}\right)}{c \sqrt[4]{d}}+\frac{2 \sqrt[4]{a} \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^3+a x^4}}\right)}{c}-\frac{(1-i) \sqrt[4]{-b c+a d} \tanh ^{-1}\left(\frac{\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \sqrt[4]{-b c+a d} x^2}{\sqrt[4]{d}}+\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt[4]{d} \sqrt{-b x^3+a x^4}}{\sqrt[4]{-b c+a d}}}{x \sqrt[4]{-b x^3+a x^4}}\right)}{c \sqrt[4]{d}}",1,"(-2*a^(1/4)*ArcTan[(a^(1/4)*x)/(-(b*x^3) + a*x^4)^(1/4)])/c - (Sqrt[2]*(b*c - a*d)^(1/4)*ArcTan[(Sqrt[2]*d^(1/4)*(b*c - a*d)^(1/4)*x*(-(b*x^3) + a*x^4)^(1/4))/(Sqrt[b*c - a*d]*x^2 - Sqrt[d]*Sqrt[-(b*x^3) + a*x^4])])/(c*d^(1/4)) + (2*a^(1/4)*ArcTanh[(a^(1/4)*x)/(-(b*x^3) + a*x^4)^(1/4)])/c - (Sqrt[2]*(b*c - a*d)^(1/4)*ArcTanh[(((b*c - a*d)^(1/4)*x^2)/(Sqrt[2]*d^(1/4)) + (d^(1/4)*Sqrt[-(b*x^3) + a*x^4])/(Sqrt[2]*(b*c - a*d)^(1/4)))/(x*(-(b*x^3) + a*x^4)^(1/4))])/(c*d^(1/4))","A",1
2846,1,291,291,1.1953121,"\int \frac{(-a+x) (-b+x) \left(-2 a b x+(a+b) x^2\right)}{(x (-a+x) (-b+x))^{2/3} \left(a^2 b^2 d-2 a b (a+b) d x+\left(a^2+4 a b+b^2\right) d x^2-2 (a+b) d x^3+(-1+d) x^4\right)} \, dx","IntegrateAlgebraic[((-a + x)*(-b + x)*(-2*a*b*x + (a + b)*x^2))/((x*(-a + x)*(-b + x))^(2/3)*(a^2*b^2*d - 2*a*b*(a + b)*d*x + (a^2 + 4*a*b + b^2)*d*x^2 - 2*(a + b)*d*x^3 + (-1 + d)*x^4)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{d} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}}\right)}{2 d^{2/3}}+\frac{\log \left(x-\sqrt[6]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}\right)}{2 d^{2/3}}+\frac{\log \left(x+\sqrt[6]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}\right)}{2 d^{2/3}}-\frac{\log \left(x^2-\sqrt[6]{d} x \sqrt[3]{a b x+(-a-b) x^2+x^3}+\sqrt[3]{d} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}\right)}{4 d^{2/3}}-\frac{\log \left(x^2+\sqrt[6]{d} x \sqrt[3]{a b x+(-a-b) x^2+x^3}+\sqrt[3]{d} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}\right)}{4 d^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \sqrt[3]{d} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}}\right)}{2 d^{2/3}}+\frac{\log \left(x-\sqrt[6]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}\right)}{2 d^{2/3}}+\frac{\log \left(x+\sqrt[6]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}\right)}{2 d^{2/3}}-\frac{\log \left(x^2-\sqrt[6]{d} x \sqrt[3]{a b x+(-a-b) x^2+x^3}+\sqrt[3]{d} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}\right)}{4 d^{2/3}}-\frac{\log \left(x^2+\sqrt[6]{d} x \sqrt[3]{a b x+(-a-b) x^2+x^3}+\sqrt[3]{d} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}\right)}{4 d^{2/3}}",1,"-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*d^(1/3)*(a*b*x + (-a - b)*x^2 + x^3)^(2/3))])/d^(2/3) + Log[x - d^(1/6)*(a*b*x + (-a - b)*x^2 + x^3)^(1/3)]/(2*d^(2/3)) + Log[x + d^(1/6)*(a*b*x + (-a - b)*x^2 + x^3)^(1/3)]/(2*d^(2/3)) - Log[x^2 - d^(1/6)*x*(a*b*x + (-a - b)*x^2 + x^3)^(1/3) + d^(1/3)*(a*b*x + (-a - b)*x^2 + x^3)^(2/3)]/(4*d^(2/3)) - Log[x^2 + d^(1/6)*x*(a*b*x + (-a - b)*x^2 + x^3)^(1/3) + d^(1/3)*(a*b*x + (-a - b)*x^2 + x^3)^(2/3)]/(4*d^(2/3))","A",1
2847,1,291,291,3.1109158,"\int \frac{1+x^3+x^6}{\sqrt[4]{x^3+x^5} \left(1-x^6\right)} \, dx","IntegrateAlgebraic[(1 + x^3 + x^6)/((x^3 + x^5)^(1/4)*(1 - x^6)),x]","\frac{1}{3} \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^5}}\right)+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^5}}\right)}{2 \sqrt[4]{2}}-\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^3+x^5}}{\sqrt{2} x^2-\sqrt{x^3+x^5}}\right)}{6\ 2^{3/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{x^3+x^5}}{-x^2+\sqrt{x^3+x^5}}\right)}{\sqrt{2}}+\frac{1}{3} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^5}}\right)+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^5}}\right)}{2 \sqrt[4]{2}}+\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^3+x^5}}{2^{3/4}}}{x \sqrt[4]{x^3+x^5}}\right)}{6\ 2^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{x^3+x^5}}{\sqrt{2}}}{x \sqrt[4]{x^3+x^5}}\right)}{\sqrt{2}}","\frac{1}{3} \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^5}}\right)+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^5}}\right)}{2 \sqrt[4]{2}}-\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^3+x^5}}{\sqrt{2} x^2-\sqrt{x^3+x^5}}\right)}{6\ 2^{3/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{x^3+x^5}}{-x^2+\sqrt{x^3+x^5}}\right)}{\sqrt{2}}+\frac{1}{3} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^5}}\right)+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^5}}\right)}{2 \sqrt[4]{2}}+\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^3+x^5}}{2^{3/4}}}{x \sqrt[4]{x^3+x^5}}\right)}{6\ 2^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{x^3+x^5}}{\sqrt{2}}}{x \sqrt[4]{x^3+x^5}}\right)}{\sqrt{2}}",1,"ArcTan[x/(x^3 + x^5)^(1/4)]/3 + ArcTan[(2^(1/4)*x)/(x^3 + x^5)^(1/4)]/(2*2^(1/4)) - ArcTan[(2^(3/4)*x*(x^3 + x^5)^(1/4))/(Sqrt[2]*x^2 - Sqrt[x^3 + x^5])]/(6*2^(3/4)) + ArcTan[(Sqrt[2]*x*(x^3 + x^5)^(1/4))/(-x^2 + Sqrt[x^3 + x^5])]/Sqrt[2] + ArcTanh[x/(x^3 + x^5)^(1/4)]/3 + ArcTanh[(2^(1/4)*x)/(x^3 + x^5)^(1/4)]/(2*2^(1/4)) + ArcTanh[(x^2/2^(1/4) + Sqrt[x^3 + x^5]/2^(3/4))/(x*(x^3 + x^5)^(1/4))]/(6*2^(3/4)) + ArcTanh[(x^2/Sqrt[2] + Sqrt[x^3 + x^5]/Sqrt[2])/(x*(x^3 + x^5)^(1/4))]/Sqrt[2]","A",1
2848,1,293,293,2.5608919,"\int \frac{-a-b x+\left(b+a k^2\right) x^2}{\sqrt{(1-x) x \left(1-k^2 x\right)} \left(1-2 x+k^2 x^2\right)} \, dx","IntegrateAlgebraic[(-a - b*x + (b + a*k^2)*x^2)/(Sqrt[(1 - x)*x*(1 - k^2*x)]*(1 - 2*x + k^2*x^2)),x]","\frac{\left(2 i b+2 i a k^2-i b k^2-2 b \sqrt{-1+k^2}-2 a k^2 \sqrt{-1+k^2}\right) \tan ^{-1}\left(\frac{\sqrt{-2+k^2-2 i \sqrt{-1+k^2}} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{k^2 (-1+x) x}\right)}{2 k^2 \sqrt{-1+k^2} \sqrt{-2+k^2-2 i \sqrt{-1+k^2}}}+\frac{\left(-2 i b-2 i a k^2+i b k^2-2 b \sqrt{-1+k^2}-2 a k^2 \sqrt{-1+k^2}\right) \tan ^{-1}\left(\frac{\sqrt{-2+k^2+2 i \sqrt{-1+k^2}} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{k^2 (-1+x) x}\right)}{2 k^2 \sqrt{-1+k^2} \sqrt{-2+k^2+2 i \sqrt{-1+k^2}}}","\frac{\left(2 i b+2 i a k^2-i b k^2-2 b \sqrt{-1+k^2}-2 a k^2 \sqrt{-1+k^2}\right) \tan ^{-1}\left(\frac{\sqrt{-2+k^2-2 i \sqrt{-1+k^2}} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{k^2 (-1+x) x}\right)}{2 k^2 \sqrt{-1+k^2} \sqrt{-2+k^2-2 i \sqrt{-1+k^2}}}+\frac{\left(-2 i b-2 i a k^2+i b k^2-2 b \sqrt{-1+k^2}-2 a k^2 \sqrt{-1+k^2}\right) \tan ^{-1}\left(\frac{\sqrt{-2+k^2+2 i \sqrt{-1+k^2}} \sqrt{x+\left(-1-k^2\right) x^2+k^2 x^3}}{k^2 (-1+x) x}\right)}{2 k^2 \sqrt{-1+k^2} \sqrt{-2+k^2+2 i \sqrt{-1+k^2}}}",1,"(((2*I)*b + (2*I)*a*k^2 - I*b*k^2 - 2*b*Sqrt[-1 + k^2] - 2*a*k^2*Sqrt[-1 + k^2])*ArcTan[(Sqrt[-2 + k^2 - (2*I)*Sqrt[-1 + k^2]]*Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3])/(k^2*(-1 + x)*x)])/(2*k^2*Sqrt[-1 + k^2]*Sqrt[-2 + k^2 - (2*I)*Sqrt[-1 + k^2]]) + (((-2*I)*b - (2*I)*a*k^2 + I*b*k^2 - 2*b*Sqrt[-1 + k^2] - 2*a*k^2*Sqrt[-1 + k^2])*ArcTan[(Sqrt[-2 + k^2 + (2*I)*Sqrt[-1 + k^2]]*Sqrt[x + (-1 - k^2)*x^2 + k^2*x^3])/(k^2*(-1 + x)*x)])/(2*k^2*Sqrt[-1 + k^2]*Sqrt[-2 + k^2 + (2*I)*Sqrt[-1 + k^2]])","A",1
2849,1,273,293,1.1114103,"\int \frac{1-2 x^4+x^8}{\sqrt[4]{-1+x^4} \left(1-2 x^4+2 x^8\right)} \, dx","IntegrateAlgebraic[(1 - 2*x^4 + x^8)/((-1 + x^4)^(1/4)*(1 - 2*x^4 + 2*x^8)),x]","\frac{1}{4} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)+\frac{1}{8} \sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{-1+x^4}}{-x^2+\sqrt{-1+x^4}}\right)+\frac{1}{8} \sqrt{\frac{1}{2} \left(2-\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{-1+x^4}}{-x^2+\sqrt{-1+x^4}}\right)+\frac{1}{4} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)+\frac{1}{8} \sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{-1+x^4}}{x^2+\sqrt{-1+x^4}}\right)+\frac{1}{8} \sqrt{\frac{1}{2} \left(2-\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{-1+x^4}}{x^2+\sqrt{-1+x^4}}\right)","\frac{1}{4} \tan ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)+\frac{1}{8} \sqrt{\frac{1}{2} \left(2-\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{-1+x^4}}{-x^2+\sqrt{-1+x^4}}\right)-\frac{1}{8} \sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\left(\sqrt{\frac{2}{2-\sqrt{2}}}-\frac{2}{\sqrt{2-\sqrt{2}}}\right) x \sqrt[4]{-1+x^4}}{-x^2+\sqrt{-1+x^4}}\right)+\frac{1}{4} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{-1+x^4}}\right)+\frac{1}{8} \sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x \sqrt[4]{-1+x^4}}{x^2+\sqrt{-1+x^4}}\right)+\frac{1}{8} \sqrt{\frac{1}{2} \left(2-\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} x \sqrt[4]{-1+x^4}}{x^2+\sqrt{-1+x^4}}\right)",1,"ArcTan[x/(-1 + x^4)^(1/4)]/4 + (Sqrt[(2 + Sqrt[2])/2]*ArcTan[(Sqrt[2 - Sqrt[2]]*x*(-1 + x^4)^(1/4))/(-x^2 + Sqrt[-1 + x^4])])/8 + (Sqrt[(2 - Sqrt[2])/2]*ArcTan[(Sqrt[2 + Sqrt[2]]*x*(-1 + x^4)^(1/4))/(-x^2 + Sqrt[-1 + x^4])])/8 + ArcTanh[x/(-1 + x^4)^(1/4)]/4 + (Sqrt[(2 + Sqrt[2])/2]*ArcTanh[(Sqrt[2 - Sqrt[2]]*x*(-1 + x^4)^(1/4))/(x^2 + Sqrt[-1 + x^4])])/8 + (Sqrt[(2 - Sqrt[2])/2]*ArcTanh[(Sqrt[2 + Sqrt[2]]*x*(-1 + x^4)^(1/4))/(x^2 + Sqrt[-1 + x^4])])/8","A",1
2850,1,293,293,0.4923892,"\int \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}} \, dx","IntegrateAlgebraic[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]],x]","\frac{\left(-6 b c-16 a c^3 x+48 a^2 c x^2\right) \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+\left(-15 b+8 a c^2 x\right) \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+\sqrt{b+a^2 x^2} \left(\left(-16 c^3+48 a c x\right) \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+8 c^2 \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}\right)}{60 a^2 c x+60 a c \sqrt{b+a^2 x^2}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{\sqrt{c}}\right)}{4 a c^{3/2}}","\frac{\left(-6 b c-16 a c^3 x+48 a^2 c x^2\right) \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+\left(-15 b+8 a c^2 x\right) \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+\sqrt{b+a^2 x^2} \left(\left(-16 c^3+48 a c x\right) \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+8 c^2 \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}\right)}{60 a^2 c x+60 a c \sqrt{b+a^2 x^2}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{\sqrt{c}}\right)}{4 a c^{3/2}}",1,"((-6*b*c - 16*a*c^3*x + 48*a^2*c*x^2)*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] + (-15*b + 8*a*c^2*x)*Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] + Sqrt[b + a^2*x^2]*((-16*c^3 + 48*a*c*x)*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] + 8*c^2*Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]))/(60*a^2*c*x + 60*a*c*Sqrt[b + a^2*x^2]) + (b*ArcTanh[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/Sqrt[c]])/(4*a*c^(3/2))","A",1
2851,1,426,294,1.175362,"\int \frac{x^3}{\sqrt[3]{x^2 (-a+x)} \left(-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+(-1+d) x^4\right)} \, dx","IntegrateAlgebraic[x^3/((x^2*(-a + x))^(1/3)*(-a^4 + 4*a^3*x - 6*a^2*x^2 + 4*a*x^3 + (-1 + d)*x^4)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} x^2 \left(-a x^2+x^3\right)^{2/3}}{-\sqrt[3]{d} x^4+\left(-a x^2+x^3\right)^{4/3}}\right)}{4 a d^{5/6}}-\frac{\log \left(-\sqrt[12]{d} x+\sqrt[3]{-a x^2+x^3}\right)}{4 a d^{5/6}}-\frac{\log \left(\sqrt[12]{d} x+\sqrt[3]{-a x^2+x^3}\right)}{4 a d^{5/6}}+\frac{\log \left(\sqrt[6]{d} x^2+\left(-a x^2+x^3\right)^{2/3}\right)}{4 a d^{5/6}}+\frac{\log \left(\sqrt[6]{d} x^2-\sqrt[12]{d} x \sqrt[3]{-a x^2+x^3}+\left(-a x^2+x^3\right)^{2/3}\right)}{8 a d^{5/6}}+\frac{\log \left(\sqrt[6]{d} x^2+\sqrt[12]{d} x \sqrt[3]{-a x^2+x^3}+\left(-a x^2+x^3\right)^{2/3}\right)}{8 a d^{5/6}}-\frac{\log \left(\sqrt[6]{d} x^2-\sqrt{3} \sqrt[12]{d} x \sqrt[3]{-a x^2+x^3}+\left(-a x^2+x^3\right)^{2/3}\right)}{8 a d^{5/6}}-\frac{\log \left(\sqrt[6]{d} x^2+\sqrt{3} \sqrt[12]{d} x \sqrt[3]{-a x^2+x^3}+\left(-a x^2+x^3\right)^{2/3}\right)}{8 a d^{5/6}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} x^2}{\sqrt[6]{d} x^2-2 \left(-a x^2+x^3\right)^{2/3}}\right)}{4 a d^{5/6}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} x^2}{\sqrt[6]{d} x^2+2 \left(-a x^2+x^3\right)^{2/3}}\right)}{4 a d^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{d} \left(-a x^2+x^3\right)^{2/3}}{\sqrt[3]{d} x^2+(-a+x) \sqrt[3]{-a x^2+x^3}}\right)}{4 a d^{5/6}}-\frac{\log \left(-\sqrt[12]{d} x+\sqrt[3]{-a x^2+x^3}\right)}{4 a d^{5/6}}-\frac{\log \left(\sqrt[12]{d} x+\sqrt[3]{-a x^2+x^3}\right)}{4 a d^{5/6}}+\frac{\log \left(\sqrt[6]{d} x^2+\left(-a x^2+x^3\right)^{2/3}\right)}{4 a d^{5/6}}",1,"-1/4*(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/6)*x^2*(-(a*x^2) + x^3)^(2/3))/(-(d^(1/3)*x^4) + (-(a*x^2) + x^3)^(4/3))])/(a*d^(5/6)) - Log[-(d^(1/12)*x) + (-(a*x^2) + x^3)^(1/3)]/(4*a*d^(5/6)) - Log[d^(1/12)*x + (-(a*x^2) + x^3)^(1/3)]/(4*a*d^(5/6)) + Log[d^(1/6)*x^2 + (-(a*x^2) + x^3)^(2/3)]/(4*a*d^(5/6)) + Log[d^(1/6)*x^2 - d^(1/12)*x*(-(a*x^2) + x^3)^(1/3) + (-(a*x^2) + x^3)^(2/3)]/(8*a*d^(5/6)) + Log[d^(1/6)*x^2 + d^(1/12)*x*(-(a*x^2) + x^3)^(1/3) + (-(a*x^2) + x^3)^(2/3)]/(8*a*d^(5/6)) - Log[d^(1/6)*x^2 - Sqrt[3]*d^(1/12)*x*(-(a*x^2) + x^3)^(1/3) + (-(a*x^2) + x^3)^(2/3)]/(8*a*d^(5/6)) - Log[d^(1/6)*x^2 + Sqrt[3]*d^(1/12)*x*(-(a*x^2) + x^3)^(1/3) + (-(a*x^2) + x^3)^(2/3)]/(8*a*d^(5/6))","A",1
2852,0,0,294,4.2254222,"\int \frac{\sqrt{c+\sqrt{a x^2+x \sqrt{-b+a^2 x^2}}}}{\sqrt{-b+a^2 x^2}} \, dx","IntegrateAlgebraic[Sqrt[c + Sqrt[a*x^2 + x*Sqrt[-b + a^2*x^2]]]/Sqrt[-b + a^2*x^2],x]","\int \frac{\sqrt{c+\sqrt{a x^2+x \sqrt{-b+a^2 x^2}}}}{\sqrt{-b+a^2 x^2}} \, dx","\frac{2 \sqrt{c+\sqrt{x \left(a x+\sqrt{-b+a^2 x^2}\right)}}}{a}-\frac{\sqrt{\sqrt{2} \sqrt{b}-2 \sqrt{a} c} \left(-\sqrt{b}+\sqrt{2} \sqrt{a} c\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt{c+\sqrt{x \left(a x+\sqrt{-b+a^2 x^2}\right)}}}{\sqrt{\sqrt{2} \sqrt{b}-2 \sqrt{a} c}}\right)}{a^{5/4} \left(-\sqrt{2} \sqrt{b}+2 \sqrt{a} c\right)}-\frac{\left(\sqrt{b}+\sqrt{2} \sqrt{a} c\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt{c+\sqrt{x \left(a x+\sqrt{-b+a^2 x^2}\right)}}}{\sqrt{\sqrt{2} \sqrt{b}+2 \sqrt{a} c}}\right)}{a^{5/4} \sqrt{\sqrt{2} \sqrt{b}+2 \sqrt{a} c}}",1,"Could not integrate","F",-1
2853,1,296,296,7.1105957,"\int \frac{-a (a b+a c-2 b c)+2 \left(a^2-b c\right) x+(-2 a+b+c) x^2}{((-a+x) (-b+x) (-c+x))^{2/3} \left(-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right)} \, dx","IntegrateAlgebraic[(-(a*(a*b + a*c - 2*b*c)) + 2*(a^2 - b*c)*x + (-2*a + b + c)*x^2)/(((-a + x)*(-b + x)*(-c + x))^(2/3)*(-(b*c) + a^2*d + (b + c - 2*a*d)*x + (-1 + d)*x^2)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a \sqrt[3]{d}-\sqrt{3} \sqrt[3]{d} x}{a \sqrt[3]{d}-\sqrt[3]{d} x-2 \sqrt[3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}\right)}{d^{2/3}}-\frac{\log \left(a \sqrt[3]{d}-\sqrt[3]{d} x+\sqrt[3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}\right)}{d^{2/3}}+\frac{\log \left(a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}+\left(-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3\right)^{2/3}\right)}{2 d^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a \sqrt[3]{d}-\sqrt{3} \sqrt[3]{d} x}{a \sqrt[3]{d}-\sqrt[3]{d} x-2 \sqrt[3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}\right)}{d^{2/3}}-\frac{\log \left(a \sqrt[3]{d}-\sqrt[3]{d} x+\sqrt[3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}\right)}{d^{2/3}}+\frac{\log \left(a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}+\left(-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3\right)^{2/3}\right)}{2 d^{2/3}}",1,"-((Sqrt[3]*ArcTan[(Sqrt[3]*a*d^(1/3) - Sqrt[3]*d^(1/3)*x)/(a*d^(1/3) - d^(1/3)*x - 2*(-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3)^(1/3))])/d^(2/3)) - Log[a*d^(1/3) - d^(1/3)*x + (-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3)^(1/3)]/d^(2/3) + Log[a^2*d^(2/3) - 2*a*d^(2/3)*x + d^(2/3)*x^2 + (-(a*d^(1/3)) + d^(1/3)*x)*(-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3)^(1/3) + (-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3)^(2/3)]/(2*d^(2/3))","A",1
2854,1,295,297,1.0741037,"\int \frac{x^2 (-2 a b+(a+b) x)}{\sqrt[3]{x (-a+x) (-b+x)} \left(-a^2 b^2+2 a b (a+b) x-\left(a^2+4 a b+b^2\right) x^2+2 (a+b) x^3+(-1+d) x^4\right)} \, dx","IntegrateAlgebraic[(x^2*(-2*a*b + (a + b)*x))/((x*(-a + x)*(-b + x))^(1/3)*(-(a^2*b^2) + 2*a*b*(a + b)*x - (a^2 + 4*a*b + b^2)*x^2 + 2*(a + b)*x^3 + (-1 + d)*x^4)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\frac{x^2}{\sqrt{3}}+\frac{2 \left(a b x+(-a-b) x^2+x^3\right)^{2/3}}{\sqrt{3} \sqrt[3]{d}}}{x^2}\right)}{2 d^{2/3}}-\frac{\log \left(-\sqrt[6]{d} x+\sqrt[3]{a b x+(-a-b) x^2+x^3}\right)}{2 d^{2/3}}-\frac{\log \left(\sqrt[6]{d} x+\sqrt[3]{a b x+(-a-b) x^2+x^3}\right)}{2 d^{2/3}}+\frac{\log \left(\sqrt[3]{d} x^2-\sqrt[6]{d} x \sqrt[3]{a b x+(-a-b) x^2+x^3}+\left(a b x+(-a-b) x^2+x^3\right)^{2/3}\right)}{4 d^{2/3}}+\frac{\log \left(\sqrt[3]{d} x^2+\sqrt[6]{d} x \sqrt[3]{a b x+(-a-b) x^2+x^3}+\left(a b x+(-a-b) x^2+x^3\right)^{2/3}\right)}{4 d^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} x^2}{\sqrt[3]{d} x^2+2 \left(a b x+(-a-b) x^2+x^3\right)^{2/3}}\right)}{2 d^{2/3}}-\frac{\log \left(-\sqrt[6]{d} x+\sqrt[3]{a b x+(-a-b) x^2+x^3}\right)}{2 d^{2/3}}-\frac{\log \left(\sqrt[6]{d} x+\sqrt[3]{a b x+(-a-b) x^2+x^3}\right)}{2 d^{2/3}}+\frac{\log \left(\sqrt[3]{d} x^2-\sqrt[6]{d} x \sqrt[3]{a b x+(-a-b) x^2+x^3}+\left(a b x+(-a-b) x^2+x^3\right)^{2/3}\right)}{4 d^{2/3}}+\frac{\log \left(\sqrt[3]{d} x^2+\sqrt[6]{d} x \sqrt[3]{a b x+(-a-b) x^2+x^3}+\left(a b x+(-a-b) x^2+x^3\right)^{2/3}\right)}{4 d^{2/3}}",1,"(Sqrt[3]*ArcTan[(x^2/Sqrt[3] + (2*(a*b*x + (-a - b)*x^2 + x^3)^(2/3))/(Sqrt[3]*d^(1/3)))/x^2])/(2*d^(2/3)) - Log[-(d^(1/6)*x) + (a*b*x + (-a - b)*x^2 + x^3)^(1/3)]/(2*d^(2/3)) - Log[d^(1/6)*x + (a*b*x + (-a - b)*x^2 + x^3)^(1/3)]/(2*d^(2/3)) + Log[d^(1/3)*x^2 - d^(1/6)*x*(a*b*x + (-a - b)*x^2 + x^3)^(1/3) + (a*b*x + (-a - b)*x^2 + x^3)^(2/3)]/(4*d^(2/3)) + Log[d^(1/3)*x^2 + d^(1/6)*x*(a*b*x + (-a - b)*x^2 + x^3)^(1/3) + (a*b*x + (-a - b)*x^2 + x^3)^(2/3)]/(4*d^(2/3))","A",1
2855,1,297,297,1.1980773,"\int \frac{1+x^6}{\sqrt[4]{x^3+x^5} \left(1-x^6\right)} \, dx","IntegrateAlgebraic[(1 + x^6)/((x^3 + x^5)^(1/4)*(1 - x^6)),x]","\frac{2}{3} \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^5}}\right)+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^5}}\right)}{3 \sqrt[4]{2}}-\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^3+x^5}}{\sqrt{2} x^2-\sqrt{x^3+x^5}}\right)}{3\ 2^{3/4}}+\frac{1}{3} \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{x^3+x^5}}{-x^2+\sqrt{x^3+x^5}}\right)+\frac{2}{3} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^5}}\right)+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^5}}\right)}{3 \sqrt[4]{2}}+\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^3+x^5}}{2^{3/4}}}{x \sqrt[4]{x^3+x^5}}\right)}{3\ 2^{3/4}}+\frac{1}{3} \sqrt{2} \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{x^3+x^5}}{\sqrt{2}}}{x \sqrt[4]{x^3+x^5}}\right)","\frac{2}{3} \tan ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^5}}\right)+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^5}}\right)}{3 \sqrt[4]{2}}-\frac{\tan ^{-1}\left(\frac{2^{3/4} x \sqrt[4]{x^3+x^5}}{\sqrt{2} x^2-\sqrt{x^3+x^5}}\right)}{3\ 2^{3/4}}+\frac{1}{3} \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt[4]{x^3+x^5}}{-x^2+\sqrt{x^3+x^5}}\right)+\frac{2}{3} \tanh ^{-1}\left(\frac{x}{\sqrt[4]{x^3+x^5}}\right)+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt[4]{x^3+x^5}}\right)}{3 \sqrt[4]{2}}+\frac{\tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt[4]{2}}+\frac{\sqrt{x^3+x^5}}{2^{3/4}}}{x \sqrt[4]{x^3+x^5}}\right)}{3\ 2^{3/4}}+\frac{1}{3} \sqrt{2} \tanh ^{-1}\left(\frac{\frac{x^2}{\sqrt{2}}+\frac{\sqrt{x^3+x^5}}{\sqrt{2}}}{x \sqrt[4]{x^3+x^5}}\right)",1,"(2*ArcTan[x/(x^3 + x^5)^(1/4)])/3 + ArcTan[(2^(1/4)*x)/(x^3 + x^5)^(1/4)]/(3*2^(1/4)) - ArcTan[(2^(3/4)*x*(x^3 + x^5)^(1/4))/(Sqrt[2]*x^2 - Sqrt[x^3 + x^5])]/(3*2^(3/4)) + (Sqrt[2]*ArcTan[(Sqrt[2]*x*(x^3 + x^5)^(1/4))/(-x^2 + Sqrt[x^3 + x^5])])/3 + (2*ArcTanh[x/(x^3 + x^5)^(1/4)])/3 + ArcTanh[(2^(1/4)*x)/(x^3 + x^5)^(1/4)]/(3*2^(1/4)) + ArcTanh[(x^2/2^(1/4) + Sqrt[x^3 + x^5]/2^(3/4))/(x*(x^3 + x^5)^(1/4))]/(3*2^(3/4)) + (Sqrt[2]*ArcTanh[(x^2/Sqrt[2] + Sqrt[x^3 + x^5]/Sqrt[2])/(x*(x^3 + x^5)^(1/4))])/3","A",1
2856,1,624,297,4.5198236,"\int \frac{b^{16}+a^{16} x^{16}}{\sqrt{-b^4+a^4 x^4} \left(-b^{16}+a^{16} x^{16}\right)} \, dx","IntegrateAlgebraic[(b^16 + a^16*x^16)/(Sqrt[-b^4 + a^4*x^4]*(-b^16 + a^16*x^16)),x]","-\frac{x}{4 \sqrt{-b^4+a^4 x^4}}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \tan ^{-1}\left(\frac{(1+i) a b x}{i b^2+a^2 x^2+\sqrt{-b^4+a^4 x^4}}\right)}{a b}+\frac{\left(\frac{1}{32}-\frac{i}{32}\right) \log \left(i b^4-(1+i) a b^3 x-(1-i) a^3 b x^3-i a^4 x^4+\left(b^2-(1-i) a b x-i a^2 x^2\right) \sqrt{-b^4+a^4 x^4}\right)}{a b}-\frac{\left(\frac{1}{32}-\frac{i}{32}\right) \log \left(-a b^5-(1-i) a^2 b^4 x+(1+i) a^4 b^2 x^3+a^5 b x^4+\left(i a b^3+(1+i) a^2 b^2 x+a^3 b x^2\right) \sqrt{-b^4+a^4 x^4}\right)}{a b}-\frac{1}{8} \text{RootSum}\left[16 a^8 b^8+32 i a^6 b^6 \text{$\#$1}^2+8 a^4 b^4 \text{$\#$1}^4-8 i a^2 b^2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-8 a^6 b^6 \log (x)+8 a^6 b^6 \log \left(i b^2+a^2 x^2+\sqrt{-b^4+a^4 x^4}-x \text{$\#$1}\right)-4 i a^4 b^4 \log (x) \text{$\#$1}^2+4 i a^4 b^4 \log \left(i b^2+a^2 x^2+\sqrt{-b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^2-2 a^2 b^2 \log (x) \text{$\#$1}^4+2 a^2 b^2 \log \left(i b^2+a^2 x^2+\sqrt{-b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-i \log (x) \text{$\#$1}^6+i \log \left(i b^2+a^2 x^2+\sqrt{-b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^6}{8 a^6 b^6 \text{$\#$1}-4 i a^4 b^4 \text{$\#$1}^3-6 a^2 b^2 \text{$\#$1}^5-i \text{$\#$1}^7}\&\right]","-\frac{x}{4 \sqrt{-b^4+a^4 x^4}}+\frac{\tan ^{-1}\left(\frac{2^{3/4} a b x \sqrt{-b^4+a^4 x^4}}{b^4+\sqrt{2} a^2 b^2 x^2-a^4 x^4}\right)}{4\ 2^{3/4} a b}-\frac{\tan ^{-1}\left(\frac{\frac{b^3}{2 a}+a b x^2-\frac{a^3 x^4}{2 b}}{x \sqrt{-b^4+a^4 x^4}}\right)}{16 a b}+\frac{\tanh ^{-1}\left(\frac{\frac{b^3}{2 a}-a b x^2-\frac{a^3 x^4}{2 b}}{x \sqrt{-b^4+a^4 x^4}}\right)}{16 a b}+\frac{\tanh ^{-1}\left(\frac{\frac{b^3}{2^{3/4} a}-\frac{a b x^2}{\sqrt[4]{2}}-\frac{a^3 x^4}{2^{3/4} b}}{x \sqrt{-b^4+a^4 x^4}}\right)}{4\ 2^{3/4} a b}",1,"-1/4*x/Sqrt[-b^4 + a^4*x^4] - ((1/8 - I/8)*ArcTan[((1 + I)*a*b*x)/(I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4])])/(a*b) + ((1/32 - I/32)*Log[I*b^4 - (1 + I)*a*b^3*x - (1 - I)*a^3*b*x^3 - I*a^4*x^4 + (b^2 - (1 - I)*a*b*x - I*a^2*x^2)*Sqrt[-b^4 + a^4*x^4]])/(a*b) - ((1/32 - I/32)*Log[-(a*b^5) - (1 - I)*a^2*b^4*x + (1 + I)*a^4*b^2*x^3 + a^5*b*x^4 + (I*a*b^3 + (1 + I)*a^2*b^2*x + a^3*b*x^2)*Sqrt[-b^4 + a^4*x^4]])/(a*b) - RootSum[16*a^8*b^8 + (32*I)*a^6*b^6*#1^2 + 8*a^4*b^4*#1^4 - (8*I)*a^2*b^2*#1^6 + #1^8 & , (-8*a^6*b^6*Log[x] + 8*a^6*b^6*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - x*#1] - (4*I)*a^4*b^4*Log[x]*#1^2 + (4*I)*a^4*b^4*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - x*#1]*#1^2 - 2*a^2*b^2*Log[x]*#1^4 + 2*a^2*b^2*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - x*#1]*#1^4 - I*Log[x]*#1^6 + I*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - x*#1]*#1^6)/(8*a^6*b^6*#1 - (4*I)*a^4*b^4*#1^3 - 6*a^2*b^2*#1^5 - I*#1^7) & ]/8","C",1
2857,1,298,299,0.4893902,"\int \frac{\sqrt[3]{-b^2 x^2+a^3 x^3}}{b+a x^2} \, dx","IntegrateAlgebraic[(-(b^2*x^2) + a^3*x^3)^(1/3)/(b + a*x^2),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{-b^2 x^2+a^3 x^3}}\right)-\log \left(-a x+\sqrt[3]{-b^2 x^2+a^3 x^3}\right)+\frac{1}{2} \log \left(a^2 x^2+a x \sqrt[3]{-b^2 x^2+a^3 x^3}+\left(-b^2 x^2+a^3 x^3\right)^{2/3}\right)+\frac{1}{2} \text{RootSum}\left[a^6+a b^3-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{a^5 \log (x)+b^3 \log (x)-a^5 \log \left(\sqrt[3]{-b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)-b^3 \log \left(\sqrt[3]{-b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)-a^2 \log (x) \text{$\#$1}^3+a^2 \log \left(\sqrt[3]{-b^2 x^2+a^3 x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-a^3 \text{$\#$1}^2+\text{$\#$1}^5}\&\right]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{-b^2 x^2+a^3 x^3}}\right)-\log \left(-a x+\sqrt[3]{-b^2 x^2+a^3 x^3}\right)+\frac{1}{2} \log \left(a^2 x^2+a x \sqrt[3]{-b^2 x^2+a^3 x^3}+\left(-b^2 x^2+a^3 x^3\right)^{2/3}\right)+\frac{1}{2} \text{RootSum}\left[a^6+a b^3-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-a^5 \log (x)-b^3 \log (x)+a^5 \log \left(\sqrt[3]{-b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)+b^3 \log \left(\sqrt[3]{-b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)+a^2 \log (x) \text{$\#$1}^3-a^2 \log \left(\sqrt[3]{-b^2 x^2+a^3 x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{a^3 \text{$\#$1}^2-\text{$\#$1}^5}\&\right]",1,"-(Sqrt[3]*ArcTan[(Sqrt[3]*a*x)/(a*x + 2*(-(b^2*x^2) + a^3*x^3)^(1/3))]) - Log[-(a*x) + (-(b^2*x^2) + a^3*x^3)^(1/3)] + Log[a^2*x^2 + a*x*(-(b^2*x^2) + a^3*x^3)^(1/3) + (-(b^2*x^2) + a^3*x^3)^(2/3)]/2 + RootSum[a^6 + a*b^3 - 2*a^3*#1^3 + #1^6 & , (a^5*Log[x] + b^3*Log[x] - a^5*Log[(-(b^2*x^2) + a^3*x^3)^(1/3) - x*#1] - b^3*Log[(-(b^2*x^2) + a^3*x^3)^(1/3) - x*#1] - a^2*Log[x]*#1^3 + a^2*Log[(-(b^2*x^2) + a^3*x^3)^(1/3) - x*#1]*#1^3)/(-(a^3*#1^2) + #1^5) & ]/2","A",1
2858,1,298,299,0.0023163,"\int \frac{\sqrt[3]{-b^2 x^2+a^3 x^3}}{b+a x^2} \, dx","IntegrateAlgebraic[(-(b^2*x^2) + a^3*x^3)^(1/3)/(b + a*x^2),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{-b^2 x^2+a^3 x^3}}\right)-\log \left(-a x+\sqrt[3]{-b^2 x^2+a^3 x^3}\right)+\frac{1}{2} \log \left(a^2 x^2+a x \sqrt[3]{-b^2 x^2+a^3 x^3}+\left(-b^2 x^2+a^3 x^3\right)^{2/3}\right)+\frac{1}{2} \text{RootSum}\left[a^6+a b^3-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{a^5 \log (x)+b^3 \log (x)-a^5 \log \left(\sqrt[3]{-b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)-b^3 \log \left(\sqrt[3]{-b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)-a^2 \log (x) \text{$\#$1}^3+a^2 \log \left(\sqrt[3]{-b^2 x^2+a^3 x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-a^3 \text{$\#$1}^2+\text{$\#$1}^5}\&\right]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{-b^2 x^2+a^3 x^3}}\right)-\log \left(-a x+\sqrt[3]{-b^2 x^2+a^3 x^3}\right)+\frac{1}{2} \log \left(a^2 x^2+a x \sqrt[3]{-b^2 x^2+a^3 x^3}+\left(-b^2 x^2+a^3 x^3\right)^{2/3}\right)+\frac{1}{2} \text{RootSum}\left[a^6+a b^3-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-a^5 \log (x)-b^3 \log (x)+a^5 \log \left(\sqrt[3]{-b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)+b^3 \log \left(\sqrt[3]{-b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)+a^2 \log (x) \text{$\#$1}^3-a^2 \log \left(\sqrt[3]{-b^2 x^2+a^3 x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{a^3 \text{$\#$1}^2-\text{$\#$1}^5}\&\right]",1,"-(Sqrt[3]*ArcTan[(Sqrt[3]*a*x)/(a*x + 2*(-(b^2*x^2) + a^3*x^3)^(1/3))]) - Log[-(a*x) + (-(b^2*x^2) + a^3*x^3)^(1/3)] + Log[a^2*x^2 + a*x*(-(b^2*x^2) + a^3*x^3)^(1/3) + (-(b^2*x^2) + a^3*x^3)^(2/3)]/2 + RootSum[a^6 + a*b^3 - 2*a^3*#1^3 + #1^6 & , (a^5*Log[x] + b^3*Log[x] - a^5*Log[(-(b^2*x^2) + a^3*x^3)^(1/3) - x*#1] - b^3*Log[(-(b^2*x^2) + a^3*x^3)^(1/3) - x*#1] - a^2*Log[x]*#1^3 + a^2*Log[(-(b^2*x^2) + a^3*x^3)^(1/3) - x*#1]*#1^3)/(-(a^3*#1^2) + #1^5) & ]/2","A",1
2859,1,178,299,1.3458316,"\int \frac{b^8+x^4+a^8 x^8}{\sqrt{-b^4+a^4 x^4} \left(-b^8+a^8 x^8\right)} \, dx","IntegrateAlgebraic[(b^8 + x^4 + a^8*x^8)/(Sqrt[-b^4 + a^4*x^4]*(-b^8 + a^8*x^8)),x]","\frac{\left(1+2 a^4 b^4\right) x \sqrt{-b^4+a^4 x^4}}{4 a^4 b^4 \left(b^4-a^4 x^4\right)}+\frac{\left(\frac{1}{16}+\frac{i}{16}\right) \left(-1+2 a^4 b^4\right) \tan ^{-1}\left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{-b^4+a^4 x^4}}{a b x}\right)}{a^5 b^5}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \left(-1+2 a^4 b^4\right) \tan ^{-1}\left(\frac{(1+i) a b x}{i b^2+a^2 x^2+\sqrt{-b^4+a^4 x^4}}\right)}{a^5 b^5}","\frac{\left(1+2 a^4 b^4\right) x \sqrt{-b^4+a^4 x^4}}{4 a^4 b^4 \left(b^4-a^4 x^4\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \left(-1+2 a^4 b^4\right) \tan ^{-1}\left(\frac{(1+i) a b x}{i b^2+a^2 x^2+\sqrt{-b^4+a^4 x^4}}\right)}{a^5 b^5}+\frac{\left(\frac{1}{16}-\frac{i}{16}\right) \left(-1+2 a^4 b^4\right) \tanh ^{-1}\left(\frac{(1+i) b^2+(1-i) a^2 x^2+(1-i) \sqrt{-b^4+a^4 x^4}}{2 \sqrt{3-2 \sqrt{2}} a b x}\right)}{a^5 b^5}-\frac{\left(\frac{1}{16}-\frac{i}{16}\right) \left(-1+2 a^4 b^4\right) \tanh ^{-1}\left(\frac{(1+i) b^2+(1-i) a^2 x^2+(1-i) \sqrt{-b^4+a^4 x^4}}{2 \sqrt{3+2 \sqrt{2}} a b x}\right)}{a^5 b^5}",1,"((1 + 2*a^4*b^4)*x*Sqrt[-b^4 + a^4*x^4])/(4*a^4*b^4*(b^4 - a^4*x^4)) + ((1/16 + I/16)*(-1 + 2*a^4*b^4)*ArcTan[((1/2 + I/2)*Sqrt[-b^4 + a^4*x^4])/(a*b*x)])/(a^5*b^5) - ((1/8 - I/8)*(-1 + 2*a^4*b^4)*ArcTan[((1 + I)*a*b*x)/(I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4])])/(a^5*b^5)","A",1
2860,1,299,300,0.4464231,"\int \frac{\sqrt[3]{-b^2 x^2+a^3 x^3}}{-b+a x^2} \, dx","IntegrateAlgebraic[(-(b^2*x^2) + a^3*x^3)^(1/3)/(-b + a*x^2),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{-b^2 x^2+a^3 x^3}}\right)-\log \left(-a x+\sqrt[3]{-b^2 x^2+a^3 x^3}\right)+\frac{1}{2} \log \left(a^2 x^2+a x \sqrt[3]{-b^2 x^2+a^3 x^3}+\left(-b^2 x^2+a^3 x^3\right)^{2/3}\right)+\frac{1}{2} \text{RootSum}\left[a^6-a b^3-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{a^5 \log (x)-b^3 \log (x)-a^5 \log \left(\sqrt[3]{-b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)+b^3 \log \left(\sqrt[3]{-b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)-a^2 \log (x) \text{$\#$1}^3+a^2 \log \left(\sqrt[3]{-b^2 x^2+a^3 x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-a^3 \text{$\#$1}^2+\text{$\#$1}^5}\&\right]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{-b^2 x^2+a^3 x^3}}\right)-\log \left(-a x+\sqrt[3]{-b^2 x^2+a^3 x^3}\right)+\frac{1}{2} \log \left(a^2 x^2+a x \sqrt[3]{-b^2 x^2+a^3 x^3}+\left(-b^2 x^2+a^3 x^3\right)^{2/3}\right)+\frac{1}{2} \text{RootSum}\left[a^6-a b^3-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-a^5 \log (x)+b^3 \log (x)+a^5 \log \left(\sqrt[3]{-b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)-b^3 \log \left(\sqrt[3]{-b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)+a^2 \log (x) \text{$\#$1}^3-a^2 \log \left(\sqrt[3]{-b^2 x^2+a^3 x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{a^3 \text{$\#$1}^2-\text{$\#$1}^5}\&\right]",1,"-(Sqrt[3]*ArcTan[(Sqrt[3]*a*x)/(a*x + 2*(-(b^2*x^2) + a^3*x^3)^(1/3))]) - Log[-(a*x) + (-(b^2*x^2) + a^3*x^3)^(1/3)] + Log[a^2*x^2 + a*x*(-(b^2*x^2) + a^3*x^3)^(1/3) + (-(b^2*x^2) + a^3*x^3)^(2/3)]/2 + RootSum[a^6 - a*b^3 - 2*a^3*#1^3 + #1^6 & , (a^5*Log[x] - b^3*Log[x] - a^5*Log[(-(b^2*x^2) + a^3*x^3)^(1/3) - x*#1] + b^3*Log[(-(b^2*x^2) + a^3*x^3)^(1/3) - x*#1] - a^2*Log[x]*#1^3 + a^2*Log[(-(b^2*x^2) + a^3*x^3)^(1/3) - x*#1]*#1^3)/(-(a^3*#1^2) + #1^5) & ]/2","A",1
2861,1,299,300,0.0018065,"\int \frac{\sqrt[3]{-b^2 x^2+a^3 x^3}}{-b+a x^2} \, dx","IntegrateAlgebraic[(-(b^2*x^2) + a^3*x^3)^(1/3)/(-b + a*x^2),x]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{-b^2 x^2+a^3 x^3}}\right)-\log \left(-a x+\sqrt[3]{-b^2 x^2+a^3 x^3}\right)+\frac{1}{2} \log \left(a^2 x^2+a x \sqrt[3]{-b^2 x^2+a^3 x^3}+\left(-b^2 x^2+a^3 x^3\right)^{2/3}\right)+\frac{1}{2} \text{RootSum}\left[a^6-a b^3-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{a^5 \log (x)-b^3 \log (x)-a^5 \log \left(\sqrt[3]{-b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)+b^3 \log \left(\sqrt[3]{-b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)-a^2 \log (x) \text{$\#$1}^3+a^2 \log \left(\sqrt[3]{-b^2 x^2+a^3 x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-a^3 \text{$\#$1}^2+\text{$\#$1}^5}\&\right]","-\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{-b^2 x^2+a^3 x^3}}\right)-\log \left(-a x+\sqrt[3]{-b^2 x^2+a^3 x^3}\right)+\frac{1}{2} \log \left(a^2 x^2+a x \sqrt[3]{-b^2 x^2+a^3 x^3}+\left(-b^2 x^2+a^3 x^3\right)^{2/3}\right)+\frac{1}{2} \text{RootSum}\left[a^6-a b^3-2 a^3 \text{$\#$1}^3+\text{$\#$1}^6\&,\frac{-a^5 \log (x)+b^3 \log (x)+a^5 \log \left(\sqrt[3]{-b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)-b^3 \log \left(\sqrt[3]{-b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)+a^2 \log (x) \text{$\#$1}^3-a^2 \log \left(\sqrt[3]{-b^2 x^2+a^3 x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{a^3 \text{$\#$1}^2-\text{$\#$1}^5}\&\right]",1,"-(Sqrt[3]*ArcTan[(Sqrt[3]*a*x)/(a*x + 2*(-(b^2*x^2) + a^3*x^3)^(1/3))]) - Log[-(a*x) + (-(b^2*x^2) + a^3*x^3)^(1/3)] + Log[a^2*x^2 + a*x*(-(b^2*x^2) + a^3*x^3)^(1/3) + (-(b^2*x^2) + a^3*x^3)^(2/3)]/2 + RootSum[a^6 - a*b^3 - 2*a^3*#1^3 + #1^6 & , (a^5*Log[x] - b^3*Log[x] - a^5*Log[(-(b^2*x^2) + a^3*x^3)^(1/3) - x*#1] + b^3*Log[(-(b^2*x^2) + a^3*x^3)^(1/3) - x*#1] - a^2*Log[x]*#1^3 + a^2*Log[(-(b^2*x^2) + a^3*x^3)^(1/3) - x*#1]*#1^3)/(-(a^3*#1^2) + #1^5) & ]/2","A",1
2862,1,301,301,5.843769,"\int \frac{-b+x}{\left((-a+x) (-b+x)^2\right)^{2/3} (a-b d+(-1+d) x)} \, dx","IntegrateAlgebraic[(-b + x)/(((-a + x)*(-b + x)^2)^(2/3)*(a - b*d + (-1 + d)*x)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} b \sqrt[3]{d}-\sqrt{3} \sqrt[3]{d} x}{b \sqrt[3]{d}-\sqrt[3]{d} x-2 \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}\right)}{(a-b) d^{2/3}}-\frac{\log \left(b \sqrt[3]{d}-\sqrt[3]{d} x+\sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}\right)}{(a-b) d^{2/3}}+\frac{\log \left(b^2 d^{2/3}-2 b d^{2/3} x+d^{2/3} x^2+\left(-b \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}+\left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}\right)}{2 (a-b) d^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} b \sqrt[3]{d}-\sqrt{3} \sqrt[3]{d} x}{b \sqrt[3]{d}-\sqrt[3]{d} x-2 \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}\right)}{(a-b) d^{2/3}}-\frac{\log \left(b \sqrt[3]{d}-\sqrt[3]{d} x+\sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}\right)}{(a-b) d^{2/3}}+\frac{\log \left(b^2 d^{2/3}-2 b d^{2/3} x+d^{2/3} x^2+\left(-b \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}+\left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}\right)}{2 (a-b) d^{2/3}}",1,"-((Sqrt[3]*ArcTan[(Sqrt[3]*b*d^(1/3) - Sqrt[3]*d^(1/3)*x)/(b*d^(1/3) - d^(1/3)*x - 2*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3))])/((a - b)*d^(2/3))) - Log[b*d^(1/3) - d^(1/3)*x + (-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3)]/((a - b)*d^(2/3)) + Log[b^2*d^(2/3) - 2*b*d^(2/3)*x + d^(2/3)*x^2 + (-(b*d^(1/3)) + d^(1/3)*x)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3) + (-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(2/3)]/(2*(a - b)*d^(2/3))","A",1
2863,1,301,301,0.6430614,"\int \frac{\sqrt{-b+x^2} \left(c+x^4\right) \sqrt{x+\sqrt{-b+x^2}}}{x} \, dx","IntegrateAlgebraic[(Sqrt[-b + x^2]*(c + x^4)*Sqrt[x + Sqrt[-b + x^2]])/x,x]","\frac{2 \sqrt{-b+x^2} \left(-1368 b^4 x+10395 b^2 c x+3705 b^3 x^3-32340 b c x^3+1335 b^2 x^5+18480 c x^5-8100 b x^7+5040 x^9\right)+2 \left(304 b^5-2310 b^3 c-3078 b^4 x^2+24255 b^2 c x^2+3735 b^3 x^4-41580 b c x^4+4755 b^2 x^6+18480 c x^6-10620 b x^8+5040 x^{10}\right)}{3465 \left(x+\sqrt{-b+x^2}\right)^{9/2}}+\sqrt{2} b^{3/4} c \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x+\sqrt{-b+x^2}}}{-\sqrt{b}+x+\sqrt{-b+x^2}}\right)+\sqrt{2} b^{3/4} c \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{x}{\sqrt{2} \sqrt[4]{b}}+\frac{\sqrt{-b+x^2}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt{x+\sqrt{-b+x^2}}}\right)","\frac{2 \sqrt{-b+x^2} \left(-1368 b^4 x+10395 b^2 c x+3705 b^3 x^3-32340 b c x^3+1335 b^2 x^5+18480 c x^5-8100 b x^7+5040 x^9\right)+2 \left(304 b^5-2310 b^3 c-3078 b^4 x^2+24255 b^2 c x^2+3735 b^3 x^4-41580 b c x^4+4755 b^2 x^6+18480 c x^6-10620 b x^8+5040 x^{10}\right)}{3465 \left(x+\sqrt{-b+x^2}\right)^{9/2}}+\sqrt{2} b^{3/4} c \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x+\sqrt{-b+x^2}}}{-\sqrt{b}+x+\sqrt{-b+x^2}}\right)+\sqrt{2} b^{3/4} c \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{x}{\sqrt{2} \sqrt[4]{b}}+\frac{\sqrt{-b+x^2}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt{x+\sqrt{-b+x^2}}}\right)",1,"(2*Sqrt[-b + x^2]*(-1368*b^4*x + 10395*b^2*c*x + 3705*b^3*x^3 - 32340*b*c*x^3 + 1335*b^2*x^5 + 18480*c*x^5 - 8100*b*x^7 + 5040*x^9) + 2*(304*b^5 - 2310*b^3*c - 3078*b^4*x^2 + 24255*b^2*c*x^2 + 3735*b^3*x^4 - 41580*b*c*x^4 + 4755*b^2*x^6 + 18480*c*x^6 - 10620*b*x^8 + 5040*x^10))/(3465*(x + Sqrt[-b + x^2])^(9/2)) + Sqrt[2]*b^(3/4)*c*ArcTan[(Sqrt[2]*b^(1/4)*Sqrt[x + Sqrt[-b + x^2]])/(-Sqrt[b] + x + Sqrt[-b + x^2])] + Sqrt[2]*b^(3/4)*c*ArcTanh[(b^(1/4)/Sqrt[2] + x/(Sqrt[2]*b^(1/4)) + Sqrt[-b + x^2]/(Sqrt[2]*b^(1/4)))/Sqrt[x + Sqrt[-b + x^2]]]","A",1
2864,1,612,303,2.7792218,"\int \frac{x^8}{\sqrt{-b^4+a^4 x^4} \left(-b^{16}+a^{16} x^{16}\right)} \, dx","IntegrateAlgebraic[x^8/(Sqrt[-b^4 + a^4*x^4]*(-b^16 + a^16*x^16)),x]","-\frac{x}{8 a^8 b^8 \sqrt{-b^4+a^4 x^4}}-\frac{\left(\frac{1}{16}-\frac{i}{16}\right) \tan ^{-1}\left(\frac{(1+i) a b x}{i b^2+a^2 x^2+\sqrt{-b^4+a^4 x^4}}\right)}{a^9 b^9}+\frac{\left(\frac{1}{32}-\frac{i}{32}\right) \tanh ^{-1}\left(\frac{b^4-(1+i) a b^3 x-(1-i) a^3 b x^3-a^4 x^4+\left(-i b^2-(1-i) a b x-a^2 x^2\right) \sqrt{-b^4+a^4 x^4}}{i b^4-(1-i) a b^3 x+(1+i) a^3 b x^3-i a^4 x^4+\left(b^2+(1+i) a b x-i a^2 x^2\right) \sqrt{-b^4+a^4 x^4}}\right)}{a^9 b^9}+\frac{\text{RootSum}\left[16 a^8 b^8+32 i a^6 b^6 \text{$\#$1}^2+8 a^4 b^4 \text{$\#$1}^4-8 i a^2 b^2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-8 a^6 b^6 \log (x)+8 a^6 b^6 \log \left(i b^2+a^2 x^2+\sqrt{-b^4+a^4 x^4}-x \text{$\#$1}\right)-4 i a^4 b^4 \log (x) \text{$\#$1}^2+4 i a^4 b^4 \log \left(i b^2+a^2 x^2+\sqrt{-b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^2-2 a^2 b^2 \log (x) \text{$\#$1}^4+2 a^2 b^2 \log \left(i b^2+a^2 x^2+\sqrt{-b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-i \log (x) \text{$\#$1}^6+i \log \left(i b^2+a^2 x^2+\sqrt{-b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^6}{8 a^6 b^6 \text{$\#$1}-4 i a^4 b^4 \text{$\#$1}^3-6 a^2 b^2 \text{$\#$1}^5-i \text{$\#$1}^7}\&\right]}{16 a^8 b^8}","-\frac{x}{8 a^8 b^8 \sqrt{-b^4+a^4 x^4}}-\frac{\tan ^{-1}\left(\frac{2^{3/4} a b x \sqrt{-b^4+a^4 x^4}}{b^4+\sqrt{2} a^2 b^2 x^2-a^4 x^4}\right)}{8\ 2^{3/4} a^9 b^9}-\frac{\tan ^{-1}\left(\frac{\frac{b^3}{2 a}+a b x^2-\frac{a^3 x^4}{2 b}}{x \sqrt{-b^4+a^4 x^4}}\right)}{32 a^9 b^9}+\frac{\tanh ^{-1}\left(\frac{\frac{b^3}{2 a}-a b x^2-\frac{a^3 x^4}{2 b}}{x \sqrt{-b^4+a^4 x^4}}\right)}{32 a^9 b^9}-\frac{\tanh ^{-1}\left(\frac{\frac{b^3}{2^{3/4} a}-\frac{a b x^2}{\sqrt[4]{2}}-\frac{a^3 x^4}{2^{3/4} b}}{x \sqrt{-b^4+a^4 x^4}}\right)}{8\ 2^{3/4} a^9 b^9}",1,"-1/8*x/(a^8*b^8*Sqrt[-b^4 + a^4*x^4]) - ((1/16 - I/16)*ArcTan[((1 + I)*a*b*x)/(I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4])])/(a^9*b^9) + ((1/32 - I/32)*ArcTanh[(b^4 - (1 + I)*a*b^3*x - (1 - I)*a^3*b*x^3 - a^4*x^4 + ((-I)*b^2 - (1 - I)*a*b*x - a^2*x^2)*Sqrt[-b^4 + a^4*x^4])/(I*b^4 - (1 - I)*a*b^3*x + (1 + I)*a^3*b*x^3 - I*a^4*x^4 + (b^2 + (1 + I)*a*b*x - I*a^2*x^2)*Sqrt[-b^4 + a^4*x^4])])/(a^9*b^9) + RootSum[16*a^8*b^8 + (32*I)*a^6*b^6*#1^2 + 8*a^4*b^4*#1^4 - (8*I)*a^2*b^2*#1^6 + #1^8 & , (-8*a^6*b^6*Log[x] + 8*a^6*b^6*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - x*#1] - (4*I)*a^4*b^4*Log[x]*#1^2 + (4*I)*a^4*b^4*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - x*#1]*#1^2 - 2*a^2*b^2*Log[x]*#1^4 + 2*a^2*b^2*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - x*#1]*#1^4 - I*Log[x]*#1^6 + I*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - x*#1]*#1^6)/(8*a^6*b^6*#1 - (4*I)*a^4*b^4*#1^3 - 6*a^2*b^2*#1^5 - I*#1^7) & ]/(16*a^8*b^8)","C",1
2865,1,307,304,0.7227522,"\int \frac{x^2}{\left(-b+a x^2\right) \sqrt[3]{-x+x^3}} \, dx","IntegrateAlgebraic[x^2/((-b + a*x^2)*(-x + x^3)^(1/3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)}{2 a}+\frac{\sqrt{3} \sqrt[3]{b} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a-b} x}{\sqrt[3]{a-b} x-2 \sqrt[3]{b} \sqrt[3]{-x+x^3}}\right)}{2 a \sqrt[3]{a-b}}-\frac{\log \left(-a x+a \sqrt[3]{-x+x^3}\right)}{2 a}-\frac{\sqrt[3]{b} \log \left(\sqrt[3]{a-b} x+\sqrt[3]{b} \sqrt[3]{-x+x^3}\right)}{2 a \sqrt[3]{a-b}}+\frac{\log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)}{4 a}+\frac{\sqrt[3]{b} \log \left((a-b)^{2/3} x^2-\sqrt[3]{a-b} \sqrt[3]{b} x \sqrt[3]{-x+x^3}+b^{2/3} \left(-x+x^3\right)^{2/3}\right)}{4 a \sqrt[3]{a-b}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)}{2 a}+\frac{\sqrt{3} \sqrt[3]{b} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a-b} x}{\sqrt[3]{a-b} x-2 \sqrt[3]{b} \sqrt[3]{-x+x^3}}\right)}{2 a \sqrt[3]{a-b}}-\frac{\log \left(-x+\sqrt[3]{-x+x^3}\right)}{2 a}-\frac{\sqrt[3]{b} \log \left(\sqrt[3]{a-b} x+\sqrt[3]{b} \sqrt[3]{-x+x^3}\right)}{2 a \sqrt[3]{a-b}}+\frac{\log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)}{4 a}+\frac{\sqrt[3]{b} \log \left((a-b)^{2/3} x^2-\sqrt[3]{a-b} \sqrt[3]{b} x \sqrt[3]{-x+x^3}+b^{2/3} \left(-x+x^3\right)^{2/3}\right)}{4 a \sqrt[3]{a-b}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-x + x^3)^(1/3))])/(2*a) + (Sqrt[3]*b^(1/3)*ArcTan[(Sqrt[3]*(a - b)^(1/3)*x)/((a - b)^(1/3)*x - 2*b^(1/3)*(-x + x^3)^(1/3))])/(2*a*(a - b)^(1/3)) - Log[-(a*x) + a*(-x + x^3)^(1/3)]/(2*a) - (b^(1/3)*Log[(a - b)^(1/3)*x + b^(1/3)*(-x + x^3)^(1/3)])/(2*a*(a - b)^(1/3)) + Log[x^2 + x*(-x + x^3)^(1/3) + (-x + x^3)^(2/3)]/(4*a) + (b^(1/3)*Log[(a - b)^(2/3)*x^2 - (a - b)^(1/3)*b^(1/3)*x*(-x + x^3)^(1/3) + b^(2/3)*(-x + x^3)^(2/3)])/(4*a*(a - b)^(1/3))","A",1
2866,1,304,304,0.6026818,"\int \frac{\left(1+x^4\right) \sqrt{x+\sqrt{1+x^2}}}{-1+x^4} \, dx","IntegrateAlgebraic[((1 + x^4)*Sqrt[x + Sqrt[1 + x^2]])/(-1 + x^4),x]","-\frac{1}{\sqrt{x+\sqrt{1+x^2}}}+\frac{1}{3} \left(x+\sqrt{1+x^2}\right)^{3/2}+\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\sqrt{-1+\sqrt{2}} \sqrt{x+\sqrt{1+x^2}}\right)-\sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\sqrt{1+\sqrt{2}} \sqrt{x+\sqrt{1+x^2}}\right)-\sqrt{2} \tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x}{\sqrt{2}}+\frac{\sqrt{1+x^2}}{\sqrt{2}}}{\sqrt{x+\sqrt{1+x^2}}}\right)-\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\sqrt{-1+\sqrt{2}} \sqrt{x+\sqrt{1+x^2}}\right)+\sqrt{-1+\sqrt{2}} \tanh ^{-1}\left(\sqrt{1+\sqrt{2}} \sqrt{x+\sqrt{1+x^2}}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{\frac{1}{\sqrt{2}}+\frac{x}{\sqrt{2}}+\frac{\sqrt{1+x^2}}{\sqrt{2}}}{\sqrt{x+\sqrt{1+x^2}}}\right)","-\frac{1}{\sqrt{x+\sqrt{1+x^2}}}+\frac{1}{3} \left(x+\sqrt{1+x^2}\right)^{3/2}-\sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{x+\sqrt{1+x^2}}}{\sqrt{-1+\sqrt{2}}}\right)+\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{x+\sqrt{1+x^2}}}{\sqrt{1+\sqrt{2}}}\right)-\sqrt{2} \tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x}{\sqrt{2}}+\frac{\sqrt{1+x^2}}{\sqrt{2}}}{\sqrt{x+\sqrt{1+x^2}}}\right)+\sqrt{-1+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{x+\sqrt{1+x^2}}}{\sqrt{-1+\sqrt{2}}}\right)-\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{x+\sqrt{1+x^2}}}{\sqrt{1+\sqrt{2}}}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{\frac{1}{\sqrt{2}}+\frac{x}{\sqrt{2}}+\frac{\sqrt{1+x^2}}{\sqrt{2}}}{\sqrt{x+\sqrt{1+x^2}}}\right)",1,"-(1/Sqrt[x + Sqrt[1 + x^2]]) + (x + Sqrt[1 + x^2])^(3/2)/3 + Sqrt[1 + Sqrt[2]]*ArcTan[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] - Sqrt[-1 + Sqrt[2]]*ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] - Sqrt[2]*ArcTan[(-(1/Sqrt[2]) + x/Sqrt[2] + Sqrt[1 + x^2]/Sqrt[2])/Sqrt[x + Sqrt[1 + x^2]]] - Sqrt[1 + Sqrt[2]]*ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[-1 + Sqrt[2]]*ArcTanh[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[2]*ArcTanh[(1/Sqrt[2] + x/Sqrt[2] + Sqrt[1 + x^2]/Sqrt[2])/Sqrt[x + Sqrt[1 + x^2]]]","A",1
2867,1,305,305,6.4198457,"\int \frac{(-a+x) (-b+x)}{\left((-a+x) (-b+x)^2\right)^{2/3} \left(-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2\right)} \, dx","IntegrateAlgebraic[((-a + x)*(-b + x))/(((-a + x)*(-b + x)^2)^(2/3)*(-b^2 + a^2*d + 2*(b - a*d)*x + (-1 + d)*x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a \sqrt[3]{d}-\sqrt{3} \sqrt[3]{d} x}{a \sqrt[3]{d}-\sqrt[3]{d} x-2 \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}\right)}{2 (a-b) d^{2/3}}+\frac{\log \left(a \sqrt[3]{d}-\sqrt[3]{d} x+\sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}\right)}{2 (a-b) d^{2/3}}-\frac{\log \left(a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}+\left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}\right)}{4 (a-b) d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a \sqrt[3]{d}-\sqrt{3} \sqrt[3]{d} x}{a \sqrt[3]{d}-\sqrt[3]{d} x-2 \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}\right)}{2 (a-b) d^{2/3}}+\frac{\log \left(a \sqrt[3]{d}-\sqrt[3]{d} x+\sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}\right)}{2 (a-b) d^{2/3}}-\frac{\log \left(a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}+\left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}\right)}{4 (a-b) d^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*a*d^(1/3) - Sqrt[3]*d^(1/3)*x)/(a*d^(1/3) - d^(1/3)*x - 2*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3))])/(2*(a - b)*d^(2/3)) + Log[a*d^(1/3) - d^(1/3)*x + (-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3)]/(2*(a - b)*d^(2/3)) - Log[a^2*d^(2/3) - 2*a*d^(2/3)*x + d^(2/3)*x^2 + (-(a*d^(1/3)) + d^(1/3)*x)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3) + (-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(2/3)]/(4*(a - b)*d^(2/3))","A",1
2868,1,305,305,6.3869264,"\int \frac{a b-(a+b) x+x^2}{\left((-a+x) (-b+x)^2\right)^{2/3} \left(-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2\right)} \, dx","IntegrateAlgebraic[(a*b - (a + b)*x + x^2)/(((-a + x)*(-b + x)^2)^(2/3)*(-b^2 + a^2*d + 2*(b - a*d)*x + (-1 + d)*x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a \sqrt[3]{d}-\sqrt{3} \sqrt[3]{d} x}{a \sqrt[3]{d}-\sqrt[3]{d} x-2 \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}\right)}{2 (a-b) d^{2/3}}+\frac{\log \left(a \sqrt[3]{d}-\sqrt[3]{d} x+\sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}\right)}{2 (a-b) d^{2/3}}-\frac{\log \left(a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}+\left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}\right)}{4 (a-b) d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a \sqrt[3]{d}-\sqrt{3} \sqrt[3]{d} x}{a \sqrt[3]{d}-\sqrt[3]{d} x-2 \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}\right)}{2 (a-b) d^{2/3}}+\frac{\log \left(a \sqrt[3]{d}-\sqrt[3]{d} x+\sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}\right)}{2 (a-b) d^{2/3}}-\frac{\log \left(a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}+\left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}\right)}{4 (a-b) d^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*a*d^(1/3) - Sqrt[3]*d^(1/3)*x)/(a*d^(1/3) - d^(1/3)*x - 2*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3))])/(2*(a - b)*d^(2/3)) + Log[a*d^(1/3) - d^(1/3)*x + (-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3)]/(2*(a - b)*d^(2/3)) - Log[a^2*d^(2/3) - 2*a*d^(2/3)*x + d^(2/3)*x^2 + (-(a*d^(1/3)) + d^(1/3)*x)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3) + (-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(2/3)]/(4*(a - b)*d^(2/3))","A",1
2869,1,305,305,0.4206704,"\int \frac{x}{x^2-\sqrt{b+a x} \sqrt{c+\sqrt{b+a x}}} \, dx","IntegrateAlgebraic[x/(x^2 - Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]]),x]","-4 \text{RootSum}\left[b^2-2 b c^2+c^4+a^2 c \text{$\#$1}+4 b c \text{$\#$1}^2-4 c^3 \text{$\#$1}^2-a^2 \text{$\#$1}^3-2 b \text{$\#$1}^4+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-b c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}+c^3 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}+b \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3-3 c^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3+3 c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^5-\log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^7}{a^2 c+8 b c \text{$\#$1}-8 c^3 \text{$\#$1}-3 a^2 \text{$\#$1}^2-8 b \text{$\#$1}^3+24 c^2 \text{$\#$1}^3-24 c \text{$\#$1}^5+8 \text{$\#$1}^7}\&\right]","-4 \text{RootSum}\left[b^2-2 b c^2+c^4+a^2 c \text{$\#$1}+4 b c \text{$\#$1}^2-4 c^3 \text{$\#$1}^2-a^2 \text{$\#$1}^3-2 b \text{$\#$1}^4+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-b c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}+c^3 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}+b \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3-3 c^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3+3 c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^5-\log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^7}{a^2 c+8 b c \text{$\#$1}-8 c^3 \text{$\#$1}-3 a^2 \text{$\#$1}^2-8 b \text{$\#$1}^3+24 c^2 \text{$\#$1}^3-24 c \text{$\#$1}^5+8 \text{$\#$1}^7}\&\right]",1,"-4*RootSum[b^2 - 2*b*c^2 + c^4 + a^2*c*#1 + 4*b*c*#1^2 - 4*c^3*#1^2 - a^2*#1^3 - 2*b*#1^4 + 6*c^2*#1^4 - 4*c*#1^6 + #1^8 & , (-(b*c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1) + c^3*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1 + b*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^3 - 3*c^2*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^3 + 3*c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^5 - Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^7)/(a^2*c + 8*b*c*#1 - 8*c^3*#1 - 3*a^2*#1^2 - 8*b*#1^3 + 24*c^2*#1^3 - 24*c*#1^5 + 8*#1^7) & ]","A",1
2870,1,305,305,0.0019479,"\int \frac{x}{x^2-\sqrt{b+a x} \sqrt{c+\sqrt{b+a x}}} \, dx","IntegrateAlgebraic[x/(x^2 - Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]]),x]","-4 \text{RootSum}\left[b^2-2 b c^2+c^4+a^2 c \text{$\#$1}+4 b c \text{$\#$1}^2-4 c^3 \text{$\#$1}^2-a^2 \text{$\#$1}^3-2 b \text{$\#$1}^4+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-b c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}+c^3 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}+b \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3-3 c^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3+3 c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^5-\log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^7}{a^2 c+8 b c \text{$\#$1}-8 c^3 \text{$\#$1}-3 a^2 \text{$\#$1}^2-8 b \text{$\#$1}^3+24 c^2 \text{$\#$1}^3-24 c \text{$\#$1}^5+8 \text{$\#$1}^7}\&\right]","-4 \text{RootSum}\left[b^2-2 b c^2+c^4+a^2 c \text{$\#$1}+4 b c \text{$\#$1}^2-4 c^3 \text{$\#$1}^2-a^2 \text{$\#$1}^3-2 b \text{$\#$1}^4+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-b c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}+c^3 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}+b \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3-3 c^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3+3 c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^5-\log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^7}{a^2 c+8 b c \text{$\#$1}-8 c^3 \text{$\#$1}-3 a^2 \text{$\#$1}^2-8 b \text{$\#$1}^3+24 c^2 \text{$\#$1}^3-24 c \text{$\#$1}^5+8 \text{$\#$1}^7}\&\right]",1,"-4*RootSum[b^2 - 2*b*c^2 + c^4 + a^2*c*#1 + 4*b*c*#1^2 - 4*c^3*#1^2 - a^2*#1^3 - 2*b*#1^4 + 6*c^2*#1^4 - 4*c*#1^6 + #1^8 & , (-(b*c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1) + c^3*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1 + b*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^3 - 3*c^2*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^3 + 3*c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^5 - Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^7)/(a^2*c + 8*b*c*#1 - 8*c^3*#1 - 3*a^2*#1^2 - 8*b*#1^3 + 24*c^2*#1^3 - 24*c*#1^5 + 8*#1^7) & ]","A",1
2871,1,306,306,0.5849203,"\int \frac{a b-2 b x+x^2}{\sqrt[3]{x (-a+x) (-b+x)^2} \left(b d-(a+d) x+x^2\right)} \, dx","IntegrateAlgebraic[(a*b - 2*b*x + x^2)/((x*(-a + x)*(-b + x)^2)^(1/3)*(b*d - (a + d)*x + x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{-2 b \sqrt[3]{d}+2 \sqrt[3]{d} x+\sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)}{\sqrt[3]{d}}+\frac{\log \left(b \sqrt{d}-\sqrt{d} x+\sqrt[6]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}\right)}{\sqrt[3]{d}}-\frac{\log \left(b^2 d-2 b d x+d x^2+\left(-b d^{2/3}+d^{2/3} x\right) \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}+\sqrt[3]{d} \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{-2 b \sqrt[3]{d}+2 \sqrt[3]{d} x+\sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)}{\sqrt[3]{d}}+\frac{\log \left(b \sqrt{d}-\sqrt{d} x+\sqrt[6]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}\right)}{\sqrt[3]{d}}-\frac{\log \left(b^2 d-2 b d x+d x^2+\left(-b d^{2/3}+d^{2/3} x\right) \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}+\sqrt[3]{d} \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3))/(-2*b*d^(1/3) + 2*d^(1/3)*x + (-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3))])/d^(1/3) + Log[b*Sqrt[d] - Sqrt[d]*x + d^(1/6)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3)]/d^(1/3) - Log[b^2*d - 2*b*d*x + d*x^2 + (-(b*d^(2/3)) + d^(2/3)*x)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3) + d^(1/3)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(2/3)]/(2*d^(1/3))","A",1
2872,1,306,306,0.6820449,"\int \frac{-a b^2+(4 a-b) b x-3 a x^2+x^3}{\sqrt[3]{x (-a+x) (-b+x)^2} \left(-a^2 d+\left(b^2+2 a d\right) x-(2 b+d) x^2+x^3\right)} \, dx","IntegrateAlgebraic[(-(a*b^2) + (4*a - b)*b*x - 3*a*x^2 + x^3)/((x*(-a + x)*(-b + x)^2)^(1/3)*(-(a^2*d) + (b^2 + 2*a*d)*x - (2*b + d)*x^2 + x^3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{-2 a \sqrt[3]{d}+2 \sqrt[3]{d} x+\sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)}{\sqrt[3]{d}}+\frac{\log \left(a \sqrt[3]{d}-\sqrt[3]{d} x+\sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}\right)}{\sqrt[3]{d}}-\frac{\log \left(a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}+\left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{-2 a \sqrt[3]{d}+2 \sqrt[3]{d} x+\sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)}{\sqrt[3]{d}}+\frac{\log \left(a \sqrt[3]{d}-\sqrt[3]{d} x+\sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}\right)}{\sqrt[3]{d}}-\frac{\log \left(a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}+\left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{2/3}\right)}{2 \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3))/(-2*a*d^(1/3) + 2*d^(1/3)*x + (-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3))])/d^(1/3) + Log[a*d^(1/3) - d^(1/3)*x + (-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3)]/d^(1/3) - Log[a^2*d^(2/3) - 2*a*d^(2/3)*x + d^(2/3)*x^2 + (-(a*d^(1/3)) + d^(1/3)*x)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3) + (-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(2/3)]/(2*d^(1/3))","A",1
2873,1,306,306,4.7388383,"\int \frac{-d+c x^7}{x \sqrt[3]{-b+a x^3}} \, dx","IntegrateAlgebraic[(-d + c*x^7)/(x*(-b + a*x^3)^(1/3)),x]","\frac{c \left(-b+a x^3\right)^{2/3} \left(4 b x+3 a x^4\right)}{18 a^2}-\frac{2 b^2 c \tan ^{-1}\left(\frac{\frac{x}{\sqrt{3}}+\frac{2 \sqrt[3]{-b+a x^3}}{\sqrt{3} \sqrt[3]{a}}}{x}\right)}{9 \sqrt{3} a^{7/3}}+\frac{d \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-b+a x^3}}{\sqrt{3} \sqrt[3]{b}}\right)}{\sqrt{3} \sqrt[3]{b}}+\frac{d \log \left(\sqrt[3]{b}+\sqrt[3]{-b+a x^3}\right)}{3 \sqrt[3]{b}}-\frac{2 b^2 c \log \left(-\sqrt[3]{a} x+\sqrt[3]{-b+a x^3}\right)}{27 a^{7/3}}-\frac{d \log \left(b^{2/3}-\sqrt[3]{b} \sqrt[3]{-b+a x^3}+\left(-b+a x^3\right)^{2/3}\right)}{6 \sqrt[3]{b}}+\frac{b^2 c \log \left(a^{2/3} x^2+\sqrt[3]{a} x \sqrt[3]{-b+a x^3}+\left(-b+a x^3\right)^{2/3}\right)}{27 a^{7/3}}","\frac{c \left(-b+a x^3\right)^{2/3} \left(4 b x+3 a x^4\right)}{18 a^2}-\frac{2 b^2 c \tan ^{-1}\left(\frac{\frac{x}{\sqrt{3}}+\frac{2 \sqrt[3]{-b+a x^3}}{\sqrt{3} \sqrt[3]{a}}}{x}\right)}{9 \sqrt{3} a^{7/3}}+\frac{d \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-b+a x^3}}{\sqrt{3} \sqrt[3]{b}}\right)}{\sqrt{3} \sqrt[3]{b}}+\frac{d \log \left(\sqrt[3]{b}+\sqrt[3]{-b+a x^3}\right)}{3 \sqrt[3]{b}}-\frac{2 b^2 c \log \left(-\sqrt[3]{a} x+\sqrt[3]{-b+a x^3}\right)}{27 a^{7/3}}-\frac{d \log \left(b^{2/3}-\sqrt[3]{b} \sqrt[3]{-b+a x^3}+\left(-b+a x^3\right)^{2/3}\right)}{6 \sqrt[3]{b}}+\frac{b^2 c \log \left(a^{2/3} x^2+\sqrt[3]{a} x \sqrt[3]{-b+a x^3}+\left(-b+a x^3\right)^{2/3}\right)}{27 a^{7/3}}",1,"(c*(-b + a*x^3)^(2/3)*(4*b*x + 3*a*x^4))/(18*a^2) - (2*b^2*c*ArcTan[(x/Sqrt[3] + (2*(-b + a*x^3)^(1/3))/(Sqrt[3]*a^(1/3)))/x])/(9*Sqrt[3]*a^(7/3)) + (d*ArcTan[1/Sqrt[3] - (2*(-b + a*x^3)^(1/3))/(Sqrt[3]*b^(1/3))])/(Sqrt[3]*b^(1/3)) + (d*Log[b^(1/3) + (-b + a*x^3)^(1/3)])/(3*b^(1/3)) - (2*b^2*c*Log[-(a^(1/3)*x) + (-b + a*x^3)^(1/3)])/(27*a^(7/3)) - (d*Log[b^(2/3) - b^(1/3)*(-b + a*x^3)^(1/3) + (-b + a*x^3)^(2/3)])/(6*b^(1/3)) + (b^2*c*Log[a^(2/3)*x^2 + a^(1/3)*x*(-b + a*x^3)^(1/3) + (-b + a*x^3)^(2/3)])/(27*a^(7/3))","A",1
2874,1,306,306,0.6528537,"\int \frac{\sqrt{c+\sqrt{b+a x}}}{x-\sqrt{b+a x}} \, dx","IntegrateAlgebraic[Sqrt[c + Sqrt[b + a*x]]/(x - Sqrt[b + a*x]),x]","4 \sqrt{c+\sqrt{b+a x}}+\frac{2 \left(\sqrt{2} a^2+2 \sqrt{2} b+\sqrt{2} a \sqrt{a^2+4 b}+\sqrt{2} a c+\sqrt{2} \sqrt{a^2+4 b} c\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c+\sqrt{b+a x}}}{\sqrt{-a-\sqrt{a^2+4 b}-2 c}}\right)}{\sqrt{a^2+4 b} \sqrt{-a-\sqrt{a^2+4 b}-2 c}}+\frac{2 \left(-\sqrt{2} a^2-2 \sqrt{2} b+\sqrt{2} a \sqrt{a^2+4 b}-\sqrt{2} a c+\sqrt{2} \sqrt{a^2+4 b} c\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c+\sqrt{b+a x}}}{\sqrt{-a+\sqrt{a^2+4 b}-2 c}}\right)}{\sqrt{a^2+4 b} \sqrt{-a+\sqrt{a^2+4 b}-2 c}}","4 \sqrt{c+\sqrt{b+a x}}+\frac{2 \left(\sqrt{2} a^2+2 \sqrt{2} b+\sqrt{2} a \sqrt{a^2+4 b}+\sqrt{2} a c+\sqrt{2} \sqrt{a^2+4 b} c\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c+\sqrt{b+a x}}}{\sqrt{-a-\sqrt{a^2+4 b}-2 c}}\right)}{\sqrt{a^2+4 b} \sqrt{-a-\sqrt{a^2+4 b}-2 c}}+\frac{2 \left(-\sqrt{2} a^2-2 \sqrt{2} b+\sqrt{2} a \sqrt{a^2+4 b}-\sqrt{2} a c+\sqrt{2} \sqrt{a^2+4 b} c\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c+\sqrt{b+a x}}}{\sqrt{-a+\sqrt{a^2+4 b}-2 c}}\right)}{\sqrt{a^2+4 b} \sqrt{-a+\sqrt{a^2+4 b}-2 c}}",1,"4*Sqrt[c + Sqrt[b + a*x]] + (2*(Sqrt[2]*a^2 + 2*Sqrt[2]*b + Sqrt[2]*a*Sqrt[a^2 + 4*b] + Sqrt[2]*a*c + Sqrt[2]*Sqrt[a^2 + 4*b]*c)*ArcTan[(Sqrt[2]*Sqrt[c + Sqrt[b + a*x]])/Sqrt[-a - Sqrt[a^2 + 4*b] - 2*c]])/(Sqrt[a^2 + 4*b]*Sqrt[-a - Sqrt[a^2 + 4*b] - 2*c]) + (2*(-(Sqrt[2]*a^2) - 2*Sqrt[2]*b + Sqrt[2]*a*Sqrt[a^2 + 4*b] - Sqrt[2]*a*c + Sqrt[2]*Sqrt[a^2 + 4*b]*c)*ArcTan[(Sqrt[2]*Sqrt[c + Sqrt[b + a*x]])/Sqrt[-a + Sqrt[a^2 + 4*b] - 2*c]])/(Sqrt[a^2 + 4*b]*Sqrt[-a + Sqrt[a^2 + 4*b] - 2*c])","A",1
2875,1,308,308,0.3995941,"\int \frac{1+x}{(1+2 x) \sqrt[3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}} \, dx","IntegrateAlgebraic[(1 + x)/((1 + 2*x)*(27 + 27*x + 36*x^2 + 28*x^3 + 9*x^4 + x^5)^(1/3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{5 \sqrt{3} \sqrt[3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}}{12 \sqrt[3]{10}-2 \sqrt[3]{10} x-2 \sqrt[3]{10} x^2+5 \sqrt[3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}}\right)}{5 \sqrt[3]{10}}+\frac{\log \left(-6 \sqrt[3]{10}+\sqrt[3]{10} x+\sqrt[3]{10} x^2+5 \sqrt[3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}\right)}{5 \sqrt[3]{10}}-\frac{\log \left(36\ 10^{2/3}-12\ 10^{2/3} x-11\ 10^{2/3} x^2+2\ 10^{2/3} x^3+10^{2/3} x^4+\left(30 \sqrt[3]{10}-5 \sqrt[3]{10} x-5 \sqrt[3]{10} x^2\right) \sqrt[3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}+25 \left(27+27 x+36 x^2+28 x^3+9 x^4+x^5\right)^{2/3}\right)}{10 \sqrt[3]{10}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{5 \sqrt{3} \sqrt[3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}}{12 \sqrt[3]{10}-2 \sqrt[3]{10} x-2 \sqrt[3]{10} x^2+5 \sqrt[3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}}\right)}{5 \sqrt[3]{10}}+\frac{\log \left(-6 \sqrt[3]{10}+\sqrt[3]{10} x+\sqrt[3]{10} x^2+5 \sqrt[3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}\right)}{5 \sqrt[3]{10}}-\frac{\log \left(36\ 10^{2/3}-12\ 10^{2/3} x-11\ 10^{2/3} x^2+2\ 10^{2/3} x^3+10^{2/3} x^4+\left(30 \sqrt[3]{10}-5 \sqrt[3]{10} x-5 \sqrt[3]{10} x^2\right) \sqrt[3]{27+27 x+36 x^2+28 x^3+9 x^4+x^5}+25 \left(27+27 x+36 x^2+28 x^3+9 x^4+x^5\right)^{2/3}\right)}{10 \sqrt[3]{10}}",1,"(Sqrt[3]*ArcTan[(5*Sqrt[3]*(27 + 27*x + 36*x^2 + 28*x^3 + 9*x^4 + x^5)^(1/3))/(12*10^(1/3) - 2*10^(1/3)*x - 2*10^(1/3)*x^2 + 5*(27 + 27*x + 36*x^2 + 28*x^3 + 9*x^4 + x^5)^(1/3))])/(5*10^(1/3)) + Log[-6*10^(1/3) + 10^(1/3)*x + 10^(1/3)*x^2 + 5*(27 + 27*x + 36*x^2 + 28*x^3 + 9*x^4 + x^5)^(1/3)]/(5*10^(1/3)) - Log[36*10^(2/3) - 12*10^(2/3)*x - 11*10^(2/3)*x^2 + 2*10^(2/3)*x^3 + 10^(2/3)*x^4 + (30*10^(1/3) - 5*10^(1/3)*x - 5*10^(1/3)*x^2)*(27 + 27*x + 36*x^2 + 28*x^3 + 9*x^4 + x^5)^(1/3) + 25*(27 + 27*x + 36*x^2 + 28*x^3 + 9*x^4 + x^5)^(2/3)]/(10*10^(1/3))","A",1
2876,1,313,310,3.0679258,"\int \frac{-b x+a x^3}{\left(-b+2 a x^3\right) \sqrt[3]{b^2 x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[(-(b*x) + a*x^3)/((-b + 2*a*x^3)*(b^2*x^2 + a^3*x^3)^(1/3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{2 a}-\frac{\log \left(a^2 x-a \sqrt[3]{b^2 x^2+a^3 x^3}\right)}{2 a}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{b^2 x^2+a^3 x^3}+\left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{4 a}+\frac{1}{6} \text{RootSum}\left[a^9+2 a b^5-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{a^3 \log (x)+2 b^2 \log (x)-a^3 \log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)-2 b^2 \log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^3+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-a^3 \text{$\#$1}+\text{$\#$1}^4}\&\right]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{2 a}-\frac{\log \left(-a x+\sqrt[3]{b^2 x^2+a^3 x^3}\right)}{2 a}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{b^2 x^2+a^3 x^3}+\left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{4 a}+\frac{1}{6} \text{RootSum}\left[a^9+2 a b^5-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-a^3 \log (x)-2 b^2 \log (x)+a^3 \log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)+2 b^2 \log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{a^3 \text{$\#$1}-\text{$\#$1}^4}\&\right]",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*a*x)/(a*x + 2*(b^2*x^2 + a^3*x^3)^(1/3))])/(2*a) - Log[a^2*x - a*(b^2*x^2 + a^3*x^3)^(1/3)]/(2*a) + Log[a^2*x^2 + a*x*(b^2*x^2 + a^3*x^3)^(1/3) + (b^2*x^2 + a^3*x^3)^(2/3)]/(4*a) + RootSum[a^9 + 2*a*b^5 - 3*a^6*#1^3 + 3*a^3*#1^6 - #1^9 & , (a^3*Log[x] + 2*b^2*Log[x] - a^3*Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1] - 2*b^2*Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1] - Log[x]*#1^3 + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1]*#1^3)/(-(a^3*#1) + #1^4) & ]/6","A",1
2877,1,313,310,0.0023059,"\int \frac{-b x+a x^3}{\left(-b+2 a x^3\right) \sqrt[3]{b^2 x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[(-(b*x) + a*x^3)/((-b + 2*a*x^3)*(b^2*x^2 + a^3*x^3)^(1/3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{2 a}-\frac{\log \left(a^2 x-a \sqrt[3]{b^2 x^2+a^3 x^3}\right)}{2 a}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{b^2 x^2+a^3 x^3}+\left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{4 a}+\frac{1}{6} \text{RootSum}\left[a^9+2 a b^5-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{a^3 \log (x)+2 b^2 \log (x)-a^3 \log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)-2 b^2 \log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^3+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-a^3 \text{$\#$1}+\text{$\#$1}^4}\&\right]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{2 a}-\frac{\log \left(-a x+\sqrt[3]{b^2 x^2+a^3 x^3}\right)}{2 a}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{b^2 x^2+a^3 x^3}+\left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{4 a}+\frac{1}{6} \text{RootSum}\left[a^9+2 a b^5-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-a^3 \log (x)-2 b^2 \log (x)+a^3 \log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)+2 b^2 \log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{a^3 \text{$\#$1}-\text{$\#$1}^4}\&\right]",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*a*x)/(a*x + 2*(b^2*x^2 + a^3*x^3)^(1/3))])/(2*a) - Log[a^2*x - a*(b^2*x^2 + a^3*x^3)^(1/3)]/(2*a) + Log[a^2*x^2 + a*x*(b^2*x^2 + a^3*x^3)^(1/3) + (b^2*x^2 + a^3*x^3)^(2/3)]/(4*a) + RootSum[a^9 + 2*a*b^5 - 3*a^6*#1^3 + 3*a^3*#1^6 - #1^9 & , (a^3*Log[x] + 2*b^2*Log[x] - a^3*Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1] - 2*b^2*Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1] - Log[x]*#1^3 + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1]*#1^3)/(-(a^3*#1) + #1^4) & ]/6","A",1
2878,1,313,310,3.0313923,"\int \frac{b x+a x^3}{\left(b+2 a x^3\right) \sqrt[3]{b^2 x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[(b*x + a*x^3)/((b + 2*a*x^3)*(b^2*x^2 + a^3*x^3)^(1/3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{2 a}-\frac{\log \left(a^2 x-a \sqrt[3]{b^2 x^2+a^3 x^3}\right)}{2 a}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{b^2 x^2+a^3 x^3}+\left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{4 a}+\frac{1}{6} \text{RootSum}\left[a^9-2 a b^5-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{a^3 \log (x)+2 b^2 \log (x)-a^3 \log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)-2 b^2 \log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^3+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-a^3 \text{$\#$1}+\text{$\#$1}^4}\&\right]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{2 a}-\frac{\log \left(-a x+\sqrt[3]{b^2 x^2+a^3 x^3}\right)}{2 a}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{b^2 x^2+a^3 x^3}+\left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{4 a}+\frac{1}{6} \text{RootSum}\left[a^9-2 a b^5-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-a^3 \log (x)-2 b^2 \log (x)+a^3 \log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)+2 b^2 \log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{a^3 \text{$\#$1}-\text{$\#$1}^4}\&\right]",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*a*x)/(a*x + 2*(b^2*x^2 + a^3*x^3)^(1/3))])/(2*a) - Log[a^2*x - a*(b^2*x^2 + a^3*x^3)^(1/3)]/(2*a) + Log[a^2*x^2 + a*x*(b^2*x^2 + a^3*x^3)^(1/3) + (b^2*x^2 + a^3*x^3)^(2/3)]/(4*a) + RootSum[a^9 - 2*a*b^5 - 3*a^6*#1^3 + 3*a^3*#1^6 - #1^9 & , (a^3*Log[x] + 2*b^2*Log[x] - a^3*Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1] - 2*b^2*Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1] - Log[x]*#1^3 + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1]*#1^3)/(-(a^3*#1) + #1^4) & ]/6","A",1
2879,1,310,310,4.1747379,"\int \frac{x (-b+x) (a b+(-2 a+b) x)}{\sqrt[3]{x (-a+x) (-b+x)} \left(-a^4+4 a^3 x+\left(-6 a^2+b^2 d\right) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right)} \, dx","IntegrateAlgebraic[(x*(-b + x)*(a*b + (-2*a + b)*x))/((x*(-a + x)*(-b + x))^(1/3)*(-a^4 + 4*a^3*x + (-6*a^2 + b^2*d)*x^2 + 2*(2*a - b*d)*x^3 + (-1 + d)*x^4)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}}{2 a-2 x+\sqrt[6]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}}\right)}{2 d^{5/6}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}}{-2 a+2 x+\sqrt[6]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}}\right)}{2 d^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}}{a-x}\right)}{d^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\left(a \sqrt[6]{d}-\sqrt[6]{d} x\right) \sqrt[3]{a b x+(-a-b) x^2+x^3}}{a^2-2 a x+x^2+\sqrt[3]{d} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}}\right)}{2 d^{5/6}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}}{2 a-2 x+\sqrt[6]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}}\right)}{2 d^{5/6}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}}{-2 a+2 x+\sqrt[6]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}}\right)}{2 d^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{d} \sqrt[3]{a b x+(-a-b) x^2+x^3}}{a-x}\right)}{d^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\left(a \sqrt[6]{d}-\sqrt[6]{d} x\right) \sqrt[3]{a b x+(-a-b) x^2+x^3}}{a^2-2 a x+x^2+\sqrt[3]{d} \left(a b x+(-a-b) x^2+x^3\right)^{2/3}}\right)}{2 d^{5/6}}",1,"-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/6)*(a*b*x + (-a - b)*x^2 + x^3)^(1/3))/(2*a - 2*x + d^(1/6)*(a*b*x + (-a - b)*x^2 + x^3)^(1/3))])/d^(5/6) + (Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/6)*(a*b*x + (-a - b)*x^2 + x^3)^(1/3))/(-2*a + 2*x + d^(1/6)*(a*b*x + (-a - b)*x^2 + x^3)^(1/3))])/(2*d^(5/6)) + ArcTanh[(d^(1/6)*(a*b*x + (-a - b)*x^2 + x^3)^(1/3))/(a - x)]/d^(5/6) + ArcTanh[((a*d^(1/6) - d^(1/6)*x)*(a*b*x + (-a - b)*x^2 + x^3)^(1/3))/(a^2 - 2*a*x + x^2 + d^(1/3)*(a*b*x + (-a - b)*x^2 + x^3)^(2/3))]/(2*d^(5/6))","A",1
2880,1,311,311,3.1549509,"\int \frac{-a b-a c+2 b c+(2 a-b-c) x}{\sqrt[3]{(-a+x) (-b+x) (-c+x)} \left(-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right)} \, dx","IntegrateAlgebraic[(-(a*b) - a*c + 2*b*c + (2*a - b - c)*x)/(((-a + x)*(-b + x)*(-c + x))^(1/3)*(-(b*c) + a^2*d + (b + c - 2*a*d)*x + (-1 + d)*x^2)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{-2 a \sqrt[3]{d}+2 \sqrt[3]{d} x+\sqrt[3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}\right)}{\sqrt[3]{d}}+\frac{\log \left(a \sqrt[3]{d}-\sqrt[3]{d} x+\sqrt[3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}\right)}{\sqrt[3]{d}}-\frac{\log \left(a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}+\left(-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3\right)^{2/3}\right)}{2 \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{-2 a \sqrt[3]{d}+2 \sqrt[3]{d} x+\sqrt[3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}\right)}{\sqrt[3]{d}}+\frac{\log \left(a \sqrt[3]{d}-\sqrt[3]{d} x+\sqrt[3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}\right)}{\sqrt[3]{d}}-\frac{\log \left(a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left(-a \sqrt[3]{d}+\sqrt[3]{d} x\right) \sqrt[3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}+\left(-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3\right)^{2/3}\right)}{2 \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3)^(1/3))/(-2*a*d^(1/3) + 2*d^(1/3)*x + (-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3)^(1/3))])/d^(1/3) + Log[a*d^(1/3) - d^(1/3)*x + (-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3)^(1/3)]/d^(1/3) - Log[a^2*d^(2/3) - 2*a*d^(2/3)*x + d^(2/3)*x^2 + (-(a*d^(1/3)) + d^(1/3)*x)*(-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3)^(1/3) + (-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3)^(2/3)]/(2*d^(1/3))","A",1
2881,1,309,311,0.7793667,"\int \frac{x \left(-a b+x^2\right)}{\sqrt[3]{x^2 (-a+x) (-b+x)} \left(a^2 b^2-2 a b (a+b) x+\left(a^2+4 a b+b^2-d\right) x^2-2 (a+b) x^3+x^4\right)} \, dx","IntegrateAlgebraic[(x*(-(a*b) + x^2))/((x^2*(-a + x)*(-b + x))^(1/3)*(a^2*b^2 - 2*a*b*(a + b)*x + (a^2 + 4*a*b + b^2 - d)*x^2 - 2*(a + b)*x^3 + x^4)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\frac{x^2}{\sqrt{3}}+\frac{2 \left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}}{\sqrt{3} \sqrt[3]{d}}}{x^2}\right)}{2 d^{2/3}}+\frac{\log \left(-\sqrt[6]{d} x+\sqrt[3]{a b x^2+(-a-b) x^3+x^4}\right)}{2 d^{2/3}}+\frac{\log \left(\sqrt[6]{d} x+\sqrt[3]{a b x^2+(-a-b) x^3+x^4}\right)}{2 d^{2/3}}-\frac{\log \left(\sqrt[3]{d} x^2-\sqrt[6]{d} x \sqrt[3]{a b x^2+(-a-b) x^3+x^4}+\left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}\right)}{4 d^{2/3}}-\frac{\log \left(\sqrt[3]{d} x^2+\sqrt[6]{d} x \sqrt[3]{a b x^2+(-a-b) x^3+x^4}+\left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}\right)}{4 d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} x^2}{\sqrt[3]{d} x^2+2 \left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}}\right)}{2 d^{2/3}}+\frac{\log \left(-\sqrt[6]{d} x+\sqrt[3]{a b x^2+(-a-b) x^3+x^4}\right)}{2 d^{2/3}}+\frac{\log \left(\sqrt[6]{d} x+\sqrt[3]{a b x^2+(-a-b) x^3+x^4}\right)}{2 d^{2/3}}-\frac{\log \left(\sqrt[3]{d} x^2-\sqrt[6]{d} x \sqrt[3]{a b x^2+(-a-b) x^3+x^4}+\left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}\right)}{4 d^{2/3}}-\frac{\log \left(\sqrt[3]{d} x^2+\sqrt[6]{d} x \sqrt[3]{a b x^2+(-a-b) x^3+x^4}+\left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}\right)}{4 d^{2/3}}",1,"-1/2*(Sqrt[3]*ArcTan[(x^2/Sqrt[3] + (2*(a*b*x^2 + (-a - b)*x^3 + x^4)^(2/3))/(Sqrt[3]*d^(1/3)))/x^2])/d^(2/3) + Log[-(d^(1/6)*x) + (a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3)]/(2*d^(2/3)) + Log[d^(1/6)*x + (a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3)]/(2*d^(2/3)) - Log[d^(1/3)*x^2 - d^(1/6)*x*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3) + (a*b*x^2 + (-a - b)*x^3 + x^4)^(2/3)]/(4*d^(2/3)) - Log[d^(1/3)*x^2 + d^(1/6)*x*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3) + (a*b*x^2 + (-a - b)*x^3 + x^4)^(2/3)]/(4*d^(2/3))","A",1
2882,1,311,311,0.8283545,"\int \frac{-b^6+a^6 x^6}{\sqrt{-b^2 x+a^2 x^3} \left(b^6+a^6 x^6\right)} \, dx","IntegrateAlgebraic[(-b^6 + a^6*x^6)/(Sqrt[-(b^2*x) + a^2*x^3]*(b^6 + a^6*x^6)),x]","-\frac{\tan ^{-1}\left(\frac{2 \sqrt{a} \sqrt{b} \sqrt{-b^2 x+a^2 x^3}}{-b^2-2 a b x+a^2 x^2}\right)}{6 \sqrt{a} \sqrt{b}}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{b} \sqrt{-b^2 x+a^2 x^3}}{-b^2-a b x+a^2 x^2}\right)}{3 \sqrt{a} \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{-\frac{b^{3/2}}{2 \sqrt{a}}+\sqrt{a} \sqrt{b} x+\frac{a^{3/2} x^2}{2 \sqrt{b}}}{\sqrt{-b^2 x+a^2 x^3}}\right)}{6 \sqrt{a} \sqrt{b}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{-\frac{b^{3/2}}{\sqrt{2} \sqrt{a}}+\frac{\sqrt{a} \sqrt{b} x}{\sqrt{2}}+\frac{a^{3/2} x^2}{\sqrt{2} \sqrt{b}}}{\sqrt{-b^2 x+a^2 x^3}}\right)}{3 \sqrt{a} \sqrt{b}}","-\frac{\tan ^{-1}\left(\frac{2 \sqrt{a} \sqrt{b} \sqrt{-b^2 x+a^2 x^3}}{-b^2-2 a b x+a^2 x^2}\right)}{6 \sqrt{a} \sqrt{b}}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{b} \sqrt{-b^2 x+a^2 x^3}}{-b^2-a b x+a^2 x^2}\right)}{3 \sqrt{a} \sqrt{b}}-\frac{\tanh ^{-1}\left(\frac{-\frac{b^{3/2}}{2 \sqrt{a}}+\sqrt{a} \sqrt{b} x+\frac{a^{3/2} x^2}{2 \sqrt{b}}}{\sqrt{-b^2 x+a^2 x^3}}\right)}{6 \sqrt{a} \sqrt{b}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{-\frac{b^{3/2}}{\sqrt{2} \sqrt{a}}+\frac{\sqrt{a} \sqrt{b} x}{\sqrt{2}}+\frac{a^{3/2} x^2}{\sqrt{2} \sqrt{b}}}{\sqrt{-b^2 x+a^2 x^3}}\right)}{3 \sqrt{a} \sqrt{b}}",1,"-1/6*ArcTan[(2*Sqrt[a]*Sqrt[b]*Sqrt[-(b^2*x) + a^2*x^3])/(-b^2 - 2*a*b*x + a^2*x^2)]/(Sqrt[a]*Sqrt[b]) - (Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[b]*Sqrt[-(b^2*x) + a^2*x^3])/(-b^2 - a*b*x + a^2*x^2)])/(3*Sqrt[a]*Sqrt[b]) - ArcTanh[(-1/2*b^(3/2)/Sqrt[a] + Sqrt[a]*Sqrt[b]*x + (a^(3/2)*x^2)/(2*Sqrt[b]))/Sqrt[-(b^2*x) + a^2*x^3]]/(6*Sqrt[a]*Sqrt[b]) - (Sqrt[2]*ArcTanh[(-(b^(3/2)/(Sqrt[2]*Sqrt[a])) + (Sqrt[a]*Sqrt[b]*x)/Sqrt[2] + (a^(3/2)*x^2)/(Sqrt[2]*Sqrt[b]))/Sqrt[-(b^2*x) + a^2*x^3]])/(3*Sqrt[a]*Sqrt[b])","A",1
2883,1,440,311,1.8508174,"\int \frac{\sqrt{1+x^4} \sqrt{x^2+\sqrt{1+x^4}}}{-1+x^4} \, dx","IntegrateAlgebraic[(Sqrt[1 + x^4]*Sqrt[x^2 + Sqrt[1 + x^4]])/(-1 + x^4),x]","\sqrt{\frac{1}{2} \left(-1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{-\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)-\sqrt{\frac{1}{2} \left(-1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{-\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^4}}{\sqrt{2}}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)-\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{-\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)-\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{-\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)","\sqrt{\frac{1}{2} \left(-1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(-1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-\sqrt{\frac{1}{2} \left(-1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2 \left(-1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)",1,"Sqrt[(-1 + Sqrt[2])/2]*ArcTan[(-Sqrt[-1/2 + 1/Sqrt[2]] + Sqrt[-1/2 + 1/Sqrt[2]]*x^2 + Sqrt[-1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] - Sqrt[(-1 + Sqrt[2])/2]*ArcTan[(-Sqrt[1/2 + 1/Sqrt[2]] + Sqrt[1/2 + 1/Sqrt[2]]*x^2 + Sqrt[1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] + Sqrt[2]*ArcTanh[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] - Sqrt[(1 + Sqrt[2])/2]*ArcTanh[(-Sqrt[-1/2 + 1/Sqrt[2]] + Sqrt[-1/2 + 1/Sqrt[2]]*x^2 + Sqrt[-1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] - Sqrt[(1 + Sqrt[2])/2]*ArcTanh[(-Sqrt[1/2 + 1/Sqrt[2]] + Sqrt[1/2 + 1/Sqrt[2]]*x^2 + Sqrt[1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])]","A",1
2884,1,311,311,1.4313637,"\int \frac{x^4}{1-x \sqrt{1+x^2} \sqrt{x-\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[x^4/(1 - x*Sqrt[1 + x^2]*Sqrt[x - Sqrt[1 + x^2]]),x]","\frac{1}{28 \left(x-\sqrt{1+x^2}\right)^{7/2}}-\frac{1}{4 \left(x-\sqrt{1+x^2}\right)^{3/2}}+\frac{4}{\sqrt{x-\sqrt{1+x^2}}}-\frac{3}{4} \sqrt{x-\sqrt{1+x^2}}+\frac{1}{20} \left(x-\sqrt{1+x^2}\right)^{5/2}+\frac{1}{4 \left(-x+\sqrt{1+x^2}\right)^2}+\frac{3}{2} \log \left(-x+\sqrt{1+x^2}\right)-\text{RootSum}\left[-1+4 \text{$\#$1}^3+\text{$\#$1}^8\&,\frac{-4 \log \left(\sqrt{x-\sqrt{1+x^2}}-\text{$\#$1}\right)+3 \log \left(\sqrt{x-\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}-\log \left(\sqrt{x-\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^2-\log \left(\sqrt{x-\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^5+\log \left(\sqrt{x-\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^6}{3 \text{$\#$1}+2 \text{$\#$1}^6}\&\right]","\frac{1}{28 \left(x-\sqrt{1+x^2}\right)^{7/2}}-\frac{1}{4 \left(x-\sqrt{1+x^2}\right)^{3/2}}+\frac{4}{\sqrt{x-\sqrt{1+x^2}}}-\frac{3}{4} \sqrt{x-\sqrt{1+x^2}}+\frac{1}{20} \left(x-\sqrt{1+x^2}\right)^{5/2}+\frac{1}{4 \left(-x+\sqrt{1+x^2}\right)^2}+\frac{3}{2} \log \left(-x+\sqrt{1+x^2}\right)-\text{RootSum}\left[-1+4 \text{$\#$1}^3+\text{$\#$1}^8\&,\frac{-4 \log \left(\sqrt{x-\sqrt{1+x^2}}-\text{$\#$1}\right)+3 \log \left(\sqrt{x-\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}-\log \left(\sqrt{x-\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^2-\log \left(\sqrt{x-\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^5+\log \left(\sqrt{x-\sqrt{1+x^2}}-\text{$\#$1}\right) \text{$\#$1}^6}{3 \text{$\#$1}+2 \text{$\#$1}^6}\&\right]",1,"1/(28*(x - Sqrt[1 + x^2])^(7/2)) - 1/(4*(x - Sqrt[1 + x^2])^(3/2)) + 4/Sqrt[x - Sqrt[1 + x^2]] - (3*Sqrt[x - Sqrt[1 + x^2]])/4 + (x - Sqrt[1 + x^2])^(5/2)/20 + 1/(4*(-x + Sqrt[1 + x^2])^2) + (3*Log[-x + Sqrt[1 + x^2]])/2 - RootSum[-1 + 4*#1^3 + #1^8 & , (-4*Log[Sqrt[x - Sqrt[1 + x^2]] - #1] + 3*Log[Sqrt[x - Sqrt[1 + x^2]] - #1]*#1 - Log[Sqrt[x - Sqrt[1 + x^2]] - #1]*#1^2 - Log[Sqrt[x - Sqrt[1 + x^2]] - #1]*#1^5 + Log[Sqrt[x - Sqrt[1 + x^2]] - #1]*#1^6)/(3*#1 + 2*#1^6) & ]","A",1
2885,1,312,312,3.0296925,"\int \frac{\sqrt{x^2+\sqrt{1+x^4}}}{(1+x)^2 \sqrt{1+x^4}} \, dx","IntegrateAlgebraic[Sqrt[x^2 + Sqrt[1 + x^4]]/((1 + x)^2*Sqrt[1 + x^4]),x]","\frac{1+x^2+2 x^4+x \left(-1-x^2\right) \left(x^2+\sqrt{1+x^4}\right)+\sqrt{1+x^4} \left(1+2 x^2-x \left(x^2+\sqrt{1+x^4}\right)\right)}{2 \left(-1+x^2\right) \left(x^2+\sqrt{1+x^4}\right)^{3/2}}+\frac{\tan ^{-1}\left(\sqrt{1+\sqrt{2}} \sqrt{x^2+\sqrt{1+x^4}}\right)}{2 \sqrt{-1+\sqrt{2}}}-\frac{1}{2} \sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{-2+2 \sqrt{2}} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-\frac{\tanh ^{-1}\left(\sqrt{-1+\sqrt{2}} \sqrt{x^2+\sqrt{1+x^4}}\right)}{2 \sqrt{1+\sqrt{2}}}+\frac{1}{2} \sqrt{-1+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2+2 \sqrt{2}} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)","\frac{1+x^2+2 x^4+x \left(-1-x^2\right) \left(x^2+\sqrt{1+x^4}\right)+\sqrt{1+x^4} \left(1+2 x^2-x \left(x^2+\sqrt{1+x^4}\right)\right)}{2 \left(-1+x^2\right) \left(x^2+\sqrt{1+x^4}\right)^{3/2}}+\frac{\tan ^{-1}\left(\frac{\sqrt{x^2+\sqrt{1+x^4}}}{\sqrt{-1+\sqrt{2}}}\right)}{2 \sqrt{-1+\sqrt{2}}}-\frac{1}{2} \sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2 \left(-1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt{x^2+\sqrt{1+x^4}}}{\sqrt{1+\sqrt{2}}}\right)}{2 \sqrt{1+\sqrt{2}}}+\frac{1}{2} \sqrt{-1+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)",1,"(1 + x^2 + 2*x^4 + x*(-1 - x^2)*(x^2 + Sqrt[1 + x^4]) + Sqrt[1 + x^4]*(1 + 2*x^2 - x*(x^2 + Sqrt[1 + x^4])))/(2*(-1 + x^2)*(x^2 + Sqrt[1 + x^4])^(3/2)) + ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[x^2 + Sqrt[1 + x^4]]]/(2*Sqrt[-1 + Sqrt[2]]) - (Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[-2 + 2*Sqrt[2]]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2 + Sqrt[1 + x^4])])/2 - ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[x^2 + Sqrt[1 + x^4]]]/(2*Sqrt[1 + Sqrt[2]]) + (Sqrt[-1 + Sqrt[2]]*ArcTanh[(Sqrt[2 + 2*Sqrt[2]]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2 + Sqrt[1 + x^4])])/2","A",1
2886,1,313,313,0.6942515,"\int \frac{(-b+x) \left(a b-2 b x+x^2\right)}{\sqrt[3]{x (-a+x) (-b+x)^2} \left(-b^2 d+2 b d x-\left(-a^2+d\right) x^2-2 a x^3+x^4\right)} \, dx","IntegrateAlgebraic[((-b + x)*(a*b - 2*b*x + x^2))/((x*(-a + x)*(-b + x)^2)^(1/3)*(-(b^2*d) + 2*b*d*x - (-a^2 + d)*x^2 - 2*a*x^3 + x^4)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{-2 a x+2 x^2+\sqrt[3]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)}{2 d^{2/3}}+\frac{\log \left(a x-x^2+\sqrt[3]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}\right)}{2 d^{2/3}}-\frac{\log \left(a^2 x^2-2 a x^3+x^4+\left(-a \sqrt[3]{d} x+\sqrt[3]{d} x^2\right) \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}+d^{2/3} \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{2/3}\right)}{4 d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{-2 a x+2 x^2+\sqrt[3]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)}{2 d^{2/3}}+\frac{\log \left(a x-x^2+\sqrt[3]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}\right)}{2 d^{2/3}}-\frac{\log \left(a^2 x^2-2 a x^3+x^4+\left(-a \sqrt[3]{d} x+\sqrt[3]{d} x^2\right) \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}+d^{2/3} \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{2/3}\right)}{4 d^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3))/(-2*a*x + 2*x^2 + d^(1/3)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3))])/(2*d^(2/3)) + Log[a*x - x^2 + d^(1/3)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3)]/(2*d^(2/3)) - Log[a^2*x^2 - 2*a*x^3 + x^4 + (-(a*d^(1/3)*x) + d^(1/3)*x^2)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3) + d^(2/3)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(2/3)]/(4*d^(2/3))","A",1
2887,1,247,313,1.0935756,"\int \frac{b^8+a^8 x^8}{\sqrt{-b^4+a^4 x^4} \left(-b^8+a^8 x^8\right)} \, dx","IntegrateAlgebraic[(b^8 + a^8*x^8)/(Sqrt[-b^4 + a^4*x^4]*(-b^8 + a^8*x^8)),x]","-\frac{x}{2 \sqrt{-b^4+a^4 x^4}}-\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \tan ^{-1}\left(\frac{(1+i) a b x}{i b^2+a^2 x^2+\sqrt{-b^4+a^4 x^4}}\right)}{a b}+\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \tanh ^{-1}\left(\frac{b^4-(1+i) a b^3 x-(1-i) a^3 b x^3-a^4 x^4+\left(-i b^2-(1-i) a b x-a^2 x^2\right) \sqrt{-b^4+a^4 x^4}}{i b^4-(1-i) a b^3 x+(1+i) a^3 b x^3-i a^4 x^4+\left(b^2+(1+i) a b x-i a^2 x^2\right) \sqrt{-b^4+a^4 x^4}}\right)}{a b}","-\frac{x}{2 \sqrt{-b^4+a^4 x^4}}-\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \tan ^{-1}\left(\frac{(1+i) a b x}{i b^2+a^2 x^2+\sqrt{-b^4+a^4 x^4}}\right)}{a b}+\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \tanh ^{-1}\left(\frac{\frac{\left(\frac{1}{2}+\frac{i}{2}\right) b}{\sqrt{3-2 \sqrt{2}} a}+\frac{\left(\frac{1}{2}-\frac{i}{2}\right) a x^2}{\sqrt{3-2 \sqrt{2}} b}+\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \sqrt{-b^4+a^4 x^4}}{\sqrt{3-2 \sqrt{2}} a b}}{x}\right)}{a b}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \tanh ^{-1}\left(\frac{\frac{\left(\frac{1}{2}+\frac{i}{2}\right) b}{\sqrt{3+2 \sqrt{2}} a}+\frac{\left(\frac{1}{2}-\frac{i}{2}\right) a x^2}{\sqrt{3+2 \sqrt{2}} b}+\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \sqrt{-b^4+a^4 x^4}}{\sqrt{3+2 \sqrt{2}} a b}}{x}\right)}{a b}",1,"-1/2*x/Sqrt[-b^4 + a^4*x^4] - ((1/4 - I/4)*ArcTan[((1 + I)*a*b*x)/(I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4])])/(a*b) + ((1/8 - I/8)*ArcTanh[(b^4 - (1 + I)*a*b^3*x - (1 - I)*a^3*b*x^3 - a^4*x^4 + ((-I)*b^2 - (1 - I)*a*b*x - a^2*x^2)*Sqrt[-b^4 + a^4*x^4])/(I*b^4 - (1 - I)*a*b^3*x + (1 + I)*a^3*b*x^3 - I*a^4*x^4 + (b^2 + (1 + I)*a*b*x - I*a^2*x^2)*Sqrt[-b^4 + a^4*x^4])])/(a*b)","A",1
2888,1,231,313,8.7692215,"\int \frac{1}{\sqrt{-1+x^2} \left(\sqrt{x}+\sqrt{-1+x^2}\right)^2} \, dx","IntegrateAlgebraic[1/(Sqrt[-1 + x^2]*(Sqrt[x] + Sqrt[-1 + x^2])^2),x]","\frac{-2 \sqrt{x}+4 x^{3/2}+2 \sqrt{-1+x^2}-4 x \sqrt{-1+x^2}}{5 \left(-1-x+x^2\right)}+\frac{1}{5} \sqrt{-\frac{22}{5}+2 \sqrt{5}} \tan ^{-1}\left(\sqrt{\frac{1}{2}+\frac{\sqrt{5}}{2}} \sqrt{x}\right)-\frac{1}{5} \sqrt{-\frac{22}{5}+2 \sqrt{5}} \tan ^{-1}\left(\frac{\sqrt{-2+\sqrt{5}} \sqrt{-1+x^2}}{1+x}\right)-\frac{1}{5} \sqrt{\frac{22}{5}+2 \sqrt{5}} \tanh ^{-1}\left(\sqrt{-\frac{1}{2}+\frac{\sqrt{5}}{2}} \sqrt{x}\right)+\frac{1}{5} \sqrt{\frac{22}{5}+2 \sqrt{5}} \tanh ^{-1}\left(\frac{\sqrt{2+\sqrt{5}} \sqrt{-1+x^2}}{1+x}\right)","\frac{-2 \sqrt{x}+4 x^{3/2}+2 \sqrt{-1+x^2}-4 x \sqrt{-1+x^2}}{5 \left(-1-x+x^2\right)}+\frac{2}{5} \sqrt{\frac{2}{-1+\sqrt{5}}} \tan ^{-1}\left(\sqrt{\frac{2}{-1+\sqrt{5}}} \sqrt{x}\right)-\frac{4}{5} \sqrt{\frac{2}{-5+5 \sqrt{5}}} \tan ^{-1}\left(\sqrt{\frac{2}{-1+\sqrt{5}}} \sqrt{x}\right)-\frac{1}{5} \sqrt{-\frac{22}{5}+2 \sqrt{5}} \tan ^{-1}\left(\frac{\sqrt{-1+x^2}}{\sqrt{2+\sqrt{5}} (1+x)}\right)-\frac{2}{5} \sqrt{\frac{2}{1+\sqrt{5}}} \tanh ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} \sqrt{x}\right)-\frac{4}{5} \sqrt{\frac{2}{5+5 \sqrt{5}}} \tanh ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} \sqrt{x}\right)+\frac{1}{5} \sqrt{\frac{22}{5}+2 \sqrt{5}} \tanh ^{-1}\left(\frac{\sqrt{-1+x^2}}{\sqrt{-2+\sqrt{5}} (1+x)}\right)",1,"(-2*Sqrt[x] + 4*x^(3/2) + 2*Sqrt[-1 + x^2] - 4*x*Sqrt[-1 + x^2])/(5*(-1 - x + x^2)) + (Sqrt[-22/5 + 2*Sqrt[5]]*ArcTan[Sqrt[1/2 + Sqrt[5]/2]*Sqrt[x]])/5 - (Sqrt[-22/5 + 2*Sqrt[5]]*ArcTan[(Sqrt[-2 + Sqrt[5]]*Sqrt[-1 + x^2])/(1 + x)])/5 - (Sqrt[22/5 + 2*Sqrt[5]]*ArcTanh[Sqrt[-1/2 + Sqrt[5]/2]*Sqrt[x]])/5 + (Sqrt[22/5 + 2*Sqrt[5]]*ArcTanh[(Sqrt[2 + Sqrt[5]]*Sqrt[-1 + x^2])/(1 + x)])/5","A",1
2889,1,318,315,2.9107118,"\int \frac{b+a x^2}{\left(d+c x^2\right) \sqrt[3]{x+x^3}} \, dx","IntegrateAlgebraic[(b + a*x^2)/((d + c*x^2)*(x + x^3)^(1/3)),x]","\frac{\sqrt{3} a \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)}{2 c}-\frac{\sqrt{3} (b c-a d) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{c-d} x}{\sqrt[3]{c-d} x-2 \sqrt[3]{d} \sqrt[3]{x+x^3}}\right)}{2 c \sqrt[3]{c-d} d^{2/3}}-\frac{a \log \left(-c x+c \sqrt[3]{x+x^3}\right)}{2 c}+\frac{(b c-a d) \log \left(\sqrt[3]{c-d} x+\sqrt[3]{d} \sqrt[3]{x+x^3}\right)}{2 c \sqrt[3]{c-d} d^{2/3}}+\frac{a \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)}{4 c}+\frac{(-b c+a d) \log \left((c-d)^{2/3} x^2-\sqrt[3]{c-d} \sqrt[3]{d} x \sqrt[3]{x+x^3}+d^{2/3} \left(x+x^3\right)^{2/3}\right)}{4 c \sqrt[3]{c-d} d^{2/3}}","\frac{\sqrt{3} a \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)}{2 c}-\frac{\sqrt{3} (b c-a d) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{c-d} x}{\sqrt[3]{c-d} x-2 \sqrt[3]{d} \sqrt[3]{x+x^3}}\right)}{2 c \sqrt[3]{c-d} d^{2/3}}-\frac{a \log \left(-x+\sqrt[3]{x+x^3}\right)}{2 c}+\frac{(b c-a d) \log \left(\sqrt[3]{c-d} x+\sqrt[3]{d} \sqrt[3]{x+x^3}\right)}{2 c \sqrt[3]{c-d} d^{2/3}}+\frac{a \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)}{4 c}+\frac{(-b c+a d) \log \left((c-d)^{2/3} x^2-\sqrt[3]{c-d} \sqrt[3]{d} x \sqrt[3]{x+x^3}+d^{2/3} \left(x+x^3\right)^{2/3}\right)}{4 c \sqrt[3]{c-d} d^{2/3}}",1,"(Sqrt[3]*a*ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^3)^(1/3))])/(2*c) - (Sqrt[3]*(b*c - a*d)*ArcTan[(Sqrt[3]*(c - d)^(1/3)*x)/((c - d)^(1/3)*x - 2*d^(1/3)*(x + x^3)^(1/3))])/(2*c*(c - d)^(1/3)*d^(2/3)) - (a*Log[-(c*x) + c*(x + x^3)^(1/3)])/(2*c) + ((b*c - a*d)*Log[(c - d)^(1/3)*x + d^(1/3)*(x + x^3)^(1/3)])/(2*c*(c - d)^(1/3)*d^(2/3)) + (a*Log[x^2 + x*(x + x^3)^(1/3) + (x + x^3)^(2/3)])/(4*c) + ((-(b*c) + a*d)*Log[(c - d)^(2/3)*x^2 - (c - d)^(1/3)*d^(1/3)*x*(x + x^3)^(1/3) + d^(2/3)*(x + x^3)^(2/3)])/(4*c*(c - d)^(1/3)*d^(2/3))","A",1
2890,1,315,315,1.8904984,"\int \frac{x^3 \left(4 a b-3 (a+b) x+2 x^2\right)}{\left(x^2 (-a+x) (-b+x)\right)^{2/3} \left(-a b+(a+b) x-x^2+d x^4\right)} \, dx","IntegrateAlgebraic[(x^3*(4*a*b - 3*(a + b)*x + 2*x^2))/((x^2*(-a + x)*(-b + x))^(2/3)*(-(a*b) + (a + b)*x - x^2 + d*x^4)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}}{2 a b-2 a x-2 b x+2 x^2+\sqrt[3]{d} \left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}}\right)}{d^{2/3}}+\frac{\log \left(a b-a x-b x+x^2-\sqrt[3]{d} \left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}\right)}{d^{2/3}}-\frac{\log \left(a^2 b^2-2 a^2 b x-2 a b^2 x+a^2 x^2+4 a b x^2+b^2 x^2-2 a x^3-2 b x^3+x^4+\left(a b \sqrt[3]{d}-a \sqrt[3]{d} x-b \sqrt[3]{d} x+\sqrt[3]{d} x^2\right) \left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}+d^{2/3} \left(a b x^2+(-a-b) x^3+x^4\right)^{4/3}\right)}{2 d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}}{2 a b-2 a x-2 b x+2 x^2+\sqrt[3]{d} \left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}}\right)}{d^{2/3}}+\frac{\log \left(a b-a x-b x+x^2-\sqrt[3]{d} \left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}\right)}{d^{2/3}}-\frac{\log \left(a^2 b^2-2 a^2 b x-2 a b^2 x+a^2 x^2+4 a b x^2+b^2 x^2-2 a x^3-2 b x^3+x^4+\left(a b \sqrt[3]{d}-a \sqrt[3]{d} x-b \sqrt[3]{d} x+\sqrt[3]{d} x^2\right) \left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}+d^{2/3} \left(a b x^2+(-a-b) x^3+x^4\right)^{4/3}\right)}{2 d^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(2/3))/(2*a*b - 2*a*x - 2*b*x + 2*x^2 + d^(1/3)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(2/3))])/d^(2/3) + Log[a*b - a*x - b*x + x^2 - d^(1/3)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(2/3)]/d^(2/3) - Log[a^2*b^2 - 2*a^2*b*x - 2*a*b^2*x + a^2*x^2 + 4*a*b*x^2 + b^2*x^2 - 2*a*x^3 - 2*b*x^3 + x^4 + (a*b*d^(1/3) - a*d^(1/3)*x - b*d^(1/3)*x + d^(1/3)*x^2)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(2/3) + d^(2/3)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(4/3)]/(2*d^(2/3))","A",1
2891,1,313,316,0.0023757,"\int \frac{b x+a x^3}{\left(b+2 a x^3\right) \sqrt[3]{b^2 x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[(b*x + a*x^3)/((b + 2*a*x^3)*(b^2*x^2 + a^3*x^3)^(1/3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{2 a}-\frac{\log \left(a^2 x-a \sqrt[3]{b^2 x^2+a^3 x^3}\right)}{2 a}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{b^2 x^2+a^3 x^3}+\left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{4 a}+\frac{1}{6} \text{RootSum}\left[a^9-2 a b^5-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{a^3 \log (x)+2 b^2 \log (x)-a^3 \log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)-2 b^2 \log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)-\log (x) \text{$\#$1}^3+\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{-a^3 \text{$\#$1}+\text{$\#$1}^4}\&\right]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\frac{x}{\sqrt{3}}+\frac{2 \sqrt[3]{b^2 x^2+a^3 x^3}}{\sqrt{3} a}}{x}\right)}{2 a}-\frac{\log \left(-a x+\sqrt[3]{b^2 x^2+a^3 x^3}\right)}{2 a}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{b^2 x^2+a^3 x^3}+\left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{4 a}+\frac{1}{6} \text{RootSum}\left[a^9-2 a b^5-3 a^6 \text{$\#$1}^3+3 a^3 \text{$\#$1}^6-\text{$\#$1}^9\&,\frac{-a^3 \log (x)-2 b^2 \log (x)+a^3 \log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)+2 b^2 \log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right)+\log (x) \text{$\#$1}^3-\log \left(\sqrt[3]{b^2 x^2+a^3 x^3}-x \text{$\#$1}\right) \text{$\#$1}^3}{a^3 \text{$\#$1}-\text{$\#$1}^4}\&\right]",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*a*x)/(a*x + 2*(b^2*x^2 + a^3*x^3)^(1/3))])/(2*a) - Log[a^2*x - a*(b^2*x^2 + a^3*x^3)^(1/3)]/(2*a) + Log[a^2*x^2 + a*x*(b^2*x^2 + a^3*x^3)^(1/3) + (b^2*x^2 + a^3*x^3)^(2/3)]/(4*a) + RootSum[a^9 - 2*a*b^5 - 3*a^6*#1^3 + 3*a^3*#1^6 - #1^9 & , (a^3*Log[x] + 2*b^2*Log[x] - a^3*Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1] - 2*b^2*Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1] - Log[x]*#1^3 + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1]*#1^3)/(-(a^3*#1) + #1^4) & ]/6","A",1
2892,1,176,316,2.3458914,"\int \frac{1}{\sqrt{-1+2 x} (4+3 x)+(1+x) \sqrt{-3+4 x}} \, dx","IntegrateAlgebraic[(Sqrt[-1 + 2*x]*(4 + 3*x) + (1 + x)*Sqrt[-3 + 4*x])^(-1),x]","4 \text{RootSum}\left[2+3 \sqrt{2}+50 \text{$\#$1}^2+97 \sqrt{2} \text{$\#$1}^2-50 \text{$\#$1}^4+97 \sqrt{2} \text{$\#$1}^4-2 \text{$\#$1}^6+3 \sqrt{2} \text{$\#$1}^6\&,\frac{-\log \left(-\sqrt{-3+4 x}+\sqrt{-2+4 x}-\text{$\#$1}\right)+\log \left(-\sqrt{-3+4 x}+\sqrt{-2+4 x}-\text{$\#$1}\right) \text{$\#$1}^4}{50 \text{$\#$1}+97 \sqrt{2} \text{$\#$1}-100 \text{$\#$1}^3+194 \sqrt{2} \text{$\#$1}^3-6 \text{$\#$1}^5+9 \sqrt{2} \text{$\#$1}^5}\&\right]","-\text{RootSum}\left[5-10 \text{$\#$1}+6 \text{$\#$1}^2-4 \text{$\#$1}^3+12 \text{$\#$1}^4-40 \text{$\#$1}^5+40 \text{$\#$1}^6\&,\frac{-\log \left(-1+\sqrt{-3+4 x}\right)+\log \left(-1+\sqrt{-1+2 x}+\text{$\#$1}-\sqrt{-3+4 x} \text{$\#$1}\right)+6 \log \left(-1+\sqrt{-3+4 x}\right) \text{$\#$1}-6 \log \left(-1+\sqrt{-1+2 x}+\text{$\#$1}-\sqrt{-3+4 x} \text{$\#$1}\right) \text{$\#$1}-12 \log \left(-1+\sqrt{-3+4 x}\right) \text{$\#$1}^2+12 \log \left(-1+\sqrt{-1+2 x}+\text{$\#$1}-\sqrt{-3+4 x} \text{$\#$1}\right) \text{$\#$1}^2+12 \log \left(-1+\sqrt{-3+4 x}\right) \text{$\#$1}^3-12 \log \left(-1+\sqrt{-1+2 x}+\text{$\#$1}-\sqrt{-3+4 x} \text{$\#$1}\right) \text{$\#$1}^3-4 \log \left(-1+\sqrt{-3+4 x}\right) \text{$\#$1}^4+4 \log \left(-1+\sqrt{-1+2 x}+\text{$\#$1}-\sqrt{-3+4 x} \text{$\#$1}\right) \text{$\#$1}^4}{-5+6 \text{$\#$1}-6 \text{$\#$1}^2+24 \text{$\#$1}^3-100 \text{$\#$1}^4+120 \text{$\#$1}^5}\&\right]",1,"4*RootSum[2 + 3*Sqrt[2] + 50*#1^2 + 97*Sqrt[2]*#1^2 - 50*#1^4 + 97*Sqrt[2]*#1^4 - 2*#1^6 + 3*Sqrt[2]*#1^6 & , (-Log[-Sqrt[-3 + 4*x] + Sqrt[-2 + 4*x] - #1] + Log[-Sqrt[-3 + 4*x] + Sqrt[-2 + 4*x] - #1]*#1^4)/(50*#1 + 97*Sqrt[2]*#1 - 100*#1^3 + 194*Sqrt[2]*#1^3 - 6*#1^5 + 9*Sqrt[2]*#1^5) & ]","A",1
2893,1,250,316,0.5498488,"\int \frac{f+e x}{(h+g x) \sqrt{d+\sqrt{c+\sqrt{b+a x}}}} \, dx","IntegrateAlgebraic[(f + e*x)/((h + g*x)*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]]),x]","-\frac{8 e \left(20 c-24 d^2-15 \sqrt{b+a x}\right) \sqrt{c+\sqrt{b+a x}} \sqrt{d+\sqrt{c+\sqrt{b+a x}}}}{105 a g}+\frac{16 \left(26 c d e-24 d^3 e-9 d e \sqrt{b+a x}\right) \sqrt{d+\sqrt{c+\sqrt{b+a x}}}}{105 a g}+\frac{(f g-e h) \text{RootSum}\left[b g-c^2 g+2 c d^2 g-d^4 g-a h-4 c d g \text{$\#$1}^2+4 d^3 g \text{$\#$1}^2+2 c g \text{$\#$1}^4-6 d^2 g \text{$\#$1}^4+4 d g \text{$\#$1}^6-g \text{$\#$1}^8\&,\frac{\log \left(\sqrt{d+\sqrt{c+\sqrt{b+a x}}}-\text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{g^2}","\frac{32 \left(13 c d-12 d^3\right) e \sqrt{d+\sqrt{c+\sqrt{b+a x}}}}{105 a g}-\frac{32 \left(5 c-6 d^2\right) e \sqrt{c+\sqrt{b+a x}} \sqrt{d+\sqrt{c+\sqrt{b+a x}}}}{105 a g}+\sqrt{b+a x} \left(-\frac{48 d e \sqrt{d+\sqrt{c+\sqrt{b+a x}}}}{35 a g}+\frac{8 e \sqrt{c+\sqrt{b+a x}} \sqrt{d+\sqrt{c+\sqrt{b+a x}}}}{7 a g}\right)+\frac{(f g-e h) \text{RootSum}\left[b g-c^2 g+2 c d^2 g-d^4 g-a h-4 c d g \text{$\#$1}^2+4 d^3 g \text{$\#$1}^2+2 c g \text{$\#$1}^4-6 d^2 g \text{$\#$1}^4+4 d g \text{$\#$1}^6-g \text{$\#$1}^8\&,\frac{\log \left(\sqrt{d+\sqrt{c+\sqrt{b+a x}}}-\text{$\#$1}\right)}{\text{$\#$1}}\&\right]}{g^2}",1,"(-8*e*(20*c - 24*d^2 - 15*Sqrt[b + a*x])*Sqrt[c + Sqrt[b + a*x]]*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]])/(105*a*g) + (16*(26*c*d*e - 24*d^3*e - 9*d*e*Sqrt[b + a*x])*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]])/(105*a*g) + ((f*g - e*h)*RootSum[b*g - c^2*g + 2*c*d^2*g - d^4*g - a*h - 4*c*d*g*#1^2 + 4*d^3*g*#1^2 + 2*c*g*#1^4 - 6*d^2*g*#1^4 + 4*d*g*#1^6 - g*#1^8 & , Log[Sqrt[d + Sqrt[c + Sqrt[b + a*x]]] - #1]/#1 & ])/g^2","A",1
2894,1,316,316,0.607567,"\int \left(1+x^2\right)^{3/2} \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[(1 + x^2)^(3/2)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]],x]","\frac{\left(15903121112+5227043711 x+220397520304 x^2-1415707308 x^3+407581982720 x^4-11794907136 x^5+248171986944 x^6-2099249152 x^7+66913566720 x^8\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-1176816782+66830366096 x+1984342244 x^2+96561463296 x^3+6568280064 x^4+10550149120 x^5+1130364928 x^6+1968046080 x^7\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(2167822549+88760534448 x+3694527828 x^2+308588576768 x^3-10745282560 x^4+214715203584 x^5-2099249152 x^6+66913566720 x^7\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(21890925968-875910940 x+92024406016 x^2+6003097600 x^3+9566126080 x^4+1130364928 x^5+1968046080 x^6\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{39729930240 \left(x+\sqrt{1+x^2}\right)^{7/2}}-\frac{545 \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{8192}","\frac{\left(15903121112+5227043711 x+220397520304 x^2-1415707308 x^3+407581982720 x^4-11794907136 x^5+248171986944 x^6-2099249152 x^7+66913566720 x^8\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-1176816782+66830366096 x+1984342244 x^2+96561463296 x^3+6568280064 x^4+10550149120 x^5+1130364928 x^6+1968046080 x^7\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(2167822549+88760534448 x+3694527828 x^2+308588576768 x^3-10745282560 x^4+214715203584 x^5-2099249152 x^6+66913566720 x^7\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(21890925968-875910940 x+92024406016 x^2+6003097600 x^3+9566126080 x^4+1130364928 x^5+1968046080 x^6\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{39729930240 \left(x+\sqrt{1+x^2}\right)^{7/2}}-\frac{545 \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{8192}",1,"((15903121112 + 5227043711*x + 220397520304*x^2 - 1415707308*x^3 + 407581982720*x^4 - 11794907136*x^5 + 248171986944*x^6 - 2099249152*x^7 + 66913566720*x^8)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-1176816782 + 66830366096*x + 1984342244*x^2 + 96561463296*x^3 + 6568280064*x^4 + 10550149120*x^5 + 1130364928*x^6 + 1968046080*x^7)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((2167822549 + 88760534448*x + 3694527828*x^2 + 308588576768*x^3 - 10745282560*x^4 + 214715203584*x^5 - 2099249152*x^6 + 66913566720*x^7)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (21890925968 - 875910940*x + 92024406016*x^2 + 6003097600*x^3 + 9566126080*x^4 + 1130364928*x^5 + 1968046080*x^6)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(39729930240*(x + Sqrt[1 + x^2])^(7/2)) - (545*ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]])/8192","A",1
2895,1,317,317,5.3385705,"\int \frac{(a-5 b+4 x) \left(-a^3+3 a^2 x-3 a x^2+x^3\right)}{((-a+x) (-b+x))^{2/3} \left(b-a^5 d-\left(1-5 a^4 d\right) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right)} \, dx","IntegrateAlgebraic[((a - 5*b + 4*x)*(-a^3 + 3*a^2*x - 3*a*x^2 + x^3))/(((-a + x)*(-b + x))^(2/3)*(b - a^5*d - (1 - 5*a^4*d)*x - 10*a^3*d*x^2 + 10*a^2*d*x^3 - 5*a*d*x^4 + d*x^5)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a^2 \sqrt[3]{d}-2 \sqrt{3} a \sqrt[3]{d} x+\sqrt{3} \sqrt[3]{d} x^2}{a^2 \sqrt[3]{d}-2 a \sqrt[3]{d} x+\sqrt[3]{d} x^2+2 \sqrt[3]{a b+(-a-b) x+x^2}}\right)}{d^{2/3}}+\frac{\log \left(a^3 \sqrt[3]{d}-2 a^2 \sqrt[3]{d} x+a \sqrt[3]{d} x^2-a \sqrt[3]{a b+(-a-b) x+x^2}\right)}{d^{2/3}}-\frac{\log \left(a^6 d^{2/3}-4 a^5 d^{2/3} x+6 a^4 d^{2/3} x^2-4 a^3 d^{2/3} x^3+a^2 d^{2/3} x^4+a^2 \left(a b+(-a-b) x+x^2\right)^{2/3}+\sqrt[3]{a b+(-a-b) x+x^2} \left(a^4 \sqrt[3]{d}-2 a^3 \sqrt[3]{d} x+a^2 \sqrt[3]{d} x^2\right)\right)}{2 d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a^2 \sqrt[3]{d}-2 \sqrt{3} a \sqrt[3]{d} x+\sqrt{3} \sqrt[3]{d} x^2}{a^2 \sqrt[3]{d}-2 a \sqrt[3]{d} x+\sqrt[3]{d} x^2+2 \sqrt[3]{a b+(-a-b) x+x^2}}\right)}{d^{2/3}}+\frac{\log \left(a^3 \sqrt[3]{d}-2 a^2 \sqrt[3]{d} x+a \sqrt[3]{d} x^2-a \sqrt[3]{a b+(-a-b) x+x^2}\right)}{d^{2/3}}-\frac{\log \left(a^6 d^{2/3}-4 a^5 d^{2/3} x+6 a^4 d^{2/3} x^2-4 a^3 d^{2/3} x^3+a^2 d^{2/3} x^4+a^2 \left(a b+(-a-b) x+x^2\right)^{2/3}+\sqrt[3]{a b+(-a-b) x+x^2} \left(a^4 \sqrt[3]{d}-2 a^3 \sqrt[3]{d} x+a^2 \sqrt[3]{d} x^2\right)\right)}{2 d^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*a^2*d^(1/3) - 2*Sqrt[3]*a*d^(1/3)*x + Sqrt[3]*d^(1/3)*x^2)/(a^2*d^(1/3) - 2*a*d^(1/3)*x + d^(1/3)*x^2 + 2*(a*b + (-a - b)*x + x^2)^(1/3))])/d^(2/3) + Log[a^3*d^(1/3) - 2*a^2*d^(1/3)*x + a*d^(1/3)*x^2 - a*(a*b + (-a - b)*x + x^2)^(1/3)]/d^(2/3) - Log[a^6*d^(2/3) - 4*a^5*d^(2/3)*x + 6*a^4*d^(2/3)*x^2 - 4*a^3*d^(2/3)*x^3 + a^2*d^(2/3)*x^4 + a^2*(a*b + (-a - b)*x + x^2)^(2/3) + (a*b + (-a - b)*x + x^2)^(1/3)*(a^4*d^(1/3) - 2*a^3*d^(1/3)*x + a^2*d^(1/3)*x^2)]/(2*d^(2/3))","A",1
2896,1,318,318,5.0526173,"\int \frac{\left(2-2 x+2 x^2-3 x^3+3 x^4\right) \sqrt[3]{-x-x^3-x^4+x^6}}{(1+x) \left(-1+2 x-2 x^2+x^3\right) \left(-1-x^3+x^5\right)} \, dx","IntegrateAlgebraic[((2 - 2*x + 2*x^2 - 3*x^3 + 3*x^4)*(-x - x^3 - x^4 + x^6)^(1/3))/((1 + x)*(-1 + 2*x - 2*x^2 + x^3)*(-1 - x^3 + x^5)),x]","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-x-x^3-x^4+x^6}}{-2 x+\sqrt[3]{-x-x^3-x^4+x^6}}\right)-\sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-x-x^3-x^4+x^6}}{-2 \sqrt[3]{2} x+\sqrt[3]{-x-x^3-x^4+x^6}}\right)-\log \left(x+\sqrt[3]{-x-x^3-x^4+x^6}\right)+\sqrt[3]{2} \log \left(\sqrt[3]{2} x+\sqrt[3]{-x-x^3-x^4+x^6}\right)+\frac{1}{2} \log \left(x^2-x \sqrt[3]{-x-x^3-x^4+x^6}+\left(-x-x^3-x^4+x^6\right)^{2/3}\right)-\frac{\log \left(2^{2/3} x^2-\sqrt[3]{2} x \sqrt[3]{-x-x^3-x^4+x^6}+\left(-x-x^3-x^4+x^6\right)^{2/3}\right)}{2^{2/3}}","\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-x-x^3-x^4+x^6}}{-2 x+\sqrt[3]{-x-x^3-x^4+x^6}}\right)-\sqrt[3]{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-x-x^3-x^4+x^6}}{-2 \sqrt[3]{2} x+\sqrt[3]{-x-x^3-x^4+x^6}}\right)-\log \left(x+\sqrt[3]{-x-x^3-x^4+x^6}\right)+\sqrt[3]{2} \log \left(\sqrt[3]{2} x+\sqrt[3]{-x-x^3-x^4+x^6}\right)+\frac{1}{2} \log \left(x^2-x \sqrt[3]{-x-x^3-x^4+x^6}+\left(-x-x^3-x^4+x^6\right)^{2/3}\right)-\frac{\log \left(2^{2/3} x^2-\sqrt[3]{2} x \sqrt[3]{-x-x^3-x^4+x^6}+\left(-x-x^3-x^4+x^6\right)^{2/3}\right)}{2^{2/3}}",1,"Sqrt[3]*ArcTan[(Sqrt[3]*(-x - x^3 - x^4 + x^6)^(1/3))/(-2*x + (-x - x^3 - x^4 + x^6)^(1/3))] - 2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*(-x - x^3 - x^4 + x^6)^(1/3))/(-2*2^(1/3)*x + (-x - x^3 - x^4 + x^6)^(1/3))] - Log[x + (-x - x^3 - x^4 + x^6)^(1/3)] + 2^(1/3)*Log[2^(1/3)*x + (-x - x^3 - x^4 + x^6)^(1/3)] + Log[x^2 - x*(-x - x^3 - x^4 + x^6)^(1/3) + (-x - x^3 - x^4 + x^6)^(2/3)]/2 - Log[2^(2/3)*x^2 - 2^(1/3)*x*(-x - x^3 - x^4 + x^6)^(1/3) + (-x - x^3 - x^4 + x^6)^(2/3)]/2^(2/3)","A",1
2897,1,365,319,1.9749237,"\int \frac{1}{(-b+a x) \sqrt[3]{-b^2 x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[1/((-b + a*x)*(-(b^2*x^2) + a^3*x^3)^(1/3)),x]","\frac{\sqrt{-3+3 i \sqrt{3}} \tan ^{-1}\left(\frac{3 \sqrt[3]{a} \sqrt[3]{a^2-b} x}{\sqrt{3} \sqrt[3]{a} \sqrt[3]{a^2-b} x-3 i \sqrt[3]{-b^2 x^2+a^3 x^3}-\sqrt{3} \sqrt[3]{-b^2 x^2+a^3 x^3}}\right)}{\sqrt{2} \sqrt[3]{a} \sqrt[3]{a^2-b} b}-\frac{i \left(-i+\sqrt{3}\right) \log \left(2 \sqrt[3]{a} \sqrt[3]{a^2-b} x+\left(1+i \sqrt{3}\right) \sqrt[3]{-b^2 x^2+a^3 x^3}\right)}{2 \sqrt[3]{a} \sqrt[3]{a^2-b} b}+\frac{\left(1+i \sqrt{3}\right) \log \left(-2 i a^{2/3} \left(a^2-b\right)^{2/3} x^2+\sqrt[3]{a} \sqrt[3]{a^2-b} \left(i x-\sqrt{3} x\right) \sqrt[3]{-b^2 x^2+a^3 x^3}+\left(i+\sqrt{3}\right) \left(-b^2 x^2+a^3 x^3\right)^{2/3}\right)}{4 \sqrt[3]{a} \sqrt[3]{a^2-b} b}","\frac{\sqrt{-3+3 i \sqrt{3}} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a} \sqrt[3]{a^2-b} x}{\sqrt[3]{a} \sqrt[3]{a^2-b} x-2 \sqrt[3]{-1} \sqrt[3]{-b^2 x^2+a^3 x^3}}\right)}{\sqrt{2} \sqrt[3]{a} \sqrt[3]{a^2-b} b}-\frac{i \left(-i+\sqrt{3}\right) \log \left(\sqrt[3]{a} \sqrt[3]{a^2-b} x+\sqrt[3]{-1} \sqrt[3]{-b^2 x^2+a^3 x^3}\right)}{2 \sqrt[3]{a} \sqrt[3]{a^2-b} b}+\frac{\left(1+i \sqrt{3}\right) \log \left(a^{2/3} \left(a^2-b\right)^{2/3} x^2-\sqrt[3]{-1} \sqrt[3]{a} \sqrt[3]{a^2-b} x \sqrt[3]{-b^2 x^2+a^3 x^3}+(-1)^{2/3} \left(-b^2 x^2+a^3 x^3\right)^{2/3}\right)}{4 \sqrt[3]{a} \sqrt[3]{a^2-b} b}",1,"(Sqrt[-3 + (3*I)*Sqrt[3]]*ArcTan[(3*a^(1/3)*(a^2 - b)^(1/3)*x)/(Sqrt[3]*a^(1/3)*(a^2 - b)^(1/3)*x - (3*I)*(-(b^2*x^2) + a^3*x^3)^(1/3) - Sqrt[3]*(-(b^2*x^2) + a^3*x^3)^(1/3))])/(Sqrt[2]*a^(1/3)*(a^2 - b)^(1/3)*b) - ((I/2)*(-I + Sqrt[3])*Log[2*a^(1/3)*(a^2 - b)^(1/3)*x + (1 + I*Sqrt[3])*(-(b^2*x^2) + a^3*x^3)^(1/3)])/(a^(1/3)*(a^2 - b)^(1/3)*b) + ((1 + I*Sqrt[3])*Log[(-2*I)*a^(2/3)*(a^2 - b)^(2/3)*x^2 + a^(1/3)*(a^2 - b)^(1/3)*(I*x - Sqrt[3]*x)*(-(b^2*x^2) + a^3*x^3)^(1/3) + (I + Sqrt[3])*(-(b^2*x^2) + a^3*x^3)^(2/3)])/(4*a^(1/3)*(a^2 - b)^(1/3)*b)","A",1
2898,1,319,319,3.1170377,"\int \frac{(-b+x) (-4 a+b+3 x)}{\sqrt[3]{(-a+x) (-b+x)^2} \left(b^4+a d-\left(4 b^3+d\right) x+6 b^2 x^2-4 b x^3+x^4\right)} \, dx","IntegrateAlgebraic[((-b + x)*(-4*a + b + 3*x))/(((-a + x)*(-b + x)^2)^(1/3)*(b^4 + a*d - (4*b^3 + d)*x + 6*b^2*x^2 - 4*b*x^3 + x^4)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{2 b^2-4 b x+2 x^2+\sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}\right)}{d^{2/3}}+\frac{\log \left(b^2-2 b x+x^2-\sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}\right)}{d^{2/3}}-\frac{\log \left(b^4-4 b^3 x+6 b^2 x^2-4 b x^3+x^4+\left(b^2 \sqrt[3]{d}-2 b \sqrt[3]{d} x+\sqrt[3]{d} x^2\right) \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}+d^{2/3} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}\right)}{2 d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{2 b^2-4 b x+2 x^2+\sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}\right)}{d^{2/3}}+\frac{\log \left(b^2-2 b x+x^2-\sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}\right)}{d^{2/3}}-\frac{\log \left(b^4-4 b^3 x+6 b^2 x^2-4 b x^3+x^4+\left(b^2 \sqrt[3]{d}-2 b \sqrt[3]{d} x+\sqrt[3]{d} x^2\right) \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}+d^{2/3} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}\right)}{2 d^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3))/(2*b^2 - 4*b*x + 2*x^2 + d^(1/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3))])/d^(2/3) + Log[b^2 - 2*b*x + x^2 - d^(1/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3)]/d^(2/3) - Log[b^4 - 4*b^3*x + 6*b^2*x^2 - 4*b*x^3 + x^4 + (b^2*d^(1/3) - 2*b*d^(1/3)*x + d^(1/3)*x^2)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3) + d^(2/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(2/3)]/(2*d^(2/3))","A",1
2899,1,253,319,0.920447,"\int \frac{x^4}{\sqrt{-b^4+a^4 x^4} \left(-b^8+a^8 x^8\right)} \, dx","IntegrateAlgebraic[x^4/(Sqrt[-b^4 + a^4*x^4]*(-b^8 + a^8*x^8)),x]","-\frac{x}{4 a^4 b^4 \sqrt{-b^4+a^4 x^4}}+\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \tan ^{-1}\left(\frac{(1+i) a b x}{i b^2+a^2 x^2+\sqrt{-b^4+a^4 x^4}}\right)}{a^5 b^5}-\frac{\left(\frac{1}{16}-\frac{i}{16}\right) \tanh ^{-1}\left(\frac{b^4-(1+i) a b^3 x-(1-i) a^3 b x^3-a^4 x^4+\left(-i b^2-(1-i) a b x-a^2 x^2\right) \sqrt{-b^4+a^4 x^4}}{i b^4-(1-i) a b^3 x+(1+i) a^3 b x^3-i a^4 x^4+\left(b^2+(1+i) a b x-i a^2 x^2\right) \sqrt{-b^4+a^4 x^4}}\right)}{a^5 b^5}","-\frac{x}{4 a^4 b^4 \sqrt{-b^4+a^4 x^4}}+\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \tan ^{-1}\left(\frac{(1+i) a b x}{i b^2+a^2 x^2+\sqrt{-b^4+a^4 x^4}}\right)}{a^5 b^5}-\frac{\left(\frac{1}{16}-\frac{i}{16}\right) \tanh ^{-1}\left(\frac{\frac{\left(\frac{1}{2}+\frac{i}{2}\right) b}{\sqrt{3-2 \sqrt{2}} a}+\frac{\left(\frac{1}{2}-\frac{i}{2}\right) a x^2}{\sqrt{3-2 \sqrt{2}} b}+\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \sqrt{-b^4+a^4 x^4}}{\sqrt{3-2 \sqrt{2}} a b}}{x}\right)}{a^5 b^5}+\frac{\left(\frac{1}{16}-\frac{i}{16}\right) \tanh ^{-1}\left(\frac{\frac{\left(\frac{1}{2}+\frac{i}{2}\right) b}{\sqrt{3+2 \sqrt{2}} a}+\frac{\left(\frac{1}{2}-\frac{i}{2}\right) a x^2}{\sqrt{3+2 \sqrt{2}} b}+\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \sqrt{-b^4+a^4 x^4}}{\sqrt{3+2 \sqrt{2}} a b}}{x}\right)}{a^5 b^5}",1,"-1/4*x/(a^4*b^4*Sqrt[-b^4 + a^4*x^4]) + ((1/8 - I/8)*ArcTan[((1 + I)*a*b*x)/(I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4])])/(a^5*b^5) - ((1/16 - I/16)*ArcTanh[(b^4 - (1 + I)*a*b^3*x - (1 - I)*a^3*b*x^3 - a^4*x^4 + ((-I)*b^2 - (1 - I)*a*b*x - a^2*x^2)*Sqrt[-b^4 + a^4*x^4])/(I*b^4 - (1 - I)*a*b^3*x + (1 + I)*a^3*b*x^3 - I*a^4*x^4 + (b^2 + (1 + I)*a*b*x - I*a^2*x^2)*Sqrt[-b^4 + a^4*x^4])])/(a^5*b^5)","A",1
2900,1,319,319,0.972994,"\int \frac{1}{\sqrt{-b+a^2 x^2} \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}} \, dx","IntegrateAlgebraic[1/(Sqrt[-b + a^2*x^2]*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)),x]","\frac{-6 c \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}+8 \sqrt[4]{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}}{3 a c^2 \sqrt{a x+\sqrt{-b+a^2 x^2}}}+\frac{8 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[3]{c}}\right)}{3 \sqrt{3} a c^{7/3}}+\frac{8 \log \left(-\sqrt[3]{c}+\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{9 a c^{7/3}}-\frac{4 \log \left(c^{2/3}+\sqrt[3]{c} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}\right)}{9 a c^{7/3}}","\frac{-6 c \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}+8 \sqrt[4]{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}}{3 a c^2 \sqrt{a x+\sqrt{-b+a^2 x^2}}}+\frac{8 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[3]{c}}\right)}{3 \sqrt{3} a c^{7/3}}+\frac{8 \log \left(-\sqrt[3]{c}+\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{9 a c^{7/3}}-\frac{4 \log \left(c^{2/3}+\sqrt[3]{c} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}\right)}{9 a c^{7/3}}",1,"(-6*c*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3) + 8*(a*x + Sqrt[-b + a^2*x^2])^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3))/(3*a*c^2*Sqrt[a*x + Sqrt[-b + a^2*x^2]]) + (8*ArcTan[1/Sqrt[3] + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3))/(Sqrt[3]*c^(1/3))])/(3*Sqrt[3]*a*c^(7/3)) + (8*Log[-c^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)])/(9*a*c^(7/3)) - (4*Log[c^(2/3) + c^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3)])/(9*a*c^(7/3))","A",1
2901,1,341,321,0.6993912,"\int \frac{\left(1+x^2\right)^2 \sqrt{x^2+\sqrt{1+x^4}}}{\sqrt{1+x^4} \left(-1+x^2+x^4\right)} \, dx","IntegrateAlgebraic[((1 + x^2)^2*Sqrt[x^2 + Sqrt[1 + x^4]])/(Sqrt[1 + x^4]*(-1 + x^2 + x^4)),x]","\sqrt{2} \tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^4}}{\sqrt{2}}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)+\frac{\text{RootSum}\left[1-2 \text{$\#$1}^2-6 \text{$\#$1}^4+2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(1+x^2+\sqrt{1+x^4}\right)-\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right)+2 \log \left(1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^2-2 \log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^2-\log \left(1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^6+\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^6}{-\text{$\#$1}-6 \text{$\#$1}^3+3 \text{$\#$1}^5+2 \text{$\#$1}^7}\&\right]}{\sqrt{2}}","\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-\frac{\text{RootSum}\left[1+2 \text{$\#$1}^2-6 \text{$\#$1}^4-2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(-1+x^2+\sqrt{1+x^4}\right)-\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}+x^2 \text{$\#$1}+\sqrt{1+x^4} \text{$\#$1}\right)-2 \log \left(-1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^4+2 \log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}+x^2 \text{$\#$1}+\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^4-\log \left(-1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^6+\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}+x^2 \text{$\#$1}+\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^6}{\text{$\#$1}-6 \text{$\#$1}^3-3 \text{$\#$1}^5+2 \text{$\#$1}^7}\&\right]}{\sqrt{2}}",1,"Sqrt[2]*ArcTanh[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] + RootSum[1 - 2*#1^2 - 6*#1^4 + 2*#1^6 + #1^8 & , (Log[1 + x^2 + Sqrt[1 + x^4]] - Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1] + 2*Log[1 + x^2 + Sqrt[1 + x^4]]*#1^2 - 2*Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^2 - Log[1 + x^2 + Sqrt[1 + x^4]]*#1^6 + Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^6)/(-#1 - 6*#1^3 + 3*#1^5 + 2*#1^7) & ]/Sqrt[2]","A",1
2902,1,341,321,0.0018252,"\int \frac{\left(1+x^2\right)^2 \sqrt{x^2+\sqrt{1+x^4}}}{\sqrt{1+x^4} \left(-1+x^2+x^4\right)} \, dx","IntegrateAlgebraic[((1 + x^2)^2*Sqrt[x^2 + Sqrt[1 + x^4]])/(Sqrt[1 + x^4]*(-1 + x^2 + x^4)),x]","\sqrt{2} \tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^4}}{\sqrt{2}}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)+\frac{\text{RootSum}\left[1-2 \text{$\#$1}^2-6 \text{$\#$1}^4+2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(1+x^2+\sqrt{1+x^4}\right)-\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right)+2 \log \left(1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^2-2 \log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^2-\log \left(1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^6+\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^6}{-\text{$\#$1}-6 \text{$\#$1}^3+3 \text{$\#$1}^5+2 \text{$\#$1}^7}\&\right]}{\sqrt{2}}","\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-\frac{\text{RootSum}\left[1+2 \text{$\#$1}^2-6 \text{$\#$1}^4-2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(-1+x^2+\sqrt{1+x^4}\right)-\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}+x^2 \text{$\#$1}+\sqrt{1+x^4} \text{$\#$1}\right)-2 \log \left(-1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^4+2 \log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}+x^2 \text{$\#$1}+\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^4-\log \left(-1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^6+\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}+x^2 \text{$\#$1}+\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^6}{\text{$\#$1}-6 \text{$\#$1}^3-3 \text{$\#$1}^5+2 \text{$\#$1}^7}\&\right]}{\sqrt{2}}",1,"Sqrt[2]*ArcTanh[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] + RootSum[1 - 2*#1^2 - 6*#1^4 + 2*#1^6 + #1^8 & , (Log[1 + x^2 + Sqrt[1 + x^4]] - Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1] + 2*Log[1 + x^2 + Sqrt[1 + x^4]]*#1^2 - 2*Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^2 - Log[1 + x^2 + Sqrt[1 + x^4]]*#1^6 + Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^6)/(-#1 - 6*#1^3 + 3*#1^5 + 2*#1^7) & ]/Sqrt[2]","A",1
2903,1,322,323,5.5880105,"\int \frac{x^2 \sqrt[4]{b x^3+a x^4}}{-b+a x^4} \, dx","IntegrateAlgebraic[(x^2*(b*x^3 + a*x^4)^(1/4))/(-b + a*x^4),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{a^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{a^{3/4}}+\frac{1}{4} \text{RootSum}\left[a^4-a b^3-4 a^3 \text{$\#$1}^4+6 a^2 \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{a^3 \log (x)-b^3 \log (x)-a^3 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)+b^3 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)-3 a^2 \log (x) \text{$\#$1}^4+3 a^2 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+3 a \log (x) \text{$\#$1}^8-3 a \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^8-\log (x) \text{$\#$1}^{12}+\log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^{12}}{-a^3 \text{$\#$1}^3+3 a^2 \text{$\#$1}^7-3 a \text{$\#$1}^{11}+\text{$\#$1}^{15}}\&\right]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{a^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{a^{3/4}}+\frac{1}{4} \text{RootSum}\left[a^4-a b^3-4 a^3 \text{$\#$1}^4+6 a^2 \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-a^3 \log (x)+b^3 \log (x)+a^3 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)-b^3 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)+3 a^2 \log (x) \text{$\#$1}^4-3 a^2 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-3 a \log (x) \text{$\#$1}^8+3 a \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^8+\log (x) \text{$\#$1}^{12}-\log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^{12}}{a^3 \text{$\#$1}^3-3 a^2 \text{$\#$1}^7+3 a \text{$\#$1}^{11}-\text{$\#$1}^{15}}\&\right]",1,"(-2*ArcTan[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/a^(3/4) + (2*ArcTanh[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/a^(3/4) + RootSum[a^4 - a*b^3 - 4*a^3*#1^4 + 6*a^2*#1^8 - 4*a*#1^12 + #1^16 & , (a^3*Log[x] - b^3*Log[x] - a^3*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] + b^3*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] - 3*a^2*Log[x]*#1^4 + 3*a^2*Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^4 + 3*a*Log[x]*#1^8 - 3*a*Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^8 - Log[x]*#1^12 + Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^12)/(-(a^3*#1^3) + 3*a^2*#1^7 - 3*a*#1^11 + #1^15) & ]/4","A",1
2904,1,322,323,0.0015226,"\int \frac{x^2 \sqrt[4]{b x^3+a x^4}}{-b+a x^4} \, dx","IntegrateAlgebraic[(x^2*(b*x^3 + a*x^4)^(1/4))/(-b + a*x^4),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{a^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{a^{3/4}}+\frac{1}{4} \text{RootSum}\left[a^4-a b^3-4 a^3 \text{$\#$1}^4+6 a^2 \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{a^3 \log (x)-b^3 \log (x)-a^3 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)+b^3 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)-3 a^2 \log (x) \text{$\#$1}^4+3 a^2 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+3 a \log (x) \text{$\#$1}^8-3 a \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^8-\log (x) \text{$\#$1}^{12}+\log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^{12}}{-a^3 \text{$\#$1}^3+3 a^2 \text{$\#$1}^7-3 a \text{$\#$1}^{11}+\text{$\#$1}^{15}}\&\right]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{a^{3/4}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{a^{3/4}}+\frac{1}{4} \text{RootSum}\left[a^4-a b^3-4 a^3 \text{$\#$1}^4+6 a^2 \text{$\#$1}^8-4 a \text{$\#$1}^{12}+\text{$\#$1}^{16}\&,\frac{-a^3 \log (x)+b^3 \log (x)+a^3 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)-b^3 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)+3 a^2 \log (x) \text{$\#$1}^4-3 a^2 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-3 a \log (x) \text{$\#$1}^8+3 a \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^8+\log (x) \text{$\#$1}^{12}-\log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^{12}}{a^3 \text{$\#$1}^3-3 a^2 \text{$\#$1}^7+3 a \text{$\#$1}^{11}-\text{$\#$1}^{15}}\&\right]",1,"(-2*ArcTan[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/a^(3/4) + (2*ArcTanh[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/a^(3/4) + RootSum[a^4 - a*b^3 - 4*a^3*#1^4 + 6*a^2*#1^8 - 4*a*#1^12 + #1^16 & , (a^3*Log[x] - b^3*Log[x] - a^3*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] + b^3*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] - 3*a^2*Log[x]*#1^4 + 3*a^2*Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^4 + 3*a*Log[x]*#1^8 - 3*a*Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^8 - Log[x]*#1^12 + Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^12)/(-(a^3*#1^3) + 3*a^2*#1^7 - 3*a*#1^11 + #1^15) & ]/4","A",1
2905,1,686,323,0.7739893,"\int \frac{\left(1+x^2\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1-x^2\right)^2} \, dx","IntegrateAlgebraic[((1 + x^2)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^2)^2,x]","-\frac{x \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{-1+x^2}+\frac{1}{2} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}-3 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]-\frac{1}{8} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+7 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{1}{2} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^3+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^5}{-2+4 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]-\frac{1}{8} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-9 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","-\frac{x \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{-1+x^2}+\frac{1}{8} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-4 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{-2 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]+\frac{1}{8} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-4 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}-2 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]",1,"-((x*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(-1 + x^2)) + RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1 - 3*#1^3 + #1^5) & ]/2 - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 7*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(2*#1^3 - 3*#1^5 + #1^7) & ]/8 + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-(Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1) - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^3 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^5)/(-2 + 4*#1^2 - 3*#1^4 + #1^6) & ]/2 - RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 9*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ]/8","B",1
2906,1,686,323,0.0018999,"\int \frac{\left(1+x^2\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1-x^2\right)^2} \, dx","IntegrateAlgebraic[((1 + x^2)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^2)^2,x]","-\frac{x \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{-1+x^2}+\frac{1}{2} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}-3 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]-\frac{1}{8} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+7 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{1}{2} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^3+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^5}{-2+4 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]-\frac{1}{8} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-9 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","-\frac{x \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{-1+x^2}+\frac{1}{8} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-4 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{-2 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]+\frac{1}{8} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-4 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}-2 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]",1,"-((x*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(-1 + x^2)) + RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1 - 3*#1^3 + #1^5) & ]/2 - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 7*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(2*#1^3 - 3*#1^5 + #1^7) & ]/8 + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-(Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1) - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^3 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^5)/(-2 + 4*#1^2 - 3*#1^4 + #1^6) & ]/2 - RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 9*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ]/8","B",1
2907,1,324,324,4.3475215,"\int \frac{x^3 (-b+x) \left(2 a b-3 a x+x^2\right)}{\sqrt[3]{x^2 (-a+x) (-b+x)} \left(-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+\left(-1+b^2 d\right) x^4-2 b d x^5+d x^6\right)} \, dx","IntegrateAlgebraic[(x^3*(-b + x)*(2*a*b - 3*a*x + x^2))/((x^2*(-a + x)*(-b + x))^(1/3)*(-a^4 + 4*a^3*x - 6*a^2*x^2 + 4*a*x^3 + (-1 + b^2*d)*x^4 - 2*b*d*x^5 + d*x^6)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}{2 a-2 x+\sqrt[6]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{2 d^{5/6}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}{-2 a+2 x+\sqrt[6]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{2 d^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}{a-x}\right)}{d^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\left(a \sqrt[6]{d}-\sqrt[6]{d} x\right) \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}{a^2-2 a x+x^2+\sqrt[3]{d} \left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}}\right)}{2 d^{5/6}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}{2 a-2 x+\sqrt[6]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{2 d^{5/6}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}{-2 a+2 x+\sqrt[6]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}\right)}{2 d^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{d} \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}{a-x}\right)}{d^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\left(a \sqrt[6]{d}-\sqrt[6]{d} x\right) \sqrt[3]{a b x^2+(-a-b) x^3+x^4}}{a^2-2 a x+x^2+\sqrt[3]{d} \left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}}\right)}{2 d^{5/6}}",1,"-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/6)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))/(2*a - 2*x + d^(1/6)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))])/d^(5/6) + (Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/6)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))/(-2*a + 2*x + d^(1/6)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))])/(2*d^(5/6)) + ArcTanh[(d^(1/6)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))/(a - x)]/d^(5/6) + ArcTanh[((a*d^(1/6) - d^(1/6)*x)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(1/3))/(a^2 - 2*a*x + x^2 + d^(1/3)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(2/3))]/(2*d^(5/6))","A",1
2908,1,325,325,2.045583,"\int \frac{\left(1+x^3\right)^{2/3} \left(-1+x^6\right)}{x^6 \left(-1-2 x^3+2 x^6\right)} \, dx","IntegrateAlgebraic[((1 + x^3)^(2/3)*(-1 + x^6))/(x^6*(-1 - 2*x^3 + 2*x^6)),x]","\frac{\left(1+x^3\right)^{2/3} \left(-1+4 x^3\right)}{5 x^5}-\frac{1}{2} \sqrt[3]{\frac{1}{3} \left(45+26 \sqrt{3}\right)} \tan ^{-1}\left(\frac{3^{2/3} x}{\sqrt[6]{3} x-2 \sqrt[3]{1+x^3}}\right)-\frac{\tan ^{-1}\left(\frac{3^{2/3} x}{\sqrt[6]{3} x+2 \sqrt[3]{1+x^3}}\right)}{2 \sqrt[3]{45+26 \sqrt{3}}}+\frac{\log \left(-3 x+3^{5/6} \sqrt[3]{1+x^3}\right)}{2\ 3^{2/3} \sqrt[3]{26+15 \sqrt{3}}}+\frac{\sqrt[3]{26+15 \sqrt{3}} \log \left(3 x+3^{5/6} \sqrt[3]{1+x^3}\right)}{2\ 3^{2/3}}-\frac{\sqrt[3]{26+15 \sqrt{3}} \log \left(-3 x^2+3^{5/6} x \sqrt[3]{1+x^3}-3^{2/3} \left(1+x^3\right)^{2/3}\right)}{4\ 3^{2/3}}-\frac{\log \left(3 x^2+3^{5/6} x \sqrt[3]{1+x^3}+3^{2/3} \left(1+x^3\right)^{2/3}\right)}{4\ 3^{2/3} \sqrt[3]{26+15 \sqrt{3}}}","\frac{\left(1+x^3\right)^{2/3} \left(-1+4 x^3\right)}{5 x^5}-\frac{1}{2} \sqrt[3]{\frac{1}{3} \left(45+26 \sqrt{3}\right)} \tan ^{-1}\left(\frac{3^{2/3} x}{\sqrt[6]{3} x-2 \sqrt[3]{1+x^3}}\right)-\frac{\tan ^{-1}\left(\frac{3^{2/3} x}{\sqrt[6]{3} x+2 \sqrt[3]{1+x^3}}\right)}{2 \sqrt[3]{45+26 \sqrt{3}}}+\frac{\log \left(-3 x+3^{5/6} \sqrt[3]{1+x^3}\right)}{2\ 3^{2/3} \sqrt[3]{26+15 \sqrt{3}}}+\frac{\sqrt[3]{26+15 \sqrt{3}} \log \left(3 x+3^{5/6} \sqrt[3]{1+x^3}\right)}{2\ 3^{2/3}}-\frac{\sqrt[3]{26+15 \sqrt{3}} \log \left(-3 x^2+3^{5/6} x \sqrt[3]{1+x^3}-3^{2/3} \left(1+x^3\right)^{2/3}\right)}{4\ 3^{2/3}}-\frac{\log \left(3 x^2+3^{5/6} x \sqrt[3]{1+x^3}+3^{2/3} \left(1+x^3\right)^{2/3}\right)}{4\ 3^{2/3} \sqrt[3]{26+15 \sqrt{3}}}",1,"((1 + x^3)^(2/3)*(-1 + 4*x^3))/(5*x^5) - (((45 + 26*Sqrt[3])/3)^(1/3)*ArcTan[(3^(2/3)*x)/(3^(1/6)*x - 2*(1 + x^3)^(1/3))])/2 - ArcTan[(3^(2/3)*x)/(3^(1/6)*x + 2*(1 + x^3)^(1/3))]/(2*(45 + 26*Sqrt[3])^(1/3)) + Log[-3*x + 3^(5/6)*(1 + x^3)^(1/3)]/(2*3^(2/3)*(26 + 15*Sqrt[3])^(1/3)) + ((26 + 15*Sqrt[3])^(1/3)*Log[3*x + 3^(5/6)*(1 + x^3)^(1/3)])/(2*3^(2/3)) - ((26 + 15*Sqrt[3])^(1/3)*Log[-3*x^2 + 3^(5/6)*x*(1 + x^3)^(1/3) - 3^(2/3)*(1 + x^3)^(2/3)])/(4*3^(2/3)) - Log[3*x^2 + 3^(5/6)*x*(1 + x^3)^(1/3) + 3^(2/3)*(1 + x^3)^(2/3)]/(4*3^(2/3)*(26 + 15*Sqrt[3])^(1/3))","A",1
2909,1,303,326,0.4249997,"\int \frac{1+x^4}{\left(-1+x^4\right) \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[(1 + x^4)/((-1 + x^4)*Sqrt[x + Sqrt[1 + x^2]]),x]","-\frac{1}{3 \left(x+\sqrt{1+x^2}\right)^{3/2}}+\sqrt{x+\sqrt{1+x^2}}-\sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\sqrt{-1+\sqrt{2}} \sqrt{x+\sqrt{1+x^2}}\right)+\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\sqrt{1+\sqrt{2}} \sqrt{x+\sqrt{1+x^2}}\right)-\sqrt{2} \tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x}{\sqrt{2}}+\frac{\sqrt{1+x^2}}{\sqrt{2}}}{\sqrt{x+\sqrt{1+x^2}}}\right)-\sqrt{-1+\sqrt{2}} \tanh ^{-1}\left(\sqrt{-1+\sqrt{2}} \sqrt{x+\sqrt{1+x^2}}\right)+\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\sqrt{1+\sqrt{2}} \sqrt{x+\sqrt{1+x^2}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\frac{1}{\sqrt{2}}+\frac{x}{\sqrt{2}}+\frac{\sqrt{1+x^2}}{\sqrt{2}}}{\sqrt{x+\sqrt{1+x^2}}}\right)","\frac{4}{3} x \sqrt{-x+\sqrt{1+x^2}}+\frac{2}{3} \sqrt{1+x^2} \sqrt{-x+\sqrt{1+x^2}}+\sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{-x+\sqrt{1+x^2}}}{\sqrt{-1+\sqrt{2}}}\right)-\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{-x+\sqrt{1+x^2}}}{\sqrt{1+\sqrt{2}}}\right)+\sqrt{2} \tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}-\frac{x}{\sqrt{2}}+\frac{\sqrt{1+x^2}}{\sqrt{2}}}{\sqrt{-x+\sqrt{1+x^2}}}\right)-\sqrt{-1+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{-x+\sqrt{1+x^2}}}{\sqrt{-1+\sqrt{2}}}\right)+\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{-x+\sqrt{1+x^2}}}{\sqrt{1+\sqrt{2}}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{-x+\sqrt{1+x^2}}}{1-x+\sqrt{1+x^2}}\right)",1,"-1/3*1/(x + Sqrt[1 + x^2])^(3/2) + Sqrt[x + Sqrt[1 + x^2]] - Sqrt[-1 + Sqrt[2]]*ArcTan[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + Sqrt[2]]*ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] - Sqrt[2]*ArcTan[(-(1/Sqrt[2]) + x/Sqrt[2] + Sqrt[1 + x^2]/Sqrt[2])/Sqrt[x + Sqrt[1 + x^2]]] - Sqrt[-1 + Sqrt[2]]*ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + Sqrt[2]]*ArcTanh[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] - Sqrt[2]*ArcTanh[(1/Sqrt[2] + x/Sqrt[2] + Sqrt[1 + x^2]/Sqrt[2])/Sqrt[x + Sqrt[1 + x^2]]]","A",1
2910,1,327,327,0.9174671,"\int \frac{(b+2 a x) \sqrt[4]{b x^3+a x^4}}{-b+a x+x^2} \, dx","IntegrateAlgebraic[((b + 2*a*x)*(b*x^3 + a*x^4)^(1/4))/(-b + a*x + x^2),x]","2 a \sqrt[4]{b x^3+a x^4}+\left(4 a^{9/4}-3 \sqrt[4]{a} b\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)+\left(-4 a^{9/4}+3 \sqrt[4]{a} b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)-\text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{4 a^4 \log (x)-4 a^2 b \log (x)+b^2 \log (x)-4 a^4 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)+4 a^2 b \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)-b^2 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)-2 a^3 \log (x) \text{$\#$1}^4+3 a b \log (x) \text{$\#$1}^4+2 a^3 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-3 a b \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 a \text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]","2 a \sqrt[4]{b x^3+a x^4}+\left(4 a^{9/4}-3 \sqrt[4]{a} b\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)+\left(-4 a^{9/4}+3 \sqrt[4]{a} b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)-\text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-4 a^4 \log (x)+4 a^2 b \log (x)-b^2 \log (x)+4 a^4 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)-4 a^2 b \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)+b^2 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)+2 a^3 \log (x) \text{$\#$1}^4-3 a b \log (x) \text{$\#$1}^4-2 a^3 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+3 a b \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{3 a \text{$\#$1}^3-2 \text{$\#$1}^7}\&\right]",1,"2*a*(b*x^3 + a*x^4)^(1/4) + (4*a^(9/4) - 3*a^(1/4)*b)*ArcTan[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)] + (-4*a^(9/4) + 3*a^(1/4)*b)*ArcTanh[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)] - RootSum[2*a^2 - b - 3*a*#1^4 + #1^8 & , (4*a^4*Log[x] - 4*a^2*b*Log[x] + b^2*Log[x] - 4*a^4*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] + 4*a^2*b*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] - b^2*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] - 2*a^3*Log[x]*#1^4 + 3*a*b*Log[x]*#1^4 + 2*a^3*Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^4 - 3*a*b*Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1^3 + 2*#1^7) & ]","A",1
2911,1,327,327,0.001585,"\int \frac{(b+2 a x) \sqrt[4]{b x^3+a x^4}}{-b+a x+x^2} \, dx","IntegrateAlgebraic[((b + 2*a*x)*(b*x^3 + a*x^4)^(1/4))/(-b + a*x + x^2),x]","2 a \sqrt[4]{b x^3+a x^4}+\left(4 a^{9/4}-3 \sqrt[4]{a} b\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)+\left(-4 a^{9/4}+3 \sqrt[4]{a} b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)-\text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{4 a^4 \log (x)-4 a^2 b \log (x)+b^2 \log (x)-4 a^4 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)+4 a^2 b \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)-b^2 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)-2 a^3 \log (x) \text{$\#$1}^4+3 a b \log (x) \text{$\#$1}^4+2 a^3 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-3 a b \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 a \text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]","2 a \sqrt[4]{b x^3+a x^4}+\left(4 a^{9/4}-3 \sqrt[4]{a} b\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)+\left(-4 a^{9/4}+3 \sqrt[4]{a} b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)-\text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-4 a^4 \log (x)+4 a^2 b \log (x)-b^2 \log (x)+4 a^4 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)-4 a^2 b \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)+b^2 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)+2 a^3 \log (x) \text{$\#$1}^4-3 a b \log (x) \text{$\#$1}^4-2 a^3 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+3 a b \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{3 a \text{$\#$1}^3-2 \text{$\#$1}^7}\&\right]",1,"2*a*(b*x^3 + a*x^4)^(1/4) + (4*a^(9/4) - 3*a^(1/4)*b)*ArcTan[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)] + (-4*a^(9/4) + 3*a^(1/4)*b)*ArcTanh[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)] - RootSum[2*a^2 - b - 3*a*#1^4 + #1^8 & , (4*a^4*Log[x] - 4*a^2*b*Log[x] + b^2*Log[x] - 4*a^4*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] + 4*a^2*b*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] - b^2*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] - 2*a^3*Log[x]*#1^4 + 3*a*b*Log[x]*#1^4 + 2*a^3*Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^4 - 3*a*b*Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1^3 + 2*#1^7) & ]","A",1
2912,1,312,328,0.6482092,"\int \frac{x^3}{\sqrt[3]{-x^2+x^4} \left(1+x^6\right)} \, dx","IntegrateAlgebraic[x^3/((-x^2 + x^4)^(1/3)*(1 + x^6)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{-x^2+2 \left(-x^2+x^4\right)^{2/3}}\right)}{2 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{-x^2+\sqrt[3]{2} \left(-x^2+x^4\right)^{2/3}}\right)}{4 \sqrt[3]{2} \sqrt{3}}+\frac{1}{6} \log \left(x^2+\left(-x^2+x^4\right)^{2/3}\right)+\frac{\log \left(-2 x^2+2^{2/3} \sqrt{3} x \sqrt[3]{-x^2+x^4}-\sqrt[3]{2} \left(-x^2+x^4\right)^{2/3}\right)}{24 \sqrt[3]{2}}-\frac{\log \left(2 x^2+\sqrt[3]{2} \left(-x^2+x^4\right)^{2/3}\right)}{12 \sqrt[3]{2}}+\frac{\log \left(2 x^2+2^{2/3} \sqrt{3} x \sqrt[3]{-x^2+x^4}+\sqrt[3]{2} \left(-x^2+x^4\right)^{2/3}\right)}{24 \sqrt[3]{2}}-\frac{1}{12} \log \left(x^4-x^2 \left(-x^2+x^4\right)^{2/3}+\left(-x^2+x^4\right)^{4/3}\right)","\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{-x^2+\sqrt[3]{2} \left(-x^2+x^4\right)^{2/3}}\right)}{4 \sqrt[3]{2} \sqrt{3}}+\frac{1}{6} \log \left(x^2+\left(-x^2+x^4\right)^{2/3}\right)-\frac{1}{12} i \left(-i+\sqrt{3}\right) \log \left(-x^2-i \sqrt{3} x^2+2 \left(-x^2+x^4\right)^{2/3}\right)+\frac{1}{12} i \left(i+\sqrt{3}\right) \log \left(-x^2+i \sqrt{3} x^2+2 \left(-x^2+x^4\right)^{2/3}\right)+\frac{\log \left(-2 x^2+2^{2/3} \sqrt{3} x \sqrt[3]{-x^2+x^4}-\sqrt[3]{2} \left(-x^2+x^4\right)^{2/3}\right)}{24 \sqrt[3]{2}}-\frac{\log \left(2 x^2+\sqrt[3]{2} \left(-x^2+x^4\right)^{2/3}\right)}{12 \sqrt[3]{2}}+\frac{\log \left(2 x^2+2^{2/3} \sqrt{3} x \sqrt[3]{-x^2+x^4}+\sqrt[3]{2} \left(-x^2+x^4\right)^{2/3}\right)}{24 \sqrt[3]{2}}",1,"-1/2*ArcTan[(Sqrt[3]*x^2)/(-x^2 + 2*(-x^2 + x^4)^(2/3))]/Sqrt[3] + ArcTan[(Sqrt[3]*x^2)/(-x^2 + 2^(1/3)*(-x^2 + x^4)^(2/3))]/(4*2^(1/3)*Sqrt[3]) + Log[x^2 + (-x^2 + x^4)^(2/3)]/6 + Log[-2*x^2 + 2^(2/3)*Sqrt[3]*x*(-x^2 + x^4)^(1/3) - 2^(1/3)*(-x^2 + x^4)^(2/3)]/(24*2^(1/3)) - Log[2*x^2 + 2^(1/3)*(-x^2 + x^4)^(2/3)]/(12*2^(1/3)) + Log[2*x^2 + 2^(2/3)*Sqrt[3]*x*(-x^2 + x^4)^(1/3) + 2^(1/3)*(-x^2 + x^4)^(2/3)]/(24*2^(1/3)) - Log[x^4 - x^2*(-x^2 + x^4)^(2/3) + (-x^2 + x^4)^(4/3)]/12","A",1
2913,1,323,328,2.3121277,"\int \frac{x^4 \sqrt{b+a^2 x^2}}{x^2-\sqrt{a x-\sqrt{b+a^2 x^2}}} \, dx","IntegrateAlgebraic[(x^4*Sqrt[b + a^2*x^2])/(x^2 - Sqrt[a*x - Sqrt[b + a^2*x^2]]),x]","-a^2 x+a \sqrt{b+a^2 x^2}+\frac{b^2}{6 a \left(a x-\sqrt{b+a^2 x^2}\right)^{3/2}}-\frac{b \sqrt{a x-\sqrt{b+a^2 x^2}}}{a}-\frac{\left(a x-\sqrt{b+a^2 x^2}\right)^{5/2}}{10 a}+\frac{b^4}{64 a^3 \left(-a x+\sqrt{b+a^2 x^2}\right)^4}-\frac{\left(-a x+\sqrt{b+a^2 x^2}\right)^4}{64 a^3}+\frac{b^2 \log \left(-a x+\sqrt{b+a^2 x^2}\right)}{8 a^3}+2 a \text{RootSum}\left[b^2-2 b \text{$\#$1}^4-4 a^2 \text{$\#$1}^5+\text{$\#$1}^8\&,\frac{b \log \left(\sqrt{a x-\sqrt{b+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^2+a^2 \log \left(\sqrt{a x-\sqrt{b+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^3}{2 b+5 a^2 \text{$\#$1}-2 \text{$\#$1}^4}\&\right]","-a^2 x+a \sqrt{b+a^2 x^2}+\frac{b^2}{6 a \left(a x-\sqrt{b+a^2 x^2}\right)^{3/2}}-\frac{b \sqrt{a x-\sqrt{b+a^2 x^2}}}{a}-\frac{\left(a x-\sqrt{b+a^2 x^2}\right)^{5/2}}{10 a}+\frac{b^4}{64 a^3 \left(-a x+\sqrt{b+a^2 x^2}\right)^4}-\frac{\left(-a x+\sqrt{b+a^2 x^2}\right)^4}{64 a^3}+\frac{b^2 \log \left(-a x+\sqrt{b+a^2 x^2}\right)}{8 a^3}+\frac{2 \text{RootSum}\left[b^2-2 b \text{$\#$1}^4-4 a^2 \text{$\#$1}^5+\text{$\#$1}^8\&,\frac{a^4 b \log \left(\sqrt{a x-\sqrt{b+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^2+a^6 \log \left(\sqrt{a x-\sqrt{b+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^3}{2 b+5 a^2 \text{$\#$1}-2 \text{$\#$1}^4}\&\right]}{a^3}",1,"-(a^2*x) + a*Sqrt[b + a^2*x^2] + b^2/(6*a*(a*x - Sqrt[b + a^2*x^2])^(3/2)) - (b*Sqrt[a*x - Sqrt[b + a^2*x^2]])/a - (a*x - Sqrt[b + a^2*x^2])^(5/2)/(10*a) + b^4/(64*a^3*(-(a*x) + Sqrt[b + a^2*x^2])^4) - (-(a*x) + Sqrt[b + a^2*x^2])^4/(64*a^3) + (b^2*Log[-(a*x) + Sqrt[b + a^2*x^2]])/(8*a^3) + 2*a*RootSum[b^2 - 2*b*#1^4 - 4*a^2*#1^5 + #1^8 & , (b*Log[Sqrt[a*x - Sqrt[b + a^2*x^2]] - #1]*#1^2 + a^2*Log[Sqrt[a*x - Sqrt[b + a^2*x^2]] - #1]*#1^3)/(2*b + 5*a^2*#1 - 2*#1^4) & ]","A",1
2914,1,711,329,1.0081811,"\int \frac{-1+x^2}{\sqrt[4]{\frac{b+a x}{d+c x}}} \, dx","IntegrateAlgebraic[(-1 + x^2)/((b + a*x)/(d + c*x))^(1/4),x]","\frac{96 a^4 b c^3 \left(\frac{b+a x}{d+c x}\right)^{3/4}-113 a^2 b^3 c^3 \left(\frac{b+a x}{d+c x}\right)^{3/4}-96 a^5 c^2 d \left(\frac{b+a x}{d+c x}\right)^{3/4}+123 a^3 b^2 c^2 d \left(\frac{b+a x}{d+c x}\right)^{3/4}-3 a^4 b c d^2 \left(\frac{b+a x}{d+c x}\right)^{3/4}-7 a^5 d^3 \left(\frac{b+a x}{d+c x}\right)^{3/4}-192 a^3 b c^4 \left(\frac{b+a x}{d+c x}\right)^{7/4}+126 a b^3 c^4 \left(\frac{b+a x}{d+c x}\right)^{7/4}+192 a^4 c^3 d \left(\frac{b+a x}{d+c x}\right)^{7/4}-42 a^2 b^2 c^3 d \left(\frac{b+a x}{d+c x}\right)^{7/4}-102 a^3 b c^2 d^2 \left(\frac{b+a x}{d+c x}\right)^{7/4}+18 a^4 c d^3 \left(\frac{b+a x}{d+c x}\right)^{7/4}+96 a^2 b c^5 \left(\frac{b+a x}{d+c x}\right)^{11/4}-45 b^3 c^5 \left(\frac{b+a x}{d+c x}\right)^{11/4}-96 a^3 c^4 d \left(\frac{b+a x}{d+c x}\right)^{11/4}+15 a b^2 c^4 d \left(\frac{b+a x}{d+c x}\right)^{11/4}+9 a^2 b c^3 d^2 \left(\frac{b+a x}{d+c x}\right)^{11/4}+21 a^3 c^2 d^3 \left(\frac{b+a x}{d+c x}\right)^{11/4}}{96 a^3 c^2 \left(a-\frac{c (b+a x)}{d+c x}\right)^3}+\frac{\left(-32 a^2 b c^3+15 b^3 c^3+32 a^3 c^2 d-5 a b^2 c^2 d-3 a^2 b c d^2-7 a^3 d^3\right) \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt[4]{\frac{b+a x}{d+c x}}}{\sqrt[4]{a}}\right)}{64 a^{13/4} c^{11/4}}+\frac{\left(32 a^2 b c^3-15 b^3 c^3-32 a^3 c^2 d+5 a b^2 c^2 d+3 a^2 b c d^2+7 a^3 d^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt[4]{\frac{b+a x}{d+c x}}}{\sqrt[4]{a}}\right)}{64 a^{13/4} c^{11/4}}","\frac{\left(\frac{b+a x}{d+c x}\right)^{3/4} \left(-96 a^2 c^2 d+45 b^2 c^2 d-6 a b c d^2-7 a^2 d^3-96 a^2 c^3 x+45 b^2 c^3 x-42 a b c^2 d x-3 a^2 c d^2 x-36 a b c^3 x^2+36 a^2 c^2 d x^2+32 a^2 c^3 x^3\right)}{96 a^3 c^2}+\frac{\left(-32 a^2 b c^3+15 b^3 c^3+32 a^3 c^2 d-5 a b^2 c^2 d-3 a^2 b c d^2-7 a^3 d^3\right) \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt[4]{\frac{b+a x}{d+c x}}}{\sqrt[4]{a}}\right)}{64 a^{13/4} c^{11/4}}+\frac{\left(32 a^2 b c^3-15 b^3 c^3-32 a^3 c^2 d+5 a b^2 c^2 d+3 a^2 b c d^2+7 a^3 d^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt[4]{\frac{b+a x}{d+c x}}}{\sqrt[4]{a}}\right)}{64 a^{13/4} c^{11/4}}",1,"(96*a^4*b*c^3*((b + a*x)/(d + c*x))^(3/4) - 113*a^2*b^3*c^3*((b + a*x)/(d + c*x))^(3/4) - 96*a^5*c^2*d*((b + a*x)/(d + c*x))^(3/4) + 123*a^3*b^2*c^2*d*((b + a*x)/(d + c*x))^(3/4) - 3*a^4*b*c*d^2*((b + a*x)/(d + c*x))^(3/4) - 7*a^5*d^3*((b + a*x)/(d + c*x))^(3/4) - 192*a^3*b*c^4*((b + a*x)/(d + c*x))^(7/4) + 126*a*b^3*c^4*((b + a*x)/(d + c*x))^(7/4) + 192*a^4*c^3*d*((b + a*x)/(d + c*x))^(7/4) - 42*a^2*b^2*c^3*d*((b + a*x)/(d + c*x))^(7/4) - 102*a^3*b*c^2*d^2*((b + a*x)/(d + c*x))^(7/4) + 18*a^4*c*d^3*((b + a*x)/(d + c*x))^(7/4) + 96*a^2*b*c^5*((b + a*x)/(d + c*x))^(11/4) - 45*b^3*c^5*((b + a*x)/(d + c*x))^(11/4) - 96*a^3*c^4*d*((b + a*x)/(d + c*x))^(11/4) + 15*a*b^2*c^4*d*((b + a*x)/(d + c*x))^(11/4) + 9*a^2*b*c^3*d^2*((b + a*x)/(d + c*x))^(11/4) + 21*a^3*c^2*d^3*((b + a*x)/(d + c*x))^(11/4))/(96*a^3*c^2*(a - (c*(b + a*x))/(d + c*x))^3) + ((-32*a^2*b*c^3 + 15*b^3*c^3 + 32*a^3*c^2*d - 5*a*b^2*c^2*d - 3*a^2*b*c*d^2 - 7*a^3*d^3)*ArcTan[(c^(1/4)*((b + a*x)/(d + c*x))^(1/4))/a^(1/4)])/(64*a^(13/4)*c^(11/4)) + ((32*a^2*b*c^3 - 15*b^3*c^3 - 32*a^3*c^2*d + 5*a*b^2*c^2*d + 3*a^2*b*c*d^2 + 7*a^3*d^3)*ArcTanh[(c^(1/4)*((b + a*x)/(d + c*x))^(1/4))/a^(1/4)])/(64*a^(13/4)*c^(11/4))","B",1
2915,1,52,329,1.2611229,"\int \frac{-b+a x^4}{\sqrt{b+a x^4} \left(b-c^2 x^2+a x^4\right)} \, dx","IntegrateAlgebraic[(-b + a*x^4)/(Sqrt[b + a*x^4]*(b - c^2*x^2 + a*x^4)),x]","-\frac{\log \left(c x+\sqrt{b+a x^4}\right)}{2 c}+\frac{\log \left(c^2 x-c \sqrt{b+a x^4}\right)}{2 c}","-\frac{\left(c+\sqrt{2 \sqrt{a} \sqrt{b}+c^2}\right) \sqrt{-\sqrt{a} \sqrt{b}-c^2+c \sqrt{2 \sqrt{a} \sqrt{b}+c^2}} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{-\sqrt{a} \sqrt{b}-c^2+c \sqrt{2 \sqrt{a} \sqrt{b}+c^2}} x}{\sqrt{b}+\sqrt{a} x^2+\sqrt{b+a x^4}}\right)}{\sqrt{2} \sqrt{a} \sqrt{b} c}-\frac{\left(-c+\sqrt{2 \sqrt{a} \sqrt{b}+c^2}\right) \sqrt{\sqrt{a} \sqrt{b}+c^2+c \sqrt{2 \sqrt{a} \sqrt{b}+c^2}} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\sqrt{a} \sqrt{b}+c^2+c \sqrt{2 \sqrt{a} \sqrt{b}+c^2}} x}{\sqrt{b}+\sqrt{a} x^2+\sqrt{b+a x^4}}\right)}{\sqrt{2} \sqrt{a} \sqrt{b} c}",1,"-1/2*Log[c*x + Sqrt[b + a*x^4]]/c + Log[c^2*x - c*Sqrt[b + a*x^4]]/(2*c)","A",1
2916,1,308,330,7.8480885,"\int \frac{\left(1+x^2\right) \sqrt[3]{-1-x^2+x^4+x^6}}{x} \, dx","IntegrateAlgebraic[((1 + x^2)*(-1 - x^2 + x^4 + x^6)^(1/3))/x,x]","\frac{\left(-1+x^2\right)^{2/3} \left(1+x^2\right)^{4/3} \left(\frac{1}{12} \sqrt[3]{-1+x^2} \left(4 \left(1+x^2\right)^{2/3}+3 \left(1+x^2\right)^{5/3}\right)+\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{1+x^2}}{2 \sqrt[3]{-1+x^2}-\sqrt[3]{1+x^2}}\right)+\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{1+x^2}}{2 \sqrt[3]{-1+x^2}+\sqrt[3]{1+x^2}}\right)}{6 \sqrt{3}}+\frac{1}{18} \log \left(\sqrt[3]{-1+x^2}-\sqrt[3]{1+x^2}\right)-\frac{1}{2} \log \left(\sqrt[3]{-1+x^2}+\sqrt[3]{1+x^2}\right)+\frac{1}{4} \log \left(\left(-1+x^2\right)^{2/3}-\sqrt[3]{-1+x^2} \sqrt[3]{1+x^2}+\left(1+x^2\right)^{2/3}\right)-\frac{1}{36} \log \left(\left(-1+x^2\right)^{2/3}+\sqrt[3]{-1+x^2} \sqrt[3]{1+x^2}+\left(1+x^2\right)^{2/3}\right)\right)}{\left(\left(-1+x^2\right) \left(1+x^2\right)^2\right)^{2/3}}","\frac{\left(-1+x^2\right)^{2/3} \left(1+x^2\right)^{4/3} \left(\frac{1}{2} \sqrt[3]{-1+x^2} \left(1+x^2\right)^{2/3}+\frac{1}{12} \sqrt[3]{-1+x^2} \left(-2 \left(1+x^2\right)^{2/3}+3 \left(1+x^2\right)^{5/3}\right)+\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{1+x^2}}{2 \sqrt[3]{-1+x^2}-\sqrt[3]{1+x^2}}\right)+\frac{\tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{1+x^2}}{2 \sqrt[3]{-1+x^2}+\sqrt[3]{1+x^2}}\right)}{6 \sqrt{3}}+\frac{1}{18} \log \left(\sqrt[3]{-1+x^2}-\sqrt[3]{1+x^2}\right)-\frac{1}{2} \log \left(\sqrt[3]{-1+x^2}+\sqrt[3]{1+x^2}\right)+\frac{1}{4} \log \left(\left(-1+x^2\right)^{2/3}-\sqrt[3]{-1+x^2} \sqrt[3]{1+x^2}+\left(1+x^2\right)^{2/3}\right)-\frac{1}{36} \log \left(\left(-1+x^2\right)^{2/3}+\sqrt[3]{-1+x^2} \sqrt[3]{1+x^2}+\left(1+x^2\right)^{2/3}\right)\right)}{\left(\left(-1+x^2\right) \left(1+x^2\right)^2\right)^{2/3}}",1,"((-1 + x^2)^(2/3)*(1 + x^2)^(4/3)*(((-1 + x^2)^(1/3)*(4*(1 + x^2)^(2/3) + 3*(1 + x^2)^(5/3)))/12 + (Sqrt[3]*ArcTan[(Sqrt[3]*(1 + x^2)^(1/3))/(2*(-1 + x^2)^(1/3) - (1 + x^2)^(1/3))])/2 + ArcTan[(Sqrt[3]*(1 + x^2)^(1/3))/(2*(-1 + x^2)^(1/3) + (1 + x^2)^(1/3))]/(6*Sqrt[3]) + Log[(-1 + x^2)^(1/3) - (1 + x^2)^(1/3)]/18 - Log[(-1 + x^2)^(1/3) + (1 + x^2)^(1/3)]/2 + Log[(-1 + x^2)^(2/3) - (-1 + x^2)^(1/3)*(1 + x^2)^(1/3) + (1 + x^2)^(2/3)]/4 - Log[(-1 + x^2)^(2/3) + (-1 + x^2)^(1/3)*(1 + x^2)^(1/3) + (1 + x^2)^(2/3)]/36))/((-1 + x^2)*(1 + x^2)^2)^(2/3)","A",1
2917,1,333,330,3.6752614,"\int \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}} \sqrt[3]{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}} \, dx","IntegrateAlgebraic[Sqrt[-(a/b^2) + (a^2*x^2)/b^2]*(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])^(1/3),x]","\frac{\left(-10-a x^2\right) \sqrt[3]{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{16 b}+\frac{7}{16} x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}} \sqrt[3]{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}+\frac{5 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{2} \sqrt[3]{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{\sqrt{3}}\right)}{8 \sqrt[3]{2} \sqrt{3} b}-\frac{5 \log \left(-b+\sqrt[3]{2} b \sqrt[3]{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}\right)}{24 \sqrt[3]{2} b}+\frac{5 \log \left(1+\sqrt[3]{2} \sqrt[3]{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}+2^{2/3} \left(a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}\right)^{2/3}\right)}{48 \sqrt[3]{2} b}","\frac{\left(-10-a x^2\right) \sqrt[3]{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{16 b}+\frac{7}{16} x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}} \sqrt[3]{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}+\frac{5 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{2} \sqrt[3]{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{\sqrt{3}}\right)}{8 \sqrt[3]{2} \sqrt{3} b}-\frac{5 \log \left(-1+\sqrt[3]{2} \sqrt[3]{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}\right)}{24 \sqrt[3]{2} b}+\frac{5 \log \left(1+\sqrt[3]{2} \sqrt[3]{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}+2^{2/3} \left(a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}\right)^{2/3}\right)}{48 \sqrt[3]{2} b}",1,"((-10 - a*x^2)*(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])^(1/3))/(16*b) + (7*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]*(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])^(1/3))/16 + (5*ArcTan[1/Sqrt[3] + (2*2^(1/3)*(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])^(1/3))/Sqrt[3]])/(8*2^(1/3)*Sqrt[3]*b) - (5*Log[-b + 2^(1/3)*b*(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])^(1/3)])/(24*2^(1/3)*b) + (5*Log[1 + 2^(1/3)*(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])^(1/3) + 2^(2/3)*(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])^(2/3)])/(48*2^(1/3)*b)","A",1
2918,1,334,331,3.0125242,"\int \frac{-b+a x^2}{\left(-d+c x^2\right) \sqrt[3]{-x+x^3}} \, dx","IntegrateAlgebraic[(-b + a*x^2)/((-d + c*x^2)*(-x + x^3)^(1/3)),x]","\frac{\sqrt{3} a \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)}{2 c}-\frac{\sqrt{3} (b c-a d) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{c-d} x}{\sqrt[3]{c-d} x-2 \sqrt[3]{d} \sqrt[3]{-x+x^3}}\right)}{2 c \sqrt[3]{c-d} d^{2/3}}-\frac{a \log \left(-c x+c \sqrt[3]{-x+x^3}\right)}{2 c}+\frac{(b c-a d) \log \left(\sqrt[3]{c-d} x+\sqrt[3]{d} \sqrt[3]{-x+x^3}\right)}{2 c \sqrt[3]{c-d} d^{2/3}}+\frac{a \log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)}{4 c}+\frac{(-b c+a d) \log \left((c-d)^{2/3} x^2-\sqrt[3]{c-d} \sqrt[3]{d} x \sqrt[3]{-x+x^3}+d^{2/3} \left(-x+x^3\right)^{2/3}\right)}{4 c \sqrt[3]{c-d} d^{2/3}}","\frac{\sqrt{3} a \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)}{2 c}-\frac{\sqrt{3} (b c-a d) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{c-d} x}{\sqrt[3]{c-d} x-2 \sqrt[3]{d} \sqrt[3]{-x+x^3}}\right)}{2 c \sqrt[3]{c-d} d^{2/3}}-\frac{a \log \left(-x+\sqrt[3]{-x+x^3}\right)}{2 c}+\frac{(b c-a d) \log \left(\sqrt[3]{c-d} x+\sqrt[3]{d} \sqrt[3]{-x+x^3}\right)}{2 c \sqrt[3]{c-d} d^{2/3}}+\frac{a \log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)}{4 c}+\frac{(-b c+a d) \log \left((c-d)^{2/3} x^2-\sqrt[3]{c-d} \sqrt[3]{d} x \sqrt[3]{-x+x^3}+d^{2/3} \left(-x+x^3\right)^{2/3}\right)}{4 c \sqrt[3]{c-d} d^{2/3}}",1,"(Sqrt[3]*a*ArcTan[(Sqrt[3]*x)/(x + 2*(-x + x^3)^(1/3))])/(2*c) - (Sqrt[3]*(b*c - a*d)*ArcTan[(Sqrt[3]*(c - d)^(1/3)*x)/((c - d)^(1/3)*x - 2*d^(1/3)*(-x + x^3)^(1/3))])/(2*c*(c - d)^(1/3)*d^(2/3)) - (a*Log[-(c*x) + c*(-x + x^3)^(1/3)])/(2*c) + ((b*c - a*d)*Log[(c - d)^(1/3)*x + d^(1/3)*(-x + x^3)^(1/3)])/(2*c*(c - d)^(1/3)*d^(2/3)) + (a*Log[x^2 + x*(-x + x^3)^(1/3) + (-x + x^3)^(2/3)])/(4*c) + ((-(b*c) + a*d)*Log[(c - d)^(2/3)*x^2 - (c - d)^(1/3)*d^(1/3)*x*(-x + x^3)^(1/3) + d^(2/3)*(-x + x^3)^(2/3)])/(4*c*(c - d)^(1/3)*d^(2/3))","A",1
2919,1,385,332,2.3414459,"\int \frac{1}{(b+a x) \sqrt[3]{-b^3+a^3 x^3}} \, dx","IntegrateAlgebraic[1/((b + a*x)*(-b^3 + a^3*x^3)^(1/3)),x]","\frac{(-1)^{5/6} \sqrt{3} \tanh ^{-1}\left(\frac{\frac{i b+\sqrt{3} b}{2^{2/3} \sqrt{3}}+\frac{\left(-i a-\sqrt{3} a\right) x}{2^{2/3} \sqrt{3}}+\frac{i \sqrt[3]{-b^3+a^3 x^3}}{\sqrt{3}}}{\sqrt[3]{-b^3+a^3 x^3}}\right)}{2 \sqrt[3]{2} a b}+\frac{\sqrt[3]{-\frac{1}{2}} \log \left(\left(-1+i \sqrt{3}\right) \sqrt{a} b^{3/2}+a^{3/2} \sqrt{b} x-i \sqrt{3} a^{3/2} \sqrt{b} x+2\ 2^{2/3} \sqrt{a} \sqrt{b} \sqrt[3]{-b^3+a^3 x^3}\right)}{2 a b}-\frac{\sqrt[3]{-\frac{1}{2}} \log \left(-a b^3-i \sqrt{3} a b^3+2 a^2 b^2 x+2 i \sqrt{3} a^2 b^2 x-a^3 b x^2-i \sqrt{3} a^3 b x^2+\left(-2 (-2)^{2/3} a b^2+2 (-2)^{2/3} a^2 b x\right) \sqrt[3]{-b^3+a^3 x^3}+4 \sqrt[3]{2} a b \left(-b^3+a^3 x^3\right)^{2/3}\right)}{4 a b}","\frac{(-1)^{5/6} \sqrt{3} \tanh ^{-1}\left(\frac{\frac{i b+\sqrt{3} b}{2^{2/3} \sqrt{3}}+\frac{\left(-i a-\sqrt{3} a\right) x}{2^{2/3} \sqrt{3}}+\frac{i \sqrt[3]{-b^3+a^3 x^3}}{\sqrt{3}}}{\sqrt[3]{-b^3+a^3 x^3}}\right)}{2 \sqrt[3]{2} a b}+\frac{\sqrt[3]{-\frac{1}{2}} \log \left(-(-1)^{2/3} \sqrt{a} b^{3/2}+(-1)^{2/3} a^{3/2} \sqrt{b} x-2^{2/3} \sqrt{a} \sqrt{b} \sqrt[3]{-b^3+a^3 x^3}\right)}{2 a b}-\frac{\sqrt[3]{-\frac{1}{2}} \log \left(\sqrt[3]{-1} a b^3-2 \sqrt[3]{-1} a^2 b^2 x+\sqrt[3]{-1} a^3 b x^2+\left((-2)^{2/3} a b^2-(-2)^{2/3} a^2 b x\right) \sqrt[3]{-b^3+a^3 x^3}-2 \sqrt[3]{2} a b \left(-b^3+a^3 x^3\right)^{2/3}\right)}{4 a b}",1,"((-1)^(5/6)*Sqrt[3]*ArcTanh[((I*b + Sqrt[3]*b)/(2^(2/3)*Sqrt[3]) + (((-I)*a - Sqrt[3]*a)*x)/(2^(2/3)*Sqrt[3]) + (I*(-b^3 + a^3*x^3)^(1/3))/Sqrt[3])/(-b^3 + a^3*x^3)^(1/3)])/(2*2^(1/3)*a*b) + ((-1/2)^(1/3)*Log[(-1 + I*Sqrt[3])*Sqrt[a]*b^(3/2) + a^(3/2)*Sqrt[b]*x - I*Sqrt[3]*a^(3/2)*Sqrt[b]*x + 2*2^(2/3)*Sqrt[a]*Sqrt[b]*(-b^3 + a^3*x^3)^(1/3)])/(2*a*b) - ((-1/2)^(1/3)*Log[-(a*b^3) - I*Sqrt[3]*a*b^3 + 2*a^2*b^2*x + (2*I)*Sqrt[3]*a^2*b^2*x - a^3*b*x^2 - I*Sqrt[3]*a^3*b*x^2 + (-2*(-2)^(2/3)*a*b^2 + 2*(-2)^(2/3)*a^2*b*x)*(-b^3 + a^3*x^3)^(1/3) + 4*2^(1/3)*a*b*(-b^3 + a^3*x^3)^(2/3)])/(4*a*b)","A",1
2920,1,331,333,1.3557862,"\int \frac{-b^{12}+a^{12} x^{12}}{\sqrt{b^4+a^4 x^4} \left(b^{12}+a^{12} x^{12}\right)} \, dx","IntegrateAlgebraic[(-b^12 + a^12*x^12)/(Sqrt[b^4 + a^4*x^4]*(b^12 + a^12*x^12)),x]","-\frac{x}{3 \sqrt{b^4+a^4 x^4}}+\frac{1}{6} \text{RootSum}\left[16 a^8 b^8-32 a^6 b^6 \text{$\#$1}^2-24 a^4 b^4 \text{$\#$1}^4-8 a^2 b^2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-8 a^6 b^6 \log (x)+8 a^6 b^6 \log \left(b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}-x \text{$\#$1}\right)+4 a^4 b^4 \log (x) \text{$\#$1}^2-4 a^4 b^4 \log \left(b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^2+2 a^2 b^2 \log (x) \text{$\#$1}^4-2 a^2 b^2 \log \left(b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-\log (x) \text{$\#$1}^6+\log \left(b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^6}{-8 a^6 b^6 \text{$\#$1}-12 a^4 b^4 \text{$\#$1}^3-6 a^2 b^2 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","-\frac{x}{3 \sqrt{b^4+a^4 x^4}}+\frac{1}{6} \text{RootSum}\left[16 a^8 b^8-32 a^6 b^6 \text{$\#$1}^2-24 a^4 b^4 \text{$\#$1}^4-8 a^2 b^2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{8 a^6 b^6 \log (x)-8 a^6 b^6 \log \left(b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}-x \text{$\#$1}\right)-4 a^4 b^4 \log (x) \text{$\#$1}^2+4 a^4 b^4 \log \left(b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^2-2 a^2 b^2 \log (x) \text{$\#$1}^4+2 a^2 b^2 \log \left(b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+\log (x) \text{$\#$1}^6-\log \left(b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^6}{8 a^6 b^6 \text{$\#$1}+12 a^4 b^4 \text{$\#$1}^3+6 a^2 b^2 \text{$\#$1}^5-\text{$\#$1}^7}\&\right]",1,"-1/3*x/Sqrt[b^4 + a^4*x^4] + RootSum[16*a^8*b^8 - 32*a^6*b^6*#1^2 - 24*a^4*b^4*#1^4 - 8*a^2*b^2*#1^6 + #1^8 & , (-8*a^6*b^6*Log[x] + 8*a^6*b^6*Log[b^2 + a^2*x^2 + Sqrt[b^4 + a^4*x^4] - x*#1] + 4*a^4*b^4*Log[x]*#1^2 - 4*a^4*b^4*Log[b^2 + a^2*x^2 + Sqrt[b^4 + a^4*x^4] - x*#1]*#1^2 + 2*a^2*b^2*Log[x]*#1^4 - 2*a^2*b^2*Log[b^2 + a^2*x^2 + Sqrt[b^4 + a^4*x^4] - x*#1]*#1^4 - Log[x]*#1^6 + Log[b^2 + a^2*x^2 + Sqrt[b^4 + a^4*x^4] - x*#1]*#1^6)/(-8*a^6*b^6*#1 - 12*a^4*b^4*#1^3 - 6*a^2*b^2*#1^5 + #1^7) & ]/6","A",1
2921,1,334,334,1.7137538,"\int \frac{\left(c+b x+a x^2\right)^{5/2}}{c+b x} \, dx","IntegrateAlgebraic[(c + b*x + a*x^2)^(5/2)/(c + b*x),x]","\frac{\sqrt{c+b x+a x^2} \left(-45 b^8+390 a b^6 c+24 a^2 b^4 c^2+160 a^3 b^2 c^3+1920 a^4 c^4+30 a b^7 x+1308 a^2 b^5 c x-80 a^3 b^3 c^2 x-960 a^4 b c^3 x+744 a^2 b^6 x^2+48 a^3 b^4 c x^2+640 a^4 b^2 c^2 x^2+1008 a^3 b^5 x^3-480 a^4 b^3 c x^3+384 a^4 b^4 x^4\right)}{1920 a^2 b^5}-\frac{a^{5/2} c^5 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}\right)}{b^6}+\frac{\left(-3 b^{10}+30 a b^8 c-80 a^2 b^6 c^2+256 a^5 c^5\right) \log \left(b+2 a x-2 \sqrt{a} \sqrt{c+b x+a x^2}\right)}{256 a^{5/2} b^6}+\frac{a^{5/2} c^5 \log \left(2 \sqrt{a} c+\sqrt{a} b x-b \sqrt{c+b x+a x^2}\right)}{b^6}","\frac{\sqrt{c+b x+a x^2} \left(-45 b^8+390 a b^6 c+24 a^2 b^4 c^2+160 a^3 b^2 c^3+1920 a^4 c^4+30 a b^7 x+1308 a^2 b^5 c x-80 a^3 b^3 c^2 x-960 a^4 b c^3 x+744 a^2 b^6 x^2+48 a^3 b^4 c x^2+640 a^4 b^2 c^2 x^2+1008 a^3 b^5 x^3-480 a^4 b^3 c x^3+384 a^4 b^4 x^4\right)}{1920 a^2 b^5}-\frac{a^{5/2} c^5 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}\right)}{b^6}+\frac{\left(-3 b^{10}+30 a b^8 c-80 a^2 b^6 c^2+256 a^5 c^5\right) \log \left(b+2 a x-2 \sqrt{a} \sqrt{c+b x+a x^2}\right)}{256 a^{5/2} b^6}+\frac{a^{5/2} c^5 \log \left(2 \sqrt{a} c+\sqrt{a} b x-b \sqrt{c+b x+a x^2}\right)}{b^6}",1,"(Sqrt[c + b*x + a*x^2]*(-45*b^8 + 390*a*b^6*c + 24*a^2*b^4*c^2 + 160*a^3*b^2*c^3 + 1920*a^4*c^4 + 30*a*b^7*x + 1308*a^2*b^5*c*x - 80*a^3*b^3*c^2*x - 960*a^4*b*c^3*x + 744*a^2*b^6*x^2 + 48*a^3*b^4*c*x^2 + 640*a^4*b^2*c^2*x^2 + 1008*a^3*b^5*x^3 - 480*a^4*b^3*c*x^3 + 384*a^4*b^4*x^4))/(1920*a^2*b^5) - (a^(5/2)*c^5*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2]])/b^6 + ((-3*b^10 + 30*a*b^8*c - 80*a^2*b^6*c^2 + 256*a^5*c^5)*Log[b + 2*a*x - 2*Sqrt[a]*Sqrt[c + b*x + a*x^2]])/(256*a^(5/2)*b^6) + (a^(5/2)*c^5*Log[2*Sqrt[a]*c + Sqrt[a]*b*x - b*Sqrt[c + b*x + a*x^2]])/b^6","A",1
2922,1,333,334,2.5069312,"\int \frac{\left(-b-a x^2+x^4\right) \sqrt[4]{b x^2+a x^4}}{b+a x^4} \, dx","IntegrateAlgebraic[((-b - a*x^2 + x^4)*(b*x^2 + a*x^4)^(1/4))/(b + a*x^4),x]","\frac{x \sqrt[4]{b x^2+a x^4}}{2 a}+\frac{\left(4 a^2-b\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{4 a^{7/4}}+\frac{\left(-4 a^2+b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{4 a^{7/4}}-\frac{\text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a^3 \log (x)+a^2 b \log (x)-a^3 \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)-a^2 b \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)-a^2 \log (x) \text{$\#$1}^4+b \log (x) \text{$\#$1}^4+a b \log (x) \text{$\#$1}^4+a^2 \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-b \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-a b \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}^3+\text{$\#$1}^7}\&\right]}{4 a}","\frac{x \sqrt[4]{b x^2+a x^4}}{2 a}+\frac{\left(4 a^2-b\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{4 a^{7/4}}+\frac{\left(-4 a^2+b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{4 a^{7/4}}-\frac{\text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a^3 \log (x)-a^2 b \log (x)+a^3 \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)+a^2 b \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)+a^2 \log (x) \text{$\#$1}^4-b \log (x) \text{$\#$1}^4-a b \log (x) \text{$\#$1}^4-a^2 \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+b \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+a b \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}^3-\text{$\#$1}^7}\&\right]}{4 a}",1,"(x*(b*x^2 + a*x^4)^(1/4))/(2*a) + ((4*a^2 - b)*ArcTan[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/(4*a^(7/4)) + ((-4*a^2 + b)*ArcTanh[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/(4*a^(7/4)) - RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (a^3*Log[x] + a^2*b*Log[x] - a^3*Log[(b*x^2 + a*x^4)^(1/4) - x*#1] - a^2*b*Log[(b*x^2 + a*x^4)^(1/4) - x*#1] - a^2*Log[x]*#1^4 + b*Log[x]*#1^4 + a*b*Log[x]*#1^4 + a^2*Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1^4 - b*Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1^4 - a*b*Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3) + #1^7) & ]/(4*a)","A",1
2923,1,333,334,0.0015527,"\int \frac{\left(-b-a x^2+x^4\right) \sqrt[4]{b x^2+a x^4}}{b+a x^4} \, dx","IntegrateAlgebraic[((-b - a*x^2 + x^4)*(b*x^2 + a*x^4)^(1/4))/(b + a*x^4),x]","\frac{x \sqrt[4]{b x^2+a x^4}}{2 a}+\frac{\left(4 a^2-b\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{4 a^{7/4}}+\frac{\left(-4 a^2+b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{4 a^{7/4}}-\frac{\text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a^3 \log (x)+a^2 b \log (x)-a^3 \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)-a^2 b \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)-a^2 \log (x) \text{$\#$1}^4+b \log (x) \text{$\#$1}^4+a b \log (x) \text{$\#$1}^4+a^2 \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-b \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-a b \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}^3+\text{$\#$1}^7}\&\right]}{4 a}","\frac{x \sqrt[4]{b x^2+a x^4}}{2 a}+\frac{\left(4 a^2-b\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{4 a^{7/4}}+\frac{\left(-4 a^2+b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^2+a x^4}}\right)}{4 a^{7/4}}-\frac{\text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a^3 \log (x)-a^2 b \log (x)+a^3 \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)+a^2 b \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right)+a^2 \log (x) \text{$\#$1}^4-b \log (x) \text{$\#$1}^4-a b \log (x) \text{$\#$1}^4-a^2 \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+b \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+a b \log \left(\sqrt[4]{b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}^3-\text{$\#$1}^7}\&\right]}{4 a}",1,"(x*(b*x^2 + a*x^4)^(1/4))/(2*a) + ((4*a^2 - b)*ArcTan[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/(4*a^(7/4)) + ((-4*a^2 + b)*ArcTanh[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/(4*a^(7/4)) - RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (a^3*Log[x] + a^2*b*Log[x] - a^3*Log[(b*x^2 + a*x^4)^(1/4) - x*#1] - a^2*b*Log[(b*x^2 + a*x^4)^(1/4) - x*#1] - a^2*Log[x]*#1^4 + b*Log[x]*#1^4 + a*b*Log[x]*#1^4 + a^2*Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1^4 - b*Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1^4 - a*b*Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3) + #1^7) & ]/(4*a)","A",1
2924,1,339,337,0.4115,"\int \frac{\sqrt{1+x} \left(-1+x^2\right)}{\left(1+x^2\right) \sqrt{x+\sqrt{1+x}}} \, dx","IntegrateAlgebraic[(Sqrt[1 + x]*(-1 + x^2))/((1 + x^2)*Sqrt[x + Sqrt[1 + x]]),x]","\frac{1}{2} \sqrt{x+\sqrt{1+x}} \left(-3+2 \sqrt{1+x}\right)-\frac{7}{4} \log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}\right)+\text{RootSum}\left[1-8 \text{$\#$1}+40 \text{$\#$1}^2-48 \text{$\#$1}^3+20 \text{$\#$1}^4+8 \text{$\#$1}^5-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right)+2 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}-2 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2+4 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^3-\log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^4+2 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^5}{-1+10 \text{$\#$1}-18 \text{$\#$1}^2+10 \text{$\#$1}^3+5 \text{$\#$1}^4-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","-\frac{3}{2} \sqrt{x+\sqrt{1+x}}+\sqrt{1+x} \sqrt{x+\sqrt{1+x}}-\frac{7}{4} \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}\right)+8 \text{RootSum}\left[625+1000 \text{$\#$1}+300 \text{$\#$1}^2+120 \text{$\#$1}^3+470 \text{$\#$1}^4+24 \text{$\#$1}^5+12 \text{$\#$1}^6+8 \text{$\#$1}^7+\text{$\#$1}^8\&,\frac{25 \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}+20 \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^2+14 \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^3+4 \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^4+\log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^5}{125+75 \text{$\#$1}+45 \text{$\#$1}^2+235 \text{$\#$1}^3+15 \text{$\#$1}^4+9 \text{$\#$1}^5+7 \text{$\#$1}^6+\text{$\#$1}^7}\&\right]",1,"(Sqrt[x + Sqrt[1 + x]]*(-3 + 2*Sqrt[1 + x]))/2 - (7*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]])/4 + RootSum[1 - 8*#1 + 40*#1^2 - 48*#1^3 + 20*#1^4 + 8*#1^5 - 4*#1^6 + #1^8 & , (-Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1] + 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1 - 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^2 + 4*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^3 - Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^4 + 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^5)/(-1 + 10*#1 - 18*#1^2 + 10*#1^3 + 5*#1^4 - 3*#1^5 + #1^7) & ]","A",1
2925,1,339,337,0.0011096,"\int \frac{\sqrt{1+x} \left(-1+x^2\right)}{\left(1+x^2\right) \sqrt{x+\sqrt{1+x}}} \, dx","IntegrateAlgebraic[(Sqrt[1 + x]*(-1 + x^2))/((1 + x^2)*Sqrt[x + Sqrt[1 + x]]),x]","\frac{1}{2} \sqrt{x+\sqrt{1+x}} \left(-3+2 \sqrt{1+x}\right)-\frac{7}{4} \log \left(-1-2 \sqrt{1+x}+2 \sqrt{x+\sqrt{1+x}}\right)+\text{RootSum}\left[1-8 \text{$\#$1}+40 \text{$\#$1}^2-48 \text{$\#$1}^3+20 \text{$\#$1}^4+8 \text{$\#$1}^5-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right)+2 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}-2 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^2+4 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^3-\log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^4+2 \log \left(-\sqrt{1+x}+\sqrt{x+\sqrt{1+x}}-\text{$\#$1}\right) \text{$\#$1}^5}{-1+10 \text{$\#$1}-18 \text{$\#$1}^2+10 \text{$\#$1}^3+5 \text{$\#$1}^4-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","-\frac{3}{2} \sqrt{x+\sqrt{1+x}}+\sqrt{1+x} \sqrt{x+\sqrt{1+x}}-\frac{7}{4} \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}\right)+8 \text{RootSum}\left[625+1000 \text{$\#$1}+300 \text{$\#$1}^2+120 \text{$\#$1}^3+470 \text{$\#$1}^4+24 \text{$\#$1}^5+12 \text{$\#$1}^6+8 \text{$\#$1}^7+\text{$\#$1}^8\&,\frac{25 \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}+20 \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^2+14 \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^3+4 \log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^4+\log \left(1+2 \sqrt{1+x}-2 \sqrt{x+\sqrt{1+x}}+\text{$\#$1}\right) \text{$\#$1}^5}{125+75 \text{$\#$1}+45 \text{$\#$1}^2+235 \text{$\#$1}^3+15 \text{$\#$1}^4+9 \text{$\#$1}^5+7 \text{$\#$1}^6+\text{$\#$1}^7}\&\right]",1,"(Sqrt[x + Sqrt[1 + x]]*(-3 + 2*Sqrt[1 + x]))/2 - (7*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]])/4 + RootSum[1 - 8*#1 + 40*#1^2 - 48*#1^3 + 20*#1^4 + 8*#1^5 - 4*#1^6 + #1^8 & , (-Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1] + 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1 - 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^2 + 4*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^3 - Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^4 + 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^5)/(-1 + 10*#1 - 18*#1^2 + 10*#1^3 + 5*#1^4 - 3*#1^5 + #1^7) & ]","A",1
2926,1,337,337,1.060844,"\int \frac{-x^2+\sqrt{1+2 x^2}+\left(1+2 x^2\right)^{5/2}}{x^2-x \left(1+2 x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(-x^2 + Sqrt[1 + 2*x^2] + (1 + 2*x^2)^(5/2))/(x^2 - x*(1 + 2*x^2)^(3/2)),x]","-x^2+\frac{1}{2} \left(2+\sqrt{2}\right) \log \left(-\sqrt{2} x+\sqrt{1+2 x^2}\right)-2 \log \left(-2 x^2+\sqrt{2} x \sqrt{1+2 x^2}\right)+\frac{1}{2} \text{RootSum}\left[1+3 \text{$\#$1}^2-2 \sqrt{2} \text{$\#$1}^2+3 \text{$\#$1}^4+2 \sqrt{2} \text{$\#$1}^4+\text{$\#$1}^6\&,\frac{-\log \left(-\sqrt{2} x+\sqrt{1+2 x^2}-\text{$\#$1}\right)+4 \log \left(-\sqrt{2} x+\sqrt{1+2 x^2}-\text{$\#$1}\right) \text{$\#$1}+2 \log \left(-\sqrt{2} x+\sqrt{1+2 x^2}-\text{$\#$1}\right) \text{$\#$1}^3+4 \sqrt{2} \log \left(-\sqrt{2} x+\sqrt{1+2 x^2}-\text{$\#$1}\right) \text{$\#$1}^3+\log \left(-\sqrt{2} x+\sqrt{1+2 x^2}-\text{$\#$1}\right) \text{$\#$1}^4+2 \log \left(-\sqrt{2} x+\sqrt{1+2 x^2}-\text{$\#$1}\right) \text{$\#$1}^5}{3 \text{$\#$1}-2 \sqrt{2} \text{$\#$1}+6 \text{$\#$1}^3+4 \sqrt{2} \text{$\#$1}^3+3 \text{$\#$1}^5}\&\right]","-x^2+\frac{1}{2} \left(2+\sqrt{2}\right) \log \left(-\sqrt{2} x+\sqrt{1+2 x^2}\right)-2 \log \left(-2 x^2+\sqrt{2} x \sqrt{1+2 x^2}\right)+\frac{1}{2} \text{RootSum}\left[1+3 \text{$\#$1}^2-2 \sqrt{2} \text{$\#$1}^2+3 \text{$\#$1}^4+2 \sqrt{2} \text{$\#$1}^4+\text{$\#$1}^6\&,\frac{-\log \left(-\sqrt{2} x+\sqrt{1+2 x^2}-\text{$\#$1}\right)+4 \log \left(-\sqrt{2} x+\sqrt{1+2 x^2}-\text{$\#$1}\right) \text{$\#$1}+2 \log \left(-\sqrt{2} x+\sqrt{1+2 x^2}-\text{$\#$1}\right) \text{$\#$1}^3+4 \sqrt{2} \log \left(-\sqrt{2} x+\sqrt{1+2 x^2}-\text{$\#$1}\right) \text{$\#$1}^3+\log \left(-\sqrt{2} x+\sqrt{1+2 x^2}-\text{$\#$1}\right) \text{$\#$1}^4+2 \log \left(-\sqrt{2} x+\sqrt{1+2 x^2}-\text{$\#$1}\right) \text{$\#$1}^5}{3 \text{$\#$1}-2 \sqrt{2} \text{$\#$1}+6 \text{$\#$1}^3+4 \sqrt{2} \text{$\#$1}^3+3 \text{$\#$1}^5}\&\right]",1,"-x^2 + ((2 + Sqrt[2])*Log[-(Sqrt[2]*x) + Sqrt[1 + 2*x^2]])/2 - 2*Log[-2*x^2 + Sqrt[2]*x*Sqrt[1 + 2*x^2]] + RootSum[1 + 3*#1^2 - 2*Sqrt[2]*#1^2 + 3*#1^4 + 2*Sqrt[2]*#1^4 + #1^6 & , (-Log[-(Sqrt[2]*x) + Sqrt[1 + 2*x^2] - #1] + 4*Log[-(Sqrt[2]*x) + Sqrt[1 + 2*x^2] - #1]*#1 + 2*Log[-(Sqrt[2]*x) + Sqrt[1 + 2*x^2] - #1]*#1^3 + 4*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[1 + 2*x^2] - #1]*#1^3 + Log[-(Sqrt[2]*x) + Sqrt[1 + 2*x^2] - #1]*#1^4 + 2*Log[-(Sqrt[2]*x) + Sqrt[1 + 2*x^2] - #1]*#1^5)/(3*#1 - 2*Sqrt[2]*#1 + 6*#1^3 + 4*Sqrt[2]*#1^3 + 3*#1^5) & ]/2","A",1
2927,1,339,339,4.4113321,"\int \frac{\left(-b+x^3\right) \left(b+x^3\right) \left(-c+x^3\right)}{\sqrt[3]{a x^2+x^3}} \, dx","IntegrateAlgebraic[((-b + x^3)*(b + x^3)*(-c + x^3))/(a*x^2 + x^3)^(1/3),x]","\frac{\left(a x^2+x^3\right)^{2/3} \left(38038000 a^8-77157360 a^2 b^2+49533120 a^5 c-28528500 a^7 x+57868020 a b^2 x-37149840 a^4 c x+24453000 a^6 x^2-49601160 b^2 x^2+31842720 a^3 c x^2-22007700 a^5 x^3-28658448 a^2 c x^3+20314800 a^4 x^4+26453952 a c x^4-19045125 a^3 x^5-24800580 c x^5+18042750 a^2 x^6-17222625 a x^7+16533720 x^8\right)}{148803480 x}+\frac{\left(-135850 \sqrt{3} a^9+275562 \sqrt{3} a^3 b^2-176904 \sqrt{3} a^6 c+1594323 \sqrt{3} b^2 c\right) \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{a x^2+x^3}}\right)}{1594323}+\frac{\left(135850 a^9-275562 a^3 b^2+176904 a^6 c-1594323 b^2 c\right) \log \left(-x+\sqrt[3]{a x^2+x^3}\right)}{1594323}+\frac{\left(-135850 a^9+275562 a^3 b^2-176904 a^6 c+1594323 b^2 c\right) \log \left(x^2+x \sqrt[3]{a x^2+x^3}+\left(a x^2+x^3\right)^{2/3}\right)}{3188646}","\frac{\left(a x^2+x^3\right)^{2/3} \left(38038000 a^8-77157360 a^2 b^2+49533120 a^5 c-28528500 a^7 x+57868020 a b^2 x-37149840 a^4 c x+24453000 a^6 x^2-49601160 b^2 x^2+31842720 a^3 c x^2-22007700 a^5 x^3-28658448 a^2 c x^3+20314800 a^4 x^4+26453952 a c x^4-19045125 a^3 x^5-24800580 c x^5+18042750 a^2 x^6-17222625 a x^7+16533720 x^8\right)}{148803480 x}+\frac{\left(-135850 \sqrt{3} a^9+275562 \sqrt{3} a^3 b^2-176904 \sqrt{3} a^6 c+1594323 \sqrt{3} b^2 c\right) \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{a x^2+x^3}}\right)}{1594323}+\frac{\left(135850 a^9-275562 a^3 b^2+176904 a^6 c-1594323 b^2 c\right) \log \left(-x+\sqrt[3]{a x^2+x^3}\right)}{1594323}+\frac{\left(-135850 a^9+275562 a^3 b^2-176904 a^6 c+1594323 b^2 c\right) \log \left(x^2+x \sqrt[3]{a x^2+x^3}+\left(a x^2+x^3\right)^{2/3}\right)}{3188646}",1,"((a*x^2 + x^3)^(2/3)*(38038000*a^8 - 77157360*a^2*b^2 + 49533120*a^5*c - 28528500*a^7*x + 57868020*a*b^2*x - 37149840*a^4*c*x + 24453000*a^6*x^2 - 49601160*b^2*x^2 + 31842720*a^3*c*x^2 - 22007700*a^5*x^3 - 28658448*a^2*c*x^3 + 20314800*a^4*x^4 + 26453952*a*c*x^4 - 19045125*a^3*x^5 - 24800580*c*x^5 + 18042750*a^2*x^6 - 17222625*a*x^7 + 16533720*x^8))/(148803480*x) + ((-135850*Sqrt[3]*a^9 + 275562*Sqrt[3]*a^3*b^2 - 176904*Sqrt[3]*a^6*c + 1594323*Sqrt[3]*b^2*c)*ArcTan[(Sqrt[3]*x)/(x + 2*(a*x^2 + x^3)^(1/3))])/1594323 + ((135850*a^9 - 275562*a^3*b^2 + 176904*a^6*c - 1594323*b^2*c)*Log[-x + (a*x^2 + x^3)^(1/3)])/1594323 + ((-135850*a^9 + 275562*a^3*b^2 - 176904*a^6*c + 1594323*b^2*c)*Log[x^2 + x*(a*x^2 + x^3)^(1/3) + (a*x^2 + x^3)^(2/3)])/3188646","A",1
2928,1,340,340,0.817584,"\int \frac{x (-a+x) (-b+x) \left(a b-2 b x+x^2\right)}{\left(x (-a+x) (-b+x)^2\right)^{2/3} \left(-b^2+2 b x-\left(1-a^2 d\right) x^2-2 a d x^3+d x^4\right)} \, dx","IntegrateAlgebraic[(x*(-a + x)*(-b + x)*(a*b - 2*b*x + x^2))/((x*(-a + x)*(-b + x)^2)^(2/3)*(-b^2 + 2*b*x - (1 - a^2*d)*x^2 - 2*a*d*x^3 + d*x^4)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{2/3}}{2 b^2-4 b x+2 x^2+\sqrt[3]{d} \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{2/3}}\right)}{2 d^{2/3}}+\frac{\log \left(b^2-2 b x+x^2-\sqrt[3]{d} \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{2/3}\right)}{2 d^{2/3}}-\frac{\log \left(b^4-4 b^3 x+6 b^2 x^2-4 b x^3+x^4+\left(b^2 \sqrt[3]{d}-2 b \sqrt[3]{d} x+\sqrt[3]{d} x^2\right) \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{2/3}+d^{2/3} \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{4/3}\right)}{4 d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{2/3}}{2 b^2-4 b x+2 x^2+\sqrt[3]{d} \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{2/3}}\right)}{2 d^{2/3}}+\frac{\log \left(b^2-2 b x+x^2-\sqrt[3]{d} \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{2/3}\right)}{2 d^{2/3}}-\frac{\log \left(b^4-4 b^3 x+6 b^2 x^2-4 b x^3+x^4+\left(b^2 \sqrt[3]{d}-2 b \sqrt[3]{d} x+\sqrt[3]{d} x^2\right) \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{2/3}+d^{2/3} \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{4/3}\right)}{4 d^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(2/3))/(2*b^2 - 4*b*x + 2*x^2 + d^(1/3)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(2/3))])/(2*d^(2/3)) + Log[b^2 - 2*b*x + x^2 - d^(1/3)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(2/3)]/(2*d^(2/3)) - Log[b^4 - 4*b^3*x + 6*b^2*x^2 - 4*b*x^3 + x^4 + (b^2*d^(1/3) - 2*b*d^(1/3)*x + d^(1/3)*x^2)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(2/3) + d^(2/3)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(4/3)]/(4*d^(2/3))","A",1
2929,1,341,342,2.5377747,"\int \frac{\left(-b-a x^2+x^4\right) \sqrt[4]{-b x^2+a x^4}}{b+a x^4} \, dx","IntegrateAlgebraic[((-b - a*x^2 + x^4)*(-(b*x^2) + a*x^4)^(1/4))/(b + a*x^4),x]","\frac{x \sqrt[4]{-b x^2+a x^4}}{2 a}+\frac{\left(4 a^2+b\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{4 a^{7/4}}+\frac{\left(-4 a^2-b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{4 a^{7/4}}-\frac{\text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a^3 \log (x)+a^2 b \log (x)-a^3 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)-a^2 b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)-a^2 \log (x) \text{$\#$1}^4-b \log (x) \text{$\#$1}^4-a b \log (x) \text{$\#$1}^4+a^2 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+a b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}^3+\text{$\#$1}^7}\&\right]}{4 a}","\frac{x \sqrt[4]{-b x^2+a x^4}}{2 a}+\frac{\left(4 a^2+b\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{4 a^{7/4}}+\frac{\left(-4 a^2-b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{4 a^{7/4}}-\frac{\text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a^3 \log (x)-a^2 b \log (x)+a^3 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)+a^2 b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)+a^2 \log (x) \text{$\#$1}^4+b \log (x) \text{$\#$1}^4+a b \log (x) \text{$\#$1}^4-a^2 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-a b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}^3-\text{$\#$1}^7}\&\right]}{4 a}",1,"(x*(-(b*x^2) + a*x^4)^(1/4))/(2*a) + ((4*a^2 + b)*ArcTan[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/(4*a^(7/4)) + ((-4*a^2 - b)*ArcTanh[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/(4*a^(7/4)) - RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (a^3*Log[x] + a^2*b*Log[x] - a^3*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1] - a^2*b*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1] - a^2*Log[x]*#1^4 - b*Log[x]*#1^4 - a*b*Log[x]*#1^4 + a^2*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4 + b*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4 + a*b*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3) + #1^7) & ]/(4*a)","A",1
2930,1,341,342,0.0016085,"\int \frac{\left(-b-a x^2+x^4\right) \sqrt[4]{-b x^2+a x^4}}{b+a x^4} \, dx","IntegrateAlgebraic[((-b - a*x^2 + x^4)*(-(b*x^2) + a*x^4)^(1/4))/(b + a*x^4),x]","\frac{x \sqrt[4]{-b x^2+a x^4}}{2 a}+\frac{\left(4 a^2+b\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{4 a^{7/4}}+\frac{\left(-4 a^2-b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{4 a^{7/4}}-\frac{\text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a^3 \log (x)+a^2 b \log (x)-a^3 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)-a^2 b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)-a^2 \log (x) \text{$\#$1}^4-b \log (x) \text{$\#$1}^4-a b \log (x) \text{$\#$1}^4+a^2 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+a b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}^3+\text{$\#$1}^7}\&\right]}{4 a}","\frac{x \sqrt[4]{-b x^2+a x^4}}{2 a}+\frac{\left(4 a^2+b\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{4 a^{7/4}}+\frac{\left(-4 a^2-b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{4 a^{7/4}}-\frac{\text{RootSum}\left[a^2+a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a^3 \log (x)-a^2 b \log (x)+a^3 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)+a^2 b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)+a^2 \log (x) \text{$\#$1}^4+b \log (x) \text{$\#$1}^4+a b \log (x) \text{$\#$1}^4-a^2 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-a b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}^3-\text{$\#$1}^7}\&\right]}{4 a}",1,"(x*(-(b*x^2) + a*x^4)^(1/4))/(2*a) + ((4*a^2 + b)*ArcTan[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/(4*a^(7/4)) + ((-4*a^2 - b)*ArcTanh[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/(4*a^(7/4)) - RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (a^3*Log[x] + a^2*b*Log[x] - a^3*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1] - a^2*b*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1] - a^2*Log[x]*#1^4 - b*Log[x]*#1^4 - a*b*Log[x]*#1^4 + a^2*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4 + b*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4 + a*b*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3) + #1^7) & ]/(4*a)","A",1
2931,1,347,343,0.5252001,"\int \frac{a b c-b^2 x+a^2 x^2}{\sqrt{c+b x+a x^2} \left(c+b x^2\right)} \, dx","IntegrateAlgebraic[(a*b*c - b^2*x + a^2*x^2)/(Sqrt[c + b*x + a*x^2]*(c + b*x^2)),x]","-\frac{a^{3/2} \log \left(b^2+2 a b x-2 \sqrt{a} b \sqrt{c+b x+a x^2}\right)}{b}+\frac{\text{RootSum}\left[b^2 c+b c^2-4 \sqrt{a} b c \text{$\#$1}+4 a c \text{$\#$1}^2-2 b c \text{$\#$1}^2+b \text{$\#$1}^4\&,\frac{-a^2 b c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)+b^3 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)+a b^3 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)+2 a^{5/2} c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}-2 a^{3/2} b^2 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}-b^3 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}^2}{-\sqrt{a} b c+2 a c \text{$\#$1}-b c \text{$\#$1}+b \text{$\#$1}^3}\&\right]}{2 b}","-\frac{a^{3/2} \log \left(b+2 a x-2 \sqrt{a} \sqrt{c+b x+a x^2}\right)}{b}+\frac{\text{RootSum}\left[b^2 c+b c^2-4 \sqrt{a} b c \text{$\#$1}+4 a c \text{$\#$1}^2-2 b c \text{$\#$1}^2+b \text{$\#$1}^4\&,\frac{-a^2 b c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)+b^3 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)+a b^3 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)+2 a^{5/2} c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}-2 a^{3/2} b^2 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}-b^3 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}^2}{-\sqrt{a} b c+2 a c \text{$\#$1}-b c \text{$\#$1}+b \text{$\#$1}^3}\&\right]}{2 b}",1,"-((a^(3/2)*Log[b^2 + 2*a*b*x - 2*Sqrt[a]*b*Sqrt[c + b*x + a*x^2]])/b) + RootSum[b^2*c + b*c^2 - 4*Sqrt[a]*b*c*#1 + 4*a*c*#1^2 - 2*b*c*#1^2 + b*#1^4 & , (-(a^2*b*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]) + b^3*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1] + a*b^3*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1] + 2*a^(5/2)*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 - 2*a^(3/2)*b^2*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 - b^3*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1^2)/(-(Sqrt[a]*b*c) + 2*a*c*#1 - b*c*#1 + b*#1^3) & ]/(2*b)","A",1
2932,1,342,343,2.3865846,"\int \frac{\left(b+a x^2+x^4\right) \sqrt[4]{-b x^2+a x^4}}{-b+a x^4} \, dx","IntegrateAlgebraic[((b + a*x^2 + x^4)*(-(b*x^2) + a*x^4)^(1/4))/(-b + a*x^4),x]","\frac{x \sqrt[4]{-b x^2+a x^4}}{2 a}+\frac{\left(-4 a^2+b\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{4 a^{7/4}}+\frac{\left(4 a^2-b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{4 a^{7/4}}+\frac{\text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a^3 \log (x)-a^2 b \log (x)-a^3 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)+a^2 b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)-a^2 \log (x) \text{$\#$1}^4+b \log (x) \text{$\#$1}^4+a b \log (x) \text{$\#$1}^4+a^2 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-a b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}^3+\text{$\#$1}^7}\&\right]}{4 a}","\frac{x \sqrt[4]{-b x^2+a x^4}}{2 a}+\frac{\left(-4 a^2+b\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{4 a^{7/4}}+\frac{\left(4 a^2-b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{4 a^{7/4}}+\frac{\text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a^3 \log (x)+a^2 b \log (x)+a^3 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)-a^2 b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)+a^2 \log (x) \text{$\#$1}^4-b \log (x) \text{$\#$1}^4-a b \log (x) \text{$\#$1}^4-a^2 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+a b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}^3-\text{$\#$1}^7}\&\right]}{4 a}",1,"(x*(-(b*x^2) + a*x^4)^(1/4))/(2*a) + ((-4*a^2 + b)*ArcTan[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/(4*a^(7/4)) + ((4*a^2 - b)*ArcTanh[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/(4*a^(7/4)) + RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (a^3*Log[x] - a^2*b*Log[x] - a^3*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1] + a^2*b*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1] - a^2*Log[x]*#1^4 + b*Log[x]*#1^4 + a*b*Log[x]*#1^4 + a^2*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4 - b*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4 - a*b*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3) + #1^7) & ]/(4*a)","A",1
2933,1,342,343,0.0018671,"\int \frac{\left(b+a x^2+x^4\right) \sqrt[4]{-b x^2+a x^4}}{-b+a x^4} \, dx","IntegrateAlgebraic[((b + a*x^2 + x^4)*(-(b*x^2) + a*x^4)^(1/4))/(-b + a*x^4),x]","\frac{x \sqrt[4]{-b x^2+a x^4}}{2 a}+\frac{\left(-4 a^2+b\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{4 a^{7/4}}+\frac{\left(4 a^2-b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{4 a^{7/4}}+\frac{\text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{a^3 \log (x)-a^2 b \log (x)-a^3 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)+a^2 b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)-a^2 \log (x) \text{$\#$1}^4+b \log (x) \text{$\#$1}^4+a b \log (x) \text{$\#$1}^4+a^2 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-a b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-a \text{$\#$1}^3+\text{$\#$1}^7}\&\right]}{4 a}","\frac{x \sqrt[4]{-b x^2+a x^4}}{2 a}+\frac{\left(-4 a^2+b\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{4 a^{7/4}}+\frac{\left(4 a^2-b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)}{4 a^{7/4}}+\frac{\text{RootSum}\left[a^2-a b-2 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-a^3 \log (x)+a^2 b \log (x)+a^3 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)-a^2 b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)+a^2 \log (x) \text{$\#$1}^4-b \log (x) \text{$\#$1}^4-a b \log (x) \text{$\#$1}^4-a^2 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+a b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{a \text{$\#$1}^3-\text{$\#$1}^7}\&\right]}{4 a}",1,"(x*(-(b*x^2) + a*x^4)^(1/4))/(2*a) + ((-4*a^2 + b)*ArcTan[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/(4*a^(7/4)) + ((4*a^2 - b)*ArcTanh[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/(4*a^(7/4)) + RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (a^3*Log[x] - a^2*b*Log[x] - a^3*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1] + a^2*b*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1] - a^2*Log[x]*#1^4 + b*Log[x]*#1^4 + a*b*Log[x]*#1^4 + a^2*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4 - b*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4 - a*b*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3) + #1^7) & ]/(4*a)","A",1
2934,1,343,343,3.4803213,"\int \frac{(-1+(-1+2 k) x) \left(1-2 x+x^2\right)}{\sqrt[3]{(1-x) x (1-k x)} \left(-b+4 b x+(1-6 b) x^2+(4 b-2 k) x^3+\left(-b+k^2\right) x^4\right)} \, dx","IntegrateAlgebraic[((-1 + (-1 + 2*k)*x)*(1 - 2*x + x^2))/(((1 - x)*x*(1 - k*x))^(1/3)*(-b + 4*b*x + (1 - 6*b)*x^2 + (4*b - 2*k)*x^3 + (-b + k^2)*x^4)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\frac{1}{\sqrt{3}}-\frac{2 x}{\sqrt{3}}+\frac{x^2}{\sqrt{3}}+\frac{2 \left(x+(-1-k) x^2+k x^3\right)^{2/3}}{\sqrt{3} \sqrt[3]{b}}}{(-1+x)^2}\right)}{2 b^{2/3}}-\frac{\log \left(\sqrt[6]{b}-\sqrt[6]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{2 b^{2/3}}-\frac{\log \left(-\sqrt[6]{b}+\sqrt[6]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{2 b^{2/3}}+\frac{\log \left(\sqrt[3]{b}-2 \sqrt[3]{b} x+\sqrt[3]{b} x^2+\left(\sqrt[6]{b}-\sqrt[6]{b} x\right) \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{4 b^{2/3}}+\frac{\log \left(\sqrt[3]{b}-2 \sqrt[3]{b} x+\sqrt[3]{b} x^2+\left(-\sqrt[6]{b}+\sqrt[6]{b} x\right) \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{4 b^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\frac{1}{\sqrt{3}}-\frac{2 x}{\sqrt{3}}+\frac{x^2}{\sqrt{3}}+\frac{2 \left(x+(-1-k) x^2+k x^3\right)^{2/3}}{\sqrt{3} \sqrt[3]{b}}}{(-1+x)^2}\right)}{2 b^{2/3}}-\frac{\log \left(\sqrt[6]{b}-\sqrt[6]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{2 b^{2/3}}-\frac{\log \left(-\sqrt[6]{b}+\sqrt[6]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{2 b^{2/3}}+\frac{\log \left(\sqrt[3]{b}-2 \sqrt[3]{b} x+\sqrt[3]{b} x^2+\left(\sqrt[6]{b}-\sqrt[6]{b} x\right) \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{4 b^{2/3}}+\frac{\log \left(\sqrt[3]{b}-2 \sqrt[3]{b} x+\sqrt[3]{b} x^2+\left(-\sqrt[6]{b}+\sqrt[6]{b} x\right) \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{4 b^{2/3}}",1,"(Sqrt[3]*ArcTan[(1/Sqrt[3] - (2*x)/Sqrt[3] + x^2/Sqrt[3] + (2*(x + (-1 - k)*x^2 + k*x^3)^(2/3))/(Sqrt[3]*b^(1/3)))/(-1 + x)^2])/(2*b^(2/3)) - Log[b^(1/6) - b^(1/6)*x + (x + (-1 - k)*x^2 + k*x^3)^(1/3)]/(2*b^(2/3)) - Log[-b^(1/6) + b^(1/6)*x + (x + (-1 - k)*x^2 + k*x^3)^(1/3)]/(2*b^(2/3)) + Log[b^(1/3) - 2*b^(1/3)*x + b^(1/3)*x^2 + (b^(1/6) - b^(1/6)*x)*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + (x + (-1 - k)*x^2 + k*x^3)^(2/3)]/(4*b^(2/3)) + Log[b^(1/3) - 2*b^(1/3)*x + b^(1/3)*x^2 + (-b^(1/6) + b^(1/6)*x)*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + (x + (-1 - k)*x^2 + k*x^3)^(2/3)]/(4*b^(2/3))","A",1
2935,1,345,345,1.2405836,"\int \frac{1}{(b+a x) \sqrt[4]{b^2 x+a^2 x^3}} \, dx","IntegrateAlgebraic[1/((b + a*x)*(b^2*x + a^2*x^3)^(1/4)),x]","\frac{\tan ^{-1}\left(\frac{2 \sqrt[4]{a} \sqrt[4]{b} \sqrt[4]{b^2 x+a^2 x^3}}{2^{3/4} b-2^{3/4} a x+2 \sqrt[4]{a} \sqrt[4]{b} \sqrt[4]{b^2 x+a^2 x^3}}\right)}{2\ 2^{3/4} a^{3/4} b^{3/4}}-\frac{\tan ^{-1}\left(\frac{2 \sqrt[4]{a} \sqrt[4]{b} \sqrt[4]{b^2 x+a^2 x^3}}{-2^{3/4} b+2^{3/4} a x+2 \sqrt[4]{a} \sqrt[4]{b} \sqrt[4]{b^2 x+a^2 x^3}}\right)}{2\ 2^{3/4} a^{3/4} b^{3/4}}-\frac{\tanh ^{-1}\left(\frac{-\frac{b^2}{\sqrt[4]{2}}+2^{3/4} a b x-\frac{a^2 x^2}{\sqrt[4]{2}}-2 \sqrt[4]{2} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{-2 \sqrt[4]{a} b^{5/4} \sqrt[4]{b^2 x+a^2 x^3}+2 a^{5/4} \sqrt[4]{b} x \sqrt[4]{b^2 x+a^2 x^3}}\right)}{2\ 2^{3/4} a^{3/4} b^{3/4}}","\frac{\tan ^{-1}\left(\frac{2 \sqrt[4]{a} \sqrt[4]{b} \sqrt[4]{b^2 x+a^2 x^3}}{2^{3/4} b-2^{3/4} a x+2 \sqrt[4]{a} \sqrt[4]{b} \sqrt[4]{b^2 x+a^2 x^3}}\right)}{2\ 2^{3/4} a^{3/4} b^{3/4}}-\frac{\tan ^{-1}\left(\frac{2 \sqrt[4]{a} \sqrt[4]{b} \sqrt[4]{b^2 x+a^2 x^3}}{-2^{3/4} b+2^{3/4} a x+2 \sqrt[4]{a} \sqrt[4]{b} \sqrt[4]{b^2 x+a^2 x^3}}\right)}{2\ 2^{3/4} a^{3/4} b^{3/4}}-\frac{\tanh ^{-1}\left(\frac{-\frac{b^2}{\sqrt[4]{2}}+2^{3/4} a b x-\frac{a^2 x^2}{\sqrt[4]{2}}-2 \sqrt[4]{2} \sqrt{a} \sqrt{b} \sqrt{b^2 x+a^2 x^3}}{-2 \sqrt[4]{a} b^{5/4} \sqrt[4]{b^2 x+a^2 x^3}+2 a^{5/4} \sqrt[4]{b} x \sqrt[4]{b^2 x+a^2 x^3}}\right)}{2\ 2^{3/4} a^{3/4} b^{3/4}}",1,"ArcTan[(2*a^(1/4)*b^(1/4)*(b^2*x + a^2*x^3)^(1/4))/(2^(3/4)*b - 2^(3/4)*a*x + 2*a^(1/4)*b^(1/4)*(b^2*x + a^2*x^3)^(1/4))]/(2*2^(3/4)*a^(3/4)*b^(3/4)) - ArcTan[(2*a^(1/4)*b^(1/4)*(b^2*x + a^2*x^3)^(1/4))/(-(2^(3/4)*b) + 2^(3/4)*a*x + 2*a^(1/4)*b^(1/4)*(b^2*x + a^2*x^3)^(1/4))]/(2*2^(3/4)*a^(3/4)*b^(3/4)) - ArcTanh[(-(b^2/2^(1/4)) + 2^(3/4)*a*b*x - (a^2*x^2)/2^(1/4) - 2*2^(1/4)*Sqrt[a]*Sqrt[b]*Sqrt[b^2*x + a^2*x^3])/(-2*a^(1/4)*b^(5/4)*(b^2*x + a^2*x^3)^(1/4) + 2*a^(5/4)*b^(1/4)*x*(b^2*x + a^2*x^3)^(1/4))]/(2*2^(3/4)*a^(3/4)*b^(3/4))","A",1
2936,1,346,346,3.1647082,"\int \frac{\left(-b+a x^4\right) \sqrt[4]{-b x^2+a x^4}}{-b-a x^2+x^4} \, dx","IntegrateAlgebraic[((-b + a*x^4)*(-(b*x^2) + a*x^4)^(1/4))/(-b - a*x^2 + x^4),x]","\frac{1}{2} a x \sqrt[4]{-b x^2+a x^4}+\frac{1}{4} \left(-4 a^{9/4}+\sqrt[4]{a} b\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)+\frac{1}{4} \left(4 a^{9/4}-\sqrt[4]{a} b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)+\frac{1}{2} \text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 a^4 \log (x)-a^2 b \log (x)-2 a^4 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)+a^2 b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)-a^3 \log (x) \text{$\#$1}^4-b \log (x) \text{$\#$1}^4+a b \log (x) \text{$\#$1}^4+a^3 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-a b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 a \text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]","\frac{1}{2} a x \sqrt[4]{-b x^2+a x^4}+\frac{1}{4} \left(-4 a^{9/4}+\sqrt[4]{a} b\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)+\frac{1}{4} \left(4 a^{9/4}-\sqrt[4]{a} b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)+\frac{1}{2} \text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 a^4 \log (x)+a^2 b \log (x)+2 a^4 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)-a^2 b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)+a^3 \log (x) \text{$\#$1}^4+b \log (x) \text{$\#$1}^4-a b \log (x) \text{$\#$1}^4-a^3 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+a b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{3 a \text{$\#$1}^3-2 \text{$\#$1}^7}\&\right]",1,"(a*x*(-(b*x^2) + a*x^4)^(1/4))/2 + ((-4*a^(9/4) + a^(1/4)*b)*ArcTan[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/4 + ((4*a^(9/4) - a^(1/4)*b)*ArcTanh[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/4 + RootSum[2*a^2 - b - 3*a*#1^4 + #1^8 & , (2*a^4*Log[x] - a^2*b*Log[x] - 2*a^4*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1] + a^2*b*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1] - a^3*Log[x]*#1^4 - b*Log[x]*#1^4 + a*b*Log[x]*#1^4 + a^3*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4 + b*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4 - a*b*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1^3 + 2*#1^7) & ]/2","A",1
2937,1,346,346,0.0016677,"\int \frac{\left(-b+a x^4\right) \sqrt[4]{-b x^2+a x^4}}{-b-a x^2+x^4} \, dx","IntegrateAlgebraic[((-b + a*x^4)*(-(b*x^2) + a*x^4)^(1/4))/(-b - a*x^2 + x^4),x]","\frac{1}{2} a x \sqrt[4]{-b x^2+a x^4}+\frac{1}{4} \left(-4 a^{9/4}+\sqrt[4]{a} b\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)+\frac{1}{4} \left(4 a^{9/4}-\sqrt[4]{a} b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)+\frac{1}{2} \text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 a^4 \log (x)-a^2 b \log (x)-2 a^4 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)+a^2 b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)-a^3 \log (x) \text{$\#$1}^4-b \log (x) \text{$\#$1}^4+a b \log (x) \text{$\#$1}^4+a^3 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-a b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 a \text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]","\frac{1}{2} a x \sqrt[4]{-b x^2+a x^4}+\frac{1}{4} \left(-4 a^{9/4}+\sqrt[4]{a} b\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)+\frac{1}{4} \left(4 a^{9/4}-\sqrt[4]{a} b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{-b x^2+a x^4}}\right)+\frac{1}{2} \text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 a^4 \log (x)+a^2 b \log (x)+2 a^4 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)-a^2 b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right)+a^3 \log (x) \text{$\#$1}^4+b \log (x) \text{$\#$1}^4-a b \log (x) \text{$\#$1}^4-a^3 \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+a b \log \left(\sqrt[4]{-b x^2+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{3 a \text{$\#$1}^3-2 \text{$\#$1}^7}\&\right]",1,"(a*x*(-(b*x^2) + a*x^4)^(1/4))/2 + ((-4*a^(9/4) + a^(1/4)*b)*ArcTan[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/4 + ((4*a^(9/4) - a^(1/4)*b)*ArcTanh[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)])/4 + RootSum[2*a^2 - b - 3*a*#1^4 + #1^8 & , (2*a^4*Log[x] - a^2*b*Log[x] - 2*a^4*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1] + a^2*b*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1] - a^3*Log[x]*#1^4 - b*Log[x]*#1^4 + a*b*Log[x]*#1^4 + a^3*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4 + b*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4 - a*b*Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1^3 + 2*#1^7) & ]/2","A",1
2938,1,347,347,3.191205,"\int \frac{(-b+x) (-4 a+b+3 x)}{\sqrt[3]{(-a+x) (-b+x)^2} \left(a+b^4 d-\left(1+4 b^3 d\right) x+6 b^2 d x^2-4 b d x^3+d x^4\right)} \, dx","IntegrateAlgebraic[((-b + x)*(-4*a + b + 3*x))/(((-a + x)*(-b + x)^2)^(1/3)*(a + b^4*d - (1 + 4*b^3*d)*x + 6*b^2*d*x^2 - 4*b*d*x^3 + d*x^4)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{2 b^2 \sqrt[3]{d}-4 b \sqrt[3]{d} x+2 \sqrt[3]{d} x^2+\sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}\right)}{\sqrt[3]{d}}+\frac{\log \left(b^2 \sqrt{d}-2 b \sqrt{d} x+\sqrt{d} x^2-\sqrt[6]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}\right)}{\sqrt[3]{d}}-\frac{\log \left(b^4 d-4 b^3 d x+6 b^2 d x^2-4 b d x^3+d x^4+\left(b^2 d^{2/3}-2 b d^{2/3} x+d^{2/3} x^2\right) \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}+\sqrt[3]{d} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}\right)}{2 \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{2 b^2 \sqrt[3]{d}-4 b \sqrt[3]{d} x+2 \sqrt[3]{d} x^2+\sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}\right)}{\sqrt[3]{d}}+\frac{\log \left(b^2 \sqrt{d}-2 b \sqrt{d} x+\sqrt{d} x^2-\sqrt[6]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}\right)}{\sqrt[3]{d}}-\frac{\log \left(b^4 d-4 b^3 d x+6 b^2 d x^2-4 b d x^3+d x^4+\left(b^2 d^{2/3}-2 b d^{2/3} x+d^{2/3} x^2\right) \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}+\sqrt[3]{d} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}\right)}{2 \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3))/(2*b^2*d^(1/3) - 4*b*d^(1/3)*x + 2*d^(1/3)*x^2 + (-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3))])/d^(1/3) + Log[b^2*Sqrt[d] - 2*b*Sqrt[d]*x + Sqrt[d]*x^2 - d^(1/6)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3)]/d^(1/3) - Log[b^4*d - 4*b^3*d*x + 6*b^2*d*x^2 - 4*b*d*x^3 + d*x^4 + (b^2*d^(2/3) - 2*b*d^(2/3)*x + d^(2/3)*x^2)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3) + d^(1/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(2/3)]/(2*d^(1/3))","A",1
2939,1,348,348,0.4518547,"\int \frac{\left(d+c x^2\right) \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{f+e x^2} \, dx","IntegrateAlgebraic[((d + c*x^2)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/(f + e*x^2),x]","-\frac{b^2 c}{a e \sqrt{a x+\sqrt{b^2+a^2 x^2}}}+\frac{c \left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}{3 a e}+\frac{1}{2} a d \text{RootSum}\left[b^4 e-2 b^2 e \text{$\#$1}^4+4 a^2 f \text{$\#$1}^4+e \text{$\#$1}^8\&,\frac{-b^2 \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-\log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{b^2 e \text{$\#$1}-2 a^2 f \text{$\#$1}-e \text{$\#$1}^5}\&\right]-\frac{a c f \text{RootSum}\left[b^4 e-2 b^2 e \text{$\#$1}^4+4 a^2 f \text{$\#$1}^4+e \text{$\#$1}^8\&,\frac{-b^2 \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-\log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{b^2 e \text{$\#$1}-2 a^2 f \text{$\#$1}-e \text{$\#$1}^5}\&\right]}{2 e}","-\frac{b^2 c}{a e \sqrt{a x+\sqrt{b^2+a^2 x^2}}}+\frac{c \left(a x+\sqrt{b^2+a^2 x^2}\right)^{3/2}}{3 a e}+\frac{1}{2} a d \text{RootSum}\left[b^4 e-2 b^2 e \text{$\#$1}^4+4 a^2 f \text{$\#$1}^4+e \text{$\#$1}^8\&,\frac{-b^2 \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-\log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{b^2 e \text{$\#$1}-2 a^2 f \text{$\#$1}-e \text{$\#$1}^5}\&\right]-\frac{a c f \text{RootSum}\left[b^4 e-2 b^2 e \text{$\#$1}^4+4 a^2 f \text{$\#$1}^4+e \text{$\#$1}^8\&,\frac{-b^2 \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-\log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^4}{b^2 e \text{$\#$1}-2 a^2 f \text{$\#$1}-e \text{$\#$1}^5}\&\right]}{2 e}",1,"-((b^2*c)/(a*e*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])) + (c*(a*x + Sqrt[b^2 + a^2*x^2])^(3/2))/(3*a*e) + (a*d*RootSum[b^4*e - 2*b^2*e*#1^4 + 4*a^2*f*#1^4 + e*#1^8 & , (-(b^2*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]) - Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]*#1^4)/(b^2*e*#1 - 2*a^2*f*#1 - e*#1^5) & ])/2 - (a*c*f*RootSum[b^4*e - 2*b^2*e*#1^4 + 4*a^2*f*#1^4 + e*#1^8 & , (-(b^2*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]) - Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]*#1^4)/(b^2*e*#1 - 2*a^2*f*#1 - e*#1^5) & ])/(2*e)","A",1
2940,1,359,349,0.6066173,"\int \frac{\left(1+x^2\right) \sqrt{x^2+\sqrt{1+x^4}}}{\sqrt{1+x^4} \left(-1+x^2+x^4\right)} \, dx","IntegrateAlgebraic[((1 + x^2)*Sqrt[x^2 + Sqrt[1 + x^4]])/(Sqrt[1 + x^4]*(-1 + x^2 + x^4)),x]","\frac{\text{RootSum}\left[1-2 \text{$\#$1}^2-6 \text{$\#$1}^4+2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(1+x^2+\sqrt{1+x^4}\right)-\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right)+3 \log \left(1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^2-3 \log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^2+\log \left(1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^4-\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^4-\log \left(1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^6+\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^6}{-\text{$\#$1}-6 \text{$\#$1}^3+3 \text{$\#$1}^5+2 \text{$\#$1}^7}\&\right]}{2 \sqrt{2}}","-\frac{\text{RootSum}\left[1+2 \text{$\#$1}^2-6 \text{$\#$1}^4-2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(-1+x^2+\sqrt{1+x^4}\right)-\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}+x^2 \text{$\#$1}+\sqrt{1+x^4} \text{$\#$1}\right)-\log \left(-1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^2+\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}+x^2 \text{$\#$1}+\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^2-3 \log \left(-1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^4+3 \log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}+x^2 \text{$\#$1}+\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^4-\log \left(-1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^6+\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}+x^2 \text{$\#$1}+\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^6}{\text{$\#$1}-6 \text{$\#$1}^3-3 \text{$\#$1}^5+2 \text{$\#$1}^7}\&\right]}{2 \sqrt{2}}",1,"RootSum[1 - 2*#1^2 - 6*#1^4 + 2*#1^6 + #1^8 & , (Log[1 + x^2 + Sqrt[1 + x^4]] - Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1] + 3*Log[1 + x^2 + Sqrt[1 + x^4]]*#1^2 - 3*Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^2 + Log[1 + x^2 + Sqrt[1 + x^4]]*#1^4 - Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^4 - Log[1 + x^2 + Sqrt[1 + x^4]]*#1^6 + Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^6)/(-#1 - 6*#1^3 + 3*#1^5 + 2*#1^7) & ]/(2*Sqrt[2])","A",1
2941,1,359,349,0.0014491,"\int \frac{\left(1+x^2\right) \sqrt{x^2+\sqrt{1+x^4}}}{\sqrt{1+x^4} \left(-1+x^2+x^4\right)} \, dx","IntegrateAlgebraic[((1 + x^2)*Sqrt[x^2 + Sqrt[1 + x^4]])/(Sqrt[1 + x^4]*(-1 + x^2 + x^4)),x]","\frac{\text{RootSum}\left[1-2 \text{$\#$1}^2-6 \text{$\#$1}^4+2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(1+x^2+\sqrt{1+x^4}\right)-\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right)+3 \log \left(1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^2-3 \log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^2+\log \left(1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^4-\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^4-\log \left(1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^6+\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^6}{-\text{$\#$1}-6 \text{$\#$1}^3+3 \text{$\#$1}^5+2 \text{$\#$1}^7}\&\right]}{2 \sqrt{2}}","-\frac{\text{RootSum}\left[1+2 \text{$\#$1}^2-6 \text{$\#$1}^4-2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(-1+x^2+\sqrt{1+x^4}\right)-\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}+x^2 \text{$\#$1}+\sqrt{1+x^4} \text{$\#$1}\right)-\log \left(-1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^2+\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}+x^2 \text{$\#$1}+\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^2-3 \log \left(-1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^4+3 \log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}+x^2 \text{$\#$1}+\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^4-\log \left(-1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^6+\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}+x^2 \text{$\#$1}+\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^6}{\text{$\#$1}-6 \text{$\#$1}^3-3 \text{$\#$1}^5+2 \text{$\#$1}^7}\&\right]}{2 \sqrt{2}}",1,"RootSum[1 - 2*#1^2 - 6*#1^4 + 2*#1^6 + #1^8 & , (Log[1 + x^2 + Sqrt[1 + x^4]] - Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1] + 3*Log[1 + x^2 + Sqrt[1 + x^4]]*#1^2 - 3*Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^2 + Log[1 + x^2 + Sqrt[1 + x^4]]*#1^4 - Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^4 - Log[1 + x^2 + Sqrt[1 + x^4]]*#1^6 + Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^6)/(-#1 - 6*#1^3 + 3*#1^5 + 2*#1^7) & ]/(2*Sqrt[2])","A",1
2942,1,377,351,3.5615323,"\int \frac{1-x^4}{\left(1+x^2+x^4\right) \sqrt[4]{-x^3+x^5}} \, dx","IntegrateAlgebraic[(1 - x^4)/((1 + x^2 + x^4)*(-x^3 + x^5)^(1/4)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} 3^{7/8} x \sqrt[4]{-x^3+x^5}}{\sqrt[8]{17+12 \sqrt{2}} \left(-3 x^2+3^{3/4} \sqrt{-x^3+x^5}\right)}\right)}{2^{3/4} \sqrt[8]{3 \left(17+12 \sqrt{2}\right)}}+\frac{\sqrt[8]{\frac{1}{3} \left(17+12 \sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt[4]{2} 3^{7/8} \sqrt[8]{17+12 \sqrt{2}} x \sqrt[4]{-x^3+x^5}}{-3 x^2+3^{3/4} \sqrt{-x^3+x^5}}\right)}{2^{3/4}}+\frac{\tanh ^{-1}\left(\frac{3 \sqrt[8]{\frac{17}{8748}+\frac{\sqrt{2}}{729}} x^2+3^{3/4} \sqrt[8]{\frac{17}{8748}+\frac{\sqrt{2}}{729}} \sqrt{-x^3+x^5}}{x \sqrt[4]{-x^3+x^5}}\right)}{2^{3/4} \sqrt[8]{3 \left(17+12 \sqrt{2}\right)}}+\frac{\sqrt[8]{\frac{1}{3} \left(17+12 \sqrt{2}\right)} \tanh ^{-1}\left(\frac{\frac{\sqrt[8]{\frac{3}{17+12 \sqrt{2}}} x^2}{\sqrt[4]{2}}+\frac{\sqrt{-x^3+x^5}}{\sqrt[4]{2} \sqrt[8]{3 \left(17+12 \sqrt{2}\right)}}}{x \sqrt[4]{-x^3+x^5}}\right)}{2^{3/4}}","\frac{\tan ^{-1}\left(\frac{3^{7/8} \sqrt{2-\sqrt{2}} x \sqrt[4]{-x^3+x^5}}{-3 x^2+3^{3/4} \sqrt{-x^3+x^5}}\right)}{2^{3/4} \sqrt[8]{3 \left(17+12 \sqrt{2}\right)}}+\frac{\sqrt[8]{\frac{1}{3} \left(17+12 \sqrt{2}\right)} \tan ^{-1}\left(\frac{3^{7/8} \sqrt{2+\sqrt{2}} x \sqrt[4]{-x^3+x^5}}{-3 x^2+3^{3/4} \sqrt{-x^3+x^5}}\right)}{2^{3/4}}+\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[8]{3} x^2}{\sqrt{2-\sqrt{2}}}+\frac{\sqrt{-x^3+x^5}}{\sqrt[8]{3} \sqrt{2-\sqrt{2}}}}{x \sqrt[4]{-x^3+x^5}}\right)}{2^{3/4} \sqrt[8]{3 \left(17+12 \sqrt{2}\right)}}+\frac{\sqrt[8]{\frac{1}{3} \left(17+12 \sqrt{2}\right)} \tanh ^{-1}\left(\frac{\frac{\sqrt[8]{3} x^2}{\sqrt{2+\sqrt{2}}}+\frac{\sqrt{-x^3+x^5}}{\sqrt[8]{3} \sqrt{2+\sqrt{2}}}}{x \sqrt[4]{-x^3+x^5}}\right)}{2^{3/4}}",1,"ArcTan[(2^(1/4)*3^(7/8)*x*(-x^3 + x^5)^(1/4))/((17 + 12*Sqrt[2])^(1/8)*(-3*x^2 + 3^(3/4)*Sqrt[-x^3 + x^5]))]/(2^(3/4)*(3*(17 + 12*Sqrt[2]))^(1/8)) + (((17 + 12*Sqrt[2])/3)^(1/8)*ArcTan[(2^(1/4)*3^(7/8)*(17 + 12*Sqrt[2])^(1/8)*x*(-x^3 + x^5)^(1/4))/(-3*x^2 + 3^(3/4)*Sqrt[-x^3 + x^5])])/2^(3/4) + ArcTanh[(3*(17/8748 + Sqrt[2]/729)^(1/8)*x^2 + 3^(3/4)*(17/8748 + Sqrt[2]/729)^(1/8)*Sqrt[-x^3 + x^5])/(x*(-x^3 + x^5)^(1/4))]/(2^(3/4)*(3*(17 + 12*Sqrt[2]))^(1/8)) + (((17 + 12*Sqrt[2])/3)^(1/8)*ArcTanh[(((3/(17 + 12*Sqrt[2]))^(1/8)*x^2)/2^(1/4) + Sqrt[-x^3 + x^5]/(2^(1/4)*(3*(17 + 12*Sqrt[2]))^(1/8)))/(x*(-x^3 + x^5)^(1/4))])/2^(3/4)","A",1
2943,1,351,351,3.4975691,"\int \frac{\sqrt{-b+a^2 x^2}}{\sqrt[3]{a x^2+x \sqrt{-b+a^2 x^2}}} \, dx","IntegrateAlgebraic[Sqrt[-b + a^2*x^2]/(a*x^2 + x*Sqrt[-b + a^2*x^2])^(1/3),x]","\frac{\left(5 b-4 a^2 x^2\right) \left(a x^2+x \sqrt{-b+a^2 x^2}\right)^{2/3}}{8 b}+\frac{a x \sqrt{-b+a^2 x^2} \left(a x^2+x \sqrt{-b+a^2 x^2}\right)^{2/3}}{2 b}-\frac{7 b^{2/3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{2} \sqrt[3]{a} \sqrt[3]{a x^2+x \sqrt{-b+a^2 x^2}}}{\sqrt{3} \sqrt[3]{b}}\right)}{4\ 2^{2/3} \sqrt{3} a^{2/3}}-\frac{7 b^{2/3} \log \left(-1+\frac{\sqrt[3]{2} \sqrt[3]{a} \sqrt[3]{a x^2+x \sqrt{-b+a^2 x^2}}}{\sqrt[3]{b}}\right)}{12\ 2^{2/3} a^{2/3}}+\frac{7 b^{2/3} \log \left(1+\frac{\sqrt[3]{2} \sqrt[3]{a} \sqrt[3]{a x^2+x \sqrt{-b+a^2 x^2}}}{\sqrt[3]{b}}+\frac{2^{2/3} a^{2/3} \left(a x^2+x \sqrt{-b+a^2 x^2}\right)^{2/3}}{b^{2/3}}\right)}{24\ 2^{2/3} a^{2/3}}","\frac{\left(5 b-4 a^2 x^2\right) \left(a x^2+x \sqrt{-b+a^2 x^2}\right)^{2/3}}{8 b}+\frac{a x \sqrt{-b+a^2 x^2} \left(a x^2+x \sqrt{-b+a^2 x^2}\right)^{2/3}}{2 b}-\frac{7 b^{2/3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{2} \sqrt[3]{a} \sqrt[3]{a x^2+x \sqrt{-b+a^2 x^2}}}{\sqrt{3} \sqrt[3]{b}}\right)}{4\ 2^{2/3} \sqrt{3} a^{2/3}}-\frac{7 b^{2/3} \log \left(-1+\frac{\sqrt[3]{2} \sqrt[3]{a} \sqrt[3]{a x^2+x \sqrt{-b+a^2 x^2}}}{\sqrt[3]{b}}\right)}{12\ 2^{2/3} a^{2/3}}+\frac{7 b^{2/3} \log \left(1+\frac{\sqrt[3]{2} \sqrt[3]{a} \sqrt[3]{a x^2+x \sqrt{-b+a^2 x^2}}}{\sqrt[3]{b}}+\frac{2^{2/3} a^{2/3} \left(a x^2+x \sqrt{-b+a^2 x^2}\right)^{2/3}}{b^{2/3}}\right)}{24\ 2^{2/3} a^{2/3}}",1,"((5*b - 4*a^2*x^2)*(a*x^2 + x*Sqrt[-b + a^2*x^2])^(2/3))/(8*b) + (a*x*Sqrt[-b + a^2*x^2]*(a*x^2 + x*Sqrt[-b + a^2*x^2])^(2/3))/(2*b) - (7*b^(2/3)*ArcTan[1/Sqrt[3] + (2*2^(1/3)*a^(1/3)*(a*x^2 + x*Sqrt[-b + a^2*x^2])^(1/3))/(Sqrt[3]*b^(1/3))])/(4*2^(2/3)*Sqrt[3]*a^(2/3)) - (7*b^(2/3)*Log[-1 + (2^(1/3)*a^(1/3)*(a*x^2 + x*Sqrt[-b + a^2*x^2])^(1/3))/b^(1/3)])/(12*2^(2/3)*a^(2/3)) + (7*b^(2/3)*Log[1 + (2^(1/3)*a^(1/3)*(a*x^2 + x*Sqrt[-b + a^2*x^2])^(1/3))/b^(1/3) + (2^(2/3)*a^(2/3)*(a*x^2 + x*Sqrt[-b + a^2*x^2])^(2/3))/b^(2/3)])/(24*2^(2/3)*a^(2/3))","A",1
2944,1,1255,351,36.0216572,"\int \frac{\left(b+a^2 x^2\right) \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\left(-b+a^2 x^2\right) \sqrt{b+a^2 x^4}} \, dx","IntegrateAlgebraic[((b + a^2*x^2)*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/((-b + a^2*x^2)*Sqrt[b + a^2*x^4]),x]","-\frac{\log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}-i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{\sqrt{2} \sqrt{a}}+2 \sqrt{2} a^{3/2} b^4 \text{RootSum}\left[a^5 b^4+4 a^4 b^3 \text{$\#$1}^2+16 a^2 b^4 \text{$\#$1}^2+6 a^3 b^2 \text{$\#$1}^4-32 a b^3 \text{$\#$1}^4+4 a^2 b \text{$\#$1}^6+16 b^2 \text{$\#$1}^6+a \text{$\#$1}^8\&,\frac{\log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}-i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\text{$\#$1}\right)}{a^4 b^3+4 a^2 b^4+3 a^3 b^2 \text{$\#$1}^2-16 a b^3 \text{$\#$1}^2+3 a^2 b \text{$\#$1}^4+12 b^2 \text{$\#$1}^4+a \text{$\#$1}^6}\&\right]-4 \sqrt{2} \sqrt{a} b^3 \text{RootSum}\left[a^5 b^4+4 a^4 b^3 \text{$\#$1}^2+16 a^2 b^4 \text{$\#$1}^2+6 a^3 b^2 \text{$\#$1}^4-32 a b^3 \text{$\#$1}^4+4 a^2 b \text{$\#$1}^6+16 b^2 \text{$\#$1}^6+a \text{$\#$1}^8\&,\frac{\log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}-i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\text{$\#$1}\right) \text{$\#$1}^2}{a^4 b^3+4 a^2 b^4+3 a^3 b^2 \text{$\#$1}^2-16 a b^3 \text{$\#$1}^2+3 a^2 b \text{$\#$1}^4+12 b^2 \text{$\#$1}^4+a \text{$\#$1}^6}\&\right]+\frac{2 \sqrt{2} b^2 \text{RootSum}\left[a^5 b^4+4 a^4 b^3 \text{$\#$1}^2+16 a^2 b^4 \text{$\#$1}^2+6 a^3 b^2 \text{$\#$1}^4-32 a b^3 \text{$\#$1}^4+4 a^2 b \text{$\#$1}^6+16 b^2 \text{$\#$1}^6+a \text{$\#$1}^8\&,\frac{\log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}-i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\text{$\#$1}\right) \text{$\#$1}^4}{a^4 b^3+4 a^2 b^4+3 a^3 b^2 \text{$\#$1}^2-16 a b^3 \text{$\#$1}^2+3 a^2 b \text{$\#$1}^4+12 b^2 \text{$\#$1}^4+a \text{$\#$1}^6}\&\right]}{\sqrt{a}}+\frac{i b \text{RootSum}\left[a^5 b^4+4 a^4 b^3 \text{$\#$1}^2+16 a^2 b^4 \text{$\#$1}^2+6 a^3 b^2 \text{$\#$1}^4-32 a b^3 \text{$\#$1}^4+4 a^2 b \text{$\#$1}^6+16 b^2 \text{$\#$1}^6+a \text{$\#$1}^8\&,\frac{-a^3 b^3 \log \left(-i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+i \sqrt{a} \left(a x^2+\sqrt{b+a^2 x^4}\right)+\text{$\#$1}\right)-a^2 b^2 \log \left(-i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+i \sqrt{a} \left(a x^2+\sqrt{b+a^2 x^4}\right)+\text{$\#$1}\right) \text{$\#$1}^2+a b \log \left(-i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+i \sqrt{a} \left(a x^2+\sqrt{b+a^2 x^4}\right)+\text{$\#$1}\right) \text{$\#$1}^4+\log \left(-i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+i \sqrt{a} \left(a x^2+\sqrt{b+a^2 x^4}\right)+\text{$\#$1}\right) \text{$\#$1}^6}{a^4 b^3 \text{$\#$1}+4 a^2 b^4 \text{$\#$1}+3 a^3 b^2 \text{$\#$1}^3-16 a b^3 \text{$\#$1}^3+3 a^2 b \text{$\#$1}^5+12 b^2 \text{$\#$1}^5+a \text{$\#$1}^7}\&\right]}{\sqrt{2}}","\frac{\log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}+i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{\sqrt{2} \sqrt{a}}-\frac{i \sqrt{2} \text{RootSum}\left[a^{5/2} b^2+4 i a b^2 \text{$\#$1}+2 a^{3/2} b \text{$\#$1}^2-4 i b \text{$\#$1}^3+\sqrt{a} \text{$\#$1}^4\&,\frac{a b^2 \log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}+i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)-b \log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}+i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2}{i a b^2+a^{3/2} b \text{$\#$1}-3 i b \text{$\#$1}^2+\sqrt{a} \text{$\#$1}^3}\&\right]}{\sqrt{a}}",1,"-(Log[I*a^(3/2)*x^2 + I*Sqrt[a]*Sqrt[b + a^2*x^4] - I*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]]/(Sqrt[2]*Sqrt[a])) + 2*Sqrt[2]*a^(3/2)*b^4*RootSum[a^5*b^4 + 4*a^4*b^3*#1^2 + 16*a^2*b^4*#1^2 + 6*a^3*b^2*#1^4 - 32*a*b^3*#1^4 + 4*a^2*b*#1^6 + 16*b^2*#1^6 + a*#1^8 & , Log[I*a^(3/2)*x^2 + I*Sqrt[a]*Sqrt[b + a^2*x^4] - I*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + #1]/(a^4*b^3 + 4*a^2*b^4 + 3*a^3*b^2*#1^2 - 16*a*b^3*#1^2 + 3*a^2*b*#1^4 + 12*b^2*#1^4 + a*#1^6) & ] - 4*Sqrt[2]*Sqrt[a]*b^3*RootSum[a^5*b^4 + 4*a^4*b^3*#1^2 + 16*a^2*b^4*#1^2 + 6*a^3*b^2*#1^4 - 32*a*b^3*#1^4 + 4*a^2*b*#1^6 + 16*b^2*#1^6 + a*#1^8 & , (Log[I*a^(3/2)*x^2 + I*Sqrt[a]*Sqrt[b + a^2*x^4] - I*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + #1]*#1^2)/(a^4*b^3 + 4*a^2*b^4 + 3*a^3*b^2*#1^2 - 16*a*b^3*#1^2 + 3*a^2*b*#1^4 + 12*b^2*#1^4 + a*#1^6) & ] + (2*Sqrt[2]*b^2*RootSum[a^5*b^4 + 4*a^4*b^3*#1^2 + 16*a^2*b^4*#1^2 + 6*a^3*b^2*#1^4 - 32*a*b^3*#1^4 + 4*a^2*b*#1^6 + 16*b^2*#1^6 + a*#1^8 & , (Log[I*a^(3/2)*x^2 + I*Sqrt[a]*Sqrt[b + a^2*x^4] - I*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + #1]*#1^4)/(a^4*b^3 + 4*a^2*b^4 + 3*a^3*b^2*#1^2 - 16*a*b^3*#1^2 + 3*a^2*b*#1^4 + 12*b^2*#1^4 + a*#1^6) & ])/Sqrt[a] + (I*b*RootSum[a^5*b^4 + 4*a^4*b^3*#1^2 + 16*a^2*b^4*#1^2 + 6*a^3*b^2*#1^4 - 32*a*b^3*#1^4 + 4*a^2*b*#1^6 + 16*b^2*#1^6 + a*#1^8 & , (-(a^3*b^3*Log[(-I)*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + I*Sqrt[a]*(a*x^2 + Sqrt[b + a^2*x^4]) + #1]) - a^2*b^2*Log[(-I)*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + I*Sqrt[a]*(a*x^2 + Sqrt[b + a^2*x^4]) + #1]*#1^2 + a*b*Log[(-I)*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + I*Sqrt[a]*(a*x^2 + Sqrt[b + a^2*x^4]) + #1]*#1^4 + Log[(-I)*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + I*Sqrt[a]*(a*x^2 + Sqrt[b + a^2*x^4]) + #1]*#1^6)/(a^4*b^3*#1 + 4*a^2*b^4*#1 + 3*a^3*b^2*#1^3 - 16*a*b^3*#1^3 + 3*a^2*b*#1^5 + 12*b^2*#1^5 + a*#1^7) & ])/Sqrt[2]","B",1
2945,1,1255,351,0.0025868,"\int \frac{\left(b+a^2 x^2\right) \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\left(-b+a^2 x^2\right) \sqrt{b+a^2 x^4}} \, dx","IntegrateAlgebraic[((b + a^2*x^2)*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/((-b + a^2*x^2)*Sqrt[b + a^2*x^4]),x]","-\frac{\log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}-i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{\sqrt{2} \sqrt{a}}+2 \sqrt{2} a^{3/2} b^4 \text{RootSum}\left[a^5 b^4+4 a^4 b^3 \text{$\#$1}^2+16 a^2 b^4 \text{$\#$1}^2+6 a^3 b^2 \text{$\#$1}^4-32 a b^3 \text{$\#$1}^4+4 a^2 b \text{$\#$1}^6+16 b^2 \text{$\#$1}^6+a \text{$\#$1}^8\&,\frac{\log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}-i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\text{$\#$1}\right)}{a^4 b^3+4 a^2 b^4+3 a^3 b^2 \text{$\#$1}^2-16 a b^3 \text{$\#$1}^2+3 a^2 b \text{$\#$1}^4+12 b^2 \text{$\#$1}^4+a \text{$\#$1}^6}\&\right]-4 \sqrt{2} \sqrt{a} b^3 \text{RootSum}\left[a^5 b^4+4 a^4 b^3 \text{$\#$1}^2+16 a^2 b^4 \text{$\#$1}^2+6 a^3 b^2 \text{$\#$1}^4-32 a b^3 \text{$\#$1}^4+4 a^2 b \text{$\#$1}^6+16 b^2 \text{$\#$1}^6+a \text{$\#$1}^8\&,\frac{\log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}-i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\text{$\#$1}\right) \text{$\#$1}^2}{a^4 b^3+4 a^2 b^4+3 a^3 b^2 \text{$\#$1}^2-16 a b^3 \text{$\#$1}^2+3 a^2 b \text{$\#$1}^4+12 b^2 \text{$\#$1}^4+a \text{$\#$1}^6}\&\right]+\frac{2 \sqrt{2} b^2 \text{RootSum}\left[a^5 b^4+4 a^4 b^3 \text{$\#$1}^2+16 a^2 b^4 \text{$\#$1}^2+6 a^3 b^2 \text{$\#$1}^4-32 a b^3 \text{$\#$1}^4+4 a^2 b \text{$\#$1}^6+16 b^2 \text{$\#$1}^6+a \text{$\#$1}^8\&,\frac{\log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}-i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\text{$\#$1}\right) \text{$\#$1}^4}{a^4 b^3+4 a^2 b^4+3 a^3 b^2 \text{$\#$1}^2-16 a b^3 \text{$\#$1}^2+3 a^2 b \text{$\#$1}^4+12 b^2 \text{$\#$1}^4+a \text{$\#$1}^6}\&\right]}{\sqrt{a}}+\frac{i b \text{RootSum}\left[a^5 b^4+4 a^4 b^3 \text{$\#$1}^2+16 a^2 b^4 \text{$\#$1}^2+6 a^3 b^2 \text{$\#$1}^4-32 a b^3 \text{$\#$1}^4+4 a^2 b \text{$\#$1}^6+16 b^2 \text{$\#$1}^6+a \text{$\#$1}^8\&,\frac{-a^3 b^3 \log \left(-i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+i \sqrt{a} \left(a x^2+\sqrt{b+a^2 x^4}\right)+\text{$\#$1}\right)-a^2 b^2 \log \left(-i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+i \sqrt{a} \left(a x^2+\sqrt{b+a^2 x^4}\right)+\text{$\#$1}\right) \text{$\#$1}^2+a b \log \left(-i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+i \sqrt{a} \left(a x^2+\sqrt{b+a^2 x^4}\right)+\text{$\#$1}\right) \text{$\#$1}^4+\log \left(-i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+i \sqrt{a} \left(a x^2+\sqrt{b+a^2 x^4}\right)+\text{$\#$1}\right) \text{$\#$1}^6}{a^4 b^3 \text{$\#$1}+4 a^2 b^4 \text{$\#$1}+3 a^3 b^2 \text{$\#$1}^3-16 a b^3 \text{$\#$1}^3+3 a^2 b \text{$\#$1}^5+12 b^2 \text{$\#$1}^5+a \text{$\#$1}^7}\&\right]}{\sqrt{2}}","\frac{\log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}+i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{\sqrt{2} \sqrt{a}}-\frac{i \sqrt{2} \text{RootSum}\left[a^{5/2} b^2+4 i a b^2 \text{$\#$1}+2 a^{3/2} b \text{$\#$1}^2-4 i b \text{$\#$1}^3+\sqrt{a} \text{$\#$1}^4\&,\frac{a b^2 \log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}+i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)-b \log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}+i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2}{i a b^2+a^{3/2} b \text{$\#$1}-3 i b \text{$\#$1}^2+\sqrt{a} \text{$\#$1}^3}\&\right]}{\sqrt{a}}",1,"-(Log[I*a^(3/2)*x^2 + I*Sqrt[a]*Sqrt[b + a^2*x^4] - I*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]]/(Sqrt[2]*Sqrt[a])) + 2*Sqrt[2]*a^(3/2)*b^4*RootSum[a^5*b^4 + 4*a^4*b^3*#1^2 + 16*a^2*b^4*#1^2 + 6*a^3*b^2*#1^4 - 32*a*b^3*#1^4 + 4*a^2*b*#1^6 + 16*b^2*#1^6 + a*#1^8 & , Log[I*a^(3/2)*x^2 + I*Sqrt[a]*Sqrt[b + a^2*x^4] - I*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + #1]/(a^4*b^3 + 4*a^2*b^4 + 3*a^3*b^2*#1^2 - 16*a*b^3*#1^2 + 3*a^2*b*#1^4 + 12*b^2*#1^4 + a*#1^6) & ] - 4*Sqrt[2]*Sqrt[a]*b^3*RootSum[a^5*b^4 + 4*a^4*b^3*#1^2 + 16*a^2*b^4*#1^2 + 6*a^3*b^2*#1^4 - 32*a*b^3*#1^4 + 4*a^2*b*#1^6 + 16*b^2*#1^6 + a*#1^8 & , (Log[I*a^(3/2)*x^2 + I*Sqrt[a]*Sqrt[b + a^2*x^4] - I*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + #1]*#1^2)/(a^4*b^3 + 4*a^2*b^4 + 3*a^3*b^2*#1^2 - 16*a*b^3*#1^2 + 3*a^2*b*#1^4 + 12*b^2*#1^4 + a*#1^6) & ] + (2*Sqrt[2]*b^2*RootSum[a^5*b^4 + 4*a^4*b^3*#1^2 + 16*a^2*b^4*#1^2 + 6*a^3*b^2*#1^4 - 32*a*b^3*#1^4 + 4*a^2*b*#1^6 + 16*b^2*#1^6 + a*#1^8 & , (Log[I*a^(3/2)*x^2 + I*Sqrt[a]*Sqrt[b + a^2*x^4] - I*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + #1]*#1^4)/(a^4*b^3 + 4*a^2*b^4 + 3*a^3*b^2*#1^2 - 16*a*b^3*#1^2 + 3*a^2*b*#1^4 + 12*b^2*#1^4 + a*#1^6) & ])/Sqrt[a] + (I*b*RootSum[a^5*b^4 + 4*a^4*b^3*#1^2 + 16*a^2*b^4*#1^2 + 6*a^3*b^2*#1^4 - 32*a*b^3*#1^4 + 4*a^2*b*#1^6 + 16*b^2*#1^6 + a*#1^8 & , (-(a^3*b^3*Log[(-I)*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + I*Sqrt[a]*(a*x^2 + Sqrt[b + a^2*x^4]) + #1]) - a^2*b^2*Log[(-I)*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + I*Sqrt[a]*(a*x^2 + Sqrt[b + a^2*x^4]) + #1]*#1^2 + a*b*Log[(-I)*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + I*Sqrt[a]*(a*x^2 + Sqrt[b + a^2*x^4]) + #1]*#1^4 + Log[(-I)*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + I*Sqrt[a]*(a*x^2 + Sqrt[b + a^2*x^4]) + #1]*#1^6)/(a^4*b^3*#1 + 4*a^2*b^4*#1 + 3*a^3*b^2*#1^3 - 16*a*b^3*#1^3 + 3*a^2*b*#1^5 + 12*b^2*#1^5 + a*#1^7) & ])/Sqrt[2]","B",1
2946,1,352,352,4.7357169,"\int \frac{x^3}{\sqrt{a+b x+c x^2+b x^3+a x^4} \left(1-x^6\right)} \, dx","IntegrateAlgebraic[x^3/(Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]*(1 - x^6)),x]","\frac{\sqrt{-2 a-2 b-c} \tan ^{-1}\left(\frac{\sqrt{-2 a-2 b-c} x}{\sqrt{a}-2 \sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{6 (2 a+2 b+c)}+\frac{\tan ^{-1}\left(\frac{\sqrt{a-b-c} x}{\sqrt{a}-\sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{3 \sqrt{a-b-c}}-\frac{\tan ^{-1}\left(\frac{\sqrt{a+b-c} x}{\sqrt{a}+\sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{3 \sqrt{a+b-c}}-\frac{\sqrt{-2 a+2 b-c} \tan ^{-1}\left(\frac{\sqrt{-2 a+2 b-c} x}{\sqrt{a}+2 \sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{6 (2 a-2 b+c)}","\frac{\sqrt{-2 a-2 b-c} \tan ^{-1}\left(\frac{\sqrt{-2 a-2 b-c} x}{\sqrt{a}-2 \sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{6 (2 a+2 b+c)}+\frac{\tan ^{-1}\left(\frac{\sqrt{a-b-c} x}{\sqrt{a}-\sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{3 \sqrt{a-b-c}}-\frac{\tan ^{-1}\left(\frac{\sqrt{a+b-c} x}{\sqrt{a}+\sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{3 \sqrt{a+b-c}}-\frac{\sqrt{-2 a+2 b-c} \tan ^{-1}\left(\frac{\sqrt{-2 a+2 b-c} x}{\sqrt{a}+2 \sqrt{a} x+\sqrt{a} x^2-\sqrt{a+b x+c x^2+b x^3+a x^4}}\right)}{6 (2 a-2 b+c)}",1,"(Sqrt[-2*a - 2*b - c]*ArcTan[(Sqrt[-2*a - 2*b - c]*x)/(Sqrt[a] - 2*Sqrt[a]*x + Sqrt[a]*x^2 - Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])])/(6*(2*a + 2*b + c)) + ArcTan[(Sqrt[a - b - c]*x)/(Sqrt[a] - Sqrt[a]*x + Sqrt[a]*x^2 - Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])]/(3*Sqrt[a - b - c]) - ArcTan[(Sqrt[a + b - c]*x)/(Sqrt[a] + Sqrt[a]*x + Sqrt[a]*x^2 - Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])]/(3*Sqrt[a + b - c]) - (Sqrt[-2*a + 2*b - c]*ArcTan[(Sqrt[-2*a + 2*b - c]*x)/(Sqrt[a] + 2*Sqrt[a]*x + Sqrt[a]*x^2 - Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])])/(6*(2*a - 2*b + c))","A",1
2947,1,352,352,0.5309577,"\int \frac{\left(-q+2 p x^3\right) \sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left(b x^6+a \left(q+p x^3\right)^6\right)}{x^9} \, dx","IntegrateAlgebraic[((-q + 2*p*x^3)*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]*(b*x^6 + a*(q + p*x^3)^6))/x^9,x]","\frac{\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left(6 a q^7-2 a p q^6 x^2+42 a p q^6 x^3-5 a p^2 q^5 x^4-10 a p^2 q^5 x^5+24 b q x^6-15 a p^3 q^4 x^6+126 a p^2 q^5 x^6-15 a p^3 q^4 x^7-20 a p^3 q^4 x^8+24 b p x^9-15 a p^4 q^3 x^9+210 a p^3 q^4 x^9-15 a p^4 q^3 x^{10}-20 a p^4 q^3 x^{11}+210 a p^4 q^3 x^{12}-5 a p^5 q^2 x^{13}-10 a p^5 q^2 x^{14}+126 a p^5 q^2 x^{15}-2 a p^6 q x^{17}+42 a p^6 q x^{18}+6 a p^7 x^{21}\right)}{48 x^8}+\frac{1}{8} \left(8 b p q+5 a p^4 q^4\right) \log (x)+\frac{1}{8} \left(-8 b p q-5 a p^4 q^4\right) \log \left(q+p x^3+\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right)","\frac{\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left(6 a q^7-2 a p q^6 x^2+42 a p q^6 x^3-5 a p^2 q^5 x^4-10 a p^2 q^5 x^5+24 b q x^6-15 a p^3 q^4 x^6+126 a p^2 q^5 x^6-15 a p^3 q^4 x^7-20 a p^3 q^4 x^8+24 b p x^9-15 a p^4 q^3 x^9+210 a p^3 q^4 x^9-15 a p^4 q^3 x^{10}-20 a p^4 q^3 x^{11}+210 a p^4 q^3 x^{12}-5 a p^5 q^2 x^{13}-10 a p^5 q^2 x^{14}+126 a p^5 q^2 x^{15}-2 a p^6 q x^{17}+42 a p^6 q x^{18}+6 a p^7 x^{21}\right)}{48 x^8}+\frac{1}{8} \left(8 b p q+5 a p^4 q^4\right) \log (x)+\frac{1}{8} \left(-8 b p q-5 a p^4 q^4\right) \log \left(q+p x^3+\sqrt{q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right)",1,"(Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]*(6*a*q^7 - 2*a*p*q^6*x^2 + 42*a*p*q^6*x^3 - 5*a*p^2*q^5*x^4 - 10*a*p^2*q^5*x^5 + 24*b*q*x^6 - 15*a*p^3*q^4*x^6 + 126*a*p^2*q^5*x^6 - 15*a*p^3*q^4*x^7 - 20*a*p^3*q^4*x^8 + 24*b*p*x^9 - 15*a*p^4*q^3*x^9 + 210*a*p^3*q^4*x^9 - 15*a*p^4*q^3*x^10 - 20*a*p^4*q^3*x^11 + 210*a*p^4*q^3*x^12 - 5*a*p^5*q^2*x^13 - 10*a*p^5*q^2*x^14 + 126*a*p^5*q^2*x^15 - 2*a*p^6*q*x^17 + 42*a*p^6*q*x^18 + 6*a*p^7*x^21))/(48*x^8) + ((8*b*p*q + 5*a*p^4*q^4)*Log[x])/8 + ((-8*b*p*q - 5*a*p^4*q^4)*Log[q + p*x^3 + Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]])/8","A",1
2948,1,352,352,0.5777465,"\int \frac{\left(-2 q+p x^3\right) \sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left(b x^{12}+a \left(q+p x^3\right)^6\right)}{x^{17}} \, dx","IntegrateAlgebraic[((-2*q + p*x^3)*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]*(b*x^12 + a*(q + p*x^3)^6))/x^17,x]","\frac{\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left(6 a q^7+42 a p q^6 x^3-2 a p q^6 x^4+126 a p^2 q^5 x^6-10 a p^2 q^5 x^7-5 a p^2 q^5 x^8+210 a p^3 q^4 x^9-20 a p^3 q^4 x^{10}-15 a p^3 q^4 x^{11}+24 b q x^{12}+210 a p^4 q^3 x^{12}-15 a p^3 q^4 x^{12}-20 a p^4 q^3 x^{13}-15 a p^4 q^3 x^{14}+24 b p x^{15}+126 a p^5 q^2 x^{15}-15 a p^4 q^3 x^{15}-10 a p^5 q^2 x^{16}-5 a p^5 q^2 x^{17}+42 a p^6 q x^{18}-2 a p^6 q x^{19}+6 a p^7 x^{21}\right)}{48 x^{16}}+\frac{1}{4} \left(8 b p q+5 a p^4 q^4\right) \log (x)+\frac{1}{8} \left(-8 b p q-5 a p^4 q^4\right) \log \left(q+p x^3+\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}\right)","\frac{\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left(6 a q^7+42 a p q^6 x^3-2 a p q^6 x^4+126 a p^2 q^5 x^6-10 a p^2 q^5 x^7-5 a p^2 q^5 x^8+210 a p^3 q^4 x^9-20 a p^3 q^4 x^{10}-15 a p^3 q^4 x^{11}+24 b q x^{12}+210 a p^4 q^3 x^{12}-15 a p^3 q^4 x^{12}-20 a p^4 q^3 x^{13}-15 a p^4 q^3 x^{14}+24 b p x^{15}+126 a p^5 q^2 x^{15}-15 a p^4 q^3 x^{15}-10 a p^5 q^2 x^{16}-5 a p^5 q^2 x^{17}+42 a p^6 q x^{18}-2 a p^6 q x^{19}+6 a p^7 x^{21}\right)}{48 x^{16}}+\frac{1}{4} \left(8 b p q+5 a p^4 q^4\right) \log (x)+\frac{1}{8} \left(-8 b p q-5 a p^4 q^4\right) \log \left(q+p x^3+\sqrt{q^2+2 p q x^3-2 p q x^4+p^2 x^6}\right)",1,"(Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]*(6*a*q^7 + 42*a*p*q^6*x^3 - 2*a*p*q^6*x^4 + 126*a*p^2*q^5*x^6 - 10*a*p^2*q^5*x^7 - 5*a*p^2*q^5*x^8 + 210*a*p^3*q^4*x^9 - 20*a*p^3*q^4*x^10 - 15*a*p^3*q^4*x^11 + 24*b*q*x^12 + 210*a*p^4*q^3*x^12 - 15*a*p^3*q^4*x^12 - 20*a*p^4*q^3*x^13 - 15*a*p^4*q^3*x^14 + 24*b*p*x^15 + 126*a*p^5*q^2*x^15 - 15*a*p^4*q^3*x^15 - 10*a*p^5*q^2*x^16 - 5*a*p^5*q^2*x^17 + 42*a*p^6*q*x^18 - 2*a*p^6*q*x^19 + 6*a*p^7*x^21))/(48*x^16) + ((8*b*p*q + 5*a*p^4*q^4)*Log[x])/4 + ((-8*b*p*q - 5*a*p^4*q^4)*Log[q + p*x^3 + Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]])/8","A",1
2949,1,171,352,1.631913,"\int \frac{\left(-q+p x^4\right) \sqrt{q+p x^4}}{b x^4+a \left(q+p x^4\right)^2} \, dx","IntegrateAlgebraic[((-q + p*x^4)*Sqrt[q + p*x^4])/(b*x^4 + a*(q + p*x^4)^2),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} x \sqrt{q+p x^4}}{\sqrt{a} q-\sqrt{b} x^2+\sqrt{a} p x^4}\right)}{2 \sqrt{2} a^{3/4} \sqrt[4]{b}}-\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[4]{a} q}{\sqrt{2} \sqrt[4]{b}}+\frac{\sqrt[4]{b} x^2}{\sqrt{2} \sqrt[4]{a}}+\frac{\sqrt[4]{a} p x^4}{\sqrt{2} \sqrt[4]{b}}}{x \sqrt{q+p x^4}}\right)}{2 \sqrt{2} a^{3/4} \sqrt[4]{b}}","\frac{1}{4} \text{RootSum}\left[16 a p^2 q^2-32 a p^{3/2} q^{3/2} \text{$\#$1}^2+16 b \text{$\#$1}^4+24 a p q \text{$\#$1}^4-8 a \sqrt{p} \sqrt{q} \text{$\#$1}^6+a \text{$\#$1}^8\&,\frac{8 p^{3/2} q^{3/2} \log (x)-8 p^{3/2} q^{3/2} \log \left(\sqrt{q}+\sqrt{p} x^2+\sqrt{q+p x^4}-x \text{$\#$1}\right)-4 p q \log (x) \text{$\#$1}^2+4 p q \log \left(\sqrt{q}+\sqrt{p} x^2+\sqrt{q+p x^4}-x \text{$\#$1}\right) \text{$\#$1}^2-2 \sqrt{p} \sqrt{q} \log (x) \text{$\#$1}^4+2 \sqrt{p} \sqrt{q} \log \left(\sqrt{q}+\sqrt{p} x^2+\sqrt{q+p x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+\log (x) \text{$\#$1}^6-\log \left(\sqrt{q}+\sqrt{p} x^2+\sqrt{q+p x^4}-x \text{$\#$1}\right) \text{$\#$1}^6}{8 a p^{3/2} q^{3/2} \text{$\#$1}-8 b \text{$\#$1}^3-12 a p q \text{$\#$1}^3+6 a \sqrt{p} \sqrt{q} \text{$\#$1}^5-a \text{$\#$1}^7}\&\right]",1,"-1/2*ArcTan[(Sqrt[2]*a^(1/4)*b^(1/4)*x*Sqrt[q + p*x^4])/(Sqrt[a]*q - Sqrt[b]*x^2 + Sqrt[a]*p*x^4)]/(Sqrt[2]*a^(3/4)*b^(1/4)) - ArcTanh[((a^(1/4)*q)/(Sqrt[2]*b^(1/4)) + (b^(1/4)*x^2)/(Sqrt[2]*a^(1/4)) + (a^(1/4)*p*x^4)/(Sqrt[2]*b^(1/4)))/(x*Sqrt[q + p*x^4])]/(2*Sqrt[2]*a^(3/4)*b^(1/4))","A",1
2950,1,354,354,0.5923624,"\int \frac{-3 b+a x}{\sqrt[3]{b^2-a^2 x^2} \left(3 b^2+a^2 x^2\right)} \, dx","IntegrateAlgebraic[(-3*b + a*x)/((b^2 - a^2*x^2)^(1/3)*(3*b^2 + a^2*x^2)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} \sqrt[3]{b^2-a^2 x^2}}{-2^{2/3} b-2^{2/3} a x+\sqrt[3]{b} \sqrt[3]{b^2-a^2 x^2}}\right)}{2^{2/3} a b^{2/3}}+\frac{\log \left(\sqrt[3]{b} \sqrt[3]{b^2-a^2 x^2}\right)}{2^{2/3} a b^{2/3}}-\frac{\log \left(b^{2/3} \left(b^2-a^2 x^2\right)^{2/3}\right)}{2\ 2^{2/3} a b^{2/3}}-\frac{\log \left(2^{2/3} b+2^{2/3} a x+2 \sqrt[3]{b} \sqrt[3]{b^2-a^2 x^2}\right)}{2^{2/3} a b^{2/3}}+\frac{\log \left(\sqrt[3]{2} b^2+2 \sqrt[3]{2} a b x+\sqrt[3]{2} a^2 x^2-2^{2/3} b^{4/3} \sqrt[3]{b^2-a^2 x^2}-2^{2/3} a \sqrt[3]{b} x \sqrt[3]{b^2-a^2 x^2}+2 b^{2/3} \left(b^2-a^2 x^2\right)^{2/3}\right)}{2\ 2^{2/3} a b^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b} \sqrt[3]{b^2-a^2 x^2}}{-2^{2/3} b-2^{2/3} a x+\sqrt[3]{b} \sqrt[3]{b^2-a^2 x^2}}\right)}{2^{2/3} a b^{2/3}}+\frac{\log \left(\sqrt[3]{b} \sqrt[3]{b^2-a^2 x^2}\right)}{2^{2/3} a b^{2/3}}-\frac{\log \left(b^{2/3} \left(b^2-a^2 x^2\right)^{2/3}\right)}{2\ 2^{2/3} a b^{2/3}}-\frac{\log \left(2^{2/3} b+2^{2/3} a x+2 \sqrt[3]{b} \sqrt[3]{b^2-a^2 x^2}\right)}{2^{2/3} a b^{2/3}}+\frac{\log \left(\sqrt[3]{2} b^2+2 \sqrt[3]{2} a b x+\sqrt[3]{2} a^2 x^2-2^{2/3} b^{4/3} \sqrt[3]{b^2-a^2 x^2}-2^{2/3} a \sqrt[3]{b} x \sqrt[3]{b^2-a^2 x^2}+2 b^{2/3} \left(b^2-a^2 x^2\right)^{2/3}\right)}{2\ 2^{2/3} a b^{2/3}}",1,"-((Sqrt[3]*ArcTan[(Sqrt[3]*b^(1/3)*(b^2 - a^2*x^2)^(1/3))/(-(2^(2/3)*b) - 2^(2/3)*a*x + b^(1/3)*(b^2 - a^2*x^2)^(1/3))])/(2^(2/3)*a*b^(2/3))) + Log[b^(1/3)*(b^2 - a^2*x^2)^(1/3)]/(2^(2/3)*a*b^(2/3)) - Log[b^(2/3)*(b^2 - a^2*x^2)^(2/3)]/(2*2^(2/3)*a*b^(2/3)) - Log[2^(2/3)*b + 2^(2/3)*a*x + 2*b^(1/3)*(b^2 - a^2*x^2)^(1/3)]/(2^(2/3)*a*b^(2/3)) + Log[2^(1/3)*b^2 + 2*2^(1/3)*a*b*x + 2^(1/3)*a^2*x^2 - 2^(2/3)*b^(4/3)*(b^2 - a^2*x^2)^(1/3) - 2^(2/3)*a*b^(1/3)*x*(b^2 - a^2*x^2)^(1/3) + 2*b^(2/3)*(b^2 - a^2*x^2)^(2/3)]/(2*2^(2/3)*a*b^(2/3))","A",1
2951,1,354,354,0.2449563,"\int \frac{x+\sqrt{1+4 x+7 x^2+8 x^3+5 x^4+2 x^5}}{1-\sqrt{1+4 x+7 x^2+8 x^3+5 x^4+2 x^5}} \, dx","IntegrateAlgebraic[(x + Sqrt[1 + 4*x + 7*x^2 + 8*x^3 + 5*x^4 + 2*x^5])/(1 - Sqrt[1 + 4*x + 7*x^2 + 8*x^3 + 5*x^4 + 2*x^5]),x]","-x+\frac{1}{2} \log \left(1+x+x^2\right)-\frac{1}{2} \log \left(-1-x-x^2+\sqrt{1+4 x+7 x^2+8 x^3+5 x^4+2 x^5}\right)+\frac{1}{2} \text{RootSum}\left[4+\text{$\#$1}+\text{$\#$1}^2+\text{$\#$1}^3+\text{$\#$1}^4\&,\frac{4 \log \left(1+x+x^2\right)-4 \log \left(\sqrt{1+4 x+7 x^2+8 x^3+5 x^4+2 x^5}-\text{$\#$1}-x \text{$\#$1}-x^2 \text{$\#$1}\right)+\log \left(1+x+x^2\right) \text{$\#$1}-\log \left(\sqrt{1+4 x+7 x^2+8 x^3+5 x^4+2 x^5}-\text{$\#$1}-x \text{$\#$1}-x^2 \text{$\#$1}\right) \text{$\#$1}+2 \log \left(1+x+x^2\right) \text{$\#$1}^2-2 \log \left(\sqrt{1+4 x+7 x^2+8 x^3+5 x^4+2 x^5}-\text{$\#$1}-x \text{$\#$1}-x^2 \text{$\#$1}\right) \text{$\#$1}^2-\log \left(1+x+x^2\right) \text{$\#$1}^3+\log \left(\sqrt{1+4 x+7 x^2+8 x^3+5 x^4+2 x^5}-\text{$\#$1}-x \text{$\#$1}-x^2 \text{$\#$1}\right) \text{$\#$1}^3}{1+2 \text{$\#$1}+3 \text{$\#$1}^2+4 \text{$\#$1}^3}\&\right]","-x+\frac{1}{2} \log \left(1+x+x^2\right)-\frac{1}{2} \log \left(-1-x-x^2+\sqrt{1+4 x+7 x^2+8 x^3+5 x^4+2 x^5}\right)+\frac{1}{2} \text{RootSum}\left[4+\text{$\#$1}+\text{$\#$1}^2+\text{$\#$1}^3+\text{$\#$1}^4\&,\frac{4 \log \left(1+x+x^2\right)-4 \log \left(\sqrt{1+4 x+7 x^2+8 x^3+5 x^4+2 x^5}-\text{$\#$1}-x \text{$\#$1}-x^2 \text{$\#$1}\right)+\log \left(1+x+x^2\right) \text{$\#$1}-\log \left(\sqrt{1+4 x+7 x^2+8 x^3+5 x^4+2 x^5}-\text{$\#$1}-x \text{$\#$1}-x^2 \text{$\#$1}\right) \text{$\#$1}+2 \log \left(1+x+x^2\right) \text{$\#$1}^2-2 \log \left(\sqrt{1+4 x+7 x^2+8 x^3+5 x^4+2 x^5}-\text{$\#$1}-x \text{$\#$1}-x^2 \text{$\#$1}\right) \text{$\#$1}^2-\log \left(1+x+x^2\right) \text{$\#$1}^3+\log \left(\sqrt{1+4 x+7 x^2+8 x^3+5 x^4+2 x^5}-\text{$\#$1}-x \text{$\#$1}-x^2 \text{$\#$1}\right) \text{$\#$1}^3}{1+2 \text{$\#$1}+3 \text{$\#$1}^2+4 \text{$\#$1}^3}\&\right]",1,"-x + Log[1 + x + x^2]/2 - Log[-1 - x - x^2 + Sqrt[1 + 4*x + 7*x^2 + 8*x^3 + 5*x^4 + 2*x^5]]/2 + RootSum[4 + #1 + #1^2 + #1^3 + #1^4 & , (4*Log[1 + x + x^2] - 4*Log[Sqrt[1 + 4*x + 7*x^2 + 8*x^3 + 5*x^4 + 2*x^5] - #1 - x*#1 - x^2*#1] + Log[1 + x + x^2]*#1 - Log[Sqrt[1 + 4*x + 7*x^2 + 8*x^3 + 5*x^4 + 2*x^5] - #1 - x*#1 - x^2*#1]*#1 + 2*Log[1 + x + x^2]*#1^2 - 2*Log[Sqrt[1 + 4*x + 7*x^2 + 8*x^3 + 5*x^4 + 2*x^5] - #1 - x*#1 - x^2*#1]*#1^2 - Log[1 + x + x^2]*#1^3 + Log[Sqrt[1 + 4*x + 7*x^2 + 8*x^3 + 5*x^4 + 2*x^5] - #1 - x*#1 - x^2*#1]*#1^3)/(1 + 2*#1 + 3*#1^2 + 4*#1^3) & ]/2","A",1
2952,1,355,355,1.227773,"\int \frac{x^3 \left(4 a b-3 (a+b) x+2 x^2\right)}{\left(x^2 (-a+x) (-b+x)\right)^{2/3} \left(-a b d+(a+b) d x-d x^2+x^4\right)} \, dx","IntegrateAlgebraic[(x^3*(4*a*b - 3*(a + b)*x + 2*x^2))/((x^2*(-a + x)*(-b + x))^(2/3)*(-(a*b*d) + (a + b)*d*x - d*x^2 + x^4)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}}{2 a b \sqrt[3]{d}+\left(-2 a \sqrt[3]{d}-2 b \sqrt[3]{d}\right) x+2 \sqrt[3]{d} x^2+\left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}}\right)}{\sqrt[3]{d}}+\frac{\log \left(-a b \sqrt{d}+a \sqrt{d} x+b \sqrt{d} x-\sqrt{d} x^2+\sqrt[6]{d} \left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}\right)}{\sqrt[3]{d}}-\frac{\log \left(a^2 b^2 d-2 a^2 b d x-2 a b^2 d x+a^2 d x^2+4 a b d x^2+b^2 d x^2-2 a d x^3-2 b d x^3+d x^4+\left(a b d^{2/3}-a d^{2/3} x-b d^{2/3} x+d^{2/3} x^2\right) \left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}+\sqrt[3]{d} \left(a b x^2+(-a-b) x^3+x^4\right)^{4/3}\right)}{2 \sqrt[3]{d}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}}{2 a b \sqrt[3]{d}+\left(-2 a \sqrt[3]{d}-2 b \sqrt[3]{d}\right) x+2 \sqrt[3]{d} x^2+\left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}}\right)}{\sqrt[3]{d}}+\frac{\log \left(-a b \sqrt{d}+a \sqrt{d} x+b \sqrt{d} x-\sqrt{d} x^2+\sqrt[6]{d} \left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}\right)}{\sqrt[3]{d}}-\frac{\log \left(a^2 b^2 d-2 a^2 b d x-2 a b^2 d x+a^2 d x^2+4 a b d x^2+b^2 d x^2-2 a d x^3-2 b d x^3+d x^4+\left(a b d^{2/3}-a d^{2/3} x-b d^{2/3} x+d^{2/3} x^2\right) \left(a b x^2+(-a-b) x^3+x^4\right)^{2/3}+\sqrt[3]{d} \left(a b x^2+(-a-b) x^3+x^4\right)^{4/3}\right)}{2 \sqrt[3]{d}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*(a*b*x^2 + (-a - b)*x^3 + x^4)^(2/3))/(2*a*b*d^(1/3) + (-2*a*d^(1/3) - 2*b*d^(1/3))*x + 2*d^(1/3)*x^2 + (a*b*x^2 + (-a - b)*x^3 + x^4)^(2/3))])/d^(1/3) + Log[-(a*b*Sqrt[d]) + a*Sqrt[d]*x + b*Sqrt[d]*x - Sqrt[d]*x^2 + d^(1/6)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(2/3)]/d^(1/3) - Log[a^2*b^2*d - 2*a^2*b*d*x - 2*a*b^2*d*x + a^2*d*x^2 + 4*a*b*d*x^2 + b^2*d*x^2 - 2*a*d*x^3 - 2*b*d*x^3 + d*x^4 + (a*b*d^(2/3) - a*d^(2/3)*x - b*d^(2/3)*x + d^(2/3)*x^2)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(2/3) + d^(1/3)*(a*b*x^2 + (-a - b)*x^3 + x^4)^(4/3)]/(2*d^(1/3))","A",1
2953,1,360,356,0.9341802,"\int \frac{x^3}{\sqrt[3]{x^2+x^4} \left(-1+x^6\right)} \, dx","IntegrateAlgebraic[x^3/((x^2 + x^4)^(1/3)*(-1 + x^6)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}}\right)}{4 \sqrt[3]{2} \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{\frac{x^2}{\sqrt{3}}+\frac{2 \left(x^2+x^4\right)^{2/3}}{\sqrt{3}}}{x^2}\right)}{2 \sqrt{3}}-\frac{1}{6} \log \left(-x+\sqrt[3]{x^2+x^4}\right)-\frac{1}{6} \log \left(x+\sqrt[3]{x^2+x^4}\right)+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x^2+x^4}\right)}{12 \sqrt[3]{2}}+\frac{\log \left(2 x+2^{2/3} \sqrt[3]{x^2+x^4}\right)}{12 \sqrt[3]{2}}+\frac{1}{12} \log \left(x^2-x \sqrt[3]{x^2+x^4}+\left(x^2+x^4\right)^{2/3}\right)+\frac{1}{12} \log \left(x^2+x \sqrt[3]{x^2+x^4}+\left(x^2+x^4\right)^{2/3}\right)-\frac{\log \left(-2 x^2+2^{2/3} x \sqrt[3]{x^2+x^4}-\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}\right)}{24 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x^2+x^4}+\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}\right)}{24 \sqrt[3]{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+2 \left(x^2+x^4\right)^{2/3}}\right)}{2 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{\sqrt{3} x^2}{x^2+\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}}\right)}{4 \sqrt[3]{2} \sqrt{3}}-\frac{1}{6} \log \left(-x+\sqrt[3]{x^2+x^4}\right)-\frac{1}{6} \log \left(x+\sqrt[3]{x^2+x^4}\right)+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x^2+x^4}\right)}{12 \sqrt[3]{2}}+\frac{\log \left(2 x+2^{2/3} \sqrt[3]{x^2+x^4}\right)}{12 \sqrt[3]{2}}+\frac{1}{12} \log \left(x^2-x \sqrt[3]{x^2+x^4}+\left(x^2+x^4\right)^{2/3}\right)+\frac{1}{12} \log \left(x^2+x \sqrt[3]{x^2+x^4}+\left(x^2+x^4\right)^{2/3}\right)-\frac{\log \left(-2 x^2+2^{2/3} x \sqrt[3]{x^2+x^4}-\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}\right)}{24 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x^2+x^4}+\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}\right)}{24 \sqrt[3]{2}}",1,"ArcTan[(Sqrt[3]*x^2)/(x^2 + 2^(1/3)*(x^2 + x^4)^(2/3))]/(4*2^(1/3)*Sqrt[3]) + ArcTan[(x^2/Sqrt[3] + (2*(x^2 + x^4)^(2/3))/Sqrt[3])/x^2]/(2*Sqrt[3]) - Log[-x + (x^2 + x^4)^(1/3)]/6 - Log[x + (x^2 + x^4)^(1/3)]/6 + Log[-2*x + 2^(2/3)*(x^2 + x^4)^(1/3)]/(12*2^(1/3)) + Log[2*x + 2^(2/3)*(x^2 + x^4)^(1/3)]/(12*2^(1/3)) + Log[x^2 - x*(x^2 + x^4)^(1/3) + (x^2 + x^4)^(2/3)]/12 + Log[x^2 + x*(x^2 + x^4)^(1/3) + (x^2 + x^4)^(2/3)]/12 - Log[-2*x^2 + 2^(2/3)*x*(x^2 + x^4)^(1/3) - 2^(1/3)*(x^2 + x^4)^(2/3)]/(24*2^(1/3)) - Log[2*x^2 + 2^(2/3)*x*(x^2 + x^4)^(1/3) + 2^(1/3)*(x^2 + x^4)^(2/3)]/(24*2^(1/3))","A",1
2954,1,640,357,0.6073186,"\int \frac{\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1-x^2\right)^2 \sqrt{1+x^2}} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((1 - x^2)^2*Sqrt[1 + x^2]),x]","\frac{\left(-1-3 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-3 x \sqrt{1+x^2} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{8 x \left(-1+x^2\right) \sqrt{1+x^2}+4 \left(-1+x^2\right) \left(1+2 x^2\right)}-\frac{1}{4} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{-\text{$\#$1}^3+\text{$\#$1}^5}\&\right]-\frac{1}{32} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{14 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{1}{4} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{1}{32} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-16 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","\frac{\left(-1-3 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-3 x \sqrt{1+x^2} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{8 x \left(-1+x^2\right) \sqrt{1+x^2}+4 \left(-1+x^2\right) \left(1+2 x^2\right)}-\frac{1}{32} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-10 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+11 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{1}{32} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-10 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}+11 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^3}{-2+4 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]",1,"((-1 - 3*x^2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - 3*x*Sqrt[1 + x^2]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(8*x*(-1 + x^2)*Sqrt[1 + x^2] + 4*(-1 + x^2)*(1 + 2*x^2)) - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2)/(-#1^3 + #1^5) & ]/4 - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (14*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1^3 - 3*#1^5 + #1^7) & ]/32 + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ]/4 + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-16*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ]/32","A",1
2955,1,640,357,0.0016234,"\int \frac{\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1-x^2\right)^2 \sqrt{1+x^2}} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((1 - x^2)^2*Sqrt[1 + x^2]),x]","\frac{\left(-1-3 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-3 x \sqrt{1+x^2} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{8 x \left(-1+x^2\right) \sqrt{1+x^2}+4 \left(-1+x^2\right) \left(1+2 x^2\right)}-\frac{1}{4} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{-\text{$\#$1}^3+\text{$\#$1}^5}\&\right]-\frac{1}{32} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{14 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{1}{4} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{1}{32} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-16 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","\frac{\left(-1-3 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-3 x \sqrt{1+x^2} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{8 x \left(-1+x^2\right) \sqrt{1+x^2}+4 \left(-1+x^2\right) \left(1+2 x^2\right)}-\frac{1}{32} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-10 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+11 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{1}{32} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-10 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}+11 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^3}{-2+4 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]",1,"((-1 - 3*x^2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - 3*x*Sqrt[1 + x^2]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(8*x*(-1 + x^2)*Sqrt[1 + x^2] + 4*(-1 + x^2)*(1 + 2*x^2)) - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2)/(-#1^3 + #1^5) & ]/4 - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (14*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1^3 - 3*#1^5 + #1^7) & ]/32 + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ]/4 + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-16*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ]/32","A",1
2956,1,537,357,0.5030358,"\int \frac{\sqrt{1+x^2} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1-x^2\right)^2} \, dx","IntegrateAlgebraic[(Sqrt[1 + x^2]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^2)^2,x]","\frac{\left(-1-3 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-3 x \sqrt{1+x^2} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{4 x \left(-1+x^2\right) \sqrt{1+x^2}+2 \left(-1+x^2\right) \left(1+2 x^2\right)}+\text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-\frac{1}{16} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{14 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{1}{16} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-16 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","\frac{\left(-1-3 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-3 x \sqrt{1+x^2} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{4 x \left(-1+x^2\right) \sqrt{1+x^2}+2 \left(-1+x^2\right) \left(1+2 x^2\right)}-\frac{1}{16} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{1}{16} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^3}{-2+4 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]",1,"((-1 - 3*x^2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - 3*x*Sqrt[1 + x^2]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(4*x*(-1 + x^2)*Sqrt[1 + x^2] + 2*(-1 + x^2)*(1 + 2*x^2)) + RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/(2*#1^3 - 3*#1^5 + #1^7) & ] - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (14*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1^3 - 3*#1^5 + #1^7) & ]/16 + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ] + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-16*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ]/16","A",1
2957,1,537,357,0.0020477,"\int \frac{\sqrt{1+x^2} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1-x^2\right)^2} \, dx","IntegrateAlgebraic[(Sqrt[1 + x^2]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^2)^2,x]","\frac{\left(-1-3 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-3 x \sqrt{1+x^2} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{4 x \left(-1+x^2\right) \sqrt{1+x^2}+2 \left(-1+x^2\right) \left(1+2 x^2\right)}+\text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-\frac{1}{16} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{14 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{1}{16} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-16 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","\frac{\left(-1-3 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-3 x \sqrt{1+x^2} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{4 x \left(-1+x^2\right) \sqrt{1+x^2}+2 \left(-1+x^2\right) \left(1+2 x^2\right)}-\frac{1}{16} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{1}{16} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^3}{-2+4 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]",1,"((-1 - 3*x^2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - 3*x*Sqrt[1 + x^2]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(4*x*(-1 + x^2)*Sqrt[1 + x^2] + 2*(-1 + x^2)*(1 + 2*x^2)) + RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/(2*#1^3 - 3*#1^5 + #1^7) & ] - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (14*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1^3 - 3*#1^5 + #1^7) & ]/16 + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ] + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-16*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ]/16","A",1
2958,1,537,357,0.0016122,"\int \frac{\sqrt{1+x^2} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1-x^2\right)^2} \, dx","IntegrateAlgebraic[(Sqrt[1 + x^2]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^2)^2,x]","\frac{\left(-1-3 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-3 x \sqrt{1+x^2} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{4 x \left(-1+x^2\right) \sqrt{1+x^2}+2 \left(-1+x^2\right) \left(1+2 x^2\right)}+\text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-\frac{1}{16} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{14 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{1}{16} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-16 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","\frac{\left(-1-3 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-3 x \sqrt{1+x^2} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{4 x \left(-1+x^2\right) \sqrt{1+x^2}+2 \left(-1+x^2\right) \left(1+2 x^2\right)}-\frac{1}{16} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{1}{16} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^3}{-2+4 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]",1,"((-1 - 3*x^2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - 3*x*Sqrt[1 + x^2]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(4*x*(-1 + x^2)*Sqrt[1 + x^2] + 2*(-1 + x^2)*(1 + 2*x^2)) + RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/(2*#1^3 - 3*#1^5 + #1^7) & ] - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (14*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1^3 - 3*#1^5 + #1^7) & ]/16 + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ] + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-16*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ]/16","A",1
2959,1,366,362,1.8427158,"\int \frac{\left(c+b x+a x^2\right)^{5/2}}{(c+b x)^2} \, dx","IntegrateAlgebraic[(c + b*x + a*x^2)^(5/2)/(c + b*x)^2,x]","\frac{\sqrt{c+b x+a x^2} \left(15 b^6 c+28 a b^4 c^2+400 a^2 b^2 c^3-960 a^3 c^4+15 b^7 x+146 a b^5 c x+200 a^2 b^3 c^2 x-480 a^3 b c^3 x+118 a b^6 x^2-64 a^2 b^4 c x^2+160 a^3 b^2 c^2 x^2+136 a^2 b^5 x^3-80 a^3 b^3 c x^3+48 a^3 b^4 x^4\right)}{192 a b^5 (c+b x)}+\frac{5 \left(-a^{3/2} b^2 c^3+2 a^{5/2} c^4\right) \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}\right)}{2 b^6}-\frac{5 \left(-b^8+8 a b^6 c-64 a^3 b^2 c^3+128 a^4 c^4\right) \log \left(a b+2 a^2 x-2 a^{3/2} \sqrt{c+b x+a x^2}\right)}{128 a^{3/2} b^6}-\frac{5 \left(-a^{3/2} b^2 c^3+2 a^{5/2} c^4\right) \log \left(\sqrt{a} (2 c+b x)-b \sqrt{c+b x+a x^2}\right)}{2 b^6}","\frac{\sqrt{c+b x+a x^2} \left(15 b^6 c+28 a b^4 c^2+400 a^2 b^2 c^3-960 a^3 c^4+15 b^7 x+146 a b^5 c x+200 a^2 b^3 c^2 x-480 a^3 b c^3 x+118 a b^6 x^2-64 a^2 b^4 c x^2+160 a^3 b^2 c^2 x^2+136 a^2 b^5 x^3-80 a^3 b^3 c x^3+48 a^3 b^4 x^4\right)}{192 a b^5 (c+b x)}+\frac{5 \left(-a^{3/2} b^2 c^3+2 a^{5/2} c^4\right) \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}\right)}{2 b^6}-\frac{5 \left(-b^8+8 a b^6 c-64 a^3 b^2 c^3+128 a^4 c^4\right) \log \left(b+2 a x-2 \sqrt{a} \sqrt{c+b x+a x^2}\right)}{128 a^{3/2} b^6}-\frac{5 \left(-a^{3/2} b^2 c^3+2 a^{5/2} c^4\right) \log \left(\sqrt{a} (2 c+b x)-b \sqrt{c+b x+a x^2}\right)}{2 b^6}",1,"(Sqrt[c + b*x + a*x^2]*(15*b^6*c + 28*a*b^4*c^2 + 400*a^2*b^2*c^3 - 960*a^3*c^4 + 15*b^7*x + 146*a*b^5*c*x + 200*a^2*b^3*c^2*x - 480*a^3*b*c^3*x + 118*a*b^6*x^2 - 64*a^2*b^4*c*x^2 + 160*a^3*b^2*c^2*x^2 + 136*a^2*b^5*x^3 - 80*a^3*b^3*c*x^3 + 48*a^3*b^4*x^4))/(192*a*b^5*(c + b*x)) + (5*(-(a^(3/2)*b^2*c^3) + 2*a^(5/2)*c^4)*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2]])/(2*b^6) - (5*(-b^8 + 8*a*b^6*c - 64*a^3*b^2*c^3 + 128*a^4*c^4)*Log[a*b + 2*a^2*x - 2*a^(3/2)*Sqrt[c + b*x + a*x^2]])/(128*a^(3/2)*b^6) - (5*(-(a^(3/2)*b^2*c^3) + 2*a^(5/2)*c^4)*Log[Sqrt[a]*(2*c + b*x) - b*Sqrt[c + b*x + a*x^2]])/(2*b^6)","A",1
2960,1,483,362,6.8523563,"\int \frac{\left(1+x^4\right)^2}{\left(-1+x^4\right)^2 \sqrt{x^2+\sqrt{1+x^4}}} \, dx","IntegrateAlgebraic[(1 + x^4)^2/((-1 + x^4)^2*Sqrt[x^2 + Sqrt[1 + x^4]]),x]","\frac{x \left(-3+x^4\right)}{2 \left(-1+x^4\right) \sqrt{x^2+\sqrt{1+x^4}}}+\frac{1}{2} \sqrt{\frac{7}{2}+\frac{5}{\sqrt{2}}} \tan ^{-1}\left(\frac{-\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)-\frac{1}{2} \sqrt{\frac{7}{2}+\frac{5}{\sqrt{2}}} \tan ^{-1}\left(\frac{-\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)+\frac{\tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^4}}{\sqrt{2}}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)}{\sqrt{2}}-\frac{1}{2} \sqrt{-\frac{7}{2}+\frac{5}{\sqrt{2}}} \tanh ^{-1}\left(\frac{-\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)-\frac{1}{2} \sqrt{-\frac{7}{2}+\frac{5}{\sqrt{2}}} \tanh ^{-1}\left(\frac{-\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}}+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} x^2+\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{1+x^4}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)","\frac{x \left(-3+x^4\right)}{2 \left(-1+x^4\right) \sqrt{x^2+\sqrt{1+x^4}}}+\frac{1}{2} \sqrt{\frac{1}{2} \left(7+5 \sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(-1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{2} \left(7+5 \sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)}{\sqrt{2}}-\frac{1}{2} \sqrt{\frac{1}{2} \left(-7+5 \sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2 \left(-1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-\frac{1}{2} \sqrt{\frac{1}{2} \left(-7+5 \sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)",1,"(x*(-3 + x^4))/(2*(-1 + x^4)*Sqrt[x^2 + Sqrt[1 + x^4]]) + (Sqrt[7/2 + 5/Sqrt[2]]*ArcTan[(-Sqrt[-1/2 + 1/Sqrt[2]] + Sqrt[-1/2 + 1/Sqrt[2]]*x^2 + Sqrt[-1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])])/2 - (Sqrt[7/2 + 5/Sqrt[2]]*ArcTan[(-Sqrt[1/2 + 1/Sqrt[2]] + Sqrt[1/2 + 1/Sqrt[2]]*x^2 + Sqrt[1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])])/2 + ArcTanh[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])]/Sqrt[2] - (Sqrt[-7/2 + 5/Sqrt[2]]*ArcTanh[(-Sqrt[-1/2 + 1/Sqrt[2]] + Sqrt[-1/2 + 1/Sqrt[2]]*x^2 + Sqrt[-1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])])/2 - (Sqrt[-7/2 + 5/Sqrt[2]]*ArcTanh[(-Sqrt[1/2 + 1/Sqrt[2]] + Sqrt[1/2 + 1/Sqrt[2]]*x^2 + Sqrt[1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])])/2","A",1
2961,1,155,363,0.3870292,"\int \frac{\sqrt{a x+\sqrt{-b+a x}}}{1+\sqrt{-b+a x}} \, dx","IntegrateAlgebraic[Sqrt[a*x + Sqrt[-b + a*x]]/(1 + Sqrt[-b + a*x]),x]","\frac{\sqrt{a x+\sqrt{-b+a x}} \left(-3+2 \sqrt{-b+a x}\right)}{2 a}-\frac{4 \sqrt{b} \tanh ^{-1}\left(\frac{1}{\sqrt{b}}+\frac{\sqrt{-b+a x}}{\sqrt{b}}-\frac{\sqrt{a x+\sqrt{-b+a x}}}{\sqrt{b}}\right)}{a}+\frac{(-3-4 b) \log \left(a \left(-1-2 \sqrt{-b+a x}\right)+2 a \sqrt{a x+\sqrt{-b+a x}}\right)}{4 a}","-\frac{\sqrt{-b+a x} \sqrt{\frac{a x+\sqrt{-b+a x}}{\left(1+\sqrt{-b+a x}\right)^2}}}{2 a}+\frac{(-3-2 b+2 a x) \sqrt{\frac{a x+\sqrt{-b+a x}}{\left(1+\sqrt{-b+a x}\right)^2}}}{2 a}+\frac{(-3-4 b) \tanh ^{-1}\left(\frac{\sqrt{b}-\sqrt{\frac{a x+\sqrt{-b+a x}}{\left(1+\sqrt{-b+a x}\right)^2}}-\sqrt{-b+a x} \sqrt{\frac{a x+\sqrt{-b+a x}}{\left(1+\sqrt{-b+a x}\right)^2}}}{1+\sqrt{-b+a x}}\right)}{2 a}+\frac{2 \sqrt{b} \log \left(1+\sqrt{-b+a x}\right)}{a}-\frac{2 \sqrt{b} \log \left(1-2 b+2 \sqrt{b} \sqrt{\frac{a x+\sqrt{-b+a x}}{\left(1+\sqrt{-b+a x}\right)^2}}+\sqrt{-b+a x} \left(1+2 \sqrt{b} \sqrt{\frac{a x+\sqrt{-b+a x}}{\left(1+\sqrt{-b+a x}\right)^2}}\right)\right)}{a}",1,"(Sqrt[a*x + Sqrt[-b + a*x]]*(-3 + 2*Sqrt[-b + a*x]))/(2*a) - (4*Sqrt[b]*ArcTanh[1/Sqrt[b] + Sqrt[-b + a*x]/Sqrt[b] - Sqrt[a*x + Sqrt[-b + a*x]]/Sqrt[b]])/a + ((-3 - 4*b)*Log[a*(-1 - 2*Sqrt[-b + a*x]) + 2*a*Sqrt[a*x + Sqrt[-b + a*x]]])/(4*a)","A",1
2962,1,366,366,1.070537,"\int \frac{\left(b x+a x^2\right) \sqrt[4]{b x^3+a x^4}}{-b+a x+x^2} \, dx","IntegrateAlgebraic[((b*x + a*x^2)*(b*x^3 + a*x^4)^(1/4))/(-b + a*x + x^2),x]","\frac{1}{8} \left(-8 a^2+9 b+4 a x\right) \sqrt[4]{b x^3+a x^4}+\frac{\left(-32 a^4+8 a^2 b-5 b^2\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{16 a^{3/4}}+\frac{\left(32 a^4-8 a^2 b+5 b^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{16 a^{3/4}}+\text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 a^5 \log (x)-a^3 b \log (x)-2 a^5 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)+a^3 b \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)-a^4 \log (x) \text{$\#$1}^4+a^2 b \log (x) \text{$\#$1}^4-b^2 \log (x) \text{$\#$1}^4+a^4 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-a^2 b \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+b^2 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 a \text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]","\frac{1}{8} \left(-8 a^2+9 b+4 a x\right) \sqrt[4]{b x^3+a x^4}+\frac{\left(-32 a^4+8 a^2 b-5 b^2\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{16 a^{3/4}}+\frac{\left(32 a^4-8 a^2 b+5 b^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{16 a^{3/4}}+\text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 a^5 \log (x)+a^3 b \log (x)+2 a^5 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)-a^3 b \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)+a^4 \log (x) \text{$\#$1}^4-a^2 b \log (x) \text{$\#$1}^4+b^2 \log (x) \text{$\#$1}^4-a^4 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+a^2 b \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-b^2 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{3 a \text{$\#$1}^3-2 \text{$\#$1}^7}\&\right]",1,"((-8*a^2 + 9*b + 4*a*x)*(b*x^3 + a*x^4)^(1/4))/8 + ((-32*a^4 + 8*a^2*b - 5*b^2)*ArcTan[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(16*a^(3/4)) + ((32*a^4 - 8*a^2*b + 5*b^2)*ArcTanh[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(16*a^(3/4)) + RootSum[2*a^2 - b - 3*a*#1^4 + #1^8 & , (2*a^5*Log[x] - a^3*b*Log[x] - 2*a^5*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] + a^3*b*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] - a^4*Log[x]*#1^4 + a^2*b*Log[x]*#1^4 - b^2*Log[x]*#1^4 + a^4*Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^4 - a^2*b*Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^4 + b^2*Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1^3 + 2*#1^7) & ]","A",1
2963,1,366,366,0.0020831,"\int \frac{\left(b x+a x^2\right) \sqrt[4]{b x^3+a x^4}}{-b+a x+x^2} \, dx","IntegrateAlgebraic[((b*x + a*x^2)*(b*x^3 + a*x^4)^(1/4))/(-b + a*x + x^2),x]","\frac{1}{8} \left(-8 a^2+9 b+4 a x\right) \sqrt[4]{b x^3+a x^4}+\frac{\left(-32 a^4+8 a^2 b-5 b^2\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{16 a^{3/4}}+\frac{\left(32 a^4-8 a^2 b+5 b^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{16 a^{3/4}}+\text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{2 a^5 \log (x)-a^3 b \log (x)-2 a^5 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)+a^3 b \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)-a^4 \log (x) \text{$\#$1}^4+a^2 b \log (x) \text{$\#$1}^4-b^2 \log (x) \text{$\#$1}^4+a^4 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-a^2 b \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+b^2 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{-3 a \text{$\#$1}^3+2 \text{$\#$1}^7}\&\right]","\frac{1}{8} \left(-8 a^2+9 b+4 a x\right) \sqrt[4]{b x^3+a x^4}+\frac{\left(-32 a^4+8 a^2 b-5 b^2\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{16 a^{3/4}}+\frac{\left(32 a^4-8 a^2 b+5 b^2\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b x^3+a x^4}}\right)}{16 a^{3/4}}+\text{RootSum}\left[2 a^2-b-3 a \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{-2 a^5 \log (x)+a^3 b \log (x)+2 a^5 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)-a^3 b \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right)+a^4 \log (x) \text{$\#$1}^4-a^2 b \log (x) \text{$\#$1}^4+b^2 \log (x) \text{$\#$1}^4-a^4 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+a^2 b \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-b^2 \log \left(\sqrt[4]{b x^3+a x^4}-x \text{$\#$1}\right) \text{$\#$1}^4}{3 a \text{$\#$1}^3-2 \text{$\#$1}^7}\&\right]",1,"((-8*a^2 + 9*b + 4*a*x)*(b*x^3 + a*x^4)^(1/4))/8 + ((-32*a^4 + 8*a^2*b - 5*b^2)*ArcTan[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(16*a^(3/4)) + ((32*a^4 - 8*a^2*b + 5*b^2)*ArcTanh[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(16*a^(3/4)) + RootSum[2*a^2 - b - 3*a*#1^4 + #1^8 & , (2*a^5*Log[x] - a^3*b*Log[x] - 2*a^5*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] + a^3*b*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] - a^4*Log[x]*#1^4 + a^2*b*Log[x]*#1^4 - b^2*Log[x]*#1^4 + a^4*Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^4 - a^2*b*Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^4 + b^2*Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1^3 + 2*#1^7) & ]","A",1
2964,1,364,366,2.3814962,"\int \frac{-b^{10}+a^{10} x^{10}}{\sqrt{b^4+a^4 x^4} \left(b^{10}+a^{10} x^{10}\right)} \, dx","IntegrateAlgebraic[(-b^10 + a^10*x^10)/(Sqrt[b^4 + a^4*x^4]*(b^10 + a^10*x^10)),x]","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} a b x}{b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}}\right)}{5 a b}+\frac{1}{5} \text{RootSum}\left[16 a^8 b^8-48 a^6 b^6 \text{$\#$1}^2+24 a^4 b^4 \text{$\#$1}^4-12 a^2 b^2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-8 a^6 b^6 \log (x)+8 a^6 b^6 \log \left(b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}-x \text{$\#$1}\right)+8 a^4 b^4 \log (x) \text{$\#$1}^2-8 a^4 b^4 \log \left(b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^2+4 a^2 b^2 \log (x) \text{$\#$1}^4-4 a^2 b^2 \log \left(b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-\log (x) \text{$\#$1}^6+\log \left(b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^6}{-12 a^6 b^6 \text{$\#$1}+12 a^4 b^4 \text{$\#$1}^3-9 a^2 b^2 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} a b x}{b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}}\right)}{5 a b}+\frac{1}{5} \text{RootSum}\left[16 a^8 b^8-48 a^6 b^6 \text{$\#$1}^2+24 a^4 b^4 \text{$\#$1}^4-12 a^2 b^2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{8 a^6 b^6 \log (x)-8 a^6 b^6 \log \left(b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}-x \text{$\#$1}\right)-8 a^4 b^4 \log (x) \text{$\#$1}^2+8 a^4 b^4 \log \left(b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^2-4 a^2 b^2 \log (x) \text{$\#$1}^4+4 a^2 b^2 \log \left(b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+\log (x) \text{$\#$1}^6-\log \left(b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^6}{12 a^6 b^6 \text{$\#$1}-12 a^4 b^4 \text{$\#$1}^3+9 a^2 b^2 \text{$\#$1}^5-\text{$\#$1}^7}\&\right]",1,"-1/5*(Sqrt[2]*ArcTan[(Sqrt[2]*a*b*x)/(b^2 + a^2*x^2 + Sqrt[b^4 + a^4*x^4])])/(a*b) + RootSum[16*a^8*b^8 - 48*a^6*b^6*#1^2 + 24*a^4*b^4*#1^4 - 12*a^2*b^2*#1^6 + #1^8 & , (-8*a^6*b^6*Log[x] + 8*a^6*b^6*Log[b^2 + a^2*x^2 + Sqrt[b^4 + a^4*x^4] - x*#1] + 8*a^4*b^4*Log[x]*#1^2 - 8*a^4*b^4*Log[b^2 + a^2*x^2 + Sqrt[b^4 + a^4*x^4] - x*#1]*#1^2 + 4*a^2*b^2*Log[x]*#1^4 - 4*a^2*b^2*Log[b^2 + a^2*x^2 + Sqrt[b^4 + a^4*x^4] - x*#1]*#1^4 - Log[x]*#1^6 + Log[b^2 + a^2*x^2 + Sqrt[b^4 + a^4*x^4] - x*#1]*#1^6)/(-12*a^6*b^6*#1 + 12*a^4*b^4*#1^3 - 9*a^2*b^2*#1^5 + #1^7) & ]/5","A",1
2965,1,364,366,0.0030934,"\int \frac{-b^{10}+a^{10} x^{10}}{\sqrt{b^4+a^4 x^4} \left(b^{10}+a^{10} x^{10}\right)} \, dx","IntegrateAlgebraic[(-b^10 + a^10*x^10)/(Sqrt[b^4 + a^4*x^4]*(b^10 + a^10*x^10)),x]","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} a b x}{b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}}\right)}{5 a b}+\frac{1}{5} \text{RootSum}\left[16 a^8 b^8-48 a^6 b^6 \text{$\#$1}^2+24 a^4 b^4 \text{$\#$1}^4-12 a^2 b^2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-8 a^6 b^6 \log (x)+8 a^6 b^6 \log \left(b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}-x \text{$\#$1}\right)+8 a^4 b^4 \log (x) \text{$\#$1}^2-8 a^4 b^4 \log \left(b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^2+4 a^2 b^2 \log (x) \text{$\#$1}^4-4 a^2 b^2 \log \left(b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^4-\log (x) \text{$\#$1}^6+\log \left(b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^6}{-12 a^6 b^6 \text{$\#$1}+12 a^4 b^4 \text{$\#$1}^3-9 a^2 b^2 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} a b x}{b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}}\right)}{5 a b}+\frac{1}{5} \text{RootSum}\left[16 a^8 b^8-48 a^6 b^6 \text{$\#$1}^2+24 a^4 b^4 \text{$\#$1}^4-12 a^2 b^2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{8 a^6 b^6 \log (x)-8 a^6 b^6 \log \left(b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}-x \text{$\#$1}\right)-8 a^4 b^4 \log (x) \text{$\#$1}^2+8 a^4 b^4 \log \left(b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^2-4 a^2 b^2 \log (x) \text{$\#$1}^4+4 a^2 b^2 \log \left(b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+\log (x) \text{$\#$1}^6-\log \left(b^2+a^2 x^2+\sqrt{b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^6}{12 a^6 b^6 \text{$\#$1}-12 a^4 b^4 \text{$\#$1}^3+9 a^2 b^2 \text{$\#$1}^5-\text{$\#$1}^7}\&\right]",1,"-1/5*(Sqrt[2]*ArcTan[(Sqrt[2]*a*b*x)/(b^2 + a^2*x^2 + Sqrt[b^4 + a^4*x^4])])/(a*b) + RootSum[16*a^8*b^8 - 48*a^6*b^6*#1^2 + 24*a^4*b^4*#1^4 - 12*a^2*b^2*#1^6 + #1^8 & , (-8*a^6*b^6*Log[x] + 8*a^6*b^6*Log[b^2 + a^2*x^2 + Sqrt[b^4 + a^4*x^4] - x*#1] + 8*a^4*b^4*Log[x]*#1^2 - 8*a^4*b^4*Log[b^2 + a^2*x^2 + Sqrt[b^4 + a^4*x^4] - x*#1]*#1^2 + 4*a^2*b^2*Log[x]*#1^4 - 4*a^2*b^2*Log[b^2 + a^2*x^2 + Sqrt[b^4 + a^4*x^4] - x*#1]*#1^4 - Log[x]*#1^6 + Log[b^2 + a^2*x^2 + Sqrt[b^4 + a^4*x^4] - x*#1]*#1^6)/(-12*a^6*b^6*#1 + 12*a^4*b^4*#1^3 - 9*a^2*b^2*#1^5 + #1^7) & ]/5","A",1
2966,1,367,367,1.2000944,"\int \frac{\left(b+a x^2\right) \sqrt[3]{x+x^3}}{d+c x^2} \, dx","IntegrateAlgebraic[((b + a*x^2)*(x + x^3)^(1/3))/(d + c*x^2),x]","\frac{a x \sqrt[3]{x+x^3}}{2 c}-\frac{(a c+3 b c-3 a d) \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)}{2 \sqrt{3} c^2}-\frac{\sqrt{3} \sqrt[3]{c-d} (b c-a d) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{c-d} x}{\sqrt[3]{c-d} x-2 \sqrt[3]{d} \sqrt[3]{x+x^3}}\right)}{2 c^2 \sqrt[3]{d}}+\frac{(-a c-3 b c+3 a d) \log \left(-x+\sqrt[3]{x+x^3}\right)}{6 c^2}-\frac{\sqrt[3]{c-d} (b c-a d) \log \left(\sqrt[3]{c-d} x+\sqrt[3]{d} \sqrt[3]{x+x^3}\right)}{2 c^2 \sqrt[3]{d}}+\frac{(a c+3 b c-3 a d) \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)}{12 c^2}+\frac{\sqrt[3]{c-d} (b c-a d) \log \left((c-d)^{2/3} x^2-\sqrt[3]{c-d} \sqrt[3]{d} x \sqrt[3]{x+x^3}+d^{2/3} \left(x+x^3\right)^{2/3}\right)}{4 c^2 \sqrt[3]{d}}","\frac{a x \sqrt[3]{x+x^3}}{2 c}-\frac{(a c+3 b c-3 a d) \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x+x^3}}\right)}{2 \sqrt{3} c^2}-\frac{\sqrt{3} \sqrt[3]{c-d} (b c-a d) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{c-d} x}{\sqrt[3]{c-d} x-2 \sqrt[3]{d} \sqrt[3]{x+x^3}}\right)}{2 c^2 \sqrt[3]{d}}+\frac{(-a c-3 b c+3 a d) \log \left(-x+\sqrt[3]{x+x^3}\right)}{6 c^2}-\frac{\sqrt[3]{c-d} (b c-a d) \log \left(\sqrt[3]{c-d} x+\sqrt[3]{d} \sqrt[3]{x+x^3}\right)}{2 c^2 \sqrt[3]{d}}+\frac{(a c+3 b c-3 a d) \log \left(x^2+x \sqrt[3]{x+x^3}+\left(x+x^3\right)^{2/3}\right)}{12 c^2}+\frac{\sqrt[3]{c-d} (b c-a d) \log \left((c-d)^{2/3} x^2-\sqrt[3]{c-d} \sqrt[3]{d} x \sqrt[3]{x+x^3}+d^{2/3} \left(x+x^3\right)^{2/3}\right)}{4 c^2 \sqrt[3]{d}}",1,"(a*x*(x + x^3)^(1/3))/(2*c) - ((a*c + 3*b*c - 3*a*d)*ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^3)^(1/3))])/(2*Sqrt[3]*c^2) - (Sqrt[3]*(c - d)^(1/3)*(b*c - a*d)*ArcTan[(Sqrt[3]*(c - d)^(1/3)*x)/((c - d)^(1/3)*x - 2*d^(1/3)*(x + x^3)^(1/3))])/(2*c^2*d^(1/3)) + ((-(a*c) - 3*b*c + 3*a*d)*Log[-x + (x + x^3)^(1/3)])/(6*c^2) - ((c - d)^(1/3)*(b*c - a*d)*Log[(c - d)^(1/3)*x + d^(1/3)*(x + x^3)^(1/3)])/(2*c^2*d^(1/3)) + ((a*c + 3*b*c - 3*a*d)*Log[x^2 + x*(x + x^3)^(1/3) + (x + x^3)^(2/3)])/(12*c^2) + ((c - d)^(1/3)*(b*c - a*d)*Log[(c - d)^(2/3)*x^2 - (c - d)^(1/3)*d^(1/3)*x*(x + x^3)^(1/3) + d^(2/3)*(x + x^3)^(2/3)])/(4*c^2*d^(1/3))","A",1
2967,1,367,367,0.6833539,"\int \frac{x^2 \left(b+a x^3\right) \left(-b p+3 a q+2 a p x^3\right)}{\left(q+p x^3\right)^{2/3} \left(b^3 c+d q+\left(3 a b^2 c+d p\right) x^3+3 a^2 b c x^6+a^3 c x^9\right)} \, dx","IntegrateAlgebraic[(x^2*(b + a*x^3)*(-(b*p) + 3*a*q + 2*a*p*x^3))/((q + p*x^3)^(2/3)*(b^3*c + d*q + (3*a*b^2*c + d*p)*x^3 + 3*a^2*b*c*x^6 + a^3*c*x^9)),x]","\frac{1}{3} p \text{RootSum}\left[b^3 c p^3-3 a b^2 c p^2 q+3 a^2 b c p q^2-a^3 c q^3+3 a b^2 c p^2 \text{$\#$1}^3+d p^3 \text{$\#$1}^3-6 a^2 b c p q \text{$\#$1}^3+3 a^3 c q^2 \text{$\#$1}^3+3 a^2 b c p \text{$\#$1}^6-3 a^3 c q \text{$\#$1}^6+a^3 c \text{$\#$1}^9\&,\frac{-b^2 p^2 \log \left(\sqrt[3]{q+p x^3}-\text{$\#$1}\right)+2 a b p q \log \left(\sqrt[3]{q+p x^3}-\text{$\#$1}\right)-a^2 q^2 \log \left(\sqrt[3]{q+p x^3}-\text{$\#$1}\right)+a b p \log \left(\sqrt[3]{q+p x^3}-\text{$\#$1}\right) \text{$\#$1}^3-a^2 q \log \left(\sqrt[3]{q+p x^3}-\text{$\#$1}\right) \text{$\#$1}^3+2 a^2 \log \left(\sqrt[3]{q+p x^3}-\text{$\#$1}\right) \text{$\#$1}^6}{3 a b^2 c p^2 \text{$\#$1}^2+d p^3 \text{$\#$1}^2-6 a^2 b c p q \text{$\#$1}^2+3 a^3 c q^2 \text{$\#$1}^2+6 a^2 b c p \text{$\#$1}^5-6 a^3 c q \text{$\#$1}^5+3 a^3 c \text{$\#$1}^8}\&\right]","\frac{1}{3} p \text{RootSum}\left[b^3 c p^3-3 a b^2 c p^2 q+3 a^2 b c p q^2-a^3 c q^3+3 a b^2 c p^2 \text{$\#$1}^3+d p^3 \text{$\#$1}^3-6 a^2 b c p q \text{$\#$1}^3+3 a^3 c q^2 \text{$\#$1}^3+3 a^2 b c p \text{$\#$1}^6-3 a^3 c q \text{$\#$1}^6+a^3 c \text{$\#$1}^9\&,\frac{-b^2 p^2 \log \left(\sqrt[3]{q+p x^3}-\text{$\#$1}\right)+2 a b p q \log \left(\sqrt[3]{q+p x^3}-\text{$\#$1}\right)-a^2 q^2 \log \left(\sqrt[3]{q+p x^3}-\text{$\#$1}\right)+a b p \log \left(\sqrt[3]{q+p x^3}-\text{$\#$1}\right) \text{$\#$1}^3-a^2 q \log \left(\sqrt[3]{q+p x^3}-\text{$\#$1}\right) \text{$\#$1}^3+2 a^2 \log \left(\sqrt[3]{q+p x^3}-\text{$\#$1}\right) \text{$\#$1}^6}{3 a b^2 c p^2 \text{$\#$1}^2+d p^3 \text{$\#$1}^2-6 a^2 b c p q \text{$\#$1}^2+3 a^3 c q^2 \text{$\#$1}^2+6 a^2 b c p \text{$\#$1}^5-6 a^3 c q \text{$\#$1}^5+3 a^3 c \text{$\#$1}^8}\&\right]",1,"(p*RootSum[b^3*c*p^3 - 3*a*b^2*c*p^2*q + 3*a^2*b*c*p*q^2 - a^3*c*q^3 + 3*a*b^2*c*p^2*#1^3 + d*p^3*#1^3 - 6*a^2*b*c*p*q*#1^3 + 3*a^3*c*q^2*#1^3 + 3*a^2*b*c*p*#1^6 - 3*a^3*c*q*#1^6 + a^3*c*#1^9 & , (-(b^2*p^2*Log[(q + p*x^3)^(1/3) - #1]) + 2*a*b*p*q*Log[(q + p*x^3)^(1/3) - #1] - a^2*q^2*Log[(q + p*x^3)^(1/3) - #1] + a*b*p*Log[(q + p*x^3)^(1/3) - #1]*#1^3 - a^2*q*Log[(q + p*x^3)^(1/3) - #1]*#1^3 + 2*a^2*Log[(q + p*x^3)^(1/3) - #1]*#1^6)/(3*a*b^2*c*p^2*#1^2 + d*p^3*#1^2 - 6*a^2*b*c*p*q*#1^2 + 3*a^3*c*q^2*#1^2 + 6*a^2*b*c*p*#1^5 - 6*a^3*c*q*#1^5 + 3*a^3*c*#1^8) & ])/3","A",1
2968,1,455,369,6.3373431,"\int \frac{a b+a c-2 b c+(-2 a+b+c) x}{\sqrt[3]{(-a+x) (-b+x) (-c+x)} \left(a^2-b c d+(-2 a+b d+c d) x+(1-d) x^2\right)} \, dx","IntegrateAlgebraic[(a*b + a*c - 2*b*c + (-2*a + b + c)*x)/(((-a + x)*(-b + x)*(-c + x))^(1/3)*(a^2 - b*c*d + (-2*a + b*d + c*d)*x + (1 - d)*x^2)),x]","\frac{\sqrt{\frac{1}{2} \left(-3+3 i \sqrt{3}\right)} \tan ^{-1}\left(\frac{3 \sqrt[3]{d} \sqrt[3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{-3 i a+\sqrt{3} a+3 i x-\sqrt{3} x+\sqrt{3} \sqrt[3]{d} \sqrt[3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}\right)}{d^{2/3}}+\frac{\left(1+i \sqrt{3}\right) \log \left(-a+\sqrt{3} (i a-i x)+x+2 \sqrt[3]{d} \sqrt[3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}\right)}{2 d^{2/3}}-\frac{i \left(-i+\sqrt{3}\right) \log \left(-a^2+2 a x-x^2+\sqrt[3]{d} (a-x) \sqrt[3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}+2 d^{2/3} \left(-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3\right)^{2/3}+\sqrt{3} \left(-i a^2+2 i a x-i x^2+\sqrt[3]{d} (-i a+i x) \sqrt[3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}\right)\right)}{4 d^{2/3}}","\frac{\sqrt{\frac{1}{2} \left(-3+3 i \sqrt{3}\right)} \tan ^{-1}\left(\frac{3 \sqrt[3]{d} \sqrt[3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{-3 i a+\sqrt{3} a+3 i x-\sqrt{3} x+\sqrt{3} \sqrt[3]{d} \sqrt[3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}\right)}{d^{2/3}}+\frac{\left(1+i \sqrt{3}\right) \log \left((-1)^{2/3} (a-x)+\sqrt[3]{d} \sqrt[3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}\right)}{2 d^{2/3}}-\frac{i \left(-i+\sqrt{3}\right) \log \left(\sqrt[3]{-1} \left(a^2-2 a x+x^2\right)+(-1)^{2/3} \sqrt[3]{d} (a-x) \sqrt[3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}-d^{2/3} \left(-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3\right)^{2/3}\right)}{4 d^{2/3}}",1,"(Sqrt[(-3 + (3*I)*Sqrt[3])/2]*ArcTan[(3*d^(1/3)*(-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3)^(1/3))/((-3*I)*a + Sqrt[3]*a + (3*I)*x - Sqrt[3]*x + Sqrt[3]*d^(1/3)*(-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3)^(1/3))])/d^(2/3) + ((1 + I*Sqrt[3])*Log[-a + Sqrt[3]*(I*a - I*x) + x + 2*d^(1/3)*(-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3)^(1/3)])/(2*d^(2/3)) - ((I/4)*(-I + Sqrt[3])*Log[-a^2 + 2*a*x - x^2 + d^(1/3)*(a - x)*(-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3)^(1/3) + 2*d^(2/3)*(-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3)^(2/3) + Sqrt[3]*((-I)*a^2 + (2*I)*a*x - I*x^2 + d^(1/3)*((-I)*a + I*x)*(-(a*b*c) + (a*b + a*c + b*c)*x + (-a - b - c)*x^2 + x^3)^(1/3))])/d^(2/3)","A",1
2969,1,370,370,1.3196881,"\int \frac{(-2+x) \left(1-x+x^2\right)}{x^3 \left(-1+x+x^2\right) \sqrt[3]{\frac{1-x+2 x^2}{1-x+3 x^2}}} \, dx","IntegrateAlgebraic[((-2 + x)*(1 - x + x^2))/(x^3*(-1 + x + x^2)*((1 - x + 2*x^2)/(1 - x + 3*x^2))^(1/3)),x]","\frac{\left(-1+x-3 x^2\right) \left(\frac{1-x+2 x^2}{1-x+3 x^2}\right)^{2/3}}{x^2}-2\ 2^{2/3} \sqrt[6]{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2\ 2^{2/3} \sqrt[3]{\frac{1-x+2 x^2}{1-x+3 x^2}}}{3^{5/6}}\right)+\frac{7 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{\frac{1-x+2 x^2}{1-x+3 x^2}}}{\sqrt{3}}\right)}{\sqrt{3}}+\frac{7}{3} \log \left(-1+\sqrt[3]{\frac{1-x+2 x^2}{1-x+3 x^2}}\right)-\frac{2\ 2^{2/3} \log \left(-3+6^{2/3} \sqrt[3]{\frac{1-x+2 x^2}{1-x+3 x^2}}\right)}{\sqrt[3]{3}}-\frac{7}{6} \log \left(1+\sqrt[3]{\frac{1-x+2 x^2}{1-x+3 x^2}}+\left(\frac{1-x+2 x^2}{1-x+3 x^2}\right)^{2/3}\right)+\frac{2^{2/3} \log \left(3+6^{2/3} \sqrt[3]{\frac{1-x+2 x^2}{1-x+3 x^2}}+2 \sqrt[3]{6} \left(\frac{1-x+2 x^2}{1-x+3 x^2}\right)^{2/3}\right)}{\sqrt[3]{3}}","\frac{\left(-1+x-3 x^2\right) \left(\frac{1-x+2 x^2}{1-x+3 x^2}\right)^{2/3}}{x^2}-2\ 2^{2/3} \sqrt[6]{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2\ 2^{2/3} \sqrt[3]{\frac{1-x+2 x^2}{1-x+3 x^2}}}{3^{5/6}}\right)+\frac{7 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{\frac{1-x+2 x^2}{1-x+3 x^2}}}{\sqrt{3}}\right)}{\sqrt{3}}+\frac{7}{3} \log \left(-1+\sqrt[3]{\frac{1-x+2 x^2}{1-x+3 x^2}}\right)-\frac{2\ 2^{2/3} \log \left(-3+6^{2/3} \sqrt[3]{\frac{1-x+2 x^2}{1-x+3 x^2}}\right)}{\sqrt[3]{3}}-\frac{7}{6} \log \left(1+\sqrt[3]{\frac{1-x+2 x^2}{1-x+3 x^2}}+\left(\frac{1-x+2 x^2}{1-x+3 x^2}\right)^{2/3}\right)+\frac{2^{2/3} \log \left(3+6^{2/3} \sqrt[3]{\frac{1-x+2 x^2}{1-x+3 x^2}}+2 \sqrt[3]{6} \left(\frac{1-x+2 x^2}{1-x+3 x^2}\right)^{2/3}\right)}{\sqrt[3]{3}}",1,"((-1 + x - 3*x^2)*((1 - x + 2*x^2)/(1 - x + 3*x^2))^(2/3))/x^2 - 2*2^(2/3)*3^(1/6)*ArcTan[1/Sqrt[3] + (2*2^(2/3)*((1 - x + 2*x^2)/(1 - x + 3*x^2))^(1/3))/3^(5/6)] + (7*ArcTan[1/Sqrt[3] + (2*((1 - x + 2*x^2)/(1 - x + 3*x^2))^(1/3))/Sqrt[3]])/Sqrt[3] + (7*Log[-1 + ((1 - x + 2*x^2)/(1 - x + 3*x^2))^(1/3)])/3 - (2*2^(2/3)*Log[-3 + 6^(2/3)*((1 - x + 2*x^2)/(1 - x + 3*x^2))^(1/3)])/3^(1/3) - (7*Log[1 + ((1 - x + 2*x^2)/(1 - x + 3*x^2))^(1/3) + ((1 - x + 2*x^2)/(1 - x + 3*x^2))^(2/3)])/6 + (2^(2/3)*Log[3 + 6^(2/3)*((1 - x + 2*x^2)/(1 - x + 3*x^2))^(1/3) + 2*6^(1/3)*((1 - x + 2*x^2)/(1 - x + 3*x^2))^(2/3)])/3^(1/3)","A",1
2970,1,371,371,16.5618693,"\int \frac{x^3 \left(5 b+9 a x^4\right)}{\sqrt[4]{b x+a x^5} \left(1+b x^5+a x^9\right)} \, dx","IntegrateAlgebraic[(x^3*(5*b + 9*a*x^4))/((b*x + a*x^5)^(1/4)*(1 + b*x^5 + a*x^9)),x]","-\sqrt{2} \tan ^{-1}\left(\frac{-2^{2/3} x \sqrt[4]{b x+a x^5}+x^2 \sqrt[4]{b x+a x^5}}{2 \sqrt[6]{2}-\sqrt{2} x+2^{2/3} x \sqrt[4]{b x+a x^5}-x^2 \sqrt[4]{b x+a x^5}}\right)+\sqrt{2} \tan ^{-1}\left(\frac{-2^{2/3} x \sqrt[4]{b x+a x^5}+x^2 \sqrt[4]{b x+a x^5}}{-2 \sqrt[6]{2}+\sqrt{2} x+2^{2/3} x \sqrt[4]{b x+a x^5}-x^2 \sqrt[4]{b x+a x^5}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{-2 2^{5/6} x \sqrt[4]{b x+a x^5}+4 \sqrt[6]{2} x^2 \sqrt[4]{b x+a x^5}-\sqrt{2} x^3 \sqrt[4]{b x+a x^5}}{-2 \sqrt[3]{2}+2\ 2^{2/3} x-x^2-2 \sqrt[3]{2} x^2 \sqrt{b x+a x^5}+2\ 2^{2/3} x^3 \sqrt{b x+a x^5}-x^4 \sqrt{b x+a x^5}}\right)","-\sqrt{2} \tan ^{-1}\left(\frac{-2^{2/3} x \sqrt[4]{b x+a x^5}+x^2 \sqrt[4]{b x+a x^5}}{2 \sqrt[6]{2}-\sqrt{2} x+2^{2/3} x \sqrt[4]{b x+a x^5}-x^2 \sqrt[4]{b x+a x^5}}\right)+\sqrt{2} \tan ^{-1}\left(\frac{-2^{2/3} x \sqrt[4]{b x+a x^5}+x^2 \sqrt[4]{b x+a x^5}}{-2 \sqrt[6]{2}+\sqrt{2} x+2^{2/3} x \sqrt[4]{b x+a x^5}-x^2 \sqrt[4]{b x+a x^5}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{-2 2^{5/6} x \sqrt[4]{b x+a x^5}+4 \sqrt[6]{2} x^2 \sqrt[4]{b x+a x^5}-\sqrt{2} x^3 \sqrt[4]{b x+a x^5}}{-2 \sqrt[3]{2}+2\ 2^{2/3} x-x^2-2 \sqrt[3]{2} x^2 \sqrt{b x+a x^5}+2\ 2^{2/3} x^3 \sqrt{b x+a x^5}-x^4 \sqrt{b x+a x^5}}\right)",1,"-(Sqrt[2]*ArcTan[(-(2^(2/3)*x*(b*x + a*x^5)^(1/4)) + x^2*(b*x + a*x^5)^(1/4))/(2*2^(1/6) - Sqrt[2]*x + 2^(2/3)*x*(b*x + a*x^5)^(1/4) - x^2*(b*x + a*x^5)^(1/4))]) + Sqrt[2]*ArcTan[(-(2^(2/3)*x*(b*x + a*x^5)^(1/4)) + x^2*(b*x + a*x^5)^(1/4))/(-2*2^(1/6) + Sqrt[2]*x + 2^(2/3)*x*(b*x + a*x^5)^(1/4) - x^2*(b*x + a*x^5)^(1/4))] - Sqrt[2]*ArcTanh[(-2*2^(5/6)*x*(b*x + a*x^5)^(1/4) + 4*2^(1/6)*x^2*(b*x + a*x^5)^(1/4) - Sqrt[2]*x^3*(b*x + a*x^5)^(1/4))/(-2*2^(1/3) + 2*2^(2/3)*x - x^2 - 2*2^(1/3)*x^2*Sqrt[b*x + a*x^5] + 2*2^(2/3)*x^3*Sqrt[b*x + a*x^5] - x^4*Sqrt[b*x + a*x^5])]","A",1
2971,1,375,375,0.612652,"\int \frac{\sqrt{1+x^2} \left(1+x^4\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{1-x^4} \, dx","IntegrateAlgebraic[(Sqrt[1 + x^2]*(1 + x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^4),x]","\frac{\left(-78032+1254 x-193024 x^2+3072 x^3-35840 x^4\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-184-345 x-2048 x^2-2560 x^3\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(-282-175104 x+3072 x^2-35840 x^3\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(935-2048 x-2560 x^2\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{80640 x \sqrt{1+x^2}+40320 \left(1+2 x^2\right)}+\frac{251}{128} \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)-\text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{-2 \text{$\#$1}+\text{$\#$1}^3}\&\right]+\text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{2-2 \text{$\#$1}^2+\text{$\#$1}^4}\&\right]","\frac{\left(-78032+1254 x-193024 x^2+3072 x^3-35840 x^4\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-184-345 x-2048 x^2-2560 x^3\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(-282-175104 x+3072 x^2-35840 x^3\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(935-2048 x-2560 x^2\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{80640 x \sqrt{1+x^2}+40320 \left(1+2 x^2\right)}+\frac{251}{128} \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)-\text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{-2 \text{$\#$1}+\text{$\#$1}^3}\&\right]+\text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{2-2 \text{$\#$1}^2+\text{$\#$1}^4}\&\right]",1,"((-78032 + 1254*x - 193024*x^2 + 3072*x^3 - 35840*x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-184 - 345*x - 2048*x^2 - 2560*x^3)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((-282 - 175104*x + 3072*x^2 - 35840*x^3)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (935 - 2048*x - 2560*x^2)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(80640*x*Sqrt[1 + x^2] + 40320*(1 + 2*x^2)) + (251*ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]])/128 - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/(-2*#1 + #1^3) & ] + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(2 - 2*#1^2 + #1^4) & ]","A",1
2972,1,375,375,0.00194,"\int \frac{\sqrt{1+x^2} \left(1+x^4\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{1-x^4} \, dx","IntegrateAlgebraic[(Sqrt[1 + x^2]*(1 + x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^4),x]","\frac{\left(-78032+1254 x-193024 x^2+3072 x^3-35840 x^4\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-184-345 x-2048 x^2-2560 x^3\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(-282-175104 x+3072 x^2-35840 x^3\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(935-2048 x-2560 x^2\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{80640 x \sqrt{1+x^2}+40320 \left(1+2 x^2\right)}+\frac{251}{128} \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)-\text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{-2 \text{$\#$1}+\text{$\#$1}^3}\&\right]+\text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{2-2 \text{$\#$1}^2+\text{$\#$1}^4}\&\right]","\frac{\left(-78032+1254 x-193024 x^2+3072 x^3-35840 x^4\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-184-345 x-2048 x^2-2560 x^3\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(-282-175104 x+3072 x^2-35840 x^3\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(935-2048 x-2560 x^2\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{80640 x \sqrt{1+x^2}+40320 \left(1+2 x^2\right)}+\frac{251}{128} \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)-\text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{-2 \text{$\#$1}+\text{$\#$1}^3}\&\right]+\text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{2-2 \text{$\#$1}^2+\text{$\#$1}^4}\&\right]",1,"((-78032 + 1254*x - 193024*x^2 + 3072*x^3 - 35840*x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-184 - 345*x - 2048*x^2 - 2560*x^3)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((-282 - 175104*x + 3072*x^2 - 35840*x^3)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (935 - 2048*x - 2560*x^2)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(80640*x*Sqrt[1 + x^2] + 40320*(1 + 2*x^2)) + (251*ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]])/128 - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/(-2*#1 + #1^3) & ] + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(2 - 2*#1^2 + #1^4) & ]","A",1
2973,1,517,376,2.0815282,"\int \frac{b^{12}+a^{12} x^{12}}{\sqrt{-b^4+a^4 x^4} \left(-b^{12}+a^{12} x^{12}\right)} \, dx","IntegrateAlgebraic[(b^12 + a^12*x^12)/(Sqrt[-b^4 + a^4*x^4]*(-b^12 + a^12*x^12)),x]","-\frac{x}{3 \sqrt{-b^4+a^4 x^4}}+\left(\frac{1}{6}-\frac{i}{6}\right) \text{RootSum}\left[4 a^4 b^4+(4-4 i) a^3 b^3 \text{$\#$1}-(2+2 i) a b \text{$\#$1}^3-\text{$\#$1}^4\&,\frac{-2 a^2 b^2 \log (x)+2 a^2 b^2 \log \left(i b^2+a^2 x^2+\sqrt{-b^4+a^4 x^4}-x \text{$\#$1}\right)-(1-i) a b \log (x) \text{$\#$1}+(1-i) a b \log \left(i b^2+a^2 x^2+\sqrt{-b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}+i \log (x) \text{$\#$1}^2-i \log \left(i b^2+a^2 x^2+\sqrt{-b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^2}{2 a^3 b^3-3 i a b \text{$\#$1}^2-(1+i) \text{$\#$1}^3}\&\right]-\left(\frac{1}{6}-\frac{i}{6}\right) \text{RootSum}\left[4 a^4 b^4-(4-4 i) a^3 b^3 \text{$\#$1}+(2+2 i) a b \text{$\#$1}^3-\text{$\#$1}^4\&,\frac{-2 a^2 b^2 \log (x)+2 a^2 b^2 \log \left(i b^2+a^2 x^2+\sqrt{-b^4+a^4 x^4}-x \text{$\#$1}\right)+(1-i) a b \log (x) \text{$\#$1}-(1-i) a b \log \left(i b^2+a^2 x^2+\sqrt{-b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}+i \log (x) \text{$\#$1}^2-i \log \left(i b^2+a^2 x^2+\sqrt{-b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^2}{2 a^3 b^3-3 i a b \text{$\#$1}^2+(1+i) \text{$\#$1}^3}\&\right]","-\frac{x}{3 \sqrt{-b^4+a^4 x^4}}+\frac{1}{6} \text{RootSum}\left[16 a^8 b^8+32 i a^6 b^6 \text{$\#$1}^2+24 a^4 b^4 \text{$\#$1}^4-8 i a^2 b^2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{8 a^6 b^6 \log (x)-8 a^6 b^6 \log \left(i b^2+a^2 x^2+\sqrt{-b^4+a^4 x^4}-x \text{$\#$1}\right)+4 i a^4 b^4 \log (x) \text{$\#$1}^2-4 i a^4 b^4 \log \left(i b^2+a^2 x^2+\sqrt{-b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^2+2 a^2 b^2 \log (x) \text{$\#$1}^4-2 a^2 b^2 \log \left(i b^2+a^2 x^2+\sqrt{-b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+i \log (x) \text{$\#$1}^6-i \log \left(i b^2+a^2 x^2+\sqrt{-b^4+a^4 x^4}-x \text{$\#$1}\right) \text{$\#$1}^6}{8 a^6 b^6 \text{$\#$1}-12 i a^4 b^4 \text{$\#$1}^3-6 a^2 b^2 \text{$\#$1}^5-i \text{$\#$1}^7}\&\right]",1,"-1/3*x/Sqrt[-b^4 + a^4*x^4] + (1/6 - I/6)*RootSum[4*a^4*b^4 + (4 - 4*I)*a^3*b^3*#1 - (2 + 2*I)*a*b*#1^3 - #1^4 & , (-2*a^2*b^2*Log[x] + 2*a^2*b^2*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - x*#1] - (1 - I)*a*b*Log[x]*#1 + (1 - I)*a*b*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - x*#1]*#1 + I*Log[x]*#1^2 - I*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - x*#1]*#1^2)/(2*a^3*b^3 - (3*I)*a*b*#1^2 - (1 + I)*#1^3) & ] - (1/6 - I/6)*RootSum[4*a^4*b^4 - (4 - 4*I)*a^3*b^3*#1 + (2 + 2*I)*a*b*#1^3 - #1^4 & , (-2*a^2*b^2*Log[x] + 2*a^2*b^2*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - x*#1] + (1 - I)*a*b*Log[x]*#1 - (1 - I)*a*b*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - x*#1]*#1 + I*Log[x]*#1^2 - I*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - x*#1]*#1^2)/(2*a^3*b^3 - (3*I)*a*b*#1^2 + (1 + I)*#1^3) & ]","A",1
2974,1,376,376,12.0916663,"\int \frac{\sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{b+a^2 x^4} \left(d+c x^4\right)} \, dx","IntegrateAlgebraic[Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/(Sqrt[b + a^2*x^4]*(d + c*x^4)),x]","-\sqrt{2} a^{3/2} \text{RootSum}\left[b^4 c+4 b^3 c \text{$\#$1}^2-16 a^2 b^2 d \text{$\#$1}^2+6 b^2 c \text{$\#$1}^4+32 a^2 b d \text{$\#$1}^4+4 b c \text{$\#$1}^6-16 a^2 d \text{$\#$1}^6+c \text{$\#$1}^8\&,\frac{b^2 \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)-2 b \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^4}{b^3 c-4 a^2 b^2 d+3 b^2 c \text{$\#$1}^2+16 a^2 b d \text{$\#$1}^2+3 b c \text{$\#$1}^4-12 a^2 d \text{$\#$1}^4+c \text{$\#$1}^6}\&\right]","-\sqrt{2} a^{3/2} \text{RootSum}\left[b^4 c+4 b^3 c \text{$\#$1}^2-16 a^2 b^2 d \text{$\#$1}^2+6 b^2 c \text{$\#$1}^4+32 a^2 b d \text{$\#$1}^4+4 b c \text{$\#$1}^6-16 a^2 d \text{$\#$1}^6+c \text{$\#$1}^8\&,\frac{b^2 \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)-2 b \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^4}{b^3 c-4 a^2 b^2 d+3 b^2 c \text{$\#$1}^2+16 a^2 b d \text{$\#$1}^2+3 b c \text{$\#$1}^4-12 a^2 d \text{$\#$1}^4+c \text{$\#$1}^6}\&\right]",1,"-(Sqrt[2]*a^(3/2)*RootSum[b^4*c + 4*b^3*c*#1^2 - 16*a^2*b^2*d*#1^2 + 6*b^2*c*#1^4 + 32*a^2*b*d*#1^4 + 4*b*c*#1^6 - 16*a^2*d*#1^6 + c*#1^8 & , (b^2*Log[I*a*x^2 + I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1] - 2*b*Log[I*a*x^2 + I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1]*#1^2 + Log[I*a*x^2 + I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1]*#1^4)/(b^3*c - 4*a^2*b^2*d + 3*b^2*c*#1^2 + 16*a^2*b*d*#1^2 + 3*b*c*#1^4 - 12*a^2*d*#1^4 + c*#1^6) & ])","A",1
2975,1,376,376,0.0027192,"\int \frac{\sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{b+a^2 x^4} \left(d+c x^4\right)} \, dx","IntegrateAlgebraic[Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/(Sqrt[b + a^2*x^4]*(d + c*x^4)),x]","-\sqrt{2} a^{3/2} \text{RootSum}\left[b^4 c+4 b^3 c \text{$\#$1}^2-16 a^2 b^2 d \text{$\#$1}^2+6 b^2 c \text{$\#$1}^4+32 a^2 b d \text{$\#$1}^4+4 b c \text{$\#$1}^6-16 a^2 d \text{$\#$1}^6+c \text{$\#$1}^8\&,\frac{b^2 \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)-2 b \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^4}{b^3 c-4 a^2 b^2 d+3 b^2 c \text{$\#$1}^2+16 a^2 b d \text{$\#$1}^2+3 b c \text{$\#$1}^4-12 a^2 d \text{$\#$1}^4+c \text{$\#$1}^6}\&\right]","-\sqrt{2} a^{3/2} \text{RootSum}\left[b^4 c+4 b^3 c \text{$\#$1}^2-16 a^2 b^2 d \text{$\#$1}^2+6 b^2 c \text{$\#$1}^4+32 a^2 b d \text{$\#$1}^4+4 b c \text{$\#$1}^6-16 a^2 d \text{$\#$1}^6+c \text{$\#$1}^8\&,\frac{b^2 \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)-2 b \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^4}{b^3 c-4 a^2 b^2 d+3 b^2 c \text{$\#$1}^2+16 a^2 b d \text{$\#$1}^2+3 b c \text{$\#$1}^4-12 a^2 d \text{$\#$1}^4+c \text{$\#$1}^6}\&\right]",1,"-(Sqrt[2]*a^(3/2)*RootSum[b^4*c + 4*b^3*c*#1^2 - 16*a^2*b^2*d*#1^2 + 6*b^2*c*#1^4 + 32*a^2*b*d*#1^4 + 4*b*c*#1^6 - 16*a^2*d*#1^6 + c*#1^8 & , (b^2*Log[I*a*x^2 + I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1] - 2*b*Log[I*a*x^2 + I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1]*#1^2 + Log[I*a*x^2 + I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1]*#1^4)/(b^3*c - 4*a^2*b^2*d + 3*b^2*c*#1^2 + 16*a^2*b*d*#1^2 + 3*b*c*#1^4 - 12*a^2*d*#1^4 + c*#1^6) & ])","A",1
2976,1,389,380,0.2914998,"\int \frac{x}{x-\sqrt{b+a x} \sqrt{c+\sqrt{b+a x}}} \, dx","IntegrateAlgebraic[x/(x - Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]]),x]","\frac{4}{3} \sqrt{c+\sqrt{b+a x}} \left(3 a^2+c+\sqrt{b+a x}\right)+\frac{b+2 a^2 c-c^2+a x+2 a^2 \sqrt{b+a x}}{a}-4 \text{RootSum}\left[b-c^2-a c \text{$\#$1}+2 c \text{$\#$1}^2+a \text{$\#$1}^3-\text{$\#$1}^4\&,\frac{-a^2 b \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right)+a^2 c^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right)-a b \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}+a^3 c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}+a c^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}-b \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2-a^2 c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2-a^3 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3-a c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3}{a c-4 c \text{$\#$1}-3 a \text{$\#$1}^2+4 \text{$\#$1}^3}\&\right]","x+\frac{4}{3} \left(3 a^2+c\right) \sqrt{c+\sqrt{b+a x}}+\sqrt{b+a x} \left(2 a+\frac{4}{3} \sqrt{c+\sqrt{b+a x}}\right)-4 \text{RootSum}\left[b-c^2-a c \text{$\#$1}+2 c \text{$\#$1}^2+a \text{$\#$1}^3-\text{$\#$1}^4\&,\frac{-a^2 b \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right)+a^2 c^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right)-a b \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}+a^3 c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}+a c^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}-b \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2-a^2 c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2-a^3 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3-a c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3}{a c-4 c \text{$\#$1}-3 a \text{$\#$1}^2+4 \text{$\#$1}^3}\&\right]",1,"(4*Sqrt[c + Sqrt[b + a*x]]*(3*a^2 + c + Sqrt[b + a*x]))/3 + (b + 2*a^2*c - c^2 + a*x + 2*a^2*Sqrt[b + a*x])/a - 4*RootSum[b - c^2 - a*c*#1 + 2*c*#1^2 + a*#1^3 - #1^4 & , (-(a^2*b*Log[Sqrt[c + Sqrt[b + a*x]] - #1]) + a^2*c^2*Log[Sqrt[c + Sqrt[b + a*x]] - #1] - a*b*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1 + a^3*c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1 + a*c^2*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1 - b*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^2 - a^2*c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^2 - a^3*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^3 - a*c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^3)/(a*c - 4*c*#1 - 3*a*#1^2 + 4*#1^3) & ]","A",1
2977,1,389,380,0.0020999,"\int \frac{x}{x-\sqrt{b+a x} \sqrt{c+\sqrt{b+a x}}} \, dx","IntegrateAlgebraic[x/(x - Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]]),x]","\frac{4}{3} \sqrt{c+\sqrt{b+a x}} \left(3 a^2+c+\sqrt{b+a x}\right)+\frac{b+2 a^2 c-c^2+a x+2 a^2 \sqrt{b+a x}}{a}-4 \text{RootSum}\left[b-c^2-a c \text{$\#$1}+2 c \text{$\#$1}^2+a \text{$\#$1}^3-\text{$\#$1}^4\&,\frac{-a^2 b \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right)+a^2 c^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right)-a b \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}+a^3 c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}+a c^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}-b \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2-a^2 c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2-a^3 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3-a c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3}{a c-4 c \text{$\#$1}-3 a \text{$\#$1}^2+4 \text{$\#$1}^3}\&\right]","x+\frac{4}{3} \left(3 a^2+c\right) \sqrt{c+\sqrt{b+a x}}+\sqrt{b+a x} \left(2 a+\frac{4}{3} \sqrt{c+\sqrt{b+a x}}\right)-4 \text{RootSum}\left[b-c^2-a c \text{$\#$1}+2 c \text{$\#$1}^2+a \text{$\#$1}^3-\text{$\#$1}^4\&,\frac{-a^2 b \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right)+a^2 c^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right)-a b \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}+a^3 c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}+a c^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}-b \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2-a^2 c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2-a^3 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3-a c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3}{a c-4 c \text{$\#$1}-3 a \text{$\#$1}^2+4 \text{$\#$1}^3}\&\right]",1,"(4*Sqrt[c + Sqrt[b + a*x]]*(3*a^2 + c + Sqrt[b + a*x]))/3 + (b + 2*a^2*c - c^2 + a*x + 2*a^2*Sqrt[b + a*x])/a - 4*RootSum[b - c^2 - a*c*#1 + 2*c*#1^2 + a*#1^3 - #1^4 & , (-(a^2*b*Log[Sqrt[c + Sqrt[b + a*x]] - #1]) + a^2*c^2*Log[Sqrt[c + Sqrt[b + a*x]] - #1] - a*b*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1 + a^3*c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1 + a*c^2*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1 - b*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^2 - a^2*c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^2 - a^3*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^3 - a*c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^3)/(a*c - 4*c*#1 - 3*a*#1^2 + 4*#1^3) & ]","A",1
2978,1,383,383,1.7895138,"\int \frac{\left(-2 x+(1+k) x^2\right) \left(1-(1+k) x+(a+k) x^2\right)}{((1-x) x (1-k x))^{2/3} \left(1-2 (1+k) x+\left(1+4 k+k^2\right) x^2-2 \left(k+k^2\right) x^3+\left(-b+k^2\right) x^4\right)} \, dx","IntegrateAlgebraic[((-2*x + (1 + k)*x^2)*(1 - (1 + k)*x + (a + k)*x^2))/(((1 - x)*x*(1 - k*x))^(2/3)*(1 - 2*(1 + k)*x + (1 + 4*k + k^2)*x^2 - 2*(k + k^2)*x^3 + (-b + k^2)*x^4)),x]","-\frac{\sqrt{3} \left(a-\sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{b} x}{\sqrt[6]{b} x-2 \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{2 b^{5/6}}+\frac{\sqrt{3} \left(a+\sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{b} x}{\sqrt[6]{b} x+2 \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{2 b^{5/6}}+\frac{\left(a+\sqrt{b}\right) \log \left(-\sqrt[6]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{2 b^{5/6}}+\frac{\left(-a+\sqrt{b}\right) \log \left(\sqrt[6]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{2 b^{5/6}}+\frac{\left(a-\sqrt{b}\right) \log \left(\sqrt[3]{b} x^2-\sqrt[6]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{4 b^{5/6}}+\frac{\left(-a-\sqrt{b}\right) \log \left(\sqrt[3]{b} x^2+\sqrt[6]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{4 b^{5/6}}","-\frac{\sqrt{3} \left(a-\sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{b} x}{\sqrt[6]{b} x-2 \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{2 b^{5/6}}+\frac{\sqrt{3} \left(a+\sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{b} x}{\sqrt[6]{b} x+2 \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{2 b^{5/6}}+\frac{\left(a+\sqrt{b}\right) \log \left(-\sqrt[6]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{2 b^{5/6}}+\frac{\left(-a+\sqrt{b}\right) \log \left(\sqrt[6]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{2 b^{5/6}}+\frac{\left(a-\sqrt{b}\right) \log \left(\sqrt[3]{b} x^2-\sqrt[6]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{4 b^{5/6}}+\frac{\left(-a-\sqrt{b}\right) \log \left(\sqrt[3]{b} x^2+\sqrt[6]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{4 b^{5/6}}",1,"-1/2*(Sqrt[3]*(a - Sqrt[b])*ArcTan[(Sqrt[3]*b^(1/6)*x)/(b^(1/6)*x - 2*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/b^(5/6) + (Sqrt[3]*(a + Sqrt[b])*ArcTan[(Sqrt[3]*b^(1/6)*x)/(b^(1/6)*x + 2*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/(2*b^(5/6)) + ((a + Sqrt[b])*Log[-(b^(1/6)*x) + (x + (-1 - k)*x^2 + k*x^3)^(1/3)])/(2*b^(5/6)) + ((-a + Sqrt[b])*Log[b^(1/6)*x + (x + (-1 - k)*x^2 + k*x^3)^(1/3)])/(2*b^(5/6)) + ((a - Sqrt[b])*Log[b^(1/3)*x^2 - b^(1/6)*x*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + (x + (-1 - k)*x^2 + k*x^3)^(2/3)])/(4*b^(5/6)) + ((-a - Sqrt[b])*Log[b^(1/3)*x^2 + b^(1/6)*x*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + (x + (-1 - k)*x^2 + k*x^3)^(2/3)])/(4*b^(5/6))","A",1
2979,1,383,383,1.4002999,"\int \frac{(-2+(1+k) x) \left(1-(1+k) x+(a+k) x^2\right)}{\sqrt[3]{(1-x) x (1-k x)} \left(1-(2+2 k) x+\left(1+4 k+k^2\right) x^2-2 \left(k+k^2\right) x^3+\left(-b+k^2\right) x^4\right)} \, dx","IntegrateAlgebraic[((-2 + (1 + k)*x)*(1 - (1 + k)*x + (a + k)*x^2))/(((1 - x)*x*(1 - k*x))^(1/3)*(1 - (2 + 2*k)*x + (1 + 4*k + k^2)*x^2 - 2*(k + k^2)*x^3 + (-b + k^2)*x^4)),x]","-\frac{\sqrt{3} \left(a-\sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{b} x}{\sqrt[6]{b} x-2 \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{2 b^{2/3}}-\frac{\sqrt{3} \left(a+\sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{b} x}{\sqrt[6]{b} x+2 \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{2 b^{2/3}}+\frac{\left(a+\sqrt{b}\right) \log \left(-\sqrt[6]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{2 b^{2/3}}+\frac{\left(a-\sqrt{b}\right) \log \left(\sqrt[6]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{2 b^{2/3}}+\frac{\left(-a+\sqrt{b}\right) \log \left(\sqrt[3]{b} x^2-\sqrt[6]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{4 b^{2/3}}+\frac{\left(-a-\sqrt{b}\right) \log \left(\sqrt[3]{b} x^2+\sqrt[6]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{4 b^{2/3}}","-\frac{\sqrt{3} \left(a-\sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{b} x}{\sqrt[6]{b} x-2 \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{2 b^{2/3}}-\frac{\sqrt{3} \left(a+\sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{b} x}{\sqrt[6]{b} x+2 \sqrt[3]{x+(-1-k) x^2+k x^3}}\right)}{2 b^{2/3}}+\frac{\left(a+\sqrt{b}\right) \log \left(-\sqrt[6]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{2 b^{2/3}}+\frac{\left(a-\sqrt{b}\right) \log \left(\sqrt[6]{b} x+\sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{2 b^{2/3}}+\frac{\left(-a+\sqrt{b}\right) \log \left(\sqrt[3]{b} x^2-\sqrt[6]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{4 b^{2/3}}+\frac{\left(-a-\sqrt{b}\right) \log \left(\sqrt[3]{b} x^2+\sqrt[6]{b} x \sqrt[3]{x+(-1-k) x^2+k x^3}+\left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{4 b^{2/3}}",1,"-1/2*(Sqrt[3]*(a - Sqrt[b])*ArcTan[(Sqrt[3]*b^(1/6)*x)/(b^(1/6)*x - 2*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/b^(2/3) - (Sqrt[3]*(a + Sqrt[b])*ArcTan[(Sqrt[3]*b^(1/6)*x)/(b^(1/6)*x + 2*(x + (-1 - k)*x^2 + k*x^3)^(1/3))])/(2*b^(2/3)) + ((a + Sqrt[b])*Log[-(b^(1/6)*x) + (x + (-1 - k)*x^2 + k*x^3)^(1/3)])/(2*b^(2/3)) + ((a - Sqrt[b])*Log[b^(1/6)*x + (x + (-1 - k)*x^2 + k*x^3)^(1/3)])/(2*b^(2/3)) + ((-a + Sqrt[b])*Log[b^(1/3)*x^2 - b^(1/6)*x*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + (x + (-1 - k)*x^2 + k*x^3)^(2/3)])/(4*b^(2/3)) + ((-a - Sqrt[b])*Log[b^(1/3)*x^2 + b^(1/6)*x*(x + (-1 - k)*x^2 + k*x^3)^(1/3) + (x + (-1 - k)*x^2 + k*x^3)^(2/3)])/(4*b^(2/3))","A",1
2980,1,355,383,22.21462,"\int \frac{1}{\sqrt{-b+a^2 x^2} \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt[6]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}} \, dx","IntegrateAlgebraic[1/(Sqrt[-b + a^2*x^2]*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6)),x]","\frac{\left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{5/6} \left(-6 c+7 \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)}{3 a c^2 \sqrt{a x+\sqrt{-b+a^2 x^2}}}-\frac{7 \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[6]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[6]{c}}\right)}{6 \sqrt{3} a c^{13/6}}+\frac{7 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[6]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[6]{c}}\right)}{6 \sqrt{3} a c^{13/6}}-\frac{7 \tanh ^{-1}\left(\frac{\sqrt[6]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[6]{c}}\right)}{9 a c^{13/6}}-\frac{7 \tanh ^{-1}\left(\frac{\sqrt[6]{c} \sqrt[6]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[3]{c}+\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}\right)}{18 a c^{13/6}}","\frac{-6 c \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{5/6}+7 \sqrt[4]{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{5/6}}{3 a c^2 \sqrt{a x+\sqrt{-b+a^2 x^2}}}-\frac{7 \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[6]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[6]{c}}\right)}{6 \sqrt{3} a c^{13/6}}+\frac{7 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[6]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[6]{c}}\right)}{6 \sqrt{3} a c^{13/6}}-\frac{7 \tanh ^{-1}\left(\frac{\sqrt[6]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[6]{c}}\right)}{9 a c^{13/6}}-\frac{7 \tanh ^{-1}\left(\frac{\sqrt[6]{c}+\frac{\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[6]{c}}}{\sqrt[6]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}\right)}{18 a c^{13/6}}",1,"((c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(5/6)*(-6*c + 7*(a*x + Sqrt[-b + a^2*x^2])^(1/4)))/(3*a*c^2*Sqrt[a*x + Sqrt[-b + a^2*x^2]]) - (7*ArcTan[1/Sqrt[3] - (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6))/(Sqrt[3]*c^(1/6))])/(6*Sqrt[3]*a*c^(13/6)) + (7*ArcTan[1/Sqrt[3] + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6))/(Sqrt[3]*c^(1/6))])/(6*Sqrt[3]*a*c^(13/6)) - (7*ArcTanh[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6)/c^(1/6)])/(9*a*c^(13/6)) - (7*ArcTanh[(c^(1/6)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6))/(c^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3))])/(18*a*c^(13/6))","A",1
2981,1,383,383,0.8545065,"\int \frac{x}{\sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}} \, dx","IntegrateAlgebraic[x/Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]],x]","\frac{\left(6144 b+1680 b^2+6912 a b x+2450 a b^2 x-12288 a^2 x^2-9216 a^3 x^3\right) \sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\left(-3072 b-1960 b^2-5760 a b x-3675 a b^2 x+6144 a^2 x^2+7680 a^3 x^3\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\sqrt{-b+a^2 x^2} \left(\left(2304 b+2450 b^2-12288 a x-9216 a^2 x^2\right) \sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\left(-1920 b-3675 b^2+6144 a x+7680 a^2 x^2\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{26880 a^3 x \sqrt{-b+a^2 x^2}+13440 a^2 \left(-b+2 a^2 x^2\right)}+\frac{35 b^2 \tanh ^{-1}\left(\sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{128 a^2}","\frac{\left(6144 b+1680 b^2+6912 a b x+2450 a b^2 x-12288 a^2 x^2-9216 a^3 x^3\right) \sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\left(-3072 b-1960 b^2-5760 a b x-3675 a b^2 x+6144 a^2 x^2+7680 a^3 x^3\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\sqrt{-b+a^2 x^2} \left(\left(2304 b+2450 b^2-12288 a x-9216 a^2 x^2\right) \sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\left(-1920 b-3675 b^2+6144 a x+7680 a^2 x^2\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{26880 a^3 x \sqrt{-b+a^2 x^2}+13440 a^2 \left(-b+2 a^2 x^2\right)}+\frac{35 b^2 \tanh ^{-1}\left(\sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{128 a^2}",1,"((6144*b + 1680*b^2 + 6912*a*b*x + 2450*a*b^2*x - 12288*a^2*x^2 - 9216*a^3*x^3)*Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]] + (-3072*b - 1960*b^2 - 5760*a*b*x - 3675*a*b^2*x + 6144*a^2*x^2 + 7680*a^3*x^3)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]] + Sqrt[-b + a^2*x^2]*((2304*b + 2450*b^2 - 12288*a*x - 9216*a^2*x^2)*Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]] + (-1920*b - 3675*b^2 + 6144*a*x + 7680*a^2*x^2)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]))/(26880*a^3*x*Sqrt[-b + a^2*x^2] + 13440*a^2*(-b + 2*a^2*x^2)) + (35*b^2*ArcTanh[Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]])/(128*a^2)","A",1
2982,1,446,384,4.1140564,"\int \frac{\left(d+c x^2\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{5/4}}{\left(-b+a^2 x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[((d + c*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(5/4))/(-b + a^2*x^2)^(3/2),x]","\frac{\sqrt{-b+a^2 x^2} \left(-9 b^{9/8} c-5 a^2 \sqrt[8]{b} d+4 a^2 \sqrt[8]{b} c x^2\right) \sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}{5 a^3 \left(-\sqrt{b}+a x\right) \left(\sqrt{b}+a x\right)}+\frac{4 \sqrt[8]{b} c x \sqrt[4]{\frac{a x}{\sqrt{b}}+\frac{\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}{5 a^2}-\frac{5 \left(b c+a^2 d\right) \tan ^{-1}\left(\sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}\right)}{2 a^3 b^{3/8}}+\frac{5 \left(b c+a^2 d\right) \tan ^{-1}\left(\frac{-1+\sqrt{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}{\sqrt{2} \sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}\right)}{2 \sqrt{2} a^3 b^{3/8}}-\frac{5 \left(b c+a^2 d\right) \tanh ^{-1}\left(\sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}\right)}{2 a^3 b^{3/8}}+\frac{5 \left(b c+a^2 d\right) \tanh ^{-1}\left(\frac{1+\sqrt{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}{\sqrt{2} \sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}\right)}{2 \sqrt{2} a^3 b^{3/8}}","\frac{\sqrt{-b+a^2 x^2} \left(-9 b^{9/8} c-5 a^2 \sqrt[8]{b} d+4 a^2 \sqrt[8]{b} c x^2\right) \sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}{5 a^3 \left(-\sqrt{b}+a x\right) \left(\sqrt{b}+a x\right)}+\frac{4 \sqrt[8]{b} c x \sqrt[4]{\frac{a x}{\sqrt{b}}+\frac{\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}{5 a^2}-\frac{5 \left(b c+a^2 d\right) \tan ^{-1}\left(\sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}\right)}{2 a^3 b^{3/8}}-\frac{5 \left(b c+a^2 d\right) \tanh ^{-1}\left(\sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}\right)}{2 a^3 b^{3/8}}-\frac{5 (-1)^{3/4} \left(b c+a^2 d\right) \tanh ^{-1}\left(\sqrt[4]{-1} \sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}\right)}{2 a^3 b^{3/8}}-\frac{5 \sqrt[4]{-1} \left(b c+a^2 d\right) \tanh ^{-1}\left((-1)^{3/4} \sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}\right)}{2 a^3 b^{3/8}}",1,"(Sqrt[-b + a^2*x^2]*(-9*b^(9/8)*c - 5*a^2*b^(1/8)*d + 4*a^2*b^(1/8)*c*x^2)*((a*x + Sqrt[-b + a^2*x^2])/Sqrt[b])^(1/4))/(5*a^3*(-Sqrt[b] + a*x)*(Sqrt[b] + a*x)) + (4*b^(1/8)*c*x*((a*x)/Sqrt[b] + Sqrt[-b + a^2*x^2]/Sqrt[b])^(1/4))/(5*a^2) - (5*(b*c + a^2*d)*ArcTan[((a*x + Sqrt[-b + a^2*x^2])/Sqrt[b])^(1/4)])/(2*a^3*b^(3/8)) + (5*(b*c + a^2*d)*ArcTan[(-1 + Sqrt[(a*x + Sqrt[-b + a^2*x^2])/Sqrt[b]])/(Sqrt[2]*((a*x + Sqrt[-b + a^2*x^2])/Sqrt[b])^(1/4))])/(2*Sqrt[2]*a^3*b^(3/8)) - (5*(b*c + a^2*d)*ArcTanh[((a*x + Sqrt[-b + a^2*x^2])/Sqrt[b])^(1/4)])/(2*a^3*b^(3/8)) + (5*(b*c + a^2*d)*ArcTanh[(1 + Sqrt[(a*x + Sqrt[-b + a^2*x^2])/Sqrt[b]])/(Sqrt[2]*((a*x + Sqrt[-b + a^2*x^2])/Sqrt[b])^(1/4))])/(2*Sqrt[2]*a^3*b^(3/8))","A",0
2983,1,385,385,1.162826,"\int \frac{\left(-b+a x^2\right) \sqrt[3]{-x+x^3}}{-d+c x^2} \, dx","IntegrateAlgebraic[((-b + a*x^2)*(-x + x^3)^(1/3))/(-d + c*x^2),x]","\frac{a x \sqrt[3]{-x+x^3}}{2 c}+\frac{(a c+3 b c-3 a d) \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)}{2 \sqrt{3} c^2}+\frac{\sqrt{3} \sqrt[3]{c-d} (b c-a d) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{c-d} x}{\sqrt[3]{c-d} x-2 \sqrt[3]{d} \sqrt[3]{-x+x^3}}\right)}{2 c^2 \sqrt[3]{d}}+\frac{(a c+3 b c-3 a d) \log \left(-x+\sqrt[3]{-x+x^3}\right)}{6 c^2}+\frac{\sqrt[3]{c-d} (b c-a d) \log \left(\sqrt[3]{c-d} x+\sqrt[3]{d} \sqrt[3]{-x+x^3}\right)}{2 c^2 \sqrt[3]{d}}+\frac{(-a c-3 b c+3 a d) \log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)}{12 c^2}-\frac{\sqrt[3]{c-d} (b c-a d) \log \left((c-d)^{2/3} x^2-\sqrt[3]{c-d} \sqrt[3]{d} x \sqrt[3]{-x+x^3}+d^{2/3} \left(-x+x^3\right)^{2/3}\right)}{4 c^2 \sqrt[3]{d}}","\frac{a x \sqrt[3]{-x+x^3}}{2 c}+\frac{(a c+3 b c-3 a d) \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{-x+x^3}}\right)}{2 \sqrt{3} c^2}+\frac{\sqrt{3} \sqrt[3]{c-d} (b c-a d) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{c-d} x}{\sqrt[3]{c-d} x-2 \sqrt[3]{d} \sqrt[3]{-x+x^3}}\right)}{2 c^2 \sqrt[3]{d}}+\frac{(a c+3 b c-3 a d) \log \left(-x+\sqrt[3]{-x+x^3}\right)}{6 c^2}+\frac{\sqrt[3]{c-d} (b c-a d) \log \left(\sqrt[3]{c-d} x+\sqrt[3]{d} \sqrt[3]{-x+x^3}\right)}{2 c^2 \sqrt[3]{d}}+\frac{(-a c-3 b c+3 a d) \log \left(x^2+x \sqrt[3]{-x+x^3}+\left(-x+x^3\right)^{2/3}\right)}{12 c^2}-\frac{\sqrt[3]{c-d} (b c-a d) \log \left((c-d)^{2/3} x^2-\sqrt[3]{c-d} \sqrt[3]{d} x \sqrt[3]{-x+x^3}+d^{2/3} \left(-x+x^3\right)^{2/3}\right)}{4 c^2 \sqrt[3]{d}}",1,"(a*x*(-x + x^3)^(1/3))/(2*c) + ((a*c + 3*b*c - 3*a*d)*ArcTan[(Sqrt[3]*x)/(x + 2*(-x + x^3)^(1/3))])/(2*Sqrt[3]*c^2) + (Sqrt[3]*(c - d)^(1/3)*(b*c - a*d)*ArcTan[(Sqrt[3]*(c - d)^(1/3)*x)/((c - d)^(1/3)*x - 2*d^(1/3)*(-x + x^3)^(1/3))])/(2*c^2*d^(1/3)) + ((a*c + 3*b*c - 3*a*d)*Log[-x + (-x + x^3)^(1/3)])/(6*c^2) + ((c - d)^(1/3)*(b*c - a*d)*Log[(c - d)^(1/3)*x + d^(1/3)*(-x + x^3)^(1/3)])/(2*c^2*d^(1/3)) + ((-(a*c) - 3*b*c + 3*a*d)*Log[x^2 + x*(-x + x^3)^(1/3) + (-x + x^3)^(2/3)])/(12*c^2) - ((c - d)^(1/3)*(b*c - a*d)*Log[(c - d)^(2/3)*x^2 - (c - d)^(1/3)*d^(1/3)*x*(-x + x^3)^(1/3) + d^(2/3)*(-x + x^3)^(2/3)])/(4*c^2*d^(1/3))","A",1
2984,1,387,387,5.8467559,"\int \frac{(-a b+(2 a-b) x) \left(a^2-2 a x+x^2\right)}{\sqrt[3]{x (-a+x) (-b+x)} \left(a^4 d-4 a^3 d x+\left(-b^2+6 a^2 d\right) x^2+2 (b-2 a d) x^3+(-1+d) x^4\right)} \, dx","IntegrateAlgebraic[((-(a*b) + (2*a - b)*x)*(a^2 - 2*a*x + x^2))/((x*(-a + x)*(-b + x))^(1/3)*(a^4*d - 4*a^3*d*x + (-b^2 + 6*a^2*d)*x^2 + 2*(b - 2*a*d)*x^3 + (-1 + d)*x^4)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\frac{a^2}{\sqrt{3}}-\frac{2 a x}{\sqrt{3}}+\frac{x^2}{\sqrt{3}}+\frac{2 \left(a b x+(-a-b) x^2+x^3\right)^{2/3}}{\sqrt{3} \sqrt[3]{d}}}{(a-x)^2}\right)}{2 d^{2/3}}+\frac{\log \left(a \sqrt[6]{d}-\sqrt[6]{d} x+\sqrt[3]{a b x+(-a-b) x^2+x^3}\right)}{2 d^{2/3}}+\frac{\log \left(-a \sqrt[6]{d}+\sqrt[6]{d} x+\sqrt[3]{a b x+(-a-b) x^2+x^3}\right)}{2 d^{2/3}}-\frac{\log \left(a^2 \sqrt[3]{d}-2 a \sqrt[3]{d} x+\sqrt[3]{d} x^2+\left(a \sqrt[6]{d}-\sqrt[6]{d} x\right) \sqrt[3]{a b x+(-a-b) x^2+x^3}+\left(a b x+(-a-b) x^2+x^3\right)^{2/3}\right)}{4 d^{2/3}}-\frac{\log \left(a^2 \sqrt[3]{d}-2 a \sqrt[3]{d} x+\sqrt[3]{d} x^2+\left(-a \sqrt[6]{d}+\sqrt[6]{d} x\right) \sqrt[3]{a b x+(-a-b) x^2+x^3}+\left(a b x+(-a-b) x^2+x^3\right)^{2/3}\right)}{4 d^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\frac{a^2}{\sqrt{3}}-\frac{2 a x}{\sqrt{3}}+\frac{x^2}{\sqrt{3}}+\frac{2 \left(a b x+(-a-b) x^2+x^3\right)^{2/3}}{\sqrt{3} \sqrt[3]{d}}}{(a-x)^2}\right)}{2 d^{2/3}}+\frac{\log \left(a \sqrt[6]{d}-\sqrt[6]{d} x+\sqrt[3]{a b x+(-a-b) x^2+x^3}\right)}{2 d^{2/3}}+\frac{\log \left(-a \sqrt[6]{d}+\sqrt[6]{d} x+\sqrt[3]{a b x+(-a-b) x^2+x^3}\right)}{2 d^{2/3}}-\frac{\log \left(a^2 \sqrt[3]{d}-2 a \sqrt[3]{d} x+\sqrt[3]{d} x^2+\left(a \sqrt[6]{d}-\sqrt[6]{d} x\right) \sqrt[3]{a b x+(-a-b) x^2+x^3}+\left(a b x+(-a-b) x^2+x^3\right)^{2/3}\right)}{4 d^{2/3}}-\frac{\log \left(a^2 \sqrt[3]{d}-2 a \sqrt[3]{d} x+\sqrt[3]{d} x^2+\left(-a \sqrt[6]{d}+\sqrt[6]{d} x\right) \sqrt[3]{a b x+(-a-b) x^2+x^3}+\left(a b x+(-a-b) x^2+x^3\right)^{2/3}\right)}{4 d^{2/3}}",1,"-1/2*(Sqrt[3]*ArcTan[(a^2/Sqrt[3] - (2*a*x)/Sqrt[3] + x^2/Sqrt[3] + (2*(a*b*x + (-a - b)*x^2 + x^3)^(2/3))/(Sqrt[3]*d^(1/3)))/(a - x)^2])/d^(2/3) + Log[a*d^(1/6) - d^(1/6)*x + (a*b*x + (-a - b)*x^2 + x^3)^(1/3)]/(2*d^(2/3)) + Log[-(a*d^(1/6)) + d^(1/6)*x + (a*b*x + (-a - b)*x^2 + x^3)^(1/3)]/(2*d^(2/3)) - Log[a^2*d^(1/3) - 2*a*d^(1/3)*x + d^(1/3)*x^2 + (a*d^(1/6) - d^(1/6)*x)*(a*b*x + (-a - b)*x^2 + x^3)^(1/3) + (a*b*x + (-a - b)*x^2 + x^3)^(2/3)]/(4*d^(2/3)) - Log[a^2*d^(1/3) - 2*a*d^(1/3)*x + d^(1/3)*x^2 + (-(a*d^(1/6)) + d^(1/6)*x)*(a*b*x + (-a - b)*x^2 + x^3)^(1/3) + (a*b*x + (-a - b)*x^2 + x^3)^(2/3)]/(4*d^(2/3))","A",1
2985,1,815,388,1.0624544,"\int \frac{b+d x}{x^4 \sqrt[4]{\frac{b+a x}{d+c x}}} \, dx","IntegrateAlgebraic[(b + d*x)/(x^4*((b + a*x)/(d + c*x))^(1/4)),x]","\frac{7 b^5 c^3 \left(\frac{b+a x}{d+c x}\right)^{3/4}+3 a b^4 c^2 d \left(\frac{b+a x}{d+c x}\right)^{3/4}-123 a^2 b^3 c d^2 \left(\frac{b+a x}{d+c x}\right)^{3/4}-12 b^4 c^2 d^2 \left(\frac{b+a x}{d+c x}\right)^{3/4}+113 a^3 b^2 d^3 \left(\frac{b+a x}{d+c x}\right)^{3/4}+120 a b^3 c d^3 \left(\frac{b+a x}{d+c x}\right)^{3/4}-108 a^2 b^2 d^4 \left(\frac{b+a x}{d+c x}\right)^{3/4}-18 b^4 c^3 d \left(\frac{b+a x}{d+c x}\right)^{7/4}+102 a b^3 c^2 d^2 \left(\frac{b+a x}{d+c x}\right)^{7/4}+42 a^2 b^2 c d^3 \left(\frac{b+a x}{d+c x}\right)^{7/4}-24 b^3 c^2 d^3 \left(\frac{b+a x}{d+c x}\right)^{7/4}-126 a^3 b d^4 \left(\frac{b+a x}{d+c x}\right)^{7/4}-144 a b^2 c d^4 \left(\frac{b+a x}{d+c x}\right)^{7/4}+168 a^2 b d^5 \left(\frac{b+a x}{d+c x}\right)^{7/4}-21 b^3 c^3 d^2 \left(\frac{b+a x}{d+c x}\right)^{11/4}-9 a b^2 c^2 d^3 \left(\frac{b+a x}{d+c x}\right)^{11/4}-15 a^2 b c d^4 \left(\frac{b+a x}{d+c x}\right)^{11/4}+36 b^2 c^2 d^4 \left(\frac{b+a x}{d+c x}\right)^{11/4}+45 a^3 d^5 \left(\frac{b+a x}{d+c x}\right)^{11/4}+24 a b c d^5 \left(\frac{b+a x}{d+c x}\right)^{11/4}-60 a^2 d^6 \left(\frac{b+a x}{d+c x}\right)^{11/4}}{96 b^2 d^2 \left(b-\frac{d (b+a x)}{d+c x}\right)^3}+\frac{\left(7 b^3 c^3+3 a b^2 c^2 d+5 a^2 b c d^2-12 b^2 c^2 d^2-15 a^3 d^3-8 a b c d^3+20 a^2 d^4\right) \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{\frac{b+a x}{d+c x}}}{\sqrt[4]{b}}\right)}{64 b^{9/4} d^{11/4}}+\frac{\left(-7 b^3 c^3-3 a b^2 c^2 d-5 a^2 b c d^2+12 b^2 c^2 d^2+15 a^3 d^3+8 a b c d^3-20 a^2 d^4\right) \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{\frac{b+a x}{d+c x}}}{\sqrt[4]{b}}\right)}{64 b^{9/4} d^{11/4}}","\frac{\left(\frac{b+a x}{d+c x}\right)^{3/4} \left(-32 b^2 d^3-36 b^2 c d^2 x+36 a b d^3 x-48 b d^4 x+3 b^2 c^2 d x^2+42 a b c d^2 x^2-45 a^2 d^3 x^2-60 b c d^3 x^2+60 a d^4 x^2+7 b^2 c^3 x^3+6 a b c^2 d x^3-45 a^2 c d^2 x^3-12 b c^2 d^2 x^3+60 a c d^3 x^3\right)}{96 b^2 d^2 x^3}+\frac{\left(7 b^3 c^3+3 a b^2 c^2 d+5 a^2 b c d^2-12 b^2 c^2 d^2-15 a^3 d^3-8 a b c d^3+20 a^2 d^4\right) \tan ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{\frac{b+a x}{d+c x}}}{\sqrt[4]{b}}\right)}{64 b^{9/4} d^{11/4}}+\frac{\left(-7 b^3 c^3-3 a b^2 c^2 d-5 a^2 b c d^2+12 b^2 c^2 d^2+15 a^3 d^3+8 a b c d^3-20 a^2 d^4\right) \tanh ^{-1}\left(\frac{\sqrt[4]{d} \sqrt[4]{\frac{b+a x}{d+c x}}}{\sqrt[4]{b}}\right)}{64 b^{9/4} d^{11/4}}",1,"(7*b^5*c^3*((b + a*x)/(d + c*x))^(3/4) + 3*a*b^4*c^2*d*((b + a*x)/(d + c*x))^(3/4) - 123*a^2*b^3*c*d^2*((b + a*x)/(d + c*x))^(3/4) - 12*b^4*c^2*d^2*((b + a*x)/(d + c*x))^(3/4) + 113*a^3*b^2*d^3*((b + a*x)/(d + c*x))^(3/4) + 120*a*b^3*c*d^3*((b + a*x)/(d + c*x))^(3/4) - 108*a^2*b^2*d^4*((b + a*x)/(d + c*x))^(3/4) - 18*b^4*c^3*d*((b + a*x)/(d + c*x))^(7/4) + 102*a*b^3*c^2*d^2*((b + a*x)/(d + c*x))^(7/4) + 42*a^2*b^2*c*d^3*((b + a*x)/(d + c*x))^(7/4) - 24*b^3*c^2*d^3*((b + a*x)/(d + c*x))^(7/4) - 126*a^3*b*d^4*((b + a*x)/(d + c*x))^(7/4) - 144*a*b^2*c*d^4*((b + a*x)/(d + c*x))^(7/4) + 168*a^2*b*d^5*((b + a*x)/(d + c*x))^(7/4) - 21*b^3*c^3*d^2*((b + a*x)/(d + c*x))^(11/4) - 9*a*b^2*c^2*d^3*((b + a*x)/(d + c*x))^(11/4) - 15*a^2*b*c*d^4*((b + a*x)/(d + c*x))^(11/4) + 36*b^2*c^2*d^4*((b + a*x)/(d + c*x))^(11/4) + 45*a^3*d^5*((b + a*x)/(d + c*x))^(11/4) + 24*a*b*c*d^5*((b + a*x)/(d + c*x))^(11/4) - 60*a^2*d^6*((b + a*x)/(d + c*x))^(11/4))/(96*b^2*d^2*(b - (d*(b + a*x))/(d + c*x))^3) + ((7*b^3*c^3 + 3*a*b^2*c^2*d + 5*a^2*b*c*d^2 - 12*b^2*c^2*d^2 - 15*a^3*d^3 - 8*a*b*c*d^3 + 20*a^2*d^4)*ArcTan[(d^(1/4)*((b + a*x)/(d + c*x))^(1/4))/b^(1/4)])/(64*b^(9/4)*d^(11/4)) + ((-7*b^3*c^3 - 3*a*b^2*c^2*d - 5*a^2*b*c*d^2 + 12*b^2*c^2*d^2 + 15*a^3*d^3 + 8*a*b*c*d^3 - 20*a^2*d^4)*ArcTanh[(d^(1/4)*((b + a*x)/(d + c*x))^(1/4))/b^(1/4)])/(64*b^(9/4)*d^(11/4))","B",1
2986,1,388,388,13.4068309,"\int \frac{f+e x}{d+c x+\sqrt{a x+\sqrt{b^2+a^2 x^2}}} \, dx","IntegrateAlgebraic[(f + e*x)/(d + c*x + Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]),x]","\frac{e x}{c}-\frac{2 e \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{c^2}+\frac{(d e-c f) \log \left(a x+\sqrt{b^2+a^2 x^2}\right)}{c^2}+\frac{2 \text{RootSum}\left[b^2 c-2 a d \text{$\#$1}^2-2 a \text{$\#$1}^3-c \text{$\#$1}^4\&,\frac{-b^2 c e \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-a d^2 e \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}+a c d f \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}+a c f \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^2+a e \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^3-c d e \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^3+c^2 f \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^3}{2 a d \text{$\#$1}+3 a \text{$\#$1}^2+2 c \text{$\#$1}^3}\&\right]}{c^2}","\frac{e x}{c}-\frac{2 e \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{c^2}+\frac{(d e-c f) \log \left(a x+\sqrt{b^2+a^2 x^2}\right)}{c^2}+\frac{2 \text{RootSum}\left[b^2 c-2 a d \text{$\#$1}^2-2 a \text{$\#$1}^3-c \text{$\#$1}^4\&,\frac{-b^2 c e \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right)-a d^2 e \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}+a c d f \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}+a c f \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^2+a e \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^3-c d e \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^3+c^2 f \log \left(\sqrt{a x+\sqrt{b^2+a^2 x^2}}-\text{$\#$1}\right) \text{$\#$1}^3}{2 a d \text{$\#$1}+3 a \text{$\#$1}^2+2 c \text{$\#$1}^3}\&\right]}{c^2}",1,"(e*x)/c - (2*e*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/c^2 + ((d*e - c*f)*Log[a*x + Sqrt[b^2 + a^2*x^2]])/c^2 + (2*RootSum[b^2*c - 2*a*d*#1^2 - 2*a*#1^3 - c*#1^4 & , (-(b^2*c*e*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]) - a*d^2*e*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]*#1 + a*c*d*f*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]*#1 + a*c*f*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]*#1^2 + a*e*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]*#1^3 - c*d*e*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]*#1^3 + c^2*f*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]*#1^3)/(2*a*d*#1 + 3*a*#1^2 + 2*c*#1^3) & ])/c^2","A",1
2987,1,416,390,4.5139106,"\int \frac{1+x^6}{\sqrt[4]{-x^3+x^5} \left(1-x^6\right)} \, dx","IntegrateAlgebraic[(1 + x^6)/((-x^3 + x^5)^(1/4)*(1 - x^6)),x]","\frac{4 \left(-x^3+x^5\right)^{3/4}}{3 x^2 \left(-1+x^2\right)}+\frac{\sqrt[4]{2} \tan ^{-1}\left(\frac{\sqrt[4]{2} 3^{7/8} x \sqrt[4]{-x^3+x^5}}{\sqrt[8]{17+12 \sqrt{2}} \left(-3 x^2+3^{3/4} \sqrt{-x^3+x^5}\right)}\right)}{3 \sqrt[8]{3 \left(17+12 \sqrt{2}\right)}}+\frac{1}{3} \sqrt[4]{2} \sqrt[8]{\frac{1}{3} \left(17+12 \sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt[4]{2} 3^{7/8} \sqrt[8]{17+12 \sqrt{2}} x \sqrt[4]{-x^3+x^5}}{-3 x^2+3^{3/4} \sqrt{-x^3+x^5}}\right)+\frac{\sqrt[4]{2} \tanh ^{-1}\left(\frac{3 \sqrt[8]{\frac{17}{8748}+\frac{\sqrt{2}}{729}} x^2+3^{3/4} \sqrt[8]{\frac{17}{8748}+\frac{\sqrt{2}}{729}} \sqrt{-x^3+x^5}}{x \sqrt[4]{-x^3+x^5}}\right)}{3 \sqrt[8]{3 \left(17+12 \sqrt{2}\right)}}+\frac{1}{3} \sqrt[4]{2} \sqrt[8]{\frac{1}{3} \left(17+12 \sqrt{2}\right)} \tanh ^{-1}\left(\frac{\frac{\sqrt[8]{\frac{3}{17+12 \sqrt{2}}} x^2}{\sqrt[4]{2}}+\frac{\sqrt{-x^3+x^5}}{\sqrt[4]{2} \sqrt[8]{3 \left(17+12 \sqrt{2}\right)}}}{x \sqrt[4]{-x^3+x^5}}\right)","\frac{4 \left(-x^3+x^5\right)^{3/4}}{3 x^2 \left(-1+x^2\right)}+\frac{\sqrt[4]{2} \tan ^{-1}\left(\frac{3^{7/8} \sqrt{2-\sqrt{2}} x \sqrt[4]{-x^3+x^5}}{-3 x^2+3^{3/4} \sqrt{-x^3+x^5}}\right)}{3 \sqrt[8]{3 \left(17+12 \sqrt{2}\right)}}+\frac{1}{3} \sqrt[4]{2} \sqrt[8]{\frac{1}{3} \left(17+12 \sqrt{2}\right)} \tan ^{-1}\left(\frac{3^{7/8} \sqrt{2+\sqrt{2}} x \sqrt[4]{-x^3+x^5}}{-3 x^2+3^{3/4} \sqrt{-x^3+x^5}}\right)+\frac{\sqrt[4]{2} \tanh ^{-1}\left(\frac{\frac{\sqrt[8]{3} x^2}{\sqrt{2-\sqrt{2}}}+\frac{\sqrt{-x^3+x^5}}{\sqrt[8]{3} \sqrt{2-\sqrt{2}}}}{x \sqrt[4]{-x^3+x^5}}\right)}{3 \sqrt[8]{3 \left(17+12 \sqrt{2}\right)}}+\frac{1}{3} \sqrt[4]{2} \sqrt[8]{\frac{1}{3} \left(17+12 \sqrt{2}\right)} \tanh ^{-1}\left(\frac{\frac{\sqrt[8]{3} x^2}{\sqrt{2+\sqrt{2}}}+\frac{\sqrt{-x^3+x^5}}{\sqrt[8]{3} \sqrt{2+\sqrt{2}}}}{x \sqrt[4]{-x^3+x^5}}\right)",1,"(4*(-x^3 + x^5)^(3/4))/(3*x^2*(-1 + x^2)) + (2^(1/4)*ArcTan[(2^(1/4)*3^(7/8)*x*(-x^3 + x^5)^(1/4))/((17 + 12*Sqrt[2])^(1/8)*(-3*x^2 + 3^(3/4)*Sqrt[-x^3 + x^5]))])/(3*(3*(17 + 12*Sqrt[2]))^(1/8)) + (2^(1/4)*((17 + 12*Sqrt[2])/3)^(1/8)*ArcTan[(2^(1/4)*3^(7/8)*(17 + 12*Sqrt[2])^(1/8)*x*(-x^3 + x^5)^(1/4))/(-3*x^2 + 3^(3/4)*Sqrt[-x^3 + x^5])])/3 + (2^(1/4)*ArcTanh[(3*(17/8748 + Sqrt[2]/729)^(1/8)*x^2 + 3^(3/4)*(17/8748 + Sqrt[2]/729)^(1/8)*Sqrt[-x^3 + x^5])/(x*(-x^3 + x^5)^(1/4))])/(3*(3*(17 + 12*Sqrt[2]))^(1/8)) + (2^(1/4)*((17 + 12*Sqrt[2])/3)^(1/8)*ArcTanh[(((3/(17 + 12*Sqrt[2]))^(1/8)*x^2)/2^(1/4) + Sqrt[-x^3 + x^5]/(2^(1/4)*(3*(17 + 12*Sqrt[2]))^(1/8)))/(x*(-x^3 + x^5)^(1/4))])/3","A",0
2988,1,390,390,0.8939375,"\int \frac{\sqrt{-b+a^2 x^2} \left(d+c x^4\right) \sqrt{a x+\sqrt{-b+a^2 x^2}}}{x^2} \, dx","IntegrateAlgebraic[(Sqrt[-b + a^2*x^2]*(d + c*x^4)*Sqrt[a*x + Sqrt[-b + a^2*x^2]])/x^2,x]","\frac{63 a^3 b^3 d-72 a b^4 c x^2+126 a^5 b^2 d x^2+210 a^3 b^3 c x^4-2016 a^7 b d x^4+126 a^5 b^2 c x^6+2016 a^9 d x^6-504 a^7 b c x^8+224 a^9 c x^{10}+\sqrt{-b+a^2 x^2} \left(-16 b^4 c x-126 a^4 b^2 d x+154 a^2 b^3 c x^3-1008 a^6 b d x^3-42 a^4 b^2 c x^5+2016 a^8 d x^5-392 a^6 b c x^7+224 a^8 c x^9\right)}{63 a^3 x \left(a x+\sqrt{-b+a^2 x^2}\right)^{9/2}}+\frac{a \sqrt[4]{b} d \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt{b}+a x+\sqrt{-b+a^2 x^2}}\right)}{\sqrt{2}}-\frac{a \sqrt[4]{b} d \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{a x}{\sqrt{2} \sqrt[4]{b}}+\frac{\sqrt{-b+a^2 x^2}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{\sqrt{2}}","\frac{63 a^3 b^3 d-72 a b^4 c x^2+126 a^5 b^2 d x^2+210 a^3 b^3 c x^4-2016 a^7 b d x^4+126 a^5 b^2 c x^6+2016 a^9 d x^6-504 a^7 b c x^8+224 a^9 c x^{10}+\sqrt{-b+a^2 x^2} \left(-16 b^4 c x-126 a^4 b^2 d x+154 a^2 b^3 c x^3-1008 a^6 b d x^3-42 a^4 b^2 c x^5+2016 a^8 d x^5-392 a^6 b c x^7+224 a^8 c x^9\right)}{63 a^3 x \left(a x+\sqrt{-b+a^2 x^2}\right)^{9/2}}+\frac{a \sqrt[4]{b} d \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt{b}+a x+\sqrt{-b+a^2 x^2}}\right)}{\sqrt{2}}-\frac{a \sqrt[4]{b} d \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{a x}{\sqrt{2} \sqrt[4]{b}}+\frac{\sqrt{-b+a^2 x^2}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{\sqrt{2}}",1,"(63*a^3*b^3*d - 72*a*b^4*c*x^2 + 126*a^5*b^2*d*x^2 + 210*a^3*b^3*c*x^4 - 2016*a^7*b*d*x^4 + 126*a^5*b^2*c*x^6 + 2016*a^9*d*x^6 - 504*a^7*b*c*x^8 + 224*a^9*c*x^10 + Sqrt[-b + a^2*x^2]*(-16*b^4*c*x - 126*a^4*b^2*d*x + 154*a^2*b^3*c*x^3 - 1008*a^6*b*d*x^3 - 42*a^4*b^2*c*x^5 + 2016*a^8*d*x^5 - 392*a^6*b*c*x^7 + 224*a^8*c*x^9))/(63*a^3*x*(a*x + Sqrt[-b + a^2*x^2])^(9/2)) + (a*b^(1/4)*d*ArcTan[(Sqrt[2]*b^(1/4)*Sqrt[a*x + Sqrt[-b + a^2*x^2]])/(-Sqrt[b] + a*x + Sqrt[-b + a^2*x^2])])/Sqrt[2] - (a*b^(1/4)*d*ArcTanh[(b^(1/4)/Sqrt[2] + (a*x)/(Sqrt[2]*b^(1/4)) + Sqrt[-b + a^2*x^2]/(Sqrt[2]*b^(1/4)))/Sqrt[a*x + Sqrt[-b + a^2*x^2]]])/Sqrt[2]","A",1
2989,1,710,392,0.7523316,"\int \frac{\left(1+x^2\right)^{3/2} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1-x^2\right)^2} \, dx","IntegrateAlgebraic[((1 + x^2)^(3/2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^2)^2,x]","\frac{\sqrt{1+x^2} \left(-11 x+8 x^3\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-5-7 x^2+8 x^4\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{2 x \left(-1+x^2\right) \sqrt{1+x^2}+\left(-1+x^2\right) \left(1+2 x^2\right)}-4 \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)+\text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-\frac{1}{8} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{14 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-\text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{1}{8} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-16 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","\frac{\sqrt{1+x^2} \left(-11 x+8 x^3\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-5-7 x^2+8 x^4\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{2 x \left(-1+x^2\right) \sqrt{1+x^2}+\left(-1+x^2\right) \left(1+2 x^2\right)}-4 \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)+\frac{1}{8} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-6 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+5 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-\frac{1}{8} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-6 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}+5 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^3}{-2+4 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]",1,"(Sqrt[1 + x^2]*(-11*x + 8*x^3)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-5 - 7*x^2 + 8*x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(2*x*(-1 + x^2)*Sqrt[1 + x^2] + (-1 + x^2)*(1 + 2*x^2)) - 4*ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] + RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1^3 - 3*#1^5 + #1^7) & ] - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (14*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1^3 - 3*#1^5 + #1^7) & ]/8 - RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ] + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-16*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ]/8","A",1
2990,1,710,392,0.002048,"\int \frac{\left(1+x^2\right)^{3/2} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1-x^2\right)^2} \, dx","IntegrateAlgebraic[((1 + x^2)^(3/2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^2)^2,x]","\frac{\sqrt{1+x^2} \left(-11 x+8 x^3\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-5-7 x^2+8 x^4\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{2 x \left(-1+x^2\right) \sqrt{1+x^2}+\left(-1+x^2\right) \left(1+2 x^2\right)}-4 \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)+\text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-\frac{1}{8} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{14 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-\text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{1}{8} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-16 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","\frac{\sqrt{1+x^2} \left(-11 x+8 x^3\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-5-7 x^2+8 x^4\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{2 x \left(-1+x^2\right) \sqrt{1+x^2}+\left(-1+x^2\right) \left(1+2 x^2\right)}-4 \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)+\frac{1}{8} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-6 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+5 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-\frac{1}{8} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-6 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}+5 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^3}{-2+4 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]",1,"(Sqrt[1 + x^2]*(-11*x + 8*x^3)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-5 - 7*x^2 + 8*x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(2*x*(-1 + x^2)*Sqrt[1 + x^2] + (-1 + x^2)*(1 + 2*x^2)) - 4*ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] + RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1^3 - 3*#1^5 + #1^7) & ] - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (14*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1^3 - 3*#1^5 + #1^7) & ]/8 - RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ] + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-16*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ]/8","A",1
2991,1,710,392,0.0017453,"\int \frac{\left(1+x^2\right)^{3/2} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1-x^2\right)^2} \, dx","IntegrateAlgebraic[((1 + x^2)^(3/2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^2)^2,x]","\frac{\sqrt{1+x^2} \left(-11 x+8 x^3\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-5-7 x^2+8 x^4\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{2 x \left(-1+x^2\right) \sqrt{1+x^2}+\left(-1+x^2\right) \left(1+2 x^2\right)}-4 \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)+\text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-\frac{1}{8} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{14 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-\text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{1}{8} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-16 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","\frac{\sqrt{1+x^2} \left(-11 x+8 x^3\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-5-7 x^2+8 x^4\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{2 x \left(-1+x^2\right) \sqrt{1+x^2}+\left(-1+x^2\right) \left(1+2 x^2\right)}-4 \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)+\frac{1}{8} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-6 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+5 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-\frac{1}{8} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-6 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}+5 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^3}{-2+4 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]",1,"(Sqrt[1 + x^2]*(-11*x + 8*x^3)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-5 - 7*x^2 + 8*x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(2*x*(-1 + x^2)*Sqrt[1 + x^2] + (-1 + x^2)*(1 + 2*x^2)) - 4*ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] + RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1^3 - 3*#1^5 + #1^7) & ] - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (14*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1^3 - 3*#1^5 + #1^7) & ]/8 - RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ] + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-16*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ]/8","A",1
2992,1,416,394,0.731467,"\int \frac{\left(1+x^2+x^4\right) \sqrt{x^2+\sqrt{1+x^4}}}{\sqrt{1+x^4} \left(-1+x^2+x^4\right)} \, dx","IntegrateAlgebraic[((1 + x^2 + x^4)*Sqrt[x^2 + Sqrt[1 + x^4]])/(Sqrt[1 + x^4]*(-1 + x^2 + x^4)),x]","\sqrt{2} \tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^4}}{\sqrt{2}}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)+\frac{\text{RootSum}\left[1-2 \text{$\#$1}^2-6 \text{$\#$1}^4+2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(1+x^2+\sqrt{1+x^4}\right)-\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right)+\log \left(1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^2-\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^2-\log \left(1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^4+\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^4-\log \left(1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^6+\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^6}{-\text{$\#$1}-6 \text{$\#$1}^3+3 \text{$\#$1}^5+2 \text{$\#$1}^7}\&\right]}{\sqrt{2}}","\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-\frac{\text{RootSum}\left[1+2 \text{$\#$1}^2-6 \text{$\#$1}^4-2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(-1+x^2+\sqrt{1+x^4}\right)-\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}+x^2 \text{$\#$1}+\sqrt{1+x^4} \text{$\#$1}\right)+\log \left(-1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^2-\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}+x^2 \text{$\#$1}+\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^2-\log \left(-1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^4+\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}+x^2 \text{$\#$1}+\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^4-\log \left(-1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^6+\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}+x^2 \text{$\#$1}+\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^6}{\text{$\#$1}-6 \text{$\#$1}^3-3 \text{$\#$1}^5+2 \text{$\#$1}^7}\&\right]}{\sqrt{2}}",1,"Sqrt[2]*ArcTanh[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] + RootSum[1 - 2*#1^2 - 6*#1^4 + 2*#1^6 + #1^8 & , (Log[1 + x^2 + Sqrt[1 + x^4]] - Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1] + Log[1 + x^2 + Sqrt[1 + x^4]]*#1^2 - Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^2 - Log[1 + x^2 + Sqrt[1 + x^4]]*#1^4 + Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^4 - Log[1 + x^2 + Sqrt[1 + x^4]]*#1^6 + Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^6)/(-#1 - 6*#1^3 + 3*#1^5 + 2*#1^7) & ]/Sqrt[2]","A",1
2993,1,416,394,0.0015349,"\int \frac{\left(1+x^2+x^4\right) \sqrt{x^2+\sqrt{1+x^4}}}{\sqrt{1+x^4} \left(-1+x^2+x^4\right)} \, dx","IntegrateAlgebraic[((1 + x^2 + x^4)*Sqrt[x^2 + Sqrt[1 + x^4]])/(Sqrt[1 + x^4]*(-1 + x^2 + x^4)),x]","\sqrt{2} \tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^4}}{\sqrt{2}}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)+\frac{\text{RootSum}\left[1-2 \text{$\#$1}^2-6 \text{$\#$1}^4+2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(1+x^2+\sqrt{1+x^4}\right)-\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right)+\log \left(1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^2-\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^2-\log \left(1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^4+\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^4-\log \left(1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^6+\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^6}{-\text{$\#$1}-6 \text{$\#$1}^3+3 \text{$\#$1}^5+2 \text{$\#$1}^7}\&\right]}{\sqrt{2}}","\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-\frac{\text{RootSum}\left[1+2 \text{$\#$1}^2-6 \text{$\#$1}^4-2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(-1+x^2+\sqrt{1+x^4}\right)-\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}+x^2 \text{$\#$1}+\sqrt{1+x^4} \text{$\#$1}\right)+\log \left(-1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^2-\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}+x^2 \text{$\#$1}+\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^2-\log \left(-1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^4+\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}+x^2 \text{$\#$1}+\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^4-\log \left(-1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^6+\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}+x^2 \text{$\#$1}+\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^6}{\text{$\#$1}-6 \text{$\#$1}^3-3 \text{$\#$1}^5+2 \text{$\#$1}^7}\&\right]}{\sqrt{2}}",1,"Sqrt[2]*ArcTanh[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] + RootSum[1 - 2*#1^2 - 6*#1^4 + 2*#1^6 + #1^8 & , (Log[1 + x^2 + Sqrt[1 + x^4]] - Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1] + Log[1 + x^2 + Sqrt[1 + x^4]]*#1^2 - Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^2 - Log[1 + x^2 + Sqrt[1 + x^4]]*#1^4 + Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^4 - Log[1 + x^2 + Sqrt[1 + x^4]]*#1^6 + Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^6)/(-#1 - 6*#1^3 + 3*#1^5 + 2*#1^7) & ]/Sqrt[2]","A",1
2994,1,397,397,0.9251398,"\int \frac{\sqrt{-b+a^2 x^2} \left(d+c x^4\right) \sqrt{a x+\sqrt{-b+a^2 x^2}}}{x} \, dx","IntegrateAlgebraic[(Sqrt[-b + a^2*x^2]*(d + c*x^4)*Sqrt[a*x + Sqrt[-b + a^2*x^2]])/x,x]","\frac{2 \sqrt{-b+a^2 x^2} \left(-1368 a b^4 c x+10395 a^5 b^2 d x+3705 a^3 b^3 c x^3-32340 a^7 b d x^3+1335 a^5 b^2 c x^5+18480 a^9 d x^5-8100 a^7 b c x^7+5040 a^9 c x^9\right)+2 \left(304 b^5 c-2310 a^4 b^3 d-3078 a^2 b^4 c x^2+24255 a^6 b^2 d x^2+3735 a^4 b^3 c x^4-41580 a^8 b d x^4+4755 a^6 b^2 c x^6+18480 a^{10} d x^6-10620 a^8 b c x^8+5040 a^{10} c x^{10}\right)}{3465 a^4 \left(a x+\sqrt{-b+a^2 x^2}\right)^{9/2}}+\sqrt{2} b^{3/4} d \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt{b}+a x+\sqrt{-b+a^2 x^2}}\right)+\sqrt{2} b^{3/4} d \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{a x}{\sqrt{2} \sqrt[4]{b}}+\frac{\sqrt{-b+a^2 x^2}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)","\frac{2 \sqrt{-b+a^2 x^2} \left(-1368 a b^4 c x+10395 a^5 b^2 d x+3705 a^3 b^3 c x^3-32340 a^7 b d x^3+1335 a^5 b^2 c x^5+18480 a^9 d x^5-8100 a^7 b c x^7+5040 a^9 c x^9\right)+2 \left(304 b^5 c-2310 a^4 b^3 d-3078 a^2 b^4 c x^2+24255 a^6 b^2 d x^2+3735 a^4 b^3 c x^4-41580 a^8 b d x^4+4755 a^6 b^2 c x^6+18480 a^{10} d x^6-10620 a^8 b c x^8+5040 a^{10} c x^{10}\right)}{3465 a^4 \left(a x+\sqrt{-b+a^2 x^2}\right)^{9/2}}+\sqrt{2} b^{3/4} d \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt{b}+a x+\sqrt{-b+a^2 x^2}}\right)+\sqrt{2} b^{3/4} d \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{b}}{\sqrt{2}}+\frac{a x}{\sqrt{2} \sqrt[4]{b}}+\frac{\sqrt{-b+a^2 x^2}}{\sqrt{2} \sqrt[4]{b}}}{\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)",1,"(2*Sqrt[-b + a^2*x^2]*(-1368*a*b^4*c*x + 10395*a^5*b^2*d*x + 3705*a^3*b^3*c*x^3 - 32340*a^7*b*d*x^3 + 1335*a^5*b^2*c*x^5 + 18480*a^9*d*x^5 - 8100*a^7*b*c*x^7 + 5040*a^9*c*x^9) + 2*(304*b^5*c - 2310*a^4*b^3*d - 3078*a^2*b^4*c*x^2 + 24255*a^6*b^2*d*x^2 + 3735*a^4*b^3*c*x^4 - 41580*a^8*b*d*x^4 + 4755*a^6*b^2*c*x^6 + 18480*a^10*d*x^6 - 10620*a^8*b*c*x^8 + 5040*a^10*c*x^10))/(3465*a^4*(a*x + Sqrt[-b + a^2*x^2])^(9/2)) + Sqrt[2]*b^(3/4)*d*ArcTan[(Sqrt[2]*b^(1/4)*Sqrt[a*x + Sqrt[-b + a^2*x^2]])/(-Sqrt[b] + a*x + Sqrt[-b + a^2*x^2])] + Sqrt[2]*b^(3/4)*d*ArcTanh[(b^(1/4)/Sqrt[2] + (a*x)/(Sqrt[2]*b^(1/4)) + Sqrt[-b + a^2*x^2]/(Sqrt[2]*b^(1/4)))/Sqrt[a*x + Sqrt[-b + a^2*x^2]]]","A",1
2995,1,398,398,0.9155822,"\int \frac{\sqrt{a x^2+\sqrt{b+a^2 x^4}}}{d+c x^2} \, dx","IntegrateAlgebraic[Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/(d + c*x^2),x]","-\frac{\sqrt{a} \log \left(-i a x^2-i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{\sqrt{2} c}+\frac{\sqrt{a} \text{RootSum}\left[b^2 c+4 i a b d \text{$\#$1}+2 b c \text{$\#$1}^2-4 i a d \text{$\#$1}^3+c \text{$\#$1}^4\&,\frac{a b d \log \left(-i a x^2-i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)-2 i b c \log \left(-i a x^2-i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}-a d \log \left(-i a x^2-i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2}{a b d-i b c \text{$\#$1}-3 a d \text{$\#$1}^2-i c \text{$\#$1}^3}\&\right]}{\sqrt{2} c}","\frac{\sqrt{a} \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{\sqrt{2} c}-\frac{\sqrt{a} \text{RootSum}\left[b^2 c-4 i a b d \text{$\#$1}+2 b c \text{$\#$1}^2+4 i a d \text{$\#$1}^3+c \text{$\#$1}^4\&,\frac{a b d \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)+2 i b c \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}-a d \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2}{a b d+i b c \text{$\#$1}-3 a d \text{$\#$1}^2+i c \text{$\#$1}^3}\&\right]}{\sqrt{2} c}",1,"-((Sqrt[a]*Log[(-I)*a*x^2 - I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]])/(Sqrt[2]*c)) + (Sqrt[a]*RootSum[b^2*c + (4*I)*a*b*d*#1 + 2*b*c*#1^2 - (4*I)*a*d*#1^3 + c*#1^4 & , (a*b*d*Log[(-I)*a*x^2 - I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1] - (2*I)*b*c*Log[(-I)*a*x^2 - I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1]*#1 - a*d*Log[(-I)*a*x^2 - I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1]*#1^2)/(a*b*d - I*b*c*#1 - 3*a*d*#1^2 - I*c*#1^3) & ])/(Sqrt[2]*c)","A",1
2996,1,398,398,0.0018861,"\int \frac{\sqrt{a x^2+\sqrt{b+a^2 x^4}}}{d+c x^2} \, dx","IntegrateAlgebraic[Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/(d + c*x^2),x]","-\frac{\sqrt{a} \log \left(-i a x^2-i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{\sqrt{2} c}+\frac{\sqrt{a} \text{RootSum}\left[b^2 c+4 i a b d \text{$\#$1}+2 b c \text{$\#$1}^2-4 i a d \text{$\#$1}^3+c \text{$\#$1}^4\&,\frac{a b d \log \left(-i a x^2-i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)-2 i b c \log \left(-i a x^2-i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}-a d \log \left(-i a x^2-i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2}{a b d-i b c \text{$\#$1}-3 a d \text{$\#$1}^2-i c \text{$\#$1}^3}\&\right]}{\sqrt{2} c}","\frac{\sqrt{a} \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{\sqrt{2} c}-\frac{\sqrt{a} \text{RootSum}\left[b^2 c-4 i a b d \text{$\#$1}+2 b c \text{$\#$1}^2+4 i a d \text{$\#$1}^3+c \text{$\#$1}^4\&,\frac{a b d \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)+2 i b c \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}-a d \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2}{a b d+i b c \text{$\#$1}-3 a d \text{$\#$1}^2+i c \text{$\#$1}^3}\&\right]}{\sqrt{2} c}",1,"-((Sqrt[a]*Log[(-I)*a*x^2 - I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]])/(Sqrt[2]*c)) + (Sqrt[a]*RootSum[b^2*c + (4*I)*a*b*d*#1 + 2*b*c*#1^2 - (4*I)*a*d*#1^3 + c*#1^4 & , (a*b*d*Log[(-I)*a*x^2 - I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1] - (2*I)*b*c*Log[(-I)*a*x^2 - I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1]*#1 - a*d*Log[(-I)*a*x^2 - I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1]*#1^2)/(a*b*d - I*b*c*#1 - 3*a*d*#1^2 - I*c*#1^3) & ])/(Sqrt[2]*c)","A",1
2997,1,399,399,20.102986,"\int \frac{x^5 \left(7 b+10 a x^3\right)}{\sqrt[4]{b x^3+a x^6} \left(1+b x^7+a x^{10}\right)} \, dx","IntegrateAlgebraic[(x^5*(7*b + 10*a*x^3))/((b*x^3 + a*x^6)^(1/4)*(1 + b*x^7 + a*x^10)),x]","-\sqrt{2} \tan ^{-1}\left(\frac{-2^{2/3} x \sqrt[4]{b x^3+a x^6}+x^2 \sqrt[4]{b x^3+a x^6}}{2 \sqrt[6]{2}-\sqrt{2} x+2^{2/3} x \sqrt[4]{b x^3+a x^6}-x^2 \sqrt[4]{b x^3+a x^6}}\right)+\sqrt{2} \tan ^{-1}\left(\frac{-2^{2/3} x \sqrt[4]{b x^3+a x^6}+x^2 \sqrt[4]{b x^3+a x^6}}{-2 \sqrt[6]{2}+\sqrt{2} x+2^{2/3} x \sqrt[4]{b x^3+a x^6}-x^2 \sqrt[4]{b x^3+a x^6}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{-2 2^{5/6} x \sqrt[4]{b x^3+a x^6}+4 \sqrt[6]{2} x^2 \sqrt[4]{b x^3+a x^6}-\sqrt{2} x^3 \sqrt[4]{b x^3+a x^6}}{-2 \sqrt[3]{2}+2\ 2^{2/3} x-x^2-2 \sqrt[3]{2} x^2 \sqrt{b x^3+a x^6}+2\ 2^{2/3} x^3 \sqrt{b x^3+a x^6}-x^4 \sqrt{b x^3+a x^6}}\right)","-\sqrt{2} \tan ^{-1}\left(\frac{-2^{2/3} x \sqrt[4]{b x^3+a x^6}+x^2 \sqrt[4]{b x^3+a x^6}}{2 \sqrt[6]{2}-\sqrt{2} x+2^{2/3} x \sqrt[4]{b x^3+a x^6}-x^2 \sqrt[4]{b x^3+a x^6}}\right)+\sqrt{2} \tan ^{-1}\left(\frac{-2^{2/3} x \sqrt[4]{b x^3+a x^6}+x^2 \sqrt[4]{b x^3+a x^6}}{-2 \sqrt[6]{2}+\sqrt{2} x+2^{2/3} x \sqrt[4]{b x^3+a x^6}-x^2 \sqrt[4]{b x^3+a x^6}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{-2 2^{5/6} x \sqrt[4]{b x^3+a x^6}+4 \sqrt[6]{2} x^2 \sqrt[4]{b x^3+a x^6}-\sqrt{2} x^3 \sqrt[4]{b x^3+a x^6}}{-2 \sqrt[3]{2}+2\ 2^{2/3} x-x^2-2 \sqrt[3]{2} x^2 \sqrt{b x^3+a x^6}+2\ 2^{2/3} x^3 \sqrt{b x^3+a x^6}-x^4 \sqrt{b x^3+a x^6}}\right)",1,"-(Sqrt[2]*ArcTan[(-(2^(2/3)*x*(b*x^3 + a*x^6)^(1/4)) + x^2*(b*x^3 + a*x^6)^(1/4))/(2*2^(1/6) - Sqrt[2]*x + 2^(2/3)*x*(b*x^3 + a*x^6)^(1/4) - x^2*(b*x^3 + a*x^6)^(1/4))]) + Sqrt[2]*ArcTan[(-(2^(2/3)*x*(b*x^3 + a*x^6)^(1/4)) + x^2*(b*x^3 + a*x^6)^(1/4))/(-2*2^(1/6) + Sqrt[2]*x + 2^(2/3)*x*(b*x^3 + a*x^6)^(1/4) - x^2*(b*x^3 + a*x^6)^(1/4))] - Sqrt[2]*ArcTanh[(-2*2^(5/6)*x*(b*x^3 + a*x^6)^(1/4) + 4*2^(1/6)*x^2*(b*x^3 + a*x^6)^(1/4) - Sqrt[2]*x^3*(b*x^3 + a*x^6)^(1/4))/(-2*2^(1/3) + 2*2^(2/3)*x - x^2 - 2*2^(1/3)*x^2*Sqrt[b*x^3 + a*x^6] + 2*2^(2/3)*x^3*Sqrt[b*x^3 + a*x^6] - x^4*Sqrt[b*x^3 + a*x^6])]","A",1
2998,1,367,399,3.495871,"\int \frac{x^4 \left(-2 q+p x^3\right) \sqrt{q+p x^3}}{b x^8+a \left(q+p x^3\right)^4} \, dx","IntegrateAlgebraic[(x^4*(-2*q + p*x^3)*Sqrt[q + p*x^3])/(b*x^8 + a*(q + p*x^3)^4),x]","-\frac{\sqrt{2+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{a} \sqrt[8]{b} x \sqrt{q+p x^3}}{\sqrt[4]{a} q-\sqrt[4]{b} x^2+\sqrt[4]{a} p x^3}\right)}{4 a^{3/8} b^{5/8}}+\frac{\sqrt{2-\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{a} \sqrt[8]{b} x \sqrt{q+p x^3}}{\sqrt[4]{a} q-\sqrt[4]{b} x^2+\sqrt[4]{a} p x^3}\right)}{4 a^{3/8} b^{5/8}}-\frac{\sqrt{2+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{a} \sqrt[8]{b} x \sqrt{q+p x^3}}{\sqrt[4]{a} q+\sqrt[4]{b} x^2+\sqrt[4]{a} p x^3}\right)}{4 a^{3/8} b^{5/8}}+\frac{\sqrt{2-\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{a} \sqrt[8]{b} x \sqrt{q+p x^3}}{\sqrt[4]{a} q+\sqrt[4]{b} x^2+\sqrt[4]{a} p x^3}\right)}{4 a^{3/8} b^{5/8}}","-\frac{\sqrt{2+\sqrt{2}} \tan ^{-1}\left(\frac{\left(\sqrt{\frac{2}{2-\sqrt{2}}} \sqrt[8]{a} \sqrt[8]{b}-\frac{2 \sqrt[8]{a} \sqrt[8]{b}}{\sqrt{2-\sqrt{2}}}\right) x \sqrt{q+p x^3}}{-\sqrt[4]{a} q+\sqrt[4]{b} x^2-\sqrt[4]{a} p x^3}\right)}{4 a^{3/8} b^{5/8}}+\frac{\sqrt{2-\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{a} \sqrt[8]{b} x \sqrt{q+p x^3}}{\sqrt[4]{a} q-\sqrt[4]{b} x^2+\sqrt[4]{a} p x^3}\right)}{4 a^{3/8} b^{5/8}}-\frac{\sqrt{2+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{a} \sqrt[8]{b} x \sqrt{q+p x^3}}{\sqrt[4]{a} q+\sqrt[4]{b} x^2+\sqrt[4]{a} p x^3}\right)}{4 a^{3/8} b^{5/8}}+\frac{\sqrt{2-\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{a} \sqrt[8]{b} x \sqrt{q+p x^3}}{\sqrt[4]{a} q+\sqrt[4]{b} x^2+\sqrt[4]{a} p x^3}\right)}{4 a^{3/8} b^{5/8}}",1,"-1/4*(Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]]*a^(1/8)*b^(1/8)*x*Sqrt[q + p*x^3])/(a^(1/4)*q - b^(1/4)*x^2 + a^(1/4)*p*x^3)])/(a^(3/8)*b^(5/8)) + (Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]]*a^(1/8)*b^(1/8)*x*Sqrt[q + p*x^3])/(a^(1/4)*q - b^(1/4)*x^2 + a^(1/4)*p*x^3)])/(4*a^(3/8)*b^(5/8)) - (Sqrt[2 + Sqrt[2]]*ArcTanh[(Sqrt[2 - Sqrt[2]]*a^(1/8)*b^(1/8)*x*Sqrt[q + p*x^3])/(a^(1/4)*q + b^(1/4)*x^2 + a^(1/4)*p*x^3)])/(4*a^(3/8)*b^(5/8)) + (Sqrt[2 - Sqrt[2]]*ArcTanh[(Sqrt[2 + Sqrt[2]]*a^(1/8)*b^(1/8)*x*Sqrt[q + p*x^3])/(a^(1/4)*q + b^(1/4)*x^2 + a^(1/4)*p*x^3)])/(4*a^(3/8)*b^(5/8))","A",1
2999,1,502,399,0.686522,"\int \frac{\sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1+x^2\right)^2} \, dx","IntegrateAlgebraic[(Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 + x^2)^2,x]","\frac{\left(-1-2 x-x^2-2 x^3\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(1+18 x+x^2+26 x^3\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(-1-x-2 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(5+x+26 x^2\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{16 \sqrt{1+x^2} \left(2 x+2 x^3\right)+16 \left(1+3 x^2+2 x^4\right)}+\frac{1}{2} \text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{-\text{$\#$1}+\text{$\#$1}^3}\&\right]-\frac{1}{64} \text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{24 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-26 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-16 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+19 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-\text{$\#$1}+3 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","\frac{\left(-1-2 x-x^2-2 x^3\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(1+18 x+x^2+26 x^3\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(-1-x-2 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(5+x+26 x^2\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{16 \sqrt{1+x^2} \left(2 x+2 x^3\right)+16 \left(1+3 x^2+2 x^4\right)}+\frac{1}{64} \text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{8 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-6 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-16 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+13 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-\text{$\#$1}+3 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]",1,"((-1 - 2*x - x^2 - 2*x^3)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (1 + 18*x + x^2 + 26*x^3)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((-1 - x - 2*x^2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (5 + x + 26*x^2)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(16*Sqrt[1 + x^2]*(2*x + 2*x^3) + 16*(1 + 3*x^2 + 2*x^4)) + RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2)/(-#1 + #1^3) & ]/2 - RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (24*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 26*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - 16*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + 19*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(-#1 + 3*#1^3 - 3*#1^5 + #1^7) & ]/64","A",1
3000,1,502,399,0.0018259,"\int \frac{\sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1+x^2\right)^2} \, dx","IntegrateAlgebraic[(Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 + x^2)^2,x]","\frac{\left(-1-2 x-x^2-2 x^3\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(1+18 x+x^2+26 x^3\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(-1-x-2 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(5+x+26 x^2\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{16 \sqrt{1+x^2} \left(2 x+2 x^3\right)+16 \left(1+3 x^2+2 x^4\right)}+\frac{1}{2} \text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{-\text{$\#$1}+\text{$\#$1}^3}\&\right]-\frac{1}{64} \text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{24 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-26 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-16 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+19 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-\text{$\#$1}+3 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","\frac{\left(-1-2 x-x^2-2 x^3\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(1+18 x+x^2+26 x^3\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(-1-x-2 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(5+x+26 x^2\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{16 \sqrt{1+x^2} \left(2 x+2 x^3\right)+16 \left(1+3 x^2+2 x^4\right)}+\frac{1}{64} \text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{8 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-6 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-16 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+13 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-\text{$\#$1}+3 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]",1,"((-1 - 2*x - x^2 - 2*x^3)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (1 + 18*x + x^2 + 26*x^3)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((-1 - x - 2*x^2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (5 + x + 26*x^2)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(16*Sqrt[1 + x^2]*(2*x + 2*x^3) + 16*(1 + 3*x^2 + 2*x^4)) + RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2)/(-#1 + #1^3) & ]/2 - RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (24*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 26*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - 16*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + 19*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(-#1 + 3*#1^3 - 3*#1^5 + #1^7) & ]/64","A",1
3001,1,552,401,0.7729505,"\int \frac{\sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1-x^2\right)^2} \, dx","IntegrateAlgebraic[(Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^2)^2,x]","-\frac{x \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{2 \left(-1+x^2\right)}+\frac{1}{4} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{2 \text{$\#$1}-3 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]-\frac{1}{16} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+7 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-\frac{1}{4} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-2+4 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]-\frac{1}{16} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-9 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","-\frac{x \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{2 \left(-1+x^2\right)}-\frac{1}{16} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-\frac{1}{16} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-5 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]",1,"-1/2*(x*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(-1 + x^2) + RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/(2*#1 - 3*#1^3 + #1^5) & ]/4 - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 7*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(2*#1^3 - 3*#1^5 + #1^7) & ]/16 - RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(-2 + 4*#1^2 - 3*#1^4 + #1^6) & ]/4 - RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 9*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ]/16","A",1
3002,1,552,401,0.0019757,"\int \frac{\sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1-x^2\right)^2} \, dx","IntegrateAlgebraic[(Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^2)^2,x]","-\frac{x \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{2 \left(-1+x^2\right)}+\frac{1}{4} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{2 \text{$\#$1}-3 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]-\frac{1}{16} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+7 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-\frac{1}{4} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-2+4 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]-\frac{1}{16} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-9 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","-\frac{x \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{2 \left(-1+x^2\right)}-\frac{1}{16} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-\frac{1}{16} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-5 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]",1,"-1/2*(x*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(-1 + x^2) + RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/(2*#1 - 3*#1^3 + #1^5) & ]/4 - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 7*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(2*#1^3 - 3*#1^5 + #1^7) & ]/16 - RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(-2 + 4*#1^2 - 3*#1^4 + #1^6) & ]/4 - RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 9*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ]/16","A",1
3003,1,518,402,0.6310046,"\int \frac{\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1+x^2\right)^2 \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((1 + x^2)^2*Sqrt[x + Sqrt[1 + x^2]]),x]","\frac{\left(-1+2 x-x^2+2 x^3\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-3-2 x+5 x^2-2 x^3\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(1-x+2 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-1+5 x-2 x^2\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{16 \sqrt{1+x^2} \left(2 x+2 x^3\right)+16 \left(1+3 x^2+2 x^4\right)}+\frac{1}{2} \text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+5 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{-\text{$\#$1}+3 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-\frac{1}{64} \text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{36 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+136 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-4 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-\text{$\#$1}+3 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","\frac{\left(-1+2 x-x^2+2 x^3\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-3-2 x+5 x^2-2 x^3\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(1-x+2 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-1+5 x-2 x^2\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{16 \sqrt{1+x^2} \left(2 x+2 x^3\right)+16 \left(1+3 x^2+2 x^4\right)}-\frac{1}{64} \text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{4 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-24 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-4 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-\text{$\#$1}+3 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]",1,"((-1 + 2*x - x^2 + 2*x^3)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-3 - 2*x + 5*x^2 - 2*x^3)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((1 - x + 2*x^2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-1 + 5*x - 2*x^2)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(16*Sqrt[1 + x^2]*(2*x + 2*x^3) + 16*(1 + 3*x^2 + 2*x^4)) + RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 5*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2)/(-#1 + 3*#1^3 - 3*#1^5 + #1^7) & ]/2 - RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (36*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 136*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - 4*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(-#1 + 3*#1^3 - 3*#1^5 + #1^7) & ]/64","A",1
3004,1,518,402,0.0027962,"\int \frac{\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1+x^2\right)^2 \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((1 + x^2)^2*Sqrt[x + Sqrt[1 + x^2]]),x]","\frac{\left(-1+2 x-x^2+2 x^3\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-3-2 x+5 x^2-2 x^3\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(1-x+2 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-1+5 x-2 x^2\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{16 \sqrt{1+x^2} \left(2 x+2 x^3\right)+16 \left(1+3 x^2+2 x^4\right)}+\frac{1}{2} \text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+5 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{-\text{$\#$1}+3 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-\frac{1}{64} \text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{36 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+136 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-4 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-\text{$\#$1}+3 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","\frac{\left(-1+2 x-x^2+2 x^3\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-3-2 x+5 x^2-2 x^3\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(1-x+2 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-1+5 x-2 x^2\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{16 \sqrt{1+x^2} \left(2 x+2 x^3\right)+16 \left(1+3 x^2+2 x^4\right)}-\frac{1}{64} \text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{4 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-24 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-4 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-\text{$\#$1}+3 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]",1,"((-1 + 2*x - x^2 + 2*x^3)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-3 - 2*x + 5*x^2 - 2*x^3)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((1 - x + 2*x^2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-1 + 5*x - 2*x^2)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(16*Sqrt[1 + x^2]*(2*x + 2*x^3) + 16*(1 + 3*x^2 + 2*x^4)) + RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 5*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2)/(-#1 + 3*#1^3 - 3*#1^5 + #1^7) & ]/2 - RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (36*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 136*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - 4*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(-#1 + 3*#1^3 - 3*#1^5 + #1^7) & ]/64","A",1
3005,1,322,404,6.566405,"\int \frac{(1+(-2+k) x) \left(1-2 k x+k^2 x^2\right)}{\sqrt[3]{(1-x) x (1-k x)} \left(b-4 b k x+\left(-1+6 b k^2\right) x^2+\left(2-4 b k^3\right) x^3+\left(-1+b k^4\right) x^4\right)} \, dx","IntegrateAlgebraic[((1 + (-2 + k)*x)*(1 - 2*k*x + k^2*x^2))/(((1 - x)*x*(1 - k*x))^(1/3)*(b - 4*b*k*x + (-1 + 6*b*k^2)*x^2 + (2 - 4*b*k^3)*x^3 + (-1 + b*k^4)*x^4)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{b}-2 \sqrt{3} \sqrt[3]{b} k x+\sqrt{3} \sqrt[3]{b} k^2 x^2}{\sqrt[3]{b}-2 \sqrt[3]{b} k x+\sqrt[3]{b} k^2 x^2+2 \left(x+(-1-k) x^2+k x^3\right)^{2/3}}\right)}{2 b^{2/3}}-\frac{\log \left(\sqrt[3]{b} k-2 \sqrt[3]{b} k^2 x+\sqrt[3]{b} k^3 x^2-k \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{2 b^{2/3}}+\frac{\log \left(b^{2/3} k^2-4 b^{2/3} k^3 x+6 b^{2/3} k^4 x^2-4 b^{2/3} k^5 x^3+b^{2/3} k^6 x^4+\left(\sqrt[3]{b} k^2-2 \sqrt[3]{b} k^3 x+\sqrt[3]{b} k^4 x^2\right) \left(x+(-1-k) x^2+k x^3\right)^{2/3}+k^2 \left(x+(-1-k) x^2+k x^3\right)^{4/3}\right)}{4 b^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\frac{1}{\sqrt{3}}-\frac{2 k x}{\sqrt{3}}+\frac{k^2 x^2}{\sqrt{3}}+\frac{2 \left(x+(-1-k) x^2+k x^3\right)^{2/3}}{\sqrt{3} \sqrt[3]{b}}}{(-1+k x)^2}\right)}{2 b^{2/3}}-\frac{\log \left(-\sqrt[6]{b} k+\sqrt[6]{b} k^2 x-k \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{2 b^{2/3}}-\frac{\log \left(-\sqrt[6]{b} k+\sqrt[6]{b} k^2 x+k \sqrt[3]{x+(-1-k) x^2+k x^3}\right)}{2 b^{2/3}}+\frac{\log \left(\sqrt[3]{b} k^2-2 \sqrt[3]{b} k^3 x+\sqrt[3]{b} k^4 x^2+\left(\sqrt[6]{b} k^2-\sqrt[6]{b} k^3 x\right) \sqrt[3]{x+(-1-k) x^2+k x^3}+k^2 \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{4 b^{2/3}}+\frac{\log \left(\sqrt[3]{b} k^2-2 \sqrt[3]{b} k^3 x+\sqrt[3]{b} k^4 x^2+\left(-\sqrt[6]{b} k^2+\sqrt[6]{b} k^3 x\right) \sqrt[3]{x+(-1-k) x^2+k x^3}+k^2 \left(x+(-1-k) x^2+k x^3\right)^{2/3}\right)}{4 b^{2/3}}",1,"-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*b^(1/3) - 2*Sqrt[3]*b^(1/3)*k*x + Sqrt[3]*b^(1/3)*k^2*x^2)/(b^(1/3) - 2*b^(1/3)*k*x + b^(1/3)*k^2*x^2 + 2*(x + (-1 - k)*x^2 + k*x^3)^(2/3))])/b^(2/3) - Log[b^(1/3)*k - 2*b^(1/3)*k^2*x + b^(1/3)*k^3*x^2 - k*(x + (-1 - k)*x^2 + k*x^3)^(2/3)]/(2*b^(2/3)) + Log[b^(2/3)*k^2 - 4*b^(2/3)*k^3*x + 6*b^(2/3)*k^4*x^2 - 4*b^(2/3)*k^5*x^3 + b^(2/3)*k^6*x^4 + (b^(1/3)*k^2 - 2*b^(1/3)*k^3*x + b^(1/3)*k^4*x^2)*(x + (-1 - k)*x^2 + k*x^3)^(2/3) + k^2*(x + (-1 - k)*x^2 + k*x^3)^(4/3)]/(4*b^(2/3))","A",1
3006,1,558,404,0.6659952,"\int \frac{\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1-x^2\right)^2 \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((1 - x^2)^2*Sqrt[x + Sqrt[1 + x^2]]),x]","-\frac{x \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{2 \left(-1+x^2\right) \sqrt{x+\sqrt{1+x^2}}}-\frac{1}{4} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}-3 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]-\frac{1}{16} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+5 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{1}{4} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^3+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^5}{-2+4 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]-\frac{1}{16} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+5 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","-\frac{x \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{2 \left(-1+x^2\right) \sqrt{x+\sqrt{1+x^2}}}-\frac{1}{16} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-8 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+4 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{1}{16} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+7 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-8 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+4 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]",1,"-1/2*(x*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/((-1 + x^2)*Sqrt[x + Sqrt[1 + x^2]]) - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (-Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1 - 3*#1^3 + #1^5) & ]/4 - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 5*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2)/(2*#1^3 - 3*#1^5 + #1^7) & ]/16 + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1 - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^3 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^5)/(-2 + 4*#1^2 - 3*#1^4 + #1^6) & ]/4 - RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 5*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ]/16","A",1
3007,1,558,404,0.0018309,"\int \frac{\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1-x^2\right)^2 \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((1 - x^2)^2*Sqrt[x + Sqrt[1 + x^2]]),x]","-\frac{x \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{2 \left(-1+x^2\right) \sqrt{x+\sqrt{1+x^2}}}-\frac{1}{4} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}-3 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]-\frac{1}{16} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+5 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{1}{4} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^3+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^5}{-2+4 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]-\frac{1}{16} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+5 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","-\frac{x \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{2 \left(-1+x^2\right) \sqrt{x+\sqrt{1+x^2}}}-\frac{1}{16} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-8 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+4 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{1}{16} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+7 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-8 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+4 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]",1,"-1/2*(x*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/((-1 + x^2)*Sqrt[x + Sqrt[1 + x^2]]) - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (-Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1 - 3*#1^3 + #1^5) & ]/4 - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 5*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2)/(2*#1^3 - 3*#1^5 + #1^7) & ]/16 + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1 - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^3 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^5)/(-2 + 4*#1^2 - 3*#1^4 + #1^6) & ]/4 - RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 5*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ]/16","A",1
3008,1,405,405,3.5405238,"\int \frac{(a-2 b+x) \left(a^2-2 a x+x^2\right)}{\sqrt[3]{(-a+x) (-b+x)} \left(-b^2+a^4 d+2 \left(b-2 a^3 d\right) x+\left(-1+6 a^2 d\right) x^2-4 a d x^3+d x^4\right)} \, dx","IntegrateAlgebraic[((a - 2*b + x)*(a^2 - 2*a*x + x^2))/(((-a + x)*(-b + x))^(1/3)*(-b^2 + a^4*d + 2*(b - 2*a^3*d)*x + (-1 + 6*a^2*d)*x^2 - 4*a*d*x^3 + d*x^4)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\frac{a^2}{\sqrt{3}}-\frac{2 a x}{\sqrt{3}}+\frac{x^2}{\sqrt{3}}+\frac{2 \left(a b+(-a-b) x+x^2\right)^{2/3}}{\sqrt{3} \sqrt[3]{d}}}{(a-x)^2}\right)}{2 d^{2/3}}+\frac{\log \left(a^2 \sqrt[6]{d}-a \sqrt[6]{d} x-a \sqrt[3]{a b+(-a-b) x+x^2}\right)}{2 d^{2/3}}+\frac{\log \left(a^2 \sqrt[6]{d}-a \sqrt[6]{d} x+a \sqrt[3]{a b+(-a-b) x+x^2}\right)}{2 d^{2/3}}-\frac{\log \left(a^4 \sqrt[3]{d}-2 a^3 \sqrt[3]{d} x+a^2 \sqrt[3]{d} x^2+\left(a^3 \sqrt[6]{d}-a^2 \sqrt[6]{d} x\right) \sqrt[3]{a b+(-a-b) x+x^2}+a^2 \left(a b+(-a-b) x+x^2\right)^{2/3}\right)}{4 d^{2/3}}-\frac{\log \left(a^4 \sqrt[3]{d}-2 a^3 \sqrt[3]{d} x+a^2 \sqrt[3]{d} x^2+\left(-a^3 \sqrt[6]{d}+a^2 \sqrt[6]{d} x\right) \sqrt[3]{a b+(-a-b) x+x^2}+a^2 \left(a b+(-a-b) x+x^2\right)^{2/3}\right)}{4 d^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\frac{a^2}{\sqrt{3}}-\frac{2 a x}{\sqrt{3}}+\frac{x^2}{\sqrt{3}}+\frac{2 \left(a b+(-a-b) x+x^2\right)^{2/3}}{\sqrt{3} \sqrt[3]{d}}}{(a-x)^2}\right)}{2 d^{2/3}}+\frac{\log \left(a^2 \sqrt[6]{d}-a \sqrt[6]{d} x-a \sqrt[3]{a b+(-a-b) x+x^2}\right)}{2 d^{2/3}}+\frac{\log \left(a^2 \sqrt[6]{d}-a \sqrt[6]{d} x+a \sqrt[3]{a b+(-a-b) x+x^2}\right)}{2 d^{2/3}}-\frac{\log \left(a^4 \sqrt[3]{d}-2 a^3 \sqrt[3]{d} x+a^2 \sqrt[3]{d} x^2+\left(a^3 \sqrt[6]{d}-a^2 \sqrt[6]{d} x\right) \sqrt[3]{a b+(-a-b) x+x^2}+a^2 \left(a b+(-a-b) x+x^2\right)^{2/3}\right)}{4 d^{2/3}}-\frac{\log \left(a^4 \sqrt[3]{d}-2 a^3 \sqrt[3]{d} x+a^2 \sqrt[3]{d} x^2+\left(-a^3 \sqrt[6]{d}+a^2 \sqrt[6]{d} x\right) \sqrt[3]{a b+(-a-b) x+x^2}+a^2 \left(a b+(-a-b) x+x^2\right)^{2/3}\right)}{4 d^{2/3}}",1,"-1/2*(Sqrt[3]*ArcTan[(a^2/Sqrt[3] - (2*a*x)/Sqrt[3] + x^2/Sqrt[3] + (2*(a*b + (-a - b)*x + x^2)^(2/3))/(Sqrt[3]*d^(1/3)))/(a - x)^2])/d^(2/3) + Log[a^2*d^(1/6) - a*d^(1/6)*x - a*(a*b + (-a - b)*x + x^2)^(1/3)]/(2*d^(2/3)) + Log[a^2*d^(1/6) - a*d^(1/6)*x + a*(a*b + (-a - b)*x + x^2)^(1/3)]/(2*d^(2/3)) - Log[a^4*d^(1/3) - 2*a^3*d^(1/3)*x + a^2*d^(1/3)*x^2 + (a^3*d^(1/6) - a^2*d^(1/6)*x)*(a*b + (-a - b)*x + x^2)^(1/3) + a^2*(a*b + (-a - b)*x + x^2)^(2/3)]/(4*d^(2/3)) - Log[a^4*d^(1/3) - 2*a^3*d^(1/3)*x + a^2*d^(1/3)*x^2 + (-(a^3*d^(1/6)) + a^2*d^(1/6)*x)*(a*b + (-a - b)*x + x^2)^(1/3) + a^2*(a*b + (-a - b)*x + x^2)^(2/3)]/(4*d^(2/3))","A",1
3009,1,406,406,0.5518242,"\int \frac{\left(1+x^4\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1-x^4\right) \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[((1 + x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/((1 - x^4)*Sqrt[x + Sqrt[1 + x^2]]),x]","\frac{\left(-8-3 x-32 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+(2-16 x) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left((-3-32 x) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-16 \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{24 \left(x+\sqrt{1+x^2}\right)^{3/2}}+\frac{1}{8} \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)-\frac{1}{2} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^2}\&\right]+\text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+3 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]+\frac{1}{2} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^2}\&\right]","\frac{\left(-8-3 x-32 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+(2-16 x) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left((-3-32 x) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-16 \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{24 \left(x+\sqrt{1+x^2}\right)^{3/2}}+\frac{1}{8} \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)-\frac{1}{2} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^2}\&\right]+\text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+3 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]+\frac{1}{2} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^2}\&\right]",1,"((-8 - 3*x - 32*x^2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (2 - 16*x)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((-3 - 32*x)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - 16*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(24*(x + Sqrt[1 + x^2])^(3/2)) + ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]]/8 - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(-1 + #1^2) & ]/2 + RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(-1 + 3*#1^2 - 3*#1^4 + #1^6) & ] + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(-1 + #1^2) & ]/2","A",1
3010,1,406,406,0.0019281,"\int \frac{\left(1+x^4\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1-x^4\right) \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[((1 + x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/((1 - x^4)*Sqrt[x + Sqrt[1 + x^2]]),x]","\frac{\left(-8-3 x-32 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+(2-16 x) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left((-3-32 x) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-16 \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{24 \left(x+\sqrt{1+x^2}\right)^{3/2}}+\frac{1}{8} \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)-\frac{1}{2} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^2}\&\right]+\text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+3 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]+\frac{1}{2} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^2}\&\right]","\frac{\left(-8-3 x-32 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+(2-16 x) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left((-3-32 x) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-16 \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{24 \left(x+\sqrt{1+x^2}\right)^{3/2}}+\frac{1}{8} \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)-\frac{1}{2} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^2}\&\right]+\text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+3 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]+\frac{1}{2} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^2}\&\right]",1,"((-8 - 3*x - 32*x^2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (2 - 16*x)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((-3 - 32*x)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - 16*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(24*(x + Sqrt[1 + x^2])^(3/2)) + ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]]/8 - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(-1 + #1^2) & ]/2 + RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(-1 + 3*#1^2 - 3*#1^4 + #1^6) & ] + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(-1 + #1^2) & ]/2","A",1
3011,1,388,407,0.8871427,"\int \frac{\sqrt{b+a x} \sqrt{c+\sqrt{b+a x}}}{x-\sqrt{b+a x}} \, dx","IntegrateAlgebraic[(Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]])/(x - Sqrt[b + a*x]),x]","\frac{4}{3} \sqrt{c+\sqrt{b+a x}} \left(3 a+c+\sqrt{b+a x}\right)+\frac{2 \left(\sqrt{2} a^3+3 \sqrt{2} a b+\sqrt{2} a^2 \sqrt{a^2+4 b}+\sqrt{2} b \sqrt{a^2+4 b}+\sqrt{2} a^2 c+2 \sqrt{2} b c+\sqrt{2} a \sqrt{a^2+4 b} c\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c+\sqrt{b+a x}}}{\sqrt{-a-\sqrt{a^2+4 b}-2 c}}\right)}{\sqrt{a^2+4 b} \sqrt{-a-\sqrt{a^2+4 b}-2 c}}+\frac{2 \left(-\sqrt{2} a^3-3 \sqrt{2} a b+\sqrt{2} a^2 \sqrt{a^2+4 b}+\sqrt{2} b \sqrt{a^2+4 b}-\sqrt{2} a^2 c-2 \sqrt{2} b c+\sqrt{2} a \sqrt{a^2+4 b} c\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c+\sqrt{b+a x}}}{\sqrt{-a+\sqrt{a^2+4 b}-2 c}}\right)}{\sqrt{a^2+4 b} \sqrt{-a+\sqrt{a^2+4 b}-2 c}}","\frac{4}{3} (3 a+c) \sqrt{c+\sqrt{b+a x}}+\frac{4}{3} \sqrt{b+a x} \sqrt{c+\sqrt{b+a x}}+\frac{2 \left(\sqrt{2} a^3+3 \sqrt{2} a b+\sqrt{2} a^2 \sqrt{a^2+4 b}+\sqrt{2} b \sqrt{a^2+4 b}+\sqrt{2} a^2 c+2 \sqrt{2} b c+\sqrt{2} a \sqrt{a^2+4 b} c\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c+\sqrt{b+a x}}}{\sqrt{-a-\sqrt{a^2+4 b}-2 c}}\right)}{\sqrt{a^2+4 b} \sqrt{-a-\sqrt{a^2+4 b}-2 c}}+\frac{2 \left(-\sqrt{2} a^3-3 \sqrt{2} a b+\sqrt{2} a^2 \sqrt{a^2+4 b}+\sqrt{2} b \sqrt{a^2+4 b}-\sqrt{2} a^2 c-2 \sqrt{2} b c+\sqrt{2} a \sqrt{a^2+4 b} c\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c+\sqrt{b+a x}}}{\sqrt{-a+\sqrt{a^2+4 b}-2 c}}\right)}{\sqrt{a^2+4 b} \sqrt{-a+\sqrt{a^2+4 b}-2 c}}",1,"(4*Sqrt[c + Sqrt[b + a*x]]*(3*a + c + Sqrt[b + a*x]))/3 + (2*(Sqrt[2]*a^3 + 3*Sqrt[2]*a*b + Sqrt[2]*a^2*Sqrt[a^2 + 4*b] + Sqrt[2]*b*Sqrt[a^2 + 4*b] + Sqrt[2]*a^2*c + 2*Sqrt[2]*b*c + Sqrt[2]*a*Sqrt[a^2 + 4*b]*c)*ArcTan[(Sqrt[2]*Sqrt[c + Sqrt[b + a*x]])/Sqrt[-a - Sqrt[a^2 + 4*b] - 2*c]])/(Sqrt[a^2 + 4*b]*Sqrt[-a - Sqrt[a^2 + 4*b] - 2*c]) + (2*(-(Sqrt[2]*a^3) - 3*Sqrt[2]*a*b + Sqrt[2]*a^2*Sqrt[a^2 + 4*b] + Sqrt[2]*b*Sqrt[a^2 + 4*b] - Sqrt[2]*a^2*c - 2*Sqrt[2]*b*c + Sqrt[2]*a*Sqrt[a^2 + 4*b]*c)*ArcTan[(Sqrt[2]*Sqrt[c + Sqrt[b + a*x]])/Sqrt[-a + Sqrt[a^2 + 4*b] - 2*c]])/(Sqrt[a^2 + 4*b]*Sqrt[-a + Sqrt[a^2 + 4*b] - 2*c])","A",1
3012,1,426,411,1.1853724,"\int \frac{1+x}{(1-a x) \sqrt[4]{\frac{1-b x}{c+x}}} \, dx","IntegrateAlgebraic[(1 + x)/((1 - a*x)*((1 - b*x)/(c + x))^(1/4)),x]","\frac{(1+b c) \left(\frac{1-b x}{c+x}\right)^{3/4}}{a b \left(b+\frac{1-b x}{c+x}\right)}-\frac{(a+4 b+4 a b+a b c) \tan ^{-1}\left(\frac{\sqrt{b}-\sqrt{\frac{1-b x}{c+x}}}{\sqrt{2} \sqrt[4]{b} \sqrt[4]{\frac{1-b x}{c+x}}}\right)}{2 \sqrt{2} a^2 b^{5/4}}+\frac{\sqrt{2} (1+a) \sqrt[4]{1+a c} \tan ^{-1}\left(\frac{\sqrt{-a+b}-\sqrt{1+a c} \sqrt{\frac{1-b x}{c+x}}}{\sqrt{2} \sqrt[4]{-a+b} \sqrt[4]{1+a c} \sqrt[4]{\frac{1-b x}{c+x}}}\right)}{a^2 \sqrt[4]{-a+b}}-\frac{(a+4 b+4 a b+a b c) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{\frac{1-b x}{c+x}}}{\sqrt{b}+\sqrt{\frac{1-b x}{c+x}}}\right)}{2 \sqrt{2} a^2 b^{5/4}}+\frac{\sqrt{2} (1+a) \sqrt[4]{1+a c} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-a+b} \sqrt[4]{1+a c} \sqrt[4]{\frac{1-b x}{c+x}}}{\sqrt{-a+b}+\sqrt{1+a c} \sqrt{\frac{1-b x}{c+x}}}\right)}{a^2 \sqrt[4]{-a+b}}","\frac{(c+x) \left(\frac{1-b x}{c+x}\right)^{3/4}}{a b}-\frac{(a+4 b+4 a b+a b c) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{\frac{1-b x}{c+x}}}{-\sqrt{b}+\sqrt{\frac{1-b x}{c+x}}}\right)}{2 \sqrt{2} a^2 b^{5/4}}-\frac{\sqrt{2} (1+a) \sqrt[4]{1+a c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-a+b} \sqrt[4]{1+a c} \sqrt[4]{\frac{1-b x}{c+x}}}{\sqrt{-a+b}-\sqrt{1+a c} \sqrt{\frac{1-b x}{c+x}}}\right)}{a^2 \sqrt[4]{-a+b}}-\frac{(a+4 b+4 a b+a b c) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt[4]{\frac{1-b x}{c+x}}}{\sqrt{b}+\sqrt{\frac{1-b x}{c+x}}}\right)}{2 \sqrt{2} a^2 b^{5/4}}+\frac{\sqrt{2} (1+a) \sqrt[4]{1+a c} \tanh ^{-1}\left(\frac{\sqrt{-a+b}+\sqrt{1+a c} \sqrt{\frac{1-b x}{c+x}}}{\sqrt{2} \sqrt[4]{-a+b} \sqrt[4]{1+a c} \sqrt[4]{\frac{1-b x}{c+x}}}\right)}{a^2 \sqrt[4]{-a+b}}",1,"((1 + b*c)*((1 - b*x)/(c + x))^(3/4))/(a*b*(b + (1 - b*x)/(c + x))) - ((a + 4*b + 4*a*b + a*b*c)*ArcTan[(Sqrt[b] - Sqrt[(1 - b*x)/(c + x)])/(Sqrt[2]*b^(1/4)*((1 - b*x)/(c + x))^(1/4))])/(2*Sqrt[2]*a^2*b^(5/4)) + (Sqrt[2]*(1 + a)*(1 + a*c)^(1/4)*ArcTan[(Sqrt[-a + b] - Sqrt[1 + a*c]*Sqrt[(1 - b*x)/(c + x)])/(Sqrt[2]*(-a + b)^(1/4)*(1 + a*c)^(1/4)*((1 - b*x)/(c + x))^(1/4))])/(a^2*(-a + b)^(1/4)) - ((a + 4*b + 4*a*b + a*b*c)*ArcTanh[(Sqrt[2]*b^(1/4)*((1 - b*x)/(c + x))^(1/4))/(Sqrt[b] + Sqrt[(1 - b*x)/(c + x)])])/(2*Sqrt[2]*a^2*b^(5/4)) + (Sqrt[2]*(1 + a)*(1 + a*c)^(1/4)*ArcTanh[(Sqrt[2]*(-a + b)^(1/4)*(1 + a*c)^(1/4)*((1 - b*x)/(c + x))^(1/4))/(Sqrt[-a + b] + Sqrt[1 + a*c]*Sqrt[(1 - b*x)/(c + x)])])/(a^2*(-a + b)^(1/4))","A",1
3013,0,0,413,78.1738761,"\int \frac{x^5 \left(-7 b+9 a x^2\right)}{\sqrt[4]{-b x^3+a x^5} \left(1-b x^7+a x^9\right)} \, dx","IntegrateAlgebraic[(x^5*(-7*b + 9*a*x^2))/((-(b*x^3) + a*x^5)^(1/4)*(1 - b*x^7 + a*x^9)),x]","\int \frac{x^5 \left(-7 b+9 a x^2\right)}{\sqrt[4]{-b x^3+a x^5} \left(1-b x^7+a x^9\right)} \, dx","-\sqrt{2} \tan ^{-1}\left(\frac{-2^{2/3} x \sqrt[4]{-b x^3+a x^5}+x^2 \sqrt[4]{-b x^3+a x^5}}{2 \sqrt[6]{2}-\sqrt{2} x+2^{2/3} x \sqrt[4]{-b x^3+a x^5}-x^2 \sqrt[4]{-b x^3+a x^5}}\right)+\sqrt{2} \tan ^{-1}\left(\frac{-2^{2/3} x \sqrt[4]{-b x^3+a x^5}+x^2 \sqrt[4]{-b x^3+a x^5}}{-2 \sqrt[6]{2}+\sqrt{2} x+2^{2/3} x \sqrt[4]{-b x^3+a x^5}-x^2 \sqrt[4]{-b x^3+a x^5}}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{-2 2^{5/6} x \sqrt[4]{-b x^3+a x^5}+4 \sqrt[6]{2} x^2 \sqrt[4]{-b x^3+a x^5}-\sqrt{2} x^3 \sqrt[4]{-b x^3+a x^5}}{-2 \sqrt[3]{2}+2\ 2^{2/3} x-x^2-2 \sqrt[3]{2} x^2 \sqrt{-b x^3+a x^5}+2\ 2^{2/3} x^3 \sqrt{-b x^3+a x^5}-x^4 \sqrt{-b x^3+a x^5}}\right)",1,"Could not integrate","F",-1
3014,1,31,415,1.1750012,"\int \frac{b+a x^4}{\sqrt{-b+a x^4} \left(-b+c^2 x^2+a x^4\right)} \, dx","IntegrateAlgebraic[(b + a*x^4)/(Sqrt[-b + a*x^4]*(-b + c^2*x^2 + a*x^4)),x]","\frac{\tan ^{-1}\left(\frac{c x \sqrt{-b+a x^4}}{b-a x^4}\right)}{c}","\frac{i \left(\sqrt{2} c-(1-i) \sqrt{2 \sqrt{a} \sqrt{b}+i c^2}\right) \sqrt{-i \sqrt{a} \sqrt{b}+c^2-(-1)^{3/4} c \sqrt{2 \sqrt{a} \sqrt{b}+i c^2}} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{-i \sqrt{a} \sqrt{b}+c^2-(-1)^{3/4} c \sqrt{2 \sqrt{a} \sqrt{b}+i c^2}} x}{i \sqrt{b}+\sqrt{a} x^2+\sqrt{-b+a x^4}}\right)}{2 \sqrt{a} \sqrt{b} c}-\frac{i \left(\sqrt{2} c+(1-i) \sqrt{2 \sqrt{a} \sqrt{b}+i c^2}\right) \sqrt{i \sqrt{a} \sqrt{b}-c^2-(-1)^{3/4} c \sqrt{2 \sqrt{a} \sqrt{b}+i c^2}} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{i \sqrt{a} \sqrt{b}-c^2-(-1)^{3/4} c \sqrt{2 \sqrt{a} \sqrt{b}+i c^2}} x}{i \sqrt{b}+\sqrt{a} x^2+\sqrt{-b+a x^4}}\right)}{2 \sqrt{a} \sqrt{b} c}",1,"ArcTan[(c*x*Sqrt[-b + a*x^4])/(b - a*x^4)]/c","A",1
3015,1,415,415,1.6906613,"\int \frac{x (-a+x) \left(a b-2 b x+x^2\right)}{\sqrt[3]{x (-a+x) (-b+x)^2} \left(-b^2+2 b x+\left(-1+a^2 d\right) x^2-2 a d x^3+d x^4\right)} \, dx","IntegrateAlgebraic[(x*(-a + x)*(a*b - 2*b*x + x^2))/((x*(-a + x)*(-b + x)^2)^(1/3)*(-b^2 + 2*b*x + (-1 + a^2*d)*x^2 - 2*a*d*x^3 + d*x^4)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{2 b-2 x+\sqrt[6]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)}{2 d^{5/6}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{-2 b+2 x+\sqrt[6]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)}{2 d^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{b-x}\right)}{d^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\left(b \sqrt[6]{d}-\sqrt[6]{d} x\right) \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{b^2-2 b x+x^2+\sqrt[3]{d} \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{2/3}}\right)}{2 d^{5/6}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{2 b-2 x+\sqrt[6]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)}{2 d^{5/6}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{-2 b+2 x+\sqrt[6]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}\right)}{2 d^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{d} \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{b-x}\right)}{d^{5/6}}+\frac{\tanh ^{-1}\left(\frac{\left(b \sqrt[6]{d}-\sqrt[6]{d} x\right) \sqrt[3]{-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4}}{b^2-2 b x+x^2+\sqrt[3]{d} \left(-a b^2 x+\left(2 a b+b^2\right) x^2+(-a-2 b) x^3+x^4\right)^{2/3}}\right)}{2 d^{5/6}}",1,"-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/6)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3))/(2*b - 2*x + d^(1/6)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3))])/d^(5/6) + (Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/6)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3))/(-2*b + 2*x + d^(1/6)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3))])/(2*d^(5/6)) + ArcTanh[(d^(1/6)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3))/(b - x)]/d^(5/6) + ArcTanh[((b*d^(1/6) - d^(1/6)*x)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(1/3))/(b^2 - 2*b*x + x^2 + d^(1/3)*(-(a*b^2*x) + (2*a*b + b^2)*x^2 + (-a - 2*b)*x^3 + x^4)^(2/3))]/(2*d^(5/6))","A",1
3016,1,466,417,3.139626,"\int \frac{b+a x}{(-b+a x) \sqrt[3]{b^2 x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[(b + a*x)/((-b + a*x)*(b^2*x^2 + a^3*x^3)^(1/3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{a}+\frac{\sqrt{6 \left(-1+i \sqrt{3}\right)} \tan ^{-1}\left(\frac{3 \sqrt[3]{a} \sqrt[3]{a^2+b} x}{\sqrt{3} \sqrt[3]{a} \sqrt[3]{a^2+b} x-3 i \sqrt[3]{b^2 x^2+a^3 x^3}-\sqrt{3} \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{\sqrt[3]{a} \sqrt[3]{a^2+b}}-\frac{i \left(-i+\sqrt{3}\right) \log \left(2 \sqrt[3]{a} \sqrt[3]{a^2+b} x+\left(1+i \sqrt{3}\right) \sqrt[3]{b^2 x^2+a^3 x^3}\right)}{\sqrt[3]{a} \sqrt[3]{a^2+b}}-\frac{\log \left(a^2 x-a \sqrt[3]{b^2 x^2+a^3 x^3}\right)}{a}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{b^2 x^2+a^3 x^3}+\left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{2 a}+\frac{\left(1+i \sqrt{3}\right) \log \left(-2 i a^{2/3} \left(a^2+b\right)^{2/3} x^2+\sqrt[3]{a} \sqrt[3]{a^2+b} \left(i x-\sqrt{3} x\right) \sqrt[3]{b^2 x^2+a^3 x^3}+\left(i+\sqrt{3}\right) \left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{2 \sqrt[3]{a} \sqrt[3]{a^2+b}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{a}+\frac{\sqrt{6 \left(-1+i \sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a} \sqrt[3]{a^2+b} x}{\sqrt[3]{a} \sqrt[3]{a^2+b} x-2 \sqrt[3]{-1} \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{\sqrt[3]{a} \sqrt[3]{a^2+b}}-\frac{\log \left(-a x+\sqrt[3]{b^2 x^2+a^3 x^3}\right)}{a}-\frac{i \left(-i+\sqrt{3}\right) \log \left(\sqrt[3]{a} \sqrt[3]{a^2+b} x+\sqrt[3]{-1} \sqrt[3]{b^2 x^2+a^3 x^3}\right)}{\sqrt[3]{a} \sqrt[3]{a^2+b}}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{b^2 x^2+a^3 x^3}+\left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{2 a}+\frac{\left(1+i \sqrt{3}\right) \log \left(a^{2/3} \left(a^2+b\right)^{2/3} x^2-\sqrt[3]{-1} \sqrt[3]{a} \sqrt[3]{a^2+b} x \sqrt[3]{b^2 x^2+a^3 x^3}+(-1)^{2/3} \left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{2 \sqrt[3]{a} \sqrt[3]{a^2+b}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*a*x)/(a*x + 2*(b^2*x^2 + a^3*x^3)^(1/3))])/a + (Sqrt[6*(-1 + I*Sqrt[3])]*ArcTan[(3*a^(1/3)*(a^2 + b)^(1/3)*x)/(Sqrt[3]*a^(1/3)*(a^2 + b)^(1/3)*x - (3*I)*(b^2*x^2 + a^3*x^3)^(1/3) - Sqrt[3]*(b^2*x^2 + a^3*x^3)^(1/3))])/(a^(1/3)*(a^2 + b)^(1/3)) - (I*(-I + Sqrt[3])*Log[2*a^(1/3)*(a^2 + b)^(1/3)*x + (1 + I*Sqrt[3])*(b^2*x^2 + a^3*x^3)^(1/3)])/(a^(1/3)*(a^2 + b)^(1/3)) - Log[a^2*x - a*(b^2*x^2 + a^3*x^3)^(1/3)]/a + Log[a^2*x^2 + a*x*(b^2*x^2 + a^3*x^3)^(1/3) + (b^2*x^2 + a^3*x^3)^(2/3)]/(2*a) + ((1 + I*Sqrt[3])*Log[(-2*I)*a^(2/3)*(a^2 + b)^(2/3)*x^2 + a^(1/3)*(a^2 + b)^(1/3)*(I*x - Sqrt[3]*x)*(b^2*x^2 + a^3*x^3)^(1/3) + (I + Sqrt[3])*(b^2*x^2 + a^3*x^3)^(2/3)])/(2*a^(1/3)*(a^2 + b)^(1/3))","A",1
3017,1,421,421,30.8080762,"\int \frac{\sqrt{b+a^2 x^4} \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{-b+a^2 x^4} \, dx","IntegrateAlgebraic[(Sqrt[b + a^2*x^4]*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/(-b + a^2*x^4),x]","\frac{\log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}+i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{\sqrt{2} \sqrt{a}}-2 \sqrt{2} \sqrt{a} b \text{RootSum}\left[a^4 b^4+20 a^3 b^3 \text{$\#$1}^2-26 a^2 b^2 \text{$\#$1}^4+20 a b \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{a^2 b^2 \log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}+i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)-2 a b \log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}+i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}+i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^4}{5 a^3 b^3-13 a^2 b^2 \text{$\#$1}^2+15 a b \text{$\#$1}^4+\text{$\#$1}^6}\&\right]","\frac{\log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}+i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{\sqrt{2} \sqrt{a}}-2 \sqrt{2} \sqrt{a} b \text{RootSum}\left[a^4 b^4+20 a^3 b^3 \text{$\#$1}^2-26 a^2 b^2 \text{$\#$1}^4+20 a b \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{a^2 b^2 \log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}+i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)-2 a b \log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}+i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}+i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^4}{5 a^3 b^3-13 a^2 b^2 \text{$\#$1}^2+15 a b \text{$\#$1}^4+\text{$\#$1}^6}\&\right]",1,"Log[I*a^(3/2)*x^2 + I*Sqrt[a]*Sqrt[b + a^2*x^4] + I*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]]/(Sqrt[2]*Sqrt[a]) - 2*Sqrt[2]*Sqrt[a]*b*RootSum[a^4*b^4 + 20*a^3*b^3*#1^2 - 26*a^2*b^2*#1^4 + 20*a*b*#1^6 + #1^8 & , (a^2*b^2*Log[I*a^(3/2)*x^2 + I*Sqrt[a]*Sqrt[b + a^2*x^4] + I*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1] - 2*a*b*Log[I*a^(3/2)*x^2 + I*Sqrt[a]*Sqrt[b + a^2*x^4] + I*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1]*#1^2 + Log[I*a^(3/2)*x^2 + I*Sqrt[a]*Sqrt[b + a^2*x^4] + I*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1]*#1^4)/(5*a^3*b^3 - 13*a^2*b^2*#1^2 + 15*a*b*#1^4 + #1^6) & ]","A",1
3018,1,421,421,0.0026465,"\int \frac{\sqrt{b+a^2 x^4} \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{-b+a^2 x^4} \, dx","IntegrateAlgebraic[(Sqrt[b + a^2*x^4]*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/(-b + a^2*x^4),x]","\frac{\log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}+i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{\sqrt{2} \sqrt{a}}-2 \sqrt{2} \sqrt{a} b \text{RootSum}\left[a^4 b^4+20 a^3 b^3 \text{$\#$1}^2-26 a^2 b^2 \text{$\#$1}^4+20 a b \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{a^2 b^2 \log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}+i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)-2 a b \log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}+i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}+i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^4}{5 a^3 b^3-13 a^2 b^2 \text{$\#$1}^2+15 a b \text{$\#$1}^4+\text{$\#$1}^6}\&\right]","\frac{\log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}+i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{\sqrt{2} \sqrt{a}}-2 \sqrt{2} \sqrt{a} b \text{RootSum}\left[a^4 b^4+20 a^3 b^3 \text{$\#$1}^2-26 a^2 b^2 \text{$\#$1}^4+20 a b \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{a^2 b^2 \log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}+i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)-2 a b \log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}+i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(i a^{3/2} x^2+i \sqrt{a} \sqrt{b+a^2 x^4}+i \sqrt{2} a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^4}{5 a^3 b^3-13 a^2 b^2 \text{$\#$1}^2+15 a b \text{$\#$1}^4+\text{$\#$1}^6}\&\right]",1,"Log[I*a^(3/2)*x^2 + I*Sqrt[a]*Sqrt[b + a^2*x^4] + I*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]]/(Sqrt[2]*Sqrt[a]) - 2*Sqrt[2]*Sqrt[a]*b*RootSum[a^4*b^4 + 20*a^3*b^3*#1^2 - 26*a^2*b^2*#1^4 + 20*a*b*#1^6 + #1^8 & , (a^2*b^2*Log[I*a^(3/2)*x^2 + I*Sqrt[a]*Sqrt[b + a^2*x^4] + I*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1] - 2*a*b*Log[I*a^(3/2)*x^2 + I*Sqrt[a]*Sqrt[b + a^2*x^4] + I*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1]*#1^2 + Log[I*a^(3/2)*x^2 + I*Sqrt[a]*Sqrt[b + a^2*x^4] + I*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1]*#1^4)/(5*a^3*b^3 - 13*a^2*b^2*#1^2 + 15*a*b*#1^4 + #1^6) & ]","A",1
3019,1,423,423,3.2129875,"\int \frac{1}{\sqrt[3]{(-a+x) (-b+x)^2} (b-a d+(-1+d) x)} \, dx","IntegrateAlgebraic[1/(((-a + x)*(-b + x)^2)^(1/3)*(b - a*d + (-1 + d)*x)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{-2 b+2 x+\sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}\right)}{(a-b) d^{2/3}}+\frac{\log \left((a-b)^{2/3} b-(a-b)^{2/3} x+(a-b)^{2/3} \sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}\right)}{(a-b) d^{2/3}}-\frac{\log \left(a \sqrt[3]{a-b} b^2-\sqrt[3]{a-b} b^3-2 a \sqrt[3]{a-b} b x+2 \sqrt[3]{a-b} b^2 x+a \sqrt[3]{a-b} x^2-\sqrt[3]{a-b} b x^2+\sqrt[3]{a-b} \sqrt[3]{d} \left(-a b+b^2+a x-b x\right) \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}+(a-b)^{4/3} d^{2/3} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}\right)}{2 (a-b) d^{2/3}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{-2 b+2 x+\sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}\right)}{(a-b) d^{2/3}}+\frac{\log \left((a-b)^{2/3} b-(a-b)^{2/3} x+(a-b)^{2/3} \sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}\right)}{(a-b) d^{2/3}}-\frac{\log \left(a \sqrt[3]{a-b} b^2-\sqrt[3]{a-b} b^3-2 a \sqrt[3]{a-b} b x+2 \sqrt[3]{a-b} b^2 x+a \sqrt[3]{a-b} x^2-\sqrt[3]{a-b} b x^2+\sqrt[3]{a-b} \sqrt[3]{d} \left(-a b+b^2+a x-b x\right) \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}+(a-b)^{4/3} d^{2/3} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}\right)}{2 (a-b) d^{2/3}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3))/(-2*b + 2*x + d^(1/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3))])/((a - b)*d^(2/3)) + Log[(a - b)^(2/3)*b - (a - b)^(2/3)*x + (a - b)^(2/3)*d^(1/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3)]/((a - b)*d^(2/3)) - Log[a*(a - b)^(1/3)*b^2 - (a - b)^(1/3)*b^3 - 2*a*(a - b)^(1/3)*b*x + 2*(a - b)^(1/3)*b^2*x + a*(a - b)^(1/3)*x^2 - (a - b)^(1/3)*b*x^2 + (a - b)^(1/3)*d^(1/3)*(-(a*b) + b^2 + a*x - b*x)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3) + (a - b)^(4/3)*d^(2/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(2/3)]/(2*(a - b)*d^(2/3))","A",1
3020,1,423,423,1.6872678,"\int \frac{\left(-4-3 x+2 x^2\right) \left(1+x-x^2+x^4\right) \sqrt[3]{\frac{1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}}{x^5 \left(-1-x+x^2+x^4\right)} \, dx","IntegrateAlgebraic[((-4 - 3*x + 2*x^2)*(1 + x - x^2 + x^4)*((1 + x - x^2 + 2*x^4)/(1 + x - x^2 + 3*x^4))^(1/3))/(x^5*(-1 - x + x^2 + x^4)),x]","\frac{\left(-1-x+x^2-3 x^4\right) \sqrt[3]{\frac{1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}}{x^4}+\sqrt[3]{2} 3^{5/6} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2\ 2^{2/3} \sqrt[3]{\frac{1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}}{3^{5/6}}\right)-\frac{5 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{\frac{1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}}{\sqrt{3}}\right)}{\sqrt{3}}+\frac{5}{3} \log \left(-1+\sqrt[3]{\frac{1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}\right)-\sqrt[3]{6} \log \left(-3+6^{2/3} \sqrt[3]{\frac{1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}\right)-\frac{5}{6} \log \left(1+\sqrt[3]{\frac{1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}+\left(\frac{1+x-x^2+2 x^4}{1+x-x^2+3 x^4}\right)^{2/3}\right)+\frac{\sqrt[3]{3} \log \left(3+6^{2/3} \sqrt[3]{\frac{1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}+2 \sqrt[3]{6} \left(\frac{1+x-x^2+2 x^4}{1+x-x^2+3 x^4}\right)^{2/3}\right)}{2^{2/3}}","\frac{\left(-1-x+x^2-3 x^4\right) \sqrt[3]{\frac{1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}}{x^4}+\sqrt[3]{2} 3^{5/6} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2\ 2^{2/3} \sqrt[3]{\frac{1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}}{3^{5/6}}\right)-\frac{5 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{\frac{1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}}{\sqrt{3}}\right)}{\sqrt{3}}+\frac{5}{3} \log \left(-1+\sqrt[3]{\frac{1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}\right)-\sqrt[3]{6} \log \left(-3+6^{2/3} \sqrt[3]{\frac{1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}\right)-\frac{5}{6} \log \left(1+\sqrt[3]{\frac{1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}+\left(\frac{1+x-x^2+2 x^4}{1+x-x^2+3 x^4}\right)^{2/3}\right)+\frac{\sqrt[3]{3} \log \left(3+6^{2/3} \sqrt[3]{\frac{1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}+2 \sqrt[3]{6} \left(\frac{1+x-x^2+2 x^4}{1+x-x^2+3 x^4}\right)^{2/3}\right)}{2^{2/3}}",1,"((-1 - x + x^2 - 3*x^4)*((1 + x - x^2 + 2*x^4)/(1 + x - x^2 + 3*x^4))^(1/3))/x^4 + 2^(1/3)*3^(5/6)*ArcTan[1/Sqrt[3] + (2*2^(2/3)*((1 + x - x^2 + 2*x^4)/(1 + x - x^2 + 3*x^4))^(1/3))/3^(5/6)] - (5*ArcTan[1/Sqrt[3] + (2*((1 + x - x^2 + 2*x^4)/(1 + x - x^2 + 3*x^4))^(1/3))/Sqrt[3]])/Sqrt[3] + (5*Log[-1 + ((1 + x - x^2 + 2*x^4)/(1 + x - x^2 + 3*x^4))^(1/3)])/3 - 6^(1/3)*Log[-3 + 6^(2/3)*((1 + x - x^2 + 2*x^4)/(1 + x - x^2 + 3*x^4))^(1/3)] - (5*Log[1 + ((1 + x - x^2 + 2*x^4)/(1 + x - x^2 + 3*x^4))^(1/3) + ((1 + x - x^2 + 2*x^4)/(1 + x - x^2 + 3*x^4))^(2/3)])/6 + (3^(1/3)*Log[3 + 6^(2/3)*((1 + x - x^2 + 2*x^4)/(1 + x - x^2 + 3*x^4))^(1/3) + 2*6^(1/3)*((1 + x - x^2 + 2*x^4)/(1 + x - x^2 + 3*x^4))^(2/3)])/2^(2/3)","A",1
3021,1,475,425,3.0922913,"\int \frac{-b+a x}{(b+a x) \sqrt[3]{-b^2 x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[(-b + a*x)/((b + a*x)*(-(b^2*x^2) + a^3*x^3)^(1/3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{-b^2 x^2+a^3 x^3}}\right)}{a}+\frac{\sqrt{6 \left(-1+i \sqrt{3}\right)} \tan ^{-1}\left(\frac{3 \sqrt[3]{a} \sqrt[3]{a^2+b} x}{\sqrt{3} \sqrt[3]{a} \sqrt[3]{a^2+b} x-3 i \sqrt[3]{-b^2 x^2+a^3 x^3}-\sqrt{3} \sqrt[3]{-b^2 x^2+a^3 x^3}}\right)}{\sqrt[3]{a} \sqrt[3]{a^2+b}}-\frac{i \left(-i+\sqrt{3}\right) \log \left(2 \sqrt[3]{a} \sqrt[3]{a^2+b} x+\left(1+i \sqrt{3}\right) \sqrt[3]{-b^2 x^2+a^3 x^3}\right)}{\sqrt[3]{a} \sqrt[3]{a^2+b}}-\frac{\log \left(a^2 x-a \sqrt[3]{-b^2 x^2+a^3 x^3}\right)}{a}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{-b^2 x^2+a^3 x^3}+\left(-b^2 x^2+a^3 x^3\right)^{2/3}\right)}{2 a}+\frac{\left(1+i \sqrt{3}\right) \log \left(-2 i a^{2/3} \left(a^2+b\right)^{2/3} x^2+\sqrt[3]{a} \sqrt[3]{a^2+b} \left(i x-\sqrt{3} x\right) \sqrt[3]{-b^2 x^2+a^3 x^3}+\left(i+\sqrt{3}\right) \left(-b^2 x^2+a^3 x^3\right)^{2/3}\right)}{2 \sqrt[3]{a} \sqrt[3]{a^2+b}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{-b^2 x^2+a^3 x^3}}\right)}{a}+\frac{\sqrt{6 \left(-1+i \sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a} \sqrt[3]{a^2+b} x}{\sqrt[3]{a} \sqrt[3]{a^2+b} x-2 \sqrt[3]{-1} \sqrt[3]{-b^2 x^2+a^3 x^3}}\right)}{\sqrt[3]{a} \sqrt[3]{a^2+b}}-\frac{\log \left(-a x+\sqrt[3]{-b^2 x^2+a^3 x^3}\right)}{a}-\frac{i \left(-i+\sqrt{3}\right) \log \left(\sqrt[3]{a} \sqrt[3]{a^2+b} x+\sqrt[3]{-1} \sqrt[3]{-b^2 x^2+a^3 x^3}\right)}{\sqrt[3]{a} \sqrt[3]{a^2+b}}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{-b^2 x^2+a^3 x^3}+\left(-b^2 x^2+a^3 x^3\right)^{2/3}\right)}{2 a}+\frac{\left(1+i \sqrt{3}\right) \log \left(a^{2/3} \left(a^2+b\right)^{2/3} x^2-\sqrt[3]{-1} \sqrt[3]{a} \sqrt[3]{a^2+b} x \sqrt[3]{-b^2 x^2+a^3 x^3}+(-1)^{2/3} \left(-b^2 x^2+a^3 x^3\right)^{2/3}\right)}{2 \sqrt[3]{a} \sqrt[3]{a^2+b}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*a*x)/(a*x + 2*(-(b^2*x^2) + a^3*x^3)^(1/3))])/a + (Sqrt[6*(-1 + I*Sqrt[3])]*ArcTan[(3*a^(1/3)*(a^2 + b)^(1/3)*x)/(Sqrt[3]*a^(1/3)*(a^2 + b)^(1/3)*x - (3*I)*(-(b^2*x^2) + a^3*x^3)^(1/3) - Sqrt[3]*(-(b^2*x^2) + a^3*x^3)^(1/3))])/(a^(1/3)*(a^2 + b)^(1/3)) - (I*(-I + Sqrt[3])*Log[2*a^(1/3)*(a^2 + b)^(1/3)*x + (1 + I*Sqrt[3])*(-(b^2*x^2) + a^3*x^3)^(1/3)])/(a^(1/3)*(a^2 + b)^(1/3)) - Log[a^2*x - a*(-(b^2*x^2) + a^3*x^3)^(1/3)]/a + Log[a^2*x^2 + a*x*(-(b^2*x^2) + a^3*x^3)^(1/3) + (-(b^2*x^2) + a^3*x^3)^(2/3)]/(2*a) + ((1 + I*Sqrt[3])*Log[(-2*I)*a^(2/3)*(a^2 + b)^(2/3)*x^2 + a^(1/3)*(a^2 + b)^(1/3)*(I*x - Sqrt[3]*x)*(-(b^2*x^2) + a^3*x^3)^(1/3) + (I + Sqrt[3])*(-(b^2*x^2) + a^3*x^3)^(2/3)])/(2*a^(1/3)*(a^2 + b)^(1/3))","A",1
3022,1,189,428,3.5751,"\int \frac{\left(-q+p x^4\right) \sqrt{q+p x^4}}{c x^4+b x^2 \left(q+p x^4\right)+a \left(q+p x^4\right)^2} \, dx","IntegrateAlgebraic[((-q + p*x^4)*Sqrt[q + p*x^4])/(c*x^4 + b*x^2*(q + p*x^4) + a*(q + p*x^4)^2),x]","-\frac{\left(-b+\sqrt{b^2-4 a c}\right) \tan ^{-1}\left(\frac{\sqrt{b-\sqrt{b^2-4 a c}} x}{\sqrt{2} \sqrt{a} \sqrt{q+p x^4}}\right)}{\sqrt{2} \sqrt{a} \sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{b+\sqrt{b^2-4 a c}} \tan ^{-1}\left(\frac{\sqrt{b+\sqrt{b^2-4 a c}} x}{\sqrt{2} \sqrt{a} \sqrt{q+p x^4}}\right)}{\sqrt{2} \sqrt{a} \sqrt{b^2-4 a c}}","\frac{1}{4} \text{RootSum}\left[16 a p^2 q^2+16 b p q \text{$\#$1}^2-32 a p^{3/2} q^{3/2} \text{$\#$1}^2+16 c \text{$\#$1}^4-16 b \sqrt{p} \sqrt{q} \text{$\#$1}^4+24 a p q \text{$\#$1}^4+4 b \text{$\#$1}^6-8 a \sqrt{p} \sqrt{q} \text{$\#$1}^6+a \text{$\#$1}^8\&,\frac{8 p^{3/2} q^{3/2} \log (x)-8 p^{3/2} q^{3/2} \log \left(-\sqrt{q}-\sqrt{p} x^2+\sqrt{q+p x^4}-x \text{$\#$1}\right)-4 p q \log (x) \text{$\#$1}^2+4 p q \log \left(-\sqrt{q}-\sqrt{p} x^2+\sqrt{q+p x^4}-x \text{$\#$1}\right) \text{$\#$1}^2-2 \sqrt{p} \sqrt{q} \log (x) \text{$\#$1}^4+2 \sqrt{p} \sqrt{q} \log \left(-\sqrt{q}-\sqrt{p} x^2+\sqrt{q+p x^4}-x \text{$\#$1}\right) \text{$\#$1}^4+\log (x) \text{$\#$1}^6-\log \left(-\sqrt{q}-\sqrt{p} x^2+\sqrt{q+p x^4}-x \text{$\#$1}\right) \text{$\#$1}^6}{-4 b p q \text{$\#$1}+8 a p^{3/2} q^{3/2} \text{$\#$1}-8 c \text{$\#$1}^3+8 b \sqrt{p} \sqrt{q} \text{$\#$1}^3-12 a p q \text{$\#$1}^3-3 b \text{$\#$1}^5+6 a \sqrt{p} \sqrt{q} \text{$\#$1}^5-a \text{$\#$1}^7}\&\right]",1,"-(((-b + Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[b - Sqrt[b^2 - 4*a*c]]*x)/(Sqrt[2]*Sqrt[a]*Sqrt[q + p*x^4])])/(Sqrt[2]*Sqrt[a]*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]])) - (Sqrt[b + Sqrt[b^2 - 4*a*c]]*ArcTan[(Sqrt[b + Sqrt[b^2 - 4*a*c]]*x)/(Sqrt[2]*Sqrt[a]*Sqrt[q + p*x^4])])/(Sqrt[2]*Sqrt[a]*Sqrt[b^2 - 4*a*c])","A",1
3023,1,328,428,0.5078037,"\int \frac{\left(1+x^4\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\sqrt{1+x^2} \left(1-x^4\right)} \, dx","IntegrateAlgebraic[((1 + x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(Sqrt[1 + x^2]*(1 - x^4)),x]","\frac{\left(-5-4 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-4 x \sqrt{1+x^2} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{1+x^2+x \sqrt{1+x^2}}+4 \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)-\frac{1}{2} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{-2 \text{$\#$1}+\text{$\#$1}^3}\&\right]+\frac{1}{4} \text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{-\text{$\#$1}+3 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{1}{2} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{2-2 \text{$\#$1}^2+\text{$\#$1}^4}\&\right]","\frac{\left(-5-4 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-4 x \sqrt{1+x^2} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{1+x^2+x \sqrt{1+x^2}}+4 \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)-\frac{1}{2} \text{RootSum}\left[-1-8 \text{$\#$1}^2-8 \text{$\#$1}^3+14 \text{$\#$1}^4+32 \text{$\#$1}^5+24 \text{$\#$1}^6+8 \text{$\#$1}^7+\text{$\#$1}^8\&,\frac{\log \left(-1+\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{-1+\text{$\#$1}+3 \text{$\#$1}^2+\text{$\#$1}^3}\&\right]+\frac{1}{4} \text{RootSum}\left[1+16 \text{$\#$1}^4+32 \text{$\#$1}^5+24 \text{$\#$1}^6+8 \text{$\#$1}^7+\text{$\#$1}^8\&,\frac{\log \left(-1+\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{8 \text{$\#$1}^3+20 \text{$\#$1}^4+18 \text{$\#$1}^5+7 \text{$\#$1}^6+\text{$\#$1}^7}\&\right]+\frac{1}{2} \text{RootSum}\left[-1+8 \text{$\#$1}^2+8 \text{$\#$1}^3+18 \text{$\#$1}^4+32 \text{$\#$1}^5+24 \text{$\#$1}^6+8 \text{$\#$1}^7+\text{$\#$1}^8\&,\frac{\log \left(-1+\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+\log \left(-1+\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{1+4 \text{$\#$1}^2+4 \text{$\#$1}^3+\text{$\#$1}^4}\&\right]",1,"((-5 - 4*x^2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - 4*x*Sqrt[1 + x^2]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 + x^2 + x*Sqrt[1 + x^2]) + 4*ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/(-2*#1 + #1^3) & ]/2 + RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/(-#1 + 3*#1^3 - 3*#1^5 + #1^7) & ]/4 + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(2 - 2*#1^2 + #1^4) & ]/2","A",1
3024,1,328,428,0.0021385,"\int \frac{\left(1+x^4\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\sqrt{1+x^2} \left(1-x^4\right)} \, dx","IntegrateAlgebraic[((1 + x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(Sqrt[1 + x^2]*(1 - x^4)),x]","\frac{\left(-5-4 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-4 x \sqrt{1+x^2} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{1+x^2+x \sqrt{1+x^2}}+4 \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)-\frac{1}{2} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{-2 \text{$\#$1}+\text{$\#$1}^3}\&\right]+\frac{1}{4} \text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{-\text{$\#$1}+3 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{1}{2} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{2-2 \text{$\#$1}^2+\text{$\#$1}^4}\&\right]","\frac{\left(-5-4 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-4 x \sqrt{1+x^2} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{1+x^2+x \sqrt{1+x^2}}+4 \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)-\frac{1}{2} \text{RootSum}\left[-1-8 \text{$\#$1}^2-8 \text{$\#$1}^3+14 \text{$\#$1}^4+32 \text{$\#$1}^5+24 \text{$\#$1}^6+8 \text{$\#$1}^7+\text{$\#$1}^8\&,\frac{\log \left(-1+\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{-1+\text{$\#$1}+3 \text{$\#$1}^2+\text{$\#$1}^3}\&\right]+\frac{1}{4} \text{RootSum}\left[1+16 \text{$\#$1}^4+32 \text{$\#$1}^5+24 \text{$\#$1}^6+8 \text{$\#$1}^7+\text{$\#$1}^8\&,\frac{\log \left(-1+\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)}{8 \text{$\#$1}^3+20 \text{$\#$1}^4+18 \text{$\#$1}^5+7 \text{$\#$1}^6+\text{$\#$1}^7}\&\right]+\frac{1}{2} \text{RootSum}\left[-1+8 \text{$\#$1}^2+8 \text{$\#$1}^3+18 \text{$\#$1}^4+32 \text{$\#$1}^5+24 \text{$\#$1}^6+8 \text{$\#$1}^7+\text{$\#$1}^8\&,\frac{\log \left(-1+\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+\log \left(-1+\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{1+4 \text{$\#$1}^2+4 \text{$\#$1}^3+\text{$\#$1}^4}\&\right]",1,"((-5 - 4*x^2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - 4*x*Sqrt[1 + x^2]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 + x^2 + x*Sqrt[1 + x^2]) + 4*ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/(-2*#1 + #1^3) & ]/2 + RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/(-#1 + 3*#1^3 - 3*#1^5 + #1^7) & ]/4 + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(2 - 2*#1^2 + #1^4) & ]/2","A",1
3025,1,429,429,1.5903436,"\int \frac{1-x^3+x^6}{\sqrt[3]{x^2+x^4} \left(-1+x^6\right)} \, dx","IntegrateAlgebraic[(1 - x^3 + x^6)/((x^2 + x^4)^(1/3)*(-1 + x^6)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{x^2+x^4}}\right)}{2 \sqrt{3}}-\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^4}}\right)-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2^{2/3} \sqrt[3]{x^2+x^4}}\right)}{4 \sqrt[3]{2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x^2+x^4}}\right)}{4 \sqrt[3]{2} \sqrt{3}}+\frac{1}{2} \log \left(-x+\sqrt[3]{x^2+x^4}\right)-\frac{1}{6} \log \left(x+\sqrt[3]{x^2+x^4}\right)+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x^2+x^4}\right)}{12 \sqrt[3]{2}}-\frac{\log \left(2 x+2^{2/3} \sqrt[3]{x^2+x^4}\right)}{4 \sqrt[3]{2}}+\frac{1}{12} \log \left(x^2-x \sqrt[3]{x^2+x^4}+\left(x^2+x^4\right)^{2/3}\right)-\frac{1}{4} \log \left(x^2+x \sqrt[3]{x^2+x^4}+\left(x^2+x^4\right)^{2/3}\right)+\frac{\log \left(-2 x^2+2^{2/3} x \sqrt[3]{x^2+x^4}-\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}\right)}{8 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x^2+x^4}+\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}\right)}{24 \sqrt[3]{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{x^2+x^4}}\right)}{2 \sqrt{3}}-\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^4}}\right)-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2^{2/3} \sqrt[3]{x^2+x^4}}\right)}{4 \sqrt[3]{2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x^2+x^4}}\right)}{4 \sqrt[3]{2} \sqrt{3}}+\frac{1}{2} \log \left(-x+\sqrt[3]{x^2+x^4}\right)-\frac{1}{6} \log \left(x+\sqrt[3]{x^2+x^4}\right)+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x^2+x^4}\right)}{12 \sqrt[3]{2}}-\frac{\log \left(2 x+2^{2/3} \sqrt[3]{x^2+x^4}\right)}{4 \sqrt[3]{2}}+\frac{1}{12} \log \left(x^2-x \sqrt[3]{x^2+x^4}+\left(x^2+x^4\right)^{2/3}\right)-\frac{1}{4} \log \left(x^2+x \sqrt[3]{x^2+x^4}+\left(x^2+x^4\right)^{2/3}\right)+\frac{\log \left(-2 x^2+2^{2/3} x \sqrt[3]{x^2+x^4}-\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}\right)}{8 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x^2+x^4}+\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}\right)}{24 \sqrt[3]{2}}",1,"-1/2*ArcTan[(Sqrt[3]*x)/(-x + 2*(x^2 + x^4)^(1/3))]/Sqrt[3] - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(x^2 + x^4)^(1/3))])/2 - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2^(2/3)*(x^2 + x^4)^(1/3))])/(4*2^(1/3)) - ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(x^2 + x^4)^(1/3))]/(4*2^(1/3)*Sqrt[3]) + Log[-x + (x^2 + x^4)^(1/3)]/2 - Log[x + (x^2 + x^4)^(1/3)]/6 + Log[-2*x + 2^(2/3)*(x^2 + x^4)^(1/3)]/(12*2^(1/3)) - Log[2*x + 2^(2/3)*(x^2 + x^4)^(1/3)]/(4*2^(1/3)) + Log[x^2 - x*(x^2 + x^4)^(1/3) + (x^2 + x^4)^(2/3)]/12 - Log[x^2 + x*(x^2 + x^4)^(1/3) + (x^2 + x^4)^(2/3)]/4 + Log[-2*x^2 + 2^(2/3)*x*(x^2 + x^4)^(1/3) - 2^(1/3)*(x^2 + x^4)^(2/3)]/(8*2^(1/3)) - Log[2*x^2 + 2^(2/3)*x*(x^2 + x^4)^(1/3) + 2^(1/3)*(x^2 + x^4)^(2/3)]/(24*2^(1/3))","A",1
3026,1,429,429,1.3805623,"\int \frac{1+x^3+x^6}{\sqrt[3]{x^2+x^4} \left(-1+x^6\right)} \, dx","IntegrateAlgebraic[(1 + x^3 + x^6)/((x^2 + x^4)^(1/3)*(-1 + x^6)),x]","-\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{x^2+x^4}}\right)-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^4}}\right)}{2 \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2^{2/3} \sqrt[3]{x^2+x^4}}\right)}{4 \sqrt[3]{2} \sqrt{3}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x^2+x^4}}\right)}{4 \sqrt[3]{2}}+\frac{1}{6} \log \left(-x+\sqrt[3]{x^2+x^4}\right)-\frac{1}{2} \log \left(x+\sqrt[3]{x^2+x^4}\right)+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x^2+x^4}\right)}{4 \sqrt[3]{2}}-\frac{\log \left(2 x+2^{2/3} \sqrt[3]{x^2+x^4}\right)}{12 \sqrt[3]{2}}+\frac{1}{4} \log \left(x^2-x \sqrt[3]{x^2+x^4}+\left(x^2+x^4\right)^{2/3}\right)-\frac{1}{12} \log \left(x^2+x \sqrt[3]{x^2+x^4}+\left(x^2+x^4\right)^{2/3}\right)+\frac{\log \left(-2 x^2+2^{2/3} x \sqrt[3]{x^2+x^4}-\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}\right)}{24 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x^2+x^4}+\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}\right)}{8 \sqrt[3]{2}}","-\frac{1}{2} \sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2 \sqrt[3]{x^2+x^4}}\right)-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{x+2 \sqrt[3]{x^2+x^4}}\right)}{2 \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{\sqrt{3} x}{-x+2^{2/3} \sqrt[3]{x^2+x^4}}\right)}{4 \sqrt[3]{2} \sqrt{3}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} x}{x+2^{2/3} \sqrt[3]{x^2+x^4}}\right)}{4 \sqrt[3]{2}}+\frac{1}{6} \log \left(-x+\sqrt[3]{x^2+x^4}\right)-\frac{1}{2} \log \left(x+\sqrt[3]{x^2+x^4}\right)+\frac{\log \left(-2 x+2^{2/3} \sqrt[3]{x^2+x^4}\right)}{4 \sqrt[3]{2}}-\frac{\log \left(2 x+2^{2/3} \sqrt[3]{x^2+x^4}\right)}{12 \sqrt[3]{2}}+\frac{1}{4} \log \left(x^2-x \sqrt[3]{x^2+x^4}+\left(x^2+x^4\right)^{2/3}\right)-\frac{1}{12} \log \left(x^2+x \sqrt[3]{x^2+x^4}+\left(x^2+x^4\right)^{2/3}\right)+\frac{\log \left(-2 x^2+2^{2/3} x \sqrt[3]{x^2+x^4}-\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}\right)}{24 \sqrt[3]{2}}-\frac{\log \left(2 x^2+2^{2/3} x \sqrt[3]{x^2+x^4}+\sqrt[3]{2} \left(x^2+x^4\right)^{2/3}\right)}{8 \sqrt[3]{2}}",1,"-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(x^2 + x^4)^(1/3))]) - ArcTan[(Sqrt[3]*x)/(x + 2*(x^2 + x^4)^(1/3))]/(2*Sqrt[3]) - ArcTan[(Sqrt[3]*x)/(-x + 2^(2/3)*(x^2 + x^4)^(1/3))]/(4*2^(1/3)*Sqrt[3]) - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(x^2 + x^4)^(1/3))])/(4*2^(1/3)) + Log[-x + (x^2 + x^4)^(1/3)]/6 - Log[x + (x^2 + x^4)^(1/3)]/2 + Log[-2*x + 2^(2/3)*(x^2 + x^4)^(1/3)]/(4*2^(1/3)) - Log[2*x + 2^(2/3)*(x^2 + x^4)^(1/3)]/(12*2^(1/3)) + Log[x^2 - x*(x^2 + x^4)^(1/3) + (x^2 + x^4)^(2/3)]/4 - Log[x^2 + x*(x^2 + x^4)^(1/3) + (x^2 + x^4)^(2/3)]/12 + Log[-2*x^2 + 2^(2/3)*x*(x^2 + x^4)^(1/3) - 2^(1/3)*(x^2 + x^4)^(2/3)]/(24*2^(1/3)) - Log[2*x^2 + 2^(2/3)*x*(x^2 + x^4)^(1/3) + 2^(1/3)*(x^2 + x^4)^(2/3)]/(8*2^(1/3))","A",1
3027,1,431,431,1.1028384,"\int \frac{\sqrt{-b+a^2 x^2} \sqrt{a x+\sqrt{-b+a^2 x^2}}}{\sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}} \, dx","IntegrateAlgebraic[(Sqrt[-b + a^2*x^2]*Sqrt[a*x + Sqrt[-b + a^2*x^2]])/Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]],x]","\frac{\left(-840 b^2 c^2+1024 b c^6-1575 a b^2 x+1920 a b c^4 x-2048 a^2 c^6 x^2-2560 a^3 c^4 x^3\right) \sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\left(1050 b^2 c-768 b c^5-11760 a b c^3 x+2048 a c^7 x+1536 a^2 c^5 x^2+2240 a^3 c^3 x^3\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\sqrt{-b+a^2 x^2} \left(\left(-1575 b^2+640 b c^4-2048 a c^6 x-2560 a^2 c^4 x^2\right) \sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\left(-10640 b c^3+2048 c^7+1536 a c^5 x+2240 a^2 c^3 x^2\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{5040 a c^3 \left(a x+\sqrt{-b+a^2 x^2}\right)^{3/2}}+\frac{5 b^2 \tanh ^{-1}\left(\frac{\sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{c}}\right)}{16 a c^{7/2}}","\frac{\left(-840 b^2 c^2+1024 b c^6-1575 a b^2 x+1920 a b c^4 x-2048 a^2 c^6 x^2-2560 a^3 c^4 x^3\right) \sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\left(1050 b^2 c-768 b c^5-11760 a b c^3 x+2048 a c^7 x+1536 a^2 c^5 x^2+2240 a^3 c^3 x^3\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\sqrt{-b+a^2 x^2} \left(\left(-1575 b^2+640 b c^4-2048 a c^6 x-2560 a^2 c^4 x^2\right) \sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\left(-10640 b c^3+2048 c^7+1536 a c^5 x+2240 a^2 c^3 x^2\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{5040 a c^3 \left(a x+\sqrt{-b+a^2 x^2}\right)^{3/2}}+\frac{5 b^2 \tanh ^{-1}\left(\frac{\sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{c}}\right)}{16 a c^{7/2}}",1,"((-840*b^2*c^2 + 1024*b*c^6 - 1575*a*b^2*x + 1920*a*b*c^4*x - 2048*a^2*c^6*x^2 - 2560*a^3*c^4*x^3)*Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]] + (1050*b^2*c - 768*b*c^5 - 11760*a*b*c^3*x + 2048*a*c^7*x + 1536*a^2*c^5*x^2 + 2240*a^3*c^3*x^3)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]] + Sqrt[-b + a^2*x^2]*((-1575*b^2 + 640*b*c^4 - 2048*a*c^6*x - 2560*a^2*c^4*x^2)*Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]] + (-10640*b*c^3 + 2048*c^7 + 1536*a*c^5*x + 2240*a^2*c^3*x^2)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]))/(5040*a*c^3*(a*x + Sqrt[-b + a^2*x^2])^(3/2)) + (5*b^2*ArcTanh[Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]/Sqrt[c]])/(16*a*c^(7/2))","A",1
3028,1,486,432,14.4845809,"\int \frac{\left(-1+a x^8\right) \left(1+a x^8\right)^{3/4}}{1+x^8+a^2 x^{16}} \, dx","IntegrateAlgebraic[((-1 + a*x^8)*(1 + a*x^8)^(3/4))/(1 + x^8 + a^2*x^16),x]","\frac{i \left(i \sqrt{2}+\sqrt{2 \left(3+2 \sqrt{2}\right)}\right) \tan ^{-1}\left(\frac{\left((-1+i)-(1+i) (-1)^{3/4}\right) \sqrt[4]{-1+2 a} x^2+(1+i) \sqrt{1+a x^8}+(1+i) (-1)^{3/4} \sqrt{1+a x^8}}{2 \sqrt[8]{-1+2 a} x \sqrt[4]{1+a x^8}}\right)}{16 \sqrt[8]{-1+2 a}}+\frac{\left(\sqrt{2}+i \sqrt{2 \left(3-2 \sqrt{2}\right)}\right) \tan ^{-1}\left(\frac{2 \sqrt[8]{-1+2 a} x \sqrt[4]{1+a x^8}}{\left((1-i)+\sqrt{2}\right) \sqrt[4]{-1+2 a} x^2-(1+i) \sqrt{1+a x^8}-\sqrt{2} \sqrt{1+a x^8}}\right)}{16 \sqrt[8]{-1+2 a}}+\frac{\left(i+(-1)^{3/4}\right) \tanh ^{-1}\left(\frac{\left((-2+2 i)-(2+2 i) (-1)^{3/4}\right) \sqrt[4]{-1+2 a} x^2-(2+2 i) \sqrt{1+a x^8}-(2+2 i) (-1)^{3/4} \sqrt{1+a x^8}}{4 \sqrt[8]{-1+2 a} x \sqrt[4]{1+a x^8}}\right)}{8 \sqrt[8]{-1+2 a}}-\frac{i \left(-i \sqrt{2}+\sqrt{2 \left(3-2 \sqrt{2}\right)}\right) \tanh ^{-1}\left(\frac{\left((1-i)+\sqrt{2}\right) \sqrt[4]{-1+2 a} x^2+(1+i) \sqrt{1+a x^8}+\sqrt{2} \sqrt{1+a x^8}}{2 \sqrt[8]{-1+2 a} x \sqrt[4]{1+a x^8}}\right)}{16 \sqrt[8]{-1+2 a}}","\frac{\left(1+\sqrt[4]{-1}\right) \tan ^{-1}\left(\frac{(-1)^{7/8} \sqrt{2+\sqrt{2}} \sqrt[8]{-1+2 a} x \sqrt[4]{1+a x^8}}{(-1)^{3/4} \sqrt[4]{-1+2 a} x^2+\sqrt{1+a x^8}}\right)}{8 \sqrt[8]{-1+2 a}}-\frac{i \left(-i \sqrt{2}+\sqrt{2 \left(3-2 \sqrt{2}\right)}\right) \tan ^{-1}\left(\frac{(-1)^{7/8} \left(-2+\sqrt{2}\right) \sqrt[8]{-1+2 a} x \sqrt[4]{1+a x^8}}{(-1)^{3/4} \sqrt{2-\sqrt{2}} \sqrt[4]{-1+2 a} x^2+\sqrt{2-\sqrt{2}} \sqrt{1+a x^8}}\right)}{16 \sqrt[8]{-1+2 a}}+\frac{\left(\sqrt{2}+i \sqrt{2 \left(3-2 \sqrt{2}\right)}\right) \tanh ^{-1}\left(\frac{(-1)^{7/8} \sqrt[4]{-1+2 a} x^2-\sqrt[8]{-1} \sqrt{1+a x^8}}{\sqrt{2-\sqrt{2}} \sqrt[8]{-1+2 a} x \sqrt[4]{1+a x^8}}\right)}{16 \sqrt[8]{-1+2 a}}+\frac{\left(1+\sqrt[4]{-1}\right) \tanh ^{-1}\left(\frac{(-1)^{7/8} \sqrt[4]{-1+2 a} x^2-\sqrt[8]{-1} \sqrt{1+a x^8}}{\sqrt{2+\sqrt{2}} \sqrt[8]{-1+2 a} x \sqrt[4]{1+a x^8}}\right)}{8 \sqrt[8]{-1+2 a}}",1,"((I/16)*(I*Sqrt[2] + Sqrt[2*(3 + 2*Sqrt[2])])*ArcTan[(((-1 + I) - (1 + I)*(-1)^(3/4))*(-1 + 2*a)^(1/4)*x^2 + (1 + I)*Sqrt[1 + a*x^8] + (1 + I)*(-1)^(3/4)*Sqrt[1 + a*x^8])/(2*(-1 + 2*a)^(1/8)*x*(1 + a*x^8)^(1/4))])/(-1 + 2*a)^(1/8) + ((Sqrt[2] + I*Sqrt[2*(3 - 2*Sqrt[2])])*ArcTan[(2*(-1 + 2*a)^(1/8)*x*(1 + a*x^8)^(1/4))/(((1 - I) + Sqrt[2])*(-1 + 2*a)^(1/4)*x^2 - (1 + I)*Sqrt[1 + a*x^8] - Sqrt[2]*Sqrt[1 + a*x^8])])/(16*(-1 + 2*a)^(1/8)) + ((I + (-1)^(3/4))*ArcTanh[(((-2 + 2*I) - (2 + 2*I)*(-1)^(3/4))*(-1 + 2*a)^(1/4)*x^2 - (2 + 2*I)*Sqrt[1 + a*x^8] - (2 + 2*I)*(-1)^(3/4)*Sqrt[1 + a*x^8])/(4*(-1 + 2*a)^(1/8)*x*(1 + a*x^8)^(1/4))])/(8*(-1 + 2*a)^(1/8)) - ((I/16)*((-I)*Sqrt[2] + Sqrt[2*(3 - 2*Sqrt[2])])*ArcTanh[(((1 - I) + Sqrt[2])*(-1 + 2*a)^(1/4)*x^2 + (1 + I)*Sqrt[1 + a*x^8] + Sqrt[2]*Sqrt[1 + a*x^8])/(2*(-1 + 2*a)^(1/8)*x*(1 + a*x^8)^(1/4))])/(-1 + 2*a)^(1/8)","A",1
3029,1,482,433,3.109758,"\int \frac{-b+a x}{(b+a x) \sqrt[3]{b^2 x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[(-b + a*x)/((b + a*x)*(b^2*x^2 + a^3*x^3)^(1/3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{a}+\frac{\sqrt{6 \left(-1+i \sqrt{3}\right)} \tan ^{-1}\left(\frac{3 \sqrt[3]{a} \sqrt[3]{a^2-b} x}{\sqrt{3} \sqrt[3]{a} \sqrt[3]{a^2-b} x-3 i \sqrt[3]{b^2 x^2+a^3 x^3}-\sqrt{3} \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{\sqrt[3]{a} \sqrt[3]{a^2-b}}-\frac{i \left(-i+\sqrt{3}\right) \log \left(2 \sqrt[3]{a} \sqrt[3]{a^2-b} x+\left(1+i \sqrt{3}\right) \sqrt[3]{b^2 x^2+a^3 x^3}\right)}{\sqrt[3]{a} \sqrt[3]{a^2-b}}-\frac{\log \left(a^2 x-a \sqrt[3]{b^2 x^2+a^3 x^3}\right)}{a}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{b^2 x^2+a^3 x^3}+\left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{2 a}+\frac{\left(1+i \sqrt{3}\right) \log \left(-2 i a^{2/3} \left(a^2-b\right)^{2/3} x^2+\sqrt[3]{a} \sqrt[3]{a^2-b} \left(i x-\sqrt{3} x\right) \sqrt[3]{b^2 x^2+a^3 x^3}+\left(i+\sqrt{3}\right) \left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{2 \sqrt[3]{a} \sqrt[3]{a^2-b}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{a}+\frac{\sqrt{6 \left(-1+i \sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a} \sqrt[3]{a^2-b} x}{\sqrt[3]{a} \sqrt[3]{a^2-b} x-2 \sqrt[3]{-1} \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{\sqrt[3]{a} \sqrt[3]{a^2-b}}-\frac{\log \left(-a x+\sqrt[3]{b^2 x^2+a^3 x^3}\right)}{a}-\frac{i \left(-i+\sqrt{3}\right) \log \left(\sqrt[3]{a} \sqrt[3]{a^2-b} x+\sqrt[3]{-1} \sqrt[3]{b^2 x^2+a^3 x^3}\right)}{\sqrt[3]{a} \sqrt[3]{a^2-b}}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{b^2 x^2+a^3 x^3}+\left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{2 a}+\frac{\left(1+i \sqrt{3}\right) \log \left(a^{2/3} \left(a^2-b\right)^{2/3} x^2-\sqrt[3]{-1} \sqrt[3]{a} \sqrt[3]{a^2-b} x \sqrt[3]{b^2 x^2+a^3 x^3}+(-1)^{2/3} \left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{2 \sqrt[3]{a} \sqrt[3]{a^2-b}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*a*x)/(a*x + 2*(b^2*x^2 + a^3*x^3)^(1/3))])/a + (Sqrt[6*(-1 + I*Sqrt[3])]*ArcTan[(3*a^(1/3)*(a^2 - b)^(1/3)*x)/(Sqrt[3]*a^(1/3)*(a^2 - b)^(1/3)*x - (3*I)*(b^2*x^2 + a^3*x^3)^(1/3) - Sqrt[3]*(b^2*x^2 + a^3*x^3)^(1/3))])/(a^(1/3)*(a^2 - b)^(1/3)) - (I*(-I + Sqrt[3])*Log[2*a^(1/3)*(a^2 - b)^(1/3)*x + (1 + I*Sqrt[3])*(b^2*x^2 + a^3*x^3)^(1/3)])/(a^(1/3)*(a^2 - b)^(1/3)) - Log[a^2*x - a*(b^2*x^2 + a^3*x^3)^(1/3)]/a + Log[a^2*x^2 + a*x*(b^2*x^2 + a^3*x^3)^(1/3) + (b^2*x^2 + a^3*x^3)^(2/3)]/(2*a) + ((1 + I*Sqrt[3])*Log[(-2*I)*a^(2/3)*(a^2 - b)^(2/3)*x^2 + a^(1/3)*(a^2 - b)^(1/3)*(I*x - Sqrt[3]*x)*(b^2*x^2 + a^3*x^3)^(1/3) + (I + Sqrt[3])*(b^2*x^2 + a^3*x^3)^(2/3)])/(2*a^(1/3)*(a^2 - b)^(1/3))","A",1
3030,1,262,434,3.495858,"\int \frac{1}{\left(-1+x^2\right)^2 \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[1/((-1 + x^2)^2*Sqrt[x + Sqrt[1 + x^2]]),x]","\frac{x^2 \sqrt{x+\sqrt{1+x^2}}}{2 \left(-1+x^2\right)}-\frac{x \sqrt{1+x^2} \sqrt{x+\sqrt{1+x^2}}}{2 \left(-1+x^2\right)}+\frac{1}{4} \sqrt{\frac{1}{2} \left(-1+5 \sqrt{2}\right)} \tan ^{-1}\left(\sqrt{-1+\sqrt{2}} \sqrt{x+\sqrt{1+x^2}}\right)-\frac{1}{4} \sqrt{\frac{1}{2} \left(1+5 \sqrt{2}\right)} \tan ^{-1}\left(\sqrt{1+\sqrt{2}} \sqrt{x+\sqrt{1+x^2}}\right)+\frac{1}{4} \sqrt{\frac{1}{2} \left(-1+5 \sqrt{2}\right)} \tanh ^{-1}\left(\sqrt{-1+\sqrt{2}} \sqrt{x+\sqrt{1+x^2}}\right)-\frac{1}{4} \sqrt{\frac{1}{2} \left(1+5 \sqrt{2}\right)} \tanh ^{-1}\left(\sqrt{1+\sqrt{2}} \sqrt{x+\sqrt{1+x^2}}\right)","\frac{x^2 \sqrt{x+\sqrt{1+x^2}}}{2 \left(-1+x^2\right)}-\frac{x \sqrt{1+x^2} \sqrt{x+\sqrt{1+x^2}}}{2 \left(-1+x^2\right)}+\frac{\left(\sqrt{2 \left(1+5 \sqrt{2}\right)}-4 \sqrt{1+\sqrt{2}} x^2+\sqrt{2 \left(1+\sqrt{2}\right)} x^2\right) \tan ^{-1}\left(\frac{\sqrt{x+\sqrt{1+x^2}}}{\sqrt{-1+\sqrt{2}}}\right)}{8 (-1+x) (1+x)}+\frac{\left(-\sqrt{2 \left(-1+5 \sqrt{2}\right)}+4 \sqrt{-1+\sqrt{2}} x^2+\sqrt{2 \left(-1+\sqrt{2}\right)} x^2\right) \tan ^{-1}\left(\frac{\sqrt{x+\sqrt{1+x^2}}}{\sqrt{1+\sqrt{2}}}\right)}{8 (-1+x) (1+x)}+\frac{\left(\sqrt{2 \left(1+5 \sqrt{2}\right)}-4 \sqrt{1+\sqrt{2}} x^2+\sqrt{2 \left(1+\sqrt{2}\right)} x^2\right) \tanh ^{-1}\left(\frac{\sqrt{x+\sqrt{1+x^2}}}{\sqrt{-1+\sqrt{2}}}\right)}{8 (-1+x) (1+x)}+\frac{\left(-\sqrt{2 \left(-1+5 \sqrt{2}\right)}+4 \sqrt{-1+\sqrt{2}} x^2+\sqrt{2 \left(-1+\sqrt{2}\right)} x^2\right) \tanh ^{-1}\left(\frac{\sqrt{x+\sqrt{1+x^2}}}{\sqrt{1+\sqrt{2}}}\right)}{8 (-1+x) (1+x)}",1,"(x^2*Sqrt[x + Sqrt[1 + x^2]])/(2*(-1 + x^2)) - (x*Sqrt[1 + x^2]*Sqrt[x + Sqrt[1 + x^2]])/(2*(-1 + x^2)) + (Sqrt[(-1 + 5*Sqrt[2])/2]*ArcTan[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]])/4 - (Sqrt[(1 + 5*Sqrt[2])/2]*ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]])/4 + (Sqrt[(-1 + 5*Sqrt[2])/2]*ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]])/4 - (Sqrt[(1 + 5*Sqrt[2])/2]*ArcTanh[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]])/4","A",1
3031,1,437,437,0.7194201,"\int \frac{\left(c+b x+a x^2\right)^{3/2}}{1-x \sqrt{c+b x+a x^2}} \, dx","IntegrateAlgebraic[(c + b*x + a*x^2)^(3/2)/(1 - x*Sqrt[c + b*x + a*x^2]),x]","-\frac{1}{2} x (2 b+a x)+\left(\sqrt{a}+c\right) \log \left(b+2 a x-2 \sqrt{a} \sqrt{c+b x+a x^2}\right)-2 \text{RootSum}\left[b^2-\sqrt{a} c^2-4 \sqrt{a} b \text{$\#$1}+b c \text{$\#$1}+4 a \text{$\#$1}^2-b \text{$\#$1}^3+\sqrt{a} \text{$\#$1}^4\&,\frac{-a b \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)+2 a^{3/2} \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}-b^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}+2 a c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}+\sqrt{a} c^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}+\sqrt{a} b \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}^2-b c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}^2+\sqrt{a} c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}^3}{-4 \sqrt{a} b+b c+8 a \text{$\#$1}-3 b \text{$\#$1}^2+4 \sqrt{a} \text{$\#$1}^3}\&\right]","-\frac{1}{2} x (2 b+a x)+\left(\sqrt{a}+c\right) \log \left(b+2 a x-2 \sqrt{a} \sqrt{c+b x+a x^2}\right)-2 \text{RootSum}\left[b^2-\sqrt{a} c^2-4 \sqrt{a} b \text{$\#$1}+b c \text{$\#$1}+4 a \text{$\#$1}^2-b \text{$\#$1}^3+\sqrt{a} \text{$\#$1}^4\&,\frac{-a b \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)+2 a^{3/2} \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}-b^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}+2 a c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}+\sqrt{a} c^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}+\sqrt{a} b \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}^2-b c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}^2+\sqrt{a} c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}^3}{-4 \sqrt{a} b+b c+8 a \text{$\#$1}-3 b \text{$\#$1}^2+4 \sqrt{a} \text{$\#$1}^3}\&\right]",1,"-1/2*(x*(2*b + a*x)) + (Sqrt[a] + c)*Log[b + 2*a*x - 2*Sqrt[a]*Sqrt[c + b*x + a*x^2]] - 2*RootSum[b^2 - Sqrt[a]*c^2 - 4*Sqrt[a]*b*#1 + b*c*#1 + 4*a*#1^2 - b*#1^3 + Sqrt[a]*#1^4 & , (-(a*b*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]) + 2*a^(3/2)*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 - b^2*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 + 2*a*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 + Sqrt[a]*c^2*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 + Sqrt[a]*b*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1^2 - b*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1^2 + Sqrt[a]*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1^3)/(-4*Sqrt[a]*b + b*c + 8*a*#1 - 3*b*#1^2 + 4*Sqrt[a]*#1^3) & ]","A",1
3032,1,438,438,14.4246122,"\int \frac{(-b+x)^2}{\left((-a+x) (-b+x)^2\right)^{2/3} \left(-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2\right)} \, dx","IntegrateAlgebraic[(-b + x)^2/(((-a + x)*(-b + x)^2)^(2/3)*(-a^2 + b^2*d + 2*(a - b*d)*x + (-1 + d)*x^2)),x]","\frac{(-a+x)^{2/3} (-b+x) \left(\sqrt[3]{d} (b-x)^{2/3}-\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+(-a+x)^{2/3}\right) \left(\sqrt[3]{d} (b-x)^{2/3}+\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+(-a+x)^{2/3}\right) \left(-b \sqrt[3]{d}+\sqrt[3]{d} x+\sqrt[3]{b-x} (-a+x)^{2/3}\right) \left(\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a+x}}{-2 \sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}}\right)}{2 (a-b) d^{5/6}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a+x}}{2 \sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}}\right)}{2 (a-b) d^{5/6}}+\frac{\tanh ^{-1}\left(\frac{(b-x)^{2/3} \sqrt[3]{-a+x}}{\sqrt[6]{d} (-b+x)}\right)}{(a-b) d^{5/6}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{d} (b-x)^{2/3}+\frac{(-a+x)^{2/3}}{\sqrt[6]{d}}}{\sqrt[3]{b-x} \sqrt[3]{-a+x}}\right)}{2 (a-b) d^{5/6}}\right)}{\left((b-x)^2 (-a+x)\right)^{2/3} \left(-a^2+b^2 d+2 a x-2 b d x+(-1+d) x^2\right)}","\frac{(-a+x)^{2/3} (-b+x) \left(\sqrt[3]{d} (b-x)^{2/3}-\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+(-a+x)^{2/3}\right) \left(\sqrt[3]{d} (b-x)^{2/3}+\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+(-a+x)^{2/3}\right) \left(-b \sqrt[3]{d}+\sqrt[3]{d} x+\sqrt[3]{b-x} (-a+x)^{2/3}\right) \left(\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a+x}}{-2 \sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}}\right)}{2 (a-b) d^{5/6}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a+x}}{2 \sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}}\right)}{2 (a-b) d^{5/6}}+\frac{\tanh ^{-1}\left(\frac{(b-x)^{2/3} \sqrt[3]{-a+x}}{\sqrt[6]{d} (-b+x)}\right)}{(a-b) d^{5/6}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[6]{d} (b-x)^{2/3}+\frac{(-a+x)^{2/3}}{\sqrt[6]{d}}}{\sqrt[3]{b-x} \sqrt[3]{-a+x}}\right)}{2 (a-b) d^{5/6}}\right)}{\left((b-x)^2 (-a+x)\right)^{2/3} \left(-a^2+b^2 d+2 a x-2 b d x+(-1+d) x^2\right)}",1,"((-a + x)^(2/3)*(-b + x)*(d^(1/3)*(b - x)^(2/3) - d^(1/6)*(b - x)^(1/3)*(-a + x)^(1/3) + (-a + x)^(2/3))*(d^(1/3)*(b - x)^(2/3) + d^(1/6)*(b - x)^(1/3)*(-a + x)^(1/3) + (-a + x)^(2/3))*(-(b*d^(1/3)) + d^(1/3)*x + (b - x)^(1/3)*(-a + x)^(2/3))*((Sqrt[3]*ArcTan[(Sqrt[3]*(-a + x)^(1/3))/(-2*d^(1/6)*(b - x)^(1/3) + (-a + x)^(1/3))])/(2*(a - b)*d^(5/6)) - (Sqrt[3]*ArcTan[(Sqrt[3]*(-a + x)^(1/3))/(2*d^(1/6)*(b - x)^(1/3) + (-a + x)^(1/3))])/(2*(a - b)*d^(5/6)) + ArcTanh[((b - x)^(2/3)*(-a + x)^(1/3))/(d^(1/6)*(-b + x))]/((a - b)*d^(5/6)) - ArcTanh[(d^(1/6)*(b - x)^(2/3) + (-a + x)^(2/3)/d^(1/6))/((b - x)^(1/3)*(-a + x)^(1/3))]/(2*(a - b)*d^(5/6))))/(((b - x)^2*(-a + x))^(2/3)*(-a^2 + b^2*d + 2*a*x - 2*b*d*x + (-1 + d)*x^2))","A",1
3033,1,491,441,3.1815562,"\int \frac{b+a x}{(-b+a x) \sqrt[3]{-b^2 x^2+a^3 x^3}} \, dx","IntegrateAlgebraic[(b + a*x)/((-b + a*x)*(-(b^2*x^2) + a^3*x^3)^(1/3)),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{-b^2 x^2+a^3 x^3}}\right)}{a}+\frac{\sqrt{6 \left(-1+i \sqrt{3}\right)} \tan ^{-1}\left(\frac{3 \sqrt[3]{a} \sqrt[3]{a^2-b} x}{\sqrt{3} \sqrt[3]{a} \sqrt[3]{a^2-b} x-3 i \sqrt[3]{-b^2 x^2+a^3 x^3}-\sqrt{3} \sqrt[3]{-b^2 x^2+a^3 x^3}}\right)}{\sqrt[3]{a} \sqrt[3]{a^2-b}}-\frac{i \left(-i+\sqrt{3}\right) \log \left(2 \sqrt[3]{a} \sqrt[3]{a^2-b} x+\left(1+i \sqrt{3}\right) \sqrt[3]{-b^2 x^2+a^3 x^3}\right)}{\sqrt[3]{a} \sqrt[3]{a^2-b}}-\frac{\log \left(a^2 x-a \sqrt[3]{-b^2 x^2+a^3 x^3}\right)}{a}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{-b^2 x^2+a^3 x^3}+\left(-b^2 x^2+a^3 x^3\right)^{2/3}\right)}{2 a}+\frac{\left(1+i \sqrt{3}\right) \log \left(-2 i a^{2/3} \left(a^2-b\right)^{2/3} x^2+\sqrt[3]{a} \sqrt[3]{a^2-b} \left(i x-\sqrt{3} x\right) \sqrt[3]{-b^2 x^2+a^3 x^3}+\left(i+\sqrt{3}\right) \left(-b^2 x^2+a^3 x^3\right)^{2/3}\right)}{2 \sqrt[3]{a} \sqrt[3]{a^2-b}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{-b^2 x^2+a^3 x^3}}\right)}{a}+\frac{\sqrt{6 \left(-1+i \sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a} \sqrt[3]{a^2-b} x}{\sqrt[3]{a} \sqrt[3]{a^2-b} x-2 \sqrt[3]{-1} \sqrt[3]{-b^2 x^2+a^3 x^3}}\right)}{\sqrt[3]{a} \sqrt[3]{a^2-b}}-\frac{\log \left(-a x+\sqrt[3]{-b^2 x^2+a^3 x^3}\right)}{a}-\frac{i \left(-i+\sqrt{3}\right) \log \left(\sqrt[3]{a} \sqrt[3]{a^2-b} x+\sqrt[3]{-1} \sqrt[3]{-b^2 x^2+a^3 x^3}\right)}{\sqrt[3]{a} \sqrt[3]{a^2-b}}+\frac{\log \left(a^2 x^2+a x \sqrt[3]{-b^2 x^2+a^3 x^3}+\left(-b^2 x^2+a^3 x^3\right)^{2/3}\right)}{2 a}+\frac{\left(1+i \sqrt{3}\right) \log \left(a^{2/3} \left(a^2-b\right)^{2/3} x^2-\sqrt[3]{-1} \sqrt[3]{a} \sqrt[3]{a^2-b} x \sqrt[3]{-b^2 x^2+a^3 x^3}+(-1)^{2/3} \left(-b^2 x^2+a^3 x^3\right)^{2/3}\right)}{2 \sqrt[3]{a} \sqrt[3]{a^2-b}}",1,"(Sqrt[3]*ArcTan[(Sqrt[3]*a*x)/(a*x + 2*(-(b^2*x^2) + a^3*x^3)^(1/3))])/a + (Sqrt[6*(-1 + I*Sqrt[3])]*ArcTan[(3*a^(1/3)*(a^2 - b)^(1/3)*x)/(Sqrt[3]*a^(1/3)*(a^2 - b)^(1/3)*x - (3*I)*(-(b^2*x^2) + a^3*x^3)^(1/3) - Sqrt[3]*(-(b^2*x^2) + a^3*x^3)^(1/3))])/(a^(1/3)*(a^2 - b)^(1/3)) - (I*(-I + Sqrt[3])*Log[2*a^(1/3)*(a^2 - b)^(1/3)*x + (1 + I*Sqrt[3])*(-(b^2*x^2) + a^3*x^3)^(1/3)])/(a^(1/3)*(a^2 - b)^(1/3)) - Log[a^2*x - a*(-(b^2*x^2) + a^3*x^3)^(1/3)]/a + Log[a^2*x^2 + a*x*(-(b^2*x^2) + a^3*x^3)^(1/3) + (-(b^2*x^2) + a^3*x^3)^(2/3)]/(2*a) + ((1 + I*Sqrt[3])*Log[(-2*I)*a^(2/3)*(a^2 - b)^(2/3)*x^2 + a^(1/3)*(a^2 - b)^(1/3)*(I*x - Sqrt[3]*x)*(-(b^2*x^2) + a^3*x^3)^(1/3) + (I + Sqrt[3])*(-(b^2*x^2) + a^3*x^3)^(2/3)])/(2*a^(1/3)*(a^2 - b)^(1/3))","A",1
3034,1,445,445,1.376094,"\int \frac{1}{\sqrt[3]{a x+\sqrt{-b+a^2 x^2}} \sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}} \, dx","IntegrateAlgebraic[1/((a*x + Sqrt[-b + a^2*x^2])^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)),x]","-\frac{8 c \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}}{7 a}+\frac{3 b \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}}{8 a c \left(a x+\sqrt{-b+a^2 x^2}\right)^{4/3}}-\frac{13 b \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}}{32 a c^2 \left(a x+\sqrt{-b+a^2 x^2}\right)}+\frac{117 b \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}}{256 a c^3 \left(a x+\sqrt{-b+a^2 x^2}\right)^{2/3}}-\frac{585 b \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}}{1024 a c^4 \sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}+\frac{6 \sqrt[3]{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}}{7 a}-\frac{585 b \tan ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{2048 a c^{17/4}}+\frac{585 b \tanh ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{2048 a c^{17/4}}","-\frac{8 c \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}}{7 a}+\frac{3 b \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}}{8 a c \left(a x+\sqrt{-b+a^2 x^2}\right)^{4/3}}-\frac{13 b \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}}{32 a c^2 \left(a x+\sqrt{-b+a^2 x^2}\right)}+\frac{117 b \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}}{256 a c^3 \left(a x+\sqrt{-b+a^2 x^2}\right)^{2/3}}-\frac{585 b \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}}{1024 a c^4 \sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}+\frac{6 \sqrt[3]{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}}{7 a}-\frac{585 b \tan ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{2048 a c^{17/4}}+\frac{585 b \tanh ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{2048 a c^{17/4}}",1,"(-8*c*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4))/(7*a) + (3*b*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4))/(8*a*c*(a*x + Sqrt[-b + a^2*x^2])^(4/3)) - (13*b*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4))/(32*a*c^2*(a*x + Sqrt[-b + a^2*x^2])) + (117*b*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4))/(256*a*c^3*(a*x + Sqrt[-b + a^2*x^2])^(2/3)) - (585*b*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4))/(1024*a*c^4*(a*x + Sqrt[-b + a^2*x^2])^(1/3)) + (6*(a*x + Sqrt[-b + a^2*x^2])^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4))/(7*a) - (585*b*ArcTan[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)])/(2048*a*c^(17/4)) + (585*b*ArcTanh[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)])/(2048*a*c^(17/4))","A",1
3035,1,510,448,39.9387404,"\int \frac{x^4}{\sqrt[4]{-b+a x^4} \left(-b+2 a x^4+x^8\right)} \, dx","IntegrateAlgebraic[x^4/((-b + a*x^4)^(1/4)*(-b + 2*a*x^4 + x^8)),x]","-\frac{i \left(-i \sqrt{2}+\sqrt{2 \left(3-2 \sqrt{2}\right)}\right) \tan ^{-1}\left(\frac{\left((-1+i)-(1+i) (-1)^{3/4}\right) \sqrt[4]{a^2+b} x^2+(1+i) \sqrt{-b+a x^4}+(1+i) (-1)^{3/4} \sqrt{-b+a x^4}}{2 \sqrt[8]{a^2+b} x \sqrt[4]{-b+a x^4}}\right)}{16 \left(a^2+b\right)^{5/8}}+\frac{\left(\sqrt{2}-i \sqrt{2 \left(3+2 \sqrt{2}\right)}\right) \tan ^{-1}\left(\frac{2 \sqrt[8]{a^2+b} x \sqrt[4]{-b+a x^4}}{\left((1-i)+\sqrt{2}\right) \sqrt[4]{a^2+b} x^2-(1+i) \sqrt{-b+a x^4}-\sqrt{2} \sqrt{-b+a x^4}}\right)}{16 \left(a^2+b\right)^{5/8}}+\frac{\left(-i+(-1)^{3/4}\right) \tanh ^{-1}\left(\frac{\left((-2+2 i)-(2+2 i) (-1)^{3/4}\right) \sqrt[4]{a^2+b} x^2-(2+2 i) \sqrt{-b+a x^4}-(2+2 i) (-1)^{3/4} \sqrt{-b+a x^4}}{4 \sqrt[8]{a^2+b} x \sqrt[4]{-b+a x^4}}\right)}{8 \left(a^2+b\right)^{5/8}}+\frac{i \left(i \sqrt{2}+\sqrt{2 \left(3+2 \sqrt{2}\right)}\right) \tanh ^{-1}\left(\frac{\left((1-i)+\sqrt{2}\right) \sqrt[4]{a^2+b} x^2+(1+i) \sqrt{-b+a x^4}+\sqrt{2} \sqrt{-b+a x^4}}{2 \sqrt[8]{a^2+b} x \sqrt[4]{-b+a x^4}}\right)}{16 \left(a^2+b\right)^{5/8}}","\frac{\left(-1+\sqrt[4]{-1}\right) \tan ^{-1}\left(\frac{(-1)^{7/8} \sqrt{2+\sqrt{2}} \sqrt[8]{a^2+b} x \sqrt[4]{-b+a x^4}}{(-1)^{3/4} \sqrt[4]{a^2+b} x^2+\sqrt{-b+a x^4}}\right)}{8 \left(a^2+b\right)^{5/8}}+\frac{i \left(i \sqrt{2}+\sqrt{2 \left(3+2 \sqrt{2}\right)}\right) \tan ^{-1}\left(\frac{(-1)^{7/8} \left(-2+\sqrt{2}\right) \sqrt[8]{a^2+b} x \sqrt[4]{-b+a x^4}}{(-1)^{3/4} \sqrt{2-\sqrt{2}} \sqrt[4]{a^2+b} x^2+\sqrt{2-\sqrt{2}} \sqrt{-b+a x^4}}\right)}{16 \left(a^2+b\right)^{5/8}}+\frac{\left(\sqrt{2}-i \sqrt{2 \left(3+2 \sqrt{2}\right)}\right) \tanh ^{-1}\left(\frac{(-1)^{7/8} \sqrt[4]{a^2+b} x^2-\sqrt[8]{-1} \sqrt{-b+a x^4}}{\sqrt{2-\sqrt{2}} \sqrt[8]{a^2+b} x \sqrt[4]{-b+a x^4}}\right)}{16 \left(a^2+b\right)^{5/8}}+\frac{\left(-1+\sqrt[4]{-1}\right) \tanh ^{-1}\left(\frac{(-1)^{7/8} \sqrt[4]{a^2+b} x^2-\sqrt[8]{-1} \sqrt{-b+a x^4}}{\sqrt{2+\sqrt{2}} \sqrt[8]{a^2+b} x \sqrt[4]{-b+a x^4}}\right)}{8 \left(a^2+b\right)^{5/8}}",1,"((-1/16*I)*((-I)*Sqrt[2] + Sqrt[2*(3 - 2*Sqrt[2])])*ArcTan[(((-1 + I) - (1 + I)*(-1)^(3/4))*(a^2 + b)^(1/4)*x^2 + (1 + I)*Sqrt[-b + a*x^4] + (1 + I)*(-1)^(3/4)*Sqrt[-b + a*x^4])/(2*(a^2 + b)^(1/8)*x*(-b + a*x^4)^(1/4))])/(a^2 + b)^(5/8) + ((Sqrt[2] - I*Sqrt[2*(3 + 2*Sqrt[2])])*ArcTan[(2*(a^2 + b)^(1/8)*x*(-b + a*x^4)^(1/4))/(((1 - I) + Sqrt[2])*(a^2 + b)^(1/4)*x^2 - (1 + I)*Sqrt[-b + a*x^4] - Sqrt[2]*Sqrt[-b + a*x^4])])/(16*(a^2 + b)^(5/8)) + ((-I + (-1)^(3/4))*ArcTanh[(((-2 + 2*I) - (2 + 2*I)*(-1)^(3/4))*(a^2 + b)^(1/4)*x^2 - (2 + 2*I)*Sqrt[-b + a*x^4] - (2 + 2*I)*(-1)^(3/4)*Sqrt[-b + a*x^4])/(4*(a^2 + b)^(1/8)*x*(-b + a*x^4)^(1/4))])/(8*(a^2 + b)^(5/8)) + ((I/16)*(I*Sqrt[2] + Sqrt[2*(3 + 2*Sqrt[2])])*ArcTanh[(((1 - I) + Sqrt[2])*(a^2 + b)^(1/4)*x^2 + (1 + I)*Sqrt[-b + a*x^4] + Sqrt[2]*Sqrt[-b + a*x^4])/(2*(a^2 + b)^(1/8)*x*(-b + a*x^4)^(1/4))])/(a^2 + b)^(5/8)","A",1
3036,1,510,448,5.3828869,"\int \frac{\left(d+c x^2\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4}}{\left(-b+a^2 x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[((d + c*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(3/4))/(-b + a^2*x^2)^(5/2),x]","\frac{\left(-53 b^2 c+43 a^2 b d+41 a^2 b c x^2-55 a^4 d x^2\right) \left(\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}\right)^{3/4}}{96 a^3 b^{13/8} \left(-\sqrt{b}+a x\right) \left(\sqrt{b}+a x\right)}+\frac{\sqrt{-b+a^2 x^2} \left(9 b^2 c x-87 a^2 b d x-41 a^2 b c x^3+55 a^4 d x^3\right) \left(\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}\right)^{3/4}}{96 a^2 b^{13/8} \left(-\sqrt{b}+a x\right)^2 \left(\sqrt{b}+a x\right)^2}+\frac{\left(41 b c-55 a^2 d\right) \tan ^{-1}\left(\sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}\right)}{64 a^3 b^{13/8}}+\frac{\left(-41 b c+55 a^2 d\right) \tan ^{-1}\left(\frac{-1+\sqrt{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}{\sqrt{2} \sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}\right)}{64 \sqrt{2} a^3 b^{13/8}}+\frac{\left(-41 b c+55 a^2 d\right) \tanh ^{-1}\left(\sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}\right)}{64 a^3 b^{13/8}}-\frac{\left(-41 b c+55 a^2 d\right) \tanh ^{-1}\left(\frac{1+\sqrt{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}{\sqrt{2} \sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}\right)}{64 \sqrt{2} a^3 b^{13/8}}","\frac{\left(-53 b^2 c+43 a^2 b d+41 a^2 b c x^2-55 a^4 d x^2\right) \left(\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}\right)^{3/4}}{96 a^3 b^{13/8} \left(-\sqrt{b}+a x\right) \left(\sqrt{b}+a x\right)}+\frac{\sqrt{-b+a^2 x^2} \left(9 b^2 c x-87 a^2 b d x-41 a^2 b c x^3+55 a^4 d x^3\right) \left(\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}\right)^{3/4}}{96 a^2 b^{13/8} \left(-\sqrt{b}+a x\right)^2 \left(\sqrt{b}+a x\right)^2}+\frac{\left(41 b c-55 a^2 d\right) \tan ^{-1}\left(\sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}\right)}{64 a^3 b^{13/8}}+\frac{\left(-41 b c+55 a^2 d\right) \tanh ^{-1}\left(\sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}\right)}{64 a^3 b^{13/8}}-\frac{\sqrt[4]{-1} \left(-41 b c+55 a^2 d\right) \tanh ^{-1}\left(\sqrt[4]{-1} \sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}\right)}{64 a^3 b^{13/8}}-\frac{(-1)^{3/4} \left(-41 b c+55 a^2 d\right) \tanh ^{-1}\left((-1)^{3/4} \sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}\right)}{64 a^3 b^{13/8}}",1,"((-53*b^2*c + 43*a^2*b*d + 41*a^2*b*c*x^2 - 55*a^4*d*x^2)*((a*x + Sqrt[-b + a^2*x^2])/Sqrt[b])^(3/4))/(96*a^3*b^(13/8)*(-Sqrt[b] + a*x)*(Sqrt[b] + a*x)) + (Sqrt[-b + a^2*x^2]*(9*b^2*c*x - 87*a^2*b*d*x - 41*a^2*b*c*x^3 + 55*a^4*d*x^3)*((a*x + Sqrt[-b + a^2*x^2])/Sqrt[b])^(3/4))/(96*a^2*b^(13/8)*(-Sqrt[b] + a*x)^2*(Sqrt[b] + a*x)^2) + ((41*b*c - 55*a^2*d)*ArcTan[((a*x + Sqrt[-b + a^2*x^2])/Sqrt[b])^(1/4)])/(64*a^3*b^(13/8)) + ((-41*b*c + 55*a^2*d)*ArcTan[(-1 + Sqrt[(a*x + Sqrt[-b + a^2*x^2])/Sqrt[b]])/(Sqrt[2]*((a*x + Sqrt[-b + a^2*x^2])/Sqrt[b])^(1/4))])/(64*Sqrt[2]*a^3*b^(13/8)) + ((-41*b*c + 55*a^2*d)*ArcTanh[((a*x + Sqrt[-b + a^2*x^2])/Sqrt[b])^(1/4)])/(64*a^3*b^(13/8)) - ((-41*b*c + 55*a^2*d)*ArcTanh[(1 + Sqrt[(a*x + Sqrt[-b + a^2*x^2])/Sqrt[b]])/(Sqrt[2]*((a*x + Sqrt[-b + a^2*x^2])/Sqrt[b])^(1/4))])/(64*Sqrt[2]*a^3*b^(13/8))","A",0
3037,1,153,452,0.8290704,"\int \frac{x^2}{\left(1+x^4\right) \sqrt[4]{-x^2+x^6}} \, dx","IntegrateAlgebraic[x^2/((1 + x^4)*(-x^2 + x^6)^(1/4)),x]","-\frac{1}{4} \sqrt{\frac{1}{2}-\frac{i}{2}} \tan ^{-1}\left(\frac{\sqrt{-1-i} x}{\sqrt[4]{-x^2+x^6}}\right)-\frac{1}{4} \sqrt{\frac{1}{2}+\frac{i}{2}} \tan ^{-1}\left(\frac{\sqrt{-1+i} x}{\sqrt[4]{-x^2+x^6}}\right)+\frac{1}{4} \sqrt{-\frac{1}{2}-\frac{i}{2}} \tan ^{-1}\left(\frac{\sqrt{1-i} x}{\sqrt[4]{-x^2+x^6}}\right)+\frac{1}{4} \sqrt{-\frac{1}{2}+\frac{i}{2}} \tan ^{-1}\left(\frac{\sqrt{1+i} x}{\sqrt[4]{-x^2+x^6}}\right)","-\frac{1}{8} \sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x}{-\sqrt{2+\sqrt{2}} x+2^{3/4} \sqrt[4]{-x^2+x^6}}\right)-\frac{1}{8} \sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x}{\sqrt{2+\sqrt{2}} x+2^{3/4} \sqrt[4]{-x^2+x^6}}\right)+\frac{1}{8} \sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\frac{2^{3/4} \sqrt{2+\sqrt{2}} x \sqrt[4]{-x^2+x^6}}{-2 x^2+\sqrt{2} \sqrt{-x^2+x^6}}\right)-\frac{1}{8} \sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{2} x^2}{\sqrt{2-\sqrt{2}}}+\frac{\sqrt{-x^2+x^6}}{\sqrt[4]{2} \sqrt{2-\sqrt{2}}}}{x \sqrt[4]{-x^2+x^6}}\right)-\frac{1}{16} \sqrt{-1+\sqrt{2}} \log \left(-2 x^2+2^{3/4} \sqrt{2+\sqrt{2}} x \sqrt[4]{-x^2+x^6}-\sqrt{2} \sqrt{-x^2+x^6}\right)+\frac{1}{16} \sqrt{-1+\sqrt{2}} \log \left(2 \sqrt{2-\sqrt{2}} x^2+2 \sqrt[4]{2} x \sqrt[4]{-x^2+x^6}+\sqrt{4-2 \sqrt{2}} \sqrt{-x^2+x^6}\right)",1,"-1/4*(Sqrt[1/2 - I/2]*ArcTan[(Sqrt[-1 - I]*x)/(-x^2 + x^6)^(1/4)]) - (Sqrt[1/2 + I/2]*ArcTan[(Sqrt[-1 + I]*x)/(-x^2 + x^6)^(1/4)])/4 + (Sqrt[-1/2 - I/2]*ArcTan[(Sqrt[1 - I]*x)/(-x^2 + x^6)^(1/4)])/4 + (Sqrt[-1/2 + I/2]*ArcTan[(Sqrt[1 + I]*x)/(-x^2 + x^6)^(1/4)])/4","C",1
3038,1,153,452,0.0014556,"\int \frac{x^2}{\left(1+x^4\right) \sqrt[4]{-x^2+x^6}} \, dx","IntegrateAlgebraic[x^2/((1 + x^4)*(-x^2 + x^6)^(1/4)),x]","-\frac{1}{4} \sqrt{\frac{1}{2}-\frac{i}{2}} \tan ^{-1}\left(\frac{\sqrt{-1-i} x}{\sqrt[4]{-x^2+x^6}}\right)-\frac{1}{4} \sqrt{\frac{1}{2}+\frac{i}{2}} \tan ^{-1}\left(\frac{\sqrt{-1+i} x}{\sqrt[4]{-x^2+x^6}}\right)+\frac{1}{4} \sqrt{-\frac{1}{2}-\frac{i}{2}} \tan ^{-1}\left(\frac{\sqrt{1-i} x}{\sqrt[4]{-x^2+x^6}}\right)+\frac{1}{4} \sqrt{-\frac{1}{2}+\frac{i}{2}} \tan ^{-1}\left(\frac{\sqrt{1+i} x}{\sqrt[4]{-x^2+x^6}}\right)","-\frac{1}{8} \sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x}{-\sqrt{2+\sqrt{2}} x+2^{3/4} \sqrt[4]{-x^2+x^6}}\right)-\frac{1}{8} \sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x}{\sqrt{2+\sqrt{2}} x+2^{3/4} \sqrt[4]{-x^2+x^6}}\right)+\frac{1}{8} \sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\frac{2^{3/4} \sqrt{2+\sqrt{2}} x \sqrt[4]{-x^2+x^6}}{-2 x^2+\sqrt{2} \sqrt{-x^2+x^6}}\right)-\frac{1}{8} \sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{2} x^2}{\sqrt{2-\sqrt{2}}}+\frac{\sqrt{-x^2+x^6}}{\sqrt[4]{2} \sqrt{2-\sqrt{2}}}}{x \sqrt[4]{-x^2+x^6}}\right)-\frac{1}{16} \sqrt{-1+\sqrt{2}} \log \left(-2 x^2+2^{3/4} \sqrt{2+\sqrt{2}} x \sqrt[4]{-x^2+x^6}-\sqrt{2} \sqrt{-x^2+x^6}\right)+\frac{1}{16} \sqrt{-1+\sqrt{2}} \log \left(2 \sqrt{2-\sqrt{2}} x^2+2 \sqrt[4]{2} x \sqrt[4]{-x^2+x^6}+\sqrt{4-2 \sqrt{2}} \sqrt{-x^2+x^6}\right)",1,"-1/4*(Sqrt[1/2 - I/2]*ArcTan[(Sqrt[-1 - I]*x)/(-x^2 + x^6)^(1/4)]) - (Sqrt[1/2 + I/2]*ArcTan[(Sqrt[-1 + I]*x)/(-x^2 + x^6)^(1/4)])/4 + (Sqrt[-1/2 - I/2]*ArcTan[(Sqrt[1 - I]*x)/(-x^2 + x^6)^(1/4)])/4 + (Sqrt[-1/2 + I/2]*ArcTan[(Sqrt[1 + I]*x)/(-x^2 + x^6)^(1/4)])/4","C",1
3039,1,153,452,0.7756882,"\int \frac{-1+x^4}{\left(1+x^4\right) \sqrt[4]{-x^2+x^6}} \, dx","IntegrateAlgebraic[(-1 + x^4)/((1 + x^4)*(-x^2 + x^6)^(1/4)),x]","-\frac{1}{2} \sqrt{-\frac{1}{2}+\frac{i}{2}} \tan ^{-1}\left(\frac{\sqrt{-1-i} x}{\sqrt[4]{-x^2+x^6}}\right)-\frac{1}{2} \sqrt{-\frac{1}{2}-\frac{i}{2}} \tan ^{-1}\left(\frac{\sqrt{-1+i} x}{\sqrt[4]{-x^2+x^6}}\right)-\frac{1}{2} \sqrt{\frac{1}{2}+\frac{i}{2}} \tan ^{-1}\left(\frac{\sqrt{1-i} x}{\sqrt[4]{-x^2+x^6}}\right)-\frac{1}{2} \sqrt{\frac{1}{2}-\frac{i}{2}} \tan ^{-1}\left(\frac{\sqrt{1+i} x}{\sqrt[4]{-x^2+x^6}}\right)","-\frac{1}{4} \sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x}{-\sqrt{2+\sqrt{2}} x+2^{3/4} \sqrt[4]{-x^2+x^6}}\right)-\frac{1}{4} \sqrt{-1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x}{\sqrt{2+\sqrt{2}} x+2^{3/4} \sqrt[4]{-x^2+x^6}}\right)-\frac{1}{4} \sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{2^{3/4} \sqrt{2+\sqrt{2}} x \sqrt[4]{-x^2+x^6}}{-2 x^2+\sqrt{2} \sqrt{-x^2+x^6}}\right)-\frac{1}{4} \sqrt{-1+\sqrt{2}} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{2} x^2}{\sqrt{2-\sqrt{2}}}+\frac{\sqrt{-x^2+x^6}}{\sqrt[4]{2} \sqrt{2-\sqrt{2}}}}{x \sqrt[4]{-x^2+x^6}}\right)+\frac{1}{8} \sqrt{1+\sqrt{2}} \log \left(-2 x^2+2^{3/4} \sqrt{2+\sqrt{2}} x \sqrt[4]{-x^2+x^6}-\sqrt{2} \sqrt{-x^2+x^6}\right)-\frac{1}{8} \sqrt{1+\sqrt{2}} \log \left(2 \sqrt{2-\sqrt{2}} x^2+2 \sqrt[4]{2} x \sqrt[4]{-x^2+x^6}+\sqrt{4-2 \sqrt{2}} \sqrt{-x^2+x^6}\right)",1,"-1/2*(Sqrt[-1/2 + I/2]*ArcTan[(Sqrt[-1 - I]*x)/(-x^2 + x^6)^(1/4)]) - (Sqrt[-1/2 - I/2]*ArcTan[(Sqrt[-1 + I]*x)/(-x^2 + x^6)^(1/4)])/2 - (Sqrt[1/2 + I/2]*ArcTan[(Sqrt[1 - I]*x)/(-x^2 + x^6)^(1/4)])/2 - (Sqrt[1/2 - I/2]*ArcTan[(Sqrt[1 + I]*x)/(-x^2 + x^6)^(1/4)])/2","C",1
3040,1,510,452,39.0909439,"\int \frac{\left(-b+a x^4\right)^{3/4}}{-b+2 a x^4+x^8} \, dx","IntegrateAlgebraic[(-b + a*x^4)^(3/4)/(-b + 2*a*x^4 + x^8),x]","\frac{\left(\sqrt{2}-i \sqrt{2 \left(3+2 \sqrt{2}\right)}\right) \tan ^{-1}\left(\frac{\left((-1+i)-(1+i) (-1)^{3/4}\right) \sqrt[4]{a^2+b} x^2+(1+i) \sqrt{-b+a x^4}+(1+i) (-1)^{3/4} \sqrt{-b+a x^4}}{2 \sqrt[8]{a^2+b} x \sqrt[4]{-b+a x^4}}\right)}{16 \sqrt[8]{a^2+b}}-\frac{i \left(-i \sqrt{2}+\sqrt{2 \left(3-2 \sqrt{2}\right)}\right) \tan ^{-1}\left(\frac{2 \sqrt[8]{a^2+b} x \sqrt[4]{-b+a x^4}}{\left((1-i)+\sqrt{2}\right) \sqrt[4]{a^2+b} x^2-(1+i) \sqrt{-b+a x^4}-\sqrt{2} \sqrt{-b+a x^4}}\right)}{16 \sqrt[8]{a^2+b}}+\frac{\left(-i-(-1)^{3/4}\right) \tanh ^{-1}\left(\frac{\left((-2+2 i)-(2+2 i) (-1)^{3/4}\right) \sqrt[4]{a^2+b} x^2-(2+2 i) \sqrt{-b+a x^4}-(2+2 i) (-1)^{3/4} \sqrt{-b+a x^4}}{4 \sqrt[8]{a^2+b} x \sqrt[4]{-b+a x^4}}\right)}{8 \sqrt[8]{a^2+b}}+\frac{\left(\sqrt{2}+i \sqrt{2 \left(3-2 \sqrt{2}\right)}\right) \tanh ^{-1}\left(\frac{\left((1-i)+\sqrt{2}\right) \sqrt[4]{a^2+b} x^2+(1+i) \sqrt{-b+a x^4}+\sqrt{2} \sqrt{-b+a x^4}}{2 \sqrt[8]{a^2+b} x \sqrt[4]{-b+a x^4}}\right)}{16 \sqrt[8]{a^2+b}}","\frac{\left(-1-\sqrt[4]{-1}\right) \tan ^{-1}\left(\frac{(-1)^{7/8} \sqrt{2+\sqrt{2}} \sqrt[8]{a^2+b} x \sqrt[4]{-b+a x^4}}{(-1)^{3/4} \sqrt[4]{a^2+b} x^2+\sqrt{-b+a x^4}}\right)}{8 \sqrt[8]{a^2+b}}+\frac{\left(\sqrt{2}+i \sqrt{2 \left(3-2 \sqrt{2}\right)}\right) \tan ^{-1}\left(\frac{(-1)^{7/8} \left(-2+\sqrt{2}\right) \sqrt[8]{a^2+b} x \sqrt[4]{-b+a x^4}}{(-1)^{3/4} \sqrt{2-\sqrt{2}} \sqrt[4]{a^2+b} x^2+\sqrt{2-\sqrt{2}} \sqrt{-b+a x^4}}\right)}{16 \sqrt[8]{a^2+b}}-\frac{i \left(-i \sqrt{2}+\sqrt{2 \left(3-2 \sqrt{2}\right)}\right) \tanh ^{-1}\left(\frac{(-1)^{7/8} \sqrt[4]{a^2+b} x^2-\sqrt[8]{-1} \sqrt{-b+a x^4}}{\sqrt{2-\sqrt{2}} \sqrt[8]{a^2+b} x \sqrt[4]{-b+a x^4}}\right)}{16 \sqrt[8]{a^2+b}}+\frac{\left(-1-\sqrt[4]{-1}\right) \tanh ^{-1}\left(\frac{(-1)^{7/8} \sqrt[4]{a^2+b} x^2-\sqrt[8]{-1} \sqrt{-b+a x^4}}{\sqrt{2+\sqrt{2}} \sqrt[8]{a^2+b} x \sqrt[4]{-b+a x^4}}\right)}{8 \sqrt[8]{a^2+b}}",1,"((Sqrt[2] - I*Sqrt[2*(3 + 2*Sqrt[2])])*ArcTan[(((-1 + I) - (1 + I)*(-1)^(3/4))*(a^2 + b)^(1/4)*x^2 + (1 + I)*Sqrt[-b + a*x^4] + (1 + I)*(-1)^(3/4)*Sqrt[-b + a*x^4])/(2*(a^2 + b)^(1/8)*x*(-b + a*x^4)^(1/4))])/(16*(a^2 + b)^(1/8)) - ((I/16)*((-I)*Sqrt[2] + Sqrt[2*(3 - 2*Sqrt[2])])*ArcTan[(2*(a^2 + b)^(1/8)*x*(-b + a*x^4)^(1/4))/(((1 - I) + Sqrt[2])*(a^2 + b)^(1/4)*x^2 - (1 + I)*Sqrt[-b + a*x^4] - Sqrt[2]*Sqrt[-b + a*x^4])])/(a^2 + b)^(1/8) + ((-I - (-1)^(3/4))*ArcTanh[(((-2 + 2*I) - (2 + 2*I)*(-1)^(3/4))*(a^2 + b)^(1/4)*x^2 - (2 + 2*I)*Sqrt[-b + a*x^4] - (2 + 2*I)*(-1)^(3/4)*Sqrt[-b + a*x^4])/(4*(a^2 + b)^(1/8)*x*(-b + a*x^4)^(1/4))])/(8*(a^2 + b)^(1/8)) + ((Sqrt[2] + I*Sqrt[2*(3 - 2*Sqrt[2])])*ArcTanh[(((1 - I) + Sqrt[2])*(a^2 + b)^(1/4)*x^2 + (1 + I)*Sqrt[-b + a*x^4] + Sqrt[2]*Sqrt[-b + a*x^4])/(2*(a^2 + b)^(1/8)*x*(-b + a*x^4)^(1/4))])/(16*(a^2 + b)^(1/8))","A",1
3041,1,819,452,1.0853399,"\int \frac{\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1+x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/(1 + x^2)^(5/2),x]","\frac{\left(-117-7 x-151 x^2-27 x^3+30 x^4-20 x^5\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(10 x+8 x^2+34 x^3+8 x^4+24 x^5\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(-1-166 x-17 x^2+30 x^3-20 x^4\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(2+4 x+22 x^2+8 x^3+24 x^4\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{192 \sqrt{1+x^2} \left(3 x+7 x^3+4 x^5\right)+192 \left(1+6 x^2+9 x^4+4 x^6\right)}+\frac{1}{2} \text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{10 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+6 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-4 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-\text{$\#$1}+3 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{23}{256} \text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{40 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+40 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-25 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+6 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-\text{$\#$1}+3 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-\frac{3}{32} \text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{90 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+70 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-45 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+11 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-\text{$\#$1}+3 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","\frac{\left(-117-7 x-151 x^2-27 x^3+30 x^4-20 x^5\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(10 x+8 x^2+34 x^3+8 x^4+24 x^5\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(-1-166 x-17 x^2+30 x^3-20 x^4\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(2+4 x+22 x^2+8 x^3+24 x^4\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{192 \sqrt{1+x^2} \left(3 x+7 x^3+4 x^5\right)+192 \left(1+6 x^2+9 x^4+4 x^6\right)}+\frac{1}{256} \text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{40 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+8 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-7 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-\text{$\#$1}+3 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]",1,"((-117 - 7*x - 151*x^2 - 27*x^3 + 30*x^4 - 20*x^5)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (10*x + 8*x^2 + 34*x^3 + 8*x^4 + 24*x^5)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((-1 - 166*x - 17*x^2 + 30*x^3 - 20*x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (2 + 4*x + 22*x^2 + 8*x^3 + 24*x^4)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(192*Sqrt[1 + x^2]*(3*x + 7*x^3 + 4*x^5) + 192*(1 + 6*x^2 + 9*x^4 + 4*x^6)) + RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (10*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 6*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - 4*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(-#1 + 3*#1^3 - 3*#1^5 + #1^7) & ]/2 + (23*RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (40*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 40*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - 25*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + 6*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(-#1 + 3*#1^3 - 3*#1^5 + #1^7) & ])/256 - (3*RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (90*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 70*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - 45*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + 11*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(-#1 + 3*#1^3 - 3*#1^5 + #1^7) & ])/32","A",1
3042,1,819,452,0.0014728,"\int \frac{\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1+x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/(1 + x^2)^(5/2),x]","\frac{\left(-117-7 x-151 x^2-27 x^3+30 x^4-20 x^5\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(10 x+8 x^2+34 x^3+8 x^4+24 x^5\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(-1-166 x-17 x^2+30 x^3-20 x^4\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(2+4 x+22 x^2+8 x^3+24 x^4\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{192 \sqrt{1+x^2} \left(3 x+7 x^3+4 x^5\right)+192 \left(1+6 x^2+9 x^4+4 x^6\right)}+\frac{1}{2} \text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{10 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+6 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-4 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-\text{$\#$1}+3 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{23}{256} \text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{40 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+40 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-25 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+6 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-\text{$\#$1}+3 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-\frac{3}{32} \text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{90 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+70 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-45 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+11 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-\text{$\#$1}+3 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","\frac{\left(-117-7 x-151 x^2-27 x^3+30 x^4-20 x^5\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(10 x+8 x^2+34 x^3+8 x^4+24 x^5\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(-1-166 x-17 x^2+30 x^3-20 x^4\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(2+4 x+22 x^2+8 x^3+24 x^4\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{192 \sqrt{1+x^2} \left(3 x+7 x^3+4 x^5\right)+192 \left(1+6 x^2+9 x^4+4 x^6\right)}+\frac{1}{256} \text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{40 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+8 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-7 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-\text{$\#$1}+3 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]",1,"((-117 - 7*x - 151*x^2 - 27*x^3 + 30*x^4 - 20*x^5)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (10*x + 8*x^2 + 34*x^3 + 8*x^4 + 24*x^5)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((-1 - 166*x - 17*x^2 + 30*x^3 - 20*x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (2 + 4*x + 22*x^2 + 8*x^3 + 24*x^4)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(192*Sqrt[1 + x^2]*(3*x + 7*x^3 + 4*x^5) + 192*(1 + 6*x^2 + 9*x^4 + 4*x^6)) + RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (10*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 6*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - 4*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(-#1 + 3*#1^3 - 3*#1^5 + #1^7) & ]/2 + (23*RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (40*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 40*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - 25*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + 6*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(-#1 + 3*#1^3 - 3*#1^5 + #1^7) & ])/256 - (3*RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (90*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 70*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - 45*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + 11*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(-#1 + 3*#1^3 - 3*#1^5 + #1^7) & ])/32","A",1
3043,1,459,455,0.8460787,"\int \frac{x^2 \sqrt{b+a x}}{x^2-\sqrt{b+a x} \sqrt{c+\sqrt{b+a x}}} \, dx","IntegrateAlgebraic[(x^2*Sqrt[b + a*x])/(x^2 - Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]]),x]","4 a \sqrt{c+\sqrt{b+a x}}+\frac{2 \left(c^3+(b+a x)^{3/2}\right)}{3 a}-4 a \text{RootSum}\left[b^2-2 b c^2+c^4+a^2 c \text{$\#$1}+4 b c \text{$\#$1}^2-4 c^3 \text{$\#$1}^2-a^2 \text{$\#$1}^3-2 b \text{$\#$1}^4+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{b^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right)-2 b c^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right)+c^4 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right)+a^2 c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}+4 b c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2-3 c^3 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2-a^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3-2 b \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^4+3 c^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^4-c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^6}{a^2 c+8 b c \text{$\#$1}-8 c^3 \text{$\#$1}-3 a^2 \text{$\#$1}^2-8 b \text{$\#$1}^3+24 c^2 \text{$\#$1}^3-24 c \text{$\#$1}^5+8 \text{$\#$1}^7}\&\right]","\frac{2 (b+a x)^{3/2}}{3 a}+4 a \sqrt{c+\sqrt{b+a x}}-4 a \text{RootSum}\left[b^2-2 b c^2+c^4+a^2 c \text{$\#$1}+4 b c \text{$\#$1}^2-4 c^3 \text{$\#$1}^2-a^2 \text{$\#$1}^3-2 b \text{$\#$1}^4+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{b^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right)-2 b c^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right)+c^4 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right)+a^2 c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}+4 b c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2-3 c^3 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2-a^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3-2 b \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^4+3 c^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^4-c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^6}{a^2 c+8 b c \text{$\#$1}-8 c^3 \text{$\#$1}-3 a^2 \text{$\#$1}^2-8 b \text{$\#$1}^3+24 c^2 \text{$\#$1}^3-24 c \text{$\#$1}^5+8 \text{$\#$1}^7}\&\right]",1,"4*a*Sqrt[c + Sqrt[b + a*x]] + (2*(c^3 + (b + a*x)^(3/2)))/(3*a) - 4*a*RootSum[b^2 - 2*b*c^2 + c^4 + a^2*c*#1 + 4*b*c*#1^2 - 4*c^3*#1^2 - a^2*#1^3 - 2*b*#1^4 + 6*c^2*#1^4 - 4*c*#1^6 + #1^8 & , (b^2*Log[Sqrt[c + Sqrt[b + a*x]] - #1] - 2*b*c^2*Log[Sqrt[c + Sqrt[b + a*x]] - #1] + c^4*Log[Sqrt[c + Sqrt[b + a*x]] - #1] + a^2*c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1 + 4*b*c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^2 - 3*c^3*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^2 - a^2*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^3 - 2*b*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^4 + 3*c^2*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^4 - c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^6)/(a^2*c + 8*b*c*#1 - 8*c^3*#1 - 3*a^2*#1^2 - 8*b*#1^3 + 24*c^2*#1^3 - 24*c*#1^5 + 8*#1^7) & ]","A",1
3044,1,459,455,0.0022582,"\int \frac{x^2 \sqrt{b+a x}}{x^2-\sqrt{b+a x} \sqrt{c+\sqrt{b+a x}}} \, dx","IntegrateAlgebraic[(x^2*Sqrt[b + a*x])/(x^2 - Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]]),x]","4 a \sqrt{c+\sqrt{b+a x}}+\frac{2 \left(c^3+(b+a x)^{3/2}\right)}{3 a}-4 a \text{RootSum}\left[b^2-2 b c^2+c^4+a^2 c \text{$\#$1}+4 b c \text{$\#$1}^2-4 c^3 \text{$\#$1}^2-a^2 \text{$\#$1}^3-2 b \text{$\#$1}^4+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{b^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right)-2 b c^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right)+c^4 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right)+a^2 c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}+4 b c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2-3 c^3 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2-a^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3-2 b \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^4+3 c^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^4-c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^6}{a^2 c+8 b c \text{$\#$1}-8 c^3 \text{$\#$1}-3 a^2 \text{$\#$1}^2-8 b \text{$\#$1}^3+24 c^2 \text{$\#$1}^3-24 c \text{$\#$1}^5+8 \text{$\#$1}^7}\&\right]","\frac{2 (b+a x)^{3/2}}{3 a}+4 a \sqrt{c+\sqrt{b+a x}}-4 a \text{RootSum}\left[b^2-2 b c^2+c^4+a^2 c \text{$\#$1}+4 b c \text{$\#$1}^2-4 c^3 \text{$\#$1}^2-a^2 \text{$\#$1}^3-2 b \text{$\#$1}^4+6 c^2 \text{$\#$1}^4-4 c \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{b^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right)-2 b c^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right)+c^4 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right)+a^2 c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}+4 b c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2-3 c^3 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2-a^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3-2 b \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^4+3 c^2 \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^4-c \log \left(\sqrt{c+\sqrt{b+a x}}-\text{$\#$1}\right) \text{$\#$1}^6}{a^2 c+8 b c \text{$\#$1}-8 c^3 \text{$\#$1}-3 a^2 \text{$\#$1}^2-8 b \text{$\#$1}^3+24 c^2 \text{$\#$1}^3-24 c \text{$\#$1}^5+8 \text{$\#$1}^7}\&\right]",1,"4*a*Sqrt[c + Sqrt[b + a*x]] + (2*(c^3 + (b + a*x)^(3/2)))/(3*a) - 4*a*RootSum[b^2 - 2*b*c^2 + c^4 + a^2*c*#1 + 4*b*c*#1^2 - 4*c^3*#1^2 - a^2*#1^3 - 2*b*#1^4 + 6*c^2*#1^4 - 4*c*#1^6 + #1^8 & , (b^2*Log[Sqrt[c + Sqrt[b + a*x]] - #1] - 2*b*c^2*Log[Sqrt[c + Sqrt[b + a*x]] - #1] + c^4*Log[Sqrt[c + Sqrt[b + a*x]] - #1] + a^2*c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1 + 4*b*c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^2 - 3*c^3*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^2 - a^2*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^3 - 2*b*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^4 + 3*c^2*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^4 - c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^6)/(a^2*c + 8*b*c*#1 - 8*c^3*#1 - 3*a^2*#1^2 - 8*b*#1^3 + 24*c^2*#1^3 - 24*c*#1^5 + 8*#1^7) & ]","A",1
3045,1,455,455,1.172474,"\int \sqrt{b+a^2 x^2} \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}} \, dx","IntegrateAlgebraic[Sqrt[b + a^2*x^2]*Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]],x]","\frac{\left(32760 b^2 c^2-1024 b c^6+3465 a b^2 x-1920 a b c^4 x+114240 a^2 b c^2 x^2-2048 a^2 c^6 x^2-2560 a^3 c^4 x^3+40320 a^4 c^2 x^4\right) \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+\left(-2310 b^2 c+768 b c^5+38640 a b c^3 x+2048 a c^7 x+1536 a^2 c^5 x^2+2240 a^3 c^3 x^3\right) \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+\sqrt{b+a^2 x^2} \left(\left(3465 b^2-640 b c^4+94080 a b c^2 x-2048 a c^6 x-2560 a^2 c^4 x^2+40320 a^3 c^2 x^3\right) \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+\left(37520 b c^3+2048 c^7+1536 a c^5 x+2240 a^2 c^3 x^2\right) \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}\right)}{55440 a c^2 \left(a x+\sqrt{b+a^2 x^2}\right)^{3/2}}-\frac{b^2 \tanh ^{-1}\left(\frac{\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{\sqrt{c}}\right)}{16 a c^{5/2}}","\frac{\left(32760 b^2 c^2-1024 b c^6+3465 a b^2 x-1920 a b c^4 x+114240 a^2 b c^2 x^2-2048 a^2 c^6 x^2-2560 a^3 c^4 x^3+40320 a^4 c^2 x^4\right) \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+\left(-2310 b^2 c+768 b c^5+38640 a b c^3 x+2048 a c^7 x+1536 a^2 c^5 x^2+2240 a^3 c^3 x^3\right) \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+\sqrt{b+a^2 x^2} \left(\left(3465 b^2-640 b c^4+94080 a b c^2 x-2048 a c^6 x-2560 a^2 c^4 x^2+40320 a^3 c^2 x^3\right) \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+\left(37520 b c^3+2048 c^7+1536 a c^5 x+2240 a^2 c^3 x^2\right) \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}\right)}{55440 a c^2 \left(a x+\sqrt{b+a^2 x^2}\right)^{3/2}}-\frac{b^2 \tanh ^{-1}\left(\frac{\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{\sqrt{c}}\right)}{16 a c^{5/2}}",1,"((32760*b^2*c^2 - 1024*b*c^6 + 3465*a*b^2*x - 1920*a*b*c^4*x + 114240*a^2*b*c^2*x^2 - 2048*a^2*c^6*x^2 - 2560*a^3*c^4*x^3 + 40320*a^4*c^2*x^4)*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] + (-2310*b^2*c + 768*b*c^5 + 38640*a*b*c^3*x + 2048*a*c^7*x + 1536*a^2*c^5*x^2 + 2240*a^3*c^3*x^3)*Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] + Sqrt[b + a^2*x^2]*((3465*b^2 - 640*b*c^4 + 94080*a*b*c^2*x - 2048*a*c^6*x - 2560*a^2*c^4*x^2 + 40320*a^3*c^2*x^3)*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] + (37520*b*c^3 + 2048*c^7 + 1536*a*c^5*x + 2240*a^2*c^3*x^2)*Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]))/(55440*a*c^2*(a*x + Sqrt[b + a^2*x^2])^(3/2)) - (b^2*ArcTanh[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/Sqrt[c]])/(16*a*c^(5/2))","A",1
3046,1,510,456,41.0870762,"\int \frac{x^4}{\sqrt[4]{b+a x^4} \left(b+2 a x^4+2 x^8\right)} \, dx","IntegrateAlgebraic[x^4/((b + a*x^4)^(1/4)*(b + 2*a*x^4 + 2*x^8)),x]","-\frac{i \left(-i \sqrt{2}+\sqrt{2 \left(3-2 \sqrt{2}\right)}\right) \tan ^{-1}\left(\frac{\left((-1+i)-(1+i) (-1)^{3/4}\right) \sqrt[4]{a^2-2 b} x^2+(1+i) \sqrt{b+a x^4}+(1+i) (-1)^{3/4} \sqrt{b+a x^4}}{2 \sqrt[8]{a^2-2 b} x \sqrt[4]{b+a x^4}}\right)}{16 \left(a^2-2 b\right)^{5/8}}+\frac{\left(\sqrt{2}-i \sqrt{2 \left(3+2 \sqrt{2}\right)}\right) \tan ^{-1}\left(\frac{2 \sqrt[8]{a^2-2 b} x \sqrt[4]{b+a x^4}}{\left((1-i)+\sqrt{2}\right) \sqrt[4]{a^2-2 b} x^2-(1+i) \sqrt{b+a x^4}-\sqrt{2} \sqrt{b+a x^4}}\right)}{16 \left(a^2-2 b\right)^{5/8}}+\frac{\left(-i+(-1)^{3/4}\right) \tanh ^{-1}\left(\frac{\left((-2+2 i)-(2+2 i) (-1)^{3/4}\right) \sqrt[4]{a^2-2 b} x^2-(2+2 i) \sqrt{b+a x^4}-(2+2 i) (-1)^{3/4} \sqrt{b+a x^4}}{4 \sqrt[8]{a^2-2 b} x \sqrt[4]{b+a x^4}}\right)}{8 \left(a^2-2 b\right)^{5/8}}+\frac{i \left(i \sqrt{2}+\sqrt{2 \left(3+2 \sqrt{2}\right)}\right) \tanh ^{-1}\left(\frac{\left((1-i)+\sqrt{2}\right) \sqrt[4]{a^2-2 b} x^2+(1+i) \sqrt{b+a x^4}+\sqrt{2} \sqrt{b+a x^4}}{2 \sqrt[8]{a^2-2 b} x \sqrt[4]{b+a x^4}}\right)}{16 \left(a^2-2 b\right)^{5/8}}","\frac{\left(-1+\sqrt[4]{-1}\right) \tan ^{-1}\left(\frac{(-1)^{7/8} \sqrt{2+\sqrt{2}} \sqrt[8]{a^2-2 b} x \sqrt[4]{b+a x^4}}{(-1)^{3/4} \sqrt[4]{a^2-2 b} x^2+\sqrt{b+a x^4}}\right)}{8 \left(a^2-2 b\right)^{5/8}}+\frac{i \left(i \sqrt{2}+\sqrt{2 \left(3+2 \sqrt{2}\right)}\right) \tan ^{-1}\left(\frac{(-1)^{7/8} \left(-2+\sqrt{2}\right) \sqrt[8]{a^2-2 b} x \sqrt[4]{b+a x^4}}{(-1)^{3/4} \sqrt{2-\sqrt{2}} \sqrt[4]{a^2-2 b} x^2+\sqrt{2-\sqrt{2}} \sqrt{b+a x^4}}\right)}{16 \left(a^2-2 b\right)^{5/8}}+\frac{\left(\sqrt{2}-i \sqrt{2 \left(3+2 \sqrt{2}\right)}\right) \tanh ^{-1}\left(\frac{(-1)^{7/8} \sqrt[4]{a^2-2 b} x^2-\sqrt[8]{-1} \sqrt{b+a x^4}}{\sqrt{2-\sqrt{2}} \sqrt[8]{a^2-2 b} x \sqrt[4]{b+a x^4}}\right)}{16 \left(a^2-2 b\right)^{5/8}}+\frac{\left(-1+\sqrt[4]{-1}\right) \tanh ^{-1}\left(\frac{(-1)^{7/8} \sqrt[4]{a^2-2 b} x^2-\sqrt[8]{-1} \sqrt{b+a x^4}}{\sqrt{2+\sqrt{2}} \sqrt[8]{a^2-2 b} x \sqrt[4]{b+a x^4}}\right)}{8 \left(a^2-2 b\right)^{5/8}}",1,"((-1/16*I)*((-I)*Sqrt[2] + Sqrt[2*(3 - 2*Sqrt[2])])*ArcTan[(((-1 + I) - (1 + I)*(-1)^(3/4))*(a^2 - 2*b)^(1/4)*x^2 + (1 + I)*Sqrt[b + a*x^4] + (1 + I)*(-1)^(3/4)*Sqrt[b + a*x^4])/(2*(a^2 - 2*b)^(1/8)*x*(b + a*x^4)^(1/4))])/(a^2 - 2*b)^(5/8) + ((Sqrt[2] - I*Sqrt[2*(3 + 2*Sqrt[2])])*ArcTan[(2*(a^2 - 2*b)^(1/8)*x*(b + a*x^4)^(1/4))/(((1 - I) + Sqrt[2])*(a^2 - 2*b)^(1/4)*x^2 - (1 + I)*Sqrt[b + a*x^4] - Sqrt[2]*Sqrt[b + a*x^4])])/(16*(a^2 - 2*b)^(5/8)) + ((-I + (-1)^(3/4))*ArcTanh[(((-2 + 2*I) - (2 + 2*I)*(-1)^(3/4))*(a^2 - 2*b)^(1/4)*x^2 - (2 + 2*I)*Sqrt[b + a*x^4] - (2 + 2*I)*(-1)^(3/4)*Sqrt[b + a*x^4])/(4*(a^2 - 2*b)^(1/8)*x*(b + a*x^4)^(1/4))])/(8*(a^2 - 2*b)^(5/8)) + ((I/16)*(I*Sqrt[2] + Sqrt[2*(3 + 2*Sqrt[2])])*ArcTanh[(((1 - I) + Sqrt[2])*(a^2 - 2*b)^(1/4)*x^2 + (1 + I)*Sqrt[b + a*x^4] + Sqrt[2]*Sqrt[b + a*x^4])/(2*(a^2 - 2*b)^(1/8)*x*(b + a*x^4)^(1/4))])/(a^2 - 2*b)^(5/8)","A",1
3047,1,425,457,24.9369699,"\int \frac{x^4 \left(-q+p x^4\right) \sqrt{q+p x^4}}{b x^8+a \left(q+p x^4\right)^4} \, dx","IntegrateAlgebraic[(x^4*(-q + p*x^4)*Sqrt[q + p*x^4])/(b*x^8 + a*(q + p*x^4)^4),x]","-\frac{\sqrt{2+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{a} \sqrt[8]{b} x \sqrt{q+p x^4}}{\sqrt[4]{a} q-\sqrt[4]{b} x^2+\sqrt[4]{a} p x^4}\right)}{8 a^{3/8} b^{5/8}}+\frac{\sqrt{2-\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{a} \sqrt[8]{b} x \sqrt{q+p x^4}}{\sqrt[4]{a} q-\sqrt[4]{b} x^2+\sqrt[4]{a} p x^4}\right)}{8 a^{3/8} b^{5/8}}+\frac{\sqrt{2-\sqrt{2}} \tanh ^{-1}\left(\frac{\frac{\sqrt{1-\frac{1}{\sqrt{2}}} \sqrt[8]{a} q}{\sqrt[8]{b}}+\frac{\sqrt{1-\frac{1}{\sqrt{2}}} \sqrt[8]{b} x^2}{\sqrt[8]{a}}+\frac{\sqrt{1-\frac{1}{\sqrt{2}}} \sqrt[8]{a} p x^4}{\sqrt[8]{b}}}{x \sqrt{q+p x^4}}\right)}{8 a^{3/8} b^{5/8}}-\frac{\sqrt{2+\sqrt{2}} \tanh ^{-1}\left(\frac{\frac{\sqrt{1+\frac{1}{\sqrt{2}}} \sqrt[8]{a} q}{\sqrt[8]{b}}+\frac{\sqrt{1+\frac{1}{\sqrt{2}}} \sqrt[8]{b} x^2}{\sqrt[8]{a}}+\frac{\sqrt{1+\frac{1}{\sqrt{2}}} \sqrt[8]{a} p x^4}{\sqrt[8]{b}}}{x \sqrt{q+p x^4}}\right)}{8 a^{3/8} b^{5/8}}","-\frac{\sqrt{2+\sqrt{2}} \tan ^{-1}\left(\frac{\left(\sqrt{\frac{2}{2-\sqrt{2}}} \sqrt[8]{a} \sqrt[8]{b}-\frac{2 \sqrt[8]{a} \sqrt[8]{b}}{\sqrt{2-\sqrt{2}}}\right) x \sqrt{q+p x^4}}{-\sqrt[4]{a} q+\sqrt[4]{b} x^2-\sqrt[4]{a} p x^4}\right)}{8 a^{3/8} b^{5/8}}+\frac{\sqrt{2-\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{a} \sqrt[8]{b} x \sqrt{q+p x^4}}{\sqrt[4]{a} q-\sqrt[4]{b} x^2+\sqrt[4]{a} p x^4}\right)}{8 a^{3/8} b^{5/8}}-\frac{\sqrt{2+\sqrt{2}} \tanh ^{-1}\left(\frac{\frac{\sqrt[8]{a} q}{\sqrt{2-\sqrt{2}} \sqrt[8]{b}}+\frac{\sqrt[8]{b} x^2}{\sqrt{2-\sqrt{2}} \sqrt[8]{a}}+\frac{\sqrt[8]{a} p x^4}{\sqrt{2-\sqrt{2}} \sqrt[8]{b}}}{x \sqrt{q+p x^4}}\right)}{8 a^{3/8} b^{5/8}}+\frac{\sqrt{2-\sqrt{2}} \tanh ^{-1}\left(\frac{\frac{\sqrt[8]{a} q}{\sqrt{2+\sqrt{2}} \sqrt[8]{b}}+\frac{\sqrt[8]{b} x^2}{\sqrt{2+\sqrt{2}} \sqrt[8]{a}}+\frac{\sqrt[8]{a} p x^4}{\sqrt{2+\sqrt{2}} \sqrt[8]{b}}}{x \sqrt{q+p x^4}}\right)}{8 a^{3/8} b^{5/8}}",1,"-1/8*(Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]]*a^(1/8)*b^(1/8)*x*Sqrt[q + p*x^4])/(a^(1/4)*q - b^(1/4)*x^2 + a^(1/4)*p*x^4)])/(a^(3/8)*b^(5/8)) + (Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]]*a^(1/8)*b^(1/8)*x*Sqrt[q + p*x^4])/(a^(1/4)*q - b^(1/4)*x^2 + a^(1/4)*p*x^4)])/(8*a^(3/8)*b^(5/8)) + (Sqrt[2 - Sqrt[2]]*ArcTanh[((Sqrt[1 - 1/Sqrt[2]]*a^(1/8)*q)/b^(1/8) + (Sqrt[1 - 1/Sqrt[2]]*b^(1/8)*x^2)/a^(1/8) + (Sqrt[1 - 1/Sqrt[2]]*a^(1/8)*p*x^4)/b^(1/8))/(x*Sqrt[q + p*x^4])])/(8*a^(3/8)*b^(5/8)) - (Sqrt[2 + Sqrt[2]]*ArcTanh[((Sqrt[1 + 1/Sqrt[2]]*a^(1/8)*q)/b^(1/8) + (Sqrt[1 + 1/Sqrt[2]]*b^(1/8)*x^2)/a^(1/8) + (Sqrt[1 + 1/Sqrt[2]]*a^(1/8)*p*x^4)/b^(1/8))/(x*Sqrt[q + p*x^4])])/(8*a^(3/8)*b^(5/8))","A",1
3048,1,459,459,0.6462043,"\int \frac{1}{\sqrt[3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}} \, dx","IntegrateAlgebraic[(1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(-1/3),x]","-\frac{\tan ^{-1}\left(\frac{\left(\sqrt{3}+\sqrt{3} x\right) \left(1-x+x^2\right)}{1+x^3+\sqrt[3]{2} \sqrt[3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}}\right)}{2^{2/3} \sqrt{3}}-\frac{\log \left(1+x^3\right)}{3\ 2^{2/3}}+\frac{\log \left(1+2 x^3+x^6\right)}{6\ 2^{2/3}}+\frac{\log \left(-2-2 x^3+\sqrt[3]{2} \sqrt[3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}\right)}{3\ 2^{2/3}}-\frac{\log \left(4+8 x^3+4 x^6+\left(2 \sqrt[3]{2}+2 \sqrt[3]{2} x^3\right) \sqrt[3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}+2^{2/3} \left(1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}\right)^{2/3}\right)}{6\ 2^{2/3}}+\frac{1}{3} \text{RootSum}\left[7+\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{5 \log \left(1+x^3\right)-5 \log \left(\sqrt[3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}-\text{$\#$1}-x^3 \text{$\#$1}\right)+\log \left(1+x^3\right) \text{$\#$1}^3-\log \left(\sqrt[3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}-\text{$\#$1}-x^3 \text{$\#$1}\right) \text{$\#$1}^3}{\text{$\#$1}+2 \text{$\#$1}^4}\&\right]","-\frac{\tan ^{-1}\left(\frac{\left(\sqrt{3}+\sqrt{3} x\right) \left(1-x+x^2\right)}{1+x^3+\sqrt[3]{2} \sqrt[3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}}\right)}{2^{2/3} \sqrt{3}}-\frac{\log \left(1+x^3\right)}{3\ 2^{2/3}}+\frac{\log \left(1+2 x^3+x^6\right)}{6\ 2^{2/3}}+\frac{\log \left(-2-2 x^3+\sqrt[3]{2} \sqrt[3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}\right)}{3\ 2^{2/3}}-\frac{\log \left(4+8 x^3+4 x^6+\left(2 \sqrt[3]{2}+2 \sqrt[3]{2} x^3\right) \sqrt[3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}+2^{2/3} \left(1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}\right)^{2/3}\right)}{6\ 2^{2/3}}+\frac{1}{3} \text{RootSum}\left[7+\text{$\#$1}^3+\text{$\#$1}^6\&,\frac{5 \log \left(1+x^3\right)-5 \log \left(\sqrt[3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}-\text{$\#$1}-x^3 \text{$\#$1}\right)+\log \left(1+x^3\right) \text{$\#$1}^3-\log \left(\sqrt[3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}-\text{$\#$1}-x^3 \text{$\#$1}\right) \text{$\#$1}^3}{\text{$\#$1}+2 \text{$\#$1}^4}\&\right]",1,"-(ArcTan[((Sqrt[3] + Sqrt[3]*x)*(1 - x + x^2))/(1 + x^3 + 2^(1/3)*(1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(1/3))]/(2^(2/3)*Sqrt[3])) - Log[1 + x^3]/(3*2^(2/3)) + Log[1 + 2*x^3 + x^6]/(6*2^(2/3)) + Log[-2 - 2*x^3 + 2^(1/3)*(1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(1/3)]/(3*2^(2/3)) - Log[4 + 8*x^3 + 4*x^6 + (2*2^(1/3) + 2*2^(1/3)*x^3)*(1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(1/3) + 2^(2/3)*(1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(2/3)]/(6*2^(2/3)) + RootSum[7 + #1^3 + #1^6 & , (5*Log[1 + x^3] - 5*Log[(1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(1/3) - #1 - x^3*#1] + Log[1 + x^3]*#1^3 - Log[(1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(1/3) - #1 - x^3*#1]*#1^3)/(#1 + 2*#1^4) & ]/3","A",1
3049,1,510,460,40.3291018,"\int \frac{\left(b+a x^4\right)^{3/4}}{b+2 a x^4+2 x^8} \, dx","IntegrateAlgebraic[(b + a*x^4)^(3/4)/(b + 2*a*x^4 + 2*x^8),x]","\frac{\left(\sqrt{2}-i \sqrt{2 \left(3+2 \sqrt{2}\right)}\right) \tan ^{-1}\left(\frac{\left((-1+i)-(1+i) (-1)^{3/4}\right) \sqrt[4]{a^2-2 b} x^2+(1+i) \sqrt{b+a x^4}+(1+i) (-1)^{3/4} \sqrt{b+a x^4}}{2 \sqrt[8]{a^2-2 b} x \sqrt[4]{b+a x^4}}\right)}{16 \sqrt[8]{a^2-2 b}}-\frac{i \left(-i \sqrt{2}+\sqrt{2 \left(3-2 \sqrt{2}\right)}\right) \tan ^{-1}\left(\frac{2 \sqrt[8]{a^2-2 b} x \sqrt[4]{b+a x^4}}{\left((1-i)+\sqrt{2}\right) \sqrt[4]{a^2-2 b} x^2-(1+i) \sqrt{b+a x^4}-\sqrt{2} \sqrt{b+a x^4}}\right)}{16 \sqrt[8]{a^2-2 b}}+\frac{\left(-i-(-1)^{3/4}\right) \tanh ^{-1}\left(\frac{\left((-2+2 i)-(2+2 i) (-1)^{3/4}\right) \sqrt[4]{a^2-2 b} x^2-(2+2 i) \sqrt{b+a x^4}-(2+2 i) (-1)^{3/4} \sqrt{b+a x^4}}{4 \sqrt[8]{a^2-2 b} x \sqrt[4]{b+a x^4}}\right)}{8 \sqrt[8]{a^2-2 b}}+\frac{\left(\sqrt{2}+i \sqrt{2 \left(3-2 \sqrt{2}\right)}\right) \tanh ^{-1}\left(\frac{\left((1-i)+\sqrt{2}\right) \sqrt[4]{a^2-2 b} x^2+(1+i) \sqrt{b+a x^4}+\sqrt{2} \sqrt{b+a x^4}}{2 \sqrt[8]{a^2-2 b} x \sqrt[4]{b+a x^4}}\right)}{16 \sqrt[8]{a^2-2 b}}","\frac{\left(-1-\sqrt[4]{-1}\right) \tan ^{-1}\left(\frac{(-1)^{7/8} \sqrt{2+\sqrt{2}} \sqrt[8]{a^2-2 b} x \sqrt[4]{b+a x^4}}{(-1)^{3/4} \sqrt[4]{a^2-2 b} x^2+\sqrt{b+a x^4}}\right)}{8 \sqrt[8]{a^2-2 b}}+\frac{\left(\sqrt{2}+i \sqrt{2 \left(3-2 \sqrt{2}\right)}\right) \tan ^{-1}\left(\frac{(-1)^{7/8} \left(-2+\sqrt{2}\right) \sqrt[8]{a^2-2 b} x \sqrt[4]{b+a x^4}}{(-1)^{3/4} \sqrt{2-\sqrt{2}} \sqrt[4]{a^2-2 b} x^2+\sqrt{2-\sqrt{2}} \sqrt{b+a x^4}}\right)}{16 \sqrt[8]{a^2-2 b}}-\frac{i \left(-i \sqrt{2}+\sqrt{2 \left(3-2 \sqrt{2}\right)}\right) \tanh ^{-1}\left(\frac{(-1)^{7/8} \sqrt[4]{a^2-2 b} x^2-\sqrt[8]{-1} \sqrt{b+a x^4}}{\sqrt{2-\sqrt{2}} \sqrt[8]{a^2-2 b} x \sqrt[4]{b+a x^4}}\right)}{16 \sqrt[8]{a^2-2 b}}+\frac{\left(-1-\sqrt[4]{-1}\right) \tanh ^{-1}\left(\frac{(-1)^{7/8} \sqrt[4]{a^2-2 b} x^2-\sqrt[8]{-1} \sqrt{b+a x^4}}{\sqrt{2+\sqrt{2}} \sqrt[8]{a^2-2 b} x \sqrt[4]{b+a x^4}}\right)}{8 \sqrt[8]{a^2-2 b}}",1,"((Sqrt[2] - I*Sqrt[2*(3 + 2*Sqrt[2])])*ArcTan[(((-1 + I) - (1 + I)*(-1)^(3/4))*(a^2 - 2*b)^(1/4)*x^2 + (1 + I)*Sqrt[b + a*x^4] + (1 + I)*(-1)^(3/4)*Sqrt[b + a*x^4])/(2*(a^2 - 2*b)^(1/8)*x*(b + a*x^4)^(1/4))])/(16*(a^2 - 2*b)^(1/8)) - ((I/16)*((-I)*Sqrt[2] + Sqrt[2*(3 - 2*Sqrt[2])])*ArcTan[(2*(a^2 - 2*b)^(1/8)*x*(b + a*x^4)^(1/4))/(((1 - I) + Sqrt[2])*(a^2 - 2*b)^(1/4)*x^2 - (1 + I)*Sqrt[b + a*x^4] - Sqrt[2]*Sqrt[b + a*x^4])])/(a^2 - 2*b)^(1/8) + ((-I - (-1)^(3/4))*ArcTanh[(((-2 + 2*I) - (2 + 2*I)*(-1)^(3/4))*(a^2 - 2*b)^(1/4)*x^2 - (2 + 2*I)*Sqrt[b + a*x^4] - (2 + 2*I)*(-1)^(3/4)*Sqrt[b + a*x^4])/(4*(a^2 - 2*b)^(1/8)*x*(b + a*x^4)^(1/4))])/(8*(a^2 - 2*b)^(1/8)) + ((Sqrt[2] + I*Sqrt[2*(3 - 2*Sqrt[2])])*ArcTanh[(((1 - I) + Sqrt[2])*(a^2 - 2*b)^(1/4)*x^2 + (1 + I)*Sqrt[b + a*x^4] + Sqrt[2]*Sqrt[b + a*x^4])/(2*(a^2 - 2*b)^(1/8)*x*(b + a*x^4)^(1/4))])/(16*(a^2 - 2*b)^(1/8))","A",1
3050,1,460,460,2.1712187,"\int \frac{\sqrt{a x^2+\sqrt{b+a^2 x^4}}}{(d+c x) \sqrt{b+a^2 x^4}} \, dx","IntegrateAlgebraic[Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/((d + c*x)*Sqrt[b + a^2*x^4]),x]","-\frac{\sqrt{-a d^2-\sqrt{b c^4+a^2 d^4}} \tan ^{-1}\left(\frac{c \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{-a d^2-\sqrt{b c^4+a^2 d^4}}}\right)}{\sqrt{b c^4+a^2 d^4}}+\frac{\sqrt{-a d^2+\sqrt{b c^4+a^2 d^4}} \tan ^{-1}\left(\frac{c \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{-a d^2+\sqrt{b c^4+a^2 d^4}}}\right)}{\sqrt{b c^4+a^2 d^4}}+\frac{i \sqrt{a} d \text{RootSum}\left[b^2 c^2+4 i a b d^2 \text{$\#$1}+2 b c^2 \text{$\#$1}^2-4 i a d^2 \text{$\#$1}^3+c^2 \text{$\#$1}^4\&,\frac{b \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)-\log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2}{i a b d^2+b c^2 \text{$\#$1}-3 i a d^2 \text{$\#$1}^2+c^2 \text{$\#$1}^3}\&\right]}{\sqrt{2}}","-\frac{\sqrt{-a d^2-\sqrt{b c^4+a^2 d^4}} \tan ^{-1}\left(\frac{c \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{-a d^2-\sqrt{b c^4+a^2 d^4}}}\right)}{\sqrt{b c^4+a^2 d^4}}+\frac{\sqrt{-a d^2+\sqrt{b c^4+a^2 d^4}} \tan ^{-1}\left(\frac{c \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{-a d^2+\sqrt{b c^4+a^2 d^4}}}\right)}{\sqrt{b c^4+a^2 d^4}}+\frac{i \sqrt{a} d \text{RootSum}\left[b^2 c^2+4 i a b d^2 \text{$\#$1}+2 b c^2 \text{$\#$1}^2-4 i a d^2 \text{$\#$1}^3+c^2 \text{$\#$1}^4\&,\frac{b \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)-\log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2}{i a b d^2+b c^2 \text{$\#$1}-3 i a d^2 \text{$\#$1}^2+c^2 \text{$\#$1}^3}\&\right]}{\sqrt{2}}",1,"-((Sqrt[-(a*d^2) - Sqrt[b*c^4 + a^2*d^4]]*ArcTan[(c*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[-(a*d^2) - Sqrt[b*c^4 + a^2*d^4]]])/Sqrt[b*c^4 + a^2*d^4]) + (Sqrt[-(a*d^2) + Sqrt[b*c^4 + a^2*d^4]]*ArcTan[(c*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[-(a*d^2) + Sqrt[b*c^4 + a^2*d^4]]])/Sqrt[b*c^4 + a^2*d^4] + (I*Sqrt[a]*d*RootSum[b^2*c^2 + (4*I)*a*b*d^2*#1 + 2*b*c^2*#1^2 - (4*I)*a*d^2*#1^3 + c^2*#1^4 & , (b*Log[I*a*x^2 + I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1] - Log[I*a*x^2 + I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1]*#1^2)/(I*a*b*d^2 + b*c^2*#1 - (3*I)*a*d^2*#1^2 + c^2*#1^3) & ])/Sqrt[2]","A",1
3051,1,460,460,0.002374,"\int \frac{\sqrt{a x^2+\sqrt{b+a^2 x^4}}}{(d+c x) \sqrt{b+a^2 x^4}} \, dx","IntegrateAlgebraic[Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/((d + c*x)*Sqrt[b + a^2*x^4]),x]","-\frac{\sqrt{-a d^2-\sqrt{b c^4+a^2 d^4}} \tan ^{-1}\left(\frac{c \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{-a d^2-\sqrt{b c^4+a^2 d^4}}}\right)}{\sqrt{b c^4+a^2 d^4}}+\frac{\sqrt{-a d^2+\sqrt{b c^4+a^2 d^4}} \tan ^{-1}\left(\frac{c \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{-a d^2+\sqrt{b c^4+a^2 d^4}}}\right)}{\sqrt{b c^4+a^2 d^4}}+\frac{i \sqrt{a} d \text{RootSum}\left[b^2 c^2+4 i a b d^2 \text{$\#$1}+2 b c^2 \text{$\#$1}^2-4 i a d^2 \text{$\#$1}^3+c^2 \text{$\#$1}^4\&,\frac{b \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)-\log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2}{i a b d^2+b c^2 \text{$\#$1}-3 i a d^2 \text{$\#$1}^2+c^2 \text{$\#$1}^3}\&\right]}{\sqrt{2}}","-\frac{\sqrt{-a d^2-\sqrt{b c^4+a^2 d^4}} \tan ^{-1}\left(\frac{c \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{-a d^2-\sqrt{b c^4+a^2 d^4}}}\right)}{\sqrt{b c^4+a^2 d^4}}+\frac{\sqrt{-a d^2+\sqrt{b c^4+a^2 d^4}} \tan ^{-1}\left(\frac{c \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{-a d^2+\sqrt{b c^4+a^2 d^4}}}\right)}{\sqrt{b c^4+a^2 d^4}}+\frac{i \sqrt{a} d \text{RootSum}\left[b^2 c^2+4 i a b d^2 \text{$\#$1}+2 b c^2 \text{$\#$1}^2-4 i a d^2 \text{$\#$1}^3+c^2 \text{$\#$1}^4\&,\frac{b \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)-\log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2}{i a b d^2+b c^2 \text{$\#$1}-3 i a d^2 \text{$\#$1}^2+c^2 \text{$\#$1}^3}\&\right]}{\sqrt{2}}",1,"-((Sqrt[-(a*d^2) - Sqrt[b*c^4 + a^2*d^4]]*ArcTan[(c*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[-(a*d^2) - Sqrt[b*c^4 + a^2*d^4]]])/Sqrt[b*c^4 + a^2*d^4]) + (Sqrt[-(a*d^2) + Sqrt[b*c^4 + a^2*d^4]]*ArcTan[(c*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[-(a*d^2) + Sqrt[b*c^4 + a^2*d^4]]])/Sqrt[b*c^4 + a^2*d^4] + (I*Sqrt[a]*d*RootSum[b^2*c^2 + (4*I)*a*b*d^2*#1 + 2*b*c^2*#1^2 - (4*I)*a*d^2*#1^3 + c^2*#1^4 & , (b*Log[I*a*x^2 + I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1] - Log[I*a*x^2 + I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1]*#1^2)/(I*a*b*d^2 + b*c^2*#1 - (3*I)*a*d^2*#1^2 + c^2*#1^3) & ])/Sqrt[2]","A",1
3052,1,463,463,0.7238088,"\int \frac{1}{\sqrt{c+b x+a x^2} \left(c^3+a^3 b^3 x^3\right)} \, dx","IntegrateAlgebraic[1/(Sqrt[c + b*x + a*x^2]*(c^3 + a^3*b^3*x^3)),x]","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{c}}{\sqrt{-b^2+a b^2+c}}+\frac{a b x}{\sqrt{c} \sqrt{-b^2+a b^2+c}}-\frac{\sqrt{a} b \sqrt{c+b x+a x^2}}{\sqrt{c} \sqrt{-b^2+a b^2+c}}\right)}{3 \sqrt{a} c^{5/2} \sqrt{-b^2+a b^2+c}}-\frac{\text{RootSum}\left[b^2 c^2+a b^2 c^2+a^2 b^2 c^2-4 \sqrt{a} b c^2 \text{$\#$1}-2 a^{3/2} b c^2 \text{$\#$1}-a b^2 c \text{$\#$1}^2-2 a^2 b^2 c \text{$\#$1}^2+4 a c^2 \text{$\#$1}^2+2 a^{3/2} b c \text{$\#$1}^3+a^2 b^2 \text{$\#$1}^4\&,\frac{2 b c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)+a b c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)-4 \sqrt{a} c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}-a b \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}^2}{2 b c^2+a b c^2+\sqrt{a} b^2 c \text{$\#$1}+2 a^{3/2} b^2 c \text{$\#$1}-4 \sqrt{a} c^2 \text{$\#$1}-3 a b c \text{$\#$1}^2-2 a^{3/2} b^2 \text{$\#$1}^3}\&\right]}{3 \sqrt{a} c^2}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{c}}{\sqrt{-b^2+a b^2+c}}+\frac{a b x}{\sqrt{c} \sqrt{-b^2+a b^2+c}}-\frac{\sqrt{a} b \sqrt{c+b x+a x^2}}{\sqrt{c} \sqrt{-b^2+a b^2+c}}\right)}{3 \sqrt{a} c^{5/2} \sqrt{-b^2+a b^2+c}}-\frac{\text{RootSum}\left[b^2 c^2+a b^2 c^2+a^2 b^2 c^2-4 \sqrt{a} b c^2 \text{$\#$1}-2 a^{3/2} b c^2 \text{$\#$1}-a b^2 c \text{$\#$1}^2-2 a^2 b^2 c \text{$\#$1}^2+4 a c^2 \text{$\#$1}^2+2 a^{3/2} b c \text{$\#$1}^3+a^2 b^2 \text{$\#$1}^4\&,\frac{2 b c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)+a b c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)-4 \sqrt{a} c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}-a b \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}^2}{2 b c^2+a b c^2+\sqrt{a} b^2 c \text{$\#$1}+2 a^{3/2} b^2 c \text{$\#$1}-4 \sqrt{a} c^2 \text{$\#$1}-3 a b c \text{$\#$1}^2-2 a^{3/2} b^2 \text{$\#$1}^3}\&\right]}{3 \sqrt{a} c^2}",1,"(2*ArcTanh[Sqrt[c]/Sqrt[-b^2 + a*b^2 + c] + (a*b*x)/(Sqrt[c]*Sqrt[-b^2 + a*b^2 + c]) - (Sqrt[a]*b*Sqrt[c + b*x + a*x^2])/(Sqrt[c]*Sqrt[-b^2 + a*b^2 + c])])/(3*Sqrt[a]*c^(5/2)*Sqrt[-b^2 + a*b^2 + c]) - RootSum[b^2*c^2 + a*b^2*c^2 + a^2*b^2*c^2 - 4*Sqrt[a]*b*c^2*#1 - 2*a^(3/2)*b*c^2*#1 - a*b^2*c*#1^2 - 2*a^2*b^2*c*#1^2 + 4*a*c^2*#1^2 + 2*a^(3/2)*b*c*#1^3 + a^2*b^2*#1^4 & , (2*b*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1] + a*b*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1] - 4*Sqrt[a]*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 - a*b*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1^2)/(2*b*c^2 + a*b*c^2 + Sqrt[a]*b^2*c*#1 + 2*a^(3/2)*b^2*c*#1 - 4*Sqrt[a]*c^2*#1 - 3*a*b*c*#1^2 - 2*a^(3/2)*b^2*#1^3) & ]/(3*Sqrt[a]*c^2)","A",1
3053,1,463,463,1.1870922,"\int \frac{\left(-d+c x^2\right) \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{d+c x^2} \, dx","IntegrateAlgebraic[((-d + c*x^2)*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/(d + c*x^2),x]","\frac{1}{2} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{a x^2}{\sqrt{b}}+\frac{\sqrt{b+a^2 x^4}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{b}}\right)}{\sqrt{2} \sqrt{a}}+\frac{\sqrt{2} \sqrt{a} d \log \left(a x^2+\sqrt{b+a^2 x^4}-\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{c}-\frac{\sqrt{2} \sqrt{a} d \text{RootSum}\left[b^2 c-4 a b d \text{$\#$1}-2 b c \text{$\#$1}^2-4 a d \text{$\#$1}^3+c \text{$\#$1}^4\&,\frac{a b d \log \left(a x^2+\sqrt{b+a^2 x^4}-\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\text{$\#$1}\right)+2 b c \log \left(a x^2+\sqrt{b+a^2 x^4}-\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\text{$\#$1}\right) \text{$\#$1}+a d \log \left(a x^2+\sqrt{b+a^2 x^4}-\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\text{$\#$1}\right) \text{$\#$1}^2}{a b d+b c \text{$\#$1}+3 a d \text{$\#$1}^2-c \text{$\#$1}^3}\&\right]}{c}","\frac{1}{2} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\frac{\sqrt{b} \tan ^{-1}\left(\frac{a x^2}{\sqrt{b}}+\frac{\sqrt{b+a^2 x^4}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{b}}\right)}{\sqrt{2} \sqrt{a}}-\frac{\sqrt{2} \sqrt{a} d \log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{c}+\frac{\sqrt{2} \sqrt{a} d \text{RootSum}\left[b^2 c+4 a b d \text{$\#$1}-2 b c \text{$\#$1}^2+4 a d \text{$\#$1}^3+c \text{$\#$1}^4\&,\frac{a b d \log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)-2 b c \log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}+a d \log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2}{a b d-b c \text{$\#$1}+3 a d \text{$\#$1}^2+c \text{$\#$1}^3}\&\right]}{c}",1,"(x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/2 - (Sqrt[b]*ArcTan[(a*x^2)/Sqrt[b] + Sqrt[b + a^2*x^4]/Sqrt[b] - (Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[b]])/(Sqrt[2]*Sqrt[a]) + (Sqrt[2]*Sqrt[a]*d*Log[a*x^2 + Sqrt[b + a^2*x^4] - Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]])/c - (Sqrt[2]*Sqrt[a]*d*RootSum[b^2*c - 4*a*b*d*#1 - 2*b*c*#1^2 - 4*a*d*#1^3 + c*#1^4 & , (a*b*d*Log[a*x^2 + Sqrt[b + a^2*x^4] - Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + #1] + 2*b*c*Log[a*x^2 + Sqrt[b + a^2*x^4] - Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + #1]*#1 + a*d*Log[a*x^2 + Sqrt[b + a^2*x^4] - Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + #1]*#1^2)/(a*b*d + b*c*#1 + 3*a*d*#1^2 - c*#1^3) & ])/c","A",1
3054,1,463,463,0.0020903,"\int \frac{\left(-d+c x^2\right) \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{d+c x^2} \, dx","IntegrateAlgebraic[((-d + c*x^2)*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/(d + c*x^2),x]","\frac{1}{2} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{a x^2}{\sqrt{b}}+\frac{\sqrt{b+a^2 x^4}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{b}}\right)}{\sqrt{2} \sqrt{a}}+\frac{\sqrt{2} \sqrt{a} d \log \left(a x^2+\sqrt{b+a^2 x^4}-\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{c}-\frac{\sqrt{2} \sqrt{a} d \text{RootSum}\left[b^2 c-4 a b d \text{$\#$1}-2 b c \text{$\#$1}^2-4 a d \text{$\#$1}^3+c \text{$\#$1}^4\&,\frac{a b d \log \left(a x^2+\sqrt{b+a^2 x^4}-\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\text{$\#$1}\right)+2 b c \log \left(a x^2+\sqrt{b+a^2 x^4}-\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\text{$\#$1}\right) \text{$\#$1}+a d \log \left(a x^2+\sqrt{b+a^2 x^4}-\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\text{$\#$1}\right) \text{$\#$1}^2}{a b d+b c \text{$\#$1}+3 a d \text{$\#$1}^2-c \text{$\#$1}^3}\&\right]}{c}","\frac{1}{2} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\frac{\sqrt{b} \tan ^{-1}\left(\frac{a x^2}{\sqrt{b}}+\frac{\sqrt{b+a^2 x^4}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{b}}\right)}{\sqrt{2} \sqrt{a}}-\frac{\sqrt{2} \sqrt{a} d \log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{c}+\frac{\sqrt{2} \sqrt{a} d \text{RootSum}\left[b^2 c+4 a b d \text{$\#$1}-2 b c \text{$\#$1}^2+4 a d \text{$\#$1}^3+c \text{$\#$1}^4\&,\frac{a b d \log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)-2 b c \log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}+a d \log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2}{a b d-b c \text{$\#$1}+3 a d \text{$\#$1}^2+c \text{$\#$1}^3}\&\right]}{c}",1,"(x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/2 - (Sqrt[b]*ArcTan[(a*x^2)/Sqrt[b] + Sqrt[b + a^2*x^4]/Sqrt[b] - (Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[b]])/(Sqrt[2]*Sqrt[a]) + (Sqrt[2]*Sqrt[a]*d*Log[a*x^2 + Sqrt[b + a^2*x^4] - Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]])/c - (Sqrt[2]*Sqrt[a]*d*RootSum[b^2*c - 4*a*b*d*#1 - 2*b*c*#1^2 - 4*a*d*#1^3 + c*#1^4 & , (a*b*d*Log[a*x^2 + Sqrt[b + a^2*x^4] - Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + #1] + 2*b*c*Log[a*x^2 + Sqrt[b + a^2*x^4] - Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + #1]*#1 + a*d*Log[a*x^2 + Sqrt[b + a^2*x^4] - Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + #1]*#1^2)/(a*b*d + b*c*#1 + 3*a*d*#1^2 - c*#1^3) & ])/c","A",1
3055,1,511,466,1.9895979,"\int \sqrt{\frac{-1+a x-2 x^2+2 a x^3-x^4+a x^5}{1+a x-2 x^2-2 a x^3+x^4+a x^5}} \, dx","IntegrateAlgebraic[Sqrt[(-1 + a*x - 2*x^2 + 2*a*x^3 - x^4 + a*x^5)/(1 + a*x - 2*x^2 - 2*a*x^3 + x^4 + a*x^5)],x]","\frac{\left(-1-a x+x^2+a x^3\right) \sqrt{\frac{-1+a x-2 x^2+2 a x^3-x^4+a x^5}{1+a x-2 x^2-2 a x^3+x^4+a x^5}}}{a \left(1+x^2\right)}+\frac{2 (-1+a) \tan ^{-1}\left(\frac{(-1+a) \left(1+x^2\right)}{\sqrt{1-a} \sqrt{1+a} (-1+x) (1+x) \sqrt{\frac{-1+a x-2 x^2+2 a x^3-x^4+a x^5}{1+a x-2 x^2-2 a x^3+x^4+a x^5}}}\right)}{\sqrt{1-a^2}}+\frac{2 \sqrt{-1-a} \tan ^{-1}\left(\frac{\frac{1}{\sqrt{-1-a} \sqrt{-1+a}}+\frac{a}{\sqrt{-1-a} \sqrt{-1+a}}+\frac{x^2}{\sqrt{-1-a} \sqrt{-1+a}}+\frac{a x^2}{\sqrt{-1-a} \sqrt{-1+a}}}{(-1+x) (1+x) \sqrt{\frac{-1+a x-2 x^2+2 a x^3-x^4+a x^5}{1+a x-2 x^2-2 a x^3+x^4+a x^5}}}\right)}{\sqrt{-1+a}}+\frac{\log \left(-1-x^2+\left(-1+x^2\right) \sqrt{\frac{-1+a x-2 x^2+2 a x^3-x^4+a x^5}{1+a x-2 x^2-2 a x^3+x^4+a x^5}}\right)}{a}-\frac{\log \left(a+a x^2+\left(-a+a x^2\right) \sqrt{\frac{-1+a x-2 x^2+2 a x^3-x^4+a x^5}{1+a x-2 x^2-2 a x^3+x^4+a x^5}}\right)}{a}","\frac{\left(-1-a x+x^2+a x^3\right) \sqrt{\frac{-1+a x-2 x^2+2 a x^3-x^4+a x^5}{1+a x-2 x^2-2 a x^3+x^4+a x^5}}}{a \left(1+x^2\right)}+\frac{2 \sqrt{-(1+a)^2} \tan ^{-1}\left(\frac{\frac{(1+a)^2}{\sqrt{-1-2 a-a^2} \sqrt{-1+a^2}}+\frac{(1+a)^2 x^2}{\sqrt{-1-2 a-a^2} \sqrt{-1+a^2}}}{(-1+x) (1+x) \sqrt{\frac{-1+a x-2 x^2+2 a x^3-x^4+a x^5}{1+a x-2 x^2-2 a x^3+x^4+a x^5}}}\right)}{\sqrt{-1+a^2}}-\frac{2 \sqrt{-(-1+a)^2} \tan ^{-1}\left(\frac{\frac{(-1+a)^2}{\sqrt{-1+2 a-a^2} \sqrt{-1+a^2}}+\frac{(-1+a)^2 x^2}{\sqrt{-1+2 a-a^2} \sqrt{-1+a^2}}}{(-1+x) (1+x) \sqrt{\frac{-1+a x-2 x^2+2 a x^3-x^4+a x^5}{1+a x-2 x^2-2 a x^3+x^4+a x^5}}}\right)}{\sqrt{-1+a^2}}-\frac{2 \tanh ^{-1}\left(\frac{(-1+x) (1+x) \sqrt{\frac{-1+a x-2 x^2+2 a x^3-x^4+a x^5}{1+a x-2 x^2-2 a x^3+x^4+a x^5}}}{1+x^2}\right)}{a}",1,"((-1 - a*x + x^2 + a*x^3)*Sqrt[(-1 + a*x - 2*x^2 + 2*a*x^3 - x^4 + a*x^5)/(1 + a*x - 2*x^2 - 2*a*x^3 + x^4 + a*x^5)])/(a*(1 + x^2)) + (2*(-1 + a)*ArcTan[((-1 + a)*(1 + x^2))/(Sqrt[1 - a]*Sqrt[1 + a]*(-1 + x)*(1 + x)*Sqrt[(-1 + a*x - 2*x^2 + 2*a*x^3 - x^4 + a*x^5)/(1 + a*x - 2*x^2 - 2*a*x^3 + x^4 + a*x^5)])])/Sqrt[1 - a^2] + (2*Sqrt[-1 - a]*ArcTan[(1/(Sqrt[-1 - a]*Sqrt[-1 + a]) + a/(Sqrt[-1 - a]*Sqrt[-1 + a]) + x^2/(Sqrt[-1 - a]*Sqrt[-1 + a]) + (a*x^2)/(Sqrt[-1 - a]*Sqrt[-1 + a]))/((-1 + x)*(1 + x)*Sqrt[(-1 + a*x - 2*x^2 + 2*a*x^3 - x^4 + a*x^5)/(1 + a*x - 2*x^2 - 2*a*x^3 + x^4 + a*x^5)])])/Sqrt[-1 + a] + Log[-1 - x^2 + (-1 + x^2)*Sqrt[(-1 + a*x - 2*x^2 + 2*a*x^3 - x^4 + a*x^5)/(1 + a*x - 2*x^2 - 2*a*x^3 + x^4 + a*x^5)]]/a - Log[a + a*x^2 + (-a + a*x^2)*Sqrt[(-1 + a*x - 2*x^2 + 2*a*x^3 - x^4 + a*x^5)/(1 + a*x - 2*x^2 - 2*a*x^3 + x^4 + a*x^5)]]/a","A",1
3056,1,468,468,6.4194269,"\int \frac{1+x}{\left(-3+x^2\right) \sqrt[3]{1+x^2}} \, dx","IntegrateAlgebraic[(1 + x)/((-3 + x^2)*(1 + x^2)^(1/3)),x]","\frac{1}{6} \sqrt[3]{\frac{1}{2} \left(-5+3 \sqrt{3}\right)} \tan ^{-1}\left(\frac{-\frac{2^{2/3}}{\sqrt{3}}+\frac{1}{3} 2^{2/3} x-\frac{\sqrt[3]{1+x^2}}{\sqrt{3}}}{\sqrt[3]{1+x^2}}\right)-\frac{1}{6} \sqrt[3]{\frac{1}{2} \left(5+3 \sqrt{3}\right)} \tan ^{-1}\left(\frac{\frac{2^{2/3}}{\sqrt{3}}+\frac{1}{3} 2^{2/3} x+\frac{\sqrt[3]{1+x^2}}{\sqrt{3}}}{\sqrt[3]{1+x^2}}\right)+\frac{\sqrt[3]{\frac{1}{2} \left(9+5 \sqrt{3}\right)} \log \left(3\ 2^{2/3}+2^{2/3} \sqrt{3} x-6 \sqrt[3]{1+x^2}\right)}{6\ 3^{2/3}}+\frac{\sqrt[3]{\frac{1}{2} \left(9-5 \sqrt{3}\right)} \log \left(-3 2^{2/3}+2^{2/3} \sqrt{3} x+6 \sqrt[3]{1+x^2}\right)}{6\ 3^{2/3}}-\frac{\sqrt[3]{\frac{1}{2} \left(9-5 \sqrt{3}\right)} \log \left(-3 \sqrt[3]{2}+2 \sqrt[3]{2} \sqrt{3} x-\sqrt[3]{2} x^2-3\ 2^{2/3} \sqrt[3]{1+x^2}+2^{2/3} \sqrt{3} x \sqrt[3]{1+x^2}-6 \left(1+x^2\right)^{2/3}\right)}{12\ 3^{2/3}}-\frac{\sqrt[3]{\frac{1}{2} \left(9+5 \sqrt{3}\right)} \log \left(3 \sqrt[3]{2}+2 \sqrt[3]{2} \sqrt{3} x+\sqrt[3]{2} x^2+3\ 2^{2/3} \sqrt[3]{1+x^2}+2^{2/3} \sqrt{3} x \sqrt[3]{1+x^2}+6 \left(1+x^2\right)^{2/3}\right)}{12\ 3^{2/3}}","\frac{1}{6} \sqrt[3]{\frac{1}{2} \left(-5+3 \sqrt{3}\right)} \tan ^{-1}\left(\frac{-\frac{2^{2/3}}{\sqrt{3}}+\frac{1}{3} 2^{2/3} x-\frac{\sqrt[3]{1+x^2}}{\sqrt{3}}}{\sqrt[3]{1+x^2}}\right)-\frac{1}{6} \sqrt[3]{\frac{1}{2} \left(5+3 \sqrt{3}\right)} \tan ^{-1}\left(\frac{\frac{2^{2/3}}{\sqrt{3}}+\frac{1}{3} 2^{2/3} x+\frac{\sqrt[3]{1+x^2}}{\sqrt{3}}}{\sqrt[3]{1+x^2}}\right)+\frac{\sqrt[3]{\frac{1}{2} \left(9+5 \sqrt{3}\right)} \log \left(3\ 2^{2/3}+2^{2/3} \sqrt{3} x-6 \sqrt[3]{1+x^2}\right)}{6\ 3^{2/3}}+\frac{\sqrt[3]{\frac{1}{2} \left(9-5 \sqrt{3}\right)} \log \left(-3 2^{2/3}+2^{2/3} \sqrt{3} x+6 \sqrt[3]{1+x^2}\right)}{6\ 3^{2/3}}-\frac{\sqrt[3]{\frac{1}{2} \left(9-5 \sqrt{3}\right)} \log \left(-3 \sqrt[3]{2}+2 \sqrt[3]{2} \sqrt{3} x-\sqrt[3]{2} x^2-3\ 2^{2/3} \sqrt[3]{1+x^2}+2^{2/3} \sqrt{3} x \sqrt[3]{1+x^2}-6 \left(1+x^2\right)^{2/3}\right)}{12\ 3^{2/3}}-\frac{\sqrt[3]{\frac{1}{2} \left(9+5 \sqrt{3}\right)} \log \left(3 \sqrt[3]{2}+2 \sqrt[3]{2} \sqrt{3} x+\sqrt[3]{2} x^2+3\ 2^{2/3} \sqrt[3]{1+x^2}+2^{2/3} \sqrt{3} x \sqrt[3]{1+x^2}+6 \left(1+x^2\right)^{2/3}\right)}{12\ 3^{2/3}}",1,"(((-5 + 3*Sqrt[3])/2)^(1/3)*ArcTan[(-(2^(2/3)/Sqrt[3]) + (2^(2/3)*x)/3 - (1 + x^2)^(1/3)/Sqrt[3])/(1 + x^2)^(1/3)])/6 - (((5 + 3*Sqrt[3])/2)^(1/3)*ArcTan[(2^(2/3)/Sqrt[3] + (2^(2/3)*x)/3 + (1 + x^2)^(1/3)/Sqrt[3])/(1 + x^2)^(1/3)])/6 + (((9 + 5*Sqrt[3])/2)^(1/3)*Log[3*2^(2/3) + 2^(2/3)*Sqrt[3]*x - 6*(1 + x^2)^(1/3)])/(6*3^(2/3)) + (((9 - 5*Sqrt[3])/2)^(1/3)*Log[-3*2^(2/3) + 2^(2/3)*Sqrt[3]*x + 6*(1 + x^2)^(1/3)])/(6*3^(2/3)) - (((9 - 5*Sqrt[3])/2)^(1/3)*Log[-3*2^(1/3) + 2*2^(1/3)*Sqrt[3]*x - 2^(1/3)*x^2 - 3*2^(2/3)*(1 + x^2)^(1/3) + 2^(2/3)*Sqrt[3]*x*(1 + x^2)^(1/3) - 6*(1 + x^2)^(2/3)])/(12*3^(2/3)) - (((9 + 5*Sqrt[3])/2)^(1/3)*Log[3*2^(1/3) + 2*2^(1/3)*Sqrt[3]*x + 2^(1/3)*x^2 + 3*2^(2/3)*(1 + x^2)^(1/3) + 2^(2/3)*Sqrt[3]*x*(1 + x^2)^(1/3) + 6*(1 + x^2)^(2/3)])/(12*3^(2/3))","A",1
3057,1,394,469,23.5684955,"\int \frac{1-x^4}{\left(1+x^4\right) \sqrt[4]{-x^3+x^5}} \, dx","IntegrateAlgebraic[(1 - x^4)/((1 + x^4)*(-x^3 + x^5)^(1/4)),x]","-\frac{1}{2} \sqrt[4]{-4+3 \sqrt{2}} \tan ^{-1}\left(\frac{\sqrt[4]{-8+6 \sqrt{2}} x \sqrt[4]{-x^3+x^5}}{\sqrt[4]{2} x^2-\sqrt{-x^3+x^5}}\right)-\frac{1}{2} \sqrt[4]{4+3 \sqrt{2}} \tan ^{-1}\left(\frac{\sqrt[4]{8+6 \sqrt{2}} x \sqrt[4]{-x^3+x^5}}{\sqrt[4]{2} x^2-\sqrt{-x^3+x^5}}\right)+\frac{1}{2} \sqrt[4]{-4+3 \sqrt{2}} \tanh ^{-1}\left(\frac{2 \sqrt[4]{\frac{1}{8}+\frac{3}{16 \sqrt{2}}} x^2+2^{3/4} \sqrt[4]{\frac{1}{8}+\frac{3}{16 \sqrt{2}}} \sqrt{-x^3+x^5}}{x \sqrt[4]{-x^3+x^5}}\right)-\frac{1}{4} \sqrt[4]{4+3 \sqrt{2}} \log \left(2 x^2-2 \sqrt[4]{4+3 \sqrt{2}} x \sqrt[4]{-x^3+x^5}+2^{3/4} \sqrt{-x^3+x^5}\right)+\frac{1}{4} \sqrt[4]{4+3 \sqrt{2}} \log \left(\sqrt{2-\sqrt{2}} x^2+2^{3/8} x \sqrt[4]{-x^3+x^5}+\sqrt{-1+\sqrt{2}} \sqrt{-x^3+x^5}\right)","\frac{1}{2} \sqrt[4]{-4+3 \sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x}{-\sqrt{2+\sqrt{2}} x+2^{7/8} \sqrt[4]{-x^3+x^5}}\right)+\frac{1}{2} \sqrt[4]{-4+3 \sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x}{\sqrt{2+\sqrt{2}} x+2^{7/8} \sqrt[4]{-x^3+x^5}}\right)+\frac{1}{2} \sqrt[4]{4+3 \sqrt{2}} \tan ^{-1}\left(\frac{2^{7/8} \sqrt{2+\sqrt{2}} x \sqrt[4]{-x^3+x^5}}{-2 x^2+2^{3/4} \sqrt{-x^3+x^5}}\right)+\frac{1}{2} \sqrt[4]{-4+3 \sqrt{2}} \tanh ^{-1}\left(\frac{\frac{\sqrt[8]{2} x^2}{\sqrt{2-\sqrt{2}}}+\frac{\sqrt{-x^3+x^5}}{\sqrt[8]{2} \sqrt{2-\sqrt{2}}}}{x \sqrt[4]{-x^3+x^5}}\right)-\frac{1}{4} \sqrt[4]{4+3 \sqrt{2}} \log \left(-2 x^2+2^{7/8} \sqrt{2+\sqrt{2}} x \sqrt[4]{-x^3+x^5}-2^{3/4} \sqrt{-x^3+x^5}\right)+\frac{1}{4} \sqrt[4]{4+3 \sqrt{2}} \log \left(2 \sqrt{2-\sqrt{2}} x^2+2\ 2^{3/8} x \sqrt[4]{-x^3+x^5}+2^{3/4} \sqrt{2-\sqrt{2}} \sqrt{-x^3+x^5}\right)",1,"-1/2*((-4 + 3*Sqrt[2])^(1/4)*ArcTan[((-8 + 6*Sqrt[2])^(1/4)*x*(-x^3 + x^5)^(1/4))/(2^(1/4)*x^2 - Sqrt[-x^3 + x^5])]) - ((4 + 3*Sqrt[2])^(1/4)*ArcTan[((8 + 6*Sqrt[2])^(1/4)*x*(-x^3 + x^5)^(1/4))/(2^(1/4)*x^2 - Sqrt[-x^3 + x^5])])/2 + ((-4 + 3*Sqrt[2])^(1/4)*ArcTanh[(2*(1/8 + 3/(16*Sqrt[2]))^(1/4)*x^2 + 2^(3/4)*(1/8 + 3/(16*Sqrt[2]))^(1/4)*Sqrt[-x^3 + x^5])/(x*(-x^3 + x^5)^(1/4))])/2 - ((4 + 3*Sqrt[2])^(1/4)*Log[2*x^2 - 2*(4 + 3*Sqrt[2])^(1/4)*x*(-x^3 + x^5)^(1/4) + 2^(3/4)*Sqrt[-x^3 + x^5]])/4 + ((4 + 3*Sqrt[2])^(1/4)*Log[Sqrt[2 - Sqrt[2]]*x^2 + 2^(3/8)*x*(-x^3 + x^5)^(1/4) + Sqrt[-1 + Sqrt[2]]*Sqrt[-x^3 + x^5]])/4","A",0
3058,1,394,469,28.7402697,"\int \frac{-1+x^8}{\sqrt[4]{-x^2+x^6} \left(1+x^8\right)} \, dx","IntegrateAlgebraic[(-1 + x^8)/((-x^2 + x^6)^(1/4)*(1 + x^8)),x]","\frac{1}{4} \sqrt[4]{-4+3 \sqrt{2}} \tan ^{-1}\left(\frac{\sqrt[4]{-8+6 \sqrt{2}} x \sqrt[4]{-x^2+x^6}}{\sqrt[4]{2} x^2-\sqrt{-x^2+x^6}}\right)+\frac{1}{4} \sqrt[4]{4+3 \sqrt{2}} \tan ^{-1}\left(\frac{\sqrt[4]{8+6 \sqrt{2}} x \sqrt[4]{-x^2+x^6}}{\sqrt[4]{2} x^2-\sqrt{-x^2+x^6}}\right)-\frac{1}{4} \sqrt[4]{-4+3 \sqrt{2}} \tanh ^{-1}\left(\frac{2 \sqrt[4]{\frac{1}{8}+\frac{3}{16 \sqrt{2}}} x^2+2^{3/4} \sqrt[4]{\frac{1}{8}+\frac{3}{16 \sqrt{2}}} \sqrt{-x^2+x^6}}{x \sqrt[4]{-x^2+x^6}}\right)+\frac{1}{8} \sqrt[4]{4+3 \sqrt{2}} \log \left(2 x^2-2 \sqrt[4]{4+3 \sqrt{2}} x \sqrt[4]{-x^2+x^6}+2^{3/4} \sqrt{-x^2+x^6}\right)-\frac{1}{8} \sqrt[4]{4+3 \sqrt{2}} \log \left(\sqrt{2-\sqrt{2}} x^2+2^{3/8} x \sqrt[4]{-x^2+x^6}+\sqrt{-1+\sqrt{2}} \sqrt{-x^2+x^6}\right)","-\frac{1}{4} \sqrt[4]{-4+3 \sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x}{-\sqrt{2+\sqrt{2}} x+2^{7/8} \sqrt[4]{-x^2+x^6}}\right)-\frac{1}{4} \sqrt[4]{-4+3 \sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x}{\sqrt{2+\sqrt{2}} x+2^{7/8} \sqrt[4]{-x^2+x^6}}\right)-\frac{1}{4} \sqrt[4]{4+3 \sqrt{2}} \tan ^{-1}\left(\frac{2^{7/8} \sqrt{2+\sqrt{2}} x \sqrt[4]{-x^2+x^6}}{-2 x^2+2^{3/4} \sqrt{-x^2+x^6}}\right)-\frac{1}{4} \sqrt[4]{-4+3 \sqrt{2}} \tanh ^{-1}\left(\frac{\frac{\sqrt[8]{2} x^2}{\sqrt{2-\sqrt{2}}}+\frac{\sqrt{-x^2+x^6}}{\sqrt[8]{2} \sqrt{2-\sqrt{2}}}}{x \sqrt[4]{-x^2+x^6}}\right)+\frac{1}{8} \sqrt[4]{4+3 \sqrt{2}} \log \left(-2 x^2+2^{7/8} \sqrt{2+\sqrt{2}} x \sqrt[4]{-x^2+x^6}-2^{3/4} \sqrt{-x^2+x^6}\right)-\frac{1}{8} \sqrt[4]{4+3 \sqrt{2}} \log \left(2 \sqrt{2-\sqrt{2}} x^2+2\ 2^{3/8} x \sqrt[4]{-x^2+x^6}+2^{3/4} \sqrt{2-\sqrt{2}} \sqrt{-x^2+x^6}\right)",1,"((-4 + 3*Sqrt[2])^(1/4)*ArcTan[((-8 + 6*Sqrt[2])^(1/4)*x*(-x^2 + x^6)^(1/4))/(2^(1/4)*x^2 - Sqrt[-x^2 + x^6])])/4 + ((4 + 3*Sqrt[2])^(1/4)*ArcTan[((8 + 6*Sqrt[2])^(1/4)*x*(-x^2 + x^6)^(1/4))/(2^(1/4)*x^2 - Sqrt[-x^2 + x^6])])/4 - ((-4 + 3*Sqrt[2])^(1/4)*ArcTanh[(2*(1/8 + 3/(16*Sqrt[2]))^(1/4)*x^2 + 2^(3/4)*(1/8 + 3/(16*Sqrt[2]))^(1/4)*Sqrt[-x^2 + x^6])/(x*(-x^2 + x^6)^(1/4))])/4 + ((4 + 3*Sqrt[2])^(1/4)*Log[2*x^2 - 2*(4 + 3*Sqrt[2])^(1/4)*x*(-x^2 + x^6)^(1/4) + 2^(3/4)*Sqrt[-x^2 + x^6]])/8 - ((4 + 3*Sqrt[2])^(1/4)*Log[Sqrt[2 - Sqrt[2]]*x^2 + 2^(3/8)*x*(-x^2 + x^6)^(1/4) + Sqrt[-1 + Sqrt[2]]*Sqrt[-x^2 + x^6]])/8","A",0
3059,1,394,469,0.0036347,"\int \frac{-1+x^8}{\sqrt[4]{-x^2+x^6} \left(1+x^8\right)} \, dx","IntegrateAlgebraic[(-1 + x^8)/((-x^2 + x^6)^(1/4)*(1 + x^8)),x]","\frac{1}{4} \sqrt[4]{-4+3 \sqrt{2}} \tan ^{-1}\left(\frac{\sqrt[4]{-8+6 \sqrt{2}} x \sqrt[4]{-x^2+x^6}}{\sqrt[4]{2} x^2-\sqrt{-x^2+x^6}}\right)+\frac{1}{4} \sqrt[4]{4+3 \sqrt{2}} \tan ^{-1}\left(\frac{\sqrt[4]{8+6 \sqrt{2}} x \sqrt[4]{-x^2+x^6}}{\sqrt[4]{2} x^2-\sqrt{-x^2+x^6}}\right)-\frac{1}{4} \sqrt[4]{-4+3 \sqrt{2}} \tanh ^{-1}\left(\frac{2 \sqrt[4]{\frac{1}{8}+\frac{3}{16 \sqrt{2}}} x^2+2^{3/4} \sqrt[4]{\frac{1}{8}+\frac{3}{16 \sqrt{2}}} \sqrt{-x^2+x^6}}{x \sqrt[4]{-x^2+x^6}}\right)+\frac{1}{8} \sqrt[4]{4+3 \sqrt{2}} \log \left(2 x^2-2 \sqrt[4]{4+3 \sqrt{2}} x \sqrt[4]{-x^2+x^6}+2^{3/4} \sqrt{-x^2+x^6}\right)-\frac{1}{8} \sqrt[4]{4+3 \sqrt{2}} \log \left(\sqrt{2-\sqrt{2}} x^2+2^{3/8} x \sqrt[4]{-x^2+x^6}+\sqrt{-1+\sqrt{2}} \sqrt{-x^2+x^6}\right)","-\frac{1}{4} \sqrt[4]{-4+3 \sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x}{-\sqrt{2+\sqrt{2}} x+2^{7/8} \sqrt[4]{-x^2+x^6}}\right)-\frac{1}{4} \sqrt[4]{-4+3 \sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x}{\sqrt{2+\sqrt{2}} x+2^{7/8} \sqrt[4]{-x^2+x^6}}\right)-\frac{1}{4} \sqrt[4]{4+3 \sqrt{2}} \tan ^{-1}\left(\frac{2^{7/8} \sqrt{2+\sqrt{2}} x \sqrt[4]{-x^2+x^6}}{-2 x^2+2^{3/4} \sqrt{-x^2+x^6}}\right)-\frac{1}{4} \sqrt[4]{-4+3 \sqrt{2}} \tanh ^{-1}\left(\frac{\frac{\sqrt[8]{2} x^2}{\sqrt{2-\sqrt{2}}}+\frac{\sqrt{-x^2+x^6}}{\sqrt[8]{2} \sqrt{2-\sqrt{2}}}}{x \sqrt[4]{-x^2+x^6}}\right)+\frac{1}{8} \sqrt[4]{4+3 \sqrt{2}} \log \left(-2 x^2+2^{7/8} \sqrt{2+\sqrt{2}} x \sqrt[4]{-x^2+x^6}-2^{3/4} \sqrt{-x^2+x^6}\right)-\frac{1}{8} \sqrt[4]{4+3 \sqrt{2}} \log \left(2 \sqrt{2-\sqrt{2}} x^2+2\ 2^{3/8} x \sqrt[4]{-x^2+x^6}+2^{3/4} \sqrt{2-\sqrt{2}} \sqrt{-x^2+x^6}\right)",1,"((-4 + 3*Sqrt[2])^(1/4)*ArcTan[((-8 + 6*Sqrt[2])^(1/4)*x*(-x^2 + x^6)^(1/4))/(2^(1/4)*x^2 - Sqrt[-x^2 + x^6])])/4 + ((4 + 3*Sqrt[2])^(1/4)*ArcTan[((8 + 6*Sqrt[2])^(1/4)*x*(-x^2 + x^6)^(1/4))/(2^(1/4)*x^2 - Sqrt[-x^2 + x^6])])/4 - ((-4 + 3*Sqrt[2])^(1/4)*ArcTanh[(2*(1/8 + 3/(16*Sqrt[2]))^(1/4)*x^2 + 2^(3/4)*(1/8 + 3/(16*Sqrt[2]))^(1/4)*Sqrt[-x^2 + x^6])/(x*(-x^2 + x^6)^(1/4))])/4 + ((4 + 3*Sqrt[2])^(1/4)*Log[2*x^2 - 2*(4 + 3*Sqrt[2])^(1/4)*x*(-x^2 + x^6)^(1/4) + 2^(3/4)*Sqrt[-x^2 + x^6]])/8 - ((4 + 3*Sqrt[2])^(1/4)*Log[Sqrt[2 - Sqrt[2]]*x^2 + 2^(3/8)*x*(-x^2 + x^6)^(1/4) + Sqrt[-1 + Sqrt[2]]*Sqrt[-x^2 + x^6]])/8","A",0
3060,1,470,470,0.9620228,"\int \frac{\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}} \, dx","IntegrateAlgebraic[(a*x + Sqrt[-b + a^2*x^2])^(1/3)/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4),x]","\frac{\left(4620 b c+3072 a c^4 x\right) \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\left(-5775 b-2688 a c^3 x\right) \sqrt[3]{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\left(-4096 c^5+2464 a c^2 x\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{2/3} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\sqrt{-b+a^2 x^2} \left(3072 c^4 \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}-2688 c^3 \sqrt[3]{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+2464 c^2 \left(a x+\sqrt{-b+a^2 x^2}\right)^{2/3} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}\right)}{6160 a c^2 \left(a x+\sqrt{-b+a^2 x^2}\right)^{2/3}}-\frac{15 b \tan ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{32 a c^{9/4}}+\frac{15 b \tanh ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{32 a c^{9/4}}","\frac{\left(4620 b c+3072 a c^4 x\right) \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\left(-5775 b-2688 a c^3 x\right) \sqrt[3]{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\left(-4096 c^5+2464 a c^2 x\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{2/3} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\sqrt{-b+a^2 x^2} \left(3072 c^4 \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}-2688 c^3 \sqrt[3]{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+2464 c^2 \left(a x+\sqrt{-b+a^2 x^2}\right)^{2/3} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}\right)}{6160 a c^2 \left(a x+\sqrt{-b+a^2 x^2}\right)^{2/3}}-\frac{15 b \tan ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{32 a c^{9/4}}+\frac{15 b \tanh ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{32 a c^{9/4}}",1,"((4620*b*c + 3072*a*c^4*x)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + (-5775*b - 2688*a*c^3*x)*(a*x + Sqrt[-b + a^2*x^2])^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + (-4096*c^5 + 2464*a*c^2*x)*(a*x + Sqrt[-b + a^2*x^2])^(2/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + Sqrt[-b + a^2*x^2]*(3072*c^4*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) - 2688*c^3*(a*x + Sqrt[-b + a^2*x^2])^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + 2464*c^2*(a*x + Sqrt[-b + a^2*x^2])^(2/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4)))/(6160*a*c^2*(a*x + Sqrt[-b + a^2*x^2])^(2/3)) - (15*b*ArcTan[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)])/(32*a*c^(9/4)) + (15*b*ArcTanh[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)])/(32*a*c^(9/4))","A",1
3061,1,1138,471,4.756928,"\int \frac{b^2+a x}{\left(-b^2+a x\right) \sqrt{b+\sqrt{b^2+a x^2}}} \, dx","IntegrateAlgebraic[(b^2 + a*x)/((-b^2 + a*x)*Sqrt[b + Sqrt[b^2 + a*x^2]]),x]","\frac{2 x}{\sqrt{b+\sqrt{b^2+a x^2}}}+\frac{2 \sqrt{2} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2} \sqrt{b}}\right)}{\sqrt{a}}-2 i \sqrt{a} b^3 \text{RootSum}\left[16 a^4 b^4-8 a^2 b^2 \text{$\#$1}^4-16 a b^4 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{\log \left(\frac{i a x}{\sqrt{b+\sqrt{b^2+a x^2}}}-i \sqrt{a} \sqrt{b+\sqrt{b^2+a x^2}}-\text{$\#$1}\right) \text{$\#$1}}{-4 a^2 b^2-8 a b^4+\text{$\#$1}^4}\&\right]-\frac{i b^2 \text{RootSum}\left[16 a^4 b^4-8 a^2 b^2 \text{$\#$1}^4-16 a b^4 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{\log \left(\frac{i a x}{\sqrt{b+\sqrt{b^2+a x^2}}}-i \sqrt{a} \sqrt{b+\sqrt{b^2+a x^2}}-\text{$\#$1}\right) \text{$\#$1}^3}{-4 a^2 b^2-8 a b^4+\text{$\#$1}^4}\&\right]}{\sqrt{a}}+4 i a^{3/2} b^4 \text{RootSum}\left[16 a^4 b^4-8 a^2 b^2 \text{$\#$1}^4-16 a b^4 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{\log \left(\frac{i a x}{\sqrt{b+\sqrt{b^2+a x^2}}}-i \sqrt{a} \sqrt{b+\sqrt{b^2+a x^2}}-\text{$\#$1}\right)}{4 a^2 b^2 \text{$\#$1}+8 a b^4 \text{$\#$1}-\text{$\#$1}^5}\&\right]+8 i a^{5/2} b^5 \text{RootSum}\left[16 a^4 b^4-8 a^2 b^2 \text{$\#$1}^4-16 a b^4 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{\log \left(\frac{i a x}{\sqrt{b+\sqrt{b^2+a x^2}}}-i \sqrt{a} \sqrt{b+\sqrt{b^2+a x^2}}-\text{$\#$1}\right)}{4 a^2 b^2 \text{$\#$1}^3+8 a b^4 \text{$\#$1}^3-\text{$\#$1}^7}\&\right]-i b \text{RootSum}\left[16 a^4 b^4-8 a^2 b^2 \text{$\#$1}^4-16 a b^4 \text{$\#$1}^4+\text{$\#$1}^8\&,\frac{8 a^3 b^3 \log \left(\frac{i a x}{\sqrt{b+\sqrt{b^2+a x^2}}}-i \sqrt{a} \sqrt{b+\sqrt{b^2+a x^2}}-\text{$\#$1}\right)+4 a^2 b^2 \log \left(\frac{i a x}{\sqrt{b+\sqrt{b^2+a x^2}}}-i \sqrt{a} \sqrt{b+\sqrt{b^2+a x^2}}-\text{$\#$1}\right) \text{$\#$1}^2+8 a b^4 \log \left(\frac{i a x}{\sqrt{b+\sqrt{b^2+a x^2}}}-i \sqrt{a} \sqrt{b+\sqrt{b^2+a x^2}}-\text{$\#$1}\right) \text{$\#$1}^2-2 a b \log \left(\frac{i a x}{\sqrt{b+\sqrt{b^2+a x^2}}}-i \sqrt{a} \sqrt{b+\sqrt{b^2+a x^2}}-\text{$\#$1}\right) \text{$\#$1}^4-4 b^3 \log \left(\frac{i a x}{\sqrt{b+\sqrt{b^2+a x^2}}}-i \sqrt{a} \sqrt{b+\sqrt{b^2+a x^2}}-\text{$\#$1}\right) \text{$\#$1}^4-\log \left(\frac{i a x}{\sqrt{b+\sqrt{b^2+a x^2}}}-i \sqrt{a} \sqrt{b+\sqrt{b^2+a x^2}}-\text{$\#$1}\right) \text{$\#$1}^6}{4 a^2 b^2 \text{$\#$1}^3+8 a b^4 \text{$\#$1}^3-\text{$\#$1}^7}\&\right]","\frac{\sqrt{2} \left(-b+\sqrt{a} x\right) \sqrt{-\sqrt{a} x+\sqrt{b^2+a x^2}}}{\sqrt{a} b}+\frac{\sqrt{2} \sqrt{b^2+a x^2} \sqrt{-\sqrt{a} x+\sqrt{b^2+a x^2}}}{\sqrt{a} b}-\frac{2 \sqrt{2} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{-\sqrt{a} x+\sqrt{b^2+a x^2}}}{\sqrt{b}}\right)}{\sqrt{a}}+\frac{2 \left(\sqrt{2} \sqrt{a} \sqrt{b} \sqrt{b+\sqrt{a+b^2}}-\sqrt{2} b^{3/2} \sqrt{b+\sqrt{a+b^2}}+\sqrt{2} \sqrt{b} \sqrt{a+b^2} \sqrt{b+\sqrt{a+b^2}}\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} \sqrt{-\sqrt{a} x+\sqrt{b^2+a x^2}}}{\sqrt{b} \sqrt{b+\sqrt{a+b^2}}}\right)}{a^{5/4}}-\frac{2 \left(-\sqrt{2} \sqrt{a} \sqrt{b} \sqrt{-b+\sqrt{a+b^2}}+\sqrt{2} b^{3/2} \sqrt{-b+\sqrt{a+b^2}}+\sqrt{2} \sqrt{b} \sqrt{a+b^2} \sqrt{-b+\sqrt{a+b^2}}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} \sqrt{-\sqrt{a} x+\sqrt{b^2+a x^2}}}{\sqrt{b} \sqrt{-b+\sqrt{a+b^2}}}\right)}{a^{5/4}}",1,"(2*x)/Sqrt[b + Sqrt[b^2 + a*x^2]] + (2*Sqrt[2]*Sqrt[b]*ArcTan[(Sqrt[a]*x)/(Sqrt[2]*Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]]) - Sqrt[b + Sqrt[b^2 + a*x^2]]/(Sqrt[2]*Sqrt[b])])/Sqrt[a] - (2*I)*Sqrt[a]*b^3*RootSum[16*a^4*b^4 - 8*a^2*b^2*#1^4 - 16*a*b^4*#1^4 + #1^8 & , (Log[(I*a*x)/Sqrt[b + Sqrt[b^2 + a*x^2]] - I*Sqrt[a]*Sqrt[b + Sqrt[b^2 + a*x^2]] - #1]*#1)/(-4*a^2*b^2 - 8*a*b^4 + #1^4) & ] - (I*b^2*RootSum[16*a^4*b^4 - 8*a^2*b^2*#1^4 - 16*a*b^4*#1^4 + #1^8 & , (Log[(I*a*x)/Sqrt[b + Sqrt[b^2 + a*x^2]] - I*Sqrt[a]*Sqrt[b + Sqrt[b^2 + a*x^2]] - #1]*#1^3)/(-4*a^2*b^2 - 8*a*b^4 + #1^4) & ])/Sqrt[a] + (4*I)*a^(3/2)*b^4*RootSum[16*a^4*b^4 - 8*a^2*b^2*#1^4 - 16*a*b^4*#1^4 + #1^8 & , Log[(I*a*x)/Sqrt[b + Sqrt[b^2 + a*x^2]] - I*Sqrt[a]*Sqrt[b + Sqrt[b^2 + a*x^2]] - #1]/(4*a^2*b^2*#1 + 8*a*b^4*#1 - #1^5) & ] + (8*I)*a^(5/2)*b^5*RootSum[16*a^4*b^4 - 8*a^2*b^2*#1^4 - 16*a*b^4*#1^4 + #1^8 & , Log[(I*a*x)/Sqrt[b + Sqrt[b^2 + a*x^2]] - I*Sqrt[a]*Sqrt[b + Sqrt[b^2 + a*x^2]] - #1]/(4*a^2*b^2*#1^3 + 8*a*b^4*#1^3 - #1^7) & ] - I*b*RootSum[16*a^4*b^4 - 8*a^2*b^2*#1^4 - 16*a*b^4*#1^4 + #1^8 & , (8*a^3*b^3*Log[(I*a*x)/Sqrt[b + Sqrt[b^2 + a*x^2]] - I*Sqrt[a]*Sqrt[b + Sqrt[b^2 + a*x^2]] - #1] + 4*a^2*b^2*Log[(I*a*x)/Sqrt[b + Sqrt[b^2 + a*x^2]] - I*Sqrt[a]*Sqrt[b + Sqrt[b^2 + a*x^2]] - #1]*#1^2 + 8*a*b^4*Log[(I*a*x)/Sqrt[b + Sqrt[b^2 + a*x^2]] - I*Sqrt[a]*Sqrt[b + Sqrt[b^2 + a*x^2]] - #1]*#1^2 - 2*a*b*Log[(I*a*x)/Sqrt[b + Sqrt[b^2 + a*x^2]] - I*Sqrt[a]*Sqrt[b + Sqrt[b^2 + a*x^2]] - #1]*#1^4 - 4*b^3*Log[(I*a*x)/Sqrt[b + Sqrt[b^2 + a*x^2]] - I*Sqrt[a]*Sqrt[b + Sqrt[b^2 + a*x^2]] - #1]*#1^4 - Log[(I*a*x)/Sqrt[b + Sqrt[b^2 + a*x^2]] - I*Sqrt[a]*Sqrt[b + Sqrt[b^2 + a*x^2]] - #1]*#1^6)/(4*a^2*b^2*#1^3 + 8*a*b^4*#1^3 - #1^7) & ]","C",1
3062,1,534,472,43.9487048,"\int \frac{\left(-b+a x^4\right)^{3/4}}{b-2 a x^4+2 x^8} \, dx","IntegrateAlgebraic[(-b + a*x^4)^(3/4)/(b - 2*a*x^4 + 2*x^8),x]","\frac{i \left(i \sqrt{2}+\sqrt{2 \left(3+2 \sqrt{2}\right)}\right) \tan ^{-1}\left(\frac{\left((-1+i)-(1+i) (-1)^{3/4}\right) \sqrt[4]{a^2-2 b} x^2+(1+i) \sqrt{-b+a x^4}+(1+i) (-1)^{3/4} \sqrt{-b+a x^4}}{2 \sqrt[8]{a^2-2 b} x \sqrt[4]{-b+a x^4}}\right)}{16 \sqrt[8]{a^2-2 b}}+\frac{\left(\sqrt{2}+i \sqrt{2 \left(3-2 \sqrt{2}\right)}\right) \tan ^{-1}\left(\frac{2 \sqrt[8]{a^2-2 b} x \sqrt[4]{-b+a x^4}}{\left((1-i)+\sqrt{2}\right) \sqrt[4]{a^2-2 b} x^2-(1+i) \sqrt{-b+a x^4}-\sqrt{2} \sqrt{-b+a x^4}}\right)}{16 \sqrt[8]{a^2-2 b}}+\frac{\left(i+(-1)^{3/4}\right) \tanh ^{-1}\left(\frac{\left((-2+2 i)-(2+2 i) (-1)^{3/4}\right) \sqrt[4]{a^2-2 b} x^2-(2+2 i) \sqrt{-b+a x^4}-(2+2 i) (-1)^{3/4} \sqrt{-b+a x^4}}{4 \sqrt[8]{a^2-2 b} x \sqrt[4]{-b+a x^4}}\right)}{8 \sqrt[8]{a^2-2 b}}-\frac{i \left(-i \sqrt{2}+\sqrt{2 \left(3-2 \sqrt{2}\right)}\right) \tanh ^{-1}\left(\frac{\left((1-i)+\sqrt{2}\right) \sqrt[4]{a^2-2 b} x^2+(1+i) \sqrt{-b+a x^4}+\sqrt{2} \sqrt{-b+a x^4}}{2 \sqrt[8]{a^2-2 b} x \sqrt[4]{-b+a x^4}}\right)}{16 \sqrt[8]{a^2-2 b}}","\frac{\left(1+\sqrt[4]{-1}\right) \tan ^{-1}\left(\frac{(-1)^{7/8} \sqrt{2+\sqrt{2}} \sqrt[8]{a^2-2 b} x \sqrt[4]{-b+a x^4}}{(-1)^{3/4} \sqrt[4]{a^2-2 b} x^2+\sqrt{-b+a x^4}}\right)}{8 \sqrt[8]{a^2-2 b}}-\frac{i \left(-i \sqrt{2}+\sqrt{2 \left(3-2 \sqrt{2}\right)}\right) \tan ^{-1}\left(\frac{(-1)^{7/8} \left(-2+\sqrt{2}\right) \sqrt[8]{a^2-2 b} x \sqrt[4]{-b+a x^4}}{(-1)^{3/4} \sqrt{2-\sqrt{2}} \sqrt[4]{a^2-2 b} x^2+\sqrt{2-\sqrt{2}} \sqrt{-b+a x^4}}\right)}{16 \sqrt[8]{a^2-2 b}}+\frac{\left(\sqrt{2}+i \sqrt{2 \left(3-2 \sqrt{2}\right)}\right) \tanh ^{-1}\left(\frac{(-1)^{7/8} \sqrt[4]{a^2-2 b} x^2-\sqrt[8]{-1} \sqrt{-b+a x^4}}{\sqrt{2-\sqrt{2}} \sqrt[8]{a^2-2 b} x \sqrt[4]{-b+a x^4}}\right)}{16 \sqrt[8]{a^2-2 b}}+\frac{\left(1+\sqrt[4]{-1}\right) \tanh ^{-1}\left(\frac{(-1)^{7/8} \sqrt[4]{a^2-2 b} x^2-\sqrt[8]{-1} \sqrt{-b+a x^4}}{\sqrt{2+\sqrt{2}} \sqrt[8]{a^2-2 b} x \sqrt[4]{-b+a x^4}}\right)}{8 \sqrt[8]{a^2-2 b}}",1,"((I/16)*(I*Sqrt[2] + Sqrt[2*(3 + 2*Sqrt[2])])*ArcTan[(((-1 + I) - (1 + I)*(-1)^(3/4))*(a^2 - 2*b)^(1/4)*x^2 + (1 + I)*Sqrt[-b + a*x^4] + (1 + I)*(-1)^(3/4)*Sqrt[-b + a*x^4])/(2*(a^2 - 2*b)^(1/8)*x*(-b + a*x^4)^(1/4))])/(a^2 - 2*b)^(1/8) + ((Sqrt[2] + I*Sqrt[2*(3 - 2*Sqrt[2])])*ArcTan[(2*(a^2 - 2*b)^(1/8)*x*(-b + a*x^4)^(1/4))/(((1 - I) + Sqrt[2])*(a^2 - 2*b)^(1/4)*x^2 - (1 + I)*Sqrt[-b + a*x^4] - Sqrt[2]*Sqrt[-b + a*x^4])])/(16*(a^2 - 2*b)^(1/8)) + ((I + (-1)^(3/4))*ArcTanh[(((-2 + 2*I) - (2 + 2*I)*(-1)^(3/4))*(a^2 - 2*b)^(1/4)*x^2 - (2 + 2*I)*Sqrt[-b + a*x^4] - (2 + 2*I)*(-1)^(3/4)*Sqrt[-b + a*x^4])/(4*(a^2 - 2*b)^(1/8)*x*(-b + a*x^4)^(1/4))])/(8*(a^2 - 2*b)^(1/8)) - ((I/16)*((-I)*Sqrt[2] + Sqrt[2*(3 - 2*Sqrt[2])])*ArcTanh[(((1 - I) + Sqrt[2])*(a^2 - 2*b)^(1/4)*x^2 + (1 + I)*Sqrt[-b + a*x^4] + Sqrt[2]*Sqrt[-b + a*x^4])/(2*(a^2 - 2*b)^(1/8)*x*(-b + a*x^4)^(1/4))])/(a^2 - 2*b)^(1/8)","A",1
3063,1,463,472,1.2835667,"\int \frac{-d+c x^2}{\left(d+c x^2\right) \sqrt{a x^2+\sqrt{b+a^2 x^4}}} \, dx","IntegrateAlgebraic[(-d + c*x^2)/((d + c*x^2)*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]),x]","\frac{x}{2 \sqrt{a x^2+\sqrt{b+a^2 x^4}}}-\frac{2 \sqrt{2} \sqrt{a} d \tan ^{-1}\left(\frac{a x^2}{\sqrt{b}}+\frac{\sqrt{b+a^2 x^4}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{b}}\right)}{\sqrt{b} c}-\frac{\log \left(a x^2+\sqrt{b+a^2 x^4}-\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{2 \sqrt{2} \sqrt{a}}+\frac{\sqrt{2} \sqrt{a} d \text{RootSum}\left[b^2 c-4 a b d \text{$\#$1}-2 b c \text{$\#$1}^2-4 a d \text{$\#$1}^3+c \text{$\#$1}^4\&,\frac{b c \log \left(a x^2+\sqrt{b+a^2 x^4}-\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\text{$\#$1}\right)-2 a d \log \left(a x^2+\sqrt{b+a^2 x^4}-\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\text{$\#$1}\right) \text{$\#$1}+c \log \left(a x^2+\sqrt{b+a^2 x^4}-\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\text{$\#$1}\right) \text{$\#$1}^2}{a b d+b c \text{$\#$1}+3 a d \text{$\#$1}^2-c \text{$\#$1}^3}\&\right]}{c}","\frac{x}{2 \sqrt{a x^2+\sqrt{b+a^2 x^4}}}+\frac{2 \sqrt{2} \sqrt{a} d \tan ^{-1}\left(\frac{a x^2}{\sqrt{b}}+\frac{\sqrt{b+a^2 x^4}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{b}}\right)}{\sqrt{b} c}+\frac{\log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{2 \sqrt{2} \sqrt{a}}-\frac{\sqrt{2} \text{RootSum}\left[b^2 c+4 a b d \text{$\#$1}-2 b c \text{$\#$1}^2+4 a d \text{$\#$1}^3+c \text{$\#$1}^4\&,\frac{a b c d \log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)+2 a^2 d^2 \log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}+a c d \log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2}{a b d-b c \text{$\#$1}+3 a d \text{$\#$1}^2+c \text{$\#$1}^3}\&\right]}{\sqrt{a} c}",1,"x/(2*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]) - (2*Sqrt[2]*Sqrt[a]*d*ArcTan[(a*x^2)/Sqrt[b] + Sqrt[b + a^2*x^4]/Sqrt[b] - (Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[b]])/(Sqrt[b]*c) - Log[a*x^2 + Sqrt[b + a^2*x^4] - Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]]/(2*Sqrt[2]*Sqrt[a]) + (Sqrt[2]*Sqrt[a]*d*RootSum[b^2*c - 4*a*b*d*#1 - 2*b*c*#1^2 - 4*a*d*#1^3 + c*#1^4 & , (b*c*Log[a*x^2 + Sqrt[b + a^2*x^4] - Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + #1] - 2*a*d*Log[a*x^2 + Sqrt[b + a^2*x^4] - Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + #1]*#1 + c*Log[a*x^2 + Sqrt[b + a^2*x^4] - Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + #1]*#1^2)/(a*b*d + b*c*#1 + 3*a*d*#1^2 - c*#1^3) & ])/c","A",1
3064,1,463,472,0.0025986,"\int \frac{-d+c x^2}{\left(d+c x^2\right) \sqrt{a x^2+\sqrt{b+a^2 x^4}}} \, dx","IntegrateAlgebraic[(-d + c*x^2)/((d + c*x^2)*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]),x]","\frac{x}{2 \sqrt{a x^2+\sqrt{b+a^2 x^4}}}-\frac{2 \sqrt{2} \sqrt{a} d \tan ^{-1}\left(\frac{a x^2}{\sqrt{b}}+\frac{\sqrt{b+a^2 x^4}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{b}}\right)}{\sqrt{b} c}-\frac{\log \left(a x^2+\sqrt{b+a^2 x^4}-\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{2 \sqrt{2} \sqrt{a}}+\frac{\sqrt{2} \sqrt{a} d \text{RootSum}\left[b^2 c-4 a b d \text{$\#$1}-2 b c \text{$\#$1}^2-4 a d \text{$\#$1}^3+c \text{$\#$1}^4\&,\frac{b c \log \left(a x^2+\sqrt{b+a^2 x^4}-\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\text{$\#$1}\right)-2 a d \log \left(a x^2+\sqrt{b+a^2 x^4}-\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\text{$\#$1}\right) \text{$\#$1}+c \log \left(a x^2+\sqrt{b+a^2 x^4}-\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\text{$\#$1}\right) \text{$\#$1}^2}{a b d+b c \text{$\#$1}+3 a d \text{$\#$1}^2-c \text{$\#$1}^3}\&\right]}{c}","\frac{x}{2 \sqrt{a x^2+\sqrt{b+a^2 x^4}}}+\frac{2 \sqrt{2} \sqrt{a} d \tan ^{-1}\left(\frac{a x^2}{\sqrt{b}}+\frac{\sqrt{b+a^2 x^4}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{b}}\right)}{\sqrt{b} c}+\frac{\log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{2 \sqrt{2} \sqrt{a}}-\frac{\sqrt{2} \text{RootSum}\left[b^2 c+4 a b d \text{$\#$1}-2 b c \text{$\#$1}^2+4 a d \text{$\#$1}^3+c \text{$\#$1}^4\&,\frac{a b c d \log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)+2 a^2 d^2 \log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}+a c d \log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2}{a b d-b c \text{$\#$1}+3 a d \text{$\#$1}^2+c \text{$\#$1}^3}\&\right]}{\sqrt{a} c}",1,"x/(2*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]) - (2*Sqrt[2]*Sqrt[a]*d*ArcTan[(a*x^2)/Sqrt[b] + Sqrt[b + a^2*x^4]/Sqrt[b] - (Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[b]])/(Sqrt[b]*c) - Log[a*x^2 + Sqrt[b + a^2*x^4] - Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]]/(2*Sqrt[2]*Sqrt[a]) + (Sqrt[2]*Sqrt[a]*d*RootSum[b^2*c - 4*a*b*d*#1 - 2*b*c*#1^2 - 4*a*d*#1^3 + c*#1^4 & , (b*c*Log[a*x^2 + Sqrt[b + a^2*x^4] - Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + #1] - 2*a*d*Log[a*x^2 + Sqrt[b + a^2*x^4] - Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + #1]*#1 + c*Log[a*x^2 + Sqrt[b + a^2*x^4] - Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + #1]*#1^2)/(a*b*d + b*c*#1 + 3*a*d*#1^2 - c*#1^3) & ])/c","A",1
3065,1,534,476,47.9352134,"\int \frac{x^4}{\sqrt[4]{-b+a x^4} \left(b-2 a x^4+2 x^8\right)} \, dx","IntegrateAlgebraic[x^4/((-b + a*x^4)^(1/4)*(b - 2*a*x^4 + 2*x^8)),x]","\frac{\left(\sqrt{2}+i \sqrt{2 \left(3-2 \sqrt{2}\right)}\right) \tan ^{-1}\left(\frac{\left((-1+i)-(1+i) (-1)^{3/4}\right) \sqrt[4]{a^2-2 b} x^2+(1+i) \sqrt{-b+a x^4}+(1+i) (-1)^{3/4} \sqrt{-b+a x^4}}{2 \sqrt[8]{a^2-2 b} x \sqrt[4]{-b+a x^4}}\right)}{16 \left(a^2-2 b\right)^{5/8}}+\frac{i \left(i \sqrt{2}+\sqrt{2 \left(3+2 \sqrt{2}\right)}\right) \tan ^{-1}\left(\frac{2 \sqrt[8]{a^2-2 b} x \sqrt[4]{-b+a x^4}}{\left((1-i)+\sqrt{2}\right) \sqrt[4]{a^2-2 b} x^2-(1+i) \sqrt{-b+a x^4}-\sqrt{2} \sqrt{-b+a x^4}}\right)}{16 \left(a^2-2 b\right)^{5/8}}+\frac{\left(i-(-1)^{3/4}\right) \tanh ^{-1}\left(\frac{\left((-2+2 i)-(2+2 i) (-1)^{3/4}\right) \sqrt[4]{a^2-2 b} x^2-(2+2 i) \sqrt{-b+a x^4}-(2+2 i) (-1)^{3/4} \sqrt{-b+a x^4}}{4 \sqrt[8]{a^2-2 b} x \sqrt[4]{-b+a x^4}}\right)}{8 \left(a^2-2 b\right)^{5/8}}+\frac{\left(\sqrt{2}-i \sqrt{2 \left(3+2 \sqrt{2}\right)}\right) \tanh ^{-1}\left(\frac{\left((1-i)+\sqrt{2}\right) \sqrt[4]{a^2-2 b} x^2+(1+i) \sqrt{-b+a x^4}+\sqrt{2} \sqrt{-b+a x^4}}{2 \sqrt[8]{a^2-2 b} x \sqrt[4]{-b+a x^4}}\right)}{16 \left(a^2-2 b\right)^{5/8}}","\frac{\left(1-\sqrt[4]{-1}\right) \tan ^{-1}\left(\frac{(-1)^{7/8} \sqrt{2+\sqrt{2}} \sqrt[8]{a^2-2 b} x \sqrt[4]{-b+a x^4}}{(-1)^{3/4} \sqrt[4]{a^2-2 b} x^2+\sqrt{-b+a x^4}}\right)}{8 \left(a^2-2 b\right)^{5/8}}+\frac{\left(\sqrt{2}-i \sqrt{2 \left(3+2 \sqrt{2}\right)}\right) \tan ^{-1}\left(\frac{(-1)^{7/8} \left(-2+\sqrt{2}\right) \sqrt[8]{a^2-2 b} x \sqrt[4]{-b+a x^4}}{(-1)^{3/4} \sqrt{2-\sqrt{2}} \sqrt[4]{a^2-2 b} x^2+\sqrt{2-\sqrt{2}} \sqrt{-b+a x^4}}\right)}{16 \left(a^2-2 b\right)^{5/8}}+\frac{i \left(i \sqrt{2}+\sqrt{2 \left(3+2 \sqrt{2}\right)}\right) \tanh ^{-1}\left(\frac{(-1)^{7/8} \sqrt[4]{a^2-2 b} x^2-\sqrt[8]{-1} \sqrt{-b+a x^4}}{\sqrt{2-\sqrt{2}} \sqrt[8]{a^2-2 b} x \sqrt[4]{-b+a x^4}}\right)}{16 \left(a^2-2 b\right)^{5/8}}+\frac{\left(1-\sqrt[4]{-1}\right) \tanh ^{-1}\left(\frac{(-1)^{7/8} \sqrt[4]{a^2-2 b} x^2-\sqrt[8]{-1} \sqrt{-b+a x^4}}{\sqrt{2+\sqrt{2}} \sqrt[8]{a^2-2 b} x \sqrt[4]{-b+a x^4}}\right)}{8 \left(a^2-2 b\right)^{5/8}}",1,"((Sqrt[2] + I*Sqrt[2*(3 - 2*Sqrt[2])])*ArcTan[(((-1 + I) - (1 + I)*(-1)^(3/4))*(a^2 - 2*b)^(1/4)*x^2 + (1 + I)*Sqrt[-b + a*x^4] + (1 + I)*(-1)^(3/4)*Sqrt[-b + a*x^4])/(2*(a^2 - 2*b)^(1/8)*x*(-b + a*x^4)^(1/4))])/(16*(a^2 - 2*b)^(5/8)) + ((I/16)*(I*Sqrt[2] + Sqrt[2*(3 + 2*Sqrt[2])])*ArcTan[(2*(a^2 - 2*b)^(1/8)*x*(-b + a*x^4)^(1/4))/(((1 - I) + Sqrt[2])*(a^2 - 2*b)^(1/4)*x^2 - (1 + I)*Sqrt[-b + a*x^4] - Sqrt[2]*Sqrt[-b + a*x^4])])/(a^2 - 2*b)^(5/8) + ((I - (-1)^(3/4))*ArcTanh[(((-2 + 2*I) - (2 + 2*I)*(-1)^(3/4))*(a^2 - 2*b)^(1/4)*x^2 - (2 + 2*I)*Sqrt[-b + a*x^4] - (2 + 2*I)*(-1)^(3/4)*Sqrt[-b + a*x^4])/(4*(a^2 - 2*b)^(1/8)*x*(-b + a*x^4)^(1/4))])/(8*(a^2 - 2*b)^(5/8)) + ((Sqrt[2] - I*Sqrt[2*(3 + 2*Sqrt[2])])*ArcTanh[(((1 - I) + Sqrt[2])*(a^2 - 2*b)^(1/4)*x^2 + (1 + I)*Sqrt[-b + a*x^4] + Sqrt[2]*Sqrt[-b + a*x^4])/(2*(a^2 - 2*b)^(1/8)*x*(-b + a*x^4)^(1/4))])/(16*(a^2 - 2*b)^(5/8))","A",1
3066,1,477,477,8.0551802,"\int \frac{x^3 (-5 b+6 a x)}{\sqrt[4]{-b x+a x^2} \left(c-b x^5+a x^6\right)} \, dx","IntegrateAlgebraic[(x^3*(-5*b + 6*a*x))/((-(b*x) + a*x^2)^(1/4)*(c - b*x^5 + a*x^6)),x]","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{-2^{2/3} x \sqrt[4]{-b x+a x^2}+\sqrt[3]{c} x^2 \sqrt[4]{-b x+a x^2}}{2 \sqrt[6]{2} \sqrt[4]{c}-\sqrt{2} c^{7/12} x+2^{2/3} x \sqrt[4]{-b x+a x^2}-\sqrt[3]{c} x^2 \sqrt[4]{-b x+a x^2}}\right)}{\sqrt[4]{c}}+\frac{\sqrt{2} \tan ^{-1}\left(\frac{-2^{2/3} x \sqrt[4]{-b x+a x^2}+\sqrt[3]{c} x^2 \sqrt[4]{-b x+a x^2}}{-2 \sqrt[6]{2} \sqrt[4]{c}+\sqrt{2} c^{7/12} x+2^{2/3} x \sqrt[4]{-b x+a x^2}-\sqrt[3]{c} x^2 \sqrt[4]{-b x+a x^2}}\right)}{\sqrt[4]{c}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{2\ 2^{5/6} \sqrt[4]{c} x \sqrt[4]{-b x+a x^2}-4 \sqrt[6]{2} c^{7/12} x^2 \sqrt[4]{-b x+a x^2}+\sqrt{2} c^{11/12} x^3 \sqrt[4]{-b x+a x^2}}{2 \sqrt[3]{2} \sqrt{c}-2\ 2^{2/3} c^{5/6} x+c^{7/6} x^2+2 \sqrt[3]{2} x^2 \sqrt{-b x+a x^2}-2\ 2^{2/3} \sqrt[3]{c} x^3 \sqrt{-b x+a x^2}+c^{2/3} x^4 \sqrt{-b x+a x^2}}\right)}{\sqrt[4]{c}}","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{-2^{2/3} x \sqrt[4]{-b x+a x^2}+\sqrt[3]{c} x^2 \sqrt[4]{-b x+a x^2}}{2 \sqrt[6]{2} \sqrt[4]{c}-\sqrt{2} c^{7/12} x+2^{2/3} x \sqrt[4]{-b x+a x^2}-\sqrt[3]{c} x^2 \sqrt[4]{-b x+a x^2}}\right)}{\sqrt[4]{c}}+\frac{\sqrt{2} \tan ^{-1}\left(\frac{-2^{2/3} x \sqrt[4]{-b x+a x^2}+\sqrt[3]{c} x^2 \sqrt[4]{-b x+a x^2}}{-2 \sqrt[6]{2} \sqrt[4]{c}+\sqrt{2} c^{7/12} x+2^{2/3} x \sqrt[4]{-b x+a x^2}-\sqrt[3]{c} x^2 \sqrt[4]{-b x+a x^2}}\right)}{\sqrt[4]{c}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{2\ 2^{5/6} \sqrt[4]{c} x \sqrt[4]{-b x+a x^2}-4 \sqrt[6]{2} c^{7/12} x^2 \sqrt[4]{-b x+a x^2}+\sqrt{2} c^{11/12} x^3 \sqrt[4]{-b x+a x^2}}{2 \sqrt[3]{2} \sqrt{c}-2\ 2^{2/3} c^{5/6} x+c^{7/6} x^2+2 \sqrt[3]{2} x^2 \sqrt{-b x+a x^2}-2\ 2^{2/3} \sqrt[3]{c} x^3 \sqrt{-b x+a x^2}+c^{2/3} x^4 \sqrt{-b x+a x^2}}\right)}{\sqrt[4]{c}}",1,"-((Sqrt[2]*ArcTan[(-(2^(2/3)*x*(-(b*x) + a*x^2)^(1/4)) + c^(1/3)*x^2*(-(b*x) + a*x^2)^(1/4))/(2*2^(1/6)*c^(1/4) - Sqrt[2]*c^(7/12)*x + 2^(2/3)*x*(-(b*x) + a*x^2)^(1/4) - c^(1/3)*x^2*(-(b*x) + a*x^2)^(1/4))])/c^(1/4)) + (Sqrt[2]*ArcTan[(-(2^(2/3)*x*(-(b*x) + a*x^2)^(1/4)) + c^(1/3)*x^2*(-(b*x) + a*x^2)^(1/4))/(-2*2^(1/6)*c^(1/4) + Sqrt[2]*c^(7/12)*x + 2^(2/3)*x*(-(b*x) + a*x^2)^(1/4) - c^(1/3)*x^2*(-(b*x) + a*x^2)^(1/4))])/c^(1/4) - (Sqrt[2]*ArcTanh[(2*2^(5/6)*c^(1/4)*x*(-(b*x) + a*x^2)^(1/4) - 4*2^(1/6)*c^(7/12)*x^2*(-(b*x) + a*x^2)^(1/4) + Sqrt[2]*c^(11/12)*x^3*(-(b*x) + a*x^2)^(1/4))/(2*2^(1/3)*Sqrt[c] - 2*2^(2/3)*c^(5/6)*x + c^(7/6)*x^2 + 2*2^(1/3)*x^2*Sqrt[-(b*x) + a*x^2] - 2*2^(2/3)*c^(1/3)*x^3*Sqrt[-(b*x) + a*x^2] + c^(2/3)*x^4*Sqrt[-(b*x) + a*x^2])])/c^(1/4)","A",1
3067,1,481,481,15.3963108,"\int \frac{-a+x}{\sqrt[3]{(-a+x) (-b+x)^2} \left(-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2\right)} \, dx","IntegrateAlgebraic[(-a + x)/(((-a + x)*(-b + x)^2)^(1/3)*(-b^2 + a^2*d + 2*(b - a*d)*x + (-1 + d)*x^2)),x]","\frac{\left(1+i \sqrt{3}\right) (-b+x)^{2/3} \left(\sqrt[3]{d} (a-x)^{2/3}-\sqrt[6]{d} \sqrt[3]{a-x} \sqrt[3]{-b+x}+(-b+x)^{2/3}\right) \left(\sqrt[3]{d} (a-x)^{2/3}+\sqrt[6]{d} \sqrt[3]{a-x} \sqrt[3]{-b+x}+(-b+x)^{2/3}\right) \left(-a \sqrt[3]{d}+\sqrt[3]{d} x+\sqrt[3]{a-x} (-b+x)^{2/3}\right) \left(\frac{\left(-3 i+\sqrt{3}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-b+x}}{-2 \sqrt[6]{d} \sqrt[3]{a-x}+\sqrt[3]{-b+x}}\right)}{4 (a-b) d^{5/6}}+\frac{\left(3 i-\sqrt{3}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-b+x}}{2 \sqrt[6]{d} \sqrt[3]{a-x}+\sqrt[3]{-b+x}}\right)}{4 (a-b) d^{5/6}}+\frac{\left(1-i \sqrt{3}\right) \tanh ^{-1}\left(\frac{(a-x)^{2/3} \sqrt[3]{-b+x}}{\sqrt[6]{d} (-a+x)}\right)}{2 (a-b) d^{5/6}}+\frac{i \left(i+\sqrt{3}\right) \tanh ^{-1}\left(\frac{\sqrt[3]{d} (a-x)^{2/3}+(-b+x)^{2/3}}{\sqrt[6]{d} \sqrt[3]{a-x} \sqrt[3]{-b+x}}\right)}{4 (a-b) d^{5/6}}\right)}{2 \sqrt[3]{(b-x)^2 (-a+x)} \left(b^2-a^2 d-2 b x+2 a d x-(-1+d) x^2\right)}","\frac{\left(1+i \sqrt{3}\right) (-b+x)^{2/3} \left(\sqrt[3]{d} (a-x)^{2/3}-\sqrt[6]{d} \sqrt[3]{a-x} \sqrt[3]{-b+x}+(-b+x)^{2/3}\right) \left(\sqrt[3]{d} (a-x)^{2/3}+\sqrt[6]{d} \sqrt[3]{a-x} \sqrt[3]{-b+x}+(-b+x)^{2/3}\right) \left(-a \sqrt[3]{d}+\sqrt[3]{d} x+\sqrt[3]{a-x} (-b+x)^{2/3}\right) \left(\frac{\left(-3 i+\sqrt{3}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-b+x}}{-2 \sqrt[6]{d} \sqrt[3]{a-x}+\sqrt[3]{-b+x}}\right)}{4 (a-b) d^{5/6}}+\frac{\left(3 i-\sqrt{3}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-b+x}}{2 \sqrt[6]{d} \sqrt[3]{a-x}+\sqrt[3]{-b+x}}\right)}{4 (a-b) d^{5/6}}+\frac{\left(1-i \sqrt{3}\right) \tanh ^{-1}\left(\frac{(a-x)^{2/3} \sqrt[3]{-b+x}}{\sqrt[6]{d} (-a+x)}\right)}{2 (a-b) d^{5/6}}+\frac{i \left(i+\sqrt{3}\right) \tanh ^{-1}\left(\frac{\sqrt[3]{d} (a-x)^{2/3}+(-b+x)^{2/3}}{\sqrt[6]{d} \sqrt[3]{a-x} \sqrt[3]{-b+x}}\right)}{4 (a-b) d^{5/6}}\right)}{2 \sqrt[3]{(b-x)^2 (-a+x)} \left(b^2-a^2 d-2 b x+2 a d x-(-1+d) x^2\right)}",1,"((1 + I*Sqrt[3])*(-b + x)^(2/3)*(d^(1/3)*(a - x)^(2/3) - d^(1/6)*(a - x)^(1/3)*(-b + x)^(1/3) + (-b + x)^(2/3))*(d^(1/3)*(a - x)^(2/3) + d^(1/6)*(a - x)^(1/3)*(-b + x)^(1/3) + (-b + x)^(2/3))*(-(a*d^(1/3)) + d^(1/3)*x + (a - x)^(1/3)*(-b + x)^(2/3))*(((-3*I + Sqrt[3])*ArcTan[(Sqrt[3]*(-b + x)^(1/3))/(-2*d^(1/6)*(a - x)^(1/3) + (-b + x)^(1/3))])/(4*(a - b)*d^(5/6)) + ((3*I - Sqrt[3])*ArcTan[(Sqrt[3]*(-b + x)^(1/3))/(2*d^(1/6)*(a - x)^(1/3) + (-b + x)^(1/3))])/(4*(a - b)*d^(5/6)) + ((1 - I*Sqrt[3])*ArcTanh[((a - x)^(2/3)*(-b + x)^(1/3))/(d^(1/6)*(-a + x))])/(2*(a - b)*d^(5/6)) + ((I/4)*(I + Sqrt[3])*ArcTanh[(d^(1/3)*(a - x)^(2/3) + (-b + x)^(2/3))/(d^(1/6)*(a - x)^(1/3)*(-b + x)^(1/3))])/((a - b)*d^(5/6))))/(2*((b - x)^2*(-a + x))^(1/3)*(b^2 - a^2*d - 2*b*x + 2*a*d*x - (-1 + d)*x^2))","A",1
3068,1,536,482,133.153294,"\int \frac{b-a x^4+2 x^8}{\sqrt[4]{b+a x^4} \left(-b-2 a x^4+x^8\right)} \, dx","IntegrateAlgebraic[(b - a*x^4 + 2*x^8)/((b + a*x^4)^(1/4)*(-b - 2*a*x^4 + x^8)),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{3 i \left(i \sqrt{2}+\sqrt{2 \left(3+2 \sqrt{2}\right)}\right) \tan ^{-1}\left(\frac{\left((-1+i)-(1+i) (-1)^{3/4}\right) \sqrt[4]{a^2+b} x^2+(1+i) \sqrt{b+a x^4}+(1+i) (-1)^{3/4} \sqrt{b+a x^4}}{2 \sqrt[8]{a^2+b} x \sqrt[4]{b+a x^4}}\right)}{16 \sqrt[8]{a^2+b}}+\frac{3 \left(\sqrt{2}+i \sqrt{2 \left(3-2 \sqrt{2}\right)}\right) \tan ^{-1}\left(\frac{2 \sqrt[8]{a^2+b} x \sqrt[4]{b+a x^4}}{\left((1-i)+\sqrt{2}\right) \sqrt[4]{a^2+b} x^2-(1+i) \sqrt{b+a x^4}-\sqrt{2} \sqrt{b+a x^4}}\right)}{16 \sqrt[8]{a^2+b}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{3 \left(i+(-1)^{3/4}\right) \tanh ^{-1}\left(\frac{\left((-2+2 i)-(2+2 i) (-1)^{3/4}\right) \sqrt[4]{a^2+b} x^2-(2+2 i) \sqrt{b+a x^4}-(2+2 i) (-1)^{3/4} \sqrt{b+a x^4}}{4 \sqrt[8]{a^2+b} x \sqrt[4]{b+a x^4}}\right)}{8 \sqrt[8]{a^2+b}}-\frac{3 i \left(-i \sqrt{2}+\sqrt{2 \left(3-2 \sqrt{2}\right)}\right) \tanh ^{-1}\left(\frac{\left((1-i)+\sqrt{2}\right) \sqrt[4]{a^2+b} x^2+(1+i) \sqrt{b+a x^4}+\sqrt{2} \sqrt{b+a x^4}}{2 \sqrt[8]{a^2+b} x \sqrt[4]{b+a x^4}}\right)}{16 \sqrt[8]{a^2+b}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{3 \left(1+\sqrt[4]{-1}\right) \tan ^{-1}\left(\frac{(-1)^{7/8} \sqrt{2+\sqrt{2}} \sqrt[8]{a^2+b} x \sqrt[4]{b+a x^4}}{(-1)^{3/4} \sqrt[4]{a^2+b} x^2+\sqrt{b+a x^4}}\right)}{8 \sqrt[8]{a^2+b}}-\frac{3 i \left(-i \sqrt{2}+\sqrt{2 \left(3-2 \sqrt{2}\right)}\right) \tan ^{-1}\left(\frac{(-1)^{7/8} \left(-2+\sqrt{2}\right) \sqrt[8]{a^2+b} x \sqrt[4]{b+a x^4}}{(-1)^{3/4} \sqrt{2-\sqrt{2}} \sqrt[4]{a^2+b} x^2+\sqrt{2-\sqrt{2}} \sqrt{b+a x^4}}\right)}{16 \sqrt[8]{a^2+b}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a} x}{\sqrt[4]{b+a x^4}}\right)}{\sqrt[4]{a}}+\frac{3 \left(\sqrt{2}+i \sqrt{2 \left(3-2 \sqrt{2}\right)}\right) \tanh ^{-1}\left(\frac{(-1)^{7/8} \sqrt[4]{a^2+b} x^2-\sqrt[8]{-1} \sqrt{b+a x^4}}{\sqrt{2-\sqrt{2}} \sqrt[8]{a^2+b} x \sqrt[4]{b+a x^4}}\right)}{16 \sqrt[8]{a^2+b}}+\frac{3 \left(1+\sqrt[4]{-1}\right) \tanh ^{-1}\left(\frac{(-1)^{7/8} \sqrt[4]{a^2+b} x^2-\sqrt[8]{-1} \sqrt{b+a x^4}}{\sqrt{2+\sqrt{2}} \sqrt[8]{a^2+b} x \sqrt[4]{b+a x^4}}\right)}{8 \sqrt[8]{a^2+b}}",1,"ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(1/4) + (((3*I)/16)*(I*Sqrt[2] + Sqrt[2*(3 + 2*Sqrt[2])])*ArcTan[(((-1 + I) - (1 + I)*(-1)^(3/4))*(a^2 + b)^(1/4)*x^2 + (1 + I)*Sqrt[b + a*x^4] + (1 + I)*(-1)^(3/4)*Sqrt[b + a*x^4])/(2*(a^2 + b)^(1/8)*x*(b + a*x^4)^(1/4))])/(a^2 + b)^(1/8) + (3*(Sqrt[2] + I*Sqrt[2*(3 - 2*Sqrt[2])])*ArcTan[(2*(a^2 + b)^(1/8)*x*(b + a*x^4)^(1/4))/(((1 - I) + Sqrt[2])*(a^2 + b)^(1/4)*x^2 - (1 + I)*Sqrt[b + a*x^4] - Sqrt[2]*Sqrt[b + a*x^4])])/(16*(a^2 + b)^(1/8)) + ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/a^(1/4) + (3*(I + (-1)^(3/4))*ArcTanh[(((-2 + 2*I) - (2 + 2*I)*(-1)^(3/4))*(a^2 + b)^(1/4)*x^2 - (2 + 2*I)*Sqrt[b + a*x^4] - (2 + 2*I)*(-1)^(3/4)*Sqrt[b + a*x^4])/(4*(a^2 + b)^(1/8)*x*(b + a*x^4)^(1/4))])/(8*(a^2 + b)^(1/8)) - (((3*I)/16)*((-I)*Sqrt[2] + Sqrt[2*(3 - 2*Sqrt[2])])*ArcTanh[(((1 - I) + Sqrt[2])*(a^2 + b)^(1/4)*x^2 + (1 + I)*Sqrt[b + a*x^4] + Sqrt[2]*Sqrt[b + a*x^4])/(2*(a^2 + b)^(1/8)*x*(b + a*x^4)^(1/4))])/(a^2 + b)^(1/8)","A",1
3069,1,495,495,1.0628327,"\int \frac{\sqrt{-b+a^2 x^2}}{\sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}} \, dx","IntegrateAlgebraic[Sqrt[-b + a^2*x^2]/Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]],x]","\frac{\left(-1680 b^2 c^3+6144 b c^7-2450 a b^2 c x+6912 a b c^5 x-12288 a^2 c^7 x^2-9216 a^3 c^5 x^3\right) \sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\left(1960 b^2 c^2-3072 b c^6+3675 a b^2 x-5760 a b c^4 x+6144 a^2 c^6 x^2+7680 a^3 c^4 x^3\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\sqrt{-b+a^2 x^2} \left(\left(-2450 b^2 c+2304 b c^5-12288 a c^7 x-9216 a^2 c^5 x^2\right) \sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\left(3675 b^2-1920 b c^4+6144 a c^6 x+7680 a^2 c^4 x^2\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{26880 a^2 c^4 x \sqrt{-b+a^2 x^2}+13440 a c^4 \left(-b+2 a^2 x^2\right)}-\frac{35 b^2 \tanh ^{-1}\left(\frac{\sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{c}}\right)}{128 a c^{9/2}}+\frac{2 b \tanh ^{-1}\left(\frac{\sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{c}}\right)}{a \sqrt{c}}","\frac{\left(-1680 b^2 c^3+6144 b c^7-2450 a b^2 c x+6912 a b c^5 x-12288 a^2 c^7 x^2-9216 a^3 c^5 x^3\right) \sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\left(1960 b^2 c^2-3072 b c^6+3675 a b^2 x-5760 a b c^4 x+6144 a^2 c^6 x^2+7680 a^3 c^4 x^3\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\sqrt{-b+a^2 x^2} \left(\left(-2450 b^2 c+2304 b c^5-12288 a c^7 x-9216 a^2 c^5 x^2\right) \sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\left(3675 b^2-1920 b c^4+6144 a c^6 x+7680 a^2 c^4 x^2\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{26880 a^2 c^4 x \sqrt{-b+a^2 x^2}+13440 a c^4 \left(-b+2 a^2 x^2\right)}-\frac{35 b^2 \tanh ^{-1}\left(\frac{\sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{c}}\right)}{128 a c^{9/2}}+\frac{2 b \tanh ^{-1}\left(\frac{\sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{c}}\right)}{a \sqrt{c}}",1,"((-1680*b^2*c^3 + 6144*b*c^7 - 2450*a*b^2*c*x + 6912*a*b*c^5*x - 12288*a^2*c^7*x^2 - 9216*a^3*c^5*x^3)*Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]] + (1960*b^2*c^2 - 3072*b*c^6 + 3675*a*b^2*x - 5760*a*b*c^4*x + 6144*a^2*c^6*x^2 + 7680*a^3*c^4*x^3)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]] + Sqrt[-b + a^2*x^2]*((-2450*b^2*c + 2304*b*c^5 - 12288*a*c^7*x - 9216*a^2*c^5*x^2)*Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]] + (3675*b^2 - 1920*b*c^4 + 6144*a*c^6*x + 7680*a^2*c^4*x^2)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]))/(26880*a^2*c^4*x*Sqrt[-b + a^2*x^2] + 13440*a*c^4*(-b + 2*a^2*x^2)) - (35*b^2*ArcTanh[Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]/Sqrt[c]])/(128*a*c^(9/2)) + (2*b*ArcTanh[Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]/Sqrt[c]])/(a*Sqrt[c])","A",1
3070,1,496,496,1.0459016,"\int \frac{1+x}{\sqrt[3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}} \, dx","IntegrateAlgebraic[(1 + x)/(27 + 189*x + 522*x^2 + 784*x^3 + 825*x^4 + 679*x^5 + 338*x^6 + 84*x^7 + 8*x^8)^(1/3),x]","\frac{\sqrt{3} \tan ^{-1}\left(\frac{5 \sqrt{3} \sqrt[3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}}{12 \sqrt[3]{10}+22 \sqrt[3]{10} x-6 \sqrt[3]{10} x^2-4 \sqrt[3]{10} x^3+5 \sqrt[3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}}\right)}{5 \sqrt[3]{10}}-\frac{\log \left(3+7 x+2 x^2\right)}{5 \sqrt[3]{10}}+\frac{\log \left(9+42 x+61 x^2+28 x^3+4 x^4\right)}{10 \sqrt[3]{10}}+\frac{\log \left(-6 \sqrt[3]{10}-11 \sqrt[3]{10} x+3 \sqrt[3]{10} x^2+2 \sqrt[3]{10} x^3+5 \sqrt[3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}\right)}{5 \sqrt[3]{10}}-\frac{\log \left(36\ 10^{2/3}+132\ 10^{2/3} x+85\ 10^{2/3} x^2-90\ 10^{2/3} x^3-35\ 10^{2/3} x^4+12\ 10^{2/3} x^5+4\ 10^{2/3} x^6+\left(30 \sqrt[3]{10}+55 \sqrt[3]{10} x-15 \sqrt[3]{10} x^2-10 \sqrt[3]{10} x^3\right) \sqrt[3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}+25 \left(27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8\right)^{2/3}\right)}{10 \sqrt[3]{10}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{5 \sqrt{3} \sqrt[3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}}{12 \sqrt[3]{10}+22 \sqrt[3]{10} x-6 \sqrt[3]{10} x^2-4 \sqrt[3]{10} x^3+5 \sqrt[3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}}\right)}{5 \sqrt[3]{10}}-\frac{\log \left(3+7 x+2 x^2\right)}{5 \sqrt[3]{10}}+\frac{\log \left(9+42 x+61 x^2+28 x^3+4 x^4\right)}{10 \sqrt[3]{10}}+\frac{\log \left(-6 \sqrt[3]{10}-11 \sqrt[3]{10} x+3 \sqrt[3]{10} x^2+2 \sqrt[3]{10} x^3+5 \sqrt[3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}\right)}{5 \sqrt[3]{10}}-\frac{\log \left(36\ 10^{2/3}+132\ 10^{2/3} x+85\ 10^{2/3} x^2-90\ 10^{2/3} x^3-35\ 10^{2/3} x^4+12\ 10^{2/3} x^5+4\ 10^{2/3} x^6+\left(30 \sqrt[3]{10}+55 \sqrt[3]{10} x-15 \sqrt[3]{10} x^2-10 \sqrt[3]{10} x^3\right) \sqrt[3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}+25 \left(27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8\right)^{2/3}\right)}{10 \sqrt[3]{10}}",1,"(Sqrt[3]*ArcTan[(5*Sqrt[3]*(27 + 189*x + 522*x^2 + 784*x^3 + 825*x^4 + 679*x^5 + 338*x^6 + 84*x^7 + 8*x^8)^(1/3))/(12*10^(1/3) + 22*10^(1/3)*x - 6*10^(1/3)*x^2 - 4*10^(1/3)*x^3 + 5*(27 + 189*x + 522*x^2 + 784*x^3 + 825*x^4 + 679*x^5 + 338*x^6 + 84*x^7 + 8*x^8)^(1/3))])/(5*10^(1/3)) - Log[3 + 7*x + 2*x^2]/(5*10^(1/3)) + Log[9 + 42*x + 61*x^2 + 28*x^3 + 4*x^4]/(10*10^(1/3)) + Log[-6*10^(1/3) - 11*10^(1/3)*x + 3*10^(1/3)*x^2 + 2*10^(1/3)*x^3 + 5*(27 + 189*x + 522*x^2 + 784*x^3 + 825*x^4 + 679*x^5 + 338*x^6 + 84*x^7 + 8*x^8)^(1/3)]/(5*10^(1/3)) - Log[36*10^(2/3) + 132*10^(2/3)*x + 85*10^(2/3)*x^2 - 90*10^(2/3)*x^3 - 35*10^(2/3)*x^4 + 12*10^(2/3)*x^5 + 4*10^(2/3)*x^6 + (30*10^(1/3) + 55*10^(1/3)*x - 15*10^(1/3)*x^2 - 10*10^(1/3)*x^3)*(27 + 189*x + 522*x^2 + 784*x^3 + 825*x^4 + 679*x^5 + 338*x^6 + 84*x^7 + 8*x^8)^(1/3) + 25*(27 + 189*x + 522*x^2 + 784*x^3 + 825*x^4 + 679*x^5 + 338*x^6 + 84*x^7 + 8*x^8)^(2/3)]/(10*10^(1/3))","A",1
3071,1,497,497,2.8416146,"\int \frac{-1+k^2 x^4}{\sqrt{\left(1-x^2\right) \left(1-k^2 x^2\right)} \left(1+k^2 x^4\right)} \, dx","IntegrateAlgebraic[(-1 + k^2*x^4)/(Sqrt[(1 - x^2)*(1 - k^2*x^2)]*(1 + k^2*x^4)),x]","-\frac{i \left(-i \sqrt{1-2 k+k^2-2 i \sqrt{2} \sqrt{k} \sqrt{1+k^2}}-i k^2 \sqrt{1-2 k+k^2-2 i \sqrt{2} \sqrt{k} \sqrt{1+k^2}}+\sqrt{2} \sqrt{k} \sqrt{1+k^2} \sqrt{1-2 k+k^2-2 i \sqrt{2} \sqrt{k} \sqrt{1+k^2}}\right) \tan ^{-1}\left(\frac{\sqrt{1-2 k+k^2-2 i \sqrt{2} \sqrt{k} \sqrt{1+k^2}} x}{1+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{\left(-i+\sqrt{k}\right)^2 \left(i+\sqrt{k}\right)^2 (-i+k) (i+k)}+\frac{i \left(i \sqrt{1-2 k+k^2+2 i \sqrt{2} \sqrt{k} \sqrt{1+k^2}}+i k^2 \sqrt{1-2 k+k^2+2 i \sqrt{2} \sqrt{k} \sqrt{1+k^2}}+\sqrt{2} \sqrt{k} \sqrt{1+k^2} \sqrt{1-2 k+k^2+2 i \sqrt{2} \sqrt{k} \sqrt{1+k^2}}\right) \tan ^{-1}\left(\frac{\sqrt{1-2 k+k^2+2 i \sqrt{2} \sqrt{k} \sqrt{1+k^2}} x}{1+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{\left(-i+\sqrt{k}\right)^2 \left(i+\sqrt{k}\right)^2 (-i+k) (i+k)}","-\frac{i \left(-i \sqrt{1-2 k+k^2-2 i \sqrt{2} \sqrt{k} \sqrt{1+k^2}}-i k^2 \sqrt{1-2 k+k^2-2 i \sqrt{2} \sqrt{k} \sqrt{1+k^2}}+\sqrt{2} \sqrt{k} \sqrt{1+k^2} \sqrt{1-2 k+k^2-2 i \sqrt{2} \sqrt{k} \sqrt{1+k^2}}\right) \tan ^{-1}\left(\frac{\sqrt{1-2 k+k^2-2 i \sqrt{2} \sqrt{k} \sqrt{1+k^2}} x}{1+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{\left(-i+\sqrt{k}\right)^2 \left(i+\sqrt{k}\right)^2 (-i+k) (i+k)}+\frac{i \left(i \sqrt{1-2 k+k^2+2 i \sqrt{2} \sqrt{k} \sqrt{1+k^2}}+i k^2 \sqrt{1-2 k+k^2+2 i \sqrt{2} \sqrt{k} \sqrt{1+k^2}}+\sqrt{2} \sqrt{k} \sqrt{1+k^2} \sqrt{1-2 k+k^2+2 i \sqrt{2} \sqrt{k} \sqrt{1+k^2}}\right) \tan ^{-1}\left(\frac{\sqrt{1-2 k+k^2+2 i \sqrt{2} \sqrt{k} \sqrt{1+k^2}} x}{1+k x^2+\sqrt{1+\left(-1-k^2\right) x^2+k^2 x^4}}\right)}{\left(-i+\sqrt{k}\right)^2 \left(i+\sqrt{k}\right)^2 (-i+k) (i+k)}",1,"((-I)*((-I)*Sqrt[1 - 2*k + k^2 - (2*I)*Sqrt[2]*Sqrt[k]*Sqrt[1 + k^2]] - I*k^2*Sqrt[1 - 2*k + k^2 - (2*I)*Sqrt[2]*Sqrt[k]*Sqrt[1 + k^2]] + Sqrt[2]*Sqrt[k]*Sqrt[1 + k^2]*Sqrt[1 - 2*k + k^2 - (2*I)*Sqrt[2]*Sqrt[k]*Sqrt[1 + k^2]])*ArcTan[(Sqrt[1 - 2*k + k^2 - (2*I)*Sqrt[2]*Sqrt[k]*Sqrt[1 + k^2]]*x)/(1 + k*x^2 + Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4])])/((-I + Sqrt[k])^2*(I + Sqrt[k])^2*(-I + k)*(I + k)) + (I*(I*Sqrt[1 - 2*k + k^2 + (2*I)*Sqrt[2]*Sqrt[k]*Sqrt[1 + k^2]] + I*k^2*Sqrt[1 - 2*k + k^2 + (2*I)*Sqrt[2]*Sqrt[k]*Sqrt[1 + k^2]] + Sqrt[2]*Sqrt[k]*Sqrt[1 + k^2]*Sqrt[1 - 2*k + k^2 + (2*I)*Sqrt[2]*Sqrt[k]*Sqrt[1 + k^2]])*ArcTan[(Sqrt[1 - 2*k + k^2 + (2*I)*Sqrt[2]*Sqrt[k]*Sqrt[1 + k^2]]*x)/(1 + k*x^2 + Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4])])/((-I + Sqrt[k])^2*(I + Sqrt[k])^2*(-I + k)*(I + k))","A",1
3072,1,499,499,7.9754978,"\int \frac{1+2 x}{\sqrt[3]{-1+x^2} \left(3+x^2\right)} \, dx","IntegrateAlgebraic[(1 + 2*x)/((-1 + x^2)^(1/3)*(3 + x^2)),x]","-\frac{\sqrt[6]{-253+1260 i \sqrt{3}} \tan ^{-1}\left(\frac{3 \sqrt[3]{-1+x^2}}{-2^{2/3} \sqrt{3}-i 2^{2/3} x+\sqrt{3} \sqrt[3]{-1+x^2}}\right)}{6\ 2^{2/3}}-\frac{\sqrt[6]{-253-1260 i \sqrt{3}} \tan ^{-1}\left(\frac{3 \sqrt[3]{-1+x^2}}{-2^{2/3} \sqrt{3}+i 2^{2/3} x+\sqrt{3} \sqrt[3]{-1+x^2}}\right)}{6\ 2^{2/3}}+\frac{(-1)^{2/3} \sqrt[3]{-54+35 i \sqrt{3}} \log \left(3\ 2^{2/3}-i 2^{2/3} \sqrt{3} x+6 \sqrt[3]{-1+x^2}\right)}{6\ 6^{2/3}}-\frac{\sqrt[3]{-1} \sqrt[3]{-54-35 i \sqrt{3}} \log \left(3\ 2^{2/3}+i 2^{2/3} \sqrt{3} x+6 \sqrt[3]{-1+x^2}\right)}{6\ 6^{2/3}}+\frac{\sqrt[3]{54+35 i \sqrt{3}} \log \left(3 \sqrt[3]{2}+2 i \sqrt[3]{2} \sqrt{3} x-\sqrt[3]{2} x^2-3\ 2^{2/3} \sqrt[3]{-1+x^2}-i 2^{2/3} \sqrt{3} x \sqrt[3]{-1+x^2}+6 \left(-1+x^2\right)^{2/3}\right)}{12\ 6^{2/3}}+\frac{\sqrt[3]{54-35 i \sqrt{3}} \log \left(3 \sqrt[3]{2}-2 i \sqrt[3]{2} \sqrt{3} x-\sqrt[3]{2} x^2-3\ 2^{2/3} \sqrt[3]{-1+x^2}+i 2^{2/3} \sqrt{3} x \sqrt[3]{-1+x^2}+6 \left(-1+x^2\right)^{2/3}\right)}{12\ 6^{2/3}}","-\frac{\sqrt[6]{-253+1260 i \sqrt{3}} \tan ^{-1}\left(\frac{3 \sqrt[3]{-1+x^2}}{-2^{2/3} \sqrt{3}-i 2^{2/3} x+\sqrt{3} \sqrt[3]{-1+x^2}}\right)}{6\ 2^{2/3}}-\frac{\sqrt[6]{-253-1260 i \sqrt{3}} \tan ^{-1}\left(\frac{3 \sqrt[3]{-1+x^2}}{-2^{2/3} \sqrt{3}+i 2^{2/3} x+\sqrt{3} \sqrt[3]{-1+x^2}}\right)}{6\ 2^{2/3}}+\frac{(-1)^{2/3} \sqrt[3]{-54+35 i \sqrt{3}} \log \left(3\ 2^{2/3}-i 2^{2/3} \sqrt{3} x+6 \sqrt[3]{-1+x^2}\right)}{6\ 6^{2/3}}-\frac{\sqrt[3]{-1} \sqrt[3]{-54-35 i \sqrt{3}} \log \left(3\ 2^{2/3}+i 2^{2/3} \sqrt{3} x+6 \sqrt[3]{-1+x^2}\right)}{6\ 6^{2/3}}+\frac{\sqrt[3]{54+35 i \sqrt{3}} \log \left(3 \sqrt[3]{2}+2 i \sqrt[3]{2} \sqrt{3} x-\sqrt[3]{2} x^2-3\ 2^{2/3} \sqrt[3]{-1+x^2}-i 2^{2/3} \sqrt{3} x \sqrt[3]{-1+x^2}+6 \left(-1+x^2\right)^{2/3}\right)}{12\ 6^{2/3}}+\frac{\sqrt[3]{54-35 i \sqrt{3}} \log \left(3 \sqrt[3]{2}-2 i \sqrt[3]{2} \sqrt{3} x-\sqrt[3]{2} x^2-3\ 2^{2/3} \sqrt[3]{-1+x^2}+i 2^{2/3} \sqrt{3} x \sqrt[3]{-1+x^2}+6 \left(-1+x^2\right)^{2/3}\right)}{12\ 6^{2/3}}",1,"-1/6*((-253 + (1260*I)*Sqrt[3])^(1/6)*ArcTan[(3*(-1 + x^2)^(1/3))/(-(2^(2/3)*Sqrt[3]) - I*2^(2/3)*x + Sqrt[3]*(-1 + x^2)^(1/3))])/2^(2/3) - ((-253 - (1260*I)*Sqrt[3])^(1/6)*ArcTan[(3*(-1 + x^2)^(1/3))/(-(2^(2/3)*Sqrt[3]) + I*2^(2/3)*x + Sqrt[3]*(-1 + x^2)^(1/3))])/(6*2^(2/3)) + ((-1)^(2/3)*(-54 + (35*I)*Sqrt[3])^(1/3)*Log[3*2^(2/3) - I*2^(2/3)*Sqrt[3]*x + 6*(-1 + x^2)^(1/3)])/(6*6^(2/3)) - ((-1)^(1/3)*(-54 - (35*I)*Sqrt[3])^(1/3)*Log[3*2^(2/3) + I*2^(2/3)*Sqrt[3]*x + 6*(-1 + x^2)^(1/3)])/(6*6^(2/3)) + ((54 + (35*I)*Sqrt[3])^(1/3)*Log[3*2^(1/3) + (2*I)*2^(1/3)*Sqrt[3]*x - 2^(1/3)*x^2 - 3*2^(2/3)*(-1 + x^2)^(1/3) - I*2^(2/3)*Sqrt[3]*x*(-1 + x^2)^(1/3) + 6*(-1 + x^2)^(2/3)])/(12*6^(2/3)) + ((54 - (35*I)*Sqrt[3])^(1/3)*Log[3*2^(1/3) - (2*I)*2^(1/3)*Sqrt[3]*x - 2^(1/3)*x^2 - 3*2^(2/3)*(-1 + x^2)^(1/3) + I*2^(2/3)*Sqrt[3]*x*(-1 + x^2)^(1/3) + 6*(-1 + x^2)^(2/3)])/(12*6^(2/3))","A",1
3073,0,0,501,3.6946464,"\int \frac{x}{\sqrt{1-\sqrt{1-\sqrt{1-\frac{1}{x}}}}} \, dx","IntegrateAlgebraic[x/Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]],x]","\int \frac{x}{\sqrt{1-\sqrt{1-\sqrt{1-\frac{1}{x}}}}} \, dx","\frac{69 \sqrt{1-\sqrt{1-\sqrt{1-\frac{1}{x}}}}+1787 \left(1-\sqrt{1-\sqrt{1-\frac{1}{x}}}\right)^{3/2}-2139 \left(1-\sqrt{1-\sqrt{1-\frac{1}{x}}}\right)^{5/2}-3301 \left(1-\sqrt{1-\sqrt{1-\frac{1}{x}}}\right)^{7/2}+8403 \left(1-\sqrt{1-\sqrt{1-\frac{1}{x}}}\right)^{9/2}-6611 \left(1-\sqrt{1-\sqrt{1-\frac{1}{x}}}\right)^{11/2}+2275 \left(1-\sqrt{1-\sqrt{1-\frac{1}{x}}}\right)^{13/2}-291 \left(1-\sqrt{1-\sqrt{1-\frac{1}{x}}}\right)^{15/2}}{384 \left(-1-2 \left(1-\sqrt{1-\sqrt{1-\frac{1}{x}}}\right)+\left(1-\sqrt{1-\sqrt{1-\frac{1}{x}}}\right)^2\right)^2 \left(1-\sqrt{1-\frac{1}{x}}\right)^2}+\left(-\frac{19}{64} \sqrt{\frac{1}{2} \left(\frac{1}{2}+\frac{1}{\sqrt{2}}\right)}-\frac{11}{64} \sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}}\right) \tan ^{-1}\left(\frac{\sqrt{1-\sqrt{1-\sqrt{1-\frac{1}{x}}}}}{\sqrt{-1+\sqrt{2}}}\right)+\frac{59}{128} \tanh ^{-1}\left(\sqrt{1-\sqrt{1-\sqrt{1-\frac{1}{x}}}}\right)+\left(\frac{19}{64} \sqrt{\frac{1}{2} \left(-\frac{1}{2}+\frac{1}{\sqrt{2}}\right)}-\frac{11}{64} \sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}}\right) \tanh ^{-1}\left(\frac{\sqrt{1-\sqrt{1-\sqrt{1-\frac{1}{x}}}}}{\sqrt{1+\sqrt{2}}}\right)",1,"Could not integrate","F",-1
3074,1,162,501,0.9658231,"\int \frac{\sqrt[4]{-x^2+x^6} \left(1-x^4+x^8\right)}{x^4 \left(1+x^4\right)} \, dx","IntegrateAlgebraic[((-x^2 + x^6)^(1/4)*(1 - x^4 + x^8))/(x^4*(1 + x^4)),x]","\frac{2 \left(-1+x^4\right) \sqrt[4]{-x^2+x^6}}{5 x^3}+\frac{3}{4} \sqrt{1+i} \tan ^{-1}\left(\frac{\sqrt{-1-i} x}{\sqrt[4]{-x^2+x^6}}\right)+\frac{3}{4} \sqrt{1-i} \tan ^{-1}\left(\frac{\sqrt{-1+i} x}{\sqrt[4]{-x^2+x^6}}\right)-\frac{3}{4} \sqrt{-1+i} \tan ^{-1}\left(\frac{\sqrt{1-i} x}{\sqrt[4]{-x^2+x^6}}\right)-\frac{3}{4} \sqrt{-1-i} \tan ^{-1}\left(\frac{\sqrt{1+i} x}{\sqrt[4]{-x^2+x^6}}\right)","\frac{2 \left(-1+x^4\right) \sqrt[4]{-x^2+x^6}}{5 x^3}+\frac{3}{4} \sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x}{-\sqrt{2+\sqrt{2}} x+2^{3/4} \sqrt[4]{-x^2+x^6}}\right)+\frac{3}{4} \sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x}{\sqrt{2+\sqrt{2}} x+2^{3/4} \sqrt[4]{-x^2+x^6}}\right)-\frac{3}{4} \sqrt{\frac{1}{2} \left(-1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{2^{3/4} \sqrt{2+\sqrt{2}} x \sqrt[4]{-x^2+x^6}}{-2 x^2+\sqrt{2} \sqrt{-x^2+x^6}}\right)-\frac{3}{4} \sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{2} x^2}{\sqrt{2-\sqrt{2}}}+\frac{\sqrt{-x^2+x^6}}{\sqrt[4]{2} \sqrt{2-\sqrt{2}}}}{x \sqrt[4]{-x^2+x^6}}\right)-\frac{3}{8} \sqrt{\frac{1}{2} \left(-1+\sqrt{2}\right)} \log \left(-2 x^2+2^{3/4} \sqrt{2+\sqrt{2}} x \sqrt[4]{-x^2+x^6}-\sqrt{2} \sqrt{-x^2+x^6}\right)+\frac{3}{8} \sqrt{\frac{1}{2} \left(-1+\sqrt{2}\right)} \log \left(2 \sqrt{2-\sqrt{2}} x^2+2 \sqrt[4]{2} x \sqrt[4]{-x^2+x^6}+\sqrt{4-2 \sqrt{2}} \sqrt{-x^2+x^6}\right)",1,"(2*(-1 + x^4)*(-x^2 + x^6)^(1/4))/(5*x^3) + (3*Sqrt[1 + I]*ArcTan[(Sqrt[-1 - I]*x)/(-x^2 + x^6)^(1/4)])/4 + (3*Sqrt[1 - I]*ArcTan[(Sqrt[-1 + I]*x)/(-x^2 + x^6)^(1/4)])/4 - (3*Sqrt[-1 + I]*ArcTan[(Sqrt[1 - I]*x)/(-x^2 + x^6)^(1/4)])/4 - (3*Sqrt[-1 - I]*ArcTan[(Sqrt[1 + I]*x)/(-x^2 + x^6)^(1/4)])/4","C",1
3075,1,162,501,0.0012894,"\int \frac{\sqrt[4]{-x^2+x^6} \left(1-x^4+x^8\right)}{x^4 \left(1+x^4\right)} \, dx","IntegrateAlgebraic[((-x^2 + x^6)^(1/4)*(1 - x^4 + x^8))/(x^4*(1 + x^4)),x]","\frac{2 \left(-1+x^4\right) \sqrt[4]{-x^2+x^6}}{5 x^3}+\frac{3}{4} \sqrt{1+i} \tan ^{-1}\left(\frac{\sqrt{-1-i} x}{\sqrt[4]{-x^2+x^6}}\right)+\frac{3}{4} \sqrt{1-i} \tan ^{-1}\left(\frac{\sqrt{-1+i} x}{\sqrt[4]{-x^2+x^6}}\right)-\frac{3}{4} \sqrt{-1+i} \tan ^{-1}\left(\frac{\sqrt{1-i} x}{\sqrt[4]{-x^2+x^6}}\right)-\frac{3}{4} \sqrt{-1-i} \tan ^{-1}\left(\frac{\sqrt{1+i} x}{\sqrt[4]{-x^2+x^6}}\right)","\frac{2 \left(-1+x^4\right) \sqrt[4]{-x^2+x^6}}{5 x^3}+\frac{3}{4} \sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x}{-\sqrt{2+\sqrt{2}} x+2^{3/4} \sqrt[4]{-x^2+x^6}}\right)+\frac{3}{4} \sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x}{\sqrt{2+\sqrt{2}} x+2^{3/4} \sqrt[4]{-x^2+x^6}}\right)-\frac{3}{4} \sqrt{\frac{1}{2} \left(-1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{2^{3/4} \sqrt{2+\sqrt{2}} x \sqrt[4]{-x^2+x^6}}{-2 x^2+\sqrt{2} \sqrt{-x^2+x^6}}\right)-\frac{3}{4} \sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{2} x^2}{\sqrt{2-\sqrt{2}}}+\frac{\sqrt{-x^2+x^6}}{\sqrt[4]{2} \sqrt{2-\sqrt{2}}}}{x \sqrt[4]{-x^2+x^6}}\right)-\frac{3}{8} \sqrt{\frac{1}{2} \left(-1+\sqrt{2}\right)} \log \left(-2 x^2+2^{3/4} \sqrt{2+\sqrt{2}} x \sqrt[4]{-x^2+x^6}-\sqrt{2} \sqrt{-x^2+x^6}\right)+\frac{3}{8} \sqrt{\frac{1}{2} \left(-1+\sqrt{2}\right)} \log \left(2 \sqrt{2-\sqrt{2}} x^2+2 \sqrt[4]{2} x \sqrt[4]{-x^2+x^6}+\sqrt{4-2 \sqrt{2}} \sqrt{-x^2+x^6}\right)",1,"(2*(-1 + x^4)*(-x^2 + x^6)^(1/4))/(5*x^3) + (3*Sqrt[1 + I]*ArcTan[(Sqrt[-1 - I]*x)/(-x^2 + x^6)^(1/4)])/4 + (3*Sqrt[1 - I]*ArcTan[(Sqrt[-1 + I]*x)/(-x^2 + x^6)^(1/4)])/4 - (3*Sqrt[-1 + I]*ArcTan[(Sqrt[1 - I]*x)/(-x^2 + x^6)^(1/4)])/4 - (3*Sqrt[-1 - I]*ArcTan[(Sqrt[1 + I]*x)/(-x^2 + x^6)^(1/4)])/4","C",1
3076,1,162,501,0.917233,"\int \frac{\sqrt[4]{-x^2+x^6} \left(1+x^4+x^8\right)}{x^4 \left(1+x^4\right)} \, dx","IntegrateAlgebraic[((-x^2 + x^6)^(1/4)*(1 + x^4 + x^8))/(x^4*(1 + x^4)),x]","\frac{2 \left(-1+x^4\right) \sqrt[4]{-x^2+x^6}}{5 x^3}+\frac{1}{4} \sqrt{1+i} \tan ^{-1}\left(\frac{\sqrt{-1-i} x}{\sqrt[4]{-x^2+x^6}}\right)+\frac{1}{4} \sqrt{1-i} \tan ^{-1}\left(\frac{\sqrt{-1+i} x}{\sqrt[4]{-x^2+x^6}}\right)-\frac{1}{4} \sqrt{-1+i} \tan ^{-1}\left(\frac{\sqrt{1-i} x}{\sqrt[4]{-x^2+x^6}}\right)-\frac{1}{4} \sqrt{-1-i} \tan ^{-1}\left(\frac{\sqrt{1+i} x}{\sqrt[4]{-x^2+x^6}}\right)","\frac{2 \left(-1+x^4\right) \sqrt[4]{-x^2+x^6}}{5 x^3}+\frac{1}{4} \sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x}{-\sqrt{2+\sqrt{2}} x+2^{3/4} \sqrt[4]{-x^2+x^6}}\right)+\frac{1}{4} \sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x}{\sqrt{2+\sqrt{2}} x+2^{3/4} \sqrt[4]{-x^2+x^6}}\right)-\frac{1}{4} \sqrt{\frac{1}{2} \left(-1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{2^{3/4} \sqrt{2+\sqrt{2}} x \sqrt[4]{-x^2+x^6}}{-2 x^2+\sqrt{2} \sqrt{-x^2+x^6}}\right)-\frac{1}{4} \sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{2} x^2}{\sqrt{2-\sqrt{2}}}+\frac{\sqrt{-x^2+x^6}}{\sqrt[4]{2} \sqrt{2-\sqrt{2}}}}{x \sqrt[4]{-x^2+x^6}}\right)-\frac{1}{8} \sqrt{\frac{1}{2} \left(-1+\sqrt{2}\right)} \log \left(-2 x^2+2^{3/4} \sqrt{2+\sqrt{2}} x \sqrt[4]{-x^2+x^6}-\sqrt{2} \sqrt{-x^2+x^6}\right)+\frac{1}{8} \sqrt{\frac{1}{2} \left(-1+\sqrt{2}\right)} \log \left(2 \sqrt{2-\sqrt{2}} x^2+2 \sqrt[4]{2} x \sqrt[4]{-x^2+x^6}+\sqrt{4-2 \sqrt{2}} \sqrt{-x^2+x^6}\right)",1,"(2*(-1 + x^4)*(-x^2 + x^6)^(1/4))/(5*x^3) + (Sqrt[1 + I]*ArcTan[(Sqrt[-1 - I]*x)/(-x^2 + x^6)^(1/4)])/4 + (Sqrt[1 - I]*ArcTan[(Sqrt[-1 + I]*x)/(-x^2 + x^6)^(1/4)])/4 - (Sqrt[-1 + I]*ArcTan[(Sqrt[1 - I]*x)/(-x^2 + x^6)^(1/4)])/4 - (Sqrt[-1 - I]*ArcTan[(Sqrt[1 + I]*x)/(-x^2 + x^6)^(1/4)])/4","C",1
3077,1,162,501,0.0011319,"\int \frac{\sqrt[4]{-x^2+x^6} \left(1+x^4+x^8\right)}{x^4 \left(1+x^4\right)} \, dx","IntegrateAlgebraic[((-x^2 + x^6)^(1/4)*(1 + x^4 + x^8))/(x^4*(1 + x^4)),x]","\frac{2 \left(-1+x^4\right) \sqrt[4]{-x^2+x^6}}{5 x^3}+\frac{1}{4} \sqrt{1+i} \tan ^{-1}\left(\frac{\sqrt{-1-i} x}{\sqrt[4]{-x^2+x^6}}\right)+\frac{1}{4} \sqrt{1-i} \tan ^{-1}\left(\frac{\sqrt{-1+i} x}{\sqrt[4]{-x^2+x^6}}\right)-\frac{1}{4} \sqrt{-1+i} \tan ^{-1}\left(\frac{\sqrt{1-i} x}{\sqrt[4]{-x^2+x^6}}\right)-\frac{1}{4} \sqrt{-1-i} \tan ^{-1}\left(\frac{\sqrt{1+i} x}{\sqrt[4]{-x^2+x^6}}\right)","\frac{2 \left(-1+x^4\right) \sqrt[4]{-x^2+x^6}}{5 x^3}+\frac{1}{4} \sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x}{-\sqrt{2+\sqrt{2}} x+2^{3/4} \sqrt[4]{-x^2+x^6}}\right)+\frac{1}{4} \sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} x}{\sqrt{2+\sqrt{2}} x+2^{3/4} \sqrt[4]{-x^2+x^6}}\right)-\frac{1}{4} \sqrt{\frac{1}{2} \left(-1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{2^{3/4} \sqrt{2+\sqrt{2}} x \sqrt[4]{-x^2+x^6}}{-2 x^2+\sqrt{2} \sqrt{-x^2+x^6}}\right)-\frac{1}{4} \sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\frac{\sqrt[4]{2} x^2}{\sqrt{2-\sqrt{2}}}+\frac{\sqrt{-x^2+x^6}}{\sqrt[4]{2} \sqrt{2-\sqrt{2}}}}{x \sqrt[4]{-x^2+x^6}}\right)-\frac{1}{8} \sqrt{\frac{1}{2} \left(-1+\sqrt{2}\right)} \log \left(-2 x^2+2^{3/4} \sqrt{2+\sqrt{2}} x \sqrt[4]{-x^2+x^6}-\sqrt{2} \sqrt{-x^2+x^6}\right)+\frac{1}{8} \sqrt{\frac{1}{2} \left(-1+\sqrt{2}\right)} \log \left(2 \sqrt{2-\sqrt{2}} x^2+2 \sqrt[4]{2} x \sqrt[4]{-x^2+x^6}+\sqrt{4-2 \sqrt{2}} \sqrt{-x^2+x^6}\right)",1,"(2*(-1 + x^4)*(-x^2 + x^6)^(1/4))/(5*x^3) + (Sqrt[1 + I]*ArcTan[(Sqrt[-1 - I]*x)/(-x^2 + x^6)^(1/4)])/4 + (Sqrt[1 - I]*ArcTan[(Sqrt[-1 + I]*x)/(-x^2 + x^6)^(1/4)])/4 - (Sqrt[-1 + I]*ArcTan[(Sqrt[1 - I]*x)/(-x^2 + x^6)^(1/4)])/4 - (Sqrt[-1 - I]*ArcTan[(Sqrt[1 + I]*x)/(-x^2 + x^6)^(1/4)])/4","C",1
3078,1,551,506,4.0588613,"\int \frac{\sqrt[3]{b^2 x^2+a^3 x^3}}{-b+a x} \, dx","IntegrateAlgebraic[(b^2*x^2 + a^3*x^3)^(1/3)/(-b + a*x),x]","\frac{\sqrt[3]{b^2 x^2+a^3 x^3}}{a}-\frac{\left(3 a^2 b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{\sqrt{3} a^3}-\frac{\sqrt{-3-3 i \sqrt{3}} b \sqrt[3]{a^2+b} \tan ^{-1}\left(\frac{3 \sqrt[3]{a} \sqrt[3]{a^2+b} x}{\sqrt{3} \sqrt[3]{a} \sqrt[3]{a^2+b} x-3 i \sqrt[3]{b^2 x^2+a^3 x^3}-\sqrt{3} \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{\sqrt{2} a^{5/3}}+\frac{\left(-3 a^2 b-b^2\right) \log \left(-a x+\sqrt[3]{b^2 x^2+a^3 x^3}\right)}{3 a^3}+\frac{i \left(i b \sqrt[3]{a^2+b}+\sqrt{3} b \sqrt[3]{a^2+b}\right) \log \left(2 \sqrt[3]{a} \sqrt[3]{a^2+b} x+\left(1+i \sqrt{3}\right) \sqrt[3]{b^2 x^2+a^3 x^3}\right)}{2 a^{5/3}}+\frac{\left(3 a^2 b+b^2\right) \log \left(a^2 x^2+a x \sqrt[3]{b^2 x^2+a^3 x^3}+\left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{6 a^3}+\frac{\left(b \sqrt[3]{a^2+b}-i \sqrt{3} b \sqrt[3]{a^2+b}\right) \log \left(-2 i a^{2/3} \left(a^2+b\right)^{2/3} x^2+\sqrt[3]{a} \sqrt[3]{a^2+b} \left(i x-\sqrt{3} x\right) \sqrt[3]{b^2 x^2+a^3 x^3}+\left(i+\sqrt{3}\right) \left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{4 a^{5/3}}","\frac{\sqrt[3]{b^2 x^2+a^3 x^3}}{a}-\frac{\left(3 a^2 b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{\sqrt{3} a^3}-\frac{\sqrt{-3-3 i \sqrt{3}} b \sqrt[3]{a^2+b} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a} \sqrt[3]{a^2+b} x}{\sqrt[3]{a} \sqrt[3]{a^2+b} x-2 \sqrt[3]{-1} \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{\sqrt{2} a^{5/3}}+\frac{\left(-3 a^2 b-b^2\right) \log \left(-a x+\sqrt[3]{b^2 x^2+a^3 x^3}\right)}{3 a^3}+\frac{i \left(i b \sqrt[3]{a^2+b}+\sqrt{3} b \sqrt[3]{a^2+b}\right) \log \left(\sqrt[3]{a} \sqrt[3]{a^2+b} x+\sqrt[3]{-1} \sqrt[3]{b^2 x^2+a^3 x^3}\right)}{2 a^{5/3}}+\frac{\left(3 a^2 b+b^2\right) \log \left(a^2 x^2+a x \sqrt[3]{b^2 x^2+a^3 x^3}+\left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{6 a^3}+\frac{\left(b \sqrt[3]{a^2+b}-i \sqrt{3} b \sqrt[3]{a^2+b}\right) \log \left(a^{2/3} \left(a^2+b\right)^{2/3} x^2-\sqrt[3]{-1} \sqrt[3]{a} \sqrt[3]{a^2+b} x \sqrt[3]{b^2 x^2+a^3 x^3}+(-1)^{2/3} \left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{4 a^{5/3}}",1,"(b^2*x^2 + a^3*x^3)^(1/3)/a - ((3*a^2*b + b^2)*ArcTan[(Sqrt[3]*a*x)/(a*x + 2*(b^2*x^2 + a^3*x^3)^(1/3))])/(Sqrt[3]*a^3) - (Sqrt[-3 - (3*I)*Sqrt[3]]*b*(a^2 + b)^(1/3)*ArcTan[(3*a^(1/3)*(a^2 + b)^(1/3)*x)/(Sqrt[3]*a^(1/3)*(a^2 + b)^(1/3)*x - (3*I)*(b^2*x^2 + a^3*x^3)^(1/3) - Sqrt[3]*(b^2*x^2 + a^3*x^3)^(1/3))])/(Sqrt[2]*a^(5/3)) + ((-3*a^2*b - b^2)*Log[-(a*x) + (b^2*x^2 + a^3*x^3)^(1/3)])/(3*a^3) + ((I/2)*(I*b*(a^2 + b)^(1/3) + Sqrt[3]*b*(a^2 + b)^(1/3))*Log[2*a^(1/3)*(a^2 + b)^(1/3)*x + (1 + I*Sqrt[3])*(b^2*x^2 + a^3*x^3)^(1/3)])/a^(5/3) + ((3*a^2*b + b^2)*Log[a^2*x^2 + a*x*(b^2*x^2 + a^3*x^3)^(1/3) + (b^2*x^2 + a^3*x^3)^(2/3)])/(6*a^3) + ((b*(a^2 + b)^(1/3) - I*Sqrt[3]*b*(a^2 + b)^(1/3))*Log[(-2*I)*a^(2/3)*(a^2 + b)^(2/3)*x^2 + a^(1/3)*(a^2 + b)^(1/3)*(I*x - Sqrt[3]*x)*(b^2*x^2 + a^3*x^3)^(1/3) + (I + Sqrt[3])*(b^2*x^2 + a^3*x^3)^(2/3)])/(4*a^(5/3))","A",1
3079,1,507,507,1.4121504,"\int \frac{\sqrt{-b+a^2 x^2}}{\sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}} \, dx","IntegrateAlgebraic[Sqrt[-b + a^2*x^2]/(Sqrt[a*x + Sqrt[-b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]),x]","\frac{\left(945 b^3-4224 b^2 c^4-504 a b^2 c^2 x+3072 a b c^6 x-1890 a^2 b^2 x^2+7680 a^2 b c^4 x^2-4096 a^3 c^6 x^3\right) \sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\left(432 b^2 c^3-2048 b c^7+630 a b^2 c x-2304 a b c^5 x+4096 a^2 c^7 x^2+3072 a^3 c^5 x^3\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\sqrt{-b+a^2 x^2} \left(\left(-504 b^2 c^2+1024 b c^6-1890 a b^2 x+7680 a b c^4 x-4096 a^2 c^6 x^2\right) \sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\left(630 b^2 c-768 b c^5+4096 a c^7 x+3072 a^2 c^5 x^2\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{3840 a c^5 \left(a x+\sqrt{-b+a^2 x^2}\right)^{5/2}}+\frac{63 b^2 \tanh ^{-1}\left(\frac{\sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{c}}\right)}{256 a c^{11/2}}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{c}}\right)}{a c^{3/2}}","\frac{\left(945 b^3-4224 b^2 c^4-504 a b^2 c^2 x+3072 a b c^6 x-1890 a^2 b^2 x^2+7680 a^2 b c^4 x^2-4096 a^3 c^6 x^3\right) \sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\left(432 b^2 c^3-2048 b c^7+630 a b^2 c x-2304 a b c^5 x+4096 a^2 c^7 x^2+3072 a^3 c^5 x^3\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\sqrt{-b+a^2 x^2} \left(\left(-504 b^2 c^2+1024 b c^6-1890 a b^2 x+7680 a b c^4 x-4096 a^2 c^6 x^2\right) \sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\left(630 b^2 c-768 b c^5+4096 a c^7 x+3072 a^2 c^5 x^2\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{3840 a c^5 \left(a x+\sqrt{-b+a^2 x^2}\right)^{5/2}}+\frac{63 b^2 \tanh ^{-1}\left(\frac{\sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{c}}\right)}{256 a c^{11/2}}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{c+\sqrt{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{c}}\right)}{a c^{3/2}}",1,"((945*b^3 - 4224*b^2*c^4 - 504*a*b^2*c^2*x + 3072*a*b*c^6*x - 1890*a^2*b^2*x^2 + 7680*a^2*b*c^4*x^2 - 4096*a^3*c^6*x^3)*Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]] + (432*b^2*c^3 - 2048*b*c^7 + 630*a*b^2*c*x - 2304*a*b*c^5*x + 4096*a^2*c^7*x^2 + 3072*a^3*c^5*x^3)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]] + Sqrt[-b + a^2*x^2]*((-504*b^2*c^2 + 1024*b*c^6 - 1890*a*b^2*x + 7680*a*b*c^4*x - 4096*a^2*c^6*x^2)*Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]] + (630*b^2*c - 768*b*c^5 + 4096*a*c^7*x + 3072*a^2*c^5*x^2)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]))/(3840*a*c^5*(a*x + Sqrt[-b + a^2*x^2])^(5/2)) + (63*b^2*ArcTanh[Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]/Sqrt[c]])/(256*a*c^(11/2)) - (b*ArcTanh[Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]/Sqrt[c]])/(a*c^(3/2))","A",1
3080,1,514,514,1.8235236,"\int \frac{\left(-3+x^2\right) \left(1-2 x^2+x^4+x^6\right)}{x^{10} \sqrt[4]{\frac{-a+a x^2+b x^3}{-c+c x^2+d x^3}}} \, dx","IntegrateAlgebraic[((-3 + x^2)*(1 - 2*x^2 + x^4 + x^6))/(x^10*((-a + a*x^2 + b*x^3)/(-c + c*x^2 + d*x^3))^(1/4)),x]","\frac{\left(\frac{-a+a x^2+b x^3}{-c+c x^2+d x^3}\right)^{3/4} \left(32 a^2 c^3-96 a^2 c^3 x^2+36 a b c^3 x^3-36 a^2 c^2 d x^3+96 a^2 c^3 x^4-72 a b c^3 x^5+72 a^2 c^2 d x^5+64 a^2 c^3 x^6+45 b^2 c^3 x^6-42 a b c^2 d x^6-3 a^2 c d^2 x^6+36 a b c^3 x^7-36 a^2 c^2 d x^7-96 a^2 c^3 x^8-45 b^2 c^3 x^8+42 a b c^2 d x^8+3 a^2 c d^2 x^8-96 a^2 c^2 d x^9-45 b^2 c^2 d x^9+6 a b c d^2 x^9+7 a^2 d^3 x^9\right)}{96 a^3 c^2 x^9}+\frac{\left(-32 a^2 b c^3-15 b^3 c^3+32 a^3 c^2 d+5 a b^2 c^2 d+3 a^2 b c d^2+7 a^3 d^3\right) \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt[4]{\frac{-a+a x^2+b x^3}{-c+c x^2+d x^3}}}{\sqrt[4]{a}}\right)}{64 a^{13/4} c^{11/4}}+\frac{\left(32 a^2 b c^3+15 b^3 c^3-32 a^3 c^2 d-5 a b^2 c^2 d-3 a^2 b c d^2-7 a^3 d^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt[4]{\frac{-a+a x^2+b x^3}{-c+c x^2+d x^3}}}{\sqrt[4]{a}}\right)}{64 a^{13/4} c^{11/4}}","\frac{\left(\frac{-a+a x^2+b x^3}{-c+c x^2+d x^3}\right)^{3/4} \left(32 a^2 c^3-96 a^2 c^3 x^2+36 a b c^3 x^3-36 a^2 c^2 d x^3+96 a^2 c^3 x^4-72 a b c^3 x^5+72 a^2 c^2 d x^5+64 a^2 c^3 x^6+45 b^2 c^3 x^6-42 a b c^2 d x^6-3 a^2 c d^2 x^6+36 a b c^3 x^7-36 a^2 c^2 d x^7-96 a^2 c^3 x^8-45 b^2 c^3 x^8+42 a b c^2 d x^8+3 a^2 c d^2 x^8-96 a^2 c^2 d x^9-45 b^2 c^2 d x^9+6 a b c d^2 x^9+7 a^2 d^3 x^9\right)}{96 a^3 c^2 x^9}+\frac{\left(-32 a^2 b c^3-15 b^3 c^3+32 a^3 c^2 d+5 a b^2 c^2 d+3 a^2 b c d^2+7 a^3 d^3\right) \tan ^{-1}\left(\frac{\sqrt[4]{c} \sqrt[4]{\frac{-a+a x^2+b x^3}{-c+c x^2+d x^3}}}{\sqrt[4]{a}}\right)}{64 a^{13/4} c^{11/4}}+\frac{\left(32 a^2 b c^3+15 b^3 c^3-32 a^3 c^2 d-5 a b^2 c^2 d-3 a^2 b c d^2-7 a^3 d^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{c} \sqrt[4]{\frac{-a+a x^2+b x^3}{-c+c x^2+d x^3}}}{\sqrt[4]{a}}\right)}{64 a^{13/4} c^{11/4}}",1,"(((-a + a*x^2 + b*x^3)/(-c + c*x^2 + d*x^3))^(3/4)*(32*a^2*c^3 - 96*a^2*c^3*x^2 + 36*a*b*c^3*x^3 - 36*a^2*c^2*d*x^3 + 96*a^2*c^3*x^4 - 72*a*b*c^3*x^5 + 72*a^2*c^2*d*x^5 + 64*a^2*c^3*x^6 + 45*b^2*c^3*x^6 - 42*a*b*c^2*d*x^6 - 3*a^2*c*d^2*x^6 + 36*a*b*c^3*x^7 - 36*a^2*c^2*d*x^7 - 96*a^2*c^3*x^8 - 45*b^2*c^3*x^8 + 42*a*b*c^2*d*x^8 + 3*a^2*c*d^2*x^8 - 96*a^2*c^2*d*x^9 - 45*b^2*c^2*d*x^9 + 6*a*b*c*d^2*x^9 + 7*a^2*d^3*x^9))/(96*a^3*c^2*x^9) + ((-32*a^2*b*c^3 - 15*b^3*c^3 + 32*a^3*c^2*d + 5*a*b^2*c^2*d + 3*a^2*b*c*d^2 + 7*a^3*d^3)*ArcTan[(c^(1/4)*((-a + a*x^2 + b*x^3)/(-c + c*x^2 + d*x^3))^(1/4))/a^(1/4)])/(64*a^(13/4)*c^(11/4)) + ((32*a^2*b*c^3 + 15*b^3*c^3 - 32*a^3*c^2*d - 5*a*b^2*c^2*d - 3*a^2*b*c*d^2 - 7*a^3*d^3)*ArcTanh[(c^(1/4)*((-a + a*x^2 + b*x^3)/(-c + c*x^2 + d*x^3))^(1/4))/a^(1/4)])/(64*a^(13/4)*c^(11/4))","A",1
3081,1,514,514,0.5696327,"\int \frac{\left(1+x^4\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{1-x^4} \, dx","IntegrateAlgebraic[((1 + x^4)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^4),x]","\frac{\left(75+8 x-60 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+(-16-6 x) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left((8-60 x) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-6 \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{105 \sqrt{x+\sqrt{1+x^2}}}+\tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)-\frac{1}{2} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}-3 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]+\text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^2}\&\right]-\frac{1}{2} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^3+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^5}{-2+4 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]","\frac{\left(75+8 x-60 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+(-16-6 x) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left((8-60 x) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-6 \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{105 \sqrt{x+\sqrt{1+x^2}}}+\tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)-\frac{1}{2} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}-3 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]+\text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^2}\&\right]-\frac{1}{2} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^3+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^5}{-2+4 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]",1,"((75 + 8*x - 60*x^2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-16 - 6*x)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((8 - 60*x)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - 6*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(105*Sqrt[x + Sqrt[1 + x^2]]) + ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1 - 3*#1^3 + #1^5) & ]/2 + RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(-1 + #1^2) & ] - RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^3 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^5)/(-2 + 4*#1^2 - 3*#1^4 + #1^6) & ]/2","A",1
3082,1,514,514,0.0022155,"\int \frac{\left(1+x^4\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{1-x^4} \, dx","IntegrateAlgebraic[((1 + x^4)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^4),x]","\frac{\left(75+8 x-60 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+(-16-6 x) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left((8-60 x) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-6 \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{105 \sqrt{x+\sqrt{1+x^2}}}+\tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)-\frac{1}{2} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}-3 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]+\text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^2}\&\right]-\frac{1}{2} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^3+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^5}{-2+4 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]","\frac{\left(75+8 x-60 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+(-16-6 x) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left((8-60 x) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-6 \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{105 \sqrt{x+\sqrt{1+x^2}}}+\tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)-\frac{1}{2} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}-3 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]+\text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{-1+\text{$\#$1}^2}\&\right]-\frac{1}{2} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^3+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^5}{-2+4 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]",1,"((75 + 8*x - 60*x^2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-16 - 6*x)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((8 - 60*x)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - 6*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(105*Sqrt[x + Sqrt[1 + x^2]]) + ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1 - 3*#1^3 + #1^5) & ]/2 + RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(-1 + #1^2) & ] - RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^3 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^5)/(-2 + 4*#1^2 - 3*#1^4 + #1^6) & ]/2","A",1
3083,1,515,515,1.2049491,"\int \sqrt{b+a^2 x^2} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}} \, dx","IntegrateAlgebraic[Sqrt[b + a^2*x^2]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]],x]","\frac{\left(80080 b^2 c^3-2048 b c^7+1050 a b^2 c x-2304 a b c^5 x+197120 a^2 b c^3 x^2-4096 a^2 c^7 x^2-3072 a^3 c^5 x^3+35840 a^4 c^3 x^4\right) \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+\left(-840 b^2 c^2+1024 b c^6-1575 a b^2 x+1920 a b c^4 x+2048 a^2 c^6 x^2+2560 a^3 c^4 x^3\right) \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+\sqrt{b+a^2 x^2} \left(\left(1050 b^2 c-768 b c^5+179200 a b c^3 x-4096 a c^7 x-3072 a^2 c^5 x^2+35840 a^3 c^3 x^3\right) \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+\left(-1575 b^2+640 b c^4+2048 a c^6 x+2560 a^2 c^4 x^2\right) \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}\right)}{80640 a^2 c^3 x \sqrt{b+a^2 x^2}+40320 a c^3 \left(b+2 a^2 x^2\right)}+\frac{5 b^2 \tanh ^{-1}\left(\frac{\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{\sqrt{c}}\right)}{128 a c^{7/2}}-\frac{2 b \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{\sqrt{c}}\right)}{a}","\frac{\left(80080 b^2 c^3-2048 b c^7+1050 a b^2 c x-2304 a b c^5 x+197120 a^2 b c^3 x^2-4096 a^2 c^7 x^2-3072 a^3 c^5 x^3+35840 a^4 c^3 x^4\right) \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+\left(-840 b^2 c^2+1024 b c^6-1575 a b^2 x+1920 a b c^4 x+2048 a^2 c^6 x^2+2560 a^3 c^4 x^3\right) \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+\sqrt{b+a^2 x^2} \left(\left(1050 b^2 c-768 b c^5+179200 a b c^3 x-4096 a c^7 x-3072 a^2 c^5 x^2+35840 a^3 c^3 x^3\right) \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+\left(-1575 b^2+640 b c^4+2048 a c^6 x+2560 a^2 c^4 x^2\right) \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}\right)}{80640 a^2 c^3 x \sqrt{b+a^2 x^2}+40320 a c^3 \left(b+2 a^2 x^2\right)}+\frac{5 b^2 \tanh ^{-1}\left(\frac{\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{\sqrt{c}}\right)}{128 a c^{7/2}}-\frac{2 b \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{\sqrt{c}}\right)}{a}",1,"((80080*b^2*c^3 - 2048*b*c^7 + 1050*a*b^2*c*x - 2304*a*b*c^5*x + 197120*a^2*b*c^3*x^2 - 4096*a^2*c^7*x^2 - 3072*a^3*c^5*x^3 + 35840*a^4*c^3*x^4)*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] + (-840*b^2*c^2 + 1024*b*c^6 - 1575*a*b^2*x + 1920*a*b*c^4*x + 2048*a^2*c^6*x^2 + 2560*a^3*c^4*x^3)*Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] + Sqrt[b + a^2*x^2]*((1050*b^2*c - 768*b*c^5 + 179200*a*b*c^3*x - 4096*a*c^7*x - 3072*a^2*c^5*x^2 + 35840*a^3*c^3*x^3)*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] + (-1575*b^2 + 640*b*c^4 + 2048*a*c^6*x + 2560*a^2*c^4*x^2)*Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]))/(80640*a^2*c^3*x*Sqrt[b + a^2*x^2] + 40320*a*c^3*(b + 2*a^2*x^2)) + (5*b^2*ArcTanh[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/Sqrt[c]])/(128*a*c^(7/2)) - (2*b*Sqrt[c]*ArcTanh[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/Sqrt[c]])/a","A",1
3084,1,518,518,0.4349569,"\int \frac{\sqrt{\left(-81+27 x+135 x^2-150 x^3+65 x^4-13 x^5+x^6\right)^3}}{-1+x} \, dx","IntegrateAlgebraic[Sqrt[(-81 + 27*x + 135*x^2 - 150*x^3 + 65*x^4 - 13*x^5 + x^6)^3]/(-1 + x),x]","\frac{\left(-1245336401+4423205098 x-508033624 x^2-2700564848 x^3+2304529024 x^4-910869760 x^5+199009280 x^6-23296000 x^7+1146880 x^8\right) \sqrt{-531441+531441 x+2480058 x^2-4704237 x^3-885735 x^4+9880866 x^5-10219851 x^6+592677 x^7+8764767 x^8-10819710 x^9+7498953 x^{10}-3554163 x^{11}+1221371 x^{12}-309774 x^{13}+57735 x^{14}-7717 x^{15}+702 x^{16}-39 x^{17}+x^{18}}}{10321920 (-3+x)^6 \left(-1-x+x^2\right)}+128 \tan ^{-1}\left(\frac{\left(-1-x+x^2\right) \left(729-1458 x+1215 x^2-540 x^3+135 x^4-18 x^5+x^6\right)}{729-1458 x-243 x^2+3105 x^3-3753 x^4+2277 x^5-809 x^6+171 x^7-20 x^8+x^9-\sqrt{-531441+531441 x+2480058 x^2-4704237 x^3-885735 x^4+9880866 x^5-10219851 x^6+592677 x^7+8764767 x^8-10819710 x^9+7498953 x^{10}-3554163 x^{11}+1221371 x^{12}-309774 x^{13}+57735 x^{14}-7717 x^{15}+702 x^{16}-39 x^{17}+x^{18}}}\right)-\frac{19451047 \log \left(-729+729 x+972 x^2-2133 x^3+1620 x^4-657 x^5+152 x^6-19 x^7+x^8\right)}{65536}+\frac{19451047 \log \left(-729+2187 x-486 x^2-4077 x^3+5886 x^4-3897 x^5+1466 x^6-323 x^7+39 x^8-2 x^9+2 \sqrt{-531441+531441 x+2480058 x^2-4704237 x^3-885735 x^4+9880866 x^5-10219851 x^6+592677 x^7+8764767 x^8-10819710 x^9+7498953 x^{10}-3554163 x^{11}+1221371 x^{12}-309774 x^{13}+57735 x^{14}-7717 x^{15}+702 x^{16}-39 x^{17}+x^{18}}\right)}{65536}","\frac{\left(-1245336401+4423205098 x-508033624 x^2-2700564848 x^3+2304529024 x^4-910869760 x^5+199009280 x^6-23296000 x^7+1146880 x^8\right) \sqrt{-531441+531441 x+2480058 x^2-4704237 x^3-885735 x^4+9880866 x^5-10219851 x^6+592677 x^7+8764767 x^8-10819710 x^9+7498953 x^{10}-3554163 x^{11}+1221371 x^{12}-309774 x^{13}+57735 x^{14}-7717 x^{15}+702 x^{16}-39 x^{17}+x^{18}}}{10321920 (-3+x)^6 \left(-1-x+x^2\right)}+128 \tan ^{-1}\left(\frac{\left(-1-x+x^2\right) \left(729-1458 x+1215 x^2-540 x^3+135 x^4-18 x^5+x^6\right)}{729-1458 x-243 x^2+3105 x^3-3753 x^4+2277 x^5-809 x^6+171 x^7-20 x^8+x^9-\sqrt{-531441+531441 x+2480058 x^2-4704237 x^3-885735 x^4+9880866 x^5-10219851 x^6+592677 x^7+8764767 x^8-10819710 x^9+7498953 x^{10}-3554163 x^{11}+1221371 x^{12}-309774 x^{13}+57735 x^{14}-7717 x^{15}+702 x^{16}-39 x^{17}+x^{18}}}\right)-\frac{19451047 \log \left(-729+729 x+972 x^2-2133 x^3+1620 x^4-657 x^5+152 x^6-19 x^7+x^8\right)}{65536}+\frac{19451047 \log \left(-729+2187 x-486 x^2-4077 x^3+5886 x^4-3897 x^5+1466 x^6-323 x^7+39 x^8-2 x^9+2 \sqrt{-531441+531441 x+2480058 x^2-4704237 x^3-885735 x^4+9880866 x^5-10219851 x^6+592677 x^7+8764767 x^8-10819710 x^9+7498953 x^{10}-3554163 x^{11}+1221371 x^{12}-309774 x^{13}+57735 x^{14}-7717 x^{15}+702 x^{16}-39 x^{17}+x^{18}}\right)}{65536}",1,"((-1245336401 + 4423205098*x - 508033624*x^2 - 2700564848*x^3 + 2304529024*x^4 - 910869760*x^5 + 199009280*x^6 - 23296000*x^7 + 1146880*x^8)*Sqrt[-531441 + 531441*x + 2480058*x^2 - 4704237*x^3 - 885735*x^4 + 9880866*x^5 - 10219851*x^6 + 592677*x^7 + 8764767*x^8 - 10819710*x^9 + 7498953*x^10 - 3554163*x^11 + 1221371*x^12 - 309774*x^13 + 57735*x^14 - 7717*x^15 + 702*x^16 - 39*x^17 + x^18])/(10321920*(-3 + x)^6*(-1 - x + x^2)) + 128*ArcTan[((-1 - x + x^2)*(729 - 1458*x + 1215*x^2 - 540*x^3 + 135*x^4 - 18*x^5 + x^6))/(729 - 1458*x - 243*x^2 + 3105*x^3 - 3753*x^4 + 2277*x^5 - 809*x^6 + 171*x^7 - 20*x^8 + x^9 - Sqrt[-531441 + 531441*x + 2480058*x^2 - 4704237*x^3 - 885735*x^4 + 9880866*x^5 - 10219851*x^6 + 592677*x^7 + 8764767*x^8 - 10819710*x^9 + 7498953*x^10 - 3554163*x^11 + 1221371*x^12 - 309774*x^13 + 57735*x^14 - 7717*x^15 + 702*x^16 - 39*x^17 + x^18])] - (19451047*Log[-729 + 729*x + 972*x^2 - 2133*x^3 + 1620*x^4 - 657*x^5 + 152*x^6 - 19*x^7 + x^8])/65536 + (19451047*Log[-729 + 2187*x - 486*x^2 - 4077*x^3 + 5886*x^4 - 3897*x^5 + 1466*x^6 - 323*x^7 + 39*x^8 - 2*x^9 + 2*Sqrt[-531441 + 531441*x + 2480058*x^2 - 4704237*x^3 - 885735*x^4 + 9880866*x^5 - 10219851*x^6 + 592677*x^7 + 8764767*x^8 - 10819710*x^9 + 7498953*x^10 - 3554163*x^11 + 1221371*x^12 - 309774*x^13 + 57735*x^14 - 7717*x^15 + 702*x^16 - 39*x^17 + x^18]])/65536","A",1
3085,1,828,520,1.1715668,"\int \frac{\left(1+x^2+x^4\right)^2 \sqrt{x^2+\sqrt{1+x^4}}}{\sqrt{1+x^4} \left(-1+x^2+x^4\right)^2} \, dx","IntegrateAlgebraic[((1 + x^2 + x^4)^2*Sqrt[x^2 + Sqrt[1 + x^4]])/(Sqrt[1 + x^4]*(-1 + x^2 + x^4)^2),x]","\frac{-2 x \sqrt{1+x^4} \left(2+3 x^2+2 x^4\right) \sqrt{x^2+\sqrt{1+x^4}}-2 x \left(1+3 x^2+3 x^4+2 x^6\right) \sqrt{x^2+\sqrt{1+x^4}}}{10 x^2 \sqrt{1+x^4} \left(-1+x^2+x^4\right)+5 \left(-1+x^2+x^4\right) \left(1+2 x^4\right)}+\sqrt{2} \tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^4}}{\sqrt{2}}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)+2 \sqrt{2} \text{RootSum}\left[1-2 \text{$\#$1}^2-6 \text{$\#$1}^4+2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-8 \log \left(1+x^2+\sqrt{1+x^4}\right)+8 \log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right)+3 \log \left(1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^2-3 \log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^2-\log \left(1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^4+\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}-6 \text{$\#$1}^3+3 \text{$\#$1}^5+2 \text{$\#$1}^7}\&\right]+\frac{\text{RootSum}\left[1-2 \text{$\#$1}^2-6 \text{$\#$1}^4+2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{163 \log \left(1+x^2+\sqrt{1+x^4}\right)-163 \log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right)-59 \log \left(1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^2+59 \log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^2+13 \log \left(1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^4-13 \log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^4-\log \left(1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^6+\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^6}{-\text{$\#$1}-6 \text{$\#$1}^3+3 \text{$\#$1}^5+2 \text{$\#$1}^7}\&\right]}{5 \sqrt{2}}","\frac{-2 x \sqrt{1+x^4} \left(2+3 x^2+2 x^4\right) \sqrt{x^2+\sqrt{1+x^4}}-2 x \left(1+3 x^2+3 x^4+2 x^6\right) \sqrt{x^2+\sqrt{1+x^4}}}{10 x^2 \sqrt{1+x^4} \left(-1+x^2+x^4\right)+5 \left(-1+x^2+x^4\right) \left(1+2 x^4\right)}+\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-\frac{\text{RootSum}\left[1+2 \text{$\#$1}^2-6 \text{$\#$1}^4-2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(-1+x^2+\sqrt{1+x^4}\right)-\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}+x^2 \text{$\#$1}+\sqrt{1+x^4} \text{$\#$1}\right)+7 \log \left(-1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^2-7 \log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}+x^2 \text{$\#$1}+\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^2-\log \left(-1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^4+\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}+x^2 \text{$\#$1}+\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^4-3 \log \left(-1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^6+3 \log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}+x^2 \text{$\#$1}+\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^6}{\text{$\#$1}-6 \text{$\#$1}^3-3 \text{$\#$1}^5+2 \text{$\#$1}^7}\&\right]}{5 \sqrt{2}}",1,"(-2*x*Sqrt[1 + x^4]*(2 + 3*x^2 + 2*x^4)*Sqrt[x^2 + Sqrt[1 + x^4]] - 2*x*(1 + 3*x^2 + 3*x^4 + 2*x^6)*Sqrt[x^2 + Sqrt[1 + x^4]])/(10*x^2*Sqrt[1 + x^4]*(-1 + x^2 + x^4) + 5*(-1 + x^2 + x^4)*(1 + 2*x^4)) + Sqrt[2]*ArcTanh[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] + 2*Sqrt[2]*RootSum[1 - 2*#1^2 - 6*#1^4 + 2*#1^6 + #1^8 & , (-8*Log[1 + x^2 + Sqrt[1 + x^4]] + 8*Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1] + 3*Log[1 + x^2 + Sqrt[1 + x^4]]*#1^2 - 3*Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^2 - Log[1 + x^2 + Sqrt[1 + x^4]]*#1^4 + Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^4)/(-#1 - 6*#1^3 + 3*#1^5 + 2*#1^7) & ] + RootSum[1 - 2*#1^2 - 6*#1^4 + 2*#1^6 + #1^8 & , (163*Log[1 + x^2 + Sqrt[1 + x^4]] - 163*Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1] - 59*Log[1 + x^2 + Sqrt[1 + x^4]]*#1^2 + 59*Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^2 + 13*Log[1 + x^2 + Sqrt[1 + x^4]]*#1^4 - 13*Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^4 - Log[1 + x^2 + Sqrt[1 + x^4]]*#1^6 + Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^6)/(-#1 - 6*#1^3 + 3*#1^5 + 2*#1^7) & ]/(5*Sqrt[2])","A",1
3086,1,828,520,0.0017413,"\int \frac{\left(1+x^2+x^4\right)^2 \sqrt{x^2+\sqrt{1+x^4}}}{\sqrt{1+x^4} \left(-1+x^2+x^4\right)^2} \, dx","IntegrateAlgebraic[((1 + x^2 + x^4)^2*Sqrt[x^2 + Sqrt[1 + x^4]])/(Sqrt[1 + x^4]*(-1 + x^2 + x^4)^2),x]","\frac{-2 x \sqrt{1+x^4} \left(2+3 x^2+2 x^4\right) \sqrt{x^2+\sqrt{1+x^4}}-2 x \left(1+3 x^2+3 x^4+2 x^6\right) \sqrt{x^2+\sqrt{1+x^4}}}{10 x^2 \sqrt{1+x^4} \left(-1+x^2+x^4\right)+5 \left(-1+x^2+x^4\right) \left(1+2 x^4\right)}+\sqrt{2} \tanh ^{-1}\left(\frac{-\frac{1}{\sqrt{2}}+\frac{x^2}{\sqrt{2}}+\frac{\sqrt{1+x^4}}{\sqrt{2}}}{x \sqrt{x^2+\sqrt{1+x^4}}}\right)+2 \sqrt{2} \text{RootSum}\left[1-2 \text{$\#$1}^2-6 \text{$\#$1}^4+2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-8 \log \left(1+x^2+\sqrt{1+x^4}\right)+8 \log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right)+3 \log \left(1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^2-3 \log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^2-\log \left(1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^4+\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^4}{-\text{$\#$1}-6 \text{$\#$1}^3+3 \text{$\#$1}^5+2 \text{$\#$1}^7}\&\right]+\frac{\text{RootSum}\left[1-2 \text{$\#$1}^2-6 \text{$\#$1}^4+2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{163 \log \left(1+x^2+\sqrt{1+x^4}\right)-163 \log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right)-59 \log \left(1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^2+59 \log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^2+13 \log \left(1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^4-13 \log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^4-\log \left(1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^6+\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}-x^2 \text{$\#$1}-\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^6}{-\text{$\#$1}-6 \text{$\#$1}^3+3 \text{$\#$1}^5+2 \text{$\#$1}^7}\&\right]}{5 \sqrt{2}}","\frac{-2 x \sqrt{1+x^4} \left(2+3 x^2+2 x^4\right) \sqrt{x^2+\sqrt{1+x^4}}-2 x \left(1+3 x^2+3 x^4+2 x^6\right) \sqrt{x^2+\sqrt{1+x^4}}}{10 x^2 \sqrt{1+x^4} \left(-1+x^2+x^4\right)+5 \left(-1+x^2+x^4\right) \left(1+2 x^4\right)}+\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)-\frac{\text{RootSum}\left[1+2 \text{$\#$1}^2-6 \text{$\#$1}^4-2 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(-1+x^2+\sqrt{1+x^4}\right)-\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}+x^2 \text{$\#$1}+\sqrt{1+x^4} \text{$\#$1}\right)+7 \log \left(-1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^2-7 \log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}+x^2 \text{$\#$1}+\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^2-\log \left(-1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^4+\log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}+x^2 \text{$\#$1}+\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^4-3 \log \left(-1+x^2+\sqrt{1+x^4}\right) \text{$\#$1}^6+3 \log \left(\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}-\text{$\#$1}+x^2 \text{$\#$1}+\sqrt{1+x^4} \text{$\#$1}\right) \text{$\#$1}^6}{\text{$\#$1}-6 \text{$\#$1}^3-3 \text{$\#$1}^5+2 \text{$\#$1}^7}\&\right]}{5 \sqrt{2}}",1,"(-2*x*Sqrt[1 + x^4]*(2 + 3*x^2 + 2*x^4)*Sqrt[x^2 + Sqrt[1 + x^4]] - 2*x*(1 + 3*x^2 + 3*x^4 + 2*x^6)*Sqrt[x^2 + Sqrt[1 + x^4]])/(10*x^2*Sqrt[1 + x^4]*(-1 + x^2 + x^4) + 5*(-1 + x^2 + x^4)*(1 + 2*x^4)) + Sqrt[2]*ArcTanh[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] + 2*Sqrt[2]*RootSum[1 - 2*#1^2 - 6*#1^4 + 2*#1^6 + #1^8 & , (-8*Log[1 + x^2 + Sqrt[1 + x^4]] + 8*Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1] + 3*Log[1 + x^2 + Sqrt[1 + x^4]]*#1^2 - 3*Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^2 - Log[1 + x^2 + Sqrt[1 + x^4]]*#1^4 + Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^4)/(-#1 - 6*#1^3 + 3*#1^5 + 2*#1^7) & ] + RootSum[1 - 2*#1^2 - 6*#1^4 + 2*#1^6 + #1^8 & , (163*Log[1 + x^2 + Sqrt[1 + x^4]] - 163*Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1] - 59*Log[1 + x^2 + Sqrt[1 + x^4]]*#1^2 + 59*Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^2 + 13*Log[1 + x^2 + Sqrt[1 + x^4]]*#1^4 - 13*Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^4 - Log[1 + x^2 + Sqrt[1 + x^4]]*#1^6 + Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^6)/(-#1 - 6*#1^3 + 3*#1^5 + 2*#1^7) & ]/(5*Sqrt[2])","A",1
3087,1,54,524,0.8814206,"\int \frac{\left(-q+p x^4\right) \sqrt{q+p x^4}}{x^2 \left(a q+b x^2+a p x^4\right)} \, dx","IntegrateAlgebraic[((-q + p*x^4)*Sqrt[q + p*x^4])/(x^2*(a*q + b*x^2 + a*p*x^4)),x]","\frac{\sqrt{q+p x^4}}{a x}+\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a} \sqrt{q+p x^4}}\right)}{a^{3/2}}","\frac{\sqrt{q+p x^4}}{a x}+\frac{\left(\sqrt{2} b \sqrt{b+i \sqrt{b} \sqrt{-b+2 a \sqrt{p} \sqrt{q}}-a \sqrt{p} \sqrt{q}}-i \sqrt{2} \sqrt{b} \sqrt{b+i \sqrt{b} \sqrt{-b+2 a \sqrt{p} \sqrt{q}}-a \sqrt{p} \sqrt{q}} \sqrt{-b+2 a \sqrt{p} \sqrt{q}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b+i \sqrt{b} \sqrt{-b+2 a \sqrt{p} \sqrt{q}}-a \sqrt{p} \sqrt{q}} x}{\sqrt{a} \left(\sqrt{q}+\sqrt{p} x^2+\sqrt{q+p x^4}\right)}\right)}{2 a^{5/2} \sqrt{p} \sqrt{q}}-\frac{i \left(-i \sqrt{2} b \sqrt{-b+i \sqrt{b} \sqrt{-b+2 a \sqrt{p} \sqrt{q}}+a \sqrt{p} \sqrt{q}}+\sqrt{2} \sqrt{b} \sqrt{-b+i \sqrt{b} \sqrt{-b+2 a \sqrt{p} \sqrt{q}}+a \sqrt{p} \sqrt{q}} \sqrt{-b+2 a \sqrt{p} \sqrt{q}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{-b+i \sqrt{b} \sqrt{-b+2 a \sqrt{p} \sqrt{q}}+a \sqrt{p} \sqrt{q}} x}{\sqrt{a} \left(\sqrt{q}+\sqrt{p} x^2+\sqrt{q+p x^4}\right)}\right)}{2 a^{5/2} \sqrt{p} \sqrt{q}}",1,"Sqrt[q + p*x^4]/(a*x) + (Sqrt[b]*ArcTan[(Sqrt[b]*x)/(Sqrt[a]*Sqrt[q + p*x^4])])/a^(3/2)","A",1
3088,1,571,526,4.145281,"\int \frac{\sqrt[3]{b^2 x^2+a^3 x^3}}{b+a x} \, dx","IntegrateAlgebraic[(b^2*x^2 + a^3*x^3)^(1/3)/(b + a*x),x]","\frac{\sqrt[3]{b^2 x^2+a^3 x^3}}{a}+\frac{\left(3 a^2 b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{\sqrt{3} a^3}+\frac{\sqrt{-3-3 i \sqrt{3}} \sqrt[3]{a^2-b} b \tan ^{-1}\left(\frac{3 \sqrt[3]{a} \sqrt[3]{a^2-b} x}{\sqrt{3} \sqrt[3]{a} \sqrt[3]{a^2-b} x-3 i \sqrt[3]{b^2 x^2+a^3 x^3}-\sqrt{3} \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{\sqrt{2} a^{5/3}}+\frac{\left(3 a^2 b-b^2\right) \log \left(-a x+\sqrt[3]{b^2 x^2+a^3 x^3}\right)}{3 a^3}+\frac{\left(\sqrt[3]{a^2-b} b-i \sqrt{3} \sqrt[3]{a^2-b} b\right) \log \left(2 \sqrt[3]{a} \sqrt[3]{a^2-b} x+\left(1+i \sqrt{3}\right) \sqrt[3]{b^2 x^2+a^3 x^3}\right)}{2 a^{5/3}}+\frac{\left(-3 a^2 b+b^2\right) \log \left(a^2 x^2+a x \sqrt[3]{b^2 x^2+a^3 x^3}+\left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{6 a^3}+\frac{i \left(i \sqrt[3]{a^2-b} b+\sqrt{3} \sqrt[3]{a^2-b} b\right) \log \left(-2 i a^{2/3} \left(a^2-b\right)^{2/3} x^2+\sqrt[3]{a} \sqrt[3]{a^2-b} \left(i x-\sqrt{3} x\right) \sqrt[3]{b^2 x^2+a^3 x^3}+\left(i+\sqrt{3}\right) \left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{4 a^{5/3}}","\frac{\sqrt[3]{b^2 x^2+a^3 x^3}}{a}+\frac{\left(3 a^2 b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{3} a x}{a x+2 \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{\sqrt{3} a^3}+\frac{\sqrt{-3-3 i \sqrt{3}} \sqrt[3]{a^2-b} b \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a} \sqrt[3]{a^2-b} x}{\sqrt[3]{a} \sqrt[3]{a^2-b} x-2 \sqrt[3]{-1} \sqrt[3]{b^2 x^2+a^3 x^3}}\right)}{\sqrt{2} a^{5/3}}+\frac{\left(3 a^2 b-b^2\right) \log \left(-a x+\sqrt[3]{b^2 x^2+a^3 x^3}\right)}{3 a^3}+\frac{\left(\sqrt[3]{a^2-b} b-i \sqrt{3} \sqrt[3]{a^2-b} b\right) \log \left(\sqrt[3]{a} \sqrt[3]{a^2-b} x+\sqrt[3]{-1} \sqrt[3]{b^2 x^2+a^3 x^3}\right)}{2 a^{5/3}}+\frac{\left(-3 a^2 b+b^2\right) \log \left(a^2 x^2+a x \sqrt[3]{b^2 x^2+a^3 x^3}+\left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{6 a^3}+\frac{i \left(i \sqrt[3]{a^2-b} b+\sqrt{3} \sqrt[3]{a^2-b} b\right) \log \left(a^{2/3} \left(a^2-b\right)^{2/3} x^2-\sqrt[3]{-1} \sqrt[3]{a} \sqrt[3]{a^2-b} x \sqrt[3]{b^2 x^2+a^3 x^3}+(-1)^{2/3} \left(b^2 x^2+a^3 x^3\right)^{2/3}\right)}{4 a^{5/3}}",1,"(b^2*x^2 + a^3*x^3)^(1/3)/a + ((3*a^2*b - b^2)*ArcTan[(Sqrt[3]*a*x)/(a*x + 2*(b^2*x^2 + a^3*x^3)^(1/3))])/(Sqrt[3]*a^3) + (Sqrt[-3 - (3*I)*Sqrt[3]]*(a^2 - b)^(1/3)*b*ArcTan[(3*a^(1/3)*(a^2 - b)^(1/3)*x)/(Sqrt[3]*a^(1/3)*(a^2 - b)^(1/3)*x - (3*I)*(b^2*x^2 + a^3*x^3)^(1/3) - Sqrt[3]*(b^2*x^2 + a^3*x^3)^(1/3))])/(Sqrt[2]*a^(5/3)) + ((3*a^2*b - b^2)*Log[-(a*x) + (b^2*x^2 + a^3*x^3)^(1/3)])/(3*a^3) + (((a^2 - b)^(1/3)*b - I*Sqrt[3]*(a^2 - b)^(1/3)*b)*Log[2*a^(1/3)*(a^2 - b)^(1/3)*x + (1 + I*Sqrt[3])*(b^2*x^2 + a^3*x^3)^(1/3)])/(2*a^(5/3)) + ((-3*a^2*b + b^2)*Log[a^2*x^2 + a*x*(b^2*x^2 + a^3*x^3)^(1/3) + (b^2*x^2 + a^3*x^3)^(2/3)])/(6*a^3) + ((I/4)*(I*(a^2 - b)^(1/3)*b + Sqrt[3]*(a^2 - b)^(1/3)*b)*Log[(-2*I)*a^(2/3)*(a^2 - b)^(2/3)*x^2 + a^(1/3)*(a^2 - b)^(1/3)*(I*x - Sqrt[3]*x)*(b^2*x^2 + a^3*x^3)^(1/3) + (I + Sqrt[3])*(b^2*x^2 + a^3*x^3)^(2/3)])/a^(5/3)","A",1
3089,1,501,526,1.7838477,"\int \frac{\left(-b+a^2 x^2\right)^{3/2} \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{x} \, dx","IntegrateAlgebraic[((-b + a^2*x^2)^(3/2)*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/x,x]","\frac{4 \sqrt{-b+a^2 x^2} \left(418 a b^2 x-561 a^3 b x^3+132 a^5 x^5\right)+4 \left(-152 b^3+682 a^2 b^2 x^2-627 a^4 b x^4+132 a^6 x^6\right)}{429 \left(a x+\sqrt{-b+a^2 x^2}\right)^{11/4}}-\sqrt{2+\sqrt{2}} b^{13/8} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{b} \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[4]{b}+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)+\sqrt{2-\sqrt{2}} b^{13/8} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{b} \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[4]{b}+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)-\sqrt{2-\sqrt{2}} b^{13/8} \tanh ^{-1}\left(\frac{\sqrt{1-\frac{1}{\sqrt{2}}} \sqrt[8]{b}+\frac{\sqrt{1-\frac{1}{\sqrt{2}}} \sqrt{a x+\sqrt{-b+a^2 x^2}}}{\sqrt[8]{b}}}{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)+\sqrt{2+\sqrt{2}} b^{13/8} \tanh ^{-1}\left(\frac{\sqrt{1+\frac{1}{\sqrt{2}}} \sqrt[8]{b}+\frac{\sqrt{1+\frac{1}{\sqrt{2}}} \sqrt{a x+\sqrt{-b+a^2 x^2}}}{\sqrt[8]{b}}}{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)","\frac{4 \sqrt{-b+a^2 x^2} \left(418 a b^2 x-561 a^3 b x^3+132 a^5 x^5\right)+4 \left(-152 b^3+682 a^2 b^2 x^2-627 a^4 b x^4+132 a^6 x^6\right)}{429 \left(a x+\sqrt{-b+a^2 x^2}\right)^{11/4}}+\sqrt{2+\sqrt{2}} b^{13/8} \tan ^{-1}\left(\frac{\left(\sqrt{\frac{2}{2-\sqrt{2}}} \sqrt[8]{b}-\frac{2 \sqrt[8]{b}}{\sqrt{2-\sqrt{2}}}\right) \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[4]{b}+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)+\sqrt{2-\sqrt{2}} b^{13/8} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{b} \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[4]{b}+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)+\sqrt{2+\sqrt{2}} b^{13/8} \tanh ^{-1}\left(\frac{\frac{\sqrt[8]{b}}{\sqrt{2-\sqrt{2}}}+\frac{\sqrt{a x+\sqrt{-b+a^2 x^2}}}{\sqrt{2-\sqrt{2}} \sqrt[8]{b}}}{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)-\sqrt{2-\sqrt{2}} b^{13/8} \tanh ^{-1}\left(\frac{\frac{\sqrt[8]{b}}{\sqrt{2+\sqrt{2}}}+\frac{\sqrt{a x+\sqrt{-b+a^2 x^2}}}{\sqrt{2+\sqrt{2}} \sqrt[8]{b}}}{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)",1,"(4*Sqrt[-b + a^2*x^2]*(418*a*b^2*x - 561*a^3*b*x^3 + 132*a^5*x^5) + 4*(-152*b^3 + 682*a^2*b^2*x^2 - 627*a^4*b*x^4 + 132*a^6*x^6))/(429*(a*x + Sqrt[-b + a^2*x^2])^(11/4)) - Sqrt[2 + Sqrt[2]]*b^(13/8)*ArcTan[(Sqrt[2 - Sqrt[2]]*b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(-b^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]])] + Sqrt[2 - Sqrt[2]]*b^(13/8)*ArcTan[(Sqrt[2 + Sqrt[2]]*b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(-b^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]])] - Sqrt[2 - Sqrt[2]]*b^(13/8)*ArcTanh[(Sqrt[1 - 1/Sqrt[2]]*b^(1/8) + (Sqrt[1 - 1/Sqrt[2]]*Sqrt[a*x + Sqrt[-b + a^2*x^2]])/b^(1/8))/(a*x + Sqrt[-b + a^2*x^2])^(1/4)] + Sqrt[2 + Sqrt[2]]*b^(13/8)*ArcTanh[(Sqrt[1 + 1/Sqrt[2]]*b^(1/8) + (Sqrt[1 + 1/Sqrt[2]]*Sqrt[a*x + Sqrt[-b + a^2*x^2]])/b^(1/8))/(a*x + Sqrt[-b + a^2*x^2])^(1/4)]","A",1
3090,1,195,530,0.5102072,"\int \frac{\left(b^2+a x^2\right) \sqrt{b+\sqrt{b^2+a x^2}}}{-b^2+a x^2} \, dx","IntegrateAlgebraic[((b^2 + a*x^2)*Sqrt[b + Sqrt[b^2 + a*x^2]])/(-b^2 + a*x^2),x]","\frac{4 b x}{3 \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{2 x \sqrt{b^2+a x^2}}{3 \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{2 \sqrt{-1+\sqrt{2}} b^{3/2} \tan ^{-1}\left(\frac{\sqrt{-1+\sqrt{2}} \sqrt{a} x}{\sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{\sqrt{a}}-\frac{2 \sqrt{1+\sqrt{2}} b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{1+\sqrt{2}} \sqrt{a} x}{\sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{\sqrt{a}}","\frac{4 b x}{3 \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{2 x \sqrt{b^2+a x^2}}{3 \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{2 i \left(\sqrt{-1+\sqrt{2}} b^{3/2}+\sqrt{2 \left(-1+\sqrt{2}\right)} b^{3/2}\right) \tan ^{-1}\left(\frac{\frac{i a x}{\sqrt{b+\sqrt{b^2+a x^2}}}-i \sqrt{a} \sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{a} \sqrt{b}}\right)}{\sqrt{a}}+\frac{2 \sqrt{-1+\sqrt{2}} b^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{b}}\right)}{\sqrt{a}}-\frac{2 i \left(-\sqrt{1+\sqrt{2}} b^{3/2}+\sqrt{2 \left(1+\sqrt{2}\right)} b^{3/2}\right) \tanh ^{-1}\left(\frac{\frac{i a x}{\sqrt{b+\sqrt{b^2+a x^2}}}-i \sqrt{a} \sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2 \left(-1+\sqrt{2}\right)} \sqrt{a} \sqrt{b}}\right)}{\sqrt{a}}+\frac{2 \sqrt{1+\sqrt{2}} b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2 \left(-1+\sqrt{2}\right)} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2 \left(-1+\sqrt{2}\right)} \sqrt{b}}\right)}{\sqrt{a}}",1,"(4*b*x)/(3*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (2*x*Sqrt[b^2 + a*x^2])/(3*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (2*Sqrt[-1 + Sqrt[2]]*b^(3/2)*ArcTan[(Sqrt[-1 + Sqrt[2]]*Sqrt[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/Sqrt[a] - (2*Sqrt[1 + Sqrt[2]]*b^(3/2)*ArcTanh[(Sqrt[1 + Sqrt[2]]*Sqrt[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/Sqrt[a]","A",1
3091,1,510,535,1.9509556,"\int \frac{d+c x^4}{x \sqrt{-b+a^2 x^2} \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}} \, dx","IntegrateAlgebraic[(d + c*x^4)/(x*Sqrt[-b + a^2*x^2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4)),x]","\frac{4 c \sqrt{-b+a^2 x^2} \left(-416 a b^2 x+455 a^3 b x^3+260 a^5 x^5\right)+4 c \left(128 b^3-676 a^2 b^2 x^2+325 a^4 b x^4+260 a^6 x^6\right)}{715 a^4 \left(a x+\sqrt{-b+a^2 x^2}\right)^{13/4}}-\frac{\sqrt{2+\sqrt{2}} d \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{b} \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[4]{b}+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{b^{5/8}}+\frac{\sqrt{2-\sqrt{2}} d \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{b} \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[4]{b}+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{b^{5/8}}+\frac{\sqrt{2-\sqrt{2}} d \tanh ^{-1}\left(\frac{\sqrt{1-\frac{1}{\sqrt{2}}} \sqrt[8]{b}+\frac{\sqrt{1-\frac{1}{\sqrt{2}}} \sqrt{a x+\sqrt{-b+a^2 x^2}}}{\sqrt[8]{b}}}{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{b^{5/8}}-\frac{\sqrt{2+\sqrt{2}} d \tanh ^{-1}\left(\frac{\sqrt{1+\frac{1}{\sqrt{2}}} \sqrt[8]{b}+\frac{\sqrt{1+\frac{1}{\sqrt{2}}} \sqrt{a x+\sqrt{-b+a^2 x^2}}}{\sqrt[8]{b}}}{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{b^{5/8}}","\frac{4 c \sqrt{-b+a^2 x^2} \left(-416 a b^2 x+455 a^3 b x^3+260 a^5 x^5\right)+4 c \left(128 b^3-676 a^2 b^2 x^2+325 a^4 b x^4+260 a^6 x^6\right)}{715 a^4 \left(a x+\sqrt{-b+a^2 x^2}\right)^{13/4}}+\frac{\sqrt{2+\sqrt{2}} d \tan ^{-1}\left(\frac{\left(\sqrt{\frac{2}{2-\sqrt{2}}} \sqrt[8]{b}-\frac{2 \sqrt[8]{b}}{\sqrt{2-\sqrt{2}}}\right) \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[4]{b}+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{b^{5/8}}+\frac{\sqrt{2-\sqrt{2}} d \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{b} \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[4]{b}+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{b^{5/8}}-\frac{\sqrt{2+\sqrt{2}} d \tanh ^{-1}\left(\frac{\frac{\sqrt[8]{b}}{\sqrt{2-\sqrt{2}}}+\frac{\sqrt{a x+\sqrt{-b+a^2 x^2}}}{\sqrt{2-\sqrt{2}} \sqrt[8]{b}}}{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{b^{5/8}}+\frac{\sqrt{2-\sqrt{2}} d \tanh ^{-1}\left(\frac{\frac{\sqrt[8]{b}}{\sqrt{2+\sqrt{2}}}+\frac{\sqrt{a x+\sqrt{-b+a^2 x^2}}}{\sqrt{2+\sqrt{2}} \sqrt[8]{b}}}{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{b^{5/8}}",1,"(4*c*Sqrt[-b + a^2*x^2]*(-416*a*b^2*x + 455*a^3*b*x^3 + 260*a^5*x^5) + 4*c*(128*b^3 - 676*a^2*b^2*x^2 + 325*a^4*b*x^4 + 260*a^6*x^6))/(715*a^4*(a*x + Sqrt[-b + a^2*x^2])^(13/4)) - (Sqrt[2 + Sqrt[2]]*d*ArcTan[(Sqrt[2 - Sqrt[2]]*b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(-b^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]])])/b^(5/8) + (Sqrt[2 - Sqrt[2]]*d*ArcTan[(Sqrt[2 + Sqrt[2]]*b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(-b^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]])])/b^(5/8) + (Sqrt[2 - Sqrt[2]]*d*ArcTanh[(Sqrt[1 - 1/Sqrt[2]]*b^(1/8) + (Sqrt[1 - 1/Sqrt[2]]*Sqrt[a*x + Sqrt[-b + a^2*x^2]])/b^(1/8))/(a*x + Sqrt[-b + a^2*x^2])^(1/4)])/b^(5/8) - (Sqrt[2 + Sqrt[2]]*d*ArcTanh[(Sqrt[1 + 1/Sqrt[2]]*b^(1/8) + (Sqrt[1 + 1/Sqrt[2]]*Sqrt[a*x + Sqrt[-b + a^2*x^2]])/b^(1/8))/(a*x + Sqrt[-b + a^2*x^2])^(1/4)])/b^(5/8)","A",1
3092,1,514,540,2.0337819,"\int \frac{\left(-b+a^2 x^2\right)^{3/2} \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{x^2} \, dx","IntegrateAlgebraic[((-b + a^2*x^2)^(3/2)*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/x^2,x]","\frac{63 b^3+1034 a^2 b^2 x^2-1652 a^4 b x^4+112 a^6 x^6+\sqrt{-b+a^2 x^2} \left(250 a b^2 x-1596 a^3 b x^3+112 a^5 x^5\right)}{63 x \left(a x+\sqrt{-b+a^2 x^2}\right)^{11/4}}-\frac{1}{4} \sqrt{2-\sqrt{2}} a b^{9/8} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{b} \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[4]{b}+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)-\frac{1}{4} \sqrt{2+\sqrt{2}} a b^{9/8} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{b} \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[4]{b}+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)+\frac{1}{4} \sqrt{2+\sqrt{2}} a b^{9/8} \tanh ^{-1}\left(\frac{\sqrt{1-\frac{1}{\sqrt{2}}} \sqrt[8]{b}+\frac{\sqrt{1-\frac{1}{\sqrt{2}}} \sqrt{a x+\sqrt{-b+a^2 x^2}}}{\sqrt[8]{b}}}{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)+\frac{1}{4} \sqrt{2-\sqrt{2}} a b^{9/8} \tanh ^{-1}\left(\frac{\sqrt{1+\frac{1}{\sqrt{2}}} \sqrt[8]{b}+\frac{\sqrt{1+\frac{1}{\sqrt{2}}} \sqrt{a x+\sqrt{-b+a^2 x^2}}}{\sqrt[8]{b}}}{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)","\frac{63 b^3+1034 a^2 b^2 x^2-1652 a^4 b x^4+112 a^6 x^6+\sqrt{-b+a^2 x^2} \left(250 a b^2 x-1596 a^3 b x^3+112 a^5 x^5\right)}{63 x \left(a x+\sqrt{-b+a^2 x^2}\right)^{11/4}}+\frac{1}{4} \sqrt{2-\sqrt{2}} a b^{9/8} \tan ^{-1}\left(\frac{\left(\sqrt{\frac{2}{2-\sqrt{2}}} \sqrt[8]{b}-\frac{2 \sqrt[8]{b}}{\sqrt{2-\sqrt{2}}}\right) \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[4]{b}+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)-\frac{1}{4} \sqrt{2+\sqrt{2}} a b^{9/8} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{b} \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[4]{b}+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)+\frac{1}{4} \sqrt{2-\sqrt{2}} a b^{9/8} \tanh ^{-1}\left(\frac{\frac{\sqrt[8]{b}}{\sqrt{2-\sqrt{2}}}+\frac{\sqrt{a x+\sqrt{-b+a^2 x^2}}}{\sqrt{2-\sqrt{2}} \sqrt[8]{b}}}{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)+\frac{1}{4} \sqrt{2+\sqrt{2}} a b^{9/8} \tanh ^{-1}\left(\frac{\frac{\sqrt[8]{b}}{\sqrt{2+\sqrt{2}}}+\frac{\sqrt{a x+\sqrt{-b+a^2 x^2}}}{\sqrt{2+\sqrt{2}} \sqrt[8]{b}}}{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)",1,"(63*b^3 + 1034*a^2*b^2*x^2 - 1652*a^4*b*x^4 + 112*a^6*x^6 + Sqrt[-b + a^2*x^2]*(250*a*b^2*x - 1596*a^3*b*x^3 + 112*a^5*x^5))/(63*x*(a*x + Sqrt[-b + a^2*x^2])^(11/4)) - (Sqrt[2 - Sqrt[2]]*a*b^(9/8)*ArcTan[(Sqrt[2 - Sqrt[2]]*b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(-b^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]])])/4 - (Sqrt[2 + Sqrt[2]]*a*b^(9/8)*ArcTan[(Sqrt[2 + Sqrt[2]]*b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(-b^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]])])/4 + (Sqrt[2 + Sqrt[2]]*a*b^(9/8)*ArcTanh[(Sqrt[1 - 1/Sqrt[2]]*b^(1/8) + (Sqrt[1 - 1/Sqrt[2]]*Sqrt[a*x + Sqrt[-b + a^2*x^2]])/b^(1/8))/(a*x + Sqrt[-b + a^2*x^2])^(1/4)])/4 + (Sqrt[2 - Sqrt[2]]*a*b^(9/8)*ArcTanh[(Sqrt[1 + 1/Sqrt[2]]*b^(1/8) + (Sqrt[1 + 1/Sqrt[2]]*Sqrt[a*x + Sqrt[-b + a^2*x^2]])/b^(1/8))/(a*x + Sqrt[-b + a^2*x^2])^(1/4)])/4","A",1
3093,1,541,541,3.7817839,"\int \frac{b-x}{\sqrt[3]{(-a+x) (-b+x)^2} \left(a^2-b^2 d-2 (a-b d) x+(1-d) x^2\right)} \, dx","IntegrateAlgebraic[(b - x)/(((-a + x)*(-b + x)^2)^(1/3)*(a^2 - b^2*d - 2*(a - b*d)*x + (1 - d)*x^2)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} (a-b)^{2/3} \sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{-2 a (-a+b)^{2/3}+2 (-a+b)^{2/3} x+(a-b)^{2/3} \sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}\right)}{2 \sqrt[3]{a-b} (-a+b)^{2/3} d^{2/3}}-\frac{\log \left(a (-a+b)^{2/3}-(-a+b)^{2/3} x+(a-b)^{2/3} \sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}\right)}{2 \sqrt[3]{a-b} (-a+b)^{2/3} d^{2/3}}+\frac{\log \left(a^3 \sqrt[3]{-a+b}-a^2 b \sqrt[3]{-a+b}-2 a^2 \sqrt[3]{-a+b} x+2 a b \sqrt[3]{-a+b} x+a \sqrt[3]{-a+b} x^2-b \sqrt[3]{-a+b} x^2+\left(a (a-b)^{2/3} (-a+b)^{2/3} \sqrt[3]{d}-(a-b)^{2/3} (-a+b)^{2/3} \sqrt[3]{d} x\right) \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}+\left(-a \sqrt[3]{a-b} d^{2/3}+\sqrt[3]{a-b} b d^{2/3}\right) \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}\right)}{4 \sqrt[3]{a-b} (-a+b)^{2/3} d^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} (a-b)^{2/3} \sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{-2 a (-a+b)^{2/3}+2 (-a+b)^{2/3} x+(a-b)^{2/3} \sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}\right)}{2 \sqrt[3]{a-b} (-a+b)^{2/3} d^{2/3}}-\frac{\log \left(a (-a+b)^{2/3}-(-a+b)^{2/3} x+(a-b)^{2/3} \sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}\right)}{2 \sqrt[3]{a-b} (-a+b)^{2/3} d^{2/3}}+\frac{\log \left(a^3 \sqrt[3]{-a+b}-a^2 b \sqrt[3]{-a+b}-2 a^2 \sqrt[3]{-a+b} x+2 a b \sqrt[3]{-a+b} x+a \sqrt[3]{-a+b} x^2-b \sqrt[3]{-a+b} x^2+\left(a (a-b)^{2/3} (-a+b)^{2/3} \sqrt[3]{d}-(a-b)^{2/3} (-a+b)^{2/3} \sqrt[3]{d} x\right) \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}+\left(-a \sqrt[3]{a-b} d^{2/3}+\sqrt[3]{a-b} b d^{2/3}\right) \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}\right)}{4 \sqrt[3]{a-b} (-a+b)^{2/3} d^{2/3}}",1,"-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*(a - b)^(2/3)*d^(1/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3))/(-2*a*(-a + b)^(2/3) + 2*(-a + b)^(2/3)*x + (a - b)^(2/3)*d^(1/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3))])/((a - b)^(1/3)*(-a + b)^(2/3)*d^(2/3)) - Log[a*(-a + b)^(2/3) - (-a + b)^(2/3)*x + (a - b)^(2/3)*d^(1/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3)]/(2*(a - b)^(1/3)*(-a + b)^(2/3)*d^(2/3)) + Log[a^3*(-a + b)^(1/3) - a^2*b*(-a + b)^(1/3) - 2*a^2*(-a + b)^(1/3)*x + 2*a*b*(-a + b)^(1/3)*x + a*(-a + b)^(1/3)*x^2 - b*(-a + b)^(1/3)*x^2 + (a*(a - b)^(2/3)*(-a + b)^(2/3)*d^(1/3) - (a - b)^(2/3)*(-a + b)^(2/3)*d^(1/3)*x)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3) + (-(a*(a - b)^(1/3)*d^(2/3)) + (a - b)^(1/3)*b*d^(2/3))*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(2/3)]/(4*(a - b)^(1/3)*(-a + b)^(2/3)*d^(2/3))","A",1
3094,1,541,541,3.6530832,"\int \frac{-b+x}{\sqrt[3]{(-a+x) (-b+x)^2} \left(-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2\right)} \, dx","IntegrateAlgebraic[(-b + x)/(((-a + x)*(-b + x)^2)^(1/3)*(-a^2 + b^2*d + 2*(a - b*d)*x + (-1 + d)*x^2)),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} (a-b)^{2/3} \sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{-2 a (-a+b)^{2/3}+2 (-a+b)^{2/3} x+(a-b)^{2/3} \sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}\right)}{2 \sqrt[3]{a-b} (-a+b)^{2/3} d^{2/3}}-\frac{\log \left(a (-a+b)^{2/3}-(-a+b)^{2/3} x+(a-b)^{2/3} \sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}\right)}{2 \sqrt[3]{a-b} (-a+b)^{2/3} d^{2/3}}+\frac{\log \left(a^3 \sqrt[3]{-a+b}-a^2 b \sqrt[3]{-a+b}-2 a^2 \sqrt[3]{-a+b} x+2 a b \sqrt[3]{-a+b} x+a \sqrt[3]{-a+b} x^2-b \sqrt[3]{-a+b} x^2+\left(a (a-b)^{2/3} (-a+b)^{2/3} \sqrt[3]{d}-(a-b)^{2/3} (-a+b)^{2/3} \sqrt[3]{d} x\right) \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}+\left(-a \sqrt[3]{a-b} d^{2/3}+\sqrt[3]{a-b} b d^{2/3}\right) \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}\right)}{4 \sqrt[3]{a-b} (-a+b)^{2/3} d^{2/3}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt{3} (a-b)^{2/3} \sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{-2 a (-a+b)^{2/3}+2 (-a+b)^{2/3} x+(a-b)^{2/3} \sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}\right)}{2 \sqrt[3]{a-b} (-a+b)^{2/3} d^{2/3}}-\frac{\log \left(a (-a+b)^{2/3}-(-a+b)^{2/3} x+(a-b)^{2/3} \sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}\right)}{2 \sqrt[3]{a-b} (-a+b)^{2/3} d^{2/3}}+\frac{\log \left(a^3 \sqrt[3]{-a+b}-a^2 b \sqrt[3]{-a+b}-2 a^2 \sqrt[3]{-a+b} x+2 a b \sqrt[3]{-a+b} x+a \sqrt[3]{-a+b} x^2-b \sqrt[3]{-a+b} x^2+\left(a (a-b)^{2/3} (-a+b)^{2/3} \sqrt[3]{d}-(a-b)^{2/3} (-a+b)^{2/3} \sqrt[3]{d} x\right) \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}+\left(-a \sqrt[3]{a-b} d^{2/3}+\sqrt[3]{a-b} b d^{2/3}\right) \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}\right)}{4 \sqrt[3]{a-b} (-a+b)^{2/3} d^{2/3}}",1,"-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*(a - b)^(2/3)*d^(1/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3))/(-2*a*(-a + b)^(2/3) + 2*(-a + b)^(2/3)*x + (a - b)^(2/3)*d^(1/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3))])/((a - b)^(1/3)*(-a + b)^(2/3)*d^(2/3)) - Log[a*(-a + b)^(2/3) - (-a + b)^(2/3)*x + (a - b)^(2/3)*d^(1/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3)]/(2*(a - b)^(1/3)*(-a + b)^(2/3)*d^(2/3)) + Log[a^3*(-a + b)^(1/3) - a^2*b*(-a + b)^(1/3) - 2*a^2*(-a + b)^(1/3)*x + 2*a*b*(-a + b)^(1/3)*x + a*(-a + b)^(1/3)*x^2 - b*(-a + b)^(1/3)*x^2 + (a*(a - b)^(2/3)*(-a + b)^(2/3)*d^(1/3) - (a - b)^(2/3)*(-a + b)^(2/3)*d^(1/3)*x)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3) + (-(a*(a - b)^(1/3)*d^(2/3)) + (a - b)^(1/3)*b*d^(2/3))*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(2/3)]/(4*(a - b)^(1/3)*(-a + b)^(2/3)*d^(2/3))","A",1
3095,1,251,541,0.7369978,"\int \frac{b^2+a x^2}{\left(-b^2+a x^2\right) \sqrt{b+\sqrt{b^2+a x^2}}} \, dx","IntegrateAlgebraic[(b^2 + a*x^2)/((-b^2 + a*x^2)*Sqrt[b + Sqrt[b^2 + a*x^2]]),x]","\frac{2 x}{\sqrt{b+\sqrt{b^2+a x^2}}}+\frac{\sqrt{2} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{\sqrt{a}}-\frac{2 \left(\sqrt{-1+\sqrt{2}} \sqrt{b}+\sqrt{2 \left(-1+\sqrt{2}\right)} \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{-1+\sqrt{2}} \sqrt{a} x}{\sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{\sqrt{a}}-\frac{2 \left(-\sqrt{1+\sqrt{2}} \sqrt{b}+\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt{1+\sqrt{2}} \sqrt{a} x}{\sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{\sqrt{a}}","\frac{2 x}{\sqrt{b+\sqrt{b^2+a x^2}}}+\frac{2 \sqrt{2} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2} \sqrt{b}}\right)}{\sqrt{a}}-\frac{2 \sqrt{1+\sqrt{2}} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2 \left(-1+\sqrt{2}\right)} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2 \left(-1+\sqrt{2}\right)} \sqrt{b}}\right)}{\sqrt{a}}-\frac{2 \sqrt{1+\sqrt{2}} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{b}}\right)}{\sqrt{a}}+\frac{2 \sqrt{-1+\sqrt{2}} \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2 \left(-1+\sqrt{2}\right)} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2 \left(-1+\sqrt{2}\right)} \sqrt{b}}\right)}{\sqrt{a}}-\frac{2 \sqrt{-1+\sqrt{2}} \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{b}}\right)}{\sqrt{a}}",1,"(2*x)/Sqrt[b + Sqrt[b^2 + a*x^2]] + (Sqrt[2]*Sqrt[b]*ArcTan[(Sqrt[a]*x)/(Sqrt[2]*Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/Sqrt[a] - (2*(Sqrt[-1 + Sqrt[2]]*Sqrt[b] + Sqrt[2*(-1 + Sqrt[2])]*Sqrt[b])*ArcTan[(Sqrt[-1 + Sqrt[2]]*Sqrt[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/Sqrt[a] - (2*(-(Sqrt[1 + Sqrt[2]]*Sqrt[b]) + Sqrt[2*(1 + Sqrt[2])]*Sqrt[b])*ArcTanh[(Sqrt[1 + Sqrt[2]]*Sqrt[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/Sqrt[a]","A",1
3096,1,703,543,7.7163122,"\int \frac{1}{\sqrt[3]{(-a+x) (-b+x)^2} (a-b d+(-1+d) x)} \, dx","IntegrateAlgebraic[1/(((-a + x)*(-b + x)^2)^(1/3)*(a - b*d + (-1 + d)*x)),x]","\frac{\sqrt{-3-3 i \sqrt{3}} \sqrt[3]{d} \tan ^{-1}\left(\frac{\sqrt{3} (a-b)^{2/3} \sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{b (-a d+b d)^{2/3}+i \sqrt{3} b (-a d+b d)^{2/3}+\left(-(-a d+b d)^{2/3}-i \sqrt{3} (-a d+b d)^{2/3}\right) x+(a-b)^{2/3} \sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}\right)}{\sqrt{2} \sqrt[3]{a-b} (-((a-b) d))^{2/3}}+\frac{\left(\sqrt[3]{d}-i \sqrt{3} \sqrt[3]{d}\right) \log \left((-a d+b d)^{2/3} \left(-b+\sqrt{3} (-i b+i x)+x\right)+2 (a-b)^{2/3} \sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}\right)}{2 \sqrt[3]{a-b} (-((a-b) d))^{2/3}}+\frac{i \left(i \sqrt[3]{d}+\sqrt{3} \sqrt[3]{d}\right) \log \left(2 (a-b)^{4/3} d^{2/3} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}+\sqrt[3]{-a d+b d} \left(d \left(a b^2-b^3-2 a b x+2 b^2 x+a x^2-b x^2\right)+\sqrt{3} d \left(-i a b^2+i b^3+2 i a b x-2 i b^2 x-i a x^2+i b x^2\right)\right)+(-a d+b d)^{2/3} \left((a-b)^{2/3} \sqrt[3]{d} (b-x) \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}+\sqrt{3} (a-b)^{2/3} \sqrt[3]{d} (i b-i x) \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}\right)\right)}{4 \sqrt[3]{a-b} (-((a-b) d))^{2/3}}","\frac{\sqrt{-3-3 i \sqrt{3}} \sqrt[3]{d} \tan ^{-1}\left(\frac{\sqrt{3} (a-b)^{2/3} \sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{2 \sqrt[3]{-1} b (-a d+b d)^{2/3}-2 \sqrt[3]{-1} (-a d+b d)^{2/3} x+(a-b)^{2/3} \sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}\right)}{\sqrt{2} \sqrt[3]{a-b} (-((a-b) d))^{2/3}}+\frac{\left(\sqrt[3]{d}-i \sqrt{3} \sqrt[3]{d}\right) \log \left(\sqrt[3]{-1} (-a d+b d)^{2/3} (b-x)-(a-b)^{2/3} \sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}\right)}{2 \sqrt[3]{a-b} (-((a-b) d))^{2/3}}+\frac{i \left(i \sqrt[3]{d}+\sqrt{3} \sqrt[3]{d}\right) \log \left((-1)^{2/3} d \sqrt[3]{-a d+b d} \left(a b^2-b^3-2 a b x+2 b^2 x+a x^2-b x^2\right)+\sqrt[3]{-1} (a-b)^{2/3} \sqrt[3]{d} (-a d+b d)^{2/3} (-b+x) \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}-(a-b)^{4/3} d^{2/3} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}\right)}{4 \sqrt[3]{a-b} (-((a-b) d))^{2/3}}",1,"(Sqrt[-3 - (3*I)*Sqrt[3]]*d^(1/3)*ArcTan[(Sqrt[3]*(a - b)^(2/3)*d^(1/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3))/(b*(-(a*d) + b*d)^(2/3) + I*Sqrt[3]*b*(-(a*d) + b*d)^(2/3) + (-(-(a*d) + b*d)^(2/3) - I*Sqrt[3]*(-(a*d) + b*d)^(2/3))*x + (a - b)^(2/3)*d^(1/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3))])/(Sqrt[2]*(a - b)^(1/3)*(-((a - b)*d))^(2/3)) + ((d^(1/3) - I*Sqrt[3]*d^(1/3))*Log[(-(a*d) + b*d)^(2/3)*(-b + Sqrt[3]*((-I)*b + I*x) + x) + 2*(a - b)^(2/3)*d^(1/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3)])/(2*(a - b)^(1/3)*(-((a - b)*d))^(2/3)) + ((I/4)*(I*d^(1/3) + Sqrt[3]*d^(1/3))*Log[2*(a - b)^(4/3)*d^(2/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(2/3) + (-(a*d) + b*d)^(1/3)*(d*(a*b^2 - b^3 - 2*a*b*x + 2*b^2*x + a*x^2 - b*x^2) + Sqrt[3]*d*((-I)*a*b^2 + I*b^3 + (2*I)*a*b*x - (2*I)*b^2*x - I*a*x^2 + I*b*x^2)) + (-(a*d) + b*d)^(2/3)*((a - b)^(2/3)*d^(1/3)*(b - x)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3) + Sqrt[3]*(a - b)^(2/3)*d^(1/3)*(I*b - I*x)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3))])/((a - b)^(1/3)*(-((a - b)*d))^(2/3))","A",1
3097,1,544,549,1.1828524,"\int \frac{\sqrt{b+a^2 x^4} \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{d+c x^2} \, dx","IntegrateAlgebraic[(Sqrt[b + a^2*x^4]*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/(d + c*x^2),x]","\frac{a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{2 c}+\frac{\sqrt{a} \sqrt{b} \tan ^{-1}\left(\frac{a x^2}{\sqrt{b}}+\frac{\sqrt{b+a^2 x^4}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{b}}\right)}{\sqrt{2} c}+\frac{a^{3/2} d \log \left(a x^2+\sqrt{b+a^2 x^4}-\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{\sqrt{2} c^2}-\frac{\sqrt{a} \text{RootSum}\left[b^2 c-4 a b d \text{$\#$1}-2 b c \text{$\#$1}^2-4 a d \text{$\#$1}^3+c \text{$\#$1}^4\&,\frac{b^2 c^2 \log \left(a x^2+\sqrt{b+a^2 x^4}-\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\text{$\#$1}\right)+a^2 b d^2 \log \left(a x^2+\sqrt{b+a^2 x^4}-\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\text{$\#$1}\right)+b c^2 \log \left(a x^2+\sqrt{b+a^2 x^4}-\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\text{$\#$1}\right) \text{$\#$1}^2+a^2 d^2 \log \left(a x^2+\sqrt{b+a^2 x^4}-\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\text{$\#$1}\right) \text{$\#$1}^2}{a b d+b c \text{$\#$1}+3 a d \text{$\#$1}^2-c \text{$\#$1}^3}\&\right]}{\sqrt{2} c^2}","\frac{a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{2 c}-\frac{\sqrt{a} \sqrt{b} \tan ^{-1}\left(\frac{a x^2}{\sqrt{b}}+\frac{\sqrt{b+a^2 x^4}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{b}}\right)}{\sqrt{2} c}-\frac{a^{3/2} d \log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{\sqrt{2} c^2}-\frac{\sqrt{a} \text{RootSum}\left[b^2 c+4 a b d \text{$\#$1}-2 b c \text{$\#$1}^2+4 a d \text{$\#$1}^3+c \text{$\#$1}^4\&,\frac{b^2 c^2 \log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)+a^2 b d^2 \log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)+b c^2 \log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2+a^2 d^2 \log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2}{-a b d+b c \text{$\#$1}-3 a d \text{$\#$1}^2-c \text{$\#$1}^3}\&\right]}{\sqrt{2} c^2}",1,"(a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/(2*c) + (Sqrt[a]*Sqrt[b]*ArcTan[(a*x^2)/Sqrt[b] + Sqrt[b + a^2*x^4]/Sqrt[b] - (Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[b]])/(Sqrt[2]*c) + (a^(3/2)*d*Log[a*x^2 + Sqrt[b + a^2*x^4] - Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]])/(Sqrt[2]*c^2) - (Sqrt[a]*RootSum[b^2*c - 4*a*b*d*#1 - 2*b*c*#1^2 - 4*a*d*#1^3 + c*#1^4 & , (b^2*c^2*Log[a*x^2 + Sqrt[b + a^2*x^4] - Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + #1] + a^2*b*d^2*Log[a*x^2 + Sqrt[b + a^2*x^4] - Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + #1] + b*c^2*Log[a*x^2 + Sqrt[b + a^2*x^4] - Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + #1]*#1^2 + a^2*d^2*Log[a*x^2 + Sqrt[b + a^2*x^4] - Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + #1]*#1^2)/(a*b*d + b*c*#1 + 3*a*d*#1^2 - c*#1^3) & ])/(Sqrt[2]*c^2)","A",1
3098,1,544,549,0.002704,"\int \frac{\sqrt{b+a^2 x^4} \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{d+c x^2} \, dx","IntegrateAlgebraic[(Sqrt[b + a^2*x^4]*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/(d + c*x^2),x]","\frac{a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{2 c}+\frac{\sqrt{a} \sqrt{b} \tan ^{-1}\left(\frac{a x^2}{\sqrt{b}}+\frac{\sqrt{b+a^2 x^4}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{b}}\right)}{\sqrt{2} c}+\frac{a^{3/2} d \log \left(a x^2+\sqrt{b+a^2 x^4}-\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{\sqrt{2} c^2}-\frac{\sqrt{a} \text{RootSum}\left[b^2 c-4 a b d \text{$\#$1}-2 b c \text{$\#$1}^2-4 a d \text{$\#$1}^3+c \text{$\#$1}^4\&,\frac{b^2 c^2 \log \left(a x^2+\sqrt{b+a^2 x^4}-\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\text{$\#$1}\right)+a^2 b d^2 \log \left(a x^2+\sqrt{b+a^2 x^4}-\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\text{$\#$1}\right)+b c^2 \log \left(a x^2+\sqrt{b+a^2 x^4}-\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\text{$\#$1}\right) \text{$\#$1}^2+a^2 d^2 \log \left(a x^2+\sqrt{b+a^2 x^4}-\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}+\text{$\#$1}\right) \text{$\#$1}^2}{a b d+b c \text{$\#$1}+3 a d \text{$\#$1}^2-c \text{$\#$1}^3}\&\right]}{\sqrt{2} c^2}","\frac{a x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{2 c}-\frac{\sqrt{a} \sqrt{b} \tan ^{-1}\left(\frac{a x^2}{\sqrt{b}}+\frac{\sqrt{b+a^2 x^4}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{b}}\right)}{\sqrt{2} c}-\frac{a^{3/2} d \log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}\right)}{\sqrt{2} c^2}-\frac{\sqrt{a} \text{RootSum}\left[b^2 c+4 a b d \text{$\#$1}-2 b c \text{$\#$1}^2+4 a d \text{$\#$1}^3+c \text{$\#$1}^4\&,\frac{b^2 c^2 \log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)+a^2 b d^2 \log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)+b c^2 \log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2+a^2 d^2 \log \left(a x^2+\sqrt{b+a^2 x^4}+\sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2}{-a b d+b c \text{$\#$1}-3 a d \text{$\#$1}^2-c \text{$\#$1}^3}\&\right]}{\sqrt{2} c^2}",1,"(a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/(2*c) + (Sqrt[a]*Sqrt[b]*ArcTan[(a*x^2)/Sqrt[b] + Sqrt[b + a^2*x^4]/Sqrt[b] - (Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[b]])/(Sqrt[2]*c) + (a^(3/2)*d*Log[a*x^2 + Sqrt[b + a^2*x^4] - Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]])/(Sqrt[2]*c^2) - (Sqrt[a]*RootSum[b^2*c - 4*a*b*d*#1 - 2*b*c*#1^2 - 4*a*d*#1^3 + c*#1^4 & , (b^2*c^2*Log[a*x^2 + Sqrt[b + a^2*x^4] - Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + #1] + a^2*b*d^2*Log[a*x^2 + Sqrt[b + a^2*x^4] - Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + #1] + b*c^2*Log[a*x^2 + Sqrt[b + a^2*x^4] - Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + #1]*#1^2 + a^2*d^2*Log[a*x^2 + Sqrt[b + a^2*x^4] - Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + #1]*#1^2)/(a*b*d + b*c*#1 + 3*a*d*#1^2 - c*#1^3) & ])/(Sqrt[2]*c^2)","A",1
3099,1,553,553,5.7503988,"\int \frac{\sqrt{x^2+\sqrt{1+x^4}}}{1+a x} \, dx","IntegrateAlgebraic[Sqrt[x^2 + Sqrt[1 + x^4]]/(1 + a*x),x]","\frac{\sqrt{x^2+\sqrt{1+x^4}}}{a}+\frac{\left(1+a^4+\sqrt{1+a^4}\right) \tan ^{-1}\left(\frac{a \sqrt{x^2+\sqrt{1+x^4}}}{\sqrt{-1-\sqrt{1+a^4}}}\right)}{a^2 \sqrt{1+a^4} \sqrt{-1-\sqrt{1+a^4}}}+\frac{\left(-1-a^4+\sqrt{1+a^4}\right) \tan ^{-1}\left(\frac{a \sqrt{x^2+\sqrt{1+x^4}}}{\sqrt{-1+\sqrt{1+a^4}}}\right)}{a^2 \sqrt{1+a^4} \sqrt{-1+\sqrt{1+a^4}}}+\frac{\left(\sqrt{2} \sqrt{-a^2-\sqrt{1+a^4}}-\sqrt{2} a^2 \sqrt{-a^2-\sqrt{1+a^4}}+\sqrt{2} \sqrt{1+a^4} \sqrt{-a^2-\sqrt{1+a^4}}\right) \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{\sqrt{-a^2-\sqrt{1+a^4}} \left(-1+x^2+\sqrt{1+x^4}\right)}\right)}{2 a^2}+\frac{\left(-\sqrt{2} \sqrt{-a^2+\sqrt{1+a^4}}+\sqrt{2} a^2 \sqrt{-a^2+\sqrt{1+a^4}}+\sqrt{2} \sqrt{1+a^4} \sqrt{-a^2+\sqrt{1+a^4}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{-a^2+\sqrt{1+a^4}} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)}{2 a^2}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)}{a^2}","\frac{\sqrt{x^2+\sqrt{1+x^4}}}{a}+\frac{\left(1+a^4+\sqrt{1+a^4}\right) \tan ^{-1}\left(\frac{a \sqrt{x^2+\sqrt{1+x^4}}}{\sqrt{-1-\sqrt{1+a^4}}}\right)}{a^2 \sqrt{1+a^4} \sqrt{-1-\sqrt{1+a^4}}}+\frac{\left(-1-a^4+\sqrt{1+a^4}\right) \tan ^{-1}\left(\frac{a \sqrt{x^2+\sqrt{1+x^4}}}{\sqrt{-1+\sqrt{1+a^4}}}\right)}{a^2 \sqrt{1+a^4} \sqrt{-1+\sqrt{1+a^4}}}+\frac{\left(\sqrt{2} \sqrt{-a^2-\sqrt{1+a^4}}-\sqrt{2} a^2 \sqrt{-a^2-\sqrt{1+a^4}}+\sqrt{2} \sqrt{1+a^4} \sqrt{-a^2-\sqrt{1+a^4}}\right) \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{\sqrt{-a^2-\sqrt{1+a^4}} \left(-1+x^2+\sqrt{1+x^4}\right)}\right)}{2 a^2}+\frac{\left(-\sqrt{2} \sqrt{-a^2+\sqrt{1+a^4}}+\sqrt{2} a^2 \sqrt{-a^2+\sqrt{1+a^4}}+\sqrt{2} \sqrt{1+a^4} \sqrt{-a^2+\sqrt{1+a^4}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{-a^2+\sqrt{1+a^4}} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)}{2 a^2}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)}{a^2}",1,"Sqrt[x^2 + Sqrt[1 + x^4]]/a + ((1 + a^4 + Sqrt[1 + a^4])*ArcTan[(a*Sqrt[x^2 + Sqrt[1 + x^4]])/Sqrt[-1 - Sqrt[1 + a^4]]])/(a^2*Sqrt[1 + a^4]*Sqrt[-1 - Sqrt[1 + a^4]]) + ((-1 - a^4 + Sqrt[1 + a^4])*ArcTan[(a*Sqrt[x^2 + Sqrt[1 + x^4]])/Sqrt[-1 + Sqrt[1 + a^4]]])/(a^2*Sqrt[1 + a^4]*Sqrt[-1 + Sqrt[1 + a^4]]) + ((Sqrt[2]*Sqrt[-a^2 - Sqrt[1 + a^4]] - Sqrt[2]*a^2*Sqrt[-a^2 - Sqrt[1 + a^4]] + Sqrt[2]*Sqrt[1 + a^4]*Sqrt[-a^2 - Sqrt[1 + a^4]])*ArcTan[(Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/(Sqrt[-a^2 - Sqrt[1 + a^4]]*(-1 + x^2 + Sqrt[1 + x^4]))])/(2*a^2) + ((-(Sqrt[2]*Sqrt[-a^2 + Sqrt[1 + a^4]]) + Sqrt[2]*a^2*Sqrt[-a^2 + Sqrt[1 + a^4]] + Sqrt[2]*Sqrt[1 + a^4]*Sqrt[-a^2 + Sqrt[1 + a^4]])*ArcTan[(Sqrt[2]*Sqrt[-a^2 + Sqrt[1 + a^4]]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2 + Sqrt[1 + x^4])])/(2*a^2) - (Sqrt[2]*ArcTanh[(Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2 + Sqrt[1 + x^4])])/a^2","A",1
3100,1,645,561,3.1474661,"\int \frac{1}{\sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7)} \, dx","IntegrateAlgebraic[1/(Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*(C[6] + x*C[7])),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{-c_3 c_4+\sqrt{c_1} \sqrt{c_3} c_5} \sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt{c_3} c_4-\sqrt{c_1} c_5}\right) \sqrt{-\sqrt{c_3} \left(\sqrt{c_3} c_4-\sqrt{c_1} c_5\right)}}{\left(\sqrt{c_3} c_4-\sqrt{c_1} c_5\right) c_7}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{-c_3 c_4-\sqrt{c_1} \sqrt{c_3} c_5} \sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt{c_3} c_4+\sqrt{c_1} c_5}\right) \sqrt{-\sqrt{c_3} \left(\sqrt{c_3} c_4+\sqrt{c_1} c_5\right)}}{\left(\sqrt{c_3} c_4+\sqrt{c_1} c_5\right) c_7}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5} \sqrt{c_3 c_6-c_2 c_7}}{\sqrt{-c_3 c_4 c_6+c_2 c_4 c_7-c_5 \sqrt{c_1 c_6-c_0 c_7} \sqrt{c_3 c_6-c_2 c_7}}}\right) \sqrt{c_3 c_6-c_2 c_7}}{c_7 \sqrt{-c_3 c_4 c_6+c_2 c_4 c_7-c_5 \sqrt{c_1 c_6-c_0 c_7} \sqrt{c_3 c_6-c_2 c_7}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5} \sqrt{c_3 c_6-c_2 c_7}}{\sqrt{-c_3 c_4 c_6+c_2 c_4 c_7+c_5 \sqrt{c_1 c_6-c_0 c_7} \sqrt{c_3 c_6-c_2 c_7}}}\right) \sqrt{c_3 c_6-c_2 c_7}}{c_7 \sqrt{-c_3 c_4 c_6+c_2 c_4 c_7+c_5 \sqrt{c_1 c_6-c_0 c_7} \sqrt{c_3 c_6-c_2 c_7}}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{c_3} \sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt{\sqrt{c_3} c_4-\sqrt{c_1} c_5}}\right) \sqrt[4]{c_3}}{\sqrt{\sqrt{c_3} c_4-\sqrt{c_1} c_5} c_7}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{c_3} \sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt{\sqrt{c_3} c_4+\sqrt{c_1} c_5}}\right) \sqrt[4]{c_3}}{\sqrt{\sqrt{c_3} c_4+\sqrt{c_1} c_5} c_7}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5} \sqrt{c_3 c_6-c_2 c_7}}{\sqrt{-c_3 c_4 c_6+c_2 c_4 c_7-c_5 \sqrt{c_1 c_6-c_0 c_7} \sqrt{c_3 c_6-c_2 c_7}}}\right) \sqrt{c_3 c_6-c_2 c_7}}{c_7 \sqrt{-c_3 c_4 c_6+c_2 c_4 c_7-c_5 \sqrt{c_1 c_6-c_0 c_7} \sqrt{c_3 c_6-c_2 c_7}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5} \sqrt{c_3 c_6-c_2 c_7}}{\sqrt{-c_3 c_4 c_6+c_2 c_4 c_7+c_5 \sqrt{c_1 c_6-c_0 c_7} \sqrt{c_3 c_6-c_2 c_7}}}\right) \sqrt{c_3 c_6-c_2 c_7}}{c_7 \sqrt{-c_3 c_4 c_6+c_2 c_4 c_7+c_5 \sqrt{c_1 c_6-c_0 c_7} \sqrt{c_3 c_6-c_2 c_7}}}",1,"(-2*ArcTan[(Sqrt[-(C[3]*C[4]) + Sqrt[C[1]]*Sqrt[C[3]]*C[5]]*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]])/(Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5])]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5]))])/((Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5])*C[7]) - (2*ArcTan[(Sqrt[-(C[3]*C[4]) - Sqrt[C[1]]*Sqrt[C[3]]*C[5]]*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]])/(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5])]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5]))])/((Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5])*C[7]) + (2*ArcTan[(Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*Sqrt[C[3]*C[6] - C[2]*C[7]])/Sqrt[-(C[3]*C[4]*C[6]) + C[2]*C[4]*C[7] - C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]]*Sqrt[C[3]*C[6] - C[2]*C[7]]]]*Sqrt[C[3]*C[6] - C[2]*C[7]])/(C[7]*Sqrt[-(C[3]*C[4]*C[6]) + C[2]*C[4]*C[7] - C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]]*Sqrt[C[3]*C[6] - C[2]*C[7]]]) + (2*ArcTan[(Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*Sqrt[C[3]*C[6] - C[2]*C[7]])/Sqrt[-(C[3]*C[4]*C[6]) + C[2]*C[4]*C[7] + C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]]*Sqrt[C[3]*C[6] - C[2]*C[7]]]]*Sqrt[C[3]*C[6] - C[2]*C[7]])/(C[7]*Sqrt[-(C[3]*C[4]*C[6]) + C[2]*C[4]*C[7] + C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]]*Sqrt[C[3]*C[6] - C[2]*C[7]]])","A",1
3101,1,569,569,3.8617115,"\int \frac{-b+x}{\sqrt[3]{(-a+x) (-b+x)^2} \left(-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2\right)} \, dx","IntegrateAlgebraic[(-b + x)/(((-a + x)*(-b + x)^2)^(1/3)*(-b^2 + a^2*d + 2*(b - a*d)*x + (-1 + d)*x^2)),x]","\frac{\sqrt{3} (a-b)^{2/3} d^{4/3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a d-b d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{-2 a \sqrt[3]{a-b} d^{2/3}+2 \sqrt[3]{a-b} d^{2/3} x+\sqrt[3]{a d-b d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}\right)}{2 ((a-b) d)^{5/3}}+\frac{(a-b)^{2/3} d^{4/3} \log \left(a (a d-b d)^{2/3}-(a d-b d)^{2/3} x+(a-b)^{2/3} \sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}\right)}{2 ((a-b) d)^{5/3}}-\frac{(a-b)^{2/3} d^{4/3} \log \left(a^3 d \sqrt[3]{a d-b d}-a^2 b d \sqrt[3]{a d-b d}-2 a^2 d \sqrt[3]{a d-b d} x+2 a b d \sqrt[3]{a d-b d} x+a d \sqrt[3]{a d-b d} x^2-b d \sqrt[3]{a d-b d} x^2+\left(-a (a-b)^{2/3} \sqrt[3]{d} (a d-b d)^{2/3}+(a-b)^{2/3} \sqrt[3]{d} (a d-b d)^{2/3} x\right) \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}+(a-b)^{4/3} d^{2/3} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}\right)}{4 ((a-b) d)^{5/3}}","\frac{\sqrt{3} (a-b)^{2/3} d^{4/3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{a d-b d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}{-2 a \sqrt[3]{a-b} d^{2/3}+2 \sqrt[3]{a-b} d^{2/3} x+\sqrt[3]{a d-b d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}}\right)}{2 ((a-b) d)^{5/3}}+\frac{(a-b)^{2/3} d^{4/3} \log \left(a (a d-b d)^{2/3}-(a d-b d)^{2/3} x+(a-b)^{2/3} \sqrt[3]{d} \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}\right)}{2 ((a-b) d)^{5/3}}-\frac{(a-b)^{2/3} d^{4/3} \log \left(a^3 d \sqrt[3]{a d-b d}-a^2 b d \sqrt[3]{a d-b d}-2 a^2 d \sqrt[3]{a d-b d} x+2 a b d \sqrt[3]{a d-b d} x+a d \sqrt[3]{a d-b d} x^2-b d \sqrt[3]{a d-b d} x^2+\left(-a (a-b)^{2/3} \sqrt[3]{d} (a d-b d)^{2/3}+(a-b)^{2/3} \sqrt[3]{d} (a d-b d)^{2/3} x\right) \sqrt[3]{-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3}+(a-b)^{4/3} d^{2/3} \left(-a b^2+\left(2 a b+b^2\right) x+(-a-2 b) x^2+x^3\right)^{2/3}\right)}{4 ((a-b) d)^{5/3}}",1,"(Sqrt[3]*(a - b)^(2/3)*d^(4/3)*ArcTan[(Sqrt[3]*(a*d - b*d)^(1/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3))/(-2*a*(a - b)^(1/3)*d^(2/3) + 2*(a - b)^(1/3)*d^(2/3)*x + (a*d - b*d)^(1/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3))])/(2*((a - b)*d)^(5/3)) + ((a - b)^(2/3)*d^(4/3)*Log[a*(a*d - b*d)^(2/3) - (a*d - b*d)^(2/3)*x + (a - b)^(2/3)*d^(1/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3)])/(2*((a - b)*d)^(5/3)) - ((a - b)^(2/3)*d^(4/3)*Log[a^3*d*(a*d - b*d)^(1/3) - a^2*b*d*(a*d - b*d)^(1/3) - 2*a^2*d*(a*d - b*d)^(1/3)*x + 2*a*b*d*(a*d - b*d)^(1/3)*x + a*d*(a*d - b*d)^(1/3)*x^2 - b*d*(a*d - b*d)^(1/3)*x^2 + (-(a*(a - b)^(2/3)*d^(1/3)*(a*d - b*d)^(2/3)) + (a - b)^(2/3)*d^(1/3)*(a*d - b*d)^(2/3)*x)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(1/3) + (a - b)^(4/3)*d^(2/3)*(-(a*b^2) + (2*a*b + b^2)*x + (-a - 2*b)*x^2 + x^3)^(2/3)])/(4*((a - b)*d)^(5/3))","A",1
3102,1,241,569,1.5303009,"\int \frac{\left(b^2+a x^2\right)^2}{\left(-b^2+a x^2\right)^2 \sqrt{b+\sqrt{b^2+a x^2}}} \, dx","IntegrateAlgebraic[(b^2 + a*x^2)^2/((-b^2 + a*x^2)^2*Sqrt[b + Sqrt[b^2 + a*x^2]]),x]","\frac{2 \left(2 b^2 x-a x^3\right)}{\left(b^2-a x^2\right) \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{\sqrt{2} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{a} x}\right)}{\sqrt{a}}+\frac{\sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{1+\sqrt{2}} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{a} x}\right)}{\sqrt{a}}-\frac{\sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}} \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{-1+\sqrt{2}} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{a} x}\right)}{\sqrt{a}}","\frac{2 x \left(-2 b^2+a x^2\right)}{\left(-b^2+a x^2\right) \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{2 \sqrt{2} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2} \sqrt{b}}\right)}{\sqrt{a}}-\frac{\sqrt{\frac{1}{2} \left(-1+\sqrt{2}\right)} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2 \left(-1+\sqrt{2}\right)} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2 \left(-1+\sqrt{2}\right)} \sqrt{b}}\right)}{\sqrt{a}}-\frac{\sqrt{\frac{1}{2} \left(-1+\sqrt{2}\right)} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{b}}\right)}{\sqrt{a}}-\frac{\sqrt{-1+\sqrt{2}} \left(2+\sqrt{2}\right) \sqrt{b} \tanh ^{-1}\left(\frac{-\frac{a x}{\sqrt{b+\sqrt{b^2+a x^2}}}+\sqrt{a} \sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2 \left(-1+\sqrt{2}\right)} \sqrt{a} \sqrt{b}}\right)}{2 \sqrt{a}}+\frac{\sqrt{-1+\sqrt{2}} \left(2+\sqrt{2}\right) \sqrt{b} \tanh ^{-1}\left(\frac{-\frac{a x}{\sqrt{b+\sqrt{b^2+a x^2}}}+\sqrt{a} \sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{a} \sqrt{b}}\right)}{2 \sqrt{a}}",1,"(2*(2*b^2*x - a*x^3))/((b^2 - a*x^2)*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (Sqrt[2]*Sqrt[b]*ArcTan[(Sqrt[2]*Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])/(Sqrt[a]*x)])/Sqrt[a] + (Sqrt[-1/2 + 1/Sqrt[2]]*Sqrt[b]*ArcTan[(Sqrt[1 + Sqrt[2]]*Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])/(Sqrt[a]*x)])/Sqrt[a] - (Sqrt[1/2 + 1/Sqrt[2]]*Sqrt[b]*ArcTanh[(Sqrt[-1 + Sqrt[2]]*Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])/(Sqrt[a]*x)])/Sqrt[a]","A",1
3103,1,604,586,0.7640539,"\int \frac{\sqrt{a x+\sqrt{-b+a x}}}{\sqrt{-b+a x} \left(1+a^2 x^2\right)} \, dx","IntegrateAlgebraic[Sqrt[a*x + Sqrt[-b + a*x]]/(Sqrt[-b + a*x]*(1 + a^2*x^2)),x]","-\frac{\text{RootSum}\left[1+b^2+2 b^3+b^4-8 \text{$\#$1}-8 b^2 \text{$\#$1}-8 b^3 \text{$\#$1}+24 \text{$\#$1}^2+20 b^2 \text{$\#$1}^2+4 b^3 \text{$\#$1}^2-32 \text{$\#$1}^3-16 b^2 \text{$\#$1}^3+16 \text{$\#$1}^4+2 b \text{$\#$1}^4+6 b^2 \text{$\#$1}^4-8 b \text{$\#$1}^5+4 b \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-b^2 \log \left(-\sqrt{-b+a x}+\sqrt{a x+\sqrt{-b+a x}}-\text{$\#$1}\right)+2 b \log \left(-\sqrt{-b+a x}+\sqrt{a x+\sqrt{-b+a x}}-\text{$\#$1}\right) \text{$\#$1}+2 b^2 \log \left(-\sqrt{-b+a x}+\sqrt{a x+\sqrt{-b+a x}}-\text{$\#$1}\right) \text{$\#$1}-\log \left(-\sqrt{-b+a x}+\sqrt{a x+\sqrt{-b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2-6 b \log \left(-\sqrt{-b+a x}+\sqrt{a x+\sqrt{-b+a x}}-\text{$\#$1}\right) \text{$\#$1}^2+4 \log \left(-\sqrt{-b+a x}+\sqrt{a x+\sqrt{-b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3+4 b \log \left(-\sqrt{-b+a x}+\sqrt{a x+\sqrt{-b+a x}}-\text{$\#$1}\right) \text{$\#$1}^3-5 \log \left(-\sqrt{-b+a x}+\sqrt{a x+\sqrt{-b+a x}}-\text{$\#$1}\right) \text{$\#$1}^4+2 \log \left(-\sqrt{-b+a x}+\sqrt{a x+\sqrt{-b+a x}}-\text{$\#$1}\right) \text{$\#$1}^5}{-1-b^2-b^3+6 \text{$\#$1}+5 b^2 \text{$\#$1}+b^3 \text{$\#$1}-12 \text{$\#$1}^2-6 b^2 \text{$\#$1}^2+8 \text{$\#$1}^3+b \text{$\#$1}^3+3 b^2 \text{$\#$1}^3-5 b \text{$\#$1}^4+3 b \text{$\#$1}^5+\text{$\#$1}^7}\&\right]}{2 a}","-\frac{\text{RootSum}\left[1+b^2+2 b^3+b^4-8 \text{$\#$1}-8 b^2 \text{$\#$1}-8 b^3 \text{$\#$1}+24 \text{$\#$1}^2+20 b^2 \text{$\#$1}^2+4 b^3 \text{$\#$1}^2-32 \text{$\#$1}^3-16 b^2 \text{$\#$1}^3+16 \text{$\#$1}^4+2 b \text{$\#$1}^4+6 b^2 \text{$\#$1}^4-8 b \text{$\#$1}^5+4 b \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-b^2 \log \left(\sqrt{-b+a x}-\sqrt{a x+\sqrt{-b+a x}}+\text{$\#$1}\right)+2 b \log \left(\sqrt{-b+a x}-\sqrt{a x+\sqrt{-b+a x}}+\text{$\#$1}\right) \text{$\#$1}+2 b^2 \log \left(\sqrt{-b+a x}-\sqrt{a x+\sqrt{-b+a x}}+\text{$\#$1}\right) \text{$\#$1}-\log \left(\sqrt{-b+a x}-\sqrt{a x+\sqrt{-b+a x}}+\text{$\#$1}\right) \text{$\#$1}^2-6 b \log \left(\sqrt{-b+a x}-\sqrt{a x+\sqrt{-b+a x}}+\text{$\#$1}\right) \text{$\#$1}^2+4 \log \left(\sqrt{-b+a x}-\sqrt{a x+\sqrt{-b+a x}}+\text{$\#$1}\right) \text{$\#$1}^3+4 b \log \left(\sqrt{-b+a x}-\sqrt{a x+\sqrt{-b+a x}}+\text{$\#$1}\right) \text{$\#$1}^3-5 \log \left(\sqrt{-b+a x}-\sqrt{a x+\sqrt{-b+a x}}+\text{$\#$1}\right) \text{$\#$1}^4+2 \log \left(\sqrt{-b+a x}-\sqrt{a x+\sqrt{-b+a x}}+\text{$\#$1}\right) \text{$\#$1}^5}{-1-b^2-b^3+6 \text{$\#$1}+5 b^2 \text{$\#$1}+b^3 \text{$\#$1}-12 \text{$\#$1}^2-6 b^2 \text{$\#$1}^2+8 \text{$\#$1}^3+b \text{$\#$1}^3+3 b^2 \text{$\#$1}^3-5 b \text{$\#$1}^4+3 b \text{$\#$1}^5+\text{$\#$1}^7}\&\right]}{2 a}",1,"-1/2*RootSum[1 + b^2 + 2*b^3 + b^4 - 8*#1 - 8*b^2*#1 - 8*b^3*#1 + 24*#1^2 + 20*b^2*#1^2 + 4*b^3*#1^2 - 32*#1^3 - 16*b^2*#1^3 + 16*#1^4 + 2*b*#1^4 + 6*b^2*#1^4 - 8*b*#1^5 + 4*b*#1^6 + #1^8 & , (-(b^2*Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1]) + 2*b*Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1 + 2*b^2*Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1 - Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1^2 - 6*b*Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1^2 + 4*Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1^3 + 4*b*Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1^3 - 5*Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1^4 + 2*Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1^5)/(-1 - b^2 - b^3 + 6*#1 + 5*b^2*#1 + b^3*#1 - 12*#1^2 - 6*b^2*#1^2 + 8*#1^3 + b*#1^3 + 3*b^2*#1^3 - 5*b*#1^4 + 3*b*#1^5 + #1^7) & ]/a","A",1
3104,1,590,590,0.60374,"\int \frac{\left(1+x^4\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{1-x^4} \, dx","IntegrateAlgebraic[((1 + x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^4),x]","\frac{\left(6+16 x-48 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+(15-8 x) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left((16-48 x) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-8 \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{60 x+60 \sqrt{1+x^2}}-\frac{1}{4} \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)-\frac{1}{2} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{1-2 \text{$\#$1}^2+\text{$\#$1}^4}\&\right]+\frac{1}{2} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+6 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-4 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","\frac{\left(6+16 x-48 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+(15-8 x) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left((16-48 x) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-8 \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{60 x+60 \sqrt{1+x^2}}-\frac{1}{4} \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)-\frac{1}{2} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{1-2 \text{$\#$1}^2+\text{$\#$1}^4}\&\right]+\frac{1}{2} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+6 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-4 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]",1,"((6 + 16*x - 48*x^2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (15 - 8*x)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((16 - 48*x)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - 8*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(60*x + 60*Sqrt[1 + x^2]) - ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]]/4 - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(2*#1^3 - 3*#1^5 + #1^7) & ]/2 + RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(1 - 2*#1^2 + #1^4) & ] + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 6*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - 4*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ]/2","A",1
3105,1,590,590,0.0015304,"\int \frac{\left(1+x^4\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{1-x^4} \, dx","IntegrateAlgebraic[((1 + x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^4),x]","\frac{\left(6+16 x-48 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+(15-8 x) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left((16-48 x) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-8 \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{60 x+60 \sqrt{1+x^2}}-\frac{1}{4} \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)-\frac{1}{2} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{1-2 \text{$\#$1}^2+\text{$\#$1}^4}\&\right]+\frac{1}{2} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+6 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-4 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","\frac{\left(6+16 x-48 x^2\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+(15-8 x) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left((16-48 x) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-8 \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{60 x+60 \sqrt{1+x^2}}-\frac{1}{4} \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)-\frac{1}{2} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\text{RootSum}\left[2-4 \text{$\#$1}^2+6 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}}{1-2 \text{$\#$1}^2+\text{$\#$1}^4}\&\right]+\frac{1}{2} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+6 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-4 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]",1,"((6 + 16*x - 48*x^2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (15 - 8*x)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((16 - 48*x)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - 8*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(60*x + 60*Sqrt[1 + x^2]) - ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]]/4 - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(2*#1^3 - 3*#1^5 + #1^7) & ]/2 + RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(1 - 2*#1^2 + #1^4) & ] + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 6*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - 4*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ]/2","A",1
3106,1,921,603,1.2861887,"\int \frac{\left(1+x^2\right)^{5/2} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1-x^2\right)^2} \, dx","IntegrateAlgebraic[((1 + x^2)^(5/2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^2)^2,x]","\frac{\left(-803792+690 x-6024144 x^2+10470 x^3-1112160 x^4+1128 x^5+5861376 x^6-12288 x^7+143360 x^8\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-184+1525 x-6328 x^2-8025 x^3-1680 x^4-3740 x^5+8192 x^6+10240 x^7\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(-282-3314848 x+5298 x^2-3989088 x^3+7272 x^4+5789696 x^5-12288 x^6+143360 x^7\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(935-2416 x-2315 x^2-5776 x^3-8860 x^4+8192 x^5+10240 x^6\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{40320 \left(-1+x^2\right) \sqrt{1+x^2} \left(4 x+8 x^3\right)+40320 \left(-1+x^2\right) \left(1+8 x^2+8 x^4\right)}-\frac{2299}{128} \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)+4 \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-\frac{1}{4} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{14 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-4 \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{1}{4} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-16 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","\frac{\left(-803792+690 x-6024144 x^2+10470 x^3-1112160 x^4+1128 x^5+5861376 x^6-12288 x^7+143360 x^8\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-184+1525 x-6328 x^2-8025 x^3-1680 x^4-3740 x^5+8192 x^6+10240 x^7\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(-282-3314848 x+5298 x^2-3989088 x^3+7272 x^4+5789696 x^5-12288 x^6+143360 x^7\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(935-2416 x-2315 x^2-5776 x^3-8860 x^4+8192 x^5+10240 x^6\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{40320 \left(-1+x^2\right) \sqrt{1+x^2} \left(4 x+8 x^3\right)+40320 \left(-1+x^2\right) \left(1+8 x^2+8 x^4\right)}-\frac{2299}{128} \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)+\frac{1}{4} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-14 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+13 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-\frac{1}{4} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-14 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}+13 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^3}{-2+4 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]",1,"((-803792 + 690*x - 6024144*x^2 + 10470*x^3 - 1112160*x^4 + 1128*x^5 + 5861376*x^6 - 12288*x^7 + 143360*x^8)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-184 + 1525*x - 6328*x^2 - 8025*x^3 - 1680*x^4 - 3740*x^5 + 8192*x^6 + 10240*x^7)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((-282 - 3314848*x + 5298*x^2 - 3989088*x^3 + 7272*x^4 + 5789696*x^5 - 12288*x^6 + 143360*x^7)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (935 - 2416*x - 2315*x^2 - 5776*x^3 - 8860*x^4 + 8192*x^5 + 10240*x^6)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(40320*(-1 + x^2)*Sqrt[1 + x^2]*(4*x + 8*x^3) + 40320*(-1 + x^2)*(1 + 8*x^2 + 8*x^4)) - (2299*ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]])/128 + 4*RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1^3 - 3*#1^5 + #1^7) & ] - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (14*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1^3 - 3*#1^5 + #1^7) & ]/4 - 4*RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ] + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-16*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ]/4","A",1
3107,1,921,603,0.0019304,"\int \frac{\left(1+x^2\right)^{5/2} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1-x^2\right)^2} \, dx","IntegrateAlgebraic[((1 + x^2)^(5/2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^2)^2,x]","\frac{\left(-803792+690 x-6024144 x^2+10470 x^3-1112160 x^4+1128 x^5+5861376 x^6-12288 x^7+143360 x^8\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-184+1525 x-6328 x^2-8025 x^3-1680 x^4-3740 x^5+8192 x^6+10240 x^7\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(-282-3314848 x+5298 x^2-3989088 x^3+7272 x^4+5789696 x^5-12288 x^6+143360 x^7\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(935-2416 x-2315 x^2-5776 x^3-8860 x^4+8192 x^5+10240 x^6\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{40320 \left(-1+x^2\right) \sqrt{1+x^2} \left(4 x+8 x^3\right)+40320 \left(-1+x^2\right) \left(1+8 x^2+8 x^4\right)}-\frac{2299}{128} \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)+4 \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-\frac{1}{4} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{14 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-4 \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{1}{4} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-16 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+3 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","\frac{\left(-803792+690 x-6024144 x^2+10470 x^3-1112160 x^4+1128 x^5+5861376 x^6-12288 x^7+143360 x^8\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-184+1525 x-6328 x^2-8025 x^3-1680 x^4-3740 x^5+8192 x^6+10240 x^7\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(-282-3314848 x+5298 x^2-3989088 x^3+7272 x^4+5789696 x^5-12288 x^6+143360 x^7\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(935-2416 x-2315 x^2-5776 x^3-8860 x^4+8192 x^5+10240 x^6\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{40320 \left(-1+x^2\right) \sqrt{1+x^2} \left(4 x+8 x^3\right)+40320 \left(-1+x^2\right) \left(1+8 x^2+8 x^4\right)}-\frac{2299}{128} \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)+\frac{1}{4} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-14 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+13 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-\frac{1}{4} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-14 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}+13 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^3}{-2+4 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]",1,"((-803792 + 690*x - 6024144*x^2 + 10470*x^3 - 1112160*x^4 + 1128*x^5 + 5861376*x^6 - 12288*x^7 + 143360*x^8)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-184 + 1525*x - 6328*x^2 - 8025*x^3 - 1680*x^4 - 3740*x^5 + 8192*x^6 + 10240*x^7)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((-282 - 3314848*x + 5298*x^2 - 3989088*x^3 + 7272*x^4 + 5789696*x^5 - 12288*x^6 + 143360*x^7)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (935 - 2416*x - 2315*x^2 - 5776*x^3 - 8860*x^4 + 8192*x^5 + 10240*x^6)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(40320*(-1 + x^2)*Sqrt[1 + x^2]*(4*x + 8*x^3) + 40320*(-1 + x^2)*(1 + 8*x^2 + 8*x^4)) - (2299*ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]])/128 + 4*RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1^3 - 3*#1^5 + #1^7) & ] - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (14*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1^3 - 3*#1^5 + #1^7) & ]/4 - 4*RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ] + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-16*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ]/4","A",1
3108,1,600,604,8.4460171,"\int \frac{\sqrt{1+x^4}}{(1+x)^3 \sqrt{x^2+\sqrt{1+x^4}}} \, dx","IntegrateAlgebraic[Sqrt[1 + x^4]/((1 + x)^3*Sqrt[x^2 + Sqrt[1 + x^4]]),x]","\frac{x \left(-5+9 x^2-52 x^4+72 x^6-96 x^8+72 x^{10}-48 x^{12}\right) \sqrt{x^2+\sqrt{1+x^4}}+\left(2 x^2+x^4+21 x^6-12 x^8+36 x^{10}-16 x^{12}+16 x^{14}\right) \sqrt{x^2+\sqrt{1+x^4}}+\sqrt{1+x^4} \left(x \left(-22 x^2+36 x^4-72 x^6+72 x^8-48 x^{10}\right) \sqrt{x^2+\sqrt{1+x^4}}+\left(x^2+9 x^4-4 x^6+28 x^8-16 x^{10}+16 x^{12}\right) \sqrt{x^2+\sqrt{1+x^4}}\right)}{2 x \left(-1+x^2\right)^2 \left(x^2+\sqrt{1+x^4}\right)^5}-\frac{\tan ^{-1}\left(\sqrt{1+\sqrt{2}} \sqrt{x^2+\sqrt{1+x^4}}\right)}{\sqrt{-1+\sqrt{2}}}-\frac{5 \tan ^{-1}\left(\sqrt{1+\sqrt{2}} \sqrt{x^2+\sqrt{1+x^4}}\right)}{2 \sqrt{-2+2 \sqrt{2}}}-3 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)+\frac{1}{2} \sqrt{\frac{73}{2}+\frac{53}{\sqrt{2}}} \tan ^{-1}\left(\frac{\sqrt{-2+2 \sqrt{2}} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)+\frac{\tanh ^{-1}\left(\sqrt{-1+\sqrt{2}} \sqrt{x^2+\sqrt{1+x^4}}\right)}{\sqrt{1+\sqrt{2}}}-\frac{5 \tanh ^{-1}\left(\sqrt{-1+\sqrt{2}} \sqrt{x^2+\sqrt{1+x^4}}\right)}{2 \sqrt{2+2 \sqrt{2}}}+\frac{1}{2} \sqrt{-\frac{73}{2}+\frac{53}{\sqrt{2}}} \tanh ^{-1}\left(\frac{\sqrt{2+2 \sqrt{2}} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)","\frac{x \left(-5+9 x^2-52 x^4+72 x^6-96 x^8+72 x^{10}-48 x^{12}\right) \sqrt{x^2+\sqrt{1+x^4}}+\left(2 x^2+x^4+21 x^6-12 x^8+36 x^{10}-16 x^{12}+16 x^{14}\right) \sqrt{x^2+\sqrt{1+x^4}}+\sqrt{1+x^4} \left(x \left(-22 x^2+36 x^4-72 x^6+72 x^8-48 x^{10}\right) \sqrt{x^2+\sqrt{1+x^4}}+\left(x^2+9 x^4-4 x^6+28 x^8-16 x^{10}+16 x^{12}\right) \sqrt{x^2+\sqrt{1+x^4}}\right)}{2 x \left(-1+x^2\right)^2 \left(x^2+\sqrt{1+x^4}\right)^5}-\frac{\tan ^{-1}\left(\frac{\sqrt{x^2+\sqrt{1+x^4}}}{\sqrt{-1+\sqrt{2}}}\right)}{\sqrt{-1+\sqrt{2}}}-\frac{5 \tan ^{-1}\left(\frac{\sqrt{x^2+\sqrt{1+x^4}}}{\sqrt{-1+\sqrt{2}}}\right)}{2 \sqrt{2 \left(-1+\sqrt{2}\right)}}-3 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)+\frac{1}{2} \sqrt{\frac{1}{2} \left(73+53 \sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(-1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)+\frac{\tanh ^{-1}\left(\frac{\sqrt{x^2+\sqrt{1+x^4}}}{\sqrt{1+\sqrt{2}}}\right)}{\sqrt{1+\sqrt{2}}}-\frac{5 \tanh ^{-1}\left(\frac{\sqrt{x^2+\sqrt{1+x^4}}}{\sqrt{1+\sqrt{2}}}\right)}{2 \sqrt{2 \left(1+\sqrt{2}\right)}}+\frac{1}{2} \sqrt{\frac{1}{2} \left(-73+53 \sqrt{2}\right)} \tanh ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)} x \sqrt{x^2+\sqrt{1+x^4}}}{1+x^2+\sqrt{1+x^4}}\right)",1,"(x*(-5 + 9*x^2 - 52*x^4 + 72*x^6 - 96*x^8 + 72*x^10 - 48*x^12)*Sqrt[x^2 + Sqrt[1 + x^4]] + (2*x^2 + x^4 + 21*x^6 - 12*x^8 + 36*x^10 - 16*x^12 + 16*x^14)*Sqrt[x^2 + Sqrt[1 + x^4]] + Sqrt[1 + x^4]*(x*(-22*x^2 + 36*x^4 - 72*x^6 + 72*x^8 - 48*x^10)*Sqrt[x^2 + Sqrt[1 + x^4]] + (x^2 + 9*x^4 - 4*x^6 + 28*x^8 - 16*x^10 + 16*x^12)*Sqrt[x^2 + Sqrt[1 + x^4]]))/(2*x*(-1 + x^2)^2*(x^2 + Sqrt[1 + x^4])^5) - ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[x^2 + Sqrt[1 + x^4]]]/Sqrt[-1 + Sqrt[2]] - (5*ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[x^2 + Sqrt[1 + x^4]]])/(2*Sqrt[-2 + 2*Sqrt[2]]) - 3*Sqrt[2]*ArcTan[(Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2 + Sqrt[1 + x^4])] + (Sqrt[73/2 + 53/Sqrt[2]]*ArcTan[(Sqrt[-2 + 2*Sqrt[2]]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2 + Sqrt[1 + x^4])])/2 + ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[x^2 + Sqrt[1 + x^4]]]/Sqrt[1 + Sqrt[2]] - (5*ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[x^2 + Sqrt[1 + x^4]]])/(2*Sqrt[2 + 2*Sqrt[2]]) + (Sqrt[-73/2 + 53/Sqrt[2]]*ArcTanh[(Sqrt[2 + 2*Sqrt[2]]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2 + Sqrt[1 + x^4])])/2","A",1
3109,0,0,617,3.6995055,"\int \frac{x^2}{\sqrt{1-\sqrt{1-\sqrt{1-\frac{1}{x}}}}} \, dx","IntegrateAlgebraic[x^2/Sqrt[1 - Sqrt[1 - Sqrt[1 - x^(-1)]]],x]","\int \frac{x^2}{\sqrt{1-\sqrt{1-\sqrt{1-\frac{1}{x}}}}} \, dx","\frac{-4395 \sqrt{1-\sqrt{1-\sqrt{1-\frac{1}{x}}}}-148125 \left(1-\sqrt{1-\sqrt{1-\frac{1}{x}}}\right)^{3/2}+184677 \left(1-\sqrt{1-\sqrt{1-\frac{1}{x}}}\right)^{5/2}+598095 \left(1-\sqrt{1-\sqrt{1-\frac{1}{x}}}\right)^{7/2}-1429914 \left(1-\sqrt{1-\sqrt{1-\frac{1}{x}}}\right)^{9/2}+215274 \left(1-\sqrt{1-\sqrt{1-\frac{1}{x}}}\right)^{11/2}+2531670 \left(1-\sqrt{1-\sqrt{1-\frac{1}{x}}}\right)^{13/2}-3755230 \left(1-\sqrt{1-\sqrt{1-\frac{1}{x}}}\right)^{15/2}+2613381 \left(1-\sqrt{1-\sqrt{1-\frac{1}{x}}}\right)^{17/2}-1002573 \left(1-\sqrt{1-\sqrt{1-\frac{1}{x}}}\right)^{19/2}+204165 \left(1-\sqrt{1-\sqrt{1-\frac{1}{x}}}\right)^{21/2}-17265 \left(1-\sqrt{1-\sqrt{1-\frac{1}{x}}}\right)^{23/2}}{30720 \left(-1-2 \left(1-\sqrt{1-\sqrt{1-\frac{1}{x}}}\right)+\left(1-\sqrt{1-\sqrt{1-\frac{1}{x}}}\right)^2\right)^3 \left(1-\sqrt{1-\frac{1}{x}}\right)^3}+\left(-\frac{7}{32} \sqrt{\frac{1}{2} \left(\frac{1}{2}+\frac{1}{\sqrt{2}}\right)}-\frac{137 \sqrt{\frac{1}{2}+\frac{1}{\sqrt{2}}}}{1024}\right) \tan ^{-1}\left(\frac{\sqrt{1-\sqrt{1-\sqrt{1-\frac{1}{x}}}}}{\sqrt{-1+\sqrt{2}}}\right)+\frac{703 \tanh ^{-1}\left(\sqrt{1-\sqrt{1-\sqrt{1-\frac{1}{x}}}}\right)}{2048}+\left(\frac{7}{32} \sqrt{\frac{1}{2} \left(-\frac{1}{2}+\frac{1}{\sqrt{2}}\right)}-\frac{137 \sqrt{-\frac{1}{2}+\frac{1}{\sqrt{2}}}}{1024}\right) \tanh ^{-1}\left(\frac{\sqrt{1-\sqrt{1-\sqrt{1-\frac{1}{x}}}}}{\sqrt{1+\sqrt{2}}}\right)",1,"Could not integrate","F",-1
3110,1,622,622,2.9135708,"\int \frac{\left(1+x^2\right) \left(-a-b x+a x^2\right)}{x^2 \left(-c+d x+c x^2\right) \sqrt[3]{\frac{-1+x^2-x c_0}{-1+x^2-x c_1}}} \, dx","IntegrateAlgebraic[((1 + x^2)*(-a - b*x + a*x^2))/(x^2*(-c + d*x + c*x^2)*((-1 + x^2 - x*C[0])/(-1 + x^2 - x*C[1]))^(1/3)),x]","\frac{\sqrt{3} (b c+a d) \tan ^{-1}\left(\frac{\sqrt[3]{d+c c_0}-2 \sqrt[3]{-d-c c_1} \sqrt[3]{\frac{1-x^2+x c_0}{1-x^2+x c_1}}}{\sqrt{3} \sqrt[3]{d+c c_0}}\right) \sqrt[3]{-d-c c_1}}{c^2 \sqrt[3]{d+c c_0}}+\frac{\tan ^{-1}\left(\frac{1+2 \sqrt[3]{\frac{-1+x^2-x c_0}{-1+x^2-x c_1}}}{\sqrt{3}}\right) \left(3 \sqrt{3} b c+3 \sqrt{3} a d-\sqrt{3} a c c_0+\sqrt{3} a c c_1\right)}{3 c^2}+\frac{\left(\frac{-1+x^2-x c_0}{-1+x^2-x c_1}\right){}^{2/3} \left(-a+a x^2-a x c_1\right)}{c x}+\frac{(3 b c+3 a d-a c c_0+a c c_1) \log \left(-1+\sqrt[3]{\frac{-1+x^2-x c_0}{-1+x^2-x c_1}}\right)}{3 c^2}+\frac{(-3 b c-3 a d+a c c_0-a c c_1) \log \left(1+\sqrt[3]{\frac{-1+x^2-x c_0}{-1+x^2-x c_1}}+\left(\frac{-1+x^2-x c_0}{-1+x^2-x c_1}\right){}^{2/3}\right)}{6 c^2}+\frac{(b c+a d) \sqrt[3]{-d-c c_1} \log \left(\sqrt[3]{d+c c_0}+\sqrt[3]{-d-c c_1} \sqrt[3]{\frac{1-x^2+x c_0}{1-x^2+x c_1}}\right)}{c^2 \sqrt[3]{d+c c_0}}-\frac{(b c+a d) \sqrt[3]{-d-c c_1} \log \left((d+c c_0){}^{2/3}-\sqrt[3]{d+c c_0} \sqrt[3]{-d-c c_1} \sqrt[3]{\frac{1-x^2+x c_0}{1-x^2+x c_1}}+(-d-c c_1){}^{2/3} \left(\frac{1-x^2+x c_0}{1-x^2+x c_1}\right){}^{2/3}\right)}{2 c^2 \sqrt[3]{d+c c_0}}","\frac{\sqrt{3} (b c+a d) \tan ^{-1}\left(\frac{\sqrt[3]{d+c c_0}-2 \sqrt[3]{-d-c c_1} \sqrt[3]{\frac{1-x^2+x c_0}{1-x^2+x c_1}}}{\sqrt{3} \sqrt[3]{d+c c_0}}\right) \sqrt[3]{-d-c c_1}}{c^2 \sqrt[3]{d+c c_0}}+\frac{\tan ^{-1}\left(\frac{1+2 \sqrt[3]{\frac{-1+x^2-x c_0}{-1+x^2-x c_1}}}{\sqrt{3}}\right) \left(3 \sqrt{3} b c+3 \sqrt{3} a d-\sqrt{3} a c c_0+\sqrt{3} a c c_1\right)}{3 c^2}+\frac{\left(\frac{-1+x^2-x c_0}{-1+x^2-x c_1}\right){}^{2/3} \left(-a+a x^2-a x c_1\right)}{c x}+\frac{(3 b c+3 a d-a c c_0+a c c_1) \log \left(-1+\sqrt[3]{\frac{-1+x^2-x c_0}{-1+x^2-x c_1}}\right)}{3 c^2}+\frac{(-3 b c-3 a d+a c c_0-a c c_1) \log \left(1+\sqrt[3]{\frac{-1+x^2-x c_0}{-1+x^2-x c_1}}+\left(\frac{-1+x^2-x c_0}{-1+x^2-x c_1}\right){}^{2/3}\right)}{6 c^2}+\frac{(b c+a d) \sqrt[3]{-d-c c_1} \log \left(\sqrt[3]{d+c c_0}+\sqrt[3]{-d-c c_1} \sqrt[3]{\frac{1-x^2+x c_0}{1-x^2+x c_1}}\right)}{c^2 \sqrt[3]{d+c c_0}}-\frac{(b c+a d) \sqrt[3]{-d-c c_1} \log \left((d+c c_0){}^{2/3}-\sqrt[3]{d+c c_0} \sqrt[3]{-d-c c_1} \sqrt[3]{\frac{1-x^2+x c_0}{1-x^2+x c_1}}+(-d-c c_1){}^{2/3} \left(\frac{1-x^2+x c_0}{1-x^2+x c_1}\right){}^{2/3}\right)}{2 c^2 \sqrt[3]{d+c c_0}}",1,"(Sqrt[3]*(b*c + a*d)*ArcTan[((d + c*C[0])^(1/3) - 2*(-d - c*C[1])^(1/3)*((1 - x^2 + x*C[0])/(1 - x^2 + x*C[1]))^(1/3))/(Sqrt[3]*(d + c*C[0])^(1/3))]*(-d - c*C[1])^(1/3))/(c^2*(d + c*C[0])^(1/3)) + (ArcTan[(1 + 2*((-1 + x^2 - x*C[0])/(-1 + x^2 - x*C[1]))^(1/3))/Sqrt[3]]*(3*Sqrt[3]*b*c + 3*Sqrt[3]*a*d - Sqrt[3]*a*c*C[0] + Sqrt[3]*a*c*C[1]))/(3*c^2) + (((-1 + x^2 - x*C[0])/(-1 + x^2 - x*C[1]))^(2/3)*(-a + a*x^2 - a*x*C[1]))/(c*x) + ((3*b*c + 3*a*d - a*c*C[0] + a*c*C[1])*Log[-1 + ((-1 + x^2 - x*C[0])/(-1 + x^2 - x*C[1]))^(1/3)])/(3*c^2) + ((-3*b*c - 3*a*d + a*c*C[0] - a*c*C[1])*Log[1 + ((-1 + x^2 - x*C[0])/(-1 + x^2 - x*C[1]))^(1/3) + ((-1 + x^2 - x*C[0])/(-1 + x^2 - x*C[1]))^(2/3)])/(6*c^2) + ((b*c + a*d)*(-d - c*C[1])^(1/3)*Log[(d + c*C[0])^(1/3) + (-d - c*C[1])^(1/3)*((1 - x^2 + x*C[0])/(1 - x^2 + x*C[1]))^(1/3)])/(c^2*(d + c*C[0])^(1/3)) - ((b*c + a*d)*(-d - c*C[1])^(1/3)*Log[(d + c*C[0])^(2/3) - (d + c*C[0])^(1/3)*(-d - c*C[1])^(1/3)*((1 - x^2 + x*C[0])/(1 - x^2 + x*C[1]))^(1/3) + (-d - c*C[1])^(2/3)*((1 - x^2 + x*C[0])/(1 - x^2 + x*C[1]))^(2/3)])/(2*c^2*(d + c*C[0])^(1/3))","A",1
3111,1,708,639,1.0861518,"\int \frac{\left(1+x^2\right)^2 \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1-x^2\right)^2 \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[((1 + x^2)^2*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/((1 - x^2)^2*Sqrt[x + Sqrt[1 + x^2]]),x]","\frac{\left(-8-9 x-248 x^2-3 x^3-208 x^4+12 x^5+128 x^6\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(2-48 x+2 x^2-16 x^3-4 x^4+64 x^5\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(-3-96 x-9 x^2-272 x^3+12 x^4+128 x^5\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-16+4 x-48 x^2-4 x^3+64 x^4\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{24 \left(-1+x^2\right) \left(x+\sqrt{1+x^2}\right)^{7/2}}-\frac{1}{8} \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)+\text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{-2 \text{$\#$1}+\text{$\#$1}^3}\&\right]-\frac{1}{4} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+5 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-\text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^3}{2-2 \text{$\#$1}^2+\text{$\#$1}^4}\&\right]-\frac{1}{4} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+5 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","\frac{\left(-8-9 x-248 x^2-3 x^3-208 x^4+12 x^5+128 x^6\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(2-48 x+2 x^2-16 x^3-4 x^4+64 x^5\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(-3-96 x-9 x^2-272 x^3+12 x^4+128 x^5\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-16+4 x-48 x^2-4 x^3+64 x^4\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{24 \left(-1+x^2\right) \left(x+\sqrt{1+x^2}\right)^{7/2}}-\frac{1}{8} \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)+\frac{1}{4} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-8 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+4 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-\frac{1}{4} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+9 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-8 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+4 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]",1,"((-8 - 9*x - 248*x^2 - 3*x^3 - 208*x^4 + 12*x^5 + 128*x^6)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (2 - 48*x + 2*x^2 - 16*x^3 - 4*x^4 + 64*x^5)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((-3 - 96*x - 9*x^2 - 272*x^3 + 12*x^4 + 128*x^5)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-16 + 4*x - 48*x^2 - 4*x^3 + 64*x^4)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(24*(-1 + x^2)*(x + Sqrt[1 + x^2])^(7/2)) - ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]]/8 + RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (-Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2)/(-2*#1 + #1^3) & ] - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 5*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2)/(2*#1^3 - 3*#1^5 + #1^7) & ]/4 - RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-(Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1) + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^3)/(2 - 2*#1^2 + #1^4) & ] - RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 5*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ]/4","A",1
3112,1,708,639,0.002047,"\int \frac{\left(1+x^2\right)^2 \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1-x^2\right)^2 \sqrt{x+\sqrt{1+x^2}}} \, dx","IntegrateAlgebraic[((1 + x^2)^2*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/((1 - x^2)^2*Sqrt[x + Sqrt[1 + x^2]]),x]","\frac{\left(-8-9 x-248 x^2-3 x^3-208 x^4+12 x^5+128 x^6\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(2-48 x+2 x^2-16 x^3-4 x^4+64 x^5\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(-3-96 x-9 x^2-272 x^3+12 x^4+128 x^5\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-16+4 x-48 x^2-4 x^3+64 x^4\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{24 \left(-1+x^2\right) \left(x+\sqrt{1+x^2}\right)^{7/2}}-\frac{1}{8} \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)+\text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{-2 \text{$\#$1}+\text{$\#$1}^3}\&\right]-\frac{1}{4} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+5 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-\text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^3}{2-2 \text{$\#$1}^2+\text{$\#$1}^4}\&\right]-\frac{1}{4} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+5 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","\frac{\left(-8-9 x-248 x^2-3 x^3-208 x^4+12 x^5+128 x^6\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(2-48 x+2 x^2-16 x^3-4 x^4+64 x^5\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(-3-96 x-9 x^2-272 x^3+12 x^4+128 x^5\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-16+4 x-48 x^2-4 x^3+64 x^4\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{24 \left(-1+x^2\right) \left(x+\sqrt{1+x^2}\right)^{7/2}}-\frac{1}{8} \tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)+\frac{1}{4} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-8 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+4 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]-\frac{1}{4} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+9 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-8 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+4 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]",1,"((-8 - 9*x - 248*x^2 - 3*x^3 - 208*x^4 + 12*x^5 + 128*x^6)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (2 - 48*x + 2*x^2 - 16*x^3 - 4*x^4 + 64*x^5)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((-3 - 96*x - 9*x^2 - 272*x^3 + 12*x^4 + 128*x^5)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-16 + 4*x - 48*x^2 - 4*x^3 + 64*x^4)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(24*(-1 + x^2)*(x + Sqrt[1 + x^2])^(7/2)) - ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]]/8 + RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (-Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2)/(-2*#1 + #1^3) & ] - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 5*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2)/(2*#1^3 - 3*#1^5 + #1^7) & ]/4 - RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-(Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1) + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^3)/(2 - 2*#1^2 + #1^4) & ] - RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 5*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ]/4","A",1
3113,1,914,639,1.3003991,"\int \frac{\left(1+x^2\right)^2 \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1-x^2\right)^2} \, dx","IntegrateAlgebraic[((1 + x^2)^2*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^2)^2,x]","\frac{\left(75+24 x-735 x^2+8 x^3-1050 x^4-32 x^5+240 x^6\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-16-18 x-16 x^2-6 x^3+32 x^4+24 x^5\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(8-120 x+24 x^2-1170 x^3-32 x^4+240 x^5\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-6-32 x-18 x^2+32 x^3+24 x^4\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{105 \left(-1+x^2\right) \left(x+\sqrt{1+x^2}\right)^{5/2}}-\tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)+\text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{5 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-4 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}-3 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]-\frac{1}{4} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+7 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}-4 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^3+2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^5}{-2+4 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]-\frac{1}{4} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-9 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","\frac{\left(75+24 x-735 x^2+8 x^3-1050 x^4-32 x^5+240 x^6\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-16-18 x-16 x^2-6 x^3+32 x^4+24 x^5\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(8-120 x+24 x^2-1170 x^3-32 x^4+240 x^5\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-6-32 x-18 x^2+32 x^3+24 x^4\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{105 \left(-1+x^2\right) \left(x+\sqrt{1+x^2}\right)^{5/2}}-\tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)+\frac{1}{4} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+13 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-15 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+7 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{1}{4} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+5 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-15 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+7 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]",1,"((75 + 24*x - 735*x^2 + 8*x^3 - 1050*x^4 - 32*x^5 + 240*x^6)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-16 - 18*x - 16*x^2 - 6*x^3 + 32*x^4 + 24*x^5)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((8 - 120*x + 24*x^2 - 1170*x^3 - 32*x^4 + 240*x^5)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-6 - 32*x - 18*x^2 + 32*x^3 + 24*x^4)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(105*(-1 + x^2)*(x + Sqrt[1 + x^2])^(5/2)) - ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] + RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (5*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 4*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1 - 3*#1^3 + #1^5) & ] - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 7*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(2*#1^3 - 3*#1^5 + #1^7) & ]/4 + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-(Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1) - 4*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^3 + 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^5)/(-2 + 4*#1^2 - 3*#1^4 + #1^6) & ] - RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 9*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ]/4","A",1
3114,1,914,639,0.0021045,"\int \frac{\left(1+x^2\right)^2 \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}}{\left(1-x^2\right)^2} \, dx","IntegrateAlgebraic[((1 + x^2)^2*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^2)^2,x]","\frac{\left(75+24 x-735 x^2+8 x^3-1050 x^4-32 x^5+240 x^6\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-16-18 x-16 x^2-6 x^3+32 x^4+24 x^5\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(8-120 x+24 x^2-1170 x^3-32 x^4+240 x^5\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-6-32 x-18 x^2+32 x^3+24 x^4\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{105 \left(-1+x^2\right) \left(x+\sqrt{1+x^2}\right)^{5/2}}-\tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)+\text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{5 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-4 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2+2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4}{2 \text{$\#$1}-3 \text{$\#$1}^3+\text{$\#$1}^5}\&\right]-\frac{1}{4} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+7 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}-4 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^3+2 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^5}{-2+4 \text{$\#$1}^2-3 \text{$\#$1}^4+\text{$\#$1}^6}\&\right]-\frac{1}{4} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)-9 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]","\frac{\left(75+24 x-735 x^2+8 x^3-1050 x^4-32 x^5+240 x^6\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-16-18 x-16 x^2-6 x^3+32 x^4+24 x^5\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\sqrt{1+x^2} \left(\left(8-120 x+24 x^2-1170 x^3-32 x^4+240 x^5\right) \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}+\left(-6-32 x-18 x^2+32 x^3+24 x^4\right) \sqrt{x+\sqrt{1+x^2}} \sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)}{105 \left(-1+x^2\right) \left(x+\sqrt{1+x^2}\right)^{5/2}}-\tanh ^{-1}\left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}\right)+\frac{1}{4} \text{RootSum}\left[-2+4 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+13 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-15 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+7 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{2 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]+\frac{1}{4} \text{RootSum}\left[2-8 \text{$\#$1}^2+8 \text{$\#$1}^4-4 \text{$\#$1}^6+\text{$\#$1}^8\&,\frac{-\log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right)+5 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^2-15 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^4+7 \log \left(\sqrt{1+\sqrt{x+\sqrt{1+x^2}}}-\text{$\#$1}\right) \text{$\#$1}^6}{-2 \text{$\#$1}+4 \text{$\#$1}^3-3 \text{$\#$1}^5+\text{$\#$1}^7}\&\right]",1,"((75 + 24*x - 735*x^2 + 8*x^3 - 1050*x^4 - 32*x^5 + 240*x^6)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-16 - 18*x - 16*x^2 - 6*x^3 + 32*x^4 + 24*x^5)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((8 - 120*x + 24*x^2 - 1170*x^3 - 32*x^4 + 240*x^5)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-6 - 32*x - 18*x^2 + 32*x^3 + 24*x^4)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(105*(-1 + x^2)*(x + Sqrt[1 + x^2])^(5/2)) - ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] + RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (5*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 4*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1 - 3*#1^3 + #1^5) & ] - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 7*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(2*#1^3 - 3*#1^5 + #1^7) & ]/4 + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-(Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1) - 4*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^3 + 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^5)/(-2 + 4*#1^2 - 3*#1^4 + #1^6) & ] - RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 9*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ]/4","A",1
3115,1,650,650,2.1363732,"\int \frac{x^2}{\sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}} \, dx","IntegrateAlgebraic[x^2/Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]],x]","\frac{\left(-491520 b^2-591360 b^3-533610 b^4+3932160 a b x+5304320 a b^2 x+365904 a b^3 x+3932160 a^2 b x^2+1774080 a^2 b^2 x^2+1067220 a^2 b^3 x^2-5242880 a^3 x^3-2293760 a^3 b x^3-3932160 a^4 x^4-5734400 a^5 x^5\right) \sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\left(409600 b^2+1005312 b^3+800415 b^4-1966080 a b x-1935360 a b^2 x-426888 a b^3 x-3276800 a^2 b x^2-2661120 a^2 b^2 x^2-1600830 a^2 b^3 x^2+2621440 a^3 x^3-1720320 a^3 b x^3+3276800 a^4 x^4+5160960 a^5 x^5\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\sqrt{-b+a^2 x^2} \left(\left(1310720 b+2007040 b^2+365904 b^3+1966080 a b x+1774080 a b^2 x+1067220 a b^3 x-5242880 a^2 x^2-5160960 a^2 b x^2-3932160 a^3 x^3-5734400 a^4 x^4\right) \sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\left(-655360 b-860160 b^2-426888 b^3-1638400 a b x-2661120 a b^2 x-1600830 a b^3 x+2621440 a^2 x^2+860160 a^2 b x^2+3276800 a^3 x^3+5160960 a^4 x^4\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{7096320 a^3 \sqrt{-b+a^2 x^2} \left(-b+4 a^2 x^2\right)+7096320 a^3 \left(-3 a b x+4 a^3 x^3\right)}+\frac{3 b^2 \tanh ^{-1}\left(\sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{16 a^3}+\frac{231 b^3 \tanh ^{-1}\left(\sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{2048 a^3}","\frac{\left(-491520 b^2-591360 b^3-533610 b^4+3932160 a b x+5304320 a b^2 x+365904 a b^3 x+3932160 a^2 b x^2+1774080 a^2 b^2 x^2+1067220 a^2 b^3 x^2-5242880 a^3 x^3-2293760 a^3 b x^3-3932160 a^4 x^4-5734400 a^5 x^5\right) \sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\left(409600 b^2+1005312 b^3+800415 b^4-1966080 a b x-1935360 a b^2 x-426888 a b^3 x-3276800 a^2 b x^2-2661120 a^2 b^2 x^2-1600830 a^2 b^3 x^2+2621440 a^3 x^3-1720320 a^3 b x^3+3276800 a^4 x^4+5160960 a^5 x^5\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\sqrt{-b+a^2 x^2} \left(\left(1310720 b+2007040 b^2+365904 b^3+1966080 a b x+1774080 a b^2 x+1067220 a b^3 x-5242880 a^2 x^2-5160960 a^2 b x^2-3932160 a^3 x^3-5734400 a^4 x^4\right) \sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}+\left(-655360 b-860160 b^2-426888 b^3-1638400 a b x-2661120 a b^2 x-1600830 a b^3 x+2621440 a^2 x^2+860160 a^2 b x^2+3276800 a^3 x^3+5160960 a^4 x^4\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{7096320 a^3 \sqrt{-b+a^2 x^2} \left(-b+4 a^2 x^2\right)+7096320 a^3 \left(-3 a b x+4 a^3 x^3\right)}+\frac{3 b^2 \tanh ^{-1}\left(\sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{16 a^3}+\frac{231 b^3 \tanh ^{-1}\left(\sqrt{1+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{2048 a^3}",1,"((-491520*b^2 - 591360*b^3 - 533610*b^4 + 3932160*a*b*x + 5304320*a*b^2*x + 365904*a*b^3*x + 3932160*a^2*b*x^2 + 1774080*a^2*b^2*x^2 + 1067220*a^2*b^3*x^2 - 5242880*a^3*x^3 - 2293760*a^3*b*x^3 - 3932160*a^4*x^4 - 5734400*a^5*x^5)*Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]] + (409600*b^2 + 1005312*b^3 + 800415*b^4 - 1966080*a*b*x - 1935360*a*b^2*x - 426888*a*b^3*x - 3276800*a^2*b*x^2 - 2661120*a^2*b^2*x^2 - 1600830*a^2*b^3*x^2 + 2621440*a^3*x^3 - 1720320*a^3*b*x^3 + 3276800*a^4*x^4 + 5160960*a^5*x^5)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]] + Sqrt[-b + a^2*x^2]*((1310720*b + 2007040*b^2 + 365904*b^3 + 1966080*a*b*x + 1774080*a*b^2*x + 1067220*a*b^3*x - 5242880*a^2*x^2 - 5160960*a^2*b*x^2 - 3932160*a^3*x^3 - 5734400*a^4*x^4)*Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]] + (-655360*b - 860160*b^2 - 426888*b^3 - 1638400*a*b*x - 2661120*a*b^2*x - 1600830*a*b^3*x + 2621440*a^2*x^2 + 860160*a^2*b*x^2 + 3276800*a^3*x^3 + 5160960*a^4*x^4)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]))/(7096320*a^3*Sqrt[-b + a^2*x^2]*(-b + 4*a^2*x^2) + 7096320*a^3*(-3*a*b*x + 4*a^3*x^3)) + (3*b^2*ArcTanh[Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]])/(16*a^3) + (231*b^3*ArcTanh[Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]])/(2048*a^3)","A",1
3116,1,669,669,2.0151135,"\int \sqrt[3]{\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}} \, dx","IntegrateAlgebraic[(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(1/3),x]","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{\sqrt[3]{1-a} \left(-2+2 x^2\right) \sqrt[3]{\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}}{\sqrt{3}}\right)}{2 \sqrt[3]{1-a}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{\sqrt[3]{1+a} \left(-2+2 x^2\right) \sqrt[3]{\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}}{\sqrt{3}}\right)}{2 \sqrt[3]{1+a}}+\frac{\log \left(1+\left(-\sqrt[3]{1-a}+\sqrt[3]{1-a} x^2\right) \sqrt[3]{\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}\right)}{2 \sqrt[3]{1-a}}+\frac{\log \left(-1+\left(-\sqrt[3]{1+a}+\sqrt[3]{1+a} x^2\right) \sqrt[3]{\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}\right)}{2 \sqrt[3]{1+a}}-\frac{\log \left(1+\left(\sqrt[3]{1-a}-\sqrt[3]{1-a} x^2\right) \sqrt[3]{\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}+\left((1-a)^{2/3}-2 (1-a)^{2/3} x^2+(1-a)^{2/3} x^4\right) \left(\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}\right)^{2/3}\right)}{4 \sqrt[3]{1-a}}-\frac{\log \left(1+\left(-\sqrt[3]{1+a}+\sqrt[3]{1+a} x^2\right) \sqrt[3]{\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}+\left((1+a)^{2/3}-2 (1+a)^{2/3} x^2+(1+a)^{2/3} x^4\right) \left(\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}\right)^{2/3}\right)}{4 \sqrt[3]{1+a}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{\sqrt[3]{1-a} \left(-2+2 x^2\right) \sqrt[3]{\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}}{\sqrt{3}}\right)}{2 \sqrt[3]{1-a}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{\sqrt[3]{1+a} \left(-2+2 x^2\right) \sqrt[3]{\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}}{\sqrt{3}}\right)}{2 \sqrt[3]{1+a}}+\frac{\log \left(1+\left(-\sqrt[3]{1-a}+\sqrt[3]{1-a} x^2\right) \sqrt[3]{\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}\right)}{2 \sqrt[3]{1-a}}+\frac{\log \left(-1+\left(-\sqrt[3]{1+a}+\sqrt[3]{1+a} x^2\right) \sqrt[3]{\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}\right)}{2 \sqrt[3]{1+a}}-\frac{\log \left(1+\left(\sqrt[3]{1-a}-\sqrt[3]{1-a} x^2\right) \sqrt[3]{\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}+\left((1-a)^{2/3}-2 (1-a)^{2/3} x^2+(1-a)^{2/3} x^4\right) \left(\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}\right)^{2/3}\right)}{4 \sqrt[3]{1-a}}-\frac{\log \left(1+\left(-\sqrt[3]{1+a}+\sqrt[3]{1+a} x^2\right) \sqrt[3]{\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}+\left((1+a)^{2/3}-2 (1+a)^{2/3} x^2+(1+a)^{2/3} x^4\right) \left(\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}\right)^{2/3}\right)}{4 \sqrt[3]{1+a}}",1,"-1/2*(Sqrt[3]*ArcTan[1/Sqrt[3] - ((1 - a)^(1/3)*(-2 + 2*x^2)*(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(1/3))/Sqrt[3]])/(1 - a)^(1/3) - (Sqrt[3]*ArcTan[1/Sqrt[3] + ((1 + a)^(1/3)*(-2 + 2*x^2)*(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(1/3))/Sqrt[3]])/(2*(1 + a)^(1/3)) + Log[1 + (-(1 - a)^(1/3) + (1 - a)^(1/3)*x^2)*(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(1/3)]/(2*(1 - a)^(1/3)) + Log[-1 + (-(1 + a)^(1/3) + (1 + a)^(1/3)*x^2)*(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(1/3)]/(2*(1 + a)^(1/3)) - Log[1 + ((1 - a)^(1/3) - (1 - a)^(1/3)*x^2)*(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(1/3) + ((1 - a)^(2/3) - 2*(1 - a)^(2/3)*x^2 + (1 - a)^(2/3)*x^4)*(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(2/3)]/(4*(1 - a)^(1/3)) - Log[1 + (-(1 + a)^(1/3) + (1 + a)^(1/3)*x^2)*(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(1/3) + ((1 + a)^(2/3) - 2*(1 + a)^(2/3)*x^2 + (1 + a)^(2/3)*x^4)*(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(2/3)]/(4*(1 + a)^(1/3))","A",1
3117,1,674,674,1.8913628,"\int \frac{\sqrt{-b+a^2 x^2} \sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}} \, dx","IntegrateAlgebraic[(Sqrt[-b + a^2*x^2]*(a*x + Sqrt[-b + a^2*x^2])^(1/3))/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4),x]","\frac{\left(-1860655104 b^2 c^4+2013265920 b c^{10}+2409402996 a b^2 c x-4541644800 a b c^7 x-4026531840 a^2 c^{10} x^2+6055526400 a^3 c^7 x^3\right) \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\left(1976946048 b^2 c^3-1761607680 b c^9-3011753745 a b^2 x+4314562560 a b c^6 x+3523215360 a^2 c^9 x^2-5752750080 a^3 c^6 x^3\right) \sqrt[3]{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\left(-2141691552 b^2 c^2+1614807040 b c^8-28943523840 a b c^5 x+2684354560 a c^{11} x-3229614080 a^2 c^8 x^2+5513052160 a^3 c^5 x^3\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{2/3} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\sqrt{-b+a^2 x^2} \left(\left(2409402996 b^2 c-1513881600 b c^7-4026531840 a c^{10} x+6055526400 a^2 c^7 x^2\right) \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\left(-3011753745 b^2+1438187520 b c^6+3523215360 a c^9 x-5752750080 a^2 c^6 x^2\right) \sqrt[3]{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\left(-26186997760 b c^5+2684354560 c^{11}-3229614080 a c^8 x+5513052160 a^2 c^5 x^2\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{2/3} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}\right)}{12404367360 a c^5 \left(a x+\sqrt{-b+a^2 x^2}\right)^{5/3}}-\frac{1989 b^2 \tan ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{16384 a c^{21/4}}+\frac{1989 b^2 \tanh ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{16384 a c^{21/4}}","\frac{\left(-1860655104 b^2 c^4+2013265920 b c^{10}+2409402996 a b^2 c x-4541644800 a b c^7 x-4026531840 a^2 c^{10} x^2+6055526400 a^3 c^7 x^3\right) \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\left(1976946048 b^2 c^3-1761607680 b c^9-3011753745 a b^2 x+4314562560 a b c^6 x+3523215360 a^2 c^9 x^2-5752750080 a^3 c^6 x^3\right) \sqrt[3]{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\left(-2141691552 b^2 c^2+1614807040 b c^8-28943523840 a b c^5 x+2684354560 a c^{11} x-3229614080 a^2 c^8 x^2+5513052160 a^3 c^5 x^3\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{2/3} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\sqrt{-b+a^2 x^2} \left(\left(2409402996 b^2 c-1513881600 b c^7-4026531840 a c^{10} x+6055526400 a^2 c^7 x^2\right) \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\left(-3011753745 b^2+1438187520 b c^6+3523215360 a c^9 x-5752750080 a^2 c^6 x^2\right) \sqrt[3]{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\left(-26186997760 b c^5+2684354560 c^{11}-3229614080 a c^8 x+5513052160 a^2 c^5 x^2\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{2/3} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}\right)}{12404367360 a c^5 \left(a x+\sqrt{-b+a^2 x^2}\right)^{5/3}}-\frac{1989 b^2 \tan ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{16384 a c^{21/4}}+\frac{1989 b^2 \tanh ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{16384 a c^{21/4}}",1,"((-1860655104*b^2*c^4 + 2013265920*b*c^10 + 2409402996*a*b^2*c*x - 4541644800*a*b*c^7*x - 4026531840*a^2*c^10*x^2 + 6055526400*a^3*c^7*x^3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + (1976946048*b^2*c^3 - 1761607680*b*c^9 - 3011753745*a*b^2*x + 4314562560*a*b*c^6*x + 3523215360*a^2*c^9*x^2 - 5752750080*a^3*c^6*x^3)*(a*x + Sqrt[-b + a^2*x^2])^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + (-2141691552*b^2*c^2 + 1614807040*b*c^8 - 28943523840*a*b*c^5*x + 2684354560*a*c^11*x - 3229614080*a^2*c^8*x^2 + 5513052160*a^3*c^5*x^3)*(a*x + Sqrt[-b + a^2*x^2])^(2/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + Sqrt[-b + a^2*x^2]*((2409402996*b^2*c - 1513881600*b*c^7 - 4026531840*a*c^10*x + 6055526400*a^2*c^7*x^2)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + (-3011753745*b^2 + 1438187520*b*c^6 + 3523215360*a*c^9*x - 5752750080*a^2*c^6*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + (-26186997760*b*c^5 + 2684354560*c^11 - 3229614080*a*c^8*x + 5513052160*a^2*c^5*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(2/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4)))/(12404367360*a*c^5*(a*x + Sqrt[-b + a^2*x^2])^(5/3)) - (1989*b^2*ArcTan[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)])/(16384*a*c^(21/4)) + (1989*b^2*ArcTanh[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)])/(16384*a*c^(21/4))","A",1
3118,1,1211,678,7.507223,"\int \frac{\sqrt[6]{a x+\sqrt{-b+a^2 x^2}}}{x^3 \sqrt{-b+a^2 x^2}} \, dx","IntegrateAlgebraic[(a*x + Sqrt[-b + a^2*x^2])^(1/6)/(x^3*Sqrt[-b + a^2*x^2]),x]","-\frac{5}{24 x^2 \left(a x+\sqrt{-b+a^2 x^2}\right)^{5/6}}+\frac{7 \left(a x+\sqrt{-b+a^2 x^2}\right)^{7/6}}{24 b x^2}+\frac{35 a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[12]{b} \sqrt[6]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[6]{b}+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}\right)}{36 \sqrt{2} b^{17/12}}-\frac{35 a^2 \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{3}} \sqrt[12]{b} \sqrt[6]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[6]{b}+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}\right)}{144 \sqrt{2} b^{17/12}}-\frac{35 a^2 \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{3}} \sqrt[12]{b} \sqrt[6]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[6]{b}+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}\right)}{48 \sqrt{6} b^{17/12}}-\frac{35 \sqrt{2+\sqrt{3}} a^2 \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{3}} \sqrt[12]{b} \sqrt[6]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[6]{b}+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}\right)}{144 b^{17/12}}+\frac{35 a^2 \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}} \sqrt[12]{b} \sqrt[6]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[6]{b}+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}\right)}{144 \sqrt{2} b^{17/12}}-\frac{35 a^2 \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}} \sqrt[12]{b} \sqrt[6]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[6]{b}+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}\right)}{48 \sqrt{6} b^{17/12}}-\frac{35 \sqrt{2-\sqrt{3}} a^2 \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}} \sqrt[12]{b} \sqrt[6]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[6]{b}+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}\right)}{144 b^{17/12}}+\frac{35 a^2 \tanh ^{-1}\left(\frac{\frac{\sqrt[12]{b}}{\sqrt{2}}+\frac{\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}{\sqrt{2} \sqrt[12]{b}}}{\sqrt[6]{a x+\sqrt{-b+a^2 x^2}}}\right)}{36 \sqrt{2} b^{17/12}}+\frac{35 a^2 \tanh ^{-1}\left(\frac{\sqrt{2-\sqrt{3}} \sqrt[12]{b}+\frac{\sqrt{2-\sqrt{3}} \sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}{\sqrt[12]{b}}}{\sqrt[6]{a x+\sqrt{-b+a^2 x^2}}}\right)}{72 \sqrt{2} b^{17/12}}-\frac{35 a^2 \tanh ^{-1}\left(\frac{\sqrt{2-\sqrt{3}} \sqrt[12]{b}+\frac{\sqrt{2-\sqrt{3}} \sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}{\sqrt[12]{b}}}{\sqrt[6]{a x+\sqrt{-b+a^2 x^2}}}\right)}{24 \sqrt{6} b^{17/12}}-\frac{35 a^2 \tanh ^{-1}\left(\frac{\sqrt{2+\sqrt{3}} \sqrt[12]{b}+\frac{\sqrt{2+\sqrt{3}} \sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}{\sqrt[12]{b}}}{\sqrt[6]{a x+\sqrt{-b+a^2 x^2}}}\right)}{72 \sqrt{2} b^{17/12}}-\frac{35 a^2 \tanh ^{-1}\left(\frac{\sqrt{2+\sqrt{3}} \sqrt[12]{b}+\frac{\sqrt{2+\sqrt{3}} \sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}{\sqrt[12]{b}}}{\sqrt[6]{a x+\sqrt{-b+a^2 x^2}}}\right)}{24 \sqrt{6} b^{17/12}}","-\frac{5}{24 x^2 \left(a x+\sqrt{-b+a^2 x^2}\right)^{5/6}}+\frac{7 \left(a x+\sqrt{-b+a^2 x^2}\right)^{7/6}}{24 b x^2}-\frac{35 \sqrt{2+\sqrt{3}} a^2 \tan ^{-1}\left(\frac{\left(\sqrt{\frac{3}{2}} \sqrt[12]{b}-\frac{\sqrt[12]{b}}{\sqrt{2}}\right) \sqrt[6]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[6]{b}+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}\right)}{72 b^{17/12}}-\frac{35 \sqrt{2-\sqrt{3}} a^2 \tan ^{-1}\left(\frac{\left(\sqrt{\frac{3}{2}} \sqrt[12]{b}+\frac{\sqrt[12]{b}}{\sqrt{2}}\right) \sqrt[6]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[6]{b}+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}\right)}{72 b^{17/12}}+\frac{35 a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[12]{b} \sqrt[6]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[6]{b}+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}\right)}{36 \sqrt{2} b^{17/12}}+\frac{35 a^2 \tanh ^{-1}\left(\frac{\frac{\sqrt[12]{b}}{\sqrt{2}}+\frac{\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}{\sqrt{2} \sqrt[12]{b}}}{\sqrt[6]{a x+\sqrt{-b+a^2 x^2}}}\right)}{36 \sqrt{2} b^{17/12}}-\frac{35 \sqrt{2+\sqrt{3}} a^2 \tanh ^{-1}\left(\frac{\frac{\sqrt{2} \sqrt[12]{b}}{-1+\sqrt{3}}+\frac{\sqrt{2} \sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}{\left(-1+\sqrt{3}\right) \sqrt[12]{b}}}{\sqrt[6]{a x+\sqrt{-b+a^2 x^2}}}\right)}{72 b^{17/12}}-\frac{35 \sqrt{2-\sqrt{3}} a^2 \tanh ^{-1}\left(\frac{\frac{\sqrt{2} \sqrt[12]{b}}{1+\sqrt{3}}+\frac{\sqrt{2} \sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}{\left(1+\sqrt{3}\right) \sqrt[12]{b}}}{\sqrt[6]{a x+\sqrt{-b+a^2 x^2}}}\right)}{72 b^{17/12}}",1,"-5/(24*x^2*(a*x + Sqrt[-b + a^2*x^2])^(5/6)) + (7*(a*x + Sqrt[-b + a^2*x^2])^(7/6))/(24*b*x^2) + (35*a^2*ArcTan[(Sqrt[2]*b^(1/12)*(a*x + Sqrt[-b + a^2*x^2])^(1/6))/(-b^(1/6) + (a*x + Sqrt[-b + a^2*x^2])^(1/3))])/(36*Sqrt[2]*b^(17/12)) - (35*a^2*ArcTan[(Sqrt[2 - Sqrt[3]]*b^(1/12)*(a*x + Sqrt[-b + a^2*x^2])^(1/6))/(-b^(1/6) + (a*x + Sqrt[-b + a^2*x^2])^(1/3))])/(144*Sqrt[2]*b^(17/12)) - (35*a^2*ArcTan[(Sqrt[2 - Sqrt[3]]*b^(1/12)*(a*x + Sqrt[-b + a^2*x^2])^(1/6))/(-b^(1/6) + (a*x + Sqrt[-b + a^2*x^2])^(1/3))])/(48*Sqrt[6]*b^(17/12)) - (35*Sqrt[2 + Sqrt[3]]*a^2*ArcTan[(Sqrt[2 - Sqrt[3]]*b^(1/12)*(a*x + Sqrt[-b + a^2*x^2])^(1/6))/(-b^(1/6) + (a*x + Sqrt[-b + a^2*x^2])^(1/3))])/(144*b^(17/12)) + (35*a^2*ArcTan[(Sqrt[2 + Sqrt[3]]*b^(1/12)*(a*x + Sqrt[-b + a^2*x^2])^(1/6))/(-b^(1/6) + (a*x + Sqrt[-b + a^2*x^2])^(1/3))])/(144*Sqrt[2]*b^(17/12)) - (35*a^2*ArcTan[(Sqrt[2 + Sqrt[3]]*b^(1/12)*(a*x + Sqrt[-b + a^2*x^2])^(1/6))/(-b^(1/6) + (a*x + Sqrt[-b + a^2*x^2])^(1/3))])/(48*Sqrt[6]*b^(17/12)) - (35*Sqrt[2 - Sqrt[3]]*a^2*ArcTan[(Sqrt[2 + Sqrt[3]]*b^(1/12)*(a*x + Sqrt[-b + a^2*x^2])^(1/6))/(-b^(1/6) + (a*x + Sqrt[-b + a^2*x^2])^(1/3))])/(144*b^(17/12)) + (35*a^2*ArcTanh[(b^(1/12)/Sqrt[2] + (a*x + Sqrt[-b + a^2*x^2])^(1/3)/(Sqrt[2]*b^(1/12)))/(a*x + Sqrt[-b + a^2*x^2])^(1/6)])/(36*Sqrt[2]*b^(17/12)) + (35*a^2*ArcTanh[(Sqrt[2 - Sqrt[3]]*b^(1/12) + (Sqrt[2 - Sqrt[3]]*(a*x + Sqrt[-b + a^2*x^2])^(1/3))/b^(1/12))/(a*x + Sqrt[-b + a^2*x^2])^(1/6)])/(72*Sqrt[2]*b^(17/12)) - (35*a^2*ArcTanh[(Sqrt[2 - Sqrt[3]]*b^(1/12) + (Sqrt[2 - Sqrt[3]]*(a*x + Sqrt[-b + a^2*x^2])^(1/3))/b^(1/12))/(a*x + Sqrt[-b + a^2*x^2])^(1/6)])/(24*Sqrt[6]*b^(17/12)) - (35*a^2*ArcTanh[(Sqrt[2 + Sqrt[3]]*b^(1/12) + (Sqrt[2 + Sqrt[3]]*(a*x + Sqrt[-b + a^2*x^2])^(1/3))/b^(1/12))/(a*x + Sqrt[-b + a^2*x^2])^(1/6)])/(72*Sqrt[2]*b^(17/12)) - (35*a^2*ArcTanh[(Sqrt[2 + Sqrt[3]]*b^(1/12) + (Sqrt[2 + Sqrt[3]]*(a*x + Sqrt[-b + a^2*x^2])^(1/3))/b^(1/12))/(a*x + Sqrt[-b + a^2*x^2])^(1/6)])/(24*Sqrt[6]*b^(17/12))","A",1
3119,1,685,685,9.5190565,"\int \frac{x^2 \left(x^2 c_3-c_4\right) \sqrt[4]{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{\left(-x+x^2 c_3+c_4\right) \left(x+x^2 c_3+c_4\right) \left(x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right)} \, dx","IntegrateAlgebraic[(x^2*(x^2*C[3] - C[4])*((x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4]))^(1/4))/((-x + x^2*C[3] + C[4])*(x + x^2*C[3] + C[4])*(x^2 + x^4*C[3]^2 + 2*x^2*C[3]*C[4] + C[4]^2)),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{1+c_0} \sqrt[4]{-1-c_1} \sqrt[4]{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{\sqrt{1+c_0}-\sqrt{-1-c_1} \sqrt{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}\right) \sqrt[4]{1+c_0}}{2 \sqrt{2} \sqrt[4]{-1-c_1}}-\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[4]{1+c_0}}{\sqrt{2} \sqrt[4]{-1-c_1}}+\frac{\sqrt[4]{-1-c_1} \sqrt{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{\sqrt{2} \sqrt[4]{1+c_0}}}{\sqrt[4]{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}\right) \sqrt[4]{1+c_0}}{2 \sqrt{2} \sqrt[4]{-1-c_1}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+c_0} \sqrt[4]{1-c_1} \sqrt[4]{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{\sqrt{-1+c_0}-\sqrt{1-c_1} \sqrt{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}\right) \sqrt[4]{-1+c_0}}{2 \sqrt{2} \sqrt[4]{1-c_1}}+\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[4]{-1+c_0}}{\sqrt{2} \sqrt[4]{1-c_1}}+\frac{\sqrt[4]{1-c_1} \sqrt{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{\sqrt{2} \sqrt[4]{-1+c_0}}}{\sqrt[4]{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}\right) \sqrt[4]{-1+c_0}}{2 \sqrt{2} \sqrt[4]{1-c_1}}+\frac{1}{4} (c_0-c_1) \text{RootSum}\left[1+c_0{}^2-2 \text{$\#$1}^4-2 c_0 c_1 \text{$\#$1}^4+\text{$\#$1}^8+c_1{}^2 \text{$\#$1}^8\&,\frac{\log \left(\sqrt[4]{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}-\text{$\#$1}\right) \text{$\#$1}}{-1-c_0 c_1+\text{$\#$1}^4+c_1{}^2 \text{$\#$1}^4}\&\right]","\frac{\tan ^{-1}\left(\frac{\frac{\sqrt[4]{1+c_0}}{\sqrt{2} \sqrt[4]{-1-c_1}}-\frac{\sqrt[4]{-1-c_1} \sqrt{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{\sqrt{2} \sqrt[4]{1+c_0}}}{\sqrt[4]{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}\right) \sqrt[4]{1+c_0}}{2 \sqrt{2} \sqrt[4]{-1-c_1}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{1+c_0} \sqrt[4]{-1-c_1} \sqrt[4]{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{\sqrt{1+c_0}+\sqrt{-1-c_1} \sqrt{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}\right) \sqrt[4]{1+c_0}}{2 \sqrt{2} \sqrt[4]{-1-c_1}}-\frac{\tan ^{-1}\left(\frac{\frac{\sqrt[4]{-1+c_0}}{\sqrt{2} \sqrt[4]{1-c_1}}-\frac{\sqrt[4]{1-c_1} \sqrt{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{\sqrt{2} \sqrt[4]{-1+c_0}}}{\sqrt[4]{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}\right) \sqrt[4]{-1+c_0}}{2 \sqrt{2} \sqrt[4]{1-c_1}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{-1+c_0} \sqrt[4]{1-c_1} \sqrt[4]{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{\sqrt{-1+c_0}+\sqrt{1-c_1} \sqrt{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}\right) \sqrt[4]{-1+c_0}}{2 \sqrt{2} \sqrt[4]{1-c_1}}+\frac{1}{4} (c_0-c_1) \text{RootSum}\left[1+c_0{}^2-2 \text{$\#$1}^4-2 c_0 c_1 \text{$\#$1}^4+\text{$\#$1}^8+c_1{}^2 \text{$\#$1}^8\&,\frac{\log \left(\sqrt[4]{\frac{x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}-\text{$\#$1}\right) \text{$\#$1}}{-1-c_0 c_1+\text{$\#$1}^4+c_1{}^2 \text{$\#$1}^4}\&\right]",1,"-1/2*(ArcTan[(Sqrt[2]*(1 + C[0])^(1/4)*(-1 - C[1])^(1/4)*((x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4]))^(1/4))/(Sqrt[1 + C[0]] - Sqrt[-1 - C[1]]*Sqrt[(x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4])])]*(1 + C[0])^(1/4))/(Sqrt[2]*(-1 - C[1])^(1/4)) - (ArcTanh[((1 + C[0])^(1/4)/(Sqrt[2]*(-1 - C[1])^(1/4)) + ((-1 - C[1])^(1/4)*Sqrt[(x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4])])/(Sqrt[2]*(1 + C[0])^(1/4)))/((x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4]))^(1/4)]*(1 + C[0])^(1/4))/(2*Sqrt[2]*(-1 - C[1])^(1/4)) + (ArcTan[(Sqrt[2]*(-1 + C[0])^(1/4)*(1 - C[1])^(1/4)*((x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4]))^(1/4))/(Sqrt[-1 + C[0]] - Sqrt[1 - C[1]]*Sqrt[(x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4])])]*(-1 + C[0])^(1/4))/(2*Sqrt[2]*(1 - C[1])^(1/4)) + (ArcTanh[((-1 + C[0])^(1/4)/(Sqrt[2]*(1 - C[1])^(1/4)) + ((1 - C[1])^(1/4)*Sqrt[(x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4])])/(Sqrt[2]*(-1 + C[0])^(1/4)))/((x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4]))^(1/4)]*(-1 + C[0])^(1/4))/(2*Sqrt[2]*(1 - C[1])^(1/4)) + ((C[0] - C[1])*RootSum[1 + C[0]^2 - 2*#1^4 - 2*C[0]*C[1]*#1^4 + #1^8 + C[1]^2*#1^8 & , (Log[((x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4]))^(1/4) - #1]*#1)/(-1 - C[0]*C[1] + #1^4 + C[1]^2*#1^4) & ])/4","A",1
3120,1,697,697,1.5246514,"\int \frac{1}{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}} \, dx","IntegrateAlgebraic[1/((a*x + Sqrt[-b + a^2*x^2])^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)),x]","\frac{\left(1944 b c^4+3640 a b x\right) \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}+\left(-2106 b c^3+6561 a c^7 x\right) \sqrt[4]{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}+\left(2340 b c^2-4374 a c^6 x\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}+\left(-2730 b c+3645 a c^5 x\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4} \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}+\sqrt{-b+a^2 x^2} \left(3640 b \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}+6561 c^7 \sqrt[4]{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}-4374 c^6 \sqrt{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}+3645 c^5 \left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4} \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}\right)}{4860 a c^5 \left(a x+\sqrt{-b+a^2 x^2}\right)^{5/4}}+\frac{182 b \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[3]{c}}\right)}{243 \sqrt{3} a c^{16/3}}+\frac{182 b \log \left(-\sqrt[3]{c}+\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{729 a c^{16/3}}-\frac{91 b \log \left(c^{2/3}+\sqrt[3]{c} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}\right)}{729 a c^{16/3}}","\frac{\left(1944 b c^4+3640 a b x\right) \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}+\left(-2106 b c^3+6561 a c^7 x\right) \sqrt[4]{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}+\left(2340 b c^2-4374 a c^6 x\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}+\left(-2730 b c+3645 a c^5 x\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4} \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}+\sqrt{-b+a^2 x^2} \left(3640 b \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}+6561 c^7 \sqrt[4]{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}-4374 c^6 \sqrt{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}+3645 c^5 \left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4} \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}\right)}{4860 a c^5 \left(a x+\sqrt{-b+a^2 x^2}\right)^{5/4}}+\frac{182 b \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[3]{c}}\right)}{243 \sqrt{3} a c^{16/3}}+\frac{182 b \log \left(-\sqrt[3]{c}+\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{729 a c^{16/3}}-\frac{91 b \log \left(c^{2/3}+\sqrt[3]{c} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}\right)}{729 a c^{16/3}}",1,"((1944*b*c^4 + 3640*a*b*x)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3) + (-2106*b*c^3 + 6561*a*c^7*x)*(a*x + Sqrt[-b + a^2*x^2])^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3) + (2340*b*c^2 - 4374*a*c^6*x)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3) + (-2730*b*c + 3645*a*c^5*x)*(a*x + Sqrt[-b + a^2*x^2])^(3/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3) + Sqrt[-b + a^2*x^2]*(3640*b*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3) + 6561*c^7*(a*x + Sqrt[-b + a^2*x^2])^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3) - 4374*c^6*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3) + 3645*c^5*(a*x + Sqrt[-b + a^2*x^2])^(3/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3)))/(4860*a*c^5*(a*x + Sqrt[-b + a^2*x^2])^(5/4)) + (182*b*ArcTan[1/Sqrt[3] + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3))/(Sqrt[3]*c^(1/3))])/(243*Sqrt[3]*a*c^(16/3)) + (182*b*Log[-c^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)])/(729*a*c^(16/3)) - (91*b*Log[c^(2/3) + c^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3)])/(729*a*c^(16/3))","A",1
3121,1,699,699,1.1588206,"\int \frac{1}{\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}} \, dx","IntegrateAlgebraic[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(-1/3),x]","\frac{\left(8910 b c^3-19683 a c^7 x\right) \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}+\left(-9900 b c^2+13122 a c^6 x\right) \sqrt[4]{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}+\left(11550 b c-10935 a c^5 x\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}+\left(-15400 b+9720 a c^4 x\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4} \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}+\sqrt{-b+a^2 x^2} \left(-19683 c^7 \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}+13122 c^6 \sqrt[4]{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}-10935 c^5 \sqrt{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}+9720 c^4 \left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4} \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}\right)}{17820 a^2 c^4 x+17820 a c^4 \sqrt{-b+a^2 x^2}}-\frac{70 b \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[3]{c}}\right)}{81 \sqrt{3} a c^{13/3}}-\frac{70 b \log \left(-\sqrt[3]{c}+\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{243 a c^{13/3}}+\frac{35 b \log \left(c^{2/3}+\sqrt[3]{c} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}\right)}{243 a c^{13/3}}","\frac{\left(8910 b c^3-19683 a c^7 x\right) \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}+\left(-9900 b c^2+13122 a c^6 x\right) \sqrt[4]{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}+\left(11550 b c-10935 a c^5 x\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}+\left(-15400 b+9720 a c^4 x\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4} \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}+\sqrt{-b+a^2 x^2} \left(-19683 c^7 \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}+13122 c^6 \sqrt[4]{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}-10935 c^5 \sqrt{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}+9720 c^4 \left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4} \left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}\right)}{17820 a^2 c^4 x+17820 a c^4 \sqrt{-b+a^2 x^2}}-\frac{70 b \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[3]{c}}\right)}{81 \sqrt{3} a c^{13/3}}-\frac{70 b \log \left(-\sqrt[3]{c}+\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{243 a c^{13/3}}+\frac{35 b \log \left(c^{2/3}+\sqrt[3]{c} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}\right)}{243 a c^{13/3}}",1,"((8910*b*c^3 - 19683*a*c^7*x)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3) + (-9900*b*c^2 + 13122*a*c^6*x)*(a*x + Sqrt[-b + a^2*x^2])^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3) + (11550*b*c - 10935*a*c^5*x)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3) + (-15400*b + 9720*a*c^4*x)*(a*x + Sqrt[-b + a^2*x^2])^(3/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3) + Sqrt[-b + a^2*x^2]*(-19683*c^7*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3) + 13122*c^6*(a*x + Sqrt[-b + a^2*x^2])^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3) - 10935*c^5*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3) + 9720*c^4*(a*x + Sqrt[-b + a^2*x^2])^(3/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3)))/(17820*a^2*c^4*x + 17820*a*c^4*Sqrt[-b + a^2*x^2]) - (70*b*ArcTan[1/Sqrt[3] + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3))/(Sqrt[3]*c^(1/3))])/(81*Sqrt[3]*a*c^(13/3)) - (70*b*Log[-c^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)])/(243*a*c^(13/3)) + (35*b*Log[c^(2/3) + c^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3)])/(243*a*c^(13/3))","A",1
3122,1,708,708,2.0348059,"\int \sqrt[4]{\frac{1+a x-4 x^2-4 a x^3+6 x^4+6 a x^5-4 x^6-4 a x^7+x^8+a x^9}{-c+b x}} \, dx","IntegrateAlgebraic[((1 + a*x - 4*x^2 - 4*a*x^3 + 6*x^4 + 6*a*x^5 - 4*x^6 - 4*a*x^7 + x^8 + a*x^9)/(-c + b*x))^(1/4),x]","\frac{7 b^2+96 a^2 b^2-6 a b c-45 a^2 c^2+3 a b^2 x+96 a^3 b^2 x-42 a^2 b c x-45 a^3 c^2 x-21 b^2 x^2-324 a^2 b^2 x^2+18 a b c x^2-36 a^3 b c x^2+135 a^2 c^2 x^2-9 a b^2 x^3-320 a^3 b^2 x^3+126 a^2 b c x^3+135 a^3 c^2 x^3+21 b^2 x^4+396 a^2 b^2 x^4-18 a b c x^4+108 a^3 b c x^4-135 a^2 c^2 x^4+9 a b^2 x^5+384 a^3 b^2 x^5-126 a^2 b c x^5-135 a^3 c^2 x^5-7 b^2 x^6-204 a^2 b^2 x^6+6 a b c x^6-108 a^3 b c x^6+45 a^2 c^2 x^6-3 a b^2 x^7-192 a^3 b^2 x^7+42 a^2 b c x^7+45 a^3 c^2 x^7+36 a^2 b^2 x^8+36 a^3 b c x^8+32 a^3 b^2 x^9}{96 a^2 b^3 \left(\frac{1+a x-4 x^2-4 a x^3+6 x^4+6 a x^5-4 x^6-4 a x^7+x^8+a x^9}{-c+b x}\right)^{3/4}}+\frac{\left(-7 b^3+32 a^2 b^3+3 a b^2 c+32 a^3 b^2 c-5 a^2 b c^2-15 a^3 c^3\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} (-1+x) (1+x)}{\sqrt[4]{b} \sqrt[4]{\frac{1+a x-4 x^2-4 a x^3+6 x^4+6 a x^5-4 x^6-4 a x^7+x^8+a x^9}{-c+b x}}}\right)}{64 a^{11/4} b^{13/4}}+\frac{\left(7 b^3-32 a^2 b^3-3 a b^2 c-32 a^3 b^2 c+5 a^2 b c^2+15 a^3 c^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} (-1+x) (1+x)}{\sqrt[4]{b} \sqrt[4]{\frac{1+a x-4 x^2-4 a x^3+6 x^4+6 a x^5-4 x^6-4 a x^7+x^8+a x^9}{-c+b x}}}\right)}{64 a^{11/4} b^{13/4}}","\frac{7 b^2+96 a^2 b^2-6 a b c-45 a^2 c^2+3 a b^2 x+96 a^3 b^2 x-42 a^2 b c x-45 a^3 c^2 x-21 b^2 x^2-324 a^2 b^2 x^2+18 a b c x^2-36 a^3 b c x^2+135 a^2 c^2 x^2-9 a b^2 x^3-320 a^3 b^2 x^3+126 a^2 b c x^3+135 a^3 c^2 x^3+21 b^2 x^4+396 a^2 b^2 x^4-18 a b c x^4+108 a^3 b c x^4-135 a^2 c^2 x^4+9 a b^2 x^5+384 a^3 b^2 x^5-126 a^2 b c x^5-135 a^3 c^2 x^5-7 b^2 x^6-204 a^2 b^2 x^6+6 a b c x^6-108 a^3 b c x^6+45 a^2 c^2 x^6-3 a b^2 x^7-192 a^3 b^2 x^7+42 a^2 b c x^7+45 a^3 c^2 x^7+36 a^2 b^2 x^8+36 a^3 b c x^8+32 a^3 b^2 x^9}{96 a^2 b^3 \left(\frac{1+a x-4 x^2-4 a x^3+6 x^4+6 a x^5-4 x^6-4 a x^7+x^8+a x^9}{-c+b x}\right)^{3/4}}+\frac{\left(-7 b^3+32 a^2 b^3+3 a b^2 c+32 a^3 b^2 c-5 a^2 b c^2-15 a^3 c^3\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} (-1+x) (1+x)}{\sqrt[4]{b} \sqrt[4]{\frac{1+a x-4 x^2-4 a x^3+6 x^4+6 a x^5-4 x^6-4 a x^7+x^8+a x^9}{-c+b x}}}\right)}{64 a^{11/4} b^{13/4}}+\frac{\left(7 b^3-32 a^2 b^3-3 a b^2 c-32 a^3 b^2 c+5 a^2 b c^2+15 a^3 c^3\right) \tanh ^{-1}\left(\frac{\sqrt[4]{a} (-1+x) (1+x)}{\sqrt[4]{b} \sqrt[4]{\frac{1+a x-4 x^2-4 a x^3+6 x^4+6 a x^5-4 x^6-4 a x^7+x^8+a x^9}{-c+b x}}}\right)}{64 a^{11/4} b^{13/4}}",1,"(7*b^2 + 96*a^2*b^2 - 6*a*b*c - 45*a^2*c^2 + 3*a*b^2*x + 96*a^3*b^2*x - 42*a^2*b*c*x - 45*a^3*c^2*x - 21*b^2*x^2 - 324*a^2*b^2*x^2 + 18*a*b*c*x^2 - 36*a^3*b*c*x^2 + 135*a^2*c^2*x^2 - 9*a*b^2*x^3 - 320*a^3*b^2*x^3 + 126*a^2*b*c*x^3 + 135*a^3*c^2*x^3 + 21*b^2*x^4 + 396*a^2*b^2*x^4 - 18*a*b*c*x^4 + 108*a^3*b*c*x^4 - 135*a^2*c^2*x^4 + 9*a*b^2*x^5 + 384*a^3*b^2*x^5 - 126*a^2*b*c*x^5 - 135*a^3*c^2*x^5 - 7*b^2*x^6 - 204*a^2*b^2*x^6 + 6*a*b*c*x^6 - 108*a^3*b*c*x^6 + 45*a^2*c^2*x^6 - 3*a*b^2*x^7 - 192*a^3*b^2*x^7 + 42*a^2*b*c*x^7 + 45*a^3*c^2*x^7 + 36*a^2*b^2*x^8 + 36*a^3*b*c*x^8 + 32*a^3*b^2*x^9)/(96*a^2*b^3*((1 + a*x - 4*x^2 - 4*a*x^3 + 6*x^4 + 6*a*x^5 - 4*x^6 - 4*a*x^7 + x^8 + a*x^9)/(-c + b*x))^(3/4)) + ((-7*b^3 + 32*a^2*b^3 + 3*a*b^2*c + 32*a^3*b^2*c - 5*a^2*b*c^2 - 15*a^3*c^3)*ArcTan[(a^(1/4)*(-1 + x)*(1 + x))/(b^(1/4)*((1 + a*x - 4*x^2 - 4*a*x^3 + 6*x^4 + 6*a*x^5 - 4*x^6 - 4*a*x^7 + x^8 + a*x^9)/(-c + b*x))^(1/4))])/(64*a^(11/4)*b^(13/4)) + ((7*b^3 - 32*a^2*b^3 - 3*a*b^2*c - 32*a^3*b^2*c + 5*a^2*b*c^2 + 15*a^3*c^3)*ArcTanh[(a^(1/4)*(-1 + x)*(1 + x))/(b^(1/4)*((1 + a*x - 4*x^2 - 4*a*x^3 + 6*x^4 + 6*a*x^5 - 4*x^6 - 4*a*x^7 + x^8 + a*x^9)/(-c + b*x))^(1/4))])/(64*a^(11/4)*b^(13/4))","A",1
3123,1,719,719,1.2395624,"\int \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}} \, dx","IntegrateAlgebraic[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3),x]","\frac{\left(2835 b c^3-19683 a c^7 x+68040 a^2 c^3 x^2\right) \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(4095 b c^2+6561 a c^6 x\right) \sqrt[4]{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(-5460 b c-4374 a c^5 x\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(9100 b+3402 a c^4 x\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\sqrt{-b+a^2 x^2} \left(\left(-19683 c^7+68040 a c^3 x\right) \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+6561 c^6 \sqrt[4]{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}-4374 c^5 \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+3402 c^4 \left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{73710 a^2 c^3 x+73710 a c^3 \sqrt{-b+a^2 x^2}}-\frac{20 b \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[3]{c}}\right)}{81 \sqrt{3} a c^{11/3}}+\frac{20 b \log \left(-\sqrt[3]{c}+\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{243 a c^{11/3}}-\frac{10 b \log \left(c^{2/3}+\sqrt[3]{c} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}\right)}{243 a c^{11/3}}","\frac{\left(2835 b c^3-19683 a c^7 x+68040 a^2 c^3 x^2\right) \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(4095 b c^2+6561 a c^6 x\right) \sqrt[4]{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(-5460 b c-4374 a c^5 x\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(9100 b+3402 a c^4 x\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\sqrt{-b+a^2 x^2} \left(\left(-19683 c^7+68040 a c^3 x\right) \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+6561 c^6 \sqrt[4]{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}-4374 c^5 \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+3402 c^4 \left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{73710 a^2 c^3 x+73710 a c^3 \sqrt{-b+a^2 x^2}}-\frac{20 b \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[3]{c}}\right)}{81 \sqrt{3} a c^{11/3}}+\frac{20 b \log \left(-\sqrt[3]{c}+\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{243 a c^{11/3}}-\frac{10 b \log \left(c^{2/3}+\sqrt[3]{c} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}\right)}{243 a c^{11/3}}",1,"((2835*b*c^3 - 19683*a*c^7*x + 68040*a^2*c^3*x^2)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (4095*b*c^2 + 6561*a*c^6*x)*(a*x + Sqrt[-b + a^2*x^2])^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (-5460*b*c - 4374*a*c^5*x)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (9100*b + 3402*a*c^4*x)*(a*x + Sqrt[-b + a^2*x^2])^(3/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + Sqrt[-b + a^2*x^2]*((-19683*c^7 + 68040*a*c^3*x)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + 6561*c^6*(a*x + Sqrt[-b + a^2*x^2])^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) - 4374*c^5*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + 3402*c^4*(a*x + Sqrt[-b + a^2*x^2])^(3/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)))/(73710*a^2*c^3*x + 73710*a*c^3*Sqrt[-b + a^2*x^2]) - (20*b*ArcTan[1/Sqrt[3] + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3))/(Sqrt[3]*c^(1/3))])/(81*Sqrt[3]*a*c^(11/3)) + (20*b*Log[-c^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)])/(243*a*c^(11/3)) - (10*b*Log[c^(2/3) + c^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3)])/(243*a*c^(11/3))","A",1
3124,1,698,725,3.2537448,"\int \frac{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{x \left(-b+a^2 x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/(x*(-b + a^2*x^2)^(3/2)),x]","\frac{\left(a x+\sqrt{-b+a^2 x^2}\right)^{5/4}}{b \left(b-a^2 x^2-a x \sqrt{-b+a^2 x^2}\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{\sqrt[8]{b}}\right)}{2 b^{11/8}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[8]{b} \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[4]{b}+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{2 \sqrt{2} b^{11/8}}+\frac{\sqrt{2+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{b} \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[4]{b}+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{b^{11/8}}-\frac{\sqrt{2-\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{b} \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[4]{b}+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{b^{11/8}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{\sqrt[8]{b}}\right)}{2 b^{11/8}}+\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[8]{b}}{\sqrt{2}}+\frac{\sqrt{a x+\sqrt{-b+a^2 x^2}}}{\sqrt{2} \sqrt[8]{b}}}{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{2 \sqrt{2} b^{11/8}}+\frac{\sqrt{2-\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{1-\frac{1}{\sqrt{2}}} \sqrt[8]{b}+\frac{\sqrt{1-\frac{1}{\sqrt{2}}} \sqrt{a x+\sqrt{-b+a^2 x^2}}}{\sqrt[8]{b}}}{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{b^{11/8}}-\frac{\sqrt{2+\sqrt{2}} \tanh ^{-1}\left(\frac{\sqrt{1+\frac{1}{\sqrt{2}}} \sqrt[8]{b}+\frac{\sqrt{1+\frac{1}{\sqrt{2}}} \sqrt{a x+\sqrt{-b+a^2 x^2}}}{\sqrt[8]{b}}}{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{b^{11/8}}","\frac{\left(a x+\sqrt{-b+a^2 x^2}\right)^{5/4}}{b \left(b-a^2 x^2-a x \sqrt{-b+a^2 x^2}\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{\sqrt[8]{b}}\right)}{2 b^{11/8}}-\frac{\sqrt{2+\sqrt{2}} \tan ^{-1}\left(\frac{\left(\sqrt{\frac{2}{2-\sqrt{2}}} \sqrt[8]{b}-\frac{2 \sqrt[8]{b}}{\sqrt{2-\sqrt{2}}}\right) \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[4]{b}+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{b^{11/8}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[8]{b} \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[4]{b}+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{2 \sqrt{2} b^{11/8}}-\frac{\sqrt{2-\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{b} \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[4]{b}+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{b^{11/8}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{\sqrt[8]{b}}\right)}{2 b^{11/8}}+\frac{\tanh ^{-1}\left(\frac{\frac{\sqrt[8]{b}}{\sqrt{2}}+\frac{\sqrt{a x+\sqrt{-b+a^2 x^2}}}{\sqrt{2} \sqrt[8]{b}}}{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{2 \sqrt{2} b^{11/8}}-\frac{\sqrt{2+\sqrt{2}} \tanh ^{-1}\left(\frac{\frac{\sqrt[8]{b}}{\sqrt{2-\sqrt{2}}}+\frac{\sqrt{a x+\sqrt{-b+a^2 x^2}}}{\sqrt{2-\sqrt{2}} \sqrt[8]{b}}}{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{b^{11/8}}+\frac{\sqrt{2-\sqrt{2}} \tanh ^{-1}\left(\frac{\frac{\sqrt[8]{b}}{\sqrt{2+\sqrt{2}}}+\frac{\sqrt{a x+\sqrt{-b+a^2 x^2}}}{\sqrt{2+\sqrt{2}} \sqrt[8]{b}}}{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{b^{11/8}}",1,"(a*x + Sqrt[-b + a^2*x^2])^(5/4)/(b*(b - a^2*x^2 - a*x*Sqrt[-b + a^2*x^2])) - ArcTan[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/b^(1/8)]/(2*b^(11/8)) - ArcTan[(Sqrt[2]*b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(-b^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]])]/(2*Sqrt[2]*b^(11/8)) + (Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]]*b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(-b^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]])])/b^(11/8) - (Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]]*b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(-b^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]])])/b^(11/8) - ArcTanh[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/b^(1/8)]/(2*b^(11/8)) + ArcTanh[(b^(1/8)/Sqrt[2] + Sqrt[a*x + Sqrt[-b + a^2*x^2]]/(Sqrt[2]*b^(1/8)))/(a*x + Sqrt[-b + a^2*x^2])^(1/4)]/(2*Sqrt[2]*b^(11/8)) + (Sqrt[2 - Sqrt[2]]*ArcTanh[(Sqrt[1 - 1/Sqrt[2]]*b^(1/8) + (Sqrt[1 - 1/Sqrt[2]]*Sqrt[a*x + Sqrt[-b + a^2*x^2]])/b^(1/8))/(a*x + Sqrt[-b + a^2*x^2])^(1/4)])/b^(11/8) - (Sqrt[2 + Sqrt[2]]*ArcTanh[(Sqrt[1 + 1/Sqrt[2]]*b^(1/8) + (Sqrt[1 + 1/Sqrt[2]]*Sqrt[a*x + Sqrt[-b + a^2*x^2]])/b^(1/8))/(a*x + Sqrt[-b + a^2*x^2])^(1/4)])/b^(11/8)","A",1
3125,1,723,723,6.7754845,"\int \frac{x \left(2 x^3 c_3-c_4\right)}{\left(-x+x^3 c_3+c_4\right) \sqrt[3]{\frac{x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}} \left(x^2+x^4 c_3+x^6 c_3{}^2+x c_4+2 x^3 c_3 c_4+c_4{}^2\right)} \, dx","IntegrateAlgebraic[(x*(2*x^3*C[3] - C[4]))/((-x + x^3*C[3] + C[4])*((x*C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C[3] + C[4]))^(1/3)*(x^2 + x^4*C[3] + x^6*C[3]^2 + x*C[4] + 2*x^3*C[3]*C[4] + C[4]^2)),x]","-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1-c_1} \sqrt[3]{\frac{x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}}{\sqrt{3} \sqrt[3]{1+c_0}}\right) \sqrt[3]{-1-c_1}}{\sqrt{3} \sqrt[3]{1+c_0}}-\frac{\sqrt[3]{-1-c_1} \log \left(\sqrt[3]{1+c_0}+\sqrt[3]{-1-c_1} \sqrt[3]{\frac{x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}\right)}{3 \sqrt[3]{1+c_0}}+\frac{\sqrt[3]{-1-c_1} \log \left((1+c_0){}^{2/3}-\sqrt[3]{1+c_0} \sqrt[3]{-1-c_1} \sqrt[3]{\frac{x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}+(-1-c_1){}^{2/3} \left(\frac{x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}\right){}^{2/3}\right)}{6 \sqrt[3]{1+c_0}}+\frac{1}{3} \text{RootSum}\left[1-c_0+c_0{}^2-2 \text{$\#$1}^3+c_0 \text{$\#$1}^3+c_1 \text{$\#$1}^3-2 c_0 c_1 \text{$\#$1}^3+\text{$\#$1}^6-c_1 \text{$\#$1}^6+c_1{}^2 \text{$\#$1}^6\&,\frac{\log \left(\sqrt[3]{\frac{x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}-\text{$\#$1}\right)+c_0 \log \left(\sqrt[3]{\frac{x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}-\text{$\#$1}\right)-2 c_1 \log \left(\sqrt[3]{\frac{x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}-\text{$\#$1}\right)+c_0 c_1 \log \left(\sqrt[3]{\frac{x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}-\text{$\#$1}\right)-\log \left(\sqrt[3]{\frac{x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}-\text{$\#$1}\right) \text{$\#$1}^3+c_1 \log \left(\sqrt[3]{\frac{x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}-\text{$\#$1}\right) \text{$\#$1}^3-c_1{}^2 \log \left(\sqrt[3]{\frac{x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}-\text{$\#$1}\right) \text{$\#$1}^3}{-2 \text{$\#$1}+c_0 \text{$\#$1}+c_1 \text{$\#$1}-2 c_0 c_1 \text{$\#$1}+2 \text{$\#$1}^4-2 c_1 \text{$\#$1}^4+2 c_1{}^2 \text{$\#$1}^4}\&\right]","-\frac{\tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-1-c_1} \sqrt[3]{\frac{x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}}{\sqrt{3} \sqrt[3]{1+c_0}}\right) \sqrt[3]{-1-c_1}}{\sqrt{3} \sqrt[3]{1+c_0}}-\frac{\sqrt[3]{-1-c_1} \log \left(\sqrt[3]{1+c_0}+\sqrt[3]{-1-c_1} \sqrt[3]{\frac{x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}\right)}{3 \sqrt[3]{1+c_0}}+\frac{\sqrt[3]{-1-c_1} \log \left((1+c_0){}^{2/3}-\sqrt[3]{1+c_0} \sqrt[3]{-1-c_1} \sqrt[3]{\frac{x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}+(-1-c_1){}^{2/3} \left(\frac{x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}\right){}^{2/3}\right)}{6 \sqrt[3]{1+c_0}}+\frac{1}{3} \text{RootSum}\left[1-c_0+c_0{}^2-2 \text{$\#$1}^3+c_0 \text{$\#$1}^3+c_1 \text{$\#$1}^3-2 c_0 c_1 \text{$\#$1}^3+\text{$\#$1}^6-c_1 \text{$\#$1}^6+c_1{}^2 \text{$\#$1}^6\&,\frac{\log \left(\sqrt[3]{\frac{x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}-\text{$\#$1}\right)+c_0 \log \left(\sqrt[3]{\frac{x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}-\text{$\#$1}\right)-2 c_1 \log \left(\sqrt[3]{\frac{x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}-\text{$\#$1}\right)+c_0 c_1 \log \left(\sqrt[3]{\frac{x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}-\text{$\#$1}\right)-\log \left(\sqrt[3]{\frac{x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}-\text{$\#$1}\right) \text{$\#$1}^3+c_1 \log \left(\sqrt[3]{\frac{x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}-\text{$\#$1}\right) \text{$\#$1}^3-c_1{}^2 \log \left(\sqrt[3]{\frac{x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}-\text{$\#$1}\right) \text{$\#$1}^3}{-2 \text{$\#$1}+c_0 \text{$\#$1}+c_1 \text{$\#$1}-2 c_0 c_1 \text{$\#$1}+2 \text{$\#$1}^4-2 c_1 \text{$\#$1}^4+2 c_1{}^2 \text{$\#$1}^4}\&\right]",1,"-((ArcTan[1/Sqrt[3] - (2*(-1 - C[1])^(1/3)*((x*C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C[3] + C[4]))^(1/3))/(Sqrt[3]*(1 + C[0])^(1/3))]*(-1 - C[1])^(1/3))/(Sqrt[3]*(1 + C[0])^(1/3))) - ((-1 - C[1])^(1/3)*Log[(1 + C[0])^(1/3) + (-1 - C[1])^(1/3)*((x*C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C[3] + C[4]))^(1/3)])/(3*(1 + C[0])^(1/3)) + ((-1 - C[1])^(1/3)*Log[(1 + C[0])^(2/3) - (1 + C[0])^(1/3)*(-1 - C[1])^(1/3)*((x*C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C[3] + C[4]))^(1/3) + (-1 - C[1])^(2/3)*((x*C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C[3] + C[4]))^(2/3)])/(6*(1 + C[0])^(1/3)) + RootSum[1 - C[0] + C[0]^2 - 2*#1^3 + C[0]*#1^3 + C[1]*#1^3 - 2*C[0]*C[1]*#1^3 + #1^6 - C[1]*#1^6 + C[1]^2*#1^6 & , (Log[((x*C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C[3] + C[4]))^(1/3) - #1] + C[0]*Log[((x*C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C[3] + C[4]))^(1/3) - #1] - 2*C[1]*Log[((x*C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C[3] + C[4]))^(1/3) - #1] + C[0]*C[1]*Log[((x*C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C[3] + C[4]))^(1/3) - #1] - Log[((x*C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C[3] + C[4]))^(1/3) - #1]*#1^3 + C[1]*Log[((x*C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C[3] + C[4]))^(1/3) - #1]*#1^3 - C[1]^2*Log[((x*C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C[3] + C[4]))^(1/3) - #1]*#1^3)/(-2*#1 + C[0]*#1 + C[1]*#1 - 2*C[0]*C[1]*#1 + 2*#1^4 - 2*C[1]*#1^4 + 2*C[1]^2*#1^4) & ]/3","A",1
3126,1,747,747,1.9526377,"\int \frac{(-d+c x) \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{d+c x} \, dx","IntegrateAlgebraic[((-d + c*x)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/(d + c*x),x]","\frac{2 (-6 a d+a c x) \sqrt{b^2+a^2 x^2}+2 \left(-b^2 c-6 a^2 d x+a^2 c x^2\right)}{3 a c \sqrt{a x+\sqrt{b^2+a^2 x^2}}}+\frac{4 a d^2 \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{a d-\sqrt{b^2 c^2+a^2 d^2}}}\right)}{c^{3/2} \sqrt{a d-\sqrt{b^2 c^2+a^2 d^2}}}-\frac{4 b^2 \sqrt{c} d \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{a d-\sqrt{b^2 c^2+a^2 d^2}}}\right)}{\sqrt{b^2 c^2+a^2 d^2} \sqrt{a d-\sqrt{b^2 c^2+a^2 d^2}}}-\frac{4 a^2 d^3 \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{a d-\sqrt{b^2 c^2+a^2 d^2}}}\right)}{c^{3/2} \sqrt{b^2 c^2+a^2 d^2} \sqrt{a d-\sqrt{b^2 c^2+a^2 d^2}}}+\frac{4 a d^2 \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{a d+\sqrt{b^2 c^2+a^2 d^2}}}\right)}{c^{3/2} \sqrt{a d+\sqrt{b^2 c^2+a^2 d^2}}}+\frac{4 b^2 \sqrt{c} d \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{a d+\sqrt{b^2 c^2+a^2 d^2}}}\right)}{\sqrt{b^2 c^2+a^2 d^2} \sqrt{a d+\sqrt{b^2 c^2+a^2 d^2}}}+\frac{4 a^2 d^3 \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{a d+\sqrt{b^2 c^2+a^2 d^2}}}\right)}{c^{3/2} \sqrt{b^2 c^2+a^2 d^2} \sqrt{a d+\sqrt{b^2 c^2+a^2 d^2}}}","\frac{2 (-6 a d+a c x) \sqrt{b^2+a^2 x^2}+2 \left(-b^2 c-6 a^2 d x+a^2 c x^2\right)}{3 a c \sqrt{a x+\sqrt{b^2+a^2 x^2}}}+\frac{4 a d^2 \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{a d-\sqrt{b^2 c^2+a^2 d^2}}}\right)}{c^{3/2} \sqrt{a d-\sqrt{b^2 c^2+a^2 d^2}}}-\frac{4 b^2 \sqrt{c} d \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{a d-\sqrt{b^2 c^2+a^2 d^2}}}\right)}{\sqrt{b^2 c^2+a^2 d^2} \sqrt{a d-\sqrt{b^2 c^2+a^2 d^2}}}-\frac{4 a^2 d^3 \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{a d-\sqrt{b^2 c^2+a^2 d^2}}}\right)}{c^{3/2} \sqrt{b^2 c^2+a^2 d^2} \sqrt{a d-\sqrt{b^2 c^2+a^2 d^2}}}+\frac{4 a d^2 \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{a d+\sqrt{b^2 c^2+a^2 d^2}}}\right)}{c^{3/2} \sqrt{a d+\sqrt{b^2 c^2+a^2 d^2}}}+\frac{4 b^2 \sqrt{c} d \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{a d+\sqrt{b^2 c^2+a^2 d^2}}}\right)}{\sqrt{b^2 c^2+a^2 d^2} \sqrt{a d+\sqrt{b^2 c^2+a^2 d^2}}}+\frac{4 a^2 d^3 \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a x+\sqrt{b^2+a^2 x^2}}}{\sqrt{a d+\sqrt{b^2 c^2+a^2 d^2}}}\right)}{c^{3/2} \sqrt{b^2 c^2+a^2 d^2} \sqrt{a d+\sqrt{b^2 c^2+a^2 d^2}}}",1,"(2*(-6*a*d + a*c*x)*Sqrt[b^2 + a^2*x^2] + 2*(-(b^2*c) - 6*a^2*d*x + a^2*c*x^2))/(3*a*c*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]) + (4*a*d^2*ArcTan[(Sqrt[c]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/Sqrt[a*d - Sqrt[b^2*c^2 + a^2*d^2]]])/(c^(3/2)*Sqrt[a*d - Sqrt[b^2*c^2 + a^2*d^2]]) - (4*b^2*Sqrt[c]*d*ArcTan[(Sqrt[c]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/Sqrt[a*d - Sqrt[b^2*c^2 + a^2*d^2]]])/(Sqrt[b^2*c^2 + a^2*d^2]*Sqrt[a*d - Sqrt[b^2*c^2 + a^2*d^2]]) - (4*a^2*d^3*ArcTan[(Sqrt[c]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/Sqrt[a*d - Sqrt[b^2*c^2 + a^2*d^2]]])/(c^(3/2)*Sqrt[b^2*c^2 + a^2*d^2]*Sqrt[a*d - Sqrt[b^2*c^2 + a^2*d^2]]) + (4*a*d^2*ArcTan[(Sqrt[c]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/Sqrt[a*d + Sqrt[b^2*c^2 + a^2*d^2]]])/(c^(3/2)*Sqrt[a*d + Sqrt[b^2*c^2 + a^2*d^2]]) + (4*b^2*Sqrt[c]*d*ArcTan[(Sqrt[c]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/Sqrt[a*d + Sqrt[b^2*c^2 + a^2*d^2]]])/(Sqrt[b^2*c^2 + a^2*d^2]*Sqrt[a*d + Sqrt[b^2*c^2 + a^2*d^2]]) + (4*a^2*d^3*ArcTan[(Sqrt[c]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/Sqrt[a*d + Sqrt[b^2*c^2 + a^2*d^2]]])/(c^(3/2)*Sqrt[b^2*c^2 + a^2*d^2]*Sqrt[a*d + Sqrt[b^2*c^2 + a^2*d^2]])","A",1
3127,1,752,752,3.743528,"\int \frac{\sqrt[3]{\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}}{x^3} \, dx","IntegrateAlgebraic[(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(1/3)/x^3,x]","-\frac{3 \left(2-a x-2 x^2-3 a^2 x^2+a x^3+3 a^2 x^4\right) \sqrt[3]{\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}}{10 x^2}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{\sqrt[3]{1-a} \left(-2+2 x^2\right) \sqrt[3]{\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}}{\sqrt{3}}\right)}{2 \sqrt[3]{1-a}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{\sqrt[3]{1+a} \left(-2+2 x^2\right) \sqrt[3]{\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}}{\sqrt{3}}\right)}{2 \sqrt[3]{1+a}}-\frac{\log \left(1+\left(-\sqrt[3]{1-a}+\sqrt[3]{1-a} x^2\right) \sqrt[3]{\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}\right)}{2 \sqrt[3]{1-a}}+\frac{\log \left(-1+\left(-\sqrt[3]{1+a}+\sqrt[3]{1+a} x^2\right) \sqrt[3]{\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}\right)}{2 \sqrt[3]{1+a}}+\frac{\log \left(1+\left(\sqrt[3]{1-a}-\sqrt[3]{1-a} x^2\right) \sqrt[3]{\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}+\left((1-a)^{2/3}-2 (1-a)^{2/3} x^2+(1-a)^{2/3} x^4\right) \left(\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}\right)^{2/3}\right)}{4 \sqrt[3]{1-a}}-\frac{\log \left(1+\left(-\sqrt[3]{1+a}+\sqrt[3]{1+a} x^2\right) \sqrt[3]{\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}+\left((1+a)^{2/3}-2 (1+a)^{2/3} x^2+(1+a)^{2/3} x^4\right) \left(\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}\right)^{2/3}\right)}{4 \sqrt[3]{1+a}}","-\frac{3 \left(2-a x-2 x^2-3 a^2 x^2+a x^3+3 a^2 x^4\right) \sqrt[3]{\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}}{10 x^2}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{\sqrt[3]{1-a} \left(-2+2 x^2\right) \sqrt[3]{\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}}{\sqrt{3}}\right)}{2 \sqrt[3]{1-a}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{\sqrt[3]{1+a} \left(-2+2 x^2\right) \sqrt[3]{\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}}{\sqrt{3}}\right)}{2 \sqrt[3]{1+a}}-\frac{\log \left(1+\left(-\sqrt[3]{1-a}+\sqrt[3]{1-a} x^2\right) \sqrt[3]{\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}\right)}{2 \sqrt[3]{1-a}}+\frac{\log \left(-1+\left(-\sqrt[3]{1+a}+\sqrt[3]{1+a} x^2\right) \sqrt[3]{\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}\right)}{2 \sqrt[3]{1+a}}+\frac{\log \left(1+\left(\sqrt[3]{1-a}-\sqrt[3]{1-a} x^2\right) \sqrt[3]{\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}+\left((1-a)^{2/3}-2 (1-a)^{2/3} x^2+(1-a)^{2/3} x^4\right) \left(\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}\right)^{2/3}\right)}{4 \sqrt[3]{1-a}}-\frac{\log \left(1+\left(-\sqrt[3]{1+a}+\sqrt[3]{1+a} x^2\right) \sqrt[3]{\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}+\left((1+a)^{2/3}-2 (1+a)^{2/3} x^2+(1+a)^{2/3} x^4\right) \left(\frac{x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}\right)^{2/3}\right)}{4 \sqrt[3]{1+a}}",1,"(-3*(2 - a*x - 2*x^2 - 3*a^2*x^2 + a*x^3 + 3*a^2*x^4)*(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(1/3))/(10*x^2) + (Sqrt[3]*ArcTan[1/Sqrt[3] - ((1 - a)^(1/3)*(-2 + 2*x^2)*(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(1/3))/Sqrt[3]])/(2*(1 - a)^(1/3)) - (Sqrt[3]*ArcTan[1/Sqrt[3] + ((1 + a)^(1/3)*(-2 + 2*x^2)*(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(1/3))/Sqrt[3]])/(2*(1 + a)^(1/3)) - Log[1 + (-(1 - a)^(1/3) + (1 - a)^(1/3)*x^2)*(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(1/3)]/(2*(1 - a)^(1/3)) + Log[-1 + (-(1 + a)^(1/3) + (1 + a)^(1/3)*x^2)*(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(1/3)]/(2*(1 + a)^(1/3)) + Log[1 + ((1 - a)^(1/3) - (1 - a)^(1/3)*x^2)*(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(1/3) + ((1 - a)^(2/3) - 2*(1 - a)^(2/3)*x^2 + (1 - a)^(2/3)*x^4)*(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(2/3)]/(4*(1 - a)^(1/3)) - Log[1 + (-(1 + a)^(1/3) + (1 + a)^(1/3)*x^2)*(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(1/3) + ((1 + a)^(2/3) - 2*(1 + a)^(2/3)*x^2 + (1 + a)^(2/3)*x^4)*(x/(-1 - a*x + 3*x^2 + 3*a*x^3 - 3*x^4 - 3*a*x^5 + x^6 + a*x^7))^(2/3)]/(4*(1 + a)^(1/3))","A",1
3128,1,757,757,3.3096996,"\int \frac{1}{a b c-(c+a b x)^2 \sqrt{c+b x+a x^2}} \, dx","IntegrateAlgebraic[(a*b*c - (c + a*b*x)^2*Sqrt[c + b*x + a*x^2])^(-1),x]","-2 \text{RootSum}\left[a b^4 c+\sqrt{a} b^2 c^3-2 a^{3/2} b^2 c^3+a^{5/2} b^2 c^3-6 a^{3/2} b^3 c \text{$\#$1}-b^3 c^2 \text{$\#$1}+2 a b^3 c^2 \text{$\#$1}-a^2 b^3 c^2 \text{$\#$1}-4 a b c^3 \text{$\#$1}+4 a^2 b c^3 \text{$\#$1}+12 a^2 b^2 c \text{$\#$1}^2+5 \sqrt{a} b^2 c^2 \text{$\#$1}^2-4 a^{3/2} b^2 c^2 \text{$\#$1}^2-a^{5/2} b^2 c^2 \text{$\#$1}^2+4 a^{3/2} c^3 \text{$\#$1}^2-8 a^{5/2} b c \text{$\#$1}^3-2 a b^3 c \text{$\#$1}^3+2 a^2 b^3 c \text{$\#$1}^3-8 a b c^2 \text{$\#$1}^3+6 a^{3/2} b^2 c \text{$\#$1}^4-a^{5/2} b^2 c \text{$\#$1}^4+4 a^{3/2} c^2 \text{$\#$1}^4-a^2 b^3 \text{$\#$1}^5-4 a^2 b c \text{$\#$1}^5+a^{5/2} b^2 \text{$\#$1}^6\&,\frac{-\sqrt{a} b c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)+b^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}+2 a c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}-3 \sqrt{a} b \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}^2+2 a \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}^3}{6 a^{3/2} b^3 c+b^3 c^2-2 a b^3 c^2+a^2 b^3 c^2+4 a b c^3-4 a^2 b c^3-24 a^2 b^2 c \text{$\#$1}-10 \sqrt{a} b^2 c^2 \text{$\#$1}+8 a^{3/2} b^2 c^2 \text{$\#$1}+2 a^{5/2} b^2 c^2 \text{$\#$1}-8 a^{3/2} c^3 \text{$\#$1}+24 a^{5/2} b c \text{$\#$1}^2+6 a b^3 c \text{$\#$1}^2-6 a^2 b^3 c \text{$\#$1}^2+24 a b c^2 \text{$\#$1}^2-24 a^{3/2} b^2 c \text{$\#$1}^3+4 a^{5/2} b^2 c \text{$\#$1}^3-16 a^{3/2} c^2 \text{$\#$1}^3+5 a^2 b^3 \text{$\#$1}^4+20 a^2 b c \text{$\#$1}^4-6 a^{5/2} b^2 \text{$\#$1}^5}\&\right]","-2 \text{RootSum}\left[a b^4 c+\sqrt{a} b^2 c^3-2 a^{3/2} b^2 c^3+a^{5/2} b^2 c^3-6 a^{3/2} b^3 c \text{$\#$1}-b^3 c^2 \text{$\#$1}+2 a b^3 c^2 \text{$\#$1}-a^2 b^3 c^2 \text{$\#$1}-4 a b c^3 \text{$\#$1}+4 a^2 b c^3 \text{$\#$1}+12 a^2 b^2 c \text{$\#$1}^2+5 \sqrt{a} b^2 c^2 \text{$\#$1}^2-4 a^{3/2} b^2 c^2 \text{$\#$1}^2-a^{5/2} b^2 c^2 \text{$\#$1}^2+4 a^{3/2} c^3 \text{$\#$1}^2-8 a^{5/2} b c \text{$\#$1}^3-2 a b^3 c \text{$\#$1}^3+2 a^2 b^3 c \text{$\#$1}^3-8 a b c^2 \text{$\#$1}^3+6 a^{3/2} b^2 c \text{$\#$1}^4-a^{5/2} b^2 c \text{$\#$1}^4+4 a^{3/2} c^2 \text{$\#$1}^4-a^2 b^3 \text{$\#$1}^5-4 a^2 b c \text{$\#$1}^5+a^{5/2} b^2 \text{$\#$1}^6\&,\frac{-\sqrt{a} b c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)+b^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}+2 a c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}-3 \sqrt{a} b \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}^2+2 a \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}^3}{6 a^{3/2} b^3 c+b^3 c^2-2 a b^3 c^2+a^2 b^3 c^2+4 a b c^3-4 a^2 b c^3-24 a^2 b^2 c \text{$\#$1}-10 \sqrt{a} b^2 c^2 \text{$\#$1}+8 a^{3/2} b^2 c^2 \text{$\#$1}+2 a^{5/2} b^2 c^2 \text{$\#$1}-8 a^{3/2} c^3 \text{$\#$1}+24 a^{5/2} b c \text{$\#$1}^2+6 a b^3 c \text{$\#$1}^2-6 a^2 b^3 c \text{$\#$1}^2+24 a b c^2 \text{$\#$1}^2-24 a^{3/2} b^2 c \text{$\#$1}^3+4 a^{5/2} b^2 c \text{$\#$1}^3-16 a^{3/2} c^2 \text{$\#$1}^3+5 a^2 b^3 \text{$\#$1}^4+20 a^2 b c \text{$\#$1}^4-6 a^{5/2} b^2 \text{$\#$1}^5}\&\right]",1,"-2*RootSum[a*b^4*c + Sqrt[a]*b^2*c^3 - 2*a^(3/2)*b^2*c^3 + a^(5/2)*b^2*c^3 - 6*a^(3/2)*b^3*c*#1 - b^3*c^2*#1 + 2*a*b^3*c^2*#1 - a^2*b^3*c^2*#1 - 4*a*b*c^3*#1 + 4*a^2*b*c^3*#1 + 12*a^2*b^2*c*#1^2 + 5*Sqrt[a]*b^2*c^2*#1^2 - 4*a^(3/2)*b^2*c^2*#1^2 - a^(5/2)*b^2*c^2*#1^2 + 4*a^(3/2)*c^3*#1^2 - 8*a^(5/2)*b*c*#1^3 - 2*a*b^3*c*#1^3 + 2*a^2*b^3*c*#1^3 - 8*a*b*c^2*#1^3 + 6*a^(3/2)*b^2*c*#1^4 - a^(5/2)*b^2*c*#1^4 + 4*a^(3/2)*c^2*#1^4 - a^2*b^3*#1^5 - 4*a^2*b*c*#1^5 + a^(5/2)*b^2*#1^6 & , (-(Sqrt[a]*b*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]) + b^2*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 + 2*a*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 - 3*Sqrt[a]*b*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1^2 + 2*a*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1^3)/(6*a^(3/2)*b^3*c + b^3*c^2 - 2*a*b^3*c^2 + a^2*b^3*c^2 + 4*a*b*c^3 - 4*a^2*b*c^3 - 24*a^2*b^2*c*#1 - 10*Sqrt[a]*b^2*c^2*#1 + 8*a^(3/2)*b^2*c^2*#1 + 2*a^(5/2)*b^2*c^2*#1 - 8*a^(3/2)*c^3*#1 + 24*a^(5/2)*b*c*#1^2 + 6*a*b^3*c*#1^2 - 6*a^2*b^3*c*#1^2 + 24*a*b*c^2*#1^2 - 24*a^(3/2)*b^2*c*#1^3 + 4*a^(5/2)*b^2*c*#1^3 - 16*a^(3/2)*c^2*#1^3 + 5*a^2*b^3*#1^4 + 20*a^2*b*c*#1^4 - 6*a^(5/2)*b^2*#1^5) & ]","A",1
3129,1,731,757,5.7087544,"\int \frac{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{x^2 \left(-b+a^2 x^2\right)^{3/2}} \, dx","IntegrateAlgebraic[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/(x^2*(-b + a^2*x^2)^(3/2)),x]","-\frac{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{x \sqrt{-b+a^2 x^2} \left(-b+2 a^2 x^2\right)+x \left(-2 a b x+2 a^3 x^3\right)}-\frac{a \tan ^{-1}\left(\frac{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{\sqrt[8]{b}}\right)}{2 b^{15/8}}+\frac{a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[8]{b} \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[4]{b}+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{2 \sqrt{2} b^{15/8}}+\frac{\sqrt{2-\sqrt{2}} a \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{b} \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[4]{b}+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{4 b^{15/8}}+\frac{\sqrt{2+\sqrt{2}} a \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{b} \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[4]{b}+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{4 b^{15/8}}-\frac{a \tanh ^{-1}\left(\frac{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{\sqrt[8]{b}}\right)}{2 b^{15/8}}-\frac{a \tanh ^{-1}\left(\frac{\frac{\sqrt[8]{b}}{\sqrt{2}}+\frac{\sqrt{a x+\sqrt{-b+a^2 x^2}}}{\sqrt{2} \sqrt[8]{b}}}{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{2 \sqrt{2} b^{15/8}}-\frac{\sqrt{2+\sqrt{2}} a \tanh ^{-1}\left(\frac{\sqrt{1-\frac{1}{\sqrt{2}}} \sqrt[8]{b}+\frac{\sqrt{1-\frac{1}{\sqrt{2}}} \sqrt{a x+\sqrt{-b+a^2 x^2}}}{\sqrt[8]{b}}}{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{4 b^{15/8}}-\frac{\sqrt{2-\sqrt{2}} a \tanh ^{-1}\left(\frac{\sqrt{1+\frac{1}{\sqrt{2}}} \sqrt[8]{b}+\frac{\sqrt{1+\frac{1}{\sqrt{2}}} \sqrt{a x+\sqrt{-b+a^2 x^2}}}{\sqrt[8]{b}}}{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{4 b^{15/8}}","-\frac{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{x \sqrt{-b+a^2 x^2} \left(-b+2 a^2 x^2\right)+x \left(-2 a b x+2 a^3 x^3\right)}-\frac{a \tan ^{-1}\left(\frac{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{\sqrt[8]{b}}\right)}{2 b^{15/8}}-\frac{\sqrt{2-\sqrt{2}} a \tan ^{-1}\left(\frac{\left(\sqrt{\frac{2}{2-\sqrt{2}}} \sqrt[8]{b}-\frac{2 \sqrt[8]{b}}{\sqrt{2-\sqrt{2}}}\right) \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[4]{b}+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{4 b^{15/8}}+\frac{a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[8]{b} \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[4]{b}+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{2 \sqrt{2} b^{15/8}}+\frac{\sqrt{2+\sqrt{2}} a \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{b} \sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{-\sqrt[4]{b}+\sqrt{a x+\sqrt{-b+a^2 x^2}}}\right)}{4 b^{15/8}}-\frac{a \tanh ^{-1}\left(\frac{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}{\sqrt[8]{b}}\right)}{2 b^{15/8}}-\frac{a \tanh ^{-1}\left(\frac{\frac{\sqrt[8]{b}}{\sqrt{2}}+\frac{\sqrt{a x+\sqrt{-b+a^2 x^2}}}{\sqrt{2} \sqrt[8]{b}}}{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{2 \sqrt{2} b^{15/8}}-\frac{\sqrt{2-\sqrt{2}} a \tanh ^{-1}\left(\frac{\frac{\sqrt[8]{b}}{\sqrt{2-\sqrt{2}}}+\frac{\sqrt{a x+\sqrt{-b+a^2 x^2}}}{\sqrt{2-\sqrt{2}} \sqrt[8]{b}}}{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{4 b^{15/8}}-\frac{\sqrt{2+\sqrt{2}} a \tanh ^{-1}\left(\frac{\frac{\sqrt[8]{b}}{\sqrt{2+\sqrt{2}}}+\frac{\sqrt{a x+\sqrt{-b+a^2 x^2}}}{\sqrt{2+\sqrt{2}} \sqrt[8]{b}}}{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{4 b^{15/8}}",1,"-((a*x + Sqrt[-b + a^2*x^2])^(1/4)/(x*Sqrt[-b + a^2*x^2]*(-b + 2*a^2*x^2) + x*(-2*a*b*x + 2*a^3*x^3))) - (a*ArcTan[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/b^(1/8)])/(2*b^(15/8)) + (a*ArcTan[(Sqrt[2]*b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(-b^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]])])/(2*Sqrt[2]*b^(15/8)) + (Sqrt[2 - Sqrt[2]]*a*ArcTan[(Sqrt[2 - Sqrt[2]]*b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(-b^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]])])/(4*b^(15/8)) + (Sqrt[2 + Sqrt[2]]*a*ArcTan[(Sqrt[2 + Sqrt[2]]*b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(-b^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]])])/(4*b^(15/8)) - (a*ArcTanh[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/b^(1/8)])/(2*b^(15/8)) - (a*ArcTanh[(b^(1/8)/Sqrt[2] + Sqrt[a*x + Sqrt[-b + a^2*x^2]]/(Sqrt[2]*b^(1/8)))/(a*x + Sqrt[-b + a^2*x^2])^(1/4)])/(2*Sqrt[2]*b^(15/8)) - (Sqrt[2 + Sqrt[2]]*a*ArcTanh[(Sqrt[1 - 1/Sqrt[2]]*b^(1/8) + (Sqrt[1 - 1/Sqrt[2]]*Sqrt[a*x + Sqrt[-b + a^2*x^2]])/b^(1/8))/(a*x + Sqrt[-b + a^2*x^2])^(1/4)])/(4*b^(15/8)) - (Sqrt[2 - Sqrt[2]]*a*ArcTanh[(Sqrt[1 + 1/Sqrt[2]]*b^(1/8) + (Sqrt[1 + 1/Sqrt[2]]*Sqrt[a*x + Sqrt[-b + a^2*x^2]])/b^(1/8))/(a*x + Sqrt[-b + a^2*x^2])^(1/4)])/(4*b^(15/8))","A",1
3130,1,773,773,6.3203884,"\int \frac{\left(-1+x^2\right) \sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5}}{1+x^2} \, dx","IntegrateAlgebraic[((-1 + x^2)*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]])/(1 + x^2),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{-c_3 c_4+\sqrt{c_1} \sqrt{c_3} c_5} \sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt{c_3} c_4-\sqrt{c_1} c_5}\right) (-c_1 c_2+c_0 c_3) c_5 \sqrt{\sqrt{c_3} \left(-\sqrt{c_3} c_4+\sqrt{c_1} c_5\right)}}{2 \sqrt{c_1} c_3{}^{3/2} \left(-\sqrt{c_3} c_4+\sqrt{c_1} c_5\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt{-c_3 c_4-\sqrt{c_1} \sqrt{c_3} c_5} \sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt{c_3} c_4+\sqrt{c_1} c_5}\right) (-c_1 c_2+c_0 c_3) c_5 \sqrt{-\sqrt{c_3} \left(\sqrt{c_3} c_4+\sqrt{c_1} c_5\right)}}{2 \sqrt{c_1} c_3{}^{3/2} \left(\sqrt{c_3} c_4+\sqrt{c_1} c_5\right)}-\frac{(c_1 c_2-c_0 c_3) \sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5}}{c_3 \left(-c_1+\frac{(c_0+x c_1) c_3}{c_2+x c_3}\right)}+\left(-c_1 c_2 c_5{}^2+c_0 c_3 c_5{}^2\right) \text{RootSum}\left[c_2{}^2 c_4{}^4+c_3{}^2 c_4{}^4-2 c_0 c_2 c_4{}^2 c_5{}^2-2 c_1 c_3 c_4{}^2 c_5{}^2+c_0{}^2 c_5{}^4+c_1{}^2 c_5{}^4-4 c_2{}^2 c_4{}^3 \text{$\#$1}^2-4 c_3{}^2 c_4{}^3 \text{$\#$1}^2+4 c_0 c_2 c_4 c_5{}^2 \text{$\#$1}^2+4 c_1 c_3 c_4 c_5{}^2 \text{$\#$1}^2+6 c_2{}^2 c_4{}^2 \text{$\#$1}^4+6 c_3{}^2 c_4{}^2 \text{$\#$1}^4-2 c_0 c_2 c_5{}^2 \text{$\#$1}^4-2 c_1 c_3 c_5{}^2 \text{$\#$1}^4-4 c_2{}^2 c_4 \text{$\#$1}^6-4 c_3{}^2 c_4 \text{$\#$1}^6+c_2{}^2 \text{$\#$1}^8+c_3{}^2 \text{$\#$1}^8\&,\frac{\log \left(\sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5}-\text{$\#$1}\right) \text{$\#$1}}{c_2{}^2 c_4{}^2+c_3{}^2 c_4{}^2-c_0 c_2 c_5{}^2-c_1 c_3 c_5{}^2-2 c_2{}^2 c_4 \text{$\#$1}^2-2 c_3{}^2 c_4 \text{$\#$1}^2+c_2{}^2 \text{$\#$1}^4+c_3{}^2 \text{$\#$1}^4}\&\right]","-\frac{\tan ^{-1}\left(\frac{\sqrt{-c_3 c_4+\sqrt{c_1} \sqrt{c_3} c_5} \sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt{c_3} c_4-\sqrt{c_1} c_5}\right) (-c_1 c_2+c_0 c_3) c_5 \sqrt{\sqrt{c_3} \left(-\sqrt{c_3} c_4+\sqrt{c_1} c_5\right)}}{2 \sqrt{c_1} c_3{}^{3/2} \left(-\sqrt{c_3} c_4+\sqrt{c_1} c_5\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt{-c_3 c_4-\sqrt{c_1} \sqrt{c_3} c_5} \sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt{c_3} c_4+\sqrt{c_1} c_5}\right) (-c_1 c_2+c_0 c_3) c_5 \sqrt{-\sqrt{c_3} \left(\sqrt{c_3} c_4+\sqrt{c_1} c_5\right)}}{2 \sqrt{c_1} c_3{}^{3/2} \left(\sqrt{c_3} c_4+\sqrt{c_1} c_5\right)}-\frac{(c_1 c_2-c_0 c_3) \sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5}}{c_3 \left(-c_1+\frac{(c_0+x c_1) c_3}{c_2+x c_3}\right)}+\left(-c_1 c_2 c_5{}^2+c_0 c_3 c_5{}^2\right) \text{RootSum}\left[c_2{}^2 c_4{}^4+c_3{}^2 c_4{}^4-2 c_0 c_2 c_4{}^2 c_5{}^2-2 c_1 c_3 c_4{}^2 c_5{}^2+c_0{}^2 c_5{}^4+c_1{}^2 c_5{}^4-4 c_2{}^2 c_4{}^3 \text{$\#$1}^2-4 c_3{}^2 c_4{}^3 \text{$\#$1}^2+4 c_0 c_2 c_4 c_5{}^2 \text{$\#$1}^2+4 c_1 c_3 c_4 c_5{}^2 \text{$\#$1}^2+6 c_2{}^2 c_4{}^2 \text{$\#$1}^4+6 c_3{}^2 c_4{}^2 \text{$\#$1}^4-2 c_0 c_2 c_5{}^2 \text{$\#$1}^4-2 c_1 c_3 c_5{}^2 \text{$\#$1}^4-4 c_2{}^2 c_4 \text{$\#$1}^6-4 c_3{}^2 c_4 \text{$\#$1}^6+c_2{}^2 \text{$\#$1}^8+c_3{}^2 \text{$\#$1}^8\&,\frac{\log \left(\sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5}-\text{$\#$1}\right) \text{$\#$1}}{c_2{}^2 c_4{}^2+c_3{}^2 c_4{}^2-c_0 c_2 c_5{}^2-c_1 c_3 c_5{}^2-2 c_2{}^2 c_4 \text{$\#$1}^2-2 c_3{}^2 c_4 \text{$\#$1}^2+c_2{}^2 \text{$\#$1}^4+c_3{}^2 \text{$\#$1}^4}\&\right]",1,"-1/2*(ArcTan[(Sqrt[-(C[3]*C[4]) + Sqrt[C[1]]*Sqrt[C[3]]*C[5]]*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]])/(Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5])]*(-(C[1]*C[2]) + C[0]*C[3])*C[5]*Sqrt[Sqrt[C[3]]*(-(Sqrt[C[3]]*C[4]) + Sqrt[C[1]]*C[5])])/(Sqrt[C[1]]*C[3]^(3/2)*(-(Sqrt[C[3]]*C[4]) + Sqrt[C[1]]*C[5])) - (ArcTan[(Sqrt[-(C[3]*C[4]) - Sqrt[C[1]]*Sqrt[C[3]]*C[5]]*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]])/(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5])]*(-(C[1]*C[2]) + C[0]*C[3])*C[5]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5]))])/(2*Sqrt[C[1]]*C[3]^(3/2)*(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5])) - ((C[1]*C[2] - C[0]*C[3])*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]])/(C[3]*(-C[1] + ((C[0] + x*C[1])*C[3])/(C[2] + x*C[3]))) + (-(C[1]*C[2]*C[5]^2) + C[0]*C[3]*C[5]^2)*RootSum[C[2]^2*C[4]^4 + C[3]^2*C[4]^4 - 2*C[0]*C[2]*C[4]^2*C[5]^2 - 2*C[1]*C[3]*C[4]^2*C[5]^2 + C[0]^2*C[5]^4 + C[1]^2*C[5]^4 - 4*C[2]^2*C[4]^3*#1^2 - 4*C[3]^2*C[4]^3*#1^2 + 4*C[0]*C[2]*C[4]*C[5]^2*#1^2 + 4*C[1]*C[3]*C[4]*C[5]^2*#1^2 + 6*C[2]^2*C[4]^2*#1^4 + 6*C[3]^2*C[4]^2*#1^4 - 2*C[0]*C[2]*C[5]^2*#1^4 - 2*C[1]*C[3]*C[5]^2*#1^4 - 4*C[2]^2*C[4]*#1^6 - 4*C[3]^2*C[4]*#1^6 + C[2]^2*#1^8 + C[3]^2*#1^8 & , (Log[Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]] - #1]*#1)/(C[2]^2*C[4]^2 + C[3]^2*C[4]^2 - C[0]*C[2]*C[5]^2 - C[1]*C[3]*C[5]^2 - 2*C[2]^2*C[4]*#1^2 - 2*C[3]^2*C[4]*#1^2 + C[2]^2*#1^4 + C[3]^2*#1^4) & ]","A",1
3131,1,787,787,1.8833107,"\int \frac{\sqrt{-b+a^2 x^2}}{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}} \, dx","IntegrateAlgebraic[Sqrt[-b + a^2*x^2]/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4),x]","\frac{\left(-1378263040 b^2 c^5+2684354560 b c^{11}+1665760096 a b^2 c^2 x-4844421120 a b c^8 x-5368709120 a^2 c^{11} x^2+6459228160 a^3 c^8 x^3\right) \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\left(1447176192 b^2 c^4-2013265920 b c^{10}-1873980108 a b^2 c x+4541644800 a b c^7 x+4026531840 a^2 c^{10} x^2-6055526400 a^3 c^7 x^3\right) \sqrt[3]{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\left(-1537624704 b^2 c^3+1761607680 b c^9+2342475135 a b^2 x-4314562560 a b c^6 x-3523215360 a^2 c^9 x^2+5752750080 a^3 c^6 x^3\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{2/3} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\sqrt{-b+a^2 x^2} \left(\left(1665760096 b^2 c^2-1614807040 b c^8-5368709120 a c^{11} x+6459228160 a^2 c^8 x^2\right) \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\left(-1873980108 b^2 c+1513881600 b c^7+4026531840 a c^{10} x-6055526400 a^2 c^7 x^2\right) \sqrt[3]{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\left(2342475135 b^2-1438187520 b c^6-3523215360 a c^9 x+5752750080 a^2 c^6 x^2\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{2/3} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}\right)}{22052208640 a^2 c^6 x \sqrt{-b+a^2 x^2}+11026104320 a c^6 \left(-b+2 a^2 x^2\right)}+\frac{13923 b^2 \tan ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{131072 a c^{25/4}}-\frac{3 b \tan ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{a \sqrt[4]{c}}-\frac{13923 b^2 \tanh ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{131072 a c^{25/4}}+\frac{3 b \tanh ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{a \sqrt[4]{c}}","\frac{\left(-1378263040 b^2 c^5+2684354560 b c^{11}+1665760096 a b^2 c^2 x-4844421120 a b c^8 x-5368709120 a^2 c^{11} x^2+6459228160 a^3 c^8 x^3\right) \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\left(1447176192 b^2 c^4-2013265920 b c^{10}-1873980108 a b^2 c x+4541644800 a b c^7 x+4026531840 a^2 c^{10} x^2-6055526400 a^3 c^7 x^3\right) \sqrt[3]{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\left(-1537624704 b^2 c^3+1761607680 b c^9+2342475135 a b^2 x-4314562560 a b c^6 x-3523215360 a^2 c^9 x^2+5752750080 a^3 c^6 x^3\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{2/3} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\sqrt{-b+a^2 x^2} \left(\left(1665760096 b^2 c^2-1614807040 b c^8-5368709120 a c^{11} x+6459228160 a^2 c^8 x^2\right) \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\left(-1873980108 b^2 c+1513881600 b c^7+4026531840 a c^{10} x-6055526400 a^2 c^7 x^2\right) \sqrt[3]{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\left(2342475135 b^2-1438187520 b c^6-3523215360 a c^9 x+5752750080 a^2 c^6 x^2\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{2/3} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}\right)}{22052208640 a^2 c^6 x \sqrt{-b+a^2 x^2}+11026104320 a c^6 \left(-b+2 a^2 x^2\right)}+\frac{13923 b^2 \tan ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{131072 a c^{25/4}}-\frac{3 b \tan ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{a \sqrt[4]{c}}-\frac{13923 b^2 \tanh ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{131072 a c^{25/4}}+\frac{3 b \tanh ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{a \sqrt[4]{c}}",1,"((-1378263040*b^2*c^5 + 2684354560*b*c^11 + 1665760096*a*b^2*c^2*x - 4844421120*a*b*c^8*x - 5368709120*a^2*c^11*x^2 + 6459228160*a^3*c^8*x^3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + (1447176192*b^2*c^4 - 2013265920*b*c^10 - 1873980108*a*b^2*c*x + 4541644800*a*b*c^7*x + 4026531840*a^2*c^10*x^2 - 6055526400*a^3*c^7*x^3)*(a*x + Sqrt[-b + a^2*x^2])^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + (-1537624704*b^2*c^3 + 1761607680*b*c^9 + 2342475135*a*b^2*x - 4314562560*a*b*c^6*x - 3523215360*a^2*c^9*x^2 + 5752750080*a^3*c^6*x^3)*(a*x + Sqrt[-b + a^2*x^2])^(2/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + Sqrt[-b + a^2*x^2]*((1665760096*b^2*c^2 - 1614807040*b*c^8 - 5368709120*a*c^11*x + 6459228160*a^2*c^8*x^2)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + (-1873980108*b^2*c + 1513881600*b*c^7 + 4026531840*a*c^10*x - 6055526400*a^2*c^7*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + (2342475135*b^2 - 1438187520*b*c^6 - 3523215360*a*c^9*x + 5752750080*a^2*c^6*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(2/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4)))/(22052208640*a^2*c^6*x*Sqrt[-b + a^2*x^2] + 11026104320*a*c^6*(-b + 2*a^2*x^2)) + (13923*b^2*ArcTan[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)])/(131072*a*c^(25/4)) - (3*b*ArcTan[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)])/(a*c^(1/4)) - (13923*b^2*ArcTanh[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)])/(131072*a*c^(25/4)) + (3*b*ArcTanh[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)])/(a*c^(1/4))","A",1
3132,1,803,803,2.4514015,"\int \frac{\sqrt{-b+a^2 x^2}}{\sqrt[3]{a x+\sqrt{-b+a^2 x^2}} \sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}} \, dx","IntegrateAlgebraic[Sqrt[-b + a^2*x^2]/((a*x + Sqrt[-b + a^2*x^2])^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)),x]","\frac{\left(363738375 b^3-3081830400 b^2 c^6+238761600 a b^2 c^3 x-1056964608 a b c^9 x-727476750 a^2 b^2 x^2+5752750080 a^2 b c^6 x^2+1409286144 a^3 c^9 x^3\right) \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\left(214016000 b^2 c^5-536870912 b c^{11}-258658400 a b^2 c^2 x+968884224 a b c^8 x+1073741824 a^2 c^{11} x^2-1291845632 a^3 c^8 x^3\right) \sqrt[3]{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\left(-224716800 b^2 c^4+402653184 b c^{10}+290990700 a b^2 c x-908328960 a b c^7 x-805306368 a^2 c^{10} x^2+1211105280 a^3 c^7 x^3\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{2/3} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\sqrt{-b+a^2 x^2} \left(\left(238761600 b^2 c^3-352321536 b c^9-727476750 a b^2 x+5752750080 a b c^6 x+1409286144 a^2 c^9 x^2\right) \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\left(-258658400 b^2 c^2+322961408 b c^8+1073741824 a c^{11} x-1291845632 a^2 c^8 x^2\right) \sqrt[3]{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\left(290990700 b^2 c-302776320 b c^7-805306368 a c^{10} x+1211105280 a^2 c^7 x^2\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{2/3} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}\right)}{1917583360 a c^7 \left(a x+\sqrt{-b+a^2 x^2}\right)^{7/3}}-\frac{49725 b^2 \tan ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{524288 a c^{29/4}}+\frac{3 b \tan ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{4 a c^{5/4}}+\frac{49725 b^2 \tanh ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{524288 a c^{29/4}}-\frac{3 b \tanh ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{4 a c^{5/4}}","\frac{\left(363738375 b^3-3081830400 b^2 c^6+238761600 a b^2 c^3 x-1056964608 a b c^9 x-727476750 a^2 b^2 x^2+5752750080 a^2 b c^6 x^2+1409286144 a^3 c^9 x^3\right) \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\left(214016000 b^2 c^5-536870912 b c^{11}-258658400 a b^2 c^2 x+968884224 a b c^8 x+1073741824 a^2 c^{11} x^2-1291845632 a^3 c^8 x^3\right) \sqrt[3]{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\left(-224716800 b^2 c^4+402653184 b c^{10}+290990700 a b^2 c x-908328960 a b c^7 x-805306368 a^2 c^{10} x^2+1211105280 a^3 c^7 x^3\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{2/3} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\sqrt{-b+a^2 x^2} \left(\left(238761600 b^2 c^3-352321536 b c^9-727476750 a b^2 x+5752750080 a b c^6 x+1409286144 a^2 c^9 x^2\right) \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\left(-258658400 b^2 c^2+322961408 b c^8+1073741824 a c^{11} x-1291845632 a^2 c^8 x^2\right) \sqrt[3]{a x+\sqrt{-b+a^2 x^2}} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}+\left(290990700 b^2 c-302776320 b c^7-805306368 a c^{10} x+1211105280 a^2 c^7 x^2\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{2/3} \left(c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}\right)^{3/4}\right)}{1917583360 a c^7 \left(a x+\sqrt{-b+a^2 x^2}\right)^{7/3}}-\frac{49725 b^2 \tan ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{524288 a c^{29/4}}+\frac{3 b \tan ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{4 a c^{5/4}}+\frac{49725 b^2 \tanh ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{524288 a c^{29/4}}-\frac{3 b \tanh ^{-1}\left(\frac{\sqrt[4]{c+\sqrt[3]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{c}}\right)}{4 a c^{5/4}}",1,"((363738375*b^3 - 3081830400*b^2*c^6 + 238761600*a*b^2*c^3*x - 1056964608*a*b*c^9*x - 727476750*a^2*b^2*x^2 + 5752750080*a^2*b*c^6*x^2 + 1409286144*a^3*c^9*x^3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + (214016000*b^2*c^5 - 536870912*b*c^11 - 258658400*a*b^2*c^2*x + 968884224*a*b*c^8*x + 1073741824*a^2*c^11*x^2 - 1291845632*a^3*c^8*x^3)*(a*x + Sqrt[-b + a^2*x^2])^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + (-224716800*b^2*c^4 + 402653184*b*c^10 + 290990700*a*b^2*c*x - 908328960*a*b*c^7*x - 805306368*a^2*c^10*x^2 + 1211105280*a^3*c^7*x^3)*(a*x + Sqrt[-b + a^2*x^2])^(2/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + Sqrt[-b + a^2*x^2]*((238761600*b^2*c^3 - 352321536*b*c^9 - 727476750*a*b^2*x + 5752750080*a*b*c^6*x + 1409286144*a^2*c^9*x^2)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + (-258658400*b^2*c^2 + 322961408*b*c^8 + 1073741824*a*c^11*x - 1291845632*a^2*c^8*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4) + (290990700*b^2*c - 302776320*b*c^7 - 805306368*a*c^10*x + 1211105280*a^2*c^7*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(2/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(3/4)))/(1917583360*a*c^7*(a*x + Sqrt[-b + a^2*x^2])^(7/3)) - (49725*b^2*ArcTan[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)])/(524288*a*c^(29/4)) + (3*b*ArcTan[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)])/(4*a*c^(5/4)) + (49725*b^2*ArcTanh[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)])/(524288*a*c^(29/4)) - (3*b*ArcTanh[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/3))^(1/4)/c^(1/4)])/(4*a*c^(5/4))","A",1
3133,1,87,827,0.7831069,"\int \frac{\left(b+a x^4\right) \sqrt{-b-c x^2+a x^4}}{\left(-b+a x^4\right)^2} \, dx","IntegrateAlgebraic[((b + a*x^4)*Sqrt[-b - c*x^2 + a*x^4])/(-b + a*x^4)^2,x]","-\frac{x \sqrt{-b-c x^2+a x^4}}{2 \left(-b+a x^4\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{c} x \sqrt{-b-c x^2+a x^4}}{b+c x^2-a x^4}\right)}{2 \sqrt{c}}","\frac{x \left(b^{3/2}-i \sqrt{a} b x^2+\sqrt{b} c x^2-a \sqrt{b} x^4-i \sqrt{a} c x^4+i a^{3/2} x^6-i b \sqrt{-b-c x^2+a x^4}-\sqrt{a} \sqrt{b} x^2 \sqrt{-b-c x^2+a x^4}-\frac{1}{2} i c x^2 \sqrt{-b-c x^2+a x^4}+i a x^4 \sqrt{-b-c x^2+a x^4}\right)}{\left(-b+a x^4\right) \left(2 i b+2 \sqrt{a} \sqrt{b} x^2+i c x^2-2 i a x^4+2 \sqrt{b} \sqrt{-b-c x^2+a x^4}-2 i \sqrt{a} x^2 \sqrt{-b-c x^2+a x^4}\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt{-2 i \sqrt{a} \sqrt{b}-(2-2 i) \sqrt[4]{a} \sqrt[4]{b} \sqrt{c}+c} x}{i \sqrt{b}+\sqrt{a} x^2+\sqrt{-b-c x^2+a x^4}}\right)}{2 \sqrt{-2 i \sqrt{a} \sqrt{b}-(2-2 i) \sqrt[4]{a} \sqrt[4]{b} \sqrt{c}+c}}+\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \sqrt[4]{a} \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt{-2 i \sqrt{a} \sqrt{b}-(2-2 i) \sqrt[4]{a} \sqrt[4]{b} \sqrt{c}+c} x}{i \sqrt{b}+\sqrt{a} x^2+\sqrt{-b-c x^2+a x^4}}\right)}{\sqrt{c} \sqrt{-2 i \sqrt{a} \sqrt{b}-(2-2 i) \sqrt[4]{a} \sqrt[4]{b} \sqrt{c}+c}}-\frac{\tan ^{-1}\left(\frac{\sqrt{-2 i \sqrt{a} \sqrt{b}+(2-2 i) \sqrt[4]{a} \sqrt[4]{b} \sqrt{c}+c} x}{i \sqrt{b}+\sqrt{a} x^2+\sqrt{-b-c x^2+a x^4}}\right)}{2 \sqrt{-2 i \sqrt{a} \sqrt{b}+(2-2 i) \sqrt[4]{a} \sqrt[4]{b} \sqrt{c}+c}}-\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \sqrt[4]{a} \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt{-2 i \sqrt{a} \sqrt{b}+(2-2 i) \sqrt[4]{a} \sqrt[4]{b} \sqrt{c}+c} x}{i \sqrt{b}+\sqrt{a} x^2+\sqrt{-b-c x^2+a x^4}}\right)}{\sqrt{c} \sqrt{-2 i \sqrt{a} \sqrt{b}+(2-2 i) \sqrt[4]{a} \sqrt[4]{b} \sqrt{c}+c}}",1,"-1/2*(x*Sqrt[-b - c*x^2 + a*x^4])/(-b + a*x^4) + ArcTan[(Sqrt[c]*x*Sqrt[-b - c*x^2 + a*x^4])/(b + c*x^2 - a*x^4)]/(2*Sqrt[c])","A",1
3134,-1,0,849,180.0423884,"\int \frac{-a-b c+(1+c) x}{\left((-a+x) (-b+x)^2\right)^{2/3} (a-b d+(-1+d) x)} \, dx","IntegrateAlgebraic[(-a - b*c + (1 + c)*x)/(((-a + x)*(-b + x)^2)^(2/3)*(a - b*d + (-1 + d)*x)),x]","\text{\$Aborted}","\frac{(b-x)^{4/3} (-a+x)^{2/3} \left(-\frac{3 a (b-x)^{2/3} \sqrt[3]{-a+x}}{(a-b)^2 (-b+x)}+\frac{3 b (b-x)^{2/3} \sqrt[3]{-a+x}}{(a-b)^2 (-b+x)}+\frac{\sqrt{3} a (-1+d) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a+x}}{-2 \sqrt[3]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}}\right)}{(a-b)^2 d^{2/3}}+\frac{\sqrt{3} b c (-1+d) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a+x}}{-2 \sqrt[3]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}}\right)}{(a-b)^2 d^{2/3}}+\frac{\sqrt{3} (a-b d) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a+x}}{-2 \sqrt[3]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}}\right)}{(a-b)^2 d^{2/3}}+\frac{\sqrt{3} c (a-b d) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a+x}}{-2 \sqrt[3]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}}\right)}{(a-b)^2 d^{2/3}}-\frac{a (-1+d) \log \left(\sqrt[3]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}\right)}{(a-b)^2 d^{2/3}}-\frac{b c (-1+d) \log \left(\sqrt[3]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}\right)}{(a-b)^2 d^{2/3}}-\frac{c (a-b d) \log \left(\sqrt[3]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}\right)}{(a-b)^2 d^{2/3}}+\frac{(-a+b d) \log \left(\sqrt[3]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}\right)}{(a-b)^2 d^{2/3}}+\frac{a (-1+d) \log \left(d^{2/3} (b-x)^{2/3}-\sqrt[3]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+(-a+x)^{2/3}\right)}{2 (a-b)^2 d^{2/3}}+\frac{b c (-1+d) \log \left(d^{2/3} (b-x)^{2/3}-\sqrt[3]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+(-a+x)^{2/3}\right)}{2 (a-b)^2 d^{2/3}}+\frac{(a-b d) \log \left(d^{2/3} (b-x)^{2/3}-\sqrt[3]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+(-a+x)^{2/3}\right)}{2 (a-b)^2 d^{2/3}}+\frac{c (a-b d) \log \left(d^{2/3} (b-x)^{2/3}-\sqrt[3]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+(-a+x)^{2/3}\right)}{2 (a-b)^2 d^{2/3}}\right)}{\left((b-x)^2 (-a+x)\right)^{2/3}}",1,"$Aborted","F",-1
3135,1,242,857,11.6649868,"\int \frac{-a-b c+(1+c) x}{(-b+x) \sqrt[3]{(-a+x) (-b+x)^2} (a-b d+(-1+d) x)} \, dx","IntegrateAlgebraic[(-a - b*c + (1 + c)*x)/((-b + x)*((-a + x)*(-b + x)^2)^(1/3)*(a - b*d + (-1 + d)*x)),x]","\frac{(b-x)^{2/3} \sqrt[3]{-a+x} \left(-\frac{3 (-a+x)^{2/3}}{2 (a-b) (b-x)^{2/3}}+\frac{\sqrt{3} (c+d) \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{d} \sqrt[3]{b-x}}{\sqrt{3} \sqrt[3]{-a+x}}\right)}{(a-b) \sqrt[3]{d}}+\frac{(c+d) \log \left(1+\frac{d^{2/3} (b-x)^{2/3}}{(-a+x)^{2/3}}-\frac{\sqrt[3]{d} \sqrt[3]{b-x}}{\sqrt[3]{-a+x}}\right)}{2 (a-b) \sqrt[3]{d}}+\frac{(-c-d) \log \left(1+\frac{\sqrt[3]{d} \sqrt[3]{b-x}}{\sqrt[3]{-a+x}}\right)}{(a-b) \sqrt[3]{d}}\right)}{\sqrt[3]{(b-x)^2 (-a+x)}}","\frac{(b-x)^{2/3} \sqrt[3]{-a+x} \left(\frac{3 a \sqrt[3]{b-x} (-a+x)^{2/3}}{2 (a-b)^2 (-b+x)}-\frac{3 b \sqrt[3]{b-x} (-a+x)^{2/3}}{2 (a-b)^2 (-b+x)}-\frac{\sqrt{3} a (-1+d) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a+x}}{-2 \sqrt[3]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}}\right)}{(a-b)^2 \sqrt[3]{d}}-\frac{\sqrt{3} b c (-1+d) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a+x}}{-2 \sqrt[3]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}}\right)}{(a-b)^2 \sqrt[3]{d}}-\frac{\sqrt{3} (a-b d) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a+x}}{-2 \sqrt[3]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}}\right)}{(a-b)^2 \sqrt[3]{d}}-\frac{\sqrt{3} c (a-b d) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a+x}}{-2 \sqrt[3]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}}\right)}{(a-b)^2 \sqrt[3]{d}}-\frac{a (-1+d) \log \left(\sqrt[3]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}\right)}{(a-b)^2 \sqrt[3]{d}}-\frac{b c (-1+d) \log \left(\sqrt[3]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}\right)}{(a-b)^2 \sqrt[3]{d}}-\frac{c (a-b d) \log \left(\sqrt[3]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}\right)}{(a-b)^2 \sqrt[3]{d}}+\frac{(-a+b d) \log \left(\sqrt[3]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}\right)}{(a-b)^2 \sqrt[3]{d}}+\frac{a (-1+d) \log \left(d^{2/3} (b-x)^{2/3}-\sqrt[3]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+(-a+x)^{2/3}\right)}{2 (a-b)^2 \sqrt[3]{d}}+\frac{b c (-1+d) \log \left(d^{2/3} (b-x)^{2/3}-\sqrt[3]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+(-a+x)^{2/3}\right)}{2 (a-b)^2 \sqrt[3]{d}}+\frac{(a-b d) \log \left(d^{2/3} (b-x)^{2/3}-\sqrt[3]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+(-a+x)^{2/3}\right)}{2 (a-b)^2 \sqrt[3]{d}}+\frac{c (a-b d) \log \left(d^{2/3} (b-x)^{2/3}-\sqrt[3]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+(-a+x)^{2/3}\right)}{2 (a-b)^2 \sqrt[3]{d}}\right)}{\sqrt[3]{(b-x)^2 (-a+x)}}",1,"((b - x)^(2/3)*(-a + x)^(1/3)*((-3*(-a + x)^(2/3))/(2*(a - b)*(b - x)^(2/3)) + (Sqrt[3]*(c + d)*ArcTan[1/Sqrt[3] - (2*d^(1/3)*(b - x)^(1/3))/(Sqrt[3]*(-a + x)^(1/3))])/((a - b)*d^(1/3)) + ((c + d)*Log[1 + (d^(2/3)*(b - x)^(2/3))/(-a + x)^(2/3) - (d^(1/3)*(b - x)^(1/3))/(-a + x)^(1/3)])/(2*(a - b)*d^(1/3)) + ((-c - d)*Log[1 + (d^(1/3)*(b - x)^(1/3))/(-a + x)^(1/3)])/((a - b)*d^(1/3))))/((b - x)^2*(-a + x))^(1/3)","A",1
3136,1,938,876,15.1301658,"\int \frac{\left(d+c x^2\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{5/4}}{x \left(-b+a^2 x^2\right)^{5/2}} \, dx","IntegrateAlgebraic[((d + c*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(5/4))/(x*(-b + a^2*x^2)^(5/2)),x]","\frac{\left(-97 b^2 c-a^2 b d+45 a^2 b c x^2-51 a^4 d x^2\right) \sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}{96 a^2 b^{15/8} \left(-\sqrt{b}+a x\right) \left(\sqrt{b}+a x\right)}+\frac{\sqrt{-b+a^2 x^2} \left(13 b^2 c x-83 a^2 b d x-45 a^2 b c x^3+51 a^4 d x^3\right) \sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}{96 a b^{15/8} \left(-\sqrt{b}+a x\right)^2 \left(\sqrt{b}+a x\right)^2}+\frac{5 \left(-3 b c+29 a^2 d\right) \tan ^{-1}\left(\sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}\right)}{64 a^2 b^{15/8}}+\frac{5 \left(-3 b c+29 a^2 d\right) \tan ^{-1}\left(\frac{-1+\sqrt{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}{\sqrt{2} \sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}\right)}{64 \sqrt{2} a^2 b^{15/8}}-\frac{\sqrt{2+\sqrt{2}} d \tan ^{-1}\left(\frac{-\sqrt{1-\frac{1}{\sqrt{2}}}+\sqrt{1-\frac{1}{\sqrt{2}}} \sqrt{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}{\sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}\right)}{b^{15/8}}-\frac{\sqrt{2-\sqrt{2}} d \tan ^{-1}\left(\frac{-\sqrt{1+\frac{1}{\sqrt{2}}}+\sqrt{1+\frac{1}{\sqrt{2}}} \sqrt{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}{\sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}\right)}{b^{15/8}}+\frac{5 \left(-3 b c+29 a^2 d\right) \tanh ^{-1}\left(\sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}\right)}{64 a^2 b^{15/8}}+\frac{5 \left(-3 b c+29 a^2 d\right) \tanh ^{-1}\left(\frac{1+\sqrt{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}{\sqrt{2} \sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}\right)}{64 \sqrt{2} a^2 b^{15/8}}-\frac{\sqrt{2+\sqrt{2}} d \tanh ^{-1}\left(\frac{\sqrt{1-\frac{1}{\sqrt{2}}}+\sqrt{1-\frac{1}{\sqrt{2}}} \sqrt{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}{\sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}\right)}{b^{15/8}}-\frac{\sqrt{2-\sqrt{2}} d \tanh ^{-1}\left(\frac{\sqrt{1+\frac{1}{\sqrt{2}}}+\sqrt{1+\frac{1}{\sqrt{2}}} \sqrt{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}{\sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}\right)}{b^{15/8}}","\frac{\left(-97 b^2 c-a^2 b d+45 a^2 b c x^2-51 a^4 d x^2\right) \sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}{96 a^2 b^{15/8} \left(-\sqrt{b}+a x\right) \left(\sqrt{b}+a x\right)}+\frac{\sqrt{-b+a^2 x^2} \left(13 b^2 c x-83 a^2 b d x-45 a^2 b c x^3+51 a^4 d x^3\right) \sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}{96 a b^{15/8} \left(-\sqrt{b}+a x\right)^2 \left(\sqrt{b}+a x\right)^2}+\frac{5 \left(-3 b c+29 a^2 d\right) \tan ^{-1}\left(\sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}\right)}{64 a^2 b^{15/8}}-\frac{\sqrt{2-\sqrt{2}} d \tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2-\sqrt{2}}}+\frac{\sqrt{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}{\sqrt{2-\sqrt{2}}}}{\sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}\right)}{b^{15/8}}-\frac{\sqrt{2+\sqrt{2}} d \tan ^{-1}\left(\frac{-\frac{1}{\sqrt{2+\sqrt{2}}}+\frac{\sqrt{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}{\sqrt{2+\sqrt{2}}}}{\sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}\right)}{b^{15/8}}+\frac{5 \left(-3 b c+29 a^2 d\right) \tanh ^{-1}\left(\sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}\right)}{64 a^2 b^{15/8}}-\frac{5 (-1)^{3/4} \left(-3 b c+29 a^2 d\right) \tanh ^{-1}\left(\sqrt[4]{-1} \sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}\right)}{64 a^2 b^{15/8}}-\frac{5 \sqrt[4]{-1} \left(-3 b c+29 a^2 d\right) \tanh ^{-1}\left((-1)^{3/4} \sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}\right)}{64 a^2 b^{15/8}}-\frac{\sqrt{2-\sqrt{2}} d \tanh ^{-1}\left(\frac{\frac{1}{\sqrt{2-\sqrt{2}}}+\frac{\sqrt{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}{\sqrt{2-\sqrt{2}}}}{\sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}\right)}{b^{15/8}}-\frac{\sqrt{2+\sqrt{2}} d \tanh ^{-1}\left(\frac{\frac{1}{\sqrt{2+\sqrt{2}}}+\frac{\sqrt{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}{\sqrt{2+\sqrt{2}}}}{\sqrt[4]{\frac{a x+\sqrt{-b+a^2 x^2}}{\sqrt{b}}}}\right)}{b^{15/8}}",1,"((-97*b^2*c - a^2*b*d + 45*a^2*b*c*x^2 - 51*a^4*d*x^2)*((a*x + Sqrt[-b + a^2*x^2])/Sqrt[b])^(1/4))/(96*a^2*b^(15/8)*(-Sqrt[b] + a*x)*(Sqrt[b] + a*x)) + (Sqrt[-b + a^2*x^2]*(13*b^2*c*x - 83*a^2*b*d*x - 45*a^2*b*c*x^3 + 51*a^4*d*x^3)*((a*x + Sqrt[-b + a^2*x^2])/Sqrt[b])^(1/4))/(96*a*b^(15/8)*(-Sqrt[b] + a*x)^2*(Sqrt[b] + a*x)^2) + (5*(-3*b*c + 29*a^2*d)*ArcTan[((a*x + Sqrt[-b + a^2*x^2])/Sqrt[b])^(1/4)])/(64*a^2*b^(15/8)) + (5*(-3*b*c + 29*a^2*d)*ArcTan[(-1 + Sqrt[(a*x + Sqrt[-b + a^2*x^2])/Sqrt[b]])/(Sqrt[2]*((a*x + Sqrt[-b + a^2*x^2])/Sqrt[b])^(1/4))])/(64*Sqrt[2]*a^2*b^(15/8)) - (Sqrt[2 + Sqrt[2]]*d*ArcTan[(-Sqrt[1 - 1/Sqrt[2]] + Sqrt[1 - 1/Sqrt[2]]*Sqrt[(a*x + Sqrt[-b + a^2*x^2])/Sqrt[b]])/((a*x + Sqrt[-b + a^2*x^2])/Sqrt[b])^(1/4)])/b^(15/8) - (Sqrt[2 - Sqrt[2]]*d*ArcTan[(-Sqrt[1 + 1/Sqrt[2]] + Sqrt[1 + 1/Sqrt[2]]*Sqrt[(a*x + Sqrt[-b + a^2*x^2])/Sqrt[b]])/((a*x + Sqrt[-b + a^2*x^2])/Sqrt[b])^(1/4)])/b^(15/8) + (5*(-3*b*c + 29*a^2*d)*ArcTanh[((a*x + Sqrt[-b + a^2*x^2])/Sqrt[b])^(1/4)])/(64*a^2*b^(15/8)) + (5*(-3*b*c + 29*a^2*d)*ArcTanh[(1 + Sqrt[(a*x + Sqrt[-b + a^2*x^2])/Sqrt[b]])/(Sqrt[2]*((a*x + Sqrt[-b + a^2*x^2])/Sqrt[b])^(1/4))])/(64*Sqrt[2]*a^2*b^(15/8)) - (Sqrt[2 + Sqrt[2]]*d*ArcTanh[(Sqrt[1 - 1/Sqrt[2]] + Sqrt[1 - 1/Sqrt[2]]*Sqrt[(a*x + Sqrt[-b + a^2*x^2])/Sqrt[b]])/((a*x + Sqrt[-b + a^2*x^2])/Sqrt[b])^(1/4)])/b^(15/8) - (Sqrt[2 - Sqrt[2]]*d*ArcTanh[(Sqrt[1 + 1/Sqrt[2]] + Sqrt[1 + 1/Sqrt[2]]*Sqrt[(a*x + Sqrt[-b + a^2*x^2])/Sqrt[b]])/((a*x + Sqrt[-b + a^2*x^2])/Sqrt[b])^(1/4)])/b^(15/8)","A",0
3137,1,884,884,3.1322674,"\int \frac{1}{\sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7){}^2} \, dx","IntegrateAlgebraic[1/(Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*(C[6] + x*C[7])^2),x]","\frac{\sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5} \left(-c_1 c_2 c_4+c_0 c_3 c_4+c_1 c_2 \sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5-c_0 c_3 \sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5\right)}{\left(-c_1 c_6+\frac{(c_0+x c_1) c_3 c_6}{c_2+x c_3}+c_0 c_7-\frac{(c_0+x c_1) c_2 c_7}{c_2+x c_3}\right) \left(c_3 c_4{}^2 c_6-c_1 c_5{}^2 c_6-c_2 c_4{}^2 c_7+c_0 c_5{}^2 c_7\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5} \sqrt{c_3 c_6-c_2 c_7}}{\sqrt{-c_3 c_4 c_6+c_2 c_4 c_7-c_5 \sqrt{c_1 c_6-c_0 c_7} \sqrt{c_3 c_6-c_2 c_7}}}\right) \left(-c_1 c_2 c_5{}^2 \sqrt{c_1 c_6-c_0 c_7}+c_0 c_3 c_5{}^2 \sqrt{c_1 c_6-c_0 c_7}+c_1 c_2 c_4 c_5 \sqrt{c_3 c_6-c_2 c_7}-c_0 c_3 c_4 c_5 \sqrt{c_3 c_6-c_2 c_7}\right)}{2 \sqrt{c_1 c_6-c_0 c_7} \sqrt{c_3 c_6-c_2 c_7} \left(-c_3 c_4{}^2 c_6+c_1 c_5{}^2 c_6+c_2 c_4{}^2 c_7-c_0 c_5{}^2 c_7\right) \sqrt{-c_3 c_4 c_6+c_2 c_4 c_7-c_5 \sqrt{c_1 c_6-c_0 c_7} \sqrt{c_3 c_6-c_2 c_7}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5} \sqrt{c_3 c_6-c_2 c_7}}{\sqrt{-c_3 c_4 c_6+c_2 c_4 c_7+c_5 \sqrt{c_1 c_6-c_0 c_7} \sqrt{c_3 c_6-c_2 c_7}}}\right) \left(-c_1 c_2 c_5{}^2 \sqrt{c_1 c_6-c_0 c_7}+c_0 c_3 c_5{}^2 \sqrt{c_1 c_6-c_0 c_7}-c_1 c_2 c_4 c_5 \sqrt{c_3 c_6-c_2 c_7}+c_0 c_3 c_4 c_5 \sqrt{c_3 c_6-c_2 c_7}\right)}{2 \sqrt{c_1 c_6-c_0 c_7} \sqrt{c_3 c_6-c_2 c_7} \left(-c_3 c_4{}^2 c_6+c_1 c_5{}^2 c_6+c_2 c_4{}^2 c_7-c_0 c_5{}^2 c_7\right) \sqrt{-c_3 c_4 c_6+c_2 c_4 c_7+c_5 \sqrt{c_1 c_6-c_0 c_7} \sqrt{c_3 c_6-c_2 c_7}}}","\frac{\sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5} \left(-c_1 c_2 c_4+c_0 c_3 c_4+c_1 c_2 \sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5-c_0 c_3 \sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5\right)}{\left(c_1 c_6-\frac{(c_0+x c_1) c_3 c_6}{c_2+x c_3}-c_0 c_7+\frac{(c_0+x c_1) c_2 c_7}{c_2+x c_3}\right) \left(-c_3 c_4{}^2 c_6+c_1 c_5{}^2 c_6+c_2 c_4{}^2 c_7-c_0 c_5{}^2 c_7\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5} \sqrt{c_3 c_6-c_2 c_7}}{\sqrt{-c_3 c_4 c_6+c_2 c_4 c_7-c_5 \sqrt{c_1 c_6-c_0 c_7} \sqrt{c_3 c_6-c_2 c_7}}}\right) \left(-c_1 c_2 c_5{}^2 \sqrt{c_1 c_6-c_0 c_7}+c_0 c_3 c_5{}^2 \sqrt{c_1 c_6-c_0 c_7}+c_1 c_2 c_4 c_5 \sqrt{c_3 c_6-c_2 c_7}-c_0 c_3 c_4 c_5 \sqrt{c_3 c_6-c_2 c_7}\right)}{2 \sqrt{c_1 c_6-c_0 c_7} \sqrt{c_3 c_6-c_2 c_7} \left(-c_3 c_4{}^2 c_6+c_1 c_5{}^2 c_6+c_2 c_4{}^2 c_7-c_0 c_5{}^2 c_7\right) \sqrt{-c_3 c_4 c_6+c_2 c_4 c_7-c_5 \sqrt{c_1 c_6-c_0 c_7} \sqrt{c_3 c_6-c_2 c_7}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5} \sqrt{c_3 c_6-c_2 c_7}}{\sqrt{-c_3 c_4 c_6+c_2 c_4 c_7+c_5 \sqrt{c_1 c_6-c_0 c_7} \sqrt{c_3 c_6-c_2 c_7}}}\right) \left(-c_1 c_2 c_5{}^2 \sqrt{c_1 c_6-c_0 c_7}+c_0 c_3 c_5{}^2 \sqrt{c_1 c_6-c_0 c_7}-c_1 c_2 c_4 c_5 \sqrt{c_3 c_6-c_2 c_7}+c_0 c_3 c_4 c_5 \sqrt{c_3 c_6-c_2 c_7}\right)}{2 \sqrt{c_1 c_6-c_0 c_7} \sqrt{c_3 c_6-c_2 c_7} \left(-c_3 c_4{}^2 c_6+c_1 c_5{}^2 c_6+c_2 c_4{}^2 c_7-c_0 c_5{}^2 c_7\right) \sqrt{-c_3 c_4 c_6+c_2 c_4 c_7+c_5 \sqrt{c_1 c_6-c_0 c_7} \sqrt{c_3 c_6-c_2 c_7}}}",1,"(Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*(-(C[1]*C[2]*C[4]) + C[0]*C[3]*C[4] + C[1]*C[2]*Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5] - C[0]*C[3]*Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]))/((-(C[1]*C[6]) + ((C[0] + x*C[1])*C[3]*C[6])/(C[2] + x*C[3]) + C[0]*C[7] - ((C[0] + x*C[1])*C[2]*C[7])/(C[2] + x*C[3]))*(C[3]*C[4]^2*C[6] - C[1]*C[5]^2*C[6] - C[2]*C[4]^2*C[7] + C[0]*C[5]^2*C[7])) + (ArcTan[(Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*Sqrt[C[3]*C[6] - C[2]*C[7]])/Sqrt[-(C[3]*C[4]*C[6]) + C[2]*C[4]*C[7] - C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]]*Sqrt[C[3]*C[6] - C[2]*C[7]]]]*(-(C[1]*C[2]*C[5]^2*Sqrt[C[1]*C[6] - C[0]*C[7]]) + C[0]*C[3]*C[5]^2*Sqrt[C[1]*C[6] - C[0]*C[7]] + C[1]*C[2]*C[4]*C[5]*Sqrt[C[3]*C[6] - C[2]*C[7]] - C[0]*C[3]*C[4]*C[5]*Sqrt[C[3]*C[6] - C[2]*C[7]]))/(2*Sqrt[C[1]*C[6] - C[0]*C[7]]*Sqrt[C[3]*C[6] - C[2]*C[7]]*(-(C[3]*C[4]^2*C[6]) + C[1]*C[5]^2*C[6] + C[2]*C[4]^2*C[7] - C[0]*C[5]^2*C[7])*Sqrt[-(C[3]*C[4]*C[6]) + C[2]*C[4]*C[7] - C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]]*Sqrt[C[3]*C[6] - C[2]*C[7]]]) + (ArcTan[(Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*Sqrt[C[3]*C[6] - C[2]*C[7]])/Sqrt[-(C[3]*C[4]*C[6]) + C[2]*C[4]*C[7] + C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]]*Sqrt[C[3]*C[6] - C[2]*C[7]]]]*(-(C[1]*C[2]*C[5]^2*Sqrt[C[1]*C[6] - C[0]*C[7]]) + C[0]*C[3]*C[5]^2*Sqrt[C[1]*C[6] - C[0]*C[7]] - C[1]*C[2]*C[4]*C[5]*Sqrt[C[3]*C[6] - C[2]*C[7]] + C[0]*C[3]*C[4]*C[5]*Sqrt[C[3]*C[6] - C[2]*C[7]]))/(2*Sqrt[C[1]*C[6] - C[0]*C[7]]*Sqrt[C[3]*C[6] - C[2]*C[7]]*(-(C[3]*C[4]^2*C[6]) + C[1]*C[5]^2*C[6] + C[2]*C[4]^2*C[7] - C[0]*C[5]^2*C[7])*Sqrt[-(C[3]*C[4]*C[6]) + C[2]*C[4]*C[7] + C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]]*Sqrt[C[3]*C[6] - C[2]*C[7]]])","A",1
3138,1,898,887,3.4837951,"\int \frac{\sqrt[6]{\frac{1-b x}{c+x}} \left(1+d x^2\right)}{(1+b x) (1+c x)} \, dx","IntegrateAlgebraic[(((1 - b*x)/(c + x))^(1/6)*(1 + d*x^2))/((1 + b*x)*(1 + c*x)),x]","\frac{(1+b c) d \sqrt[6]{\frac{1-b x}{c+x}}}{b c \left(b+\frac{1-b x}{c+x}\right)}-\frac{\left(6 b+7 c+b c^2\right) d \tan ^{-1}\left(\frac{\sqrt[6]{\frac{1-b x}{c+x}}}{\sqrt[6]{b}}\right)}{3 b^{11/6} c^2}-\frac{2 \sqrt[6]{b+c} \left(c^2+d\right) \tan ^{-1}\left(\frac{\sqrt[6]{1-c^2} \sqrt[6]{\frac{1-b x}{c+x}}}{\sqrt[6]{b+c}}\right)}{c^2 (-b+c) \sqrt[6]{1-c^2}}+\frac{\sqrt[6]{2} \sqrt{3} \left(b^2+d\right) \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2^{5/6} \sqrt[6]{-1+b c} \sqrt[6]{\frac{1-b x}{c+x}}}{\sqrt{3} \sqrt[6]{b}}\right)}{b^{11/6} (b-c) \sqrt[6]{-1+b c}}-\frac{\sqrt[6]{2} \sqrt{3} \left(b^2+d\right) \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2^{5/6} \sqrt[6]{-1+b c} \sqrt[6]{\frac{1-b x}{c+x}}}{\sqrt{3} \sqrt[6]{b}}\right)}{b^{11/6} (b-c) \sqrt[6]{-1+b c}}+\frac{\left(6 b+7 c+b c^2\right) d \tan ^{-1}\left(\frac{\sqrt[3]{b}-\sqrt[3]{\frac{1-b x}{c+x}}}{\sqrt[6]{b} \sqrt[6]{\frac{1-b x}{c+x}}}\right)}{6 b^{11/6} c^2}+\frac{\sqrt[6]{b+c} \left(c^2+d\right) \tan ^{-1}\left(\frac{\sqrt[3]{b+c}-\sqrt[3]{1-c^2} \sqrt[3]{\frac{1-b x}{c+x}}}{\sqrt[6]{b+c} \sqrt[6]{1-c^2} \sqrt[6]{\frac{1-b x}{c+x}}}\right)}{c^2 (-b+c) \sqrt[6]{1-c^2}}-\frac{2 \sqrt[6]{2} \left(b^2+d\right) \tanh ^{-1}\left(\frac{\sqrt[6]{-1+b c} \sqrt[6]{\frac{1-b x}{c+x}}}{\sqrt[6]{2} \sqrt[6]{b}}\right)}{b^{11/6} (b-c) \sqrt[6]{-1+b c}}-\frac{\left(6 b+7 c+b c^2\right) d \tanh ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{b} \sqrt[6]{\frac{1-b x}{c+x}}}{\sqrt[3]{b}+\sqrt[3]{\frac{1-b x}{c+x}}}\right)}{2 \sqrt{3} b^{11/6} c^2}-\frac{\sqrt[6]{2} \left(b^2+d\right) \tanh ^{-1}\left(\frac{2^{5/6} \sqrt[6]{b} \sqrt[6]{-1+b c} \sqrt[6]{\frac{1-b x}{c+x}}}{2 \sqrt[3]{b}+2^{2/3} \sqrt[3]{-1+b c} \sqrt[3]{\frac{1-b x}{c+x}}}\right)}{b^{11/6} (b-c) \sqrt[6]{-1+b c}}-\frac{\sqrt{3} \sqrt[6]{b+c} \left(c^2+d\right) \tanh ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{b+c} \sqrt[6]{1-c^2} \sqrt[6]{\frac{1-b x}{c+x}}}{\sqrt[3]{b+c}+\sqrt[3]{1-c^2} \sqrt[3]{\frac{1-b x}{c+x}}}\right)}{c^2 (-b+c) \sqrt[6]{1-c^2}}","\frac{d (c+x) \sqrt[6]{\frac{1-b x}{c+x}}}{b c}-\frac{\left(6 b+7 c+b c^2\right) d \tan ^{-1}\left(\frac{\sqrt[6]{\frac{1-b x}{c+x}}}{\sqrt[6]{b}}\right)}{3 b^{11/6} c^2}-\frac{2 \sqrt[6]{b+c} \left(c^2+d\right) \tan ^{-1}\left(\frac{\sqrt[6]{1-c^2} \sqrt[6]{\frac{1-b x}{c+x}}}{\sqrt[6]{b+c}}\right)}{c^2 (-b+c) \sqrt[6]{1-c^2}}-\frac{\sqrt[6]{2} \sqrt{3} \left(b^2+d\right) \tan ^{-1}\left(\frac{-\sqrt[6]{b}+2^{5/6} \sqrt[6]{-1+b c} \sqrt[6]{\frac{1-b x}{c+x}}}{\sqrt{3} \sqrt[6]{b}}\right)}{b^{11/6} (b-c) \sqrt[6]{-1+b c}}-\frac{\sqrt[6]{2} \sqrt{3} \left(b^2+d\right) \tan ^{-1}\left(\frac{\sqrt[6]{b}+2^{5/6} \sqrt[6]{-1+b c} \sqrt[6]{\frac{1-b x}{c+x}}}{\sqrt{3} \sqrt[6]{b}}\right)}{b^{11/6} (b-c) \sqrt[6]{-1+b c}}+\frac{\left(6 b+7 c+b c^2\right) d \tan ^{-1}\left(\frac{\sqrt[6]{b} \sqrt[6]{\frac{1-b x}{c+x}}}{-\sqrt[3]{b}+\sqrt[3]{\frac{1-b x}{c+x}}}\right)}{6 b^{11/6} c^2}-\frac{\sqrt[6]{b+c} \left(c^2+d\right) \tan ^{-1}\left(\frac{\sqrt[6]{b+c} \sqrt[6]{1-c^2} \sqrt[6]{\frac{1-b x}{c+x}}}{\sqrt[3]{b+c}-\sqrt[3]{1-c^2} \sqrt[3]{\frac{1-b x}{c+x}}}\right)}{c^2 (-b+c) \sqrt[6]{1-c^2}}-\frac{2 \sqrt[6]{2} \left(b^2+d\right) \tanh ^{-1}\left(\frac{\sqrt[6]{-1+b c} \sqrt[6]{\frac{1-b x}{c+x}}}{\sqrt[6]{2} \sqrt[6]{b}}\right)}{b^{11/6} (b-c) \sqrt[6]{-1+b c}}-\frac{\left(6 b+7 c+b c^2\right) d \tanh ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{b} \sqrt[6]{\frac{1-b x}{c+x}}}{\sqrt[3]{b}+\sqrt[3]{\frac{1-b x}{c+x}}}\right)}{2 \sqrt{3} b^{11/6} c^2}-\frac{\sqrt[6]{2} \left(b^2+d\right) \tanh ^{-1}\left(\frac{2^{5/6} \sqrt[6]{b} \sqrt[6]{-1+b c} \sqrt[6]{\frac{1-b x}{c+x}}}{2 \sqrt[3]{b}+2^{2/3} \sqrt[3]{-1+b c} \sqrt[3]{\frac{1-b x}{c+x}}}\right)}{b^{11/6} (b-c) \sqrt[6]{-1+b c}}-\frac{\sqrt{3} \sqrt[6]{b+c} \left(c^2+d\right) \tanh ^{-1}\left(\frac{\sqrt[3]{b+c}+\sqrt[3]{1-c^2} \sqrt[3]{\frac{1-b x}{c+x}}}{\sqrt{3} \sqrt[6]{b+c} \sqrt[6]{1-c^2} \sqrt[6]{\frac{1-b x}{c+x}}}\right)}{c^2 (-b+c) \sqrt[6]{1-c^2}}",1,"((1 + b*c)*d*((1 - b*x)/(c + x))^(1/6))/(b*c*(b + (1 - b*x)/(c + x))) - ((6*b + 7*c + b*c^2)*d*ArcTan[((1 - b*x)/(c + x))^(1/6)/b^(1/6)])/(3*b^(11/6)*c^2) - (2*(b + c)^(1/6)*(c^2 + d)*ArcTan[((1 - c^2)^(1/6)*((1 - b*x)/(c + x))^(1/6))/(b + c)^(1/6)])/(c^2*(-b + c)*(1 - c^2)^(1/6)) + (2^(1/6)*Sqrt[3]*(b^2 + d)*ArcTan[1/Sqrt[3] - (2^(5/6)*(-1 + b*c)^(1/6)*((1 - b*x)/(c + x))^(1/6))/(Sqrt[3]*b^(1/6))])/(b^(11/6)*(b - c)*(-1 + b*c)^(1/6)) - (2^(1/6)*Sqrt[3]*(b^2 + d)*ArcTan[1/Sqrt[3] + (2^(5/6)*(-1 + b*c)^(1/6)*((1 - b*x)/(c + x))^(1/6))/(Sqrt[3]*b^(1/6))])/(b^(11/6)*(b - c)*(-1 + b*c)^(1/6)) + ((6*b + 7*c + b*c^2)*d*ArcTan[(b^(1/3) - ((1 - b*x)/(c + x))^(1/3))/(b^(1/6)*((1 - b*x)/(c + x))^(1/6))])/(6*b^(11/6)*c^2) + ((b + c)^(1/6)*(c^2 + d)*ArcTan[((b + c)^(1/3) - (1 - c^2)^(1/3)*((1 - b*x)/(c + x))^(1/3))/((b + c)^(1/6)*(1 - c^2)^(1/6)*((1 - b*x)/(c + x))^(1/6))])/(c^2*(-b + c)*(1 - c^2)^(1/6)) - (2*2^(1/6)*(b^2 + d)*ArcTanh[((-1 + b*c)^(1/6)*((1 - b*x)/(c + x))^(1/6))/(2^(1/6)*b^(1/6))])/(b^(11/6)*(b - c)*(-1 + b*c)^(1/6)) - ((6*b + 7*c + b*c^2)*d*ArcTanh[(Sqrt[3]*b^(1/6)*((1 - b*x)/(c + x))^(1/6))/(b^(1/3) + ((1 - b*x)/(c + x))^(1/3))])/(2*Sqrt[3]*b^(11/6)*c^2) - (2^(1/6)*(b^2 + d)*ArcTanh[(2^(5/6)*b^(1/6)*(-1 + b*c)^(1/6)*((1 - b*x)/(c + x))^(1/6))/(2*b^(1/3) + 2^(2/3)*(-1 + b*c)^(1/3)*((1 - b*x)/(c + x))^(1/3))])/(b^(11/6)*(b - c)*(-1 + b*c)^(1/6)) - (Sqrt[3]*(b + c)^(1/6)*(c^2 + d)*ArcTanh[(Sqrt[3]*(b + c)^(1/6)*(1 - c^2)^(1/6)*((1 - b*x)/(c + x))^(1/6))/((b + c)^(1/3) + (1 - c^2)^(1/3)*((1 - b*x)/(c + x))^(1/3))])/(c^2*(-b + c)*(1 - c^2)^(1/6))","A",1
3139,1,963,963,3.7331629,"\int \frac{\sqrt{-b+a^2 x^2} \left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4}}{\left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}} \, dx","IntegrateAlgebraic[(Sqrt[-b + a^2*x^2]*(a*x + Sqrt[-b + a^2*x^2])^(3/4))/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3),x]","\frac{\left(-573080508 b^2 c^4+602654094 b c^{12}-1815934120 a b^2 x+1050127416 a b c^8 x-1205308188 a^2 c^{12} x^2-1400169888 a^3 c^8 x^3\right) \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(668593926 b^2 c^3-502211745 b c^{11}-12014883738 a b c^7 x+3486784401 a c^{15} x+1004423490 a^2 c^{11} x^2+1283489064 a^3 c^7 x^3\right) \sqrt[4]{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(-817170354 b^2 c^2+435250179 b c^{10}+4575400830 a b c^6 x-1162261467 a c^{14} x-870500358 a^2 c^{10} x^2-1188415800 a^3 c^6 x^3\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(1089560472 b^2 c-386889048 b c^9-3287950380 a b c^5 x+774840978 a c^{13} x+773778096 a^2 c^9 x^2+1109188080 a^3 c^5 x^3\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\sqrt{-b+a^2 x^2} \left(\left(-1815934120 b^2+350042472 b c^8-1205308188 a c^{12} x-1400169888 a^2 c^8 x^2\right) \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(-11373139206 b c^7+3486784401 c^{15}+1004423490 a c^{11} x+1283489064 a^2 c^7 x^2\right) \sqrt[4]{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(3981192930 b c^6-1162261467 c^{14}-870500358 a c^{10} x-1188415800 a^2 c^6 x^2\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(-2733356340 b c^5+774840978 c^{13}+773778096 a c^9 x+1109188080 a^2 c^5 x^2\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{2865402540 a c^5 \left(a x+\sqrt{-b+a^2 x^2}\right)^{5/4}}+\frac{308 b^2 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[3]{c}}\right)}{243 \sqrt{3} a c^{17/3}}-\frac{308 b^2 \log \left(-\sqrt[3]{c}+\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{729 a c^{17/3}}+\frac{154 b^2 \log \left(c^{2/3}+\sqrt[3]{c} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}\right)}{729 a c^{17/3}}","\frac{\left(-573080508 b^2 c^4+602654094 b c^{12}-1815934120 a b^2 x+1050127416 a b c^8 x-1205308188 a^2 c^{12} x^2-1400169888 a^3 c^8 x^3\right) \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(668593926 b^2 c^3-502211745 b c^{11}-12014883738 a b c^7 x+3486784401 a c^{15} x+1004423490 a^2 c^{11} x^2+1283489064 a^3 c^7 x^3\right) \sqrt[4]{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(-817170354 b^2 c^2+435250179 b c^{10}+4575400830 a b c^6 x-1162261467 a c^{14} x-870500358 a^2 c^{10} x^2-1188415800 a^3 c^6 x^3\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(1089560472 b^2 c-386889048 b c^9-3287950380 a b c^5 x+774840978 a c^{13} x+773778096 a^2 c^9 x^2+1109188080 a^3 c^5 x^3\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\sqrt{-b+a^2 x^2} \left(\left(-1815934120 b^2+350042472 b c^8-1205308188 a c^{12} x-1400169888 a^2 c^8 x^2\right) \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(-11373139206 b c^7+3486784401 c^{15}+1004423490 a c^{11} x+1283489064 a^2 c^7 x^2\right) \sqrt[4]{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(3981192930 b c^6-1162261467 c^{14}-870500358 a c^{10} x-1188415800 a^2 c^6 x^2\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(-2733356340 b c^5+774840978 c^{13}+773778096 a c^9 x+1109188080 a^2 c^5 x^2\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{2865402540 a c^5 \left(a x+\sqrt{-b+a^2 x^2}\right)^{5/4}}+\frac{308 b^2 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[3]{c}}\right)}{243 \sqrt{3} a c^{17/3}}-\frac{308 b^2 \log \left(-\sqrt[3]{c}+\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{729 a c^{17/3}}+\frac{154 b^2 \log \left(c^{2/3}+\sqrt[3]{c} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}\right)}{729 a c^{17/3}}",1,"((-573080508*b^2*c^4 + 602654094*b*c^12 - 1815934120*a*b^2*x + 1050127416*a*b*c^8*x - 1205308188*a^2*c^12*x^2 - 1400169888*a^3*c^8*x^3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (668593926*b^2*c^3 - 502211745*b*c^11 - 12014883738*a*b*c^7*x + 3486784401*a*c^15*x + 1004423490*a^2*c^11*x^2 + 1283489064*a^3*c^7*x^3)*(a*x + Sqrt[-b + a^2*x^2])^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (-817170354*b^2*c^2 + 435250179*b*c^10 + 4575400830*a*b*c^6*x - 1162261467*a*c^14*x - 870500358*a^2*c^10*x^2 - 1188415800*a^3*c^6*x^3)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (1089560472*b^2*c - 386889048*b*c^9 - 3287950380*a*b*c^5*x + 774840978*a*c^13*x + 773778096*a^2*c^9*x^2 + 1109188080*a^3*c^5*x^3)*(a*x + Sqrt[-b + a^2*x^2])^(3/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + Sqrt[-b + a^2*x^2]*((-1815934120*b^2 + 350042472*b*c^8 - 1205308188*a*c^12*x - 1400169888*a^2*c^8*x^2)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (-11373139206*b*c^7 + 3486784401*c^15 + 1004423490*a*c^11*x + 1283489064*a^2*c^7*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (3981192930*b*c^6 - 1162261467*c^14 - 870500358*a*c^10*x - 1188415800*a^2*c^6*x^2)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (-2733356340*b*c^5 + 774840978*c^13 + 773778096*a*c^9*x + 1109188080*a^2*c^5*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(3/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)))/(2865402540*a*c^5*(a*x + Sqrt[-b + a^2*x^2])^(5/4)) + (308*b^2*ArcTan[1/Sqrt[3] + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3))/(Sqrt[3]*c^(1/3))])/(243*Sqrt[3]*a*c^(17/3)) - (308*b^2*Log[-c^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)])/(729*a*c^(17/3)) + (154*b^2*Log[c^(2/3) + c^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3)])/(729*a*c^(17/3))","A",1
3140,1,2065,1178,4.9117958,"\int \frac{c_6+x c_7}{\sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5}} \, dx","IntegrateAlgebraic[(C[6] + x*C[7])/Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]],x]","\text{Result too large to show}","-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{c_3} \sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt{\sqrt{c_3} c_4+\sqrt{c_1} c_5}}\right) \left(-8 c_1{}^2 c_2 c_3{}^{3/2} c_4 c_5 c_6+8 c_0 c_1 c_3{}^{5/2} c_4 c_5 c_6-8 c_1{}^{5/2} c_2 c_3 c_5{}^2 c_6+8 c_0 c_1{}^{3/2} c_3{}^2 c_5{}^2 c_6+6 c_1{}^2 c_2{}^2 \sqrt{c_3} c_4 c_5 c_7-4 c_0 c_1 c_2 c_3{}^{3/2} c_4 c_5 c_7-2 c_0{}^2 c_3{}^{5/2} c_4 c_5 c_7+3 c_1{}^{5/2} c_2{}^2 c_5{}^2 c_7+2 c_0 c_1{}^{3/2} c_2 c_3 c_5{}^2 c_7-5 c_0{}^2 \sqrt{c_1} c_3{}^2 c_5{}^2 c_7\right)}{16 c_1{}^{3/2} c_3{}^{7/4} \left(\sqrt{c_3} c_4+\sqrt{c_1} c_5\right){}^{5/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{c_3} \sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt{\sqrt{c_3} c_4-\sqrt{c_1} c_5}}\right) \sqrt{\sqrt{c_3} c_4-\sqrt{c_1} c_5} \left(-8 c_1{}^2 c_2 c_3{}^{3/2} c_4 c_5 c_6+8 c_0 c_1 c_3{}^{5/2} c_4 c_5 c_6+8 c_1{}^{5/2} c_2 c_3 c_5{}^2 c_6-8 c_0 c_1{}^{3/2} c_3{}^2 c_5{}^2 c_6+6 c_1{}^2 c_2{}^2 \sqrt{c_3} c_4 c_5 c_7-4 c_0 c_1 c_2 c_3{}^{3/2} c_4 c_5 c_7-2 c_0{}^2 c_3{}^{5/2} c_4 c_5 c_7-3 c_1{}^{5/2} c_2{}^2 c_5{}^2 c_7-2 c_0 c_1{}^{3/2} c_2 c_3 c_5{}^2 c_7+5 c_0{}^2 \sqrt{c_1} c_3{}^2 c_5{}^2 c_7\right)}{16 c_1{}^{3/2} c_3{}^{7/4} \left(-\sqrt{c_3} c_4+\sqrt{c_1} c_5\right){}^3}+\frac{\sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5} \left(4 c_2 c_3{}^2 c_4{}^3 c_6+4 x c_3{}^3 c_4{}^3 c_6-4 c_1 c_2 c_3 c_4 c_5{}^2 c_6-4 x c_1 c_3{}^2 c_4 c_5{}^2 c_6-2 c_2{}^2 c_3 c_4{}^3 c_7+2 x^2 c_3{}^3 c_4{}^3 c_7-c_1 c_2{}^2 c_4 c_5{}^2 c_7+3 c_0 c_2 c_3 c_4 c_5{}^2 c_7-3 x c_1 c_2 c_3 c_4 c_5{}^2 c_7+3 x c_0 c_3{}^2 c_4 c_5{}^2 c_7-2 x^2 c_1 c_3{}^2 c_4 c_5{}^2 c_7\right)}{4 c_3 \left(c_3 c_4{}^2-c_1 c_5{}^2\right){}^2}+\frac{\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} \sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5} \left(-8 c_1 c_2 c_3{}^2 c_4{}^2 c_5 c_6-8 x c_1 c_3{}^3 c_4{}^2 c_5 c_6+8 c_1{}^2 c_2 c_3 c_5{}^3 c_6+8 x c_1{}^2 c_3{}^2 c_5{}^3 c_6+5 c_1 c_2{}^2 c_3 c_4{}^2 c_5 c_7-c_0 c_2 c_3{}^2 c_4{}^2 c_5 c_7+x c_1 c_2 c_3{}^2 c_4{}^2 c_5 c_7-x c_0 c_3{}^3 c_4{}^2 c_5 c_7-4 x^2 c_1 c_3{}^3 c_4{}^2 c_5 c_7+c_1{}^2 c_2{}^2 c_5{}^3 c_7-5 c_0 c_1 c_2 c_3 c_5{}^3 c_7+5 x c_1{}^2 c_2 c_3 c_5{}^3 c_7-5 x c_0 c_1 c_3{}^2 c_5{}^3 c_7+4 x^2 c_1{}^2 c_3{}^2 c_5{}^3 c_7\right)}{8 c_1 c_3 \left(c_3 c_4{}^2-c_1 c_5{}^2\right){}^2}",1,"(ArcTan[(Sqrt[-(C[3]*C[4]) - Sqrt[C[1]]*Sqrt[C[3]]*C[5]]*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]])/(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5])]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5]))]*(-8*C[1]^2*C[2]*C[3]^(3/2)*C[4]*C[5]*C[6] + 8*C[0]*C[1]*C[3]^(5/2)*C[4]*C[5]*C[6] - 8*C[1]^(5/2)*C[2]*C[3]*C[5]^2*C[6] + 8*C[0]*C[1]^(3/2)*C[3]^2*C[5]^2*C[6] + 6*C[1]^2*C[2]^2*Sqrt[C[3]]*C[4]*C[5]*C[7] - 4*C[0]*C[1]*C[2]*C[3]^(3/2)*C[4]*C[5]*C[7] - 2*C[0]^2*C[3]^(5/2)*C[4]*C[5]*C[7] + 3*C[1]^(5/2)*C[2]^2*C[5]^2*C[7] + 2*C[0]*C[1]^(3/2)*C[2]*C[3]*C[5]^2*C[7] - 5*C[0]^2*Sqrt[C[1]]*C[3]^2*C[5]^2*C[7]))/(16*C[1]^(3/2)*C[3]^2*(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5])^3) + (ArcTan[(Sqrt[-(C[3]*C[4]) + Sqrt[C[1]]*Sqrt[C[3]]*C[5]]*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]])/(Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5])]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5]))]*(-8*C[1]^2*C[2]*C[3]^(3/2)*C[4]*C[5]*C[6] + 8*C[0]*C[1]*C[3]^(5/2)*C[4]*C[5]*C[6] + 8*C[1]^(5/2)*C[2]*C[3]*C[5]^2*C[6] - 8*C[0]*C[1]^(3/2)*C[3]^2*C[5]^2*C[6] + 6*C[1]^2*C[2]^2*Sqrt[C[3]]*C[4]*C[5]*C[7] - 4*C[0]*C[1]*C[2]*C[3]^(3/2)*C[4]*C[5]*C[7] - 2*C[0]^2*C[3]^(5/2)*C[4]*C[5]*C[7] - 3*C[1]^(5/2)*C[2]^2*C[5]^2*C[7] - 2*C[0]*C[1]^(3/2)*C[2]*C[3]*C[5]^2*C[7] + 5*C[0]^2*Sqrt[C[1]]*C[3]^2*C[5]^2*C[7]))/(16*C[1]^(3/2)*C[3]^2*(-(Sqrt[C[3]]*C[4]) + Sqrt[C[1]]*C[5])^3) + (Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*(8*C[1]^3*C[2]*C[3]^2*C[4]^3*C[6] - 8*C[0]*C[1]^2*C[3]^3*C[4]^3*C[6] - (8*C[1]^2*(C[0] + x*C[1])*C[2]*C[3]^3*C[4]^3*C[6])/(C[2] + x*C[3]) + (8*C[0]*C[1]*(C[0] + x*C[1])*C[3]^4*C[4]^3*C[6])/(C[2] + x*C[3]) - 8*C[1]^3*C[2]*C[3]^2*Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[4]^2*C[5]*C[6] + 8*C[0]*C[1]^2*C[3]^3*Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[4]^2*C[5]*C[6] + 8*C[1]^2*C[2]*C[3]^3*((C[0] + x*C[1])/(C[2] + x*C[3]))^(3/2)*C[4]^2*C[5]*C[6] - 8*C[0]*C[1]*C[3]^4*((C[0] + x*C[1])/(C[2] + x*C[3]))^(3/2)*C[4]^2*C[5]*C[6] - 8*C[1]^4*C[2]*C[3]*C[4]*C[5]^2*C[6] + 8*C[0]*C[1]^3*C[3]^2*C[4]*C[5]^2*C[6] + (8*C[1]^3*(C[0] + x*C[1])*C[2]*C[3]^2*C[4]*C[5]^2*C[6])/(C[2] + x*C[3]) - (8*C[0]*C[1]^2*(C[0] + x*C[1])*C[3]^3*C[4]*C[5]^2*C[6])/(C[2] + x*C[3]) + 8*C[1]^4*C[2]*C[3]*Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]^3*C[6] - 8*C[0]*C[1]^3*C[3]^2*Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]^3*C[6] - 8*C[1]^3*C[2]*C[3]^2*((C[0] + x*C[1])/(C[2] + x*C[3]))^(3/2)*C[5]^3*C[6] + 8*C[0]*C[1]^2*C[3]^3*((C[0] + x*C[1])/(C[2] + x*C[3]))^(3/2)*C[5]^3*C[6] - 4*C[1]^3*C[2]^2*C[3]*C[4]^3*C[7] + 4*C[0]^2*C[1]*C[3]^3*C[4]^3*C[7] + (8*C[1]^2*(C[0] + x*C[1])*C[2]^2*C[3]^2*C[4]^3*C[7])/(C[2] + x*C[3]) - (8*C[0]*C[1]*(C[0] + x*C[1])*C[2]*C[3]^3*C[4]^3*C[7])/(C[2] + x*C[3]) + 5*C[1]^3*C[2]^2*C[3]*Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[4]^2*C[5]*C[7] - 2*C[0]*C[1]^2*C[2]*C[3]^2*Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[4]^2*C[5]*C[7] - 3*C[0]^2*C[1]*C[3]^3*Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[4]^2*C[5]*C[7] - 9*C[1]^2*C[2]^2*C[3]^2*((C[0] + x*C[1])/(C[2] + x*C[3]))^(3/2)*C[4]^2*C[5]*C[7] + 10*C[0]*C[1]*C[2]*C[3]^3*((C[0] + x*C[1])/(C[2] + x*C[3]))^(3/2)*C[4]^2*C[5]*C[7] - C[0]^2*C[3]^4*((C[0] + x*C[1])/(C[2] + x*C[3]))^(3/2)*C[4]^2*C[5]*C[7] - 2*C[1]^4*C[2]^2*C[4]*C[5]^2*C[7] + 12*C[0]*C[1]^3*C[2]*C[3]*C[4]*C[5]^2*C[7] - 10*C[0]^2*C[1]^2*C[3]^2*C[4]*C[5]^2*C[7] - (2*C[1]^3*(C[0] + x*C[1])*C[2]^2*C[3]*C[4]*C[5]^2*C[7])/(C[2] + x*C[3]) - (4*C[0]*C[1]^2*(C[0] + x*C[1])*C[2]*C[3]^2*C[4]*C[5]^2*C[7])/(C[2] + x*C[3]) + (6*C[0]^2*C[1]*(C[0] + x*C[1])*C[3]^3*C[4]*C[5]^2*C[7])/(C[2] + x*C[3]) + C[1]^4*C[2]^2*Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]^3*C[7] - 10*C[0]*C[1]^3*C[2]*C[3]*Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]^3*C[7] + 9*C[0]^2*C[1]^2*C[3]^2*Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]^3*C[7] + 3*C[1]^3*C[2]^2*C[3]*((C[0] + x*C[1])/(C[2] + x*C[3]))^(3/2)*C[5]^3*C[7] + 2*C[0]*C[1]^2*C[2]*C[3]^2*((C[0] + x*C[1])/(C[2] + x*C[3]))^(3/2)*C[5]^3*C[7] - 5*C[0]^2*C[1]*C[3]^3*((C[0] + x*C[1])/(C[2] + x*C[3]))^(3/2)*C[5]^3*C[7]))/(8*C[1]*C[3]*(C[1] - ((C[0] + x*C[1])*C[3])/(C[2] + x*C[3]))^2*(-(C[3]*C[4]^2) + C[1]*C[5]^2)^2)","A",1
3141,1,1186,1186,3.1569329,"\int \frac{\sqrt{-b+a^2 x^2}}{\left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}} \, dx","IntegrateAlgebraic[Sqrt[-b + a^2*x^2]/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3),x]","\frac{\left(-748701954 b^2 c^7+8135830269 b c^{15}-1204701498 a b^2 c^3 x+3515482215 a b c^{11} x-16271660538 a^2 c^{15} x^2-4687309620 a^3 c^{11} x^3\right) \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(820006902 b^2 c^6-2711943423 b c^{14}+1472412942 a b^2 c^2 x-3046751253 a b c^{10} x+5423886846 a^2 c^{14} x^2+4062335004 a^3 c^{10} x^3\right) \sqrt[4]{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(-911118780 b^2 c^5+1807962282 b c^{13}-1963217256 a b^2 c x+2708223336 a b c^9 x-3615924564 a^2 c^{13} x^2-3610964448 a^3 c^9 x^3\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(1032601284 b^2 c^4-1406192886 b c^{12}+3272028760 a b^2 x-2450297304 a b c^8 x+2812385772 a^2 c^{12} x^2+3267063072 a^3 c^8 x^3\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\sqrt{-b+a^2 x^2} \left(\left(-1204701498 b^2 c^3+1171827405 b c^{11}-16271660538 a c^{15} x-4687309620 a^2 c^{11} x^2\right) \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(1472412942 b^2 c^2-1015583751 b c^{10}+5423886846 a c^{14} x+4062335004 a^2 c^{10} x^2\right) \sqrt[4]{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(-1963217256 b^2 c+902741112 b c^9-3615924564 a c^{13} x-3610964448 a^2 c^9 x^2\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(3272028760 b^2-816765768 b c^8+2812385772 a c^{12} x+3267063072 a^2 c^8 x^2\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{11979231264 a^2 c^8 x \sqrt{-b+a^2 x^2}+5989615632 a c^8 \left(-b+2 a^2 x^2\right)}-\frac{21505 b^2 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[3]{c}}\right)}{19683 \sqrt{3} a c^{26/3}}+\frac{2 \sqrt{3} b \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[3]{c}}\right)}{a c^{2/3}}+\frac{21505 b^2 \log \left(-\sqrt[3]{c}+\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{59049 a c^{26/3}}-\frac{2 b \log \left(-\sqrt[3]{c}+\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{a c^{2/3}}-\frac{21505 b^2 \log \left(c^{2/3}+\sqrt[3]{c} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}\right)}{118098 a c^{26/3}}+\frac{b \log \left(c^{2/3}+\sqrt[3]{c} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}\right)}{a c^{2/3}}","\frac{\left(-748701954 b^2 c^7+8135830269 b c^{15}-1204701498 a b^2 c^3 x+3515482215 a b c^{11} x-16271660538 a^2 c^{15} x^2-4687309620 a^3 c^{11} x^3\right) \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(820006902 b^2 c^6-2711943423 b c^{14}+1472412942 a b^2 c^2 x-3046751253 a b c^{10} x+5423886846 a^2 c^{14} x^2+4062335004 a^3 c^{10} x^3\right) \sqrt[4]{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(-911118780 b^2 c^5+1807962282 b c^{13}-1963217256 a b^2 c x+2708223336 a b c^9 x-3615924564 a^2 c^{13} x^2-3610964448 a^3 c^9 x^3\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(1032601284 b^2 c^4-1406192886 b c^{12}+3272028760 a b^2 x-2450297304 a b c^8 x+2812385772 a^2 c^{12} x^2+3267063072 a^3 c^8 x^3\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\sqrt{-b+a^2 x^2} \left(\left(-1204701498 b^2 c^3+1171827405 b c^{11}-16271660538 a c^{15} x-4687309620 a^2 c^{11} x^2\right) \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(1472412942 b^2 c^2-1015583751 b c^{10}+5423886846 a c^{14} x+4062335004 a^2 c^{10} x^2\right) \sqrt[4]{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(-1963217256 b^2 c+902741112 b c^9-3615924564 a c^{13} x-3610964448 a^2 c^9 x^2\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(3272028760 b^2-816765768 b c^8+2812385772 a c^{12} x+3267063072 a^2 c^8 x^2\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{11979231264 a^2 c^8 x \sqrt{-b+a^2 x^2}+5989615632 a c^8 \left(-b+2 a^2 x^2\right)}-\frac{21505 b^2 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[3]{c}}\right)}{19683 \sqrt{3} a c^{26/3}}+\frac{2 \sqrt{3} b \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[3]{c}}\right)}{a c^{2/3}}+\frac{21505 b^2 \log \left(-\sqrt[3]{c}+\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{59049 a c^{26/3}}-\frac{2 b \log \left(-\sqrt[3]{c}+\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{a c^{2/3}}-\frac{21505 b^2 \log \left(c^{2/3}+\sqrt[3]{c} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}\right)}{118098 a c^{26/3}}+\frac{b \log \left(c^{2/3}+\sqrt[3]{c} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}\right)}{a c^{2/3}}",1,"((-748701954*b^2*c^7 + 8135830269*b*c^15 - 1204701498*a*b^2*c^3*x + 3515482215*a*b*c^11*x - 16271660538*a^2*c^15*x^2 - 4687309620*a^3*c^11*x^3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (820006902*b^2*c^6 - 2711943423*b*c^14 + 1472412942*a*b^2*c^2*x - 3046751253*a*b*c^10*x + 5423886846*a^2*c^14*x^2 + 4062335004*a^3*c^10*x^3)*(a*x + Sqrt[-b + a^2*x^2])^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (-911118780*b^2*c^5 + 1807962282*b*c^13 - 1963217256*a*b^2*c*x + 2708223336*a*b*c^9*x - 3615924564*a^2*c^13*x^2 - 3610964448*a^3*c^9*x^3)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (1032601284*b^2*c^4 - 1406192886*b*c^12 + 3272028760*a*b^2*x - 2450297304*a*b*c^8*x + 2812385772*a^2*c^12*x^2 + 3267063072*a^3*c^8*x^3)*(a*x + Sqrt[-b + a^2*x^2])^(3/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + Sqrt[-b + a^2*x^2]*((-1204701498*b^2*c^3 + 1171827405*b*c^11 - 16271660538*a*c^15*x - 4687309620*a^2*c^11*x^2)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (1472412942*b^2*c^2 - 1015583751*b*c^10 + 5423886846*a*c^14*x + 4062335004*a^2*c^10*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (-1963217256*b^2*c + 902741112*b*c^9 - 3615924564*a*c^13*x - 3610964448*a^2*c^9*x^2)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (3272028760*b^2 - 816765768*b*c^8 + 2812385772*a*c^12*x + 3267063072*a^2*c^8*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(3/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)))/(11979231264*a^2*c^8*x*Sqrt[-b + a^2*x^2] + 5989615632*a*c^8*(-b + 2*a^2*x^2)) - (21505*b^2*ArcTan[1/Sqrt[3] + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3))/(Sqrt[3]*c^(1/3))])/(19683*Sqrt[3]*a*c^(26/3)) + (2*Sqrt[3]*b*ArcTan[1/Sqrt[3] + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3))/(Sqrt[3]*c^(1/3))])/(a*c^(2/3)) + (21505*b^2*Log[-c^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)])/(59049*a*c^(26/3)) - (2*b*Log[-c^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)])/(a*c^(2/3)) - (21505*b^2*Log[c^(2/3) + c^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3)])/(118098*a*c^(26/3)) + (b*Log[c^(2/3) + c^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3)])/(a*c^(2/3))","A",1
3142,1,1202,1202,4.3536893,"\int \left(b+a^2 x^2\right)^{3/2} \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}} \, dx","IntegrateAlgebraic[(b + a^2*x^2)^(3/2)*Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]],x]","\frac{\left(256071816 b^5 c^2+50005263360 b^4 c^6-2317090816 b^3 c^{10}-234881024 b^2 c^{14}+1440403965 a b^5 x+22543188096 a b^4 c^4 x-7568457728 a b^3 c^8 x-734003200 a b^2 c^{12} x+512143632 a^2 b^4 c^2 x^2+682252247040 a^2 b^3 c^6 x^2-19692781568 a^2 b^2 c^{10} x^2-1879048192 a^2 b c^{14} x^2+1920538620 a^3 b^4 x^3+29797447680 a^3 b^3 c^4 x^3-33029095424 a^3 b^2 c^8 x^3-2936012800 a^3 b c^{12} x^3+1247785943040 a^4 b^2 c^6 x^4-23160946688 a^4 b c^{10} x^4-1879048192 a^4 c^{14} x^4-33035911168 a^5 b c^8 x^5-2348810240 a^5 c^{12} x^5+748215336960 a^6 b c^6 x^6-3699376128 a^6 c^{10} x^6-6297747456 a^7 c^8 x^7+200740700160 a^8 c^6 x^8\right) \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+\left(-320089770 b^5 c-5143607040 b^4 c^5+1757085696 b^3 c^9+176160768 b^2 c^{13}-219490128 a b^4 c^3 x+185450137600 a b^3 c^7 x+13851164672 a b^2 c^{11} x+1409286144 a b c^{15} x-640179540 a^2 b^4 c x^2-9932482560 a^2 b^3 c^5 x^2+15116402688 a^2 b^2 c^9 x^2+1409286144 a^2 b c^{13} x^2+267624448000 a^3 b^2 c^7 x^3+20180893696 a^3 b c^{11} x^3+1879048192 a^3 c^{15} x^3+18295554048 a^4 b c^9 x^4+1409286144 a^4 c^{13} x^4+29595238400 a^5 b c^7 x^5+2055208960 a^5 c^{11} x^5+3391094784 a^6 c^9 x^6+5904138240 a^7 c^7 x^7\right) \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+\sqrt{b+a^2 x^2} \left(\left(480134655 b^5+7644464256 b^4 c^4-1474330624 b^3 c^8-146800640 b^2 c^{12}+512143632 a b^4 c^2 x+276208558080 a b^3 c^6 x-9499574272 a b^2 c^{10} x-939524096 a b c^{14} x+1920538620 a^2 b^4 x^2+29797447680 a^2 b^3 c^4 x^2-18872795136 a^2 b^2 c^8 x^2-1761607680 a^2 b c^{12} x^2+948956037120 a^3 b^2 c^6 x^3-21311258624 a^3 b c^{10} x^3-1879048192 a^3 c^{14} x^3-29887037440 a^4 b c^8 x^4-2348810240 a^4 c^{12} x^4+647844986880 a^5 b c^6 x^5-3699376128 a^5 c^{10} x^5-6297747456 a^6 c^8 x^6+200740700160 a^7 c^6 x^7\right) \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+\left(-219490128 b^4 c^3+60891084800 b^3 c^7+4531421184 b^2 c^{11}+469762048 b c^{15}-640179540 a b^4 c x-9932482560 a b^3 c^5 x+7240286208 a b^2 c^9 x+704643072 a b c^{13} x+255040880640 a^2 b^2 c^7 x^2+19153289216 a^2 b c^{11} x^2+1879048192 a^2 c^{15} x^2+16600006656 a^3 b c^9 x^3+1409286144 a^3 c^{13} x^3+26643169280 a^4 b c^7 x^4+2055208960 a^4 c^{11} x^4+3391094784 a^5 c^9 x^5+5904138240 a^6 c^7 x^6\right) \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}\right)}{119189790720 a c^6 \left(a x+\sqrt{b+a^2 x^2}\right)^{7/2}}-\frac{33 b^4 \tanh ^{-1}\left(\frac{\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{\sqrt{c}}\right)}{8192 a c^{13/2}}-\frac{b^3 \tanh ^{-1}\left(\frac{\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{\sqrt{c}}\right)}{16 a c^{5/2}}","\frac{\left(256071816 b^5 c^2+50005263360 b^4 c^6-2317090816 b^3 c^{10}-234881024 b^2 c^{14}+1440403965 a b^5 x+22543188096 a b^4 c^4 x-7568457728 a b^3 c^8 x-734003200 a b^2 c^{12} x+512143632 a^2 b^4 c^2 x^2+682252247040 a^2 b^3 c^6 x^2-19692781568 a^2 b^2 c^{10} x^2-1879048192 a^2 b c^{14} x^2+1920538620 a^3 b^4 x^3+29797447680 a^3 b^3 c^4 x^3-33029095424 a^3 b^2 c^8 x^3-2936012800 a^3 b c^{12} x^3+1247785943040 a^4 b^2 c^6 x^4-23160946688 a^4 b c^{10} x^4-1879048192 a^4 c^{14} x^4-33035911168 a^5 b c^8 x^5-2348810240 a^5 c^{12} x^5+748215336960 a^6 b c^6 x^6-3699376128 a^6 c^{10} x^6-6297747456 a^7 c^8 x^7+200740700160 a^8 c^6 x^8\right) \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+\left(-320089770 b^5 c-5143607040 b^4 c^5+1757085696 b^3 c^9+176160768 b^2 c^{13}-219490128 a b^4 c^3 x+185450137600 a b^3 c^7 x+13851164672 a b^2 c^{11} x+1409286144 a b c^{15} x-640179540 a^2 b^4 c x^2-9932482560 a^2 b^3 c^5 x^2+15116402688 a^2 b^2 c^9 x^2+1409286144 a^2 b c^{13} x^2+267624448000 a^3 b^2 c^7 x^3+20180893696 a^3 b c^{11} x^3+1879048192 a^3 c^{15} x^3+18295554048 a^4 b c^9 x^4+1409286144 a^4 c^{13} x^4+29595238400 a^5 b c^7 x^5+2055208960 a^5 c^{11} x^5+3391094784 a^6 c^9 x^6+5904138240 a^7 c^7 x^7\right) \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+\sqrt{b+a^2 x^2} \left(\left(480134655 b^5+7644464256 b^4 c^4-1474330624 b^3 c^8-146800640 b^2 c^{12}+512143632 a b^4 c^2 x+276208558080 a b^3 c^6 x-9499574272 a b^2 c^{10} x-939524096 a b c^{14} x+1920538620 a^2 b^4 x^2+29797447680 a^2 b^3 c^4 x^2-18872795136 a^2 b^2 c^8 x^2-1761607680 a^2 b c^{12} x^2+948956037120 a^3 b^2 c^6 x^3-21311258624 a^3 b c^{10} x^3-1879048192 a^3 c^{14} x^3-29887037440 a^4 b c^8 x^4-2348810240 a^4 c^{12} x^4+647844986880 a^5 b c^6 x^5-3699376128 a^5 c^{10} x^5-6297747456 a^6 c^8 x^6+200740700160 a^7 c^6 x^7\right) \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}+\left(-219490128 b^4 c^3+60891084800 b^3 c^7+4531421184 b^2 c^{11}+469762048 b c^{15}-640179540 a b^4 c x-9932482560 a b^3 c^5 x+7240286208 a b^2 c^9 x+704643072 a b c^{13} x+255040880640 a^2 b^2 c^7 x^2+19153289216 a^2 b c^{11} x^2+1879048192 a^2 c^{15} x^2+16600006656 a^3 b c^9 x^3+1409286144 a^3 c^{13} x^3+26643169280 a^4 b c^7 x^4+2055208960 a^4 c^{11} x^4+3391094784 a^5 c^9 x^5+5904138240 a^6 c^7 x^6\right) \sqrt{a x+\sqrt{b+a^2 x^2}} \sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}\right)}{119189790720 a c^6 \left(a x+\sqrt{b+a^2 x^2}\right)^{7/2}}-\frac{33 b^4 \tanh ^{-1}\left(\frac{\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{\sqrt{c}}\right)}{8192 a c^{13/2}}-\frac{b^3 \tanh ^{-1}\left(\frac{\sqrt{c+\sqrt{a x+\sqrt{b+a^2 x^2}}}}{\sqrt{c}}\right)}{16 a c^{5/2}}",1,"((256071816*b^5*c^2 + 50005263360*b^4*c^6 - 2317090816*b^3*c^10 - 234881024*b^2*c^14 + 1440403965*a*b^5*x + 22543188096*a*b^4*c^4*x - 7568457728*a*b^3*c^8*x - 734003200*a*b^2*c^12*x + 512143632*a^2*b^4*c^2*x^2 + 682252247040*a^2*b^3*c^6*x^2 - 19692781568*a^2*b^2*c^10*x^2 - 1879048192*a^2*b*c^14*x^2 + 1920538620*a^3*b^4*x^3 + 29797447680*a^3*b^3*c^4*x^3 - 33029095424*a^3*b^2*c^8*x^3 - 2936012800*a^3*b*c^12*x^3 + 1247785943040*a^4*b^2*c^6*x^4 - 23160946688*a^4*b*c^10*x^4 - 1879048192*a^4*c^14*x^4 - 33035911168*a^5*b*c^8*x^5 - 2348810240*a^5*c^12*x^5 + 748215336960*a^6*b*c^6*x^6 - 3699376128*a^6*c^10*x^6 - 6297747456*a^7*c^8*x^7 + 200740700160*a^8*c^6*x^8)*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] + (-320089770*b^5*c - 5143607040*b^4*c^5 + 1757085696*b^3*c^9 + 176160768*b^2*c^13 - 219490128*a*b^4*c^3*x + 185450137600*a*b^3*c^7*x + 13851164672*a*b^2*c^11*x + 1409286144*a*b*c^15*x - 640179540*a^2*b^4*c*x^2 - 9932482560*a^2*b^3*c^5*x^2 + 15116402688*a^2*b^2*c^9*x^2 + 1409286144*a^2*b*c^13*x^2 + 267624448000*a^3*b^2*c^7*x^3 + 20180893696*a^3*b*c^11*x^3 + 1879048192*a^3*c^15*x^3 + 18295554048*a^4*b*c^9*x^4 + 1409286144*a^4*c^13*x^4 + 29595238400*a^5*b*c^7*x^5 + 2055208960*a^5*c^11*x^5 + 3391094784*a^6*c^9*x^6 + 5904138240*a^7*c^7*x^7)*Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] + Sqrt[b + a^2*x^2]*((480134655*b^5 + 7644464256*b^4*c^4 - 1474330624*b^3*c^8 - 146800640*b^2*c^12 + 512143632*a*b^4*c^2*x + 276208558080*a*b^3*c^6*x - 9499574272*a*b^2*c^10*x - 939524096*a*b*c^14*x + 1920538620*a^2*b^4*x^2 + 29797447680*a^2*b^3*c^4*x^2 - 18872795136*a^2*b^2*c^8*x^2 - 1761607680*a^2*b*c^12*x^2 + 948956037120*a^3*b^2*c^6*x^3 - 21311258624*a^3*b*c^10*x^3 - 1879048192*a^3*c^14*x^3 - 29887037440*a^4*b*c^8*x^4 - 2348810240*a^4*c^12*x^4 + 647844986880*a^5*b*c^6*x^5 - 3699376128*a^5*c^10*x^5 - 6297747456*a^6*c^8*x^6 + 200740700160*a^7*c^6*x^7)*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] + (-219490128*b^4*c^3 + 60891084800*b^3*c^7 + 4531421184*b^2*c^11 + 469762048*b*c^15 - 640179540*a*b^4*c*x - 9932482560*a*b^3*c^5*x + 7240286208*a*b^2*c^9*x + 704643072*a*b*c^13*x + 255040880640*a^2*b^2*c^7*x^2 + 19153289216*a^2*b*c^11*x^2 + 1879048192*a^2*c^15*x^2 + 16600006656*a^3*b*c^9*x^3 + 1409286144*a^3*c^13*x^3 + 26643169280*a^4*b*c^7*x^4 + 2055208960*a^4*c^11*x^4 + 3391094784*a^5*c^9*x^5 + 5904138240*a^6*c^7*x^6)*Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]))/(119189790720*a*c^6*(a*x + Sqrt[b + a^2*x^2])^(7/2)) - (33*b^4*ArcTanh[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/Sqrt[c]])/(8192*a*c^(13/2)) - (b^3*ArcTanh[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/Sqrt[c]])/(16*a*c^(5/2))","A",1
3143,1,1225,1225,4.0889597,"\int \frac{\sqrt{-b+a^2 x^2} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}} \, dx","IntegrateAlgebraic[(Sqrt[-b + a^2*x^2]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3))/(a*x + Sqrt[-b + a^2*x^2])^(1/4),x]","\frac{\left(-3272028760 b^3-319355415288 b^2 c^8+1032601284 a b^2 c^4 x+5423886846 a b c^{12} x+6544057520 a^2 b^2 x^2+470457082368 a^2 b c^8 x^2-7231849128 a^3 c^{12} x^3+176421405888 a^4 c^8 x^4\right) \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(-748701954 b^2 c^7-10460353203 b c^{15}-1204701498 a b^2 c^3 x-4519905705 a b c^{11} x+20920706406 a^2 c^{15} x^2+6026540940 a^3 c^{11} x^3\right) \sqrt[4]{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(820006902 b^2 c^6+3486784401 b c^{14}+1472412942 a b^2 c^2 x+3917251611 a b c^{10} x-6973568802 a^2 c^{14} x^2-5223002148 a^3 c^{10} x^3\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(-911118780 b^2 c^5-2324522934 b c^{13}-1963217256 a b^2 c x-3482001432 a b c^9 x+4649045868 a^2 c^{13} x^2+4642668576 a^3 c^9 x^3\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\sqrt{-b+a^2 x^2} \left(\left(1032601284 b^2 c^4+1807962282 b c^{12}+6544057520 a b^2 x+558667785312 a b c^8 x-7231849128 a^2 c^{12} x^2+176421405888 a^3 c^8 x^3\right) \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(-1204701498 b^2 c^3-1506635235 b c^{11}+20920706406 a c^{15} x+6026540940 a^2 c^{11} x^2\right) \sqrt[4]{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(1472412942 b^2 c^2+1305750537 b c^{10}-6973568802 a c^{14} x-5223002148 a^2 c^{10} x^2\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(-1963217256 b^2 c-1160667144 b c^9+4649045868 a c^{13} x+4642668576 a^2 c^9 x^2\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{161719622064 a c^8 \left(a x+\sqrt{-b+a^2 x^2}\right)^{9/4}}-\frac{21505 b^2 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[3]{c}}\right)}{531441 \sqrt{3} a c^{26/3}}+\frac{2 b \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[3]{c}}\right)}{\sqrt{3} a c^{2/3}}+\frac{21505 b^2 \log \left(-\sqrt[3]{c}+\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{1594323 a c^{26/3}}-\frac{2 b \log \left(-\sqrt[3]{c}+\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{3 a c^{2/3}}-\frac{21505 b^2 \log \left(c^{2/3}+\sqrt[3]{c} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}\right)}{3188646 a c^{26/3}}+\frac{b \log \left(c^{2/3}+\sqrt[3]{c} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}\right)}{3 a c^{2/3}}","\frac{\left(-3272028760 b^3-319355415288 b^2 c^8+1032601284 a b^2 c^4 x+5423886846 a b c^{12} x+6544057520 a^2 b^2 x^2+470457082368 a^2 b c^8 x^2-7231849128 a^3 c^{12} x^3+176421405888 a^4 c^8 x^4\right) \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(-748701954 b^2 c^7-10460353203 b c^{15}-1204701498 a b^2 c^3 x-4519905705 a b c^{11} x+20920706406 a^2 c^{15} x^2+6026540940 a^3 c^{11} x^3\right) \sqrt[4]{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(820006902 b^2 c^6+3486784401 b c^{14}+1472412942 a b^2 c^2 x+3917251611 a b c^{10} x-6973568802 a^2 c^{14} x^2-5223002148 a^3 c^{10} x^3\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(-911118780 b^2 c^5-2324522934 b c^{13}-1963217256 a b^2 c x-3482001432 a b c^9 x+4649045868 a^2 c^{13} x^2+4642668576 a^3 c^9 x^3\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\sqrt{-b+a^2 x^2} \left(\left(1032601284 b^2 c^4+1807962282 b c^{12}+6544057520 a b^2 x+558667785312 a b c^8 x-7231849128 a^2 c^{12} x^2+176421405888 a^3 c^8 x^3\right) \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(-1204701498 b^2 c^3-1506635235 b c^{11}+20920706406 a c^{15} x+6026540940 a^2 c^{11} x^2\right) \sqrt[4]{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(1472412942 b^2 c^2+1305750537 b c^{10}-6973568802 a c^{14} x-5223002148 a^2 c^{10} x^2\right) \sqrt{a x+\sqrt{-b+a^2 x^2}} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(-1963217256 b^2 c-1160667144 b c^9+4649045868 a c^{13} x+4642668576 a^2 c^9 x^2\right) \left(a x+\sqrt{-b+a^2 x^2}\right)^{3/4} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{161719622064 a c^8 \left(a x+\sqrt{-b+a^2 x^2}\right)^{9/4}}-\frac{21505 b^2 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[3]{c}}\right)}{531441 \sqrt{3} a c^{26/3}}+\frac{2 b \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}}{\sqrt{3} \sqrt[3]{c}}\right)}{\sqrt{3} a c^{2/3}}+\frac{21505 b^2 \log \left(-\sqrt[3]{c}+\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{1594323 a c^{26/3}}-\frac{2 b \log \left(-\sqrt[3]{c}+\sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}\right)}{3 a c^{2/3}}-\frac{21505 b^2 \log \left(c^{2/3}+\sqrt[3]{c} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}\right)}{3188646 a c^{26/3}}+\frac{b \log \left(c^{2/3}+\sqrt[3]{c} \sqrt[3]{c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}}+\left(c+\sqrt[4]{a x+\sqrt{-b+a^2 x^2}}\right)^{2/3}\right)}{3 a c^{2/3}}",1,"((-3272028760*b^3 - 319355415288*b^2*c^8 + 1032601284*a*b^2*c^4*x + 5423886846*a*b*c^12*x + 6544057520*a^2*b^2*x^2 + 470457082368*a^2*b*c^8*x^2 - 7231849128*a^3*c^12*x^3 + 176421405888*a^4*c^8*x^4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (-748701954*b^2*c^7 - 10460353203*b*c^15 - 1204701498*a*b^2*c^3*x - 4519905705*a*b*c^11*x + 20920706406*a^2*c^15*x^2 + 6026540940*a^3*c^11*x^3)*(a*x + Sqrt[-b + a^2*x^2])^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (820006902*b^2*c^6 + 3486784401*b*c^14 + 1472412942*a*b^2*c^2*x + 3917251611*a*b*c^10*x - 6973568802*a^2*c^14*x^2 - 5223002148*a^3*c^10*x^3)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (-911118780*b^2*c^5 - 2324522934*b*c^13 - 1963217256*a*b^2*c*x - 3482001432*a*b*c^9*x + 4649045868*a^2*c^13*x^2 + 4642668576*a^3*c^9*x^3)*(a*x + Sqrt[-b + a^2*x^2])^(3/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + Sqrt[-b + a^2*x^2]*((1032601284*b^2*c^4 + 1807962282*b*c^12 + 6544057520*a*b^2*x + 558667785312*a*b*c^8*x - 7231849128*a^2*c^12*x^2 + 176421405888*a^3*c^8*x^3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (-1204701498*b^2*c^3 - 1506635235*b*c^11 + 20920706406*a*c^15*x + 6026540940*a^2*c^11*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (1472412942*b^2*c^2 + 1305750537*b*c^10 - 6973568802*a*c^14*x - 5223002148*a^2*c^10*x^2)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (-1963217256*b^2*c - 1160667144*b*c^9 + 4649045868*a*c^13*x + 4642668576*a^2*c^9*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(3/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)))/(161719622064*a*c^8*(a*x + Sqrt[-b + a^2*x^2])^(9/4)) - (21505*b^2*ArcTan[1/Sqrt[3] + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3))/(Sqrt[3]*c^(1/3))])/(531441*Sqrt[3]*a*c^(26/3)) + (2*b*ArcTan[1/Sqrt[3] + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3))/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*a*c^(2/3)) + (21505*b^2*Log[-c^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)])/(1594323*a*c^(26/3)) - (2*b*Log[-c^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)])/(3*a*c^(2/3)) - (21505*b^2*Log[c^(2/3) + c^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3)])/(3188646*a*c^(26/3)) + (b*Log[c^(2/3) + c^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3)])/(3*a*c^(2/3))","A",1
3144,1,1571,1293,3.3078329,"\int \frac{a b c-b^2 x+a^2 x^2}{\sqrt{c+b x+a x^2} \left(c+b x^2\right)^2} \, dx","IntegrateAlgebraic[(a*b*c - b^2*x + a^2*x^2)/(Sqrt[c + b*x + a*x^2]*(c + b*x^2)^2),x]","\frac{\left(-a^2 b c-a b^2 c+b^3 c+a b^3 c-b^4 x+a^3 c x-a^2 b c x-a^2 b^2 c x+a b^3 c x\right) \sqrt{c+b x+a x^2}}{2 c \left(b^3+a^2 c-2 a b c+b^2 c\right) \left(c+b x^2\right)}-\frac{a \text{RootSum}\left[b^2 c+b c^2-4 \sqrt{a} b c \text{$\#$1}+4 a c \text{$\#$1}^2-2 b c \text{$\#$1}^2+b \text{$\#$1}^4\&,\frac{-a b \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)+4 b^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)+2 a^{3/2} \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}}{-\sqrt{a} b c+2 a c \text{$\#$1}-b c \text{$\#$1}+b \text{$\#$1}^3}\&\right]}{2 b}+\frac{\text{RootSum}\left[b^2 c+b c^2-4 \sqrt{a} b c \text{$\#$1}+4 a c \text{$\#$1}^2-2 b c \text{$\#$1}^2+b \text{$\#$1}^4\&,\frac{-3 a^2 b^4 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)+17 a b^5 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)+2 b^6 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)+3 a b^6 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)-4 a^4 b c^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)+23 a^3 b^2 c^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)-31 a^2 b^3 c^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)+4 a^3 b^3 c^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)+14 a b^4 c^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)-7 a^2 b^4 c^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)+a b^5 c^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)+6 a^{5/2} b^3 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}-2 a^{3/2} b^4 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}-2 \sqrt{a} b^5 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}-6 a^{3/2} b^5 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}+8 a^{9/2} c^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}-12 a^{7/2} b c^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}+4 a^{5/2} b^2 c^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}-8 a^{7/2} b^2 c^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}+12 a^{5/2} b^3 c^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}-4 a^{3/2} b^4 c^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}-b^6 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}^2-a^3 b^2 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}^2-3 a^2 b^3 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}^2+2 a b^4 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}^2+a^2 b^4 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}^2+a b^5 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}^2}{-\sqrt{a} b^4 c-a^{5/2} b c^2+2 a^{3/2} b^2 c^2-\sqrt{a} b^3 c^2+2 a b^3 c \text{$\#$1}-b^4 c \text{$\#$1}+2 a^3 c^2 \text{$\#$1}-5 a^2 b c^2 \text{$\#$1}+4 a b^2 c^2 \text{$\#$1}-b^3 c^2 \text{$\#$1}+b^4 \text{$\#$1}^3+a^2 b c \text{$\#$1}^3-2 a b^2 c \text{$\#$1}^3+b^3 c \text{$\#$1}^3}\&\right]}{8 b c}","\frac{\left(-a^2 b c-a b^2 c+b^3 c+a b^3 c-b^4 x+a^3 c x-a^2 b c x-a^2 b^2 c x+a b^3 c x\right) \sqrt{c+b x+a x^2}}{2 c \left(b^3+a^2 c-2 a b c+b^2 c\right) \left(c+b x^2\right)}-\frac{\text{RootSum}\left[b^2 c+b c^2-4 \sqrt{a} b c \text{$\#$1}+4 a c \text{$\#$1}^2-2 b c \text{$\#$1}^2+b \text{$\#$1}^4\&,\frac{-a^2 b^3 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)-a b^4 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)-2 b^5 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)-3 a b^5 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)+a^3 b c^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)-5 a^2 b^2 c^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)-4 a^3 b^2 c^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)+2 a b^3 c^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)+7 a^2 b^3 c^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)-a b^4 c^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right)+2 a^{5/2} b^2 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}+2 a^{3/2} b^3 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}+2 \sqrt{a} b^4 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}+6 a^{3/2} b^4 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}-4 a^{7/2} c^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}+4 a^{5/2} b c^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}+8 a^{7/2} b c^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}-12 a^{5/2} b^2 c^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}+4 a^{3/2} b^3 c^2 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}+b^5 \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}^2+a^3 b c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}^2+3 a^2 b^2 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}^2-2 a b^3 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}^2-a^2 b^3 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}^2-a b^4 c \log \left(-\sqrt{a} x+\sqrt{c+b x+a x^2}-\text{$\#$1}\right) \text{$\#$1}^2}{-\sqrt{a} b^4 c-a^{5/2} b c^2+2 a^{3/2} b^2 c^2-\sqrt{a} b^3 c^2+2 a b^3 c \text{$\#$1}-b^4 c \text{$\#$1}+2 a^3 c^2 \text{$\#$1}-5 a^2 b c^2 \text{$\#$1}+4 a b^2 c^2 \text{$\#$1}-b^3 c^2 \text{$\#$1}+b^4 \text{$\#$1}^3+a^2 b c \text{$\#$1}^3-2 a b^2 c \text{$\#$1}^3+b^3 c \text{$\#$1}^3}\&\right]}{8 c}",1,"((-(a^2*b*c) - a*b^2*c + b^3*c + a*b^3*c - b^4*x + a^3*c*x - a^2*b*c*x - a^2*b^2*c*x + a*b^3*c*x)*Sqrt[c + b*x + a*x^2])/(2*c*(b^3 + a^2*c - 2*a*b*c + b^2*c)*(c + b*x^2)) - (a*RootSum[b^2*c + b*c^2 - 4*Sqrt[a]*b*c*#1 + 4*a*c*#1^2 - 2*b*c*#1^2 + b*#1^4 & , (-(a*b*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]) + 4*b^2*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1] + 2*a^(3/2)*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1)/(-(Sqrt[a]*b*c) + 2*a*c*#1 - b*c*#1 + b*#1^3) & ])/(2*b) + RootSum[b^2*c + b*c^2 - 4*Sqrt[a]*b*c*#1 + 4*a*c*#1^2 - 2*b*c*#1^2 + b*#1^4 & , (-3*a^2*b^4*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1] + 17*a*b^5*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1] + 2*b^6*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1] + 3*a*b^6*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1] - 4*a^4*b*c^2*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1] + 23*a^3*b^2*c^2*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1] - 31*a^2*b^3*c^2*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1] + 4*a^3*b^3*c^2*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1] + 14*a*b^4*c^2*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1] - 7*a^2*b^4*c^2*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1] + a*b^5*c^2*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1] + 6*a^(5/2)*b^3*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 - 2*a^(3/2)*b^4*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 - 2*Sqrt[a]*b^5*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 - 6*a^(3/2)*b^5*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 + 8*a^(9/2)*c^2*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 - 12*a^(7/2)*b*c^2*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 + 4*a^(5/2)*b^2*c^2*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 - 8*a^(7/2)*b^2*c^2*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 + 12*a^(5/2)*b^3*c^2*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 - 4*a^(3/2)*b^4*c^2*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 - b^6*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1^2 - a^3*b^2*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1^2 - 3*a^2*b^3*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1^2 + 2*a*b^4*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1^2 + a^2*b^4*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1^2 + a*b^5*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1^2)/(-(Sqrt[a]*b^4*c) - a^(5/2)*b*c^2 + 2*a^(3/2)*b^2*c^2 - Sqrt[a]*b^3*c^2 + 2*a*b^3*c*#1 - b^4*c*#1 + 2*a^3*c^2*#1 - 5*a^2*b*c^2*#1 + 4*a*b^2*c^2*#1 - b^3*c^2*#1 + b^4*#1^3 + a^2*b*c*#1^3 - 2*a*b^2*c*#1^3 + b^3*c*#1^3) & ]/(8*b*c)","A",1
3145,1,262,1310,1.0215186,"\int \frac{\left(b^2+a x^2\right)^2 \sqrt{b+\sqrt{b^2+a x^2}}}{\left(-b^2+a x^2\right)^2} \, dx","IntegrateAlgebraic[((b^2 + a*x^2)^2*Sqrt[b + Sqrt[b^2 + a*x^2]])/(-b^2 + a*x^2)^2,x]","\frac{2 x \left(-4 b^2+a x^2\right) \sqrt{b^2+a x^2}}{3 \left(-b^2+a x^2\right) \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{2 x \left(-5 b^3+2 a b x^2\right)}{3 \left(-b^2+a x^2\right) \sqrt{b+\sqrt{b^2+a x^2}}}+\frac{\sqrt{-1+\sqrt{2}} \left(6+\sqrt{2}\right) b^{3/2} \tan ^{-1}\left(\frac{\sqrt{-1+\sqrt{2}} \sqrt{a} x}{\sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{2 \sqrt{a}}+\frac{\left(-6+\sqrt{2}\right) \sqrt{1+\sqrt{2}} b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{1+\sqrt{2}} \sqrt{a} x}{\sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}\right)}{2 \sqrt{a}}","\frac{-2 i a^{3/2} x \left(-9 b^7-46 a b^5 x^2-20 a^2 b^3 x^4+24 a^3 b x^6\right)+2 i \left(a b^8-13 a^2 b^6 x^2-75 a^3 b^4 x^4-52 a^4 b^2 x^6+16 a^5 x^8\right)+\sqrt{b^2+a x^2} \left(-2 i a^{3/2} x \left(-43 a b^4 x^2-60 a^2 b^2 x^4+16 a^3 x^6\right)+2 i \left(-a b^7-27 a^2 b^5 x^2-32 a^3 b^3 x^4+24 a^4 b x^6\right)\right)}{\frac{3 i a^2 x \left(-b^2+a x^2\right) \left(5 b^4+20 a b^2 x^2+16 a^2 x^4\right)}{\sqrt{b+\sqrt{b^2+a x^2}}}+3 i a^{3/2} \left(-b^2+a x^2\right) \left(-5 b^4-20 a b^2 x^2-16 a^2 x^4\right) \sqrt{b+\sqrt{b^2+a x^2}}+\sqrt{b^2+a x^2} \left(\frac{3 i a^2 x \left(-b^2+a x^2\right) \left(4 b^3+8 a b x^2\right)}{\sqrt{b+\sqrt{b^2+a x^2}}}+3 i a^{3/2} \left(-b^2+a x^2\right) \left(4 b^3+8 a b x^2\right) \sqrt{b+\sqrt{b^2+a x^2}}\right)}-\frac{2 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2 \left(-1+\sqrt{2}\right)} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2 \left(-1+\sqrt{2}\right)} \sqrt{b}}\right)}{\sqrt{-1+\sqrt{2}} \sqrt{a}}+\frac{5 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2 \left(-1+\sqrt{2}\right)} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2 \left(-1+\sqrt{2}\right)} \sqrt{b}}\right)}{\sqrt{2 \left(-1+\sqrt{2}\right)} \sqrt{a}}+\frac{3 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{b}}\right)}{\sqrt{1+\sqrt{2}} \sqrt{a}}+\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{b}}\right)}{\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{a}}+\frac{3 b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2 \left(-1+\sqrt{2}\right)} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2 \left(-1+\sqrt{2}\right)} \sqrt{b}}\right)}{\sqrt{-1+\sqrt{2}} \sqrt{a}}-\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2 \left(-1+\sqrt{2}\right)} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2 \left(-1+\sqrt{2}\right)} \sqrt{b}}\right)}{\sqrt{2 \left(-1+\sqrt{2}\right)} \sqrt{a}}-\frac{2 b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{b}}\right)}{\sqrt{1+\sqrt{2}} \sqrt{a}}-\frac{5 b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{b} \sqrt{b+\sqrt{b^2+a x^2}}}-\frac{\sqrt{b+\sqrt{b^2+a x^2}}}{\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{b}}\right)}{\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{a}}",1,"(2*x*(-4*b^2 + a*x^2)*Sqrt[b^2 + a*x^2])/(3*(-b^2 + a*x^2)*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (2*x*(-5*b^3 + 2*a*b*x^2))/(3*(-b^2 + a*x^2)*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (Sqrt[-1 + Sqrt[2]]*(6 + Sqrt[2])*b^(3/2)*ArcTan[(Sqrt[-1 + Sqrt[2]]*Sqrt[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/(2*Sqrt[a]) + ((-6 + Sqrt[2])*Sqrt[1 + Sqrt[2]]*b^(3/2)*ArcTanh[(Sqrt[1 + Sqrt[2]]*Sqrt[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/(2*Sqrt[a])","A",1
3146,1,2109,1356,8.8754545,"\int \frac{\sqrt{a x^2+\sqrt{b+a^2 x^4}}}{(d+c x)^2 \sqrt{b+a^2 x^4}} \, dx","IntegrateAlgebraic[Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/((d + c*x)^2*Sqrt[b + a^2*x^4]),x]","\text{Result too large to show}","\frac{b c^3 d}{\left(b c^4+a^2 d^4\right) \left(-d^2+c^2 x^2\right) \sqrt{a x^2+\sqrt{b+a^2 x^4}}}-\frac{b c^4 x}{\left(b c^4+a^2 d^4\right) \left(-d^2+c^2 x^2\right) \sqrt{a x^2+\sqrt{b+a^2 x^4}}}+\frac{a c d^3 \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\left(b c^4+a^2 d^4\right) \left(-d^2+c^2 x^2\right)}+\frac{i \sqrt{a} x \left(\frac{i a^{3/2} c^2 d^2 x^2}{b c^4+a^2 d^4}+\frac{i \sqrt{a} c^2 d^2 \sqrt{b+a^2 x^4}}{b c^4+a^2 d^4}\right)}{\left(-d^2+c^2 x^2\right) \sqrt{a x^2+\sqrt{b+a^2 x^4}}}-\frac{a b c^4 d \tan ^{-1}\left(\frac{c \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{-a d^2-\sqrt{b c^4+a^2 d^4}}}\right)}{\left(b c^4+a^2 d^4\right)^{3/2} \sqrt{-a d^2-\sqrt{b c^4+a^2 d^4}}}+\frac{a^3 d^5 \tan ^{-1}\left(\frac{c \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{-a d^2-\sqrt{b c^4+a^2 d^4}}}\right)}{\left(b c^4+a^2 d^4\right)^{3/2} \sqrt{-a d^2-\sqrt{b c^4+a^2 d^4}}}+\frac{a^2 d^3 \tan ^{-1}\left(\frac{c \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{-a d^2-\sqrt{b c^4+a^2 d^4}}}\right)}{\left(b c^4+a^2 d^4\right) \sqrt{-a d^2-\sqrt{b c^4+a^2 d^4}}}+\frac{a b c^4 d \tan ^{-1}\left(\frac{c \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{-a d^2+\sqrt{b c^4+a^2 d^4}}}\right)}{\left(b c^4+a^2 d^4\right)^{3/2} \sqrt{-a d^2+\sqrt{b c^4+a^2 d^4}}}-\frac{a^3 d^5 \tan ^{-1}\left(\frac{c \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{-a d^2+\sqrt{b c^4+a^2 d^4}}}\right)}{\left(b c^4+a^2 d^4\right)^{3/2} \sqrt{-a d^2+\sqrt{b c^4+a^2 d^4}}}+\frac{a^2 d^3 \tan ^{-1}\left(\frac{c \sqrt{a x^2+\sqrt{b+a^2 x^4}}}{\sqrt{-a d^2+\sqrt{b c^4+a^2 d^4}}}\right)}{\left(b c^4+a^2 d^4\right) \sqrt{-a d^2+\sqrt{b c^4+a^2 d^4}}}+\frac{a^{3/2} d^2 \text{RootSum}\left[b^2 c^2+4 i a b d^2 \text{$\#$1}+2 b c^2 \text{$\#$1}^2-4 i a d^2 \text{$\#$1}^3+c^2 \text{$\#$1}^4\&,\frac{a b d^2 \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right)+2 i b c^2 \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}-a d^2 \log \left(i a x^2+i \sqrt{b+a^2 x^4}+i \sqrt{2} \sqrt{a} x \sqrt{a x^2+\sqrt{b+a^2 x^4}}-\text{$\#$1}\right) \text{$\#$1}^2}{a b^2 c^4 d^2+a^3 b d^6-i b^2 c^6 \text{$\#$1}-i a^2 b c^2 d^4 \text{$\#$1}-3 a b c^4 d^2 \text{$\#$1}^2-3 a^3 d^6 \text{$\#$1}^2-i b c^6 \text{$\#$1}^3-i a^2 c^2 d^4 \text{$\#$1}^3}\&\right]}{\sqrt{2}}",1,"(b*c^3*d)/((b*c^4 + a^2*d^4)*(-d^2 + c^2*x^2)*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]) - (b*c^4*x)/((b*c^4 + a^2*d^4)*(-d^2 + c^2*x^2)*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]) + (a*c*d^3*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/((b*c^4 + a^2*d^4)*(-d^2 + c^2*x^2)) + (I*Sqrt[a]*x*((I*a^(3/2)*c^2*d^2*x^2)/(b*c^4 + a^2*d^4) + (I*Sqrt[a]*c^2*d^2*Sqrt[b + a^2*x^4])/(b*c^4 + a^2*d^4)))/((-d^2 + c^2*x^2)*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]) - (a*b*c^4*d*ArcTan[(c*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[-(a*d^2) - Sqrt[b*c^4 + a^2*d^4]]])/((b*c^4 + a^2*d^4)^(3/2)*Sqrt[-(a*d^2) - Sqrt[b*c^4 + a^2*d^4]]) + (a^3*d^5*ArcTan[(c*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[-(a*d^2) - Sqrt[b*c^4 + a^2*d^4]]])/((b*c^4 + a^2*d^4)^(3/2)*Sqrt[-(a*d^2) - Sqrt[b*c^4 + a^2*d^4]]) + (a^2*d^3*ArcTan[(c*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[-(a*d^2) - Sqrt[b*c^4 + a^2*d^4]]])/((b*c^4 + a^2*d^4)*Sqrt[-(a*d^2) - Sqrt[b*c^4 + a^2*d^4]]) + (a*b*c^4*d*ArcTan[(c*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[-(a*d^2) + Sqrt[b*c^4 + a^2*d^4]]])/((b*c^4 + a^2*d^4)^(3/2)*Sqrt[-(a*d^2) + Sqrt[b*c^4 + a^2*d^4]]) - (a^3*d^5*ArcTan[(c*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[-(a*d^2) + Sqrt[b*c^4 + a^2*d^4]]])/((b*c^4 + a^2*d^4)^(3/2)*Sqrt[-(a*d^2) + Sqrt[b*c^4 + a^2*d^4]]) + (a^2*d^3*ArcTan[(c*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[-(a*d^2) + Sqrt[b*c^4 + a^2*d^4]]])/((b*c^4 + a^2*d^4)*Sqrt[-(a*d^2) + Sqrt[b*c^4 + a^2*d^4]]) - (I*Sqrt[a]*RootSum[b^2*c^2 + (4*I)*a*b*d^2*#1 + 2*b*c^2*#1^2 - (4*I)*a*d^2*#1^3 + c^2*#1^4 & , (b*c^4*Log[I*a*x^2 + I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1] + 32*a^2*d^4*Log[I*a*x^2 + I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1] - (8*I)*a*c^2*d^2*Log[I*a*x^2 + I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1]*#1 - c^4*Log[I*a*x^2 + I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1]*#1^2)/(I*a*b*d^2 + b*c^2*#1 - (3*I)*a*d^2*#1^2 + c^2*#1^3) & ])/(Sqrt[2]*c^4) + (I*Sqrt[a]*RootSum[b^2*c^2 + (4*I)*a*b*d^2*#1 + 2*b*c^2*#1^2 - (4*I)*a*d^2*#1^3 + c^2*#1^4 & , (b^2*c^8*Log[I*a*x^2 + I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1] + 34*a^2*b*c^4*d^4*Log[I*a*x^2 + I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1] + 32*a^4*d^8*Log[I*a*x^2 + I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1] - (6*I)*a*b*c^6*d^2*Log[I*a*x^2 + I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1]*#1 - (8*I)*a^3*c^2*d^6*Log[I*a*x^2 + I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1]*#1 - b*c^8*Log[I*a*x^2 + I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1]*#1^2 - 2*a^2*c^4*d^4*Log[I*a*x^2 + I*Sqrt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1]*#1^2)/(I*a*b^2*c^4*d^2 + I*a^3*b*d^6 + b^2*c^6*#1 + a^2*b*c^2*d^4*#1 - (3*I)*a*b*c^4*d^2*#1^2 - (3*I)*a^3*d^6*#1^2 + b*c^6*#1^3 + a^2*c^2*d^4*#1^3) & ])/(Sqrt[2]*c^4)","A",1
3147,-1,0,1387,180.5672931,"\int \frac{-b-a c+(1+c) x}{(-a+x) \sqrt[3]{(-a+x) (-b+x)^2} (b-a d+(-1+d) x)} \, dx","IntegrateAlgebraic[(-b - a*c + (1 + c)*x)/((-a + x)*((-a + x)*(-b + x)^2)^(1/3)*(b - a*d + (-1 + d)*x)),x]","\text{\$Aborted}","\frac{\left(1+i \sqrt{3}\right) \sqrt[3]{a-x} (-b+x)^{2/3} \left(\sqrt[3]{d} \sqrt[3]{a-x}+\sqrt[3]{-b+x}\right) \left(d^{2/3} (a-x)^{2/3}-\sqrt[3]{d} \sqrt[3]{a-x} \sqrt[3]{-b+x}+(-b+x)^{2/3}\right) \left(\frac{3 \left(a \sqrt[3]{-b+x}-i \sqrt{3} a \sqrt[3]{-b+x}\right)}{2 (a-b)^2 \sqrt[3]{a-x}}+\frac{3 i \left(i b \sqrt[3]{-b+x}+\sqrt{3} b \sqrt[3]{-b+x}\right)}{2 (a-b)^2 \sqrt[3]{a-x}}+\frac{3 \left(a c \sqrt[3]{-b+x}-i \sqrt{3} a c \sqrt[3]{-b+x}\right)}{2 (a-b)^2 \sqrt[3]{a-x}}+\frac{3 i \left(i a c \sqrt[3]{-b+x}+\sqrt{3} a c \sqrt[3]{-b+x}\right)}{2 (a-b)^2 \sqrt[3]{a-x}}+\frac{\left(3 i b-\sqrt{3} b-3 i a d+\sqrt{3} a d\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-b+x}}{-2 \sqrt[3]{d} \sqrt[3]{a-x}+\sqrt[3]{-b+x}}\right)}{2 (a-b)^2 d^{2/3}}+\frac{\left(-3 i b+\sqrt{3} b+3 i b d-\sqrt{3} b d\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-b+x}}{-2 \sqrt[3]{d} \sqrt[3]{a-x}+\sqrt[3]{-b+x}}\right)}{2 (a-b)^2 d^{2/3}}+\frac{\left(-3 i a c+\sqrt{3} a c+3 i a c d-\sqrt{3} a c d\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-b+x}}{-2 \sqrt[3]{d} \sqrt[3]{a-x}+\sqrt[3]{-b+x}}\right)}{2 (a-b)^2 d^{2/3}}+\frac{\left(3 i b c-\sqrt{3} b c-3 i a c d+\sqrt{3} a c d\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-b+x}}{-2 \sqrt[3]{d} \sqrt[3]{a-x}+\sqrt[3]{-b+x}}\right)}{2 (a-b)^2 d^{2/3}}+\frac{\left(b-i \sqrt{3} b-a d+i \sqrt{3} a d\right) \log \left(\sqrt[3]{d} \sqrt[3]{a-x}+\sqrt[3]{-b+x}\right)}{2 (a-b)^2 d^{2/3}}+\frac{\left(-b+i \sqrt{3} b+b d-i \sqrt{3} b d\right) \log \left(\sqrt[3]{d} \sqrt[3]{a-x}+\sqrt[3]{-b+x}\right)}{2 (a-b)^2 d^{2/3}}+\frac{\left(-a c+i \sqrt{3} a c+a c d-i \sqrt{3} a c d\right) \log \left(\sqrt[3]{d} \sqrt[3]{a-x}+\sqrt[3]{-b+x}\right)}{2 (a-b)^2 d^{2/3}}+\frac{\left(b c-i \sqrt{3} b c-a c d+i \sqrt{3} a c d\right) \log \left(\sqrt[3]{d} \sqrt[3]{a-x}+\sqrt[3]{-b+x}\right)}{2 (a-b)^2 d^{2/3}}+\frac{\left(-b+i \sqrt{3} b+a d-i \sqrt{3} a d\right) \log \left(d^{2/3} (a-x)^{2/3}-\sqrt[3]{d} \sqrt[3]{a-x} \sqrt[3]{-b+x}+(-b+x)^{2/3}\right)}{4 (a-b)^2 d^{2/3}}+\frac{\left(b-i \sqrt{3} b-b d+i \sqrt{3} b d\right) \log \left(d^{2/3} (a-x)^{2/3}-\sqrt[3]{d} \sqrt[3]{a-x} \sqrt[3]{-b+x}+(-b+x)^{2/3}\right)}{4 (a-b)^2 d^{2/3}}+\frac{\left(-b c+i \sqrt{3} b c+a c d-i \sqrt{3} a c d\right) \log \left(d^{2/3} (a-x)^{2/3}-\sqrt[3]{d} \sqrt[3]{a-x} \sqrt[3]{-b+x}+(-b+x)^{2/3}\right)}{4 (a-b)^2 d^{2/3}}+\frac{\left(a c-i \sqrt{3} a c-a c d+i \sqrt{3} a c d\right) \log \left(d^{2/3} (a-x)^{2/3}-\sqrt[3]{d} \sqrt[3]{a-x} \sqrt[3]{-b+x}+(-b+x)^{2/3}\right)}{4 (a-b)^2 d^{2/3}}\right)}{2 \sqrt[3]{-\left((a-x) (b-x)^2\right)} (-b+a d+x-d x)}",1,"$Aborted","F",-1
3148,-1,0,1655,180.0036487,"\int \frac{-a-b c+(1+c) x}{\sqrt[3]{(-a+x) (-b+x)^2} \left(-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2\right)} \, dx","IntegrateAlgebraic[(-a - b*c + (1 + c)*x)/(((-a + x)*(-b + x)^2)^(1/3)*(-a^2 + b^2*d + 2*(a - b*d)*x + (-1 + d)*x^2)),x]","\text{\$Aborted}","\frac{(b-x)^{2/3} \sqrt[3]{-a+x} \left(-\frac{\sqrt{3} a \left(-1+\sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a+x}}{-2 \sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{2/3}}-\frac{\sqrt{3} b c \left(-1+\sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a+x}}{-2 \sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{2/3}}-\frac{\sqrt{3} \left(a-b \sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a+x}}{-2 \sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{2/3}}-\frac{\sqrt{3} c \left(a-b \sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a+x}}{-2 \sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{2/3}}+\frac{\sqrt{3} a \left(1+\sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a+x}}{2 \sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{2/3}}+\frac{\sqrt{3} b c \left(1+\sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a+x}}{2 \sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{2/3}}-\frac{\sqrt{3} \left(a+b \sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a+x}}{2 \sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{2/3}}-\frac{\sqrt{3} c \left(a+b \sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a+x}}{2 \sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{2/3}}+\frac{a \left(1+\sqrt{d}\right) \log \left(-\sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{2/3}}+\frac{b c \left(1+\sqrt{d}\right) \log \left(-\sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{2/3}}+\frac{\left(-a-b \sqrt{d}\right) \log \left(-\sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{2/3}}-\frac{c \left(a+b \sqrt{d}\right) \log \left(-\sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{2/3}}-\frac{a \left(-1+\sqrt{d}\right) \log \left(\sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{2/3}}-\frac{b c \left(-1+\sqrt{d}\right) \log \left(\sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{2/3}}-\frac{c \left(a-b \sqrt{d}\right) \log \left(\sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{2/3}}+\frac{\left(-a+b \sqrt{d}\right) \log \left(\sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{2/3}}+\frac{a \left(-1+\sqrt{d}\right) \log \left(\sqrt[3]{d} (b-x)^{2/3}-\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+(-a+x)^{2/3}\right)}{4 (a-b)^2 d^{2/3}}+\frac{b c \left(-1+\sqrt{d}\right) \log \left(\sqrt[3]{d} (b-x)^{2/3}-\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+(-a+x)^{2/3}\right)}{4 (a-b)^2 d^{2/3}}+\frac{\left(a-b \sqrt{d}\right) \log \left(\sqrt[3]{d} (b-x)^{2/3}-\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+(-a+x)^{2/3}\right)}{4 (a-b)^2 d^{2/3}}+\frac{c \left(a-b \sqrt{d}\right) \log \left(\sqrt[3]{d} (b-x)^{2/3}-\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+(-a+x)^{2/3}\right)}{4 (a-b)^2 d^{2/3}}-\frac{a \left(1+\sqrt{d}\right) \log \left(\sqrt[3]{d} (b-x)^{2/3}+\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+(-a+x)^{2/3}\right)}{4 (a-b)^2 d^{2/3}}-\frac{b c \left(1+\sqrt{d}\right) \log \left(\sqrt[3]{d} (b-x)^{2/3}+\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+(-a+x)^{2/3}\right)}{4 (a-b)^2 d^{2/3}}+\frac{\left(a+b \sqrt{d}\right) \log \left(\sqrt[3]{d} (b-x)^{2/3}+\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+(-a+x)^{2/3}\right)}{4 (a-b)^2 d^{2/3}}+\frac{c \left(a+b \sqrt{d}\right) \log \left(\sqrt[3]{d} (b-x)^{2/3}+\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+(-a+x)^{2/3}\right)}{4 (a-b)^2 d^{2/3}}\right)}{\sqrt[3]{(b-x)^2 (-a+x)}}",1,"$Aborted","F",-1
3149,-1,0,1707,180.0023141,"\int \frac{-b-a c+(1+c) x}{\sqrt[3]{(-a+x) (-b+x)^2} \left(-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2\right)} \, dx","IntegrateAlgebraic[(-b - a*c + (1 + c)*x)/(((-a + x)*(-b + x)^2)^(1/3)*(-b^2 + a^2*d + 2*(b - a*d)*x + (-1 + d)*x^2)),x]","\text{\$Aborted}","\frac{(b-x)^{2/3} \sqrt[3]{-a+x} \left(-\frac{\sqrt{3} b \left(-1+\sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{-a+x}}{-2 \sqrt[3]{b-x}+\sqrt[6]{d} \sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{5/6}}-\frac{\sqrt{3} a c \left(-1+\sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{-a+x}}{-2 \sqrt[3]{b-x}+\sqrt[6]{d} \sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{5/6}}+\frac{\sqrt{3} \left(-b+a \sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{-a+x}}{-2 \sqrt[3]{b-x}+\sqrt[6]{d} \sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{5/6}}+\frac{\sqrt{3} c \left(-b+a \sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{-a+x}}{-2 \sqrt[3]{b-x}+\sqrt[6]{d} \sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{5/6}}-\frac{\sqrt{3} b \left(1+\sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{-a+x}}{2 \sqrt[3]{b-x}+\sqrt[6]{d} \sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{5/6}}-\frac{\sqrt{3} a c \left(1+\sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{-a+x}}{2 \sqrt[3]{b-x}+\sqrt[6]{d} \sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{5/6}}+\frac{\sqrt{3} \left(b+a \sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{-a+x}}{2 \sqrt[3]{b-x}+\sqrt[6]{d} \sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{5/6}}+\frac{\sqrt{3} c \left(b+a \sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{-a+x}}{2 \sqrt[3]{b-x}+\sqrt[6]{d} \sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{5/6}}-\frac{b \left(1+\sqrt{d}\right) \log \left(\sqrt[3]{b-x}-\sqrt[6]{d} \sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{5/6}}-\frac{a c \left(1+\sqrt{d}\right) \log \left(\sqrt[3]{b-x}-\sqrt[6]{d} \sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{5/6}}+\frac{\left(b+a \sqrt{d}\right) \log \left(\sqrt[3]{b-x}-\sqrt[6]{d} \sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{5/6}}+\frac{c \left(b+a \sqrt{d}\right) \log \left(\sqrt[3]{b-x}-\sqrt[6]{d} \sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{5/6}}-\frac{b \left(-1+\sqrt{d}\right) \log \left(\sqrt[3]{b-x}+\sqrt[6]{d} \sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{5/6}}-\frac{a c \left(-1+\sqrt{d}\right) \log \left(\sqrt[3]{b-x}+\sqrt[6]{d} \sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{5/6}}+\frac{\left(-b+a \sqrt{d}\right) \log \left(\sqrt[3]{b-x}+\sqrt[6]{d} \sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{5/6}}+\frac{c \left(-b+a \sqrt{d}\right) \log \left(\sqrt[3]{b-x}+\sqrt[6]{d} \sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{5/6}}+\frac{b \left(-1+\sqrt{d}\right) \log \left((b-x)^{2/3}-\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+\sqrt[3]{d} (-a+x)^{2/3}\right)}{4 (a-b)^2 d^{5/6}}+\frac{a c \left(-1+\sqrt{d}\right) \log \left((b-x)^{2/3}-\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+\sqrt[3]{d} (-a+x)^{2/3}\right)}{4 (a-b)^2 d^{5/6}}+\frac{\left(b-a \sqrt{d}\right) \log \left((b-x)^{2/3}-\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+\sqrt[3]{d} (-a+x)^{2/3}\right)}{4 (a-b)^2 d^{5/6}}-\frac{c \left(-b+a \sqrt{d}\right) \log \left((b-x)^{2/3}-\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+\sqrt[3]{d} (-a+x)^{2/3}\right)}{4 (a-b)^2 d^{5/6}}+\frac{b \left(1+\sqrt{d}\right) \log \left((b-x)^{2/3}+\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+\sqrt[3]{d} (-a+x)^{2/3}\right)}{4 (a-b)^2 d^{5/6}}+\frac{a c \left(1+\sqrt{d}\right) \log \left((b-x)^{2/3}+\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+\sqrt[3]{d} (-a+x)^{2/3}\right)}{4 (a-b)^2 d^{5/6}}+\frac{\left(-b-a \sqrt{d}\right) \log \left((b-x)^{2/3}+\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+\sqrt[3]{d} (-a+x)^{2/3}\right)}{4 (a-b)^2 d^{5/6}}-\frac{c \left(b+a \sqrt{d}\right) \log \left((b-x)^{2/3}+\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+\sqrt[3]{d} (-a+x)^{2/3}\right)}{4 (a-b)^2 d^{5/6}}\right)}{\sqrt[3]{(b-x)^2 (-a+x)}}",1,"$Aborted","F",-1
3150,1,1344,1716,6.3316646,"\int \frac{c_8+x c_9}{\sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5} (c_6+x c_7)} \, dx","IntegrateAlgebraic[(C[8] + x*C[9])/(Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*(C[6] + x*C[7])),x]","\frac{\sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5} \left(-c_1 c_2 c_4 c_9+c_0 c_3 c_4 c_9+c_1 c_2 \sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5 c_9-c_0 c_3 \sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5 c_9\right)}{\left(c_1-\frac{(c_0+x c_1) c_3}{c_2+x c_3}\right) \left(-c_3 c_4{}^2+c_1 c_5{}^2\right) c_7}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5} \sqrt{c_3 c_6-c_2 c_7}}{\sqrt{-c_3 c_4 c_6+c_2 c_4 c_7-c_5 \sqrt{c_1 c_6-c_0 c_7} \sqrt{c_3 c_6-c_2 c_7}}}\right) \sqrt{c_3 c_6-c_2 c_7} (-c_7 c_8+c_6 c_9)}{c_7{}^2 \sqrt{-c_3 c_4 c_6+c_2 c_4 c_7-c_5 \sqrt{c_1 c_6-c_0 c_7} \sqrt{c_3 c_6-c_2 c_7}}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5} \sqrt{c_3 c_6-c_2 c_7}}{\sqrt{-c_3 c_4 c_6+c_2 c_4 c_7+c_5 \sqrt{c_1 c_6-c_0 c_7} \sqrt{c_3 c_6-c_2 c_7}}}\right) \sqrt{c_3 c_6-c_2 c_7} (-c_7 c_8+c_6 c_9)}{c_7{}^2 \sqrt{-c_3 c_4 c_6+c_2 c_4 c_7+c_5 \sqrt{c_1 c_6-c_0 c_7} \sqrt{c_3 c_6-c_2 c_7}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{-c_3 c_4+\sqrt{c_1} \sqrt{c_3} c_5} \sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt{c_3} c_4-\sqrt{c_1} c_5}\right) \left(-4 \sqrt{c_1} c_3{}^{3/2} c_4 \sqrt{-\sqrt{c_3} \left(\sqrt{c_3} c_4-\sqrt{c_1} c_5\right)} c_7 c_8+4 c_1 c_3 c_5 \sqrt{-\sqrt{c_3} \left(\sqrt{c_3} c_4-\sqrt{c_1} c_5\right)} c_7 c_8+4 \sqrt{c_1} c_3{}^{3/2} c_4 \sqrt{-\sqrt{c_3} \left(\sqrt{c_3} c_4-\sqrt{c_1} c_5\right)} c_6 c_9-4 c_1 c_3 c_5 \sqrt{-\sqrt{c_3} \left(\sqrt{c_3} c_4-\sqrt{c_1} c_5\right)} c_6 c_9+c_1 c_2 c_5 \sqrt{-\sqrt{c_3} \left(\sqrt{c_3} c_4-\sqrt{c_1} c_5\right)} c_7 c_9-c_0 c_3 c_5 \sqrt{-\sqrt{c_3} \left(\sqrt{c_3} c_4-\sqrt{c_1} c_5\right)} c_7 c_9\right)}{2 \sqrt{c_1} c_3 \left(\sqrt{c_3} c_4-\sqrt{c_1} c_5\right){}^2 c_7{}^2}+\frac{\tan ^{-1}\left(\frac{\sqrt{-c_3 c_4-\sqrt{c_1} \sqrt{c_3} c_5} \sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt{c_3} c_4+\sqrt{c_1} c_5}\right) \left(-4 \sqrt{c_1} c_3{}^{3/2} c_4 \sqrt{-\sqrt{c_3} \left(\sqrt{c_3} c_4+\sqrt{c_1} c_5\right)} c_7 c_8-4 c_1 c_3 c_5 \sqrt{-\sqrt{c_3} \left(\sqrt{c_3} c_4+\sqrt{c_1} c_5\right)} c_7 c_8+4 \sqrt{c_1} c_3{}^{3/2} c_4 \sqrt{-\sqrt{c_3} \left(\sqrt{c_3} c_4+\sqrt{c_1} c_5\right)} c_6 c_9+4 c_1 c_3 c_5 \sqrt{-\sqrt{c_3} \left(\sqrt{c_3} c_4+\sqrt{c_1} c_5\right)} c_6 c_9-c_1 c_2 c_5 \sqrt{-\sqrt{c_3} \left(\sqrt{c_3} c_4+\sqrt{c_1} c_5\right)} c_7 c_9+c_0 c_3 c_5 \sqrt{-\sqrt{c_3} \left(\sqrt{c_3} c_4+\sqrt{c_1} c_5\right)} c_7 c_9\right)}{2 \sqrt{c_1} c_3 \left(\sqrt{c_3} c_4+\sqrt{c_1} c_5\right){}^2 c_7{}^2}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{c_3} \sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt{\sqrt{c_3} c_4-\sqrt{c_1} c_5}}\right) \sqrt[4]{c_3} c_8}{\sqrt{\sqrt{c_3} c_4-\sqrt{c_1} c_5} c_7}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[4]{c_3} \sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt{\sqrt{c_3} c_4+\sqrt{c_1} c_5}}\right) \sqrt[4]{c_3} c_8}{\sqrt{\sqrt{c_3} c_4+\sqrt{c_1} c_5} c_7}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5} \sqrt{c_3 c_6-c_2 c_7}}{\sqrt{-c_3 c_4 c_6+c_2 c_4 c_7-c_5 \sqrt{c_1 c_6-c_0 c_7} \sqrt{c_3 c_6-c_2 c_7}}}\right) \sqrt{c_3 c_6-c_2 c_7} c_8}{c_7 \sqrt{-c_3 c_4 c_6+c_2 c_4 c_7-c_5 \sqrt{c_1 c_6-c_0 c_7} \sqrt{c_3 c_6-c_2 c_7}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5} \sqrt{c_3 c_6-c_2 c_7}}{\sqrt{-c_3 c_4 c_6+c_2 c_4 c_7+c_5 \sqrt{c_1 c_6-c_0 c_7} \sqrt{c_3 c_6-c_2 c_7}}}\right) \sqrt{c_3 c_6-c_2 c_7} c_8}{c_7 \sqrt{-c_3 c_4 c_6+c_2 c_4 c_7+c_5 \sqrt{c_1 c_6-c_0 c_7} \sqrt{c_3 c_6-c_2 c_7}}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5} \sqrt{-c_3 c_6+c_2 c_7}}{\sqrt{c_3 c_4 c_6-c_2 c_4 c_7-i c_5 \sqrt{c_1 c_6-c_0 c_7} \sqrt{-c_3 c_6+c_2 c_7}}}\right) c_6 \sqrt{-c_3 c_6+c_2 c_7} c_9}{c_7{}^2 \sqrt{c_3 c_4 c_6-c_2 c_4 c_7-i c_5 \sqrt{c_1 c_6-c_0 c_7} \sqrt{-c_3 c_6+c_2 c_7}}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5} \sqrt{-c_3 c_6+c_2 c_7}}{\sqrt{c_3 c_4 c_6-c_2 c_4 c_7+i c_5 \sqrt{c_1 c_6-c_0 c_7} \sqrt{-c_3 c_6+c_2 c_7}}}\right) c_6 \sqrt{-c_3 c_6+c_2 c_7} c_9}{c_7{}^2 \sqrt{c_3 c_4 c_6-c_2 c_4 c_7+i c_5 \sqrt{c_1 c_6-c_0 c_7} \sqrt{-c_3 c_6+c_2 c_7}}}+\frac{\sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5} \left(-c_1 c_2 c_4 c_9+c_0 c_3 c_4 c_9+c_1 c_2 \sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5 c_9-c_0 c_3 \sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5 c_9\right)}{\left(c_1-\frac{(c_0+x c_1) c_3}{c_2+x c_3}\right) \left(-c_3 c_4{}^2+c_1 c_5{}^2\right) c_7}+\frac{\tan ^{-1}\left(\frac{\sqrt{-c_3 c_4+\sqrt{c_1} \sqrt{c_3} c_5} \sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt{c_3} c_4-\sqrt{c_1} c_5}\right) \left(4 \sqrt{c_1} c_3{}^{3/2} c_4 \sqrt{-\sqrt{c_3} \left(\sqrt{c_3} c_4-\sqrt{c_1} c_5\right)} c_6 c_9-4 c_1 c_3 c_5 \sqrt{-\sqrt{c_3} \left(\sqrt{c_3} c_4-\sqrt{c_1} c_5\right)} c_6 c_9+c_1 c_2 c_5 \sqrt{-\sqrt{c_3} \left(\sqrt{c_3} c_4-\sqrt{c_1} c_5\right)} c_7 c_9-c_0 c_3 c_5 \sqrt{-\sqrt{c_3} \left(\sqrt{c_3} c_4-\sqrt{c_1} c_5\right)} c_7 c_9\right)}{2 \sqrt{c_1} c_3 \left(\sqrt{c_3} c_4-\sqrt{c_1} c_5\right){}^2 c_7{}^2}+\frac{\tan ^{-1}\left(\frac{\sqrt{-c_3 c_4-\sqrt{c_1} \sqrt{c_3} c_5} \sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt{c_3} c_4+\sqrt{c_1} c_5}\right) \left(4 \sqrt{c_1} c_3{}^{3/2} c_4 \sqrt{-\sqrt{c_3} \left(\sqrt{c_3} c_4+\sqrt{c_1} c_5\right)} c_6 c_9+4 c_1 c_3 c_5 \sqrt{-\sqrt{c_3} \left(\sqrt{c_3} c_4+\sqrt{c_1} c_5\right)} c_6 c_9-c_1 c_2 c_5 \sqrt{-\sqrt{c_3} \left(\sqrt{c_3} c_4+\sqrt{c_1} c_5\right)} c_7 c_9+c_0 c_3 c_5 \sqrt{-\sqrt{c_3} \left(\sqrt{c_3} c_4+\sqrt{c_1} c_5\right)} c_7 c_9\right)}{2 \sqrt{c_1} c_3 \left(\sqrt{c_3} c_4+\sqrt{c_1} c_5\right){}^2 c_7{}^2}",1,"(Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*(-(C[1]*C[2]*C[4]*C[9]) + C[0]*C[3]*C[4]*C[9] + C[1]*C[2]*Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]*C[9] - C[0]*C[3]*Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]*C[9]))/((C[1] - ((C[0] + x*C[1])*C[3])/(C[2] + x*C[3]))*(-(C[3]*C[4]^2) + C[1]*C[5]^2)*C[7]) - (2*ArcTan[(Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*Sqrt[C[3]*C[6] - C[2]*C[7]])/Sqrt[-(C[3]*C[4]*C[6]) + C[2]*C[4]*C[7] - C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]]*Sqrt[C[3]*C[6] - C[2]*C[7]]]]*Sqrt[C[3]*C[6] - C[2]*C[7]]*(-(C[7]*C[8]) + C[6]*C[9]))/(C[7]^2*Sqrt[-(C[3]*C[4]*C[6]) + C[2]*C[4]*C[7] - C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]]*Sqrt[C[3]*C[6] - C[2]*C[7]]]) - (2*ArcTan[(Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]]*Sqrt[C[3]*C[6] - C[2]*C[7]])/Sqrt[-(C[3]*C[4]*C[6]) + C[2]*C[4]*C[7] + C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]]*Sqrt[C[3]*C[6] - C[2]*C[7]]]]*Sqrt[C[3]*C[6] - C[2]*C[7]]*(-(C[7]*C[8]) + C[6]*C[9]))/(C[7]^2*Sqrt[-(C[3]*C[4]*C[6]) + C[2]*C[4]*C[7] + C[5]*Sqrt[C[1]*C[6] - C[0]*C[7]]*Sqrt[C[3]*C[6] - C[2]*C[7]]]) + (ArcTan[(Sqrt[-(C[3]*C[4]) + Sqrt[C[1]]*Sqrt[C[3]]*C[5]]*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]])/(Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5])]*(-4*Sqrt[C[1]]*C[3]^(3/2)*C[4]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5]))]*C[7]*C[8] + 4*C[1]*C[3]*C[5]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5]))]*C[7]*C[8] + 4*Sqrt[C[1]]*C[3]^(3/2)*C[4]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5]))]*C[6]*C[9] - 4*C[1]*C[3]*C[5]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5]))]*C[6]*C[9] + C[1]*C[2]*C[5]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5]))]*C[7]*C[9] - C[0]*C[3]*C[5]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5]))]*C[7]*C[9]))/(2*Sqrt[C[1]]*C[3]*(Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5])^2*C[7]^2) + (ArcTan[(Sqrt[-(C[3]*C[4]) - Sqrt[C[1]]*Sqrt[C[3]]*C[5]]*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]])/(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5])]*(-4*Sqrt[C[1]]*C[3]^(3/2)*C[4]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5]))]*C[7]*C[8] - 4*C[1]*C[3]*C[5]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5]))]*C[7]*C[8] + 4*Sqrt[C[1]]*C[3]^(3/2)*C[4]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5]))]*C[6]*C[9] + 4*C[1]*C[3]*C[5]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5]))]*C[6]*C[9] - C[1]*C[2]*C[5]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5]))]*C[7]*C[9] + C[0]*C[3]*C[5]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5]))]*C[7]*C[9]))/(2*Sqrt[C[1]]*C[3]*(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5])^2*C[7]^2)","A",1
3151,1,571,1835,18.6001437,"\int \frac{(-b+x) (-a-b c+(1+c) x)}{\left((-a+x) (-b+x)^2\right)^{2/3} \left(-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2\right)} \, dx","IntegrateAlgebraic[((-b + x)*(-a - b*c + (1 + c)*x))/(((-a + x)*(-b + x)^2)^(2/3)*(-a^2 + b^2*d + 2*(a - b*d)*x + (-1 + d)*x^2)),x]","\frac{(b-x)^{4/3} (-a+x)^{2/3} \left(\sqrt[6]{d} \sqrt[3]{b-x}-\sqrt[3]{-a+x}\right) \left(\sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}\right) \left(b d^{2/3} \sqrt[3]{b-x}-d^{2/3} \sqrt[3]{b-x} x+\sqrt[3]{d} (b-x)^{2/3} (-a+x)^{2/3}+(-a+x)^{4/3}\right) \left(\frac{\sqrt{3} \left(c+\sqrt{d}\right) \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{-a+x}}{\sqrt{3} \sqrt[6]{d} \sqrt[3]{b-x}}\right)}{2 (a-b) d^{5/6}}-\frac{\sqrt{3} \left(c-\sqrt{d}\right) \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{-a+x}}{\sqrt{3} \sqrt[6]{d} \sqrt[3]{b-x}}\right)}{2 (a-b) d^{5/6}}+\frac{\left(c-\sqrt{d}\right) \log \left(\sqrt[6]{d}-\frac{\sqrt[3]{-a+x}}{\sqrt[3]{b-x}}\right)}{2 (a-b) d^{5/6}}+\frac{\left(-c-\sqrt{d}\right) \log \left(\sqrt[6]{d}+\frac{\sqrt[3]{-a+x}}{\sqrt[3]{b-x}}\right)}{2 (a-b) d^{5/6}}+\frac{\left(c+\sqrt{d}\right) \log \left(\sqrt[3]{d}-\frac{\sqrt[6]{d} \sqrt[3]{-a+x}}{\sqrt[3]{b-x}}+\frac{(-a+x)^{2/3}}{(b-x)^{2/3}}\right)}{4 (a-b) d^{5/6}}+\frac{\left(-c+\sqrt{d}\right) \log \left(\sqrt[3]{d}+\frac{\sqrt[6]{d} \sqrt[3]{-a+x}}{\sqrt[3]{b-x}}+\frac{(-a+x)^{2/3}}{(b-x)^{2/3}}\right)}{4 (a-b) d^{5/6}}\right)}{\left((b-x)^2 (-a+x)\right)^{2/3} \left(-a^2+b^2 d+2 a x-2 b d x+(-1+d) x^2\right)}","\frac{(b-x)^{4/3} (-a+x)^{2/3} \left(\sqrt[6]{d} \sqrt[3]{b-x}-\sqrt[3]{-a+x}\right) \left(\sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}\right) \left(\sqrt[3]{d} (b-x)^{2/3}-\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+(-a+x)^{2/3}\right) \left(\sqrt[3]{d} (b-x)^{2/3}+\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+(-a+x)^{2/3}\right) \left(\frac{\sqrt{3} a \left(-1+\sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a+x}}{-2 \sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{5/6}}+\frac{\sqrt{3} b c \left(-1+\sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a+x}}{-2 \sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{5/6}}+\frac{\sqrt{3} \left(a-b \sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a+x}}{-2 \sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{5/6}}+\frac{\sqrt{3} c \left(a-b \sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a+x}}{-2 \sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{5/6}}+\frac{\sqrt{3} a \left(1+\sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a+x}}{2 \sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{5/6}}+\frac{\sqrt{3} b c \left(1+\sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a+x}}{2 \sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{5/6}}-\frac{\sqrt{3} \left(a+b \sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a+x}}{2 \sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{5/6}}-\frac{\sqrt{3} c \left(a+b \sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{-a+x}}{2 \sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{5/6}}-\frac{a \left(1+\sqrt{d}\right) \log \left(-\sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{5/6}}-\frac{b c \left(1+\sqrt{d}\right) \log \left(-\sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{5/6}}+\frac{\left(a+b \sqrt{d}\right) \log \left(-\sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{5/6}}+\frac{c \left(a+b \sqrt{d}\right) \log \left(-\sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{5/6}}-\frac{a \left(-1+\sqrt{d}\right) \log \left(\sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{5/6}}-\frac{b c \left(-1+\sqrt{d}\right) \log \left(\sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{5/6}}-\frac{c \left(a-b \sqrt{d}\right) \log \left(\sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{5/6}}+\frac{\left(-a+b \sqrt{d}\right) \log \left(\sqrt[6]{d} \sqrt[3]{b-x}+\sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{5/6}}+\frac{a \left(-1+\sqrt{d}\right) \log \left(\sqrt[3]{d} (b-x)^{2/3}-\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+(-a+x)^{2/3}\right)}{4 (a-b)^2 d^{5/6}}+\frac{b c \left(-1+\sqrt{d}\right) \log \left(\sqrt[3]{d} (b-x)^{2/3}-\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+(-a+x)^{2/3}\right)}{4 (a-b)^2 d^{5/6}}+\frac{\left(a-b \sqrt{d}\right) \log \left(\sqrt[3]{d} (b-x)^{2/3}-\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+(-a+x)^{2/3}\right)}{4 (a-b)^2 d^{5/6}}+\frac{c \left(a-b \sqrt{d}\right) \log \left(\sqrt[3]{d} (b-x)^{2/3}-\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+(-a+x)^{2/3}\right)}{4 (a-b)^2 d^{5/6}}+\frac{a \left(1+\sqrt{d}\right) \log \left(\sqrt[3]{d} (b-x)^{2/3}+\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+(-a+x)^{2/3}\right)}{4 (a-b)^2 d^{5/6}}+\frac{b c \left(1+\sqrt{d}\right) \log \left(\sqrt[3]{d} (b-x)^{2/3}+\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+(-a+x)^{2/3}\right)}{4 (a-b)^2 d^{5/6}}+\frac{\left(-a-b \sqrt{d}\right) \log \left(\sqrt[3]{d} (b-x)^{2/3}+\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+(-a+x)^{2/3}\right)}{4 (a-b)^2 d^{5/6}}-\frac{c \left(a+b \sqrt{d}\right) \log \left(\sqrt[3]{d} (b-x)^{2/3}+\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+(-a+x)^{2/3}\right)}{4 (a-b)^2 d^{5/6}}\right)}{\left((b-x)^2 (-a+x)\right)^{2/3} \left(-a^2+b^2 d+2 a x-2 b d x+(-1+d) x^2\right)}",1,"((b - x)^(4/3)*(-a + x)^(2/3)*(d^(1/6)*(b - x)^(1/3) - (-a + x)^(1/3))*(d^(1/6)*(b - x)^(1/3) + (-a + x)^(1/3))*(b*d^(2/3)*(b - x)^(1/3) - d^(2/3)*(b - x)^(1/3)*x + d^(1/3)*(b - x)^(2/3)*(-a + x)^(2/3) + (-a + x)^(4/3))*((Sqrt[3]*(c + Sqrt[d])*ArcTan[1/Sqrt[3] - (2*(-a + x)^(1/3))/(Sqrt[3]*d^(1/6)*(b - x)^(1/3))])/(2*(a - b)*d^(5/6)) - (Sqrt[3]*(c - Sqrt[d])*ArcTan[1/Sqrt[3] + (2*(-a + x)^(1/3))/(Sqrt[3]*d^(1/6)*(b - x)^(1/3))])/(2*(a - b)*d^(5/6)) + ((c - Sqrt[d])*Log[d^(1/6) - (-a + x)^(1/3)/(b - x)^(1/3)])/(2*(a - b)*d^(5/6)) + ((-c - Sqrt[d])*Log[d^(1/6) + (-a + x)^(1/3)/(b - x)^(1/3)])/(2*(a - b)*d^(5/6)) + ((c + Sqrt[d])*Log[d^(1/3) - (d^(1/6)*(-a + x)^(1/3))/(b - x)^(1/3) + (-a + x)^(2/3)/(b - x)^(2/3)])/(4*(a - b)*d^(5/6)) + ((-c + Sqrt[d])*Log[d^(1/3) + (d^(1/6)*(-a + x)^(1/3))/(b - x)^(1/3) + (-a + x)^(2/3)/(b - x)^(2/3)])/(4*(a - b)*d^(5/6))))/(((b - x)^2*(-a + x))^(2/3)*(-a^2 + b^2*d + 2*a*x - 2*b*d*x + (-1 + d)*x^2))","A",1
3152,1,587,1886,22.6501674,"\int \frac{(-b+x) (-b-a c+(1+c) x)}{\left((-a+x) (-b+x)^2\right)^{2/3} \left(-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2\right)} \, dx","IntegrateAlgebraic[((-b + x)*(-b - a*c + (1 + c)*x))/(((-a + x)*(-b + x)^2)^(2/3)*(-b^2 + a^2*d + 2*(b - a*d)*x + (-1 + d)*x^2)),x]","\frac{(b-x)^{4/3} (-a+x)^{2/3} \left(\sqrt[3]{b-x}-\sqrt[6]{d} \sqrt[3]{-a+x}\right) \left(\sqrt[3]{b-x}+\sqrt[6]{d} \sqrt[3]{-a+x}\right) \left(b \sqrt[3]{b-x}-\sqrt[3]{b-x} x-a d^{2/3} \sqrt[3]{-a+x}+d^{2/3} x \sqrt[3]{-a+x}+\sqrt[3]{d} (b-x)^{2/3} (-a+x)^{2/3}\right) \left(-\frac{\sqrt{3} \left(c+\sqrt{d}\right) \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[6]{d} \sqrt[3]{-a+x}}{\sqrt{3} \sqrt[3]{b-x}}\right)}{2 (a-b) d^{2/3}}-\frac{\sqrt{3} \left(c-\sqrt{d}\right) \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[6]{d} \sqrt[3]{-a+x}}{\sqrt{3} \sqrt[3]{b-x}}\right)}{2 (a-b) d^{2/3}}+\frac{\left(c-\sqrt{d}\right) \log \left(-1+\frac{\sqrt[6]{d} \sqrt[3]{-a+x}}{\sqrt[3]{b-x}}\right)}{2 (a-b) d^{2/3}}+\frac{\left(c+\sqrt{d}\right) \log \left(1+\frac{\sqrt[6]{d} \sqrt[3]{-a+x}}{\sqrt[3]{b-x}}\right)}{2 (a-b) d^{2/3}}+\frac{\left(-c-\sqrt{d}\right) \log \left(1-\frac{\sqrt[6]{d} \sqrt[3]{-a+x}}{\sqrt[3]{b-x}}+\frac{\sqrt[3]{d} (-a+x)^{2/3}}{(b-x)^{2/3}}\right)}{4 (a-b) d^{2/3}}+\frac{\left(-c+\sqrt{d}\right) \log \left(1+\frac{\sqrt[6]{d} \sqrt[3]{-a+x}}{\sqrt[3]{b-x}}+\frac{\sqrt[3]{d} (-a+x)^{2/3}}{(b-x)^{2/3}}\right)}{4 (a-b) d^{2/3}}\right)}{\left((b-x)^2 (-a+x)\right)^{2/3} \left(b^2-a^2 d-2 b x+2 a d x-(-1+d) x^2\right)}","\frac{(b-x)^{4/3} (-a+x)^{2/3} \left(\sqrt[3]{b-x}-\sqrt[6]{d} \sqrt[3]{-a+x}\right) \left(\sqrt[3]{b-x}+\sqrt[6]{d} \sqrt[3]{-a+x}\right) \left((b-x)^{2/3}-\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+\sqrt[3]{d} (-a+x)^{2/3}\right) \left((b-x)^{2/3}+\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+\sqrt[3]{d} (-a+x)^{2/3}\right) \left(\frac{\sqrt{3} b \left(-1+\sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{-a+x}}{-2 \sqrt[3]{b-x}+\sqrt[6]{d} \sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{2/3}}+\frac{\sqrt{3} a c \left(-1+\sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{-a+x}}{-2 \sqrt[3]{b-x}+\sqrt[6]{d} \sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{2/3}}-\frac{\sqrt{3} \left(-b+a \sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{-a+x}}{-2 \sqrt[3]{b-x}+\sqrt[6]{d} \sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{2/3}}-\frac{\sqrt{3} c \left(-b+a \sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{-a+x}}{-2 \sqrt[3]{b-x}+\sqrt[6]{d} \sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{2/3}}-\frac{\sqrt{3} b \left(1+\sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{-a+x}}{2 \sqrt[3]{b-x}+\sqrt[6]{d} \sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{2/3}}-\frac{\sqrt{3} a c \left(1+\sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{-a+x}}{2 \sqrt[3]{b-x}+\sqrt[6]{d} \sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{2/3}}+\frac{\sqrt{3} \left(b+a \sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{-a+x}}{2 \sqrt[3]{b-x}+\sqrt[6]{d} \sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{2/3}}+\frac{\sqrt{3} c \left(b+a \sqrt{d}\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{d} \sqrt[3]{-a+x}}{2 \sqrt[3]{b-x}+\sqrt[6]{d} \sqrt[3]{-a+x}}\right)}{2 (a-b)^2 d^{2/3}}+\frac{b \left(1+\sqrt{d}\right) \log \left(\sqrt[3]{b-x}-\sqrt[6]{d} \sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{2/3}}+\frac{a c \left(1+\sqrt{d}\right) \log \left(\sqrt[3]{b-x}-\sqrt[6]{d} \sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{2/3}}+\frac{\left(-b-a \sqrt{d}\right) \log \left(\sqrt[3]{b-x}-\sqrt[6]{d} \sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{2/3}}-\frac{c \left(b+a \sqrt{d}\right) \log \left(\sqrt[3]{b-x}-\sqrt[6]{d} \sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{2/3}}-\frac{b \left(-1+\sqrt{d}\right) \log \left(\sqrt[3]{b-x}+\sqrt[6]{d} \sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{2/3}}-\frac{a c \left(-1+\sqrt{d}\right) \log \left(\sqrt[3]{b-x}+\sqrt[6]{d} \sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{2/3}}+\frac{\left(-b+a \sqrt{d}\right) \log \left(\sqrt[3]{b-x}+\sqrt[6]{d} \sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{2/3}}+\frac{c \left(-b+a \sqrt{d}\right) \log \left(\sqrt[3]{b-x}+\sqrt[6]{d} \sqrt[3]{-a+x}\right)}{2 (a-b)^2 d^{2/3}}+\frac{b \left(-1+\sqrt{d}\right) \log \left((b-x)^{2/3}-\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+\sqrt[3]{d} (-a+x)^{2/3}\right)}{4 (a-b)^2 d^{2/3}}+\frac{a c \left(-1+\sqrt{d}\right) \log \left((b-x)^{2/3}-\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+\sqrt[3]{d} (-a+x)^{2/3}\right)}{4 (a-b)^2 d^{2/3}}+\frac{\left(b-a \sqrt{d}\right) \log \left((b-x)^{2/3}-\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+\sqrt[3]{d} (-a+x)^{2/3}\right)}{4 (a-b)^2 d^{2/3}}-\frac{c \left(-b+a \sqrt{d}\right) \log \left((b-x)^{2/3}-\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+\sqrt[3]{d} (-a+x)^{2/3}\right)}{4 (a-b)^2 d^{2/3}}-\frac{b \left(1+\sqrt{d}\right) \log \left((b-x)^{2/3}+\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+\sqrt[3]{d} (-a+x)^{2/3}\right)}{4 (a-b)^2 d^{2/3}}-\frac{a c \left(1+\sqrt{d}\right) \log \left((b-x)^{2/3}+\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+\sqrt[3]{d} (-a+x)^{2/3}\right)}{4 (a-b)^2 d^{2/3}}+\frac{\left(b+a \sqrt{d}\right) \log \left((b-x)^{2/3}+\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+\sqrt[3]{d} (-a+x)^{2/3}\right)}{4 (a-b)^2 d^{2/3}}+\frac{c \left(b+a \sqrt{d}\right) \log \left((b-x)^{2/3}+\sqrt[6]{d} \sqrt[3]{b-x} \sqrt[3]{-a+x}+\sqrt[3]{d} (-a+x)^{2/3}\right)}{4 (a-b)^2 d^{2/3}}\right)}{\left((b-x)^2 (-a+x)\right)^{2/3} \left(b^2-a^2 d-2 b x+2 a d x-(-1+d) x^2\right)}",1,"((b - x)^(4/3)*(-a + x)^(2/3)*((b - x)^(1/3) - d^(1/6)*(-a + x)^(1/3))*((b - x)^(1/3) + d^(1/6)*(-a + x)^(1/3))*(b*(b - x)^(1/3) - (b - x)^(1/3)*x - a*d^(2/3)*(-a + x)^(1/3) + d^(2/3)*x*(-a + x)^(1/3) + d^(1/3)*(b - x)^(2/3)*(-a + x)^(2/3))*(-1/2*(Sqrt[3]*(c + Sqrt[d])*ArcTan[1/Sqrt[3] - (2*d^(1/6)*(-a + x)^(1/3))/(Sqrt[3]*(b - x)^(1/3))])/((a - b)*d^(2/3)) - (Sqrt[3]*(c - Sqrt[d])*ArcTan[1/Sqrt[3] + (2*d^(1/6)*(-a + x)^(1/3))/(Sqrt[3]*(b - x)^(1/3))])/(2*(a - b)*d^(2/3)) + ((c - Sqrt[d])*Log[-1 + (d^(1/6)*(-a + x)^(1/3))/(b - x)^(1/3)])/(2*(a - b)*d^(2/3)) + ((c + Sqrt[d])*Log[1 + (d^(1/6)*(-a + x)^(1/3))/(b - x)^(1/3)])/(2*(a - b)*d^(2/3)) + ((-c - Sqrt[d])*Log[1 - (d^(1/6)*(-a + x)^(1/3))/(b - x)^(1/3) + (d^(1/3)*(-a + x)^(2/3))/(b - x)^(2/3)])/(4*(a - b)*d^(2/3)) + ((-c + Sqrt[d])*Log[1 + (d^(1/6)*(-a + x)^(1/3))/(b - x)^(1/3) + (d^(1/3)*(-a + x)^(2/3))/(b - x)^(2/3)])/(4*(a - b)*d^(2/3))))/(((b - x)^2*(-a + x))^(2/3)*(b^2 - a^2*d - 2*b*x + 2*a*d*x - (-1 + d)*x^2))","A",1
3153,1,4104,1916,3.4584514,"\int \frac{x^2-c x^2 \left(\frac{b+a x}{d+c x}\right)^{3/2}}{a-b \sqrt{\frac{b+a x}{d+c x}}} \, dx","IntegrateAlgebraic[(x^2 - c*x^2*((b + a*x)/(d + c*x))^(3/2))/(a - b*Sqrt[(b + a*x)/(d + c*x)]),x]","\text{Result too large to show}","\frac{6 b^6 c^3 x-6 a^3 b^3 c^4 x+6 a^2 b^6 c d x-18 a^3 b^4 c^2 d x-6 a b^5 c^2 d x+24 a^4 b^2 c^3 d x-6 a^2 b^3 c^3 d x-6 a^3 b^5 d^2 x+18 a^4 b^3 c d^2 x-18 a^5 b c^2 d^2 x+6 a^3 b^2 c^2 d^2 x-3 a^2 b^6 c^2 x^2+9 a^3 b^4 c^3 x^2-3 a b^5 c^3 x^2-6 a^4 b^2 c^4 x^2+3 a^2 b^3 c^4 x^2+3 a^3 b^5 c d x^2-9 a^4 b^3 c^2 d x^2+3 a^2 b^4 c^2 d x^2+6 a^5 b c^3 d x^2-3 a^3 b^2 c^3 d x^2-2 a^3 b^5 c^2 x^3+4 a^4 b^3 c^3 x^3+2 a^2 b^4 c^3 x^3-2 a^5 b c^4 x^3-4 a^3 b^2 c^4 x^3+2 a^4 c^5 x^3}{6 a^2 c^2 \left(-b^2+a c\right)^3}+\frac{\sqrt{\frac{b+a x}{d+c x}} \left(3 a^2 b^6 c^2 d+15 b^7 c^2 d-24 a^3 b^4 c^3 d+12 a b^5 c^3 d-3 a^4 b^2 c^4 d-3 a^2 b^3 c^4 d+4 a^3 b^5 c d^2-4 a b^6 c d^2-8 a^4 b^3 c^2 d^2-40 a^2 b^4 c^2 d^2+52 a^5 b c^3 d^2-4 a^3 b^2 c^3 d^2-15 a^4 b^4 d^3-3 a^2 b^5 d^3+48 a^5 b^2 c d^3+12 a^3 b^3 c d^3-57 a^6 c^2 d^3+15 a^4 b c^2 d^3+3 a^2 b^6 c^3 x+15 b^7 c^3 x-24 a^3 b^4 c^4 x+12 a b^5 c^4 x-3 a^4 b^2 c^5 x-3 a^2 b^3 c^5 x+2 a^3 b^5 c^2 d x-14 a b^6 c^2 d x+8 a^4 b^3 c^3 d x-32 a^2 b^4 c^3 d x+38 a^5 b c^4 d x-2 a^3 b^2 c^4 d x-5 a^4 b^4 c d^2 x-a^2 b^5 c d^2 x+16 a^5 b^2 c^2 d^2 x+20 a^3 b^3 c^2 d^2 x-35 a^6 c^3 d^2 x+5 a^4 b c^3 d^2 x-2 a^3 b^5 c^3 x^2-10 a b^6 c^3 x^2+16 a^4 b^3 c^4 x^2+8 a^2 b^4 c^4 x^2-14 a^5 b c^5 x^2+2 a^3 b^2 c^5 x^2+2 a^4 b^4 c^2 d x^2+10 a^2 b^5 c^2 d x^2-16 a^5 b^2 c^3 d x^2-8 a^3 b^3 c^3 d x^2+14 a^6 c^4 d x^2-2 a^4 b c^4 d x^2-8 a^4 b^4 c^3 x^3+8 a^2 b^5 c^3 x^3+16 a^5 b^2 c^4 x^3-16 a^3 b^3 c^4 x^3-8 a^6 c^5 x^3+8 a^4 b c^5 x^3\right)}{24 a^3 c^2 \left(-b^2+a c\right)^3}+\frac{\left(a^2 b^9 c^3+5 b^{10} c^3-9 a^3 b^7 c^4+15 a b^8 c^4-9 a^4 b^5 c^5-5 a^2 b^6 c^5+a^5 b^3 c^6+a^3 b^4 c^6+a^3 b^8 c^2 d-3 a b^9 c^2 d-a^4 b^6 c^3 d-33 a^2 b^7 c^3 d+39 a^5 b^4 c^4 d-13 a^3 b^5 c^4 d+9 a^6 b^2 c^5 d+a^4 b^3 c^5 d+3 a^4 b^7 c d^2-a^2 b^8 c d^2-11 a^5 b^5 c^2 d^2+13 a^3 b^6 c^2 d^2+5 a^6 b^3 c^3 d^2+33 a^4 b^4 c^3 d^2-45 a^7 b c^4 d^2+3 a^5 b^2 c^4 d^2-5 a^5 b^6 d^3-a^3 b^7 d^3+21 a^6 b^4 c d^3+5 a^4 b^5 c d^3-35 a^7 b^2 c^2 d^3-15 a^5 b^3 c^2 d^3+35 a^8 c^3 d^3-5 a^6 b c^3 d^3\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{b+a x}{d+c x}}}{\sqrt{a}}\right)}{8 a^{7/2} c^{5/2} \left(-b^2+a c\right)^4}+\frac{\left(b^9 c^4-a^3 b^6 c^5-a b^8 c^3 d+a^4 b^5 c^4 d-2 a^2 b^6 c^4 d+2 a^5 b^3 c^5 d+a^3 b^8 c d^2-4 a^4 b^6 c^2 d^2+6 a^5 b^4 c^3 d^2+2 a^3 b^5 c^3 d^2-6 a^6 b^2 c^4 d^2+a^4 b^3 c^4 d^2-a^4 b^7 d^3+4 a^5 b^5 c d^3-6 a^6 b^3 c^2 d^3+4 a^7 b c^3 d^3-a^5 b^2 c^3 d^3\right) \log (d+c x)}{a^3 c^3 \left(-b^2+a c\right)^4}+\frac{2 \left(b^{10} c-a^3 b^7 c^2-a b^9 d+a^4 b^6 c d-2 a^2 b^7 c d+2 a^5 b^4 c^2 d+2 a^3 b^6 d^2-2 a^6 b^3 c d^2+a^4 b^4 c d^2-a^7 b c^2 d^2-a^5 b^3 d^3+a^8 c d^3\right) \log \left(a-b \sqrt{\frac{b+a x}{d+c x}}\right)}{a^3 b \left(-b^2+a c\right)^4}",1,"(-4*a^4*b^8*c^3 + 20*a^5*b^6*c^4 - 44*a^3*b^7*c^4 + 8*a^6*b^4*c^5 + 28*a^4*b^5*c^5 - 8*a^5*b^3*c^6 - 12*a^5*b^7*c^2*d + 36*a^6*b^5*c^3*d + 60*a^4*b^6*c^3*d - 96*a^7*b^3*c^4*d + 12*a^5*b^4*c^4*d + 60*a^6*b^6*c*d^2 - 180*a^7*b^4*c^2*d^2 - 12*a^5*b^5*c^2*d^2 + 192*a^8*b^2*c^3*d^2 - 60*a^6*b^3*c^3*d^2 - 44*a^7*b^5*d^3 + 124*a^8*b^3*c*d^3 - 4*a^6*b^4*c*d^3 - 104*a^9*b*c^2*d^3 + 20*a^7*b^2*c^2*d^3 + 8*a^8*c^3*d^3 + 3*a^4*b^7*c^4*Sqrt[(b + a*x)/(d + c*x)] - 33*a^2*b^8*c^4*Sqrt[(b + a*x)/(d + c*x)] + 24*a^5*b^5*c^5*Sqrt[(b + a*x)/(d + c*x)] + 12*a^3*b^6*c^5*Sqrt[(b + a*x)/(d + c*x)] - 3*a^6*b^3*c^6*Sqrt[(b + a*x)/(d + c*x)] - 3*a^4*b^4*c^6*Sqrt[(b + a*x)/(d + c*x)] + 3*a^5*b^6*c^3*d*Sqrt[(b + a*x)/(d + c*x)] + 39*a^3*b^7*c^3*d*Sqrt[(b + a*x)/(d + c*x)] - 48*a^6*b^4*c^4*d*Sqrt[(b + a*x)/(d + c*x)] + 36*a^4*b^5*c^4*d*Sqrt[(b + a*x)/(d + c*x)] - 27*a^7*b^2*c^5*d*Sqrt[(b + a*x)/(d + c*x)] - 3*a^5*b^3*c^5*d*Sqrt[(b + a*x)/(d + c*x)] + 9*a^6*b^5*c^2*d^2*Sqrt[(b + a*x)/(d + c*x)] - 3*a^4*b^6*c^2*d^2*Sqrt[(b + a*x)/(d + c*x)] - 24*a^7*b^3*c^3*d^2*Sqrt[(b + a*x)/(d + c*x)] - 60*a^5*b^4*c^3*d^2*Sqrt[(b + a*x)/(d + c*x)] + 87*a^8*b*c^4*d^2*Sqrt[(b + a*x)/(d + c*x)] - 9*a^6*b^2*c^4*d^2*Sqrt[(b + a*x)/(d + c*x)] - 15*a^7*b^4*c*d^3*Sqrt[(b + a*x)/(d + c*x)] - 3*a^5*b^5*c*d^3*Sqrt[(b + a*x)/(d + c*x)] + 48*a^8*b^2*c^2*d^3*Sqrt[(b + a*x)/(d + c*x)] + 12*a^6*b^3*c^2*d^3*Sqrt[(b + a*x)/(d + c*x)] - 57*a^9*c^3*d^3*Sqrt[(b + a*x)/(d + c*x)] + 15*a^7*b*c^3*d^3*Sqrt[(b + a*x)/(d + c*x)] + 8*a^3*b^7*c^5*((b + a*x)/(d + c*x))^(3/2) + 40*a*b^8*c^5*((b + a*x)/(d + c*x))^(3/2) - 64*a^4*b^5*c^6*((b + a*x)/(d + c*x))^(3/2) + 16*a^2*b^6*c^6*((b + a*x)/(d + c*x))^(3/2) + 8*a^5*b^3*c^7*((b + a*x)/(d + c*x))^(3/2) - 8*a^3*b^4*c^7*((b + a*x)/(d + c*x))^(3/2) - 24*a^4*b^6*c^4*d*((b + a*x)/(d + c*x))^(3/2) - 24*a^2*b^7*c^4*d*((b + a*x)/(d + c*x))^(3/2) + 96*a^5*b^4*c^5*d*((b + a*x)/(d + c*x))^(3/2) - 144*a^3*b^5*c^5*d*((b + a*x)/(d + c*x))^(3/2) + 72*a^6*b^2*c^6*d*((b + a*x)/(d + c*x))^(3/2) + 24*a^4*b^3*c^6*d*((b + a*x)/(d + c*x))^(3/2) - 24*a^5*b^5*c^3*d^2*((b + a*x)/(d + c*x))^(3/2) - 24*a^3*b^6*c^3*d^2*((b + a*x)/(d + c*x))^(3/2) + 96*a^6*b^3*c^4*d^2*((b + a*x)/(d + c*x))^(3/2) + 144*a^4*b^4*c^4*d^2*((b + a*x)/(d + c*x))^(3/2) - 216*a^7*b*c^5*d^2*((b + a*x)/(d + c*x))^(3/2) + 24*a^5*b^2*c^5*d^2*((b + a*x)/(d + c*x))^(3/2) + 40*a^6*b^4*c^2*d^3*((b + a*x)/(d + c*x))^(3/2) + 8*a^4*b^5*c^2*d^3*((b + a*x)/(d + c*x))^(3/2) - 128*a^7*b^2*c^3*d^3*((b + a*x)/(d + c*x))^(3/2) - 16*a^5*b^3*c^3*d^3*((b + a*x)/(d + c*x))^(3/2) + 136*a^8*c^4*d^3*((b + a*x)/(d + c*x))^(3/2) - 40*a^6*b*c^4*d^3*((b + a*x)/(d + c*x))^(3/2) - 3*a^2*b^7*c^6*((b + a*x)/(d + c*x))^(5/2) - 15*b^8*c^6*((b + a*x)/(d + c*x))^(5/2) + 24*a^3*b^5*c^7*((b + a*x)/(d + c*x))^(5/2) - 12*a*b^6*c^7*((b + a*x)/(d + c*x))^(5/2) + 3*a^4*b^3*c^8*((b + a*x)/(d + c*x))^(5/2) + 3*a^2*b^4*c^8*((b + a*x)/(d + c*x))^(5/2) - 3*a^3*b^6*c^5*d*((b + a*x)/(d + c*x))^(5/2) + 9*a*b^7*c^5*d*((b + a*x)/(d + c*x))^(5/2) + 60*a^2*b^5*c^6*d*((b + a*x)/(d + c*x))^(5/2) - 69*a^5*b^2*c^7*d*((b + a*x)/(d + c*x))^(5/2) + 3*a^3*b^3*c^7*d*((b + a*x)/(d + c*x))^(5/2) + 39*a^4*b^5*c^4*d^2*((b + a*x)/(d + c*x))^(5/2) + 3*a^2*b^6*c^4*d^2*((b + a*x)/(d + c*x))^(5/2) - 120*a^5*b^3*c^5*d^2*((b + a*x)/(d + c*x))^(5/2) - 36*a^3*b^4*c^5*d^2*((b + a*x)/(d + c*x))^(5/2) + 153*a^6*b*c^6*d^2*((b + a*x)/(d + c*x))^(5/2) - 39*a^4*b^2*c^6*d^2*((b + a*x)/(d + c*x))^(5/2) - 33*a^5*b^4*c^3*d^3*((b + a*x)/(d + c*x))^(5/2) + 3*a^3*b^5*c^3*d^3*((b + a*x)/(d + c*x))^(5/2) + 96*a^6*b^2*c^4*d^3*((b + a*x)/(d + c*x))^(5/2) - 12*a^4*b^3*c^4*d^3*((b + a*x)/(d + c*x))^(5/2) - 87*a^7*c^5*d^3*((b + a*x)/(d + c*x))^(5/2) + 33*a^5*b*c^5*d^3*((b + a*x)/(d + c*x))^(5/2) - (24*a*b^7*c^6*(b + a*x)^2)/(d + c*x)^2 + (24*a^4*b^4*c^7*(b + a*x)^2)/(d + c*x)^2 - (48*a^3*b^7*c^4*d*(b + a*x)^2)/(d + c*x)^2 + (144*a^4*b^5*c^5*d*(b + a*x)^2)/(d + c*x)^2 + (24*a^2*b^6*c^5*d*(b + a*x)^2)/(d + c*x)^2 - (168*a^5*b^3*c^6*d*(b + a*x)^2)/(d + c*x)^2 + (48*a^3*b^4*c^6*d*(b + a*x)^2)/(d + c*x)^2 + (120*a^4*b^6*c^3*d^2*(b + a*x)^2)/(d + c*x)^2 - (336*a^5*b^4*c^4*d^2*(b + a*x)^2)/(d + c*x)^2 + (288*a^6*b^2*c^5*d^2*(b + a*x)^2)/(d + c*x)^2 - (48*a^4*b^3*c^5*d^2*(b + a*x)^2)/(d + c*x)^2 - (24*a^5*b*c^6*d^2*(b + a*x)^2)/(d + c*x)^2 - (72*a^5*b^5*c^2*d^3*(b + a*x)^2)/(d + c*x)^2 + (192*a^6*b^3*c^3*d^3*(b + a*x)^2)/(d + c*x)^2 - (144*a^7*b*c^4*d^3*(b + a*x)^2)/(d + c*x)^2 + (24*a^6*c^5*d^3*(b + a*x)^2)/(d + c*x)^2 + (12*a^3*b^8*c^4*(b + a*x))/(d + c*x) - (36*a^4*b^6*c^5*(b + a*x))/(d + c*x) + (60*a^2*b^7*c^5*(b + a*x))/(d + c*x) - (24*a^5*b^4*c^6*(b + a*x))/(d + c*x) - (12*a^3*b^5*c^6*(b + a*x))/(d + c*x) + (36*a^4*b^7*c^3*d*(b + a*x))/(d + c*x) - (132*a^5*b^5*c^4*d*(b + a*x))/(d + c*x) - (60*a^3*b^6*c^4*d*(b + a*x))/(d + c*x) + (240*a^6*b^3*c^5*d*(b + a*x))/(d + c*x) - (108*a^4*b^4*c^5*d*(b + a*x))/(d + c*x) + (24*a^5*b^2*c^6*d*(b + a*x))/(d + c*x) - (156*a^5*b^6*c^2*d^2*(b + a*x))/(d + c*x) + (468*a^6*b^4*c^3*d^2*(b + a*x))/(d + c*x) - (12*a^4*b^5*c^3*d^2*(b + a*x))/(d + c*x) - (456*a^7*b^2*c^4*d^2*(b + a*x))/(d + c*x) + (156*a^5*b^3*c^4*d^2*(b + a*x))/(d + c*x) + (108*a^6*b^5*c*d^3*(b + a*x))/(d + c*x) - (300*a^7*b^3*c^2*d^3*(b + a*x))/(d + c*x) + (12*a^5*b^4*c^2*d^3*(b + a*x))/(d + c*x) + (240*a^8*b*c^3*d^3*(b + a*x))/(d + c*x) - (36*a^6*b^2*c^3*d^3*(b + a*x))/(d + c*x) - (24*a^7*c^4*d^3*(b + a*x))/(d + c*x))/(24*a^3*c^3*(-b^2 + a*c)^3*(a - (c*(b + a*x))/(d + c*x))^3) + ((a^2*b^9*c^3 + 5*b^10*c^3 - 9*a^3*b^7*c^4 + 15*a*b^8*c^4 - 9*a^4*b^5*c^5 - 5*a^2*b^6*c^5 + a^5*b^3*c^6 + a^3*b^4*c^6 + a^3*b^8*c^2*d - 3*a*b^9*c^2*d - a^4*b^6*c^3*d - 33*a^2*b^7*c^3*d + 39*a^5*b^4*c^4*d - 13*a^3*b^5*c^4*d + 9*a^6*b^2*c^5*d + a^4*b^3*c^5*d + 3*a^4*b^7*c*d^2 - a^2*b^8*c*d^2 - 11*a^5*b^5*c^2*d^2 + 13*a^3*b^6*c^2*d^2 + 5*a^6*b^3*c^3*d^2 + 33*a^4*b^4*c^3*d^2 - 45*a^7*b*c^4*d^2 + 3*a^5*b^2*c^4*d^2 - 5*a^5*b^6*d^3 - a^3*b^7*d^3 + 21*a^6*b^4*c*d^3 + 5*a^4*b^5*c*d^3 - 35*a^7*b^2*c^2*d^3 - 15*a^5*b^3*c^2*d^3 + 35*a^8*c^3*d^3 - 5*a^6*b*c^3*d^3)*ArcTanh[(Sqrt[c]*Sqrt[(b + a*x)/(d + c*x)])/Sqrt[a]])/(8*a^(7/2)*c^(5/2)*(-b^2 + a*c)^4) + (2*(-b^3 + a^2*d)^2*(b^4*c - a^3*b*c^2 - a*b^3*d + a^4*c*d)*Log[a - b*Sqrt[(b + a*x)/(d + c*x)]])/(a^3*b*(-b^2 + a*c)^4) + ((-(b^9*c^4) + a^3*b^6*c^5 + a*b^8*c^3*d - a^4*b^5*c^4*d + 2*a^2*b^6*c^4*d - 2*a^5*b^3*c^5*d - a^3*b^8*c*d^2 + 4*a^4*b^6*c^2*d^2 - 6*a^5*b^4*c^3*d^2 - 2*a^3*b^5*c^3*d^2 + 6*a^6*b^2*c^4*d^2 - a^4*b^3*c^4*d^2 + a^4*b^7*d^3 - 4*a^5*b^5*c*d^3 + 6*a^6*b^3*c^2*d^3 - 4*a^7*b*c^3*d^3 + a^5*b^2*c^3*d^3)*Log[a - (c*(b + a*x))/(d + c*x)])/(a^3*c^3*(-b^2 + a*c)^4)","B",1
3154,1,9401,3329,20.1022631,"\int \frac{(c_6+x c_7){}^2}{\sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5}} \, dx","IntegrateAlgebraic[(C[6] + x*C[7])^2/Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]],x]","\text{Result too large to show}","\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{c_3} \sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt{\sqrt{c_3} c_4-\sqrt{c_1} c_5}}\right) (-c_1 c_2+c_0 c_3) c_5 \sqrt{\sqrt{c_3} c_4-\sqrt{c_1} c_5} c_6{}^2}{2 \sqrt{c_1} c_3{}^{3/4} \left(-\sqrt{c_3} c_4+\sqrt{c_1} c_5\right){}^2}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{c_3} \sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt{\sqrt{c_3} c_4+\sqrt{c_1} c_5}}\right) (-c_1 c_2+c_0 c_3) c_5 c_6{}^2}{2 \sqrt{c_1} c_3{}^{3/4} \left(\sqrt{c_3} c_4+\sqrt{c_1} c_5\right){}^{3/2}}+\frac{\sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5} \left(c_2 c_4 c_6{}^2+x c_3 c_4 c_6{}^2\right)}{c_3 c_4{}^2-c_1 c_5{}^2}+\frac{\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} \sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5} \left(c_2 c_5 c_6{}^2+x c_3 c_5 c_6{}^2\right)}{-c_3 c_4{}^2+c_1 c_5{}^2}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{c_3} \sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt{\sqrt{c_3} c_4-\sqrt{c_1} c_5}}\right) \sqrt{\sqrt{c_3} c_4-\sqrt{c_1} c_5} \left(-6 c_1{}^2 c_2{}^2 \sqrt{c_3} c_4 c_5+4 c_0 c_1 c_2 c_3{}^{3/2} c_4 c_5+2 c_0{}^2 c_3{}^{5/2} c_4 c_5+3 c_1{}^{5/2} c_2{}^2 c_5{}^2+2 c_0 c_1{}^{3/2} c_2 c_3 c_5{}^2-5 c_0{}^2 \sqrt{c_1} c_3{}^2 c_5{}^2\right) c_6 c_7}{8 c_1{}^{3/2} c_3{}^{7/4} \left(-\sqrt{c_3} c_4+\sqrt{c_1} c_5\right){}^3}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{c_3} \sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt{\sqrt{c_3} c_4+\sqrt{c_1} c_5}}\right) \left(-6 c_1{}^2 c_2{}^2 \sqrt{c_3} c_4 c_5+4 c_0 c_1 c_2 c_3{}^{3/2} c_4 c_5+2 c_0{}^2 c_3{}^{5/2} c_4 c_5-3 c_1{}^{5/2} c_2{}^2 c_5{}^2-2 c_0 c_1{}^{3/2} c_2 c_3 c_5{}^2+5 c_0{}^2 \sqrt{c_1} c_3{}^2 c_5{}^2\right) c_6 c_7}{8 c_1{}^{3/2} c_3{}^{7/4} \left(\sqrt{c_3} c_4+\sqrt{c_1} c_5\right){}^{5/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{c_3} \sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt{\sqrt{c_3} c_4-\sqrt{c_1} c_5}}\right) \sqrt{\sqrt{c_3} c_4-\sqrt{c_1} c_5} \left(-20 c_1{}^3 c_2{}^3 c_3 c_4{}^2 c_5+12 c_0 c_1{}^2 c_2{}^2 c_3{}^2 c_4{}^2 c_5+4 c_0{}^2 c_1 c_2 c_3{}^3 c_4{}^2 c_5+4 c_0{}^3 c_3{}^4 c_4{}^2 c_5+22 c_1{}^{7/2} c_2{}^3 \sqrt{c_3} c_4 c_5{}^2+6 c_0 c_1{}^{5/2} c_2{}^2 c_3{}^{3/2} c_4 c_5{}^2-14 c_0{}^2 c_1{}^{3/2} c_2 c_3{}^{5/2} c_4 c_5{}^2-14 c_0{}^3 \sqrt{c_1} c_3{}^{7/2} c_4 c_5{}^2-7 c_1{}^4 c_2{}^3 c_5{}^3-3 c_0 c_1{}^3 c_2{}^2 c_3 c_5{}^3-5 c_0{}^2 c_1{}^2 c_2 c_3{}^2 c_5{}^3+15 c_0{}^3 c_1 c_3{}^3 c_5{}^3\right) c_7{}^2}{64 c_1{}^{5/2} c_3{}^{11/4} \left(-\sqrt{c_3} c_4+\sqrt{c_1} c_5\right){}^4}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{c_3} \sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt{\sqrt{c_3} c_4+\sqrt{c_1} c_5}}\right) \left(-20 c_1{}^3 c_2{}^3 c_3 c_4{}^2 c_5+12 c_0 c_1{}^2 c_2{}^2 c_3{}^2 c_4{}^2 c_5+4 c_0{}^2 c_1 c_2 c_3{}^3 c_4{}^2 c_5+4 c_0{}^3 c_3{}^4 c_4{}^2 c_5-22 c_1{}^{7/2} c_2{}^3 \sqrt{c_3} c_4 c_5{}^2-6 c_0 c_1{}^{5/2} c_2{}^2 c_3{}^{3/2} c_4 c_5{}^2+14 c_0{}^2 c_1{}^{3/2} c_2 c_3{}^{5/2} c_4 c_5{}^2+14 c_0{}^3 \sqrt{c_1} c_3{}^{7/2} c_4 c_5{}^2-7 c_1{}^4 c_2{}^3 c_5{}^3-3 c_0 c_1{}^3 c_2{}^2 c_3 c_5{}^3-5 c_0{}^2 c_1{}^2 c_2 c_3{}^2 c_5{}^3+15 c_0{}^3 c_1 c_3{}^3 c_5{}^3\right) c_7{}^2}{64 c_1{}^{5/2} c_3{}^{11/4} \left(\sqrt{c_3} c_4+\sqrt{c_1} c_5\right){}^{7/2}}+\frac{\sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5} \left(-2 c_2{}^2 c_3 c_4{}^3 c_6 c_7+2 x^2 c_3{}^3 c_4{}^3 c_6 c_7-c_1 c_2{}^2 c_4 c_5{}^2 c_6 c_7+3 c_0 c_2 c_3 c_4 c_5{}^2 c_6 c_7-3 x c_1 c_2 c_3 c_4 c_5{}^2 c_6 c_7+3 x c_0 c_3{}^2 c_4 c_5{}^2 c_6 c_7-2 x^2 c_1 c_3{}^2 c_4 c_5{}^2 c_6 c_7\right)}{2 c_3 \left(c_3 c_4{}^2-c_1 c_5{}^2\right){}^2}+\frac{\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} \sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5} \left(5 c_1 c_2{}^2 c_3 c_4{}^2 c_5 c_6 c_7-c_0 c_2 c_3{}^2 c_4{}^2 c_5 c_6 c_7+x c_1 c_2 c_3{}^2 c_4{}^2 c_5 c_6 c_7-x c_0 c_3{}^3 c_4{}^2 c_5 c_6 c_7-4 x^2 c_1 c_3{}^3 c_4{}^2 c_5 c_6 c_7+c_1{}^2 c_2{}^2 c_5{}^3 c_6 c_7-5 c_0 c_1 c_2 c_3 c_5{}^3 c_6 c_7+5 x c_1{}^2 c_2 c_3 c_5{}^3 c_6 c_7-5 x c_0 c_1 c_3{}^2 c_5{}^3 c_6 c_7+4 x^2 c_1{}^2 c_3{}^2 c_5{}^3 c_6 c_7\right)}{4 c_1 c_3 \left(c_3 c_4{}^2-c_1 c_5{}^2\right){}^2}+\frac{\sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5} \left(32 c_1 c_2{}^3 c_3{}^2 c_4{}^5 c_7{}^2+32 x^3 c_1 c_3{}^5 c_4{}^5 c_7{}^2+39 c_1{}^2 c_2{}^3 c_3 c_4{}^3 c_5{}^2 c_7{}^2-102 c_0 c_1 c_2{}^2 c_3{}^2 c_4{}^3 c_5{}^2 c_7{}^2+63 x c_1{}^2 c_2{}^2 c_3{}^2 c_4{}^3 c_5{}^2 c_7{}^2-c_0{}^2 c_2 c_3{}^3 c_4{}^3 c_5{}^2 c_7{}^2-62 x c_0 c_1 c_2 c_3{}^3 c_4{}^3 c_5{}^2 c_7{}^2-40 x^2 c_1{}^2 c_2 c_3{}^3 c_4{}^3 c_5{}^2 c_7{}^2-x c_0{}^2 c_3{}^4 c_4{}^3 c_5{}^2 c_7{}^2+40 x^2 c_0 c_1 c_3{}^4 c_4{}^3 c_5{}^2 c_7{}^2-64 x^3 c_1{}^2 c_3{}^4 c_4{}^3 c_5{}^2 c_7{}^2-11 c_1{}^3 c_2{}^3 c_4 c_5{}^4 c_7{}^2-18 c_0 c_1{}^2 c_2{}^2 c_3 c_4 c_5{}^4 c_7{}^2-3 x c_1{}^3 c_2{}^2 c_3 c_4 c_5{}^4 c_7{}^2+61 c_0{}^2 c_1 c_2 c_3{}^2 c_4 c_5{}^4 c_7{}^2-58 x c_0 c_1{}^2 c_2 c_3{}^2 c_4 c_5{}^4 c_7{}^2+40 x^2 c_1{}^3 c_2 c_3{}^2 c_4 c_5{}^4 c_7{}^2+61 x c_0{}^2 c_1 c_3{}^3 c_4 c_5{}^4 c_7{}^2-40 x^2 c_0 c_1{}^2 c_3{}^3 c_4 c_5{}^4 c_7{}^2+32 x^3 c_1{}^3 c_3{}^3 c_4 c_5{}^4 c_7{}^2\right)}{96 c_1 c_3{}^2 \left(c_3 c_4{}^2-c_1 c_5{}^2\right){}^3}+\frac{\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} \sqrt{c_4+\sqrt{\frac{c_0+x c_1}{c_2+x c_3}} c_5} \left(-46 c_1{}^2 c_2{}^3 c_3{}^2 c_4{}^4 c_5 c_7{}^2+8 c_0 c_1 c_2{}^2 c_3{}^3 c_4{}^4 c_5 c_7{}^2-10 x c_1{}^2 c_2{}^2 c_3{}^3 c_4{}^4 c_5 c_7{}^2+6 c_0{}^2 c_2 c_3{}^4 c_4{}^4 c_5 c_7{}^2+4 x c_0 c_1 c_2 c_3{}^4 c_4{}^4 c_5 c_7{}^2+4 x^2 c_1{}^2 c_2 c_3{}^4 c_4{}^4 c_5 c_7{}^2+6 x c_0{}^2 c_3{}^5 c_4{}^4 c_5 c_7{}^2-4 x^2 c_0 c_1 c_3{}^5 c_4{}^4 c_5 c_7{}^2-32 x^3 c_1{}^2 c_3{}^5 c_4{}^4 c_5 c_7{}^2-21 c_1{}^3 c_2{}^3 c_3 c_4{}^2 c_5{}^3 c_7{}^2+106 c_0 c_1{}^2 c_2{}^2 c_3{}^2 c_4{}^2 c_5{}^3 c_7{}^2-53 x c_1{}^3 c_2{}^2 c_3{}^2 c_4{}^2 c_5{}^3 c_7{}^2-21 c_0{}^2 c_1 c_2 c_3{}^3 c_4{}^2 c_5{}^3 c_7{}^2+74 x c_0 c_1{}^2 c_2 c_3{}^3 c_4{}^2 c_5{}^3 c_7{}^2+32 x^2 c_1{}^3 c_2 c_3{}^3 c_4{}^2 c_5{}^3 c_7{}^2-21 x c_0{}^2 c_1 c_3{}^4 c_4{}^2 c_5{}^3 c_7{}^2-32 x^2 c_0 c_1{}^2 c_3{}^4 c_4{}^2 c_5{}^3 c_7{}^2+64 x^3 c_1{}^3 c_3{}^4 c_4{}^2 c_5{}^3 c_7{}^2+7 c_1{}^4 c_2{}^3 c_5{}^5 c_7{}^2+6 c_0 c_1{}^3 c_2{}^2 c_3 c_5{}^5 c_7{}^2+3 x c_1{}^4 c_2{}^2 c_3 c_5{}^5 c_7{}^2-45 c_0{}^2 c_1{}^2 c_2 c_3{}^2 c_5{}^5 c_7{}^2+42 x c_0 c_1{}^3 c_2 c_3{}^2 c_5{}^5 c_7{}^2-36 x^2 c_1{}^4 c_2 c_3{}^2 c_5{}^5 c_7{}^2-45 x c_0{}^2 c_1{}^2 c_3{}^3 c_5{}^5 c_7{}^2+36 x^2 c_0 c_1{}^3 c_3{}^3 c_5{}^5 c_7{}^2-32 x^3 c_1{}^4 c_3{}^3 c_5{}^5 c_7{}^2\right)}{96 c_1{}^2 c_3{}^2 \left(c_3 c_4{}^2-c_1 c_5{}^2\right){}^3}",1,"Result too large to show","B",1