1,1,42,52,0.0244655,"\int \left(a x^3+b x^6\right)^{5/3} \, dx","Integrate[(a*x^3 + b*x^6)^(5/3),x]","\frac{x \left(a+b x^3\right)^3 \left(8 b x^3-3 a\right)}{88 b^2 \sqrt[3]{x^3 \left(a+b x^3\right)}}","\frac{\left(a x^3+b x^6\right)^{8/3}}{11 b x^5}-\frac{3 a \left(a x^3+b x^6\right)^{8/3}}{88 b^2 x^8}",1,"(x*(a + b*x^3)^3*(-3*a + 8*b*x^3))/(88*b^2*(x^3*(a + b*x^3))^(1/3))","A",1
2,1,25,25,0.0082171,"\int \left(a x^3+b x^6\right)^{2/3} \, dx","Integrate[(a*x^3 + b*x^6)^(2/3),x]","\frac{\left(x^3 \left(a+b x^3\right)\right)^{5/3}}{5 b x^5}","\frac{\left(a x^3+b x^6\right)^{5/3}}{5 b x^5}",1,"(x^3*(a + b*x^3))^(5/3)/(5*b*x^5)","A",1
3,1,23,23,0.008236,"\int \frac{1}{\left(a x^3+b x^6\right)^{2/3}} \, dx","Integrate[(a*x^3 + b*x^6)^(-2/3),x]","-\frac{\sqrt[3]{x^3 \left(a+b x^3\right)}}{a x^2}","-\frac{\sqrt[3]{a x^3+b x^6}}{a x^2}",1,"-((x^3*(a + b*x^3))^(1/3)/(a*x^2))","A",1
4,1,46,77,0.0100088,"\int \frac{1}{\left(a x^3+b x^6\right)^{5/3}} \, dx","Integrate[(a*x^3 + b*x^6)^(-5/3),x]","\frac{-a^2+6 a b x^3+9 b^2 x^6}{4 a^3 x^2 \left(x^3 \left(a+b x^3\right)\right)^{2/3}}","\frac{9 b \sqrt[3]{a x^3+b x^6}}{4 a^3 x^2}-\frac{3 \sqrt[3]{a x^3+b x^6}}{4 a^2 x^5}+\frac{1}{2 a x^2 \left(a x^3+b x^6\right)^{2/3}}",1,"(-a^2 + 6*a*b*x^3 + 9*b^2*x^6)/(4*a^3*x^2*(x^3*(a + b*x^3))^(2/3))","A",1
5,1,48,48,0.0174518,"\int \frac{1}{-x^3+x^6} \, dx","Integrate[(-x^3 + x^6)^(-1),x]","\frac{1}{2 x^2}-\frac{1}{6} \log \left(x^2+x+1\right)+\frac{1}{3} \log (1-x)-\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{\sqrt{3}}","\frac{1}{2 x^2}-\frac{1}{6} \log \left(x^2+x+1\right)+\frac{1}{3} \log (1-x)-\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{\sqrt{3}}",1,"1/(2*x^2) - ArcTan[(1 + 2*x)/Sqrt[3]]/Sqrt[3] + Log[1 - x]/3 - Log[1 + x + x^2]/6","A",1
6,1,39,79,0.0121044,"\int x^5 \sqrt{a^2+2 a b x^3+b^2 x^6} \, dx","Integrate[x^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]","\frac{\sqrt{\left(a+b x^3\right)^2} \left(3 a x^6+2 b x^9\right)}{18 \left(a+b x^3\right)}","\frac{a x^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{6 \left(a+b x^3\right)}+\frac{b x^9 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 \left(a+b x^3\right)}",1,"(Sqrt[(a + b*x^3)^2]*(3*a*x^6 + 2*b*x^9))/(18*(a + b*x^3))","A",1
7,1,39,79,0.0077478,"\int x^4 \sqrt{a^2+2 a b x^3+b^2 x^6} \, dx","Integrate[x^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]","\frac{\sqrt{\left(a+b x^3\right)^2} \left(8 a x^5+5 b x^8\right)}{40 \left(a+b x^3\right)}","\frac{b x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left(a+b x^3\right)}+\frac{a x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left(a+b x^3\right)}",1,"(Sqrt[(a + b*x^3)^2]*(8*a*x^5 + 5*b*x^8))/(40*(a + b*x^3))","A",1
8,1,39,79,0.0068223,"\int x^3 \sqrt{a^2+2 a b x^3+b^2 x^6} \, dx","Integrate[x^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]","\frac{\sqrt{\left(a+b x^3\right)^2} \left(7 a x^4+4 b x^7\right)}{28 \left(a+b x^3\right)}","\frac{b x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left(a+b x^3\right)}+\frac{a x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left(a+b x^3\right)}",1,"(Sqrt[(a + b*x^3)^2]*(7*a*x^4 + 4*b*x^7))/(28*(a + b*x^3))","A",1
9,1,38,36,0.0088249,"\int x^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \, dx","Integrate[x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]","\frac{\sqrt{\left(a+b x^3\right)^2} \left(2 a x^3+b x^6\right)}{6 \left(a+b x^3\right)}","\frac{\left(a+b x^3\right) \sqrt{a^2+2 a b x^3+b^2 x^6}}{6 b}",1,"(Sqrt[(a + b*x^3)^2]*(2*a*x^3 + b*x^6))/(6*(a + b*x^3))","A",1
10,1,39,79,0.0087428,"\int x \sqrt{a^2+2 a b x^3+b^2 x^6} \, dx","Integrate[x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]","\frac{\sqrt{\left(a+b x^3\right)^2} \left(5 a x^2+2 b x^5\right)}{10 \left(a+b x^3\right)}","\frac{b x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left(a+b x^3\right)}+\frac{a x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left(a+b x^3\right)}",1,"(Sqrt[(a + b*x^3)^2]*(5*a*x^2 + 2*b*x^5))/(10*(a + b*x^3))","A",1
11,1,36,74,0.0071047,"\int \sqrt{a^2+2 a b x^3+b^2 x^6} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]","\frac{\sqrt{\left(a+b x^3\right)^2} \left(4 a x+b x^4\right)}{4 \left(a+b x^3\right)}","\frac{a x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{b x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left(a+b x^3\right)}",1,"(Sqrt[(a + b*x^3)^2]*(4*a*x + b*x^4))/(4*(a + b*x^3))","A",1
12,1,37,75,0.0088177,"\int \frac{\sqrt{a^2+2 a b x^3+b^2 x^6}}{x} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x,x]","\frac{\sqrt{\left(a+b x^3\right)^2} \left(3 a \log (x)+b x^3\right)}{3 \left(a+b x^3\right)}","\frac{b x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left(a+b x^3\right)}+\frac{a \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}",1,"(Sqrt[(a + b*x^3)^2]*(b*x^3 + 3*a*Log[x]))/(3*(a + b*x^3))","A",1
13,1,38,77,0.0092756,"\int \frac{\sqrt{a^2+2 a b x^3+b^2 x^6}}{x^2} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x^2,x]","\frac{\left(b x^3-2 a\right) \sqrt{\left(a+b x^3\right)^2}}{2 x \left(a+b x^3\right)}","\frac{b x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left(a+b x^3\right)}-\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left(a+b x^3\right)}",1,"((-2*a + b*x^3)*Sqrt[(a + b*x^3)^2])/(2*x*(a + b*x^3))","A",1
14,1,37,74,0.0069465,"\int \frac{\sqrt{a^2+2 a b x^3+b^2 x^6}}{x^3} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x^3,x]","-\frac{\left(a-2 b x^3\right) \sqrt{\left(a+b x^3\right)^2}}{2 x^2 \left(a+b x^3\right)}","\frac{b x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left(a+b x^3\right)}",1,"-1/2*((a - 2*b*x^3)*Sqrt[(a + b*x^3)^2])/(x^2*(a + b*x^3))","A",1
15,1,39,75,0.0090422,"\int \frac{\sqrt{a^2+2 a b x^3+b^2 x^6}}{x^4} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x^4,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(a-3 b x^3 \log (x)\right)}{3 x^3 \left(a+b x^3\right)}","\frac{b \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left(a+b x^3\right)}",1,"-1/3*(Sqrt[(a + b*x^3)^2]*(a - 3*b*x^3*Log[x]))/(x^3*(a + b*x^3))","A",1
16,1,37,77,0.0077646,"\int \frac{\sqrt{a^2+2 a b x^3+b^2 x^6}}{x^5} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x^5,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(a+4 b x^3\right)}{4 x^4 \left(a+b x^3\right)}","-\frac{b \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left(a+b x^3\right)}-\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left(a+b x^3\right)}",1,"-1/4*(Sqrt[(a + b*x^3)^2]*(a + 4*b*x^3))/(x^4*(a + b*x^3))","A",1
17,1,39,79,0.0079135,"\int \frac{\sqrt{a^2+2 a b x^3+b^2 x^6}}{x^6} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x^6,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(2 a+5 b x^3\right)}{10 x^5 \left(a+b x^3\right)}","-\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left(a+b x^3\right)}-\frac{b \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left(a+b x^3\right)}",1,"-1/10*(Sqrt[(a + b*x^3)^2]*(2*a + 5*b*x^3))/(x^5*(a + b*x^3))","A",1
18,1,37,79,0.0075565,"\int \frac{\sqrt{a^2+2 a b x^3+b^2 x^6}}{x^7} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x^7,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(a+2 b x^3\right)}{6 x^6 \left(a+b x^3\right)}","-\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left(a+b x^3\right)}-\frac{b \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left(a+b x^3\right)}",1,"-1/6*(Sqrt[(a + b*x^3)^2]*(a + 2*b*x^3))/(x^6*(a + b*x^3))","A",1
19,1,39,79,0.0077636,"\int \frac{\sqrt{a^2+2 a b x^3+b^2 x^6}}{x^8} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x^8,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(4 a+7 b x^3\right)}{28 x^7 \left(a+b x^3\right)}","-\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left(a+b x^3\right)}-\frac{b \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left(a+b x^3\right)}",1,"-1/28*(Sqrt[(a + b*x^3)^2]*(4*a + 7*b*x^3))/(x^7*(a + b*x^3))","A",1
20,1,39,79,0.0079307,"\int \frac{\sqrt{a^2+2 a b x^3+b^2 x^6}}{x^9} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x^9,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(5 a+8 b x^3\right)}{40 x^8 \left(a+b x^3\right)}","-\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left(a+b x^3\right)}-\frac{b \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left(a+b x^3\right)}",1,"-1/40*(Sqrt[(a + b*x^3)^2]*(5*a + 8*b*x^3))/(x^8*(a + b*x^3))","A",1
21,1,39,79,0.0077651,"\int \frac{\sqrt{a^2+2 a b x^3+b^2 x^6}}{x^{10}} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x^10,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(2 a+3 b x^3\right)}{18 x^9 \left(a+b x^3\right)}","-\frac{b \sqrt{a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left(a+b x^3\right)}-\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left(a+b x^3\right)}",1,"-1/18*(Sqrt[(a + b*x^3)^2]*(2*a + 3*b*x^3))/(x^9*(a + b*x^3))","A",1
22,1,39,79,0.0086654,"\int \frac{\sqrt{a^2+2 a b x^3+b^2 x^6}}{x^{11}} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x^11,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(7 a+10 b x^3\right)}{70 x^{10} \left(a+b x^3\right)}","-\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{10 x^{10} \left(a+b x^3\right)}-\frac{b \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left(a+b x^3\right)}",1,"-1/70*(Sqrt[(a + b*x^3)^2]*(7*a + 10*b*x^3))/(x^10*(a + b*x^3))","A",1
23,1,61,167,0.0201761,"\int x^9 \left(a^2+2 a b x^3+b^2 x^6\right)^{3/2} \, dx","Integrate[x^9*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2),x]","\frac{x^{10} \sqrt{\left(a+b x^3\right)^2} \left(1976 a^3+4560 a^2 b x^3+3705 a b^2 x^6+1040 b^3 x^9\right)}{19760 \left(a+b x^3\right)}","\frac{3 a b^2 x^{16} \sqrt{a^2+2 a b x^3+b^2 x^6}}{16 \left(a+b x^3\right)}+\frac{3 a^2 b x^{13} \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 \left(a+b x^3\right)}+\frac{b^3 x^{19} \sqrt{a^2+2 a b x^3+b^2 x^6}}{19 \left(a+b x^3\right)}+\frac{a^3 x^{10} \sqrt{a^2+2 a b x^3+b^2 x^6}}{10 \left(a+b x^3\right)}",1,"(x^10*Sqrt[(a + b*x^3)^2]*(1976*a^3 + 4560*a^2*b*x^3 + 3705*a*b^2*x^6 + 1040*b^3*x^9))/(19760*(a + b*x^3))","A",1
24,1,61,119,0.0150584,"\int x^8 \left(a^2+2 a b x^3+b^2 x^6\right)^{3/2} \, dx","Integrate[x^8*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2),x]","\frac{x^9 \sqrt{\left(a+b x^3\right)^2} \left(20 a^3+45 a^2 b x^3+36 a b^2 x^6+10 b^3 x^9\right)}{180 \left(a+b x^3\right)}","\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \left(a+b x^3\right)^5}{18 b^3}-\frac{2 a \sqrt{a^2+2 a b x^3+b^2 x^6} \left(a+b x^3\right)^4}{15 b^3}+\frac{a^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left(a+b x^3\right)^3}{12 b^3}",1,"(x^9*Sqrt[(a + b*x^3)^2]*(20*a^3 + 45*a^2*b*x^3 + 36*a*b^2*x^6 + 10*b^3*x^9))/(180*(a + b*x^3))","A",1
25,1,61,167,0.0159034,"\int x^7 \left(a^2+2 a b x^3+b^2 x^6\right)^{3/2} \, dx","Integrate[x^7*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2),x]","\frac{x^8 \sqrt{\left(a+b x^3\right)^2} \left(1309 a^3+2856 a^2 b x^3+2244 a b^2 x^6+616 b^3 x^9\right)}{10472 \left(a+b x^3\right)}","\frac{3 a b^2 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 \left(a+b x^3\right)}+\frac{3 a^2 b x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left(a+b x^3\right)}+\frac{b^3 x^{17} \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 \left(a+b x^3\right)}+\frac{a^3 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left(a+b x^3\right)}",1,"(x^8*Sqrt[(a + b*x^3)^2]*(1309*a^3 + 2856*a^2*b*x^3 + 2244*a*b^2*x^6 + 616*b^3*x^9))/(10472*(a + b*x^3))","A",1
26,1,61,167,0.0151551,"\int x^6 \left(a^2+2 a b x^3+b^2 x^6\right)^{3/2} \, dx","Integrate[x^6*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2),x]","\frac{x^7 \sqrt{\left(a+b x^3\right)^2} \left(1040 a^3+2184 a^2 b x^3+1680 a b^2 x^6+455 b^3 x^9\right)}{7280 \left(a+b x^3\right)}","\frac{3 a b^2 x^{13} \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 \left(a+b x^3\right)}+\frac{3 a^2 b x^{10} \sqrt{a^2+2 a b x^3+b^2 x^6}}{10 \left(a+b x^3\right)}+\frac{b^3 x^{16} \sqrt{a^2+2 a b x^3+b^2 x^6}}{16 \left(a+b x^3\right)}+\frac{a^3 x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left(a+b x^3\right)}",1,"(x^7*Sqrt[(a + b*x^3)^2]*(1040*a^3 + 2184*a^2*b*x^3 + 1680*a*b^2*x^6 + 455*b^3*x^9))/(7280*(a + b*x^3))","A",1
27,1,61,78,0.0154828,"\int x^5 \left(a^2+2 a b x^3+b^2 x^6\right)^{3/2} \, dx","Integrate[x^5*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2),x]","\frac{x^6 \sqrt{\left(a+b x^3\right)^2} \left(10 a^3+20 a^2 b x^3+15 a b^2 x^6+4 b^3 x^9\right)}{60 \left(a+b x^3\right)}","\frac{\left(a+b x^3\right)^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{15 b^2}-\frac{a \left(a+b x^3\right)^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 b^2}",1,"(x^6*Sqrt[(a + b*x^3)^2]*(10*a^3 + 20*a^2*b*x^3 + 15*a*b^2*x^6 + 4*b^3*x^9))/(60*(a + b*x^3))","A",1
28,1,61,167,0.0163132,"\int x^4 \left(a^2+2 a b x^3+b^2 x^6\right)^{3/2} \, dx","Integrate[x^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2),x]","\frac{x^5 \sqrt{\left(a+b x^3\right)^2} \left(616 a^3+1155 a^2 b x^3+840 a b^2 x^6+220 b^3 x^9\right)}{3080 \left(a+b x^3\right)}","\frac{3 a b^2 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left(a+b x^3\right)}+\frac{3 a^2 b x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left(a+b x^3\right)}+\frac{b^3 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 \left(a+b x^3\right)}+\frac{a^3 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left(a+b x^3\right)}",1,"(x^5*Sqrt[(a + b*x^3)^2]*(616*a^3 + 1155*a^2*b*x^3 + 840*a*b^2*x^6 + 220*b^3*x^9))/(3080*(a + b*x^3))","A",1
29,1,61,167,0.0152037,"\int x^3 \left(a^2+2 a b x^3+b^2 x^6\right)^{3/2} \, dx","Integrate[x^3*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2),x]","\frac{x^4 \sqrt{\left(a+b x^3\right)^2} \left(455 a^3+780 a^2 b x^3+546 a b^2 x^6+140 b^3 x^9\right)}{1820 \left(a+b x^3\right)}","\frac{3 a b^2 x^{10} \sqrt{a^2+2 a b x^3+b^2 x^6}}{10 \left(a+b x^3\right)}+\frac{3 a^2 b x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left(a+b x^3\right)}+\frac{b^3 x^{13} \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 \left(a+b x^3\right)}+\frac{a^3 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left(a+b x^3\right)}",1,"(x^4*Sqrt[(a + b*x^3)^2]*(455*a^3 + 780*a^2*b*x^3 + 546*a*b^2*x^6 + 140*b^3*x^9))/(1820*(a + b*x^3))","A",1
30,1,60,36,0.0156345,"\int x^2 \left(a^2+2 a b x^3+b^2 x^6\right)^{3/2} \, dx","Integrate[x^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2),x]","\frac{x^3 \sqrt{\left(a+b x^3\right)^2} \left(4 a^3+6 a^2 b x^3+4 a b^2 x^6+b^3 x^9\right)}{12 \left(a+b x^3\right)}","\frac{\left(a+b x^3\right) \left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}}{12 b}",1,"(x^3*Sqrt[(a + b*x^3)^2]*(4*a^3 + 6*a^2*b*x^3 + 4*a*b^2*x^6 + b^3*x^9))/(12*(a + b*x^3))","A",1
31,1,61,167,0.0154797,"\int x \left(a^2+2 a b x^3+b^2 x^6\right)^{3/2} \, dx","Integrate[x*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2),x]","\frac{x^2 \sqrt{\left(a+b x^3\right)^2} \left(220 a^3+264 a^2 b x^3+165 a b^2 x^6+40 b^3 x^9\right)}{440 \left(a+b x^3\right)}","\frac{3 a b^2 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left(a+b x^3\right)}+\frac{3 a^2 b x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left(a+b x^3\right)}+\frac{b^3 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left(a+b x^3\right)}+\frac{a^3 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left(a+b x^3\right)}",1,"(x^2*Sqrt[(a + b*x^3)^2]*(220*a^3 + 264*a^2*b*x^3 + 165*a*b^2*x^6 + 40*b^3*x^9))/(440*(a + b*x^3))","A",1
32,1,59,162,0.0144765,"\int \left(a^2+2 a b x^3+b^2 x^6\right)^{3/2} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2),x]","\frac{x \sqrt{\left(a+b x^3\right)^2} \left(140 a^3+105 a^2 b x^3+60 a b^2 x^6+14 b^3 x^9\right)}{140 \left(a+b x^3\right)}","\frac{3 a b^2 x^7 \left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}}{7 \left(a+b x^3\right)^3}+\frac{3 a^2 b x^4 \left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}}{4 \left(a+b x^3\right)^3}+\frac{b^3 x^{10} \left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}}{10 \left(a+b x^3\right)^3}+\frac{a^3 x \left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}}{\left(a+b x^3\right)^3}",1,"(x*Sqrt[(a + b*x^3)^2]*(140*a^3 + 105*a^2*b*x^3 + 60*a*b^2*x^6 + 14*b^3*x^9))/(140*(a + b*x^3))","A",1
33,1,60,160,0.0200638,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}}{x} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x,x]","\frac{\sqrt{\left(a+b x^3\right)^2} \left(18 a^3 \log (x)+b x^3 \left(18 a^2+9 a b x^3+2 b^2 x^6\right)\right)}{18 \left(a+b x^3\right)}","\frac{a b^2 x^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left(a+b x^3\right)}+\frac{a^2 b x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{b^3 x^9 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 \left(a+b x^3\right)}+\frac{a^3 \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}",1,"(Sqrt[(a + b*x^3)^2]*(b*x^3*(18*a^2 + 9*a*b*x^3 + 2*b^2*x^6) + 18*a^3*Log[x]))/(18*(a + b*x^3))","A",1
34,1,61,165,0.0160857,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}}{x^2} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^2,x]","\frac{\sqrt{\left(a+b x^3\right)^2} \left(-40 a^3+60 a^2 b x^3+24 a b^2 x^6+5 b^3 x^9\right)}{40 x \left(a+b x^3\right)}","\frac{3 a b^2 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left(a+b x^3\right)}+\frac{3 a^2 b x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left(a+b x^3\right)}+\frac{b^3 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left(a+b x^3\right)}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left(a+b x^3\right)}",1,"(Sqrt[(a + b*x^3)^2]*(-40*a^3 + 60*a^2*b*x^3 + 24*a*b^2*x^6 + 5*b^3*x^9))/(40*x*(a + b*x^3))","A",1
35,1,61,163,0.0176785,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}}{x^3} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^3,x]","\frac{\sqrt{\left(a+b x^3\right)^2} \left(-14 a^3+84 a^2 b x^3+21 a b^2 x^6+4 b^3 x^9\right)}{28 x^2 \left(a+b x^3\right)}","\frac{3 a^2 b x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{3 a b^2 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left(a+b x^3\right)}+\frac{b^3 x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left(a+b x^3\right)}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left(a+b x^3\right)}",1,"(Sqrt[(a + b*x^3)^2]*(-14*a^3 + 84*a^2*b*x^3 + 21*a*b^2*x^6 + 4*b^3*x^9))/(28*x^2*(a + b*x^3))","A",1
36,1,62,161,0.0196744,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}}{x^4} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^4,x]","\frac{\sqrt{\left(a+b x^3\right)^2} \left(-2 a^3+18 a^2 b x^3 \log (x)+6 a b^2 x^6+b^3 x^9\right)}{6 x^3 \left(a+b x^3\right)}","\frac{a b^2 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{3 a^2 b \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{b^3 x^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{6 \left(a+b x^3\right)}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left(a+b x^3\right)}",1,"(Sqrt[(a + b*x^3)^2]*(-2*a^3 + 6*a*b^2*x^6 + b^3*x^9 + 18*a^2*b*x^3*Log[x]))/(6*x^3*(a + b*x^3))","A",1
37,1,61,165,0.0150521,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}}{x^5} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^5,x]","\frac{\sqrt{\left(a+b x^3\right)^2} \left(-5 a^3-60 a^2 b x^3+30 a b^2 x^6+4 b^3 x^9\right)}{20 x^4 \left(a+b x^3\right)}","-\frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left(a+b x^3\right)}+\frac{3 a b^2 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left(a+b x^3\right)}+\frac{b^3 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left(a+b x^3\right)}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left(a+b x^3\right)}",1,"(Sqrt[(a + b*x^3)^2]*(-5*a^3 - 60*a^2*b*x^3 + 30*a*b^2*x^6 + 4*b^3*x^9))/(20*x^4*(a + b*x^3))","A",1
38,1,61,163,0.0174096,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}}{x^6} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^6,x]","\frac{\sqrt{\left(a+b x^3\right)^2} \left(-4 a^3-30 a^2 b x^3+60 a b^2 x^6+5 b^3 x^9\right)}{20 x^5 \left(a+b x^3\right)}","\frac{3 a b^2 x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left(a+b x^3\right)}+\frac{b^3 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left(a+b x^3\right)}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left(a+b x^3\right)}",1,"(Sqrt[(a + b*x^3)^2]*(-4*a^3 - 30*a^2*b*x^3 + 60*a*b^2*x^6 + 5*b^3*x^9))/(20*x^5*(a + b*x^3))","A",1
39,1,61,162,0.0152497,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}}{x^7} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^7,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(a^3+6 a^2 b x^3-18 a b^2 x^6 \log (x)-2 b^3 x^9\right)}{6 x^6 \left(a+b x^3\right)}","-\frac{a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^3 \left(a+b x^3\right)}+\frac{3 a b^2 \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{b^3 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left(a+b x^3\right)}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left(a+b x^3\right)}",1,"-1/6*(Sqrt[(a + b*x^3)^2]*(a^3 + 6*a^2*b*x^3 - 2*b^3*x^9 - 18*a*b^2*x^6*Log[x]))/(x^6*(a + b*x^3))","A",1
40,1,61,165,0.0140859,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}}{x^8} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^8,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(4 a^3+21 a^2 b x^3+84 a b^2 x^6-14 b^3 x^9\right)}{28 x^7 \left(a+b x^3\right)}","-\frac{3 a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left(a+b x^3\right)}-\frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left(a+b x^3\right)}+\frac{b^3 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left(a+b x^3\right)}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left(a+b x^3\right)}",1,"-1/28*(Sqrt[(a + b*x^3)^2]*(4*a^3 + 21*a^2*b*x^3 + 84*a*b^2*x^6 - 14*b^3*x^9))/(x^7*(a + b*x^3))","A",1
41,1,61,162,0.0137269,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}}{x^9} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^9,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(5 a^3+24 a^2 b x^3+60 a b^2 x^6-40 b^3 x^9\right)}{40 x^8 \left(a+b x^3\right)}","-\frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left(a+b x^3\right)}-\frac{3 a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left(a+b x^3\right)}+\frac{b^3 x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left(a+b x^3\right)}",1,"-1/40*(Sqrt[(a + b*x^3)^2]*(5*a^3 + 24*a^2*b*x^3 + 60*a*b^2*x^6 - 40*b^3*x^9))/(x^8*(a + b*x^3))","A",1
42,1,63,161,0.0207473,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}}{x^{10}} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^10,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(a \left(2 a^2+9 a b x^3+18 b^2 x^6\right)-18 b^3 x^9 \log (x)\right)}{18 x^9 \left(a+b x^3\right)}","-\frac{a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^6 \left(a+b x^3\right)}-\frac{a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^3 \left(a+b x^3\right)}+\frac{b^3 \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left(a+b x^3\right)}",1,"-1/18*(Sqrt[(a + b*x^3)^2]*(a*(2*a^2 + 9*a*b*x^3 + 18*b^2*x^6) - 18*b^3*x^9*Log[x]))/(x^9*(a + b*x^3))","A",1
43,1,61,165,0.0126229,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}}{x^{11}} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^11,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(14 a^3+60 a^2 b x^3+105 a b^2 x^6+140 b^3 x^9\right)}{140 x^{10} \left(a+b x^3\right)}","-\frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left(a+b x^3\right)}-\frac{3 a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left(a+b x^3\right)}-\frac{b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left(a+b x^3\right)}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{10 x^{10} \left(a+b x^3\right)}",1,"-1/140*(Sqrt[(a + b*x^3)^2]*(14*a^3 + 60*a^2*b*x^3 + 105*a*b^2*x^6 + 140*b^3*x^9))/(x^10*(a + b*x^3))","A",1
44,1,61,167,0.0158634,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}}{x^{12}} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^12,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(40 a^3+165 a^2 b x^3+264 a b^2 x^6+220 b^3 x^9\right)}{440 x^{11} \left(a+b x^3\right)}","-\frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left(a+b x^3\right)}-\frac{3 a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left(a+b x^3\right)}-\frac{b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left(a+b x^3\right)}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left(a+b x^3\right)}",1,"-1/440*(Sqrt[(a + b*x^3)^2]*(40*a^3 + 165*a^2*b*x^3 + 264*a*b^2*x^6 + 220*b^3*x^9))/(x^11*(a + b*x^3))","A",1
45,1,59,41,0.013324,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}}{x^{13}} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^13,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(a^3+4 a^2 b x^3+6 a b^2 x^6+4 b^3 x^9\right)}{12 x^{12} \left(a+b x^3\right)}","-\frac{\left(a+b x^3\right)^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 a x^{12}}",1,"-1/12*(Sqrt[(a + b*x^3)^2]*(a^3 + 4*a^2*b*x^3 + 6*a*b^2*x^6 + 4*b^3*x^9))/(x^12*(a + b*x^3))","A",1
46,1,61,167,0.0132838,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}}{x^{14}} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^14,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(140 a^3+546 a^2 b x^3+780 a b^2 x^6+455 b^3 x^9\right)}{1820 x^{13} \left(a+b x^3\right)}","-\frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{10 x^{10} \left(a+b x^3\right)}-\frac{3 a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left(a+b x^3\right)}-\frac{b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left(a+b x^3\right)}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 x^{13} \left(a+b x^3\right)}",1,"-1/1820*(Sqrt[(a + b*x^3)^2]*(140*a^3 + 546*a^2*b*x^3 + 780*a*b^2*x^6 + 455*b^3*x^9))/(x^13*(a + b*x^3))","A",1
47,1,61,167,0.0161677,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}}{x^{15}} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^15,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(220 a^3+840 a^2 b x^3+1155 a b^2 x^6+616 b^3 x^9\right)}{3080 x^{14} \left(a+b x^3\right)}","-\frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left(a+b x^3\right)}-\frac{3 a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left(a+b x^3\right)}-\frac{b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left(a+b x^3\right)}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 x^{14} \left(a+b x^3\right)}",1,"-1/3080*(Sqrt[(a + b*x^3)^2]*(220*a^3 + 840*a^2*b*x^3 + 1155*a*b^2*x^6 + 616*b^3*x^9))/(x^14*(a + b*x^3))","A",1
48,1,61,84,0.0136496,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}}{x^{16}} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^16,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(4 a^3+15 a^2 b x^3+20 a b^2 x^6+10 b^3 x^9\right)}{60 x^{15} \left(a+b x^3\right)}","\frac{b \left(a+b x^3\right)^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{60 a^2 x^{12}}-\frac{\left(a+b x^3\right)^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{15 a x^{15}}",1,"-1/60*(Sqrt[(a + b*x^3)^2]*(4*a^3 + 15*a^2*b*x^3 + 20*a*b^2*x^6 + 10*b^3*x^9))/(x^15*(a + b*x^3))","A",1
49,1,61,167,0.0138984,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}}{x^{17}} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^17,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(455 a^3+1680 a^2 b x^3+2184 a b^2 x^6+1040 b^3 x^9\right)}{7280 x^{16} \left(a+b x^3\right)}","-\frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 x^{13} \left(a+b x^3\right)}-\frac{3 a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{10 x^{10} \left(a+b x^3\right)}-\frac{b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left(a+b x^3\right)}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{16 x^{16} \left(a+b x^3\right)}",1,"-1/7280*(Sqrt[(a + b*x^3)^2]*(455*a^3 + 1680*a^2*b*x^3 + 2184*a*b^2*x^6 + 1040*b^3*x^9))/(x^16*(a + b*x^3))","A",1
50,1,83,255,0.0253145,"\int x^{13} \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2} \, dx","Integrate[x^13*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]","\frac{x^{14} \sqrt{\left(a+b x^3\right)^2} \left(147407 a^5+606970 a^4 b x^3+1031849 a^3 b^2 x^6+897260 a^2 b^3 x^9+396865 a b^4 x^{12}+71162 b^5 x^{15}\right)}{2063698 \left(a+b x^3\right)}","\frac{b^5 x^{29} \sqrt{a^2+2 a b x^3+b^2 x^6}}{29 \left(a+b x^3\right)}+\frac{5 a b^4 x^{26} \sqrt{a^2+2 a b x^3+b^2 x^6}}{26 \left(a+b x^3\right)}+\frac{10 a^2 b^3 x^{23} \sqrt{a^2+2 a b x^3+b^2 x^6}}{23 \left(a+b x^3\right)}+\frac{a^5 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 \left(a+b x^3\right)}+\frac{5 a^4 b x^{17} \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 \left(a+b x^3\right)}+\frac{a^3 b^2 x^{20} \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left(a+b x^3\right)}",1,"(x^14*Sqrt[(a + b*x^3)^2]*(147407*a^5 + 606970*a^4*b*x^3 + 1031849*a^3*b^2*x^6 + 897260*a^2*b^3*x^9 + 396865*a*b^4*x^12 + 71162*b^5*x^15))/(2063698*(a + b*x^3))","A",1
51,1,83,255,0.0202895,"\int x^{12} \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2} \, dx","Integrate[x^12*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]","\frac{x^{13} \sqrt{\left(a+b x^3\right)^2} \left(117040 a^5+475475 a^4 b x^3+800800 a^3 b^2 x^6+691600 a^2 b^3 x^9+304304 a b^4 x^{12}+54340 b^5 x^{15}\right)}{1521520 \left(a+b x^3\right)}","\frac{b^5 x^{28} \sqrt{a^2+2 a b x^3+b^2 x^6}}{28 \left(a+b x^3\right)}+\frac{a b^4 x^{25} \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left(a+b x^3\right)}+\frac{5 a^2 b^3 x^{22} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left(a+b x^3\right)}+\frac{a^5 x^{13} \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 \left(a+b x^3\right)}+\frac{5 a^4 b x^{16} \sqrt{a^2+2 a b x^3+b^2 x^6}}{16 \left(a+b x^3\right)}+\frac{10 a^3 b^2 x^{19} \sqrt{a^2+2 a b x^3+b^2 x^6}}{19 \left(a+b x^3\right)}",1,"(x^13*Sqrt[(a + b*x^3)^2]*(117040*a^5 + 475475*a^4*b*x^3 + 800800*a^3*b^2*x^6 + 691600*a^2*b^3*x^9 + 304304*a*b^4*x^12 + 54340*b^5*x^15))/(1521520*(a + b*x^3))","A",1
52,1,83,160,0.0209647,"\int x^{11} \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2} \, dx","Integrate[x^11*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]","\frac{x^{12} \sqrt{\left(a+b x^3\right)^2} \left(126 a^5+504 a^4 b x^3+840 a^3 b^2 x^6+720 a^2 b^3 x^9+315 a b^4 x^{12}+56 b^5 x^{15}\right)}{1512 \left(a+b x^3\right)}","\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \left(a+b x^3\right)^8}{27 b^4}-\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6} \left(a+b x^3\right)^7}{8 b^4}+\frac{a^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left(a+b x^3\right)^6}{7 b^4}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6} \left(a+b x^3\right)^5}{18 b^4}",1,"(x^12*Sqrt[(a + b*x^3)^2]*(126*a^5 + 504*a^4*b*x^3 + 840*a^3*b^2*x^6 + 720*a^2*b^3*x^9 + 315*a*b^4*x^12 + 56*b^5*x^15))/(1512*(a + b*x^3))","A",1
53,1,83,255,0.023895,"\int x^{10} \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2} \, dx","Integrate[x^10*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]","\frac{x^{11} \sqrt{\left(a+b x^3\right)^2} \left(71162 a^5+279565 a^4 b x^3+460460 a^3 b^2 x^6+391391 a^2 b^3 x^9+170170 a b^4 x^{12}+30107 b^5 x^{15}\right)}{782782 \left(a+b x^3\right)}","\frac{b^5 x^{26} \sqrt{a^2+2 a b x^3+b^2 x^6}}{26 \left(a+b x^3\right)}+\frac{5 a b^4 x^{23} \sqrt{a^2+2 a b x^3+b^2 x^6}}{23 \left(a+b x^3\right)}+\frac{a^2 b^3 x^{20} \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left(a+b x^3\right)}+\frac{a^5 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left(a+b x^3\right)}+\frac{5 a^4 b x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 \left(a+b x^3\right)}+\frac{10 a^3 b^2 x^{17} \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 \left(a+b x^3\right)}",1,"(x^11*Sqrt[(a + b*x^3)^2]*(71162*a^5 + 279565*a^4*b*x^3 + 460460*a^3*b^2*x^6 + 391391*a^2*b^3*x^9 + 170170*a*b^4*x^12 + 30107*b^5*x^15))/(782782*(a + b*x^3))","A",1
54,1,83,255,0.0198407,"\int x^9 \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2} \, dx","Integrate[x^9*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]","\frac{x^{10} \sqrt{\left(a+b x^3\right)^2} \left(54340 a^5+209000 a^4 b x^3+339625 a^3 b^2 x^6+286000 a^2 b^3 x^9+123500 a b^4 x^{12}+21736 b^5 x^{15}\right)}{543400 \left(a+b x^3\right)}","\frac{b^5 x^{25} \sqrt{a^2+2 a b x^3+b^2 x^6}}{25 \left(a+b x^3\right)}+\frac{5 a b^4 x^{22} \sqrt{a^2+2 a b x^3+b^2 x^6}}{22 \left(a+b x^3\right)}+\frac{10 a^2 b^3 x^{19} \sqrt{a^2+2 a b x^3+b^2 x^6}}{19 \left(a+b x^3\right)}+\frac{a^5 x^{10} \sqrt{a^2+2 a b x^3+b^2 x^6}}{10 \left(a+b x^3\right)}+\frac{5 a^4 b x^{13} \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 \left(a+b x^3\right)}+\frac{5 a^3 b^2 x^{16} \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left(a+b x^3\right)}",1,"(x^10*Sqrt[(a + b*x^3)^2]*(54340*a^5 + 209000*a^4*b*x^3 + 339625*a^3*b^2*x^6 + 286000*a^2*b^3*x^9 + 123500*a*b^4*x^12 + 21736*b^5*x^15))/(543400*(a + b*x^3))","A",1
55,1,83,119,0.0209307,"\int x^8 \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2} \, dx","Integrate[x^8*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]","\frac{x^9 \sqrt{\left(a+b x^3\right)^2} \left(56 a^5+210 a^4 b x^3+336 a^3 b^2 x^6+280 a^2 b^3 x^9+120 a b^4 x^{12}+21 b^5 x^{15}\right)}{504 \left(a+b x^3\right)}","\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \left(a+b x^3\right)^7}{24 b^3}-\frac{2 a \sqrt{a^2+2 a b x^3+b^2 x^6} \left(a+b x^3\right)^6}{21 b^3}+\frac{a^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left(a+b x^3\right)^5}{18 b^3}",1,"(x^9*Sqrt[(a + b*x^3)^2]*(56*a^5 + 210*a^4*b*x^3 + 336*a^3*b^2*x^6 + 280*a^2*b^3*x^9 + 120*a*b^4*x^12 + 21*b^5*x^15))/(504*(a + b*x^3))","A",1
56,1,83,255,0.0208054,"\int x^7 \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2} \, dx","Integrate[x^7*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]","\frac{x^8 \sqrt{\left(a+b x^3\right)^2} \left(30107 a^5+109480 a^4 b x^3+172040 a^3 b^2 x^6+141680 a^2 b^3 x^9+60214 a b^4 x^{12}+10472 b^5 x^{15}\right)}{240856 \left(a+b x^3\right)}","\frac{b^5 x^{23} \sqrt{a^2+2 a b x^3+b^2 x^6}}{23 \left(a+b x^3\right)}+\frac{a b^4 x^{20} \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left(a+b x^3\right)}+\frac{10 a^2 b^3 x^{17} \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 \left(a+b x^3\right)}+\frac{a^5 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left(a+b x^3\right)}+\frac{5 a^4 b x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left(a+b x^3\right)}+\frac{5 a^3 b^2 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left(a+b x^3\right)}",1,"(x^8*Sqrt[(a + b*x^3)^2]*(30107*a^5 + 109480*a^4*b*x^3 + 172040*a^3*b^2*x^6 + 141680*a^2*b^3*x^9 + 60214*a*b^4*x^12 + 10472*b^5*x^15))/(240856*(a + b*x^3))","A",1
57,1,83,255,0.0204089,"\int x^6 \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2} \, dx","Integrate[x^6*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]","\frac{x^7 \sqrt{\left(a+b x^3\right)^2} \left(21736 a^5+76076 a^4 b x^3+117040 a^3 b^2 x^6+95095 a^2 b^3 x^9+40040 a b^4 x^{12}+6916 b^5 x^{15}\right)}{152152 \left(a+b x^3\right)}","\frac{b^5 x^{22} \sqrt{a^2+2 a b x^3+b^2 x^6}}{22 \left(a+b x^3\right)}+\frac{5 a b^4 x^{19} \sqrt{a^2+2 a b x^3+b^2 x^6}}{19 \left(a+b x^3\right)}+\frac{5 a^2 b^3 x^{16} \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left(a+b x^3\right)}+\frac{a^5 x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left(a+b x^3\right)}+\frac{a^4 b x^{10} \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left(a+b x^3\right)}+\frac{10 a^3 b^2 x^{13} \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 \left(a+b x^3\right)}",1,"(x^7*Sqrt[(a + b*x^3)^2]*(21736*a^5 + 76076*a^4*b*x^3 + 117040*a^3*b^2*x^6 + 95095*a^2*b^3*x^9 + 40040*a*b^4*x^12 + 6916*b^5*x^15))/(152152*(a + b*x^3))","A",1
58,1,83,78,0.0208819,"\int x^5 \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2} \, dx","Integrate[x^5*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]","\frac{x^6 \sqrt{\left(a+b x^3\right)^2} \left(21 a^5+70 a^4 b x^3+105 a^3 b^2 x^6+84 a^2 b^3 x^9+35 a b^4 x^{12}+6 b^5 x^{15}\right)}{126 \left(a+b x^3\right)}","\frac{\left(a+b x^3\right)^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{21 b^2}-\frac{a \left(a+b x^3\right)^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{18 b^2}",1,"(x^6*Sqrt[(a + b*x^3)^2]*(21*a^5 + 70*a^4*b*x^3 + 105*a^3*b^2*x^6 + 84*a^2*b^3*x^9 + 35*a*b^4*x^12 + 6*b^5*x^15))/(126*(a + b*x^3))","A",1
59,1,83,255,0.0212623,"\int x^4 \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2} \, dx","Integrate[x^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]","\frac{x^5 \sqrt{\left(a+b x^3\right)^2} \left(10472 a^5+32725 a^4 b x^3+47600 a^3 b^2 x^6+37400 a^2 b^3 x^9+15400 a b^4 x^{12}+2618 b^5 x^{15}\right)}{52360 \left(a+b x^3\right)}","\frac{b^5 x^{20} \sqrt{a^2+2 a b x^3+b^2 x^6}}{20 \left(a+b x^3\right)}+\frac{5 a b^4 x^{17} \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 \left(a+b x^3\right)}+\frac{5 a^2 b^3 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left(a+b x^3\right)}+\frac{a^5 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left(a+b x^3\right)}+\frac{5 a^4 b x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left(a+b x^3\right)}+\frac{10 a^3 b^2 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left(a+b x^3\right)}",1,"(x^5*Sqrt[(a + b*x^3)^2]*(10472*a^5 + 32725*a^4*b*x^3 + 47600*a^3*b^2*x^6 + 37400*a^2*b^3*x^9 + 15400*a*b^4*x^12 + 2618*b^5*x^15))/(52360*(a + b*x^3))","A",1
60,1,83,252,0.0197945,"\int x^3 \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2} \, dx","Integrate[x^3*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]","\frac{x^4 \sqrt{\left(a+b x^3\right)^2} \left(6916 a^5+19760 a^4 b x^3+27664 a^3 b^2 x^6+21280 a^2 b^3 x^9+8645 a b^4 x^{12}+1456 b^5 x^{15}\right)}{27664 \left(a+b x^3\right)}","\frac{b^5 x^{19} \sqrt{a^2+2 a b x^3+b^2 x^6}}{19 \left(a+b x^3\right)}+\frac{5 a b^4 x^{16} \sqrt{a^2+2 a b x^3+b^2 x^6}}{16 \left(a+b x^3\right)}+\frac{10 a^2 b^3 x^{13} \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 \left(a+b x^3\right)}+\frac{a^5 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left(a+b x^3\right)}+\frac{5 a^4 b x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left(a+b x^3\right)}+\frac{a^3 b^2 x^{10} \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}",1,"(x^4*Sqrt[(a + b*x^3)^2]*(6916*a^5 + 19760*a^4*b*x^3 + 27664*a^3*b^2*x^6 + 21280*a^2*b^3*x^9 + 8645*a*b^4*x^12 + 1456*b^5*x^15))/(27664*(a + b*x^3))","A",1
61,1,82,36,0.0209942,"\int x^2 \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2} \, dx","Integrate[x^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]","\frac{x^3 \sqrt{\left(a+b x^3\right)^2} \left(6 a^5+15 a^4 b x^3+20 a^3 b^2 x^6+15 a^2 b^3 x^9+6 a b^4 x^{12}+b^5 x^{15}\right)}{18 \left(a+b x^3\right)}","\frac{\left(a+b x^3\right) \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{18 b}",1,"(x^3*Sqrt[(a + b*x^3)^2]*(6*a^5 + 15*a^4*b*x^3 + 20*a^3*b^2*x^6 + 15*a^2*b^3*x^9 + 6*a*b^4*x^12 + b^5*x^15))/(18*(a + b*x^3))","B",1
62,1,83,252,0.0203407,"\int x \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2} \, dx","Integrate[x*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]","\frac{x^2 \sqrt{\left(a+b x^3\right)^2} \left(2618 a^5+5236 a^4 b x^3+6545 a^3 b^2 x^6+4760 a^2 b^3 x^9+1870 a b^4 x^{12}+308 b^5 x^{15}\right)}{5236 \left(a+b x^3\right)}","\frac{b^5 x^{17} \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 \left(a+b x^3\right)}+\frac{5 a b^4 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 \left(a+b x^3\right)}+\frac{10 a^2 b^3 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left(a+b x^3\right)}+\frac{a^5 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left(a+b x^3\right)}+\frac{a^4 b x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{5 a^3 b^2 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left(a+b x^3\right)}",1,"(x^2*Sqrt[(a + b*x^3)^2]*(2618*a^5 + 5236*a^4*b*x^3 + 6545*a^3*b^2*x^6 + 4760*a^2*b^3*x^9 + 1870*a*b^4*x^12 + 308*b^5*x^15))/(5236*(a + b*x^3))","A",1
63,1,81,247,0.0187154,"\int \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]","\frac{x \sqrt{\left(a+b x^3\right)^2} \left(1456 a^5+1820 a^4 b x^3+2080 a^3 b^2 x^6+1456 a^2 b^3 x^9+560 a b^4 x^{12}+91 b^5 x^{15}\right)}{1456 \left(a+b x^3\right)}","\frac{b^5 x^{16} \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{16 \left(a+b x^3\right)^5}+\frac{5 a b^4 x^{13} \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{13 \left(a+b x^3\right)^5}+\frac{a^2 b^3 x^{10} \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{\left(a+b x^3\right)^5}+\frac{a^5 x \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{\left(a+b x^3\right)^5}+\frac{5 a^4 b x^4 \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{4 \left(a+b x^3\right)^5}+\frac{10 a^3 b^2 x^7 \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{7 \left(a+b x^3\right)^5}",1,"(x*Sqrt[(a + b*x^3)^2]*(1456*a^5 + 1820*a^4*b*x^3 + 2080*a^3*b^2*x^6 + 1456*a^2*b^3*x^9 + 560*a*b^4*x^12 + 91*b^5*x^15))/(1456*(a + b*x^3))","A",1
64,1,82,251,0.0246972,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{x} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x,x]","\frac{\sqrt{\left(a+b x^3\right)^2} \left(180 a^5 \log (x)+b x^3 \left(300 a^4+300 a^3 b x^3+200 a^2 b^2 x^6+75 a b^3 x^9+12 b^4 x^{12}\right)\right)}{180 \left(a+b x^3\right)}","\frac{b^5 x^{15} \sqrt{a^2+2 a b x^3+b^2 x^6}}{15 \left(a+b x^3\right)}+\frac{5 a b^4 x^{12} \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 \left(a+b x^3\right)}+\frac{10 a^2 b^3 x^9 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 \left(a+b x^3\right)}+\frac{a^5 \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{5 a^4 b x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left(a+b x^3\right)}+\frac{5 a^3 b^2 x^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left(a+b x^3\right)}",1,"(Sqrt[(a + b*x^3)^2]*(b*x^3*(300*a^4 + 300*a^3*b*x^3 + 200*a^2*b^2*x^6 + 75*a*b^3*x^9 + 12*b^4*x^12) + 180*a^5*Log[x]))/(180*(a + b*x^3))","A",1
65,1,83,251,0.0205278,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{x^2} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^2,x]","\frac{\sqrt{\left(a+b x^3\right)^2} \left(-308 a^5+770 a^4 b x^3+616 a^3 b^2 x^6+385 a^2 b^3 x^9+140 a b^4 x^{12}+22 b^5 x^{15}\right)}{308 x \left(a+b x^3\right)}","\frac{b^5 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 \left(a+b x^3\right)}+\frac{5 a b^4 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left(a+b x^3\right)}+\frac{5 a^2 b^3 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left(a+b x^3\right)}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left(a+b x^3\right)}+\frac{5 a^4 b x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left(a+b x^3\right)}+\frac{2 a^3 b^2 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}",1,"(Sqrt[(a + b*x^3)^2]*(-308*a^5 + 770*a^4*b*x^3 + 616*a^3*b^2*x^6 + 385*a^2*b^3*x^9 + 140*a*b^4*x^12 + 22*b^5*x^15))/(308*x*(a + b*x^3))","A",1
66,1,83,251,0.021277,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{x^3} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^3,x]","\frac{\sqrt{\left(a+b x^3\right)^2} \left(-91 a^5+910 a^4 b x^3+455 a^3 b^2 x^6+260 a^2 b^3 x^9+91 a b^4 x^{12}+14 b^5 x^{15}\right)}{182 x^2 \left(a+b x^3\right)}","\frac{b^5 x^{13} \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 \left(a+b x^3\right)}+\frac{a b^4 x^{10} \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left(a+b x^3\right)}+\frac{10 a^2 b^3 x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left(a+b x^3\right)}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left(a+b x^3\right)}+\frac{5 a^4 b x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{5 a^3 b^2 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left(a+b x^3\right)}",1,"(Sqrt[(a + b*x^3)^2]*(-91*a^5 + 910*a^4*b*x^3 + 455*a^3*b^2*x^6 + 260*a^2*b^3*x^9 + 91*a*b^4*x^12 + 14*b^5*x^15))/(182*x^2*(a + b*x^3))","A",1
67,1,85,252,0.0255962,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{x^4} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^4,x]","\frac{\sqrt{\left(a+b x^3\right)^2} \left(-12 a^5+180 a^4 b x^3 \log (x)+120 a^3 b^2 x^6+60 a^2 b^3 x^9+20 a b^4 x^{12}+3 b^5 x^{15}\right)}{36 x^3 \left(a+b x^3\right)}","\frac{b^5 x^{12} \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 \left(a+b x^3\right)}+\frac{5 a b^4 x^9 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 \left(a+b x^3\right)}+\frac{5 a^2 b^3 x^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left(a+b x^3\right)}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left(a+b x^3\right)}+\frac{5 a^4 b \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{10 a^3 b^2 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left(a+b x^3\right)}",1,"(Sqrt[(a + b*x^3)^2]*(-12*a^5 + 120*a^3*b^2*x^6 + 60*a^2*b^3*x^9 + 20*a*b^4*x^12 + 3*b^5*x^15 + 180*a^4*b*x^3*Log[x]))/(36*x^3*(a + b*x^3))","A",1
68,1,83,249,0.0232641,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{x^5} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^5,x]","\frac{\sqrt{\left(a+b x^3\right)^2} \left(-22 a^5-440 a^4 b x^3+440 a^3 b^2 x^6+176 a^2 b^3 x^9+55 a b^4 x^{12}+8 b^5 x^{15}\right)}{88 x^4 \left(a+b x^3\right)}","\frac{b^5 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left(a+b x^3\right)}+\frac{5 a b^4 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left(a+b x^3\right)}+\frac{2 a^2 b^3 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left(a+b x^3\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left(a+b x^3\right)}+\frac{5 a^3 b^2 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}",1,"(Sqrt[(a + b*x^3)^2]*(-22*a^5 - 440*a^4*b*x^3 + 440*a^3*b^2*x^6 + 176*a^2*b^3*x^9 + 55*a*b^4*x^12 + 8*b^5*x^15))/(88*x^4*(a + b*x^3))","A",1
69,1,83,251,0.0199235,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{x^6} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^6,x]","\frac{\sqrt{\left(a+b x^3\right)^2} \left(-14 a^5-175 a^4 b x^3+700 a^3 b^2 x^6+175 a^2 b^3 x^9+50 a b^4 x^{12}+7 b^5 x^{15}\right)}{70 x^5 \left(a+b x^3\right)}","\frac{b^5 x^{10} \sqrt{a^2+2 a b x^3+b^2 x^6}}{10 \left(a+b x^3\right)}+\frac{5 a b^4 x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left(a+b x^3\right)}+\frac{5 a^2 b^3 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left(a+b x^3\right)}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left(a+b x^3\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left(a+b x^3\right)}+\frac{10 a^3 b^2 x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}",1,"(Sqrt[(a + b*x^3)^2]*(-14*a^5 - 175*a^4*b*x^3 + 700*a^3*b^2*x^6 + 175*a^2*b^3*x^9 + 50*a*b^4*x^12 + 7*b^5*x^15))/(70*x^5*(a + b*x^3))","A",1
70,1,85,252,0.023619,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{x^7} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^7,x]","\frac{\sqrt{\left(a+b x^3\right)^2} \left(-3 a^5-30 a^4 b x^3+180 a^3 b^2 x^6 \log (x)+60 a^2 b^3 x^9+15 a b^4 x^{12}+2 b^5 x^{15}\right)}{18 x^6 \left(a+b x^3\right)}","\frac{b^5 x^9 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 \left(a+b x^3\right)}+\frac{5 a b^4 x^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{6 \left(a+b x^3\right)}+\frac{10 a^2 b^3 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left(a+b x^3\right)}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left(a+b x^3\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left(a+b x^3\right)}+\frac{10 a^3 b^2 \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}",1,"(Sqrt[(a + b*x^3)^2]*(-3*a^5 - 30*a^4*b*x^3 + 60*a^2*b^3*x^9 + 15*a*b^4*x^12 + 2*b^5*x^15 + 180*a^3*b^2*x^6*Log[x]))/(18*x^6*(a + b*x^3))","A",1
71,1,83,248,0.0207833,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{x^8} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^8,x]","\frac{\sqrt{\left(a+b x^3\right)^2} \left(-8 a^5-70 a^4 b x^3-560 a^3 b^2 x^6+280 a^2 b^3 x^9+56 a b^4 x^{12}+7 b^5 x^{15}\right)}{56 x^7 \left(a+b x^3\right)}","\frac{b^5 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left(a+b x^3\right)}+\frac{a b^4 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{5 a^2 b^3 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left(a+b x^3\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left(a+b x^3\right)}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left(a+b x^3\right)}",1,"(Sqrt[(a + b*x^3)^2]*(-8*a^5 - 70*a^4*b*x^3 - 560*a^3*b^2*x^6 + 280*a^2*b^3*x^9 + 56*a*b^4*x^12 + 7*b^5*x^15))/(56*x^7*(a + b*x^3))","A",1
72,1,83,247,0.0208353,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{x^9} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^9,x]","\frac{\sqrt{\left(a+b x^3\right)^2} \left(-7 a^5-56 a^4 b x^3-280 a^3 b^2 x^6+560 a^2 b^3 x^9+70 a b^4 x^{12}+8 b^5 x^{15}\right)}{56 x^8 \left(a+b x^3\right)}","\frac{b^5 x^7 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left(a+b x^3\right)}+\frac{5 a b^4 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left(a+b x^3\right)}+\frac{10 a^2 b^3 x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left(a+b x^3\right)}-\frac{a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^5 \left(a+b x^3\right)}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^2 \left(a+b x^3\right)}",1,"(Sqrt[(a + b*x^3)^2]*(-7*a^5 - 56*a^4*b*x^3 - 280*a^3*b^2*x^6 + 560*a^2*b^3*x^9 + 70*a*b^4*x^12 + 8*b^5*x^15))/(56*x^8*(a + b*x^3))","A",1
73,1,85,252,0.0282322,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{x^{10}} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^10,x]","\frac{\sqrt{\left(a+b x^3\right)^2} \left(-2 a^5-15 a^4 b x^3-60 a^3 b^2 x^6+180 a^2 b^3 x^9 \log (x)+30 a b^4 x^{12}+3 b^5 x^{15}\right)}{18 x^9 \left(a+b x^3\right)}","\frac{b^5 x^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{6 \left(a+b x^3\right)}+\frac{5 a b^4 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left(a+b x^3\right)}+\frac{10 a^2 b^3 \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left(a+b x^3\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left(a+b x^3\right)}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left(a+b x^3\right)}",1,"(Sqrt[(a + b*x^3)^2]*(-2*a^5 - 15*a^4*b*x^3 - 60*a^3*b^2*x^6 + 30*a*b^4*x^12 + 3*b^5*x^15 + 180*a^2*b^3*x^9*Log[x]))/(18*x^9*(a + b*x^3))","A",1
74,1,83,253,0.0170371,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{x^{11}} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^11,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(7 a^5+50 a^4 b x^3+175 a^3 b^2 x^6+700 a^2 b^3 x^9-175 a b^4 x^{12}-14 b^5 x^{15}\right)}{70 x^{10} \left(a+b x^3\right)}","\frac{b^5 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left(a+b x^3\right)}+\frac{5 a b^4 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left(a+b x^3\right)}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left(a+b x^3\right)}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{10 x^{10} \left(a+b x^3\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left(a+b x^3\right)}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^4 \left(a+b x^3\right)}",1,"-1/70*(Sqrt[(a + b*x^3)^2]*(7*a^5 + 50*a^4*b*x^3 + 175*a^3*b^2*x^6 + 700*a^2*b^3*x^9 - 175*a*b^4*x^12 - 14*b^5*x^15))/(x^10*(a + b*x^3))","A",1
75,1,83,247,0.0164702,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{x^{12}} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^12,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(8 a^5+55 a^4 b x^3+176 a^3 b^2 x^6+440 a^2 b^3 x^9-440 a b^4 x^{12}-22 b^5 x^{15}\right)}{88 x^{11} \left(a+b x^3\right)}","\frac{b^5 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left(a+b x^3\right)}+\frac{5 a b^4 x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^2 \left(a+b x^3\right)}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left(a+b x^3\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left(a+b x^3\right)}-\frac{2 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^5 \left(a+b x^3\right)}",1,"-1/88*(Sqrt[(a + b*x^3)^2]*(8*a^5 + 55*a^4*b*x^3 + 176*a^3*b^2*x^6 + 440*a^2*b^3*x^9 - 440*a*b^4*x^12 - 22*b^5*x^15))/(x^11*(a + b*x^3))","A",1
76,1,85,252,0.0192123,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{x^{13}} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^13,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(3 a^5+20 a^4 b x^3+60 a^3 b^2 x^6+120 a^2 b^3 x^9-180 a b^4 x^{12} \log (x)-12 b^5 x^{15}\right)}{36 x^{12} \left(a+b x^3\right)}","\frac{b^5 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left(a+b x^3\right)}+\frac{5 a b^4 \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left(a+b x^3\right)}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 x^{12} \left(a+b x^3\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left(a+b x^3\right)}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^6 \left(a+b x^3\right)}",1,"-1/36*(Sqrt[(a + b*x^3)^2]*(3*a^5 + 20*a^4*b*x^3 + 60*a^3*b^2*x^6 + 120*a^2*b^3*x^9 - 12*b^5*x^15 - 180*a*b^4*x^12*Log[x]))/(x^12*(a + b*x^3))","A",1
77,1,83,253,0.0173021,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{x^{14}} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^14,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(14 a^5+91 a^4 b x^3+260 a^3 b^2 x^6+455 a^2 b^3 x^9+910 a b^4 x^{12}-91 b^5 x^{15}\right)}{182 x^{13} \left(a+b x^3\right)}","\frac{b^5 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left(a+b x^3\right)}-\frac{5 a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left(a+b x^3\right)}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^4 \left(a+b x^3\right)}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 x^{13} \left(a+b x^3\right)}-\frac{a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^{10} \left(a+b x^3\right)}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left(a+b x^3\right)}",1,"-1/182*(Sqrt[(a + b*x^3)^2]*(14*a^5 + 91*a^4*b*x^3 + 260*a^3*b^2*x^6 + 455*a^2*b^3*x^9 + 910*a*b^4*x^12 - 91*b^5*x^15))/(x^13*(a + b*x^3))","A",1
78,1,83,248,0.0182906,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{x^{15}} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^15,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(22 a^5+140 a^4 b x^3+385 a^3 b^2 x^6+616 a^2 b^3 x^9+770 a b^4 x^{12}-308 b^5 x^{15}\right)}{308 x^{14} \left(a+b x^3\right)}","\frac{b^5 x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{5 a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left(a+b x^3\right)}-\frac{2 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^5 \left(a+b x^3\right)}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 x^{14} \left(a+b x^3\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left(a+b x^3\right)}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^8 \left(a+b x^3\right)}",1,"-1/308*(Sqrt[(a + b*x^3)^2]*(22*a^5 + 140*a^4*b*x^3 + 385*a^3*b^2*x^6 + 616*a^2*b^3*x^9 + 770*a*b^4*x^12 - 308*b^5*x^15))/(x^14*(a + b*x^3))","A",1
79,1,85,251,0.0275582,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{x^{16}} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^16,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(a \left(12 a^4+75 a^3 b x^3+200 a^2 b^2 x^6+300 a b^3 x^9+300 b^4 x^{12}\right)-180 b^5 x^{15} \log (x)\right)}{180 x^{15} \left(a+b x^3\right)}","\frac{b^5 \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{5 a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left(a+b x^3\right)}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^6 \left(a+b x^3\right)}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{15 x^{15} \left(a+b x^3\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 x^{12} \left(a+b x^3\right)}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left(a+b x^3\right)}",1,"-1/180*(Sqrt[(a + b*x^3)^2]*(a*(12*a^4 + 75*a^3*b*x^3 + 200*a^2*b^2*x^6 + 300*a*b^3*x^9 + 300*b^4*x^12) - 180*b^5*x^15*Log[x]))/(x^15*(a + b*x^3))","A",1
80,1,83,251,0.0171059,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{x^{17}} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^17,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(91 a^5+560 a^4 b x^3+1456 a^3 b^2 x^6+2080 a^2 b^3 x^9+1820 a b^4 x^{12}+1456 b^5 x^{15}\right)}{1456 x^{16} \left(a+b x^3\right)}","-\frac{b^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left(a+b x^3\right)}-\frac{5 a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left(a+b x^3\right)}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left(a+b x^3\right)}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{16 x^{16} \left(a+b x^3\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 x^{13} \left(a+b x^3\right)}-\frac{a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^{10} \left(a+b x^3\right)}",1,"-1/1456*(Sqrt[(a + b*x^3)^2]*(91*a^5 + 560*a^4*b*x^3 + 1456*a^3*b^2*x^6 + 2080*a^2*b^3*x^9 + 1820*a*b^4*x^12 + 1456*b^5*x^15))/(x^16*(a + b*x^3))","A",1
81,1,83,253,0.0181052,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{x^{18}} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^18,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(308 a^5+1870 a^4 b x^3+4760 a^3 b^2 x^6+6545 a^2 b^3 x^9+5236 a b^4 x^{12}+2618 b^5 x^{15}\right)}{5236 x^{17} \left(a+b x^3\right)}","-\frac{b^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left(a+b x^3\right)}-\frac{a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^5 \left(a+b x^3\right)}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^8 \left(a+b x^3\right)}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 x^{17} \left(a+b x^3\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 x^{14} \left(a+b x^3\right)}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left(a+b x^3\right)}",1,"-1/5236*(Sqrt[(a + b*x^3)^2]*(308*a^5 + 1870*a^4*b*x^3 + 4760*a^3*b^2*x^6 + 6545*a^2*b^3*x^9 + 5236*a*b^4*x^12 + 2618*b^5*x^15))/(x^17*(a + b*x^3))","A",1
82,1,81,41,0.0166796,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{x^{19}} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^19,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(a^5+6 a^4 b x^3+15 a^3 b^2 x^6+20 a^2 b^3 x^9+15 a b^4 x^{12}+6 b^5 x^{15}\right)}{18 x^{18} \left(a+b x^3\right)}","-\frac{\left(a+b x^3\right)^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{18 a x^{18}}",1,"-1/18*(Sqrt[(a + b*x^3)^2]*(a^5 + 6*a^4*b*x^3 + 15*a^3*b^2*x^6 + 20*a^2*b^3*x^9 + 15*a*b^4*x^12 + 6*b^5*x^15))/(x^18*(a + b*x^3))","A",1
83,1,83,253,0.0200814,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{x^{20}} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^20,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(1456 a^5+8645 a^4 b x^3+21280 a^3 b^2 x^6+27664 a^2 b^3 x^9+19760 a b^4 x^{12}+6916 b^5 x^{15}\right)}{27664 x^{19} \left(a+b x^3\right)}","-\frac{b^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left(a+b x^3\right)}-\frac{5 a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left(a+b x^3\right)}-\frac{a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^{10} \left(a+b x^3\right)}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{19 x^{19} \left(a+b x^3\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{16 x^{16} \left(a+b x^3\right)}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 x^{13} \left(a+b x^3\right)}",1,"-1/27664*(Sqrt[(a + b*x^3)^2]*(1456*a^5 + 8645*a^4*b*x^3 + 21280*a^3*b^2*x^6 + 27664*a^2*b^3*x^9 + 19760*a*b^4*x^12 + 6916*b^5*x^15))/(x^19*(a + b*x^3))","A",1
84,1,83,255,0.0171397,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{x^{21}} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^21,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(2618 a^5+15400 a^4 b x^3+37400 a^3 b^2 x^6+47600 a^2 b^3 x^9+32725 a b^4 x^{12}+10472 b^5 x^{15}\right)}{52360 x^{20} \left(a+b x^3\right)}","-\frac{b^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left(a+b x^3\right)}-\frac{5 a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left(a+b x^3\right)}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left(a+b x^3\right)}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{20 x^{20} \left(a+b x^3\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 x^{17} \left(a+b x^3\right)}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^{14} \left(a+b x^3\right)}",1,"-1/52360*(Sqrt[(a + b*x^3)^2]*(2618*a^5 + 15400*a^4*b*x^3 + 37400*a^3*b^2*x^6 + 47600*a^2*b^3*x^9 + 32725*a*b^4*x^12 + 10472*b^5*x^15))/(x^20*(a + b*x^3))","A",1
85,1,83,84,0.0176904,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{x^{22}} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^22,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(6 a^5+35 a^4 b x^3+84 a^3 b^2 x^6+105 a^2 b^3 x^9+70 a b^4 x^{12}+21 b^5 x^{15}\right)}{126 x^{21} \left(a+b x^3\right)}","\frac{b \left(a+b x^3\right)^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{126 a^2 x^{18}}-\frac{\left(a+b x^3\right)^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{21 a x^{21}}",1,"-1/126*(Sqrt[(a + b*x^3)^2]*(6*a^5 + 35*a^4*b*x^3 + 84*a^3*b^2*x^6 + 105*a^2*b^3*x^9 + 70*a*b^4*x^12 + 21*b^5*x^15))/(x^21*(a + b*x^3))","A",1
86,1,83,255,0.0178807,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{x^{23}} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^23,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(6916 a^5+40040 a^4 b x^3+95095 a^3 b^2 x^6+117040 a^2 b^3 x^9+76076 a b^4 x^{12}+21736 b^5 x^{15}\right)}{152152 x^{22} \left(a+b x^3\right)}","-\frac{b^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left(a+b x^3\right)}-\frac{a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^{10} \left(a+b x^3\right)}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{13 x^{13} \left(a+b x^3\right)}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{22 x^{22} \left(a+b x^3\right)}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{19 x^{19} \left(a+b x^3\right)}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^{16} \left(a+b x^3\right)}",1,"-1/152152*(Sqrt[(a + b*x^3)^2]*(6916*a^5 + 40040*a^4*b*x^3 + 95095*a^3*b^2*x^6 + 117040*a^2*b^3*x^9 + 76076*a*b^4*x^12 + 21736*b^5*x^15))/(x^22*(a + b*x^3))","A",1
87,1,83,255,0.0177558,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{x^{24}} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^24,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(10472 a^5+60214 a^4 b x^3+141680 a^3 b^2 x^6+172040 a^2 b^3 x^9+109480 a b^4 x^{12}+30107 b^5 x^{15}\right)}{240856 x^{23} \left(a+b x^3\right)}","-\frac{b^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left(a+b x^3\right)}-\frac{5 a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left(a+b x^3\right)}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^{14} \left(a+b x^3\right)}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{23 x^{23} \left(a+b x^3\right)}-\frac{a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^{20} \left(a+b x^3\right)}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 x^{17} \left(a+b x^3\right)}",1,"-1/240856*(Sqrt[(a + b*x^3)^2]*(10472*a^5 + 60214*a^4*b*x^3 + 141680*a^3*b^2*x^6 + 172040*a^2*b^3*x^9 + 109480*a*b^4*x^12 + 30107*b^5*x^15))/(x^23*(a + b*x^3))","A",1
88,1,83,128,0.0196596,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}{x^{25}} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^25,x]","-\frac{\sqrt{\left(a+b x^3\right)^2} \left(21 a^5+120 a^4 b x^3+280 a^3 b^2 x^6+336 a^2 b^3 x^9+210 a b^4 x^{12}+56 b^5 x^{15}\right)}{504 x^{24} \left(a+b x^3\right)}","-\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \left(a+b x^3\right)^5}{24 a x^{24}}+\frac{b \sqrt{a^2+2 a b x^3+b^2 x^6} \left(a+b x^3\right)^5}{84 a^2 x^{21}}-\frac{b^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left(a+b x^3\right)^5}{504 a^3 x^{18}}",1,"-1/504*(Sqrt[(a + b*x^3)^2]*(21*a^5 + 120*a^4*b*x^3 + 280*a^3*b^2*x^6 + 336*a^2*b^3*x^9 + 210*a*b^4*x^12 + 56*b^5*x^15))/(x^24*(a + b*x^3))","A",1
89,1,131,240,0.0557612,"\int \frac{x^4}{\sqrt{a^2+2 a b x^3+b^2 x^6}} \, dx","Integrate[x^4/Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]","\frac{\left(a+b x^3\right) \left(-a^{2/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+2 a^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)+2 \sqrt{3} a^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)+3 b^{2/3} x^2\right)}{6 b^{5/3} \sqrt{\left(a+b x^3\right)^2}}","\frac{x^2 \left(a+b x^3\right)}{2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{a^{2/3} \left(a+b x^3\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 b^{5/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{a^{2/3} \left(a+b x^3\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} b^{5/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{a^{2/3} \left(a+b x^3\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 b^{5/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}",1,"((a + b*x^3)*(3*b^(2/3)*x^2 + 2*Sqrt[3]*a^(2/3)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*a^(2/3)*Log[a^(1/3) + b^(1/3)*x] - a^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]))/(6*b^(5/3)*Sqrt[(a + b*x^3)^2])","A",1
90,1,128,235,0.0316241,"\int \frac{x^3}{\sqrt{a^2+2 a b x^3+b^2 x^6}} \, dx","Integrate[x^3/Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]","\frac{\left(a+b x^3\right) \left(\sqrt[3]{a} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-2 \sqrt[3]{a} \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)+2 \sqrt{3} \sqrt[3]{a} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)+6 \sqrt[3]{b} x\right)}{6 b^{4/3} \sqrt{\left(a+b x^3\right)^2}}","\frac{x \left(a+b x^3\right)}{b \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\sqrt[3]{a} \left(a+b x^3\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 b^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\sqrt[3]{a} \left(a+b x^3\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} b^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\sqrt[3]{a} \left(a+b x^3\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 b^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}",1,"((a + b*x^3)*(6*b^(1/3)*x + 2*Sqrt[3]*a^(1/3)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 2*a^(1/3)*Log[a^(1/3) + b^(1/3)*x] + a^(1/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]))/(6*b^(4/3)*Sqrt[(a + b*x^3)^2])","A",1
91,1,35,44,0.0083505,"\int \frac{x^2}{\sqrt{a^2+2 a b x^3+b^2 x^6}} \, dx","Integrate[x^2/Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]","\frac{\left(a+b x^3\right) \log \left(a+b x^3\right)}{3 b \sqrt{\left(a+b x^3\right)^2}}","\frac{\left(a+b x^3\right) \log \left(a+b x^3\right)}{3 b \sqrt{a^2+2 a b x^3+b^2 x^6}}",1,"((a + b*x^3)*Log[a + b*x^3])/(3*b*Sqrt[(a + b*x^3)^2])","A",1
92,1,109,202,0.0249167,"\int \frac{x}{\sqrt{a^2+2 a b x^3+b^2 x^6}} \, dx","Integrate[x/Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]","\frac{\left(a+b x^3\right) \left(\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)\right)}{6 \sqrt[3]{a} b^{2/3} \sqrt{\left(a+b x^3\right)^2}}","-\frac{\left(a+b x^3\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 \sqrt[3]{a} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left(a+b x^3\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} \sqrt[3]{a} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left(a+b x^3\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 \sqrt[3]{a} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}",1,"((a + b*x^3)*(-2*Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 2*Log[a^(1/3) + b^(1/3)*x] + Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]))/(6*a^(1/3)*b^(2/3)*Sqrt[(a + b*x^3)^2])","A",1
93,1,109,202,0.0225335,"\int \frac{1}{\sqrt{a^2+2 a b x^3+b^2 x^6}} \, dx","Integrate[1/Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]","-\frac{\left(a+b x^3\right) \left(\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)\right)}{6 a^{2/3} \sqrt[3]{b} \sqrt{\left(a+b x^3\right)^2}}","\frac{\left(a+b x^3\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 a^{2/3} \sqrt[3]{b} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left(a+b x^3\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{2/3} \sqrt[3]{b} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left(a+b x^3\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 a^{2/3} \sqrt[3]{b} \sqrt{a^2+2 a b x^3+b^2 x^6}}",1,"-1/6*((a + b*x^3)*(2*Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 2*Log[a^(1/3) + b^(1/3)*x] + Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]))/(a^(2/3)*b^(1/3)*Sqrt[(a + b*x^3)^2])","A",1
94,1,42,80,0.0115811,"\int \frac{1}{x \sqrt{a^2+2 a b x^3+b^2 x^6}} \, dx","Integrate[1/(x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]),x]","\frac{\left(a+b x^3\right) \left(3 \log (x)-\log \left(a+b x^3\right)\right)}{3 a \sqrt{\left(a+b x^3\right)^2}}","\frac{\log (x) \left(a+b x^3\right)}{a \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left(a+b x^3\right) \log \left(a+b x^3\right)}{3 a \sqrt{a^2+2 a b x^3+b^2 x^6}}",1,"((a + b*x^3)*(3*Log[x] - Log[a + b*x^3]))/(3*a*Sqrt[(a + b*x^3)^2])","A",1
95,1,133,238,0.0314478,"\int \frac{1}{x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}} \, dx","Integrate[1/(x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]),x]","-\frac{\left(a+b x^3\right) \left(\sqrt[3]{b} x \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-2 \sqrt[3]{b} x \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-2 \sqrt{3} \sqrt[3]{b} x \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)+6 \sqrt[3]{a}\right)}{6 a^{4/3} x \sqrt{\left(a+b x^3\right)^2}}","-\frac{a+b x^3}{a x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\sqrt[3]{b} \left(a+b x^3\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 a^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\sqrt[3]{b} \left(a+b x^3\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\sqrt[3]{b} \left(a+b x^3\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 a^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}",1,"-1/6*((a + b*x^3)*(6*a^(1/3) - 2*Sqrt[3]*b^(1/3)*x*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 2*b^(1/3)*x*Log[a^(1/3) + b^(1/3)*x] + b^(1/3)*x*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]))/(a^(4/3)*x*Sqrt[(a + b*x^3)^2])","A",1
96,1,140,243,0.0335218,"\int \frac{1}{x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}} \, dx","Integrate[1/(x^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]),x]","-\frac{\left(a+b x^3\right) \left(-b^{2/3} x^2 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+3 a^{2/3}+2 b^{2/3} x^2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-2 \sqrt{3} b^{2/3} x^2 \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)\right)}{6 a^{5/3} x^2 \sqrt{\left(a+b x^3\right)^2}}","\frac{-a-b x^3}{2 a x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{b^{2/3} \left(a+b x^3\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{3 a^{5/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{b^{2/3} \left(a+b x^3\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{5/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{b^{2/3} \left(a+b x^3\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{6 a^{5/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}",1,"-1/6*((a + b*x^3)*(3*a^(2/3) - 2*Sqrt[3]*b^(2/3)*x^2*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*b^(2/3)*x^2*Log[a^(1/3) + b^(1/3)*x] - b^(2/3)*x^2*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]))/(a^(5/3)*x^2*Sqrt[(a + b*x^3)^2])","A",1
97,1,54,125,0.0157483,"\int \frac{1}{x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}} \, dx","Integrate[1/(x^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]),x]","-\frac{\left(a+b x^3\right) \left(-b x^3 \log \left(a+b x^3\right)+a+3 b x^3 \log (x)\right)}{3 a^2 x^3 \sqrt{\left(a+b x^3\right)^2}}","\frac{-a-b x^3}{3 a x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{b \log (x) \left(a+b x^3\right)}{a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{b \left(a+b x^3\right) \log \left(a+b x^3\right)}{3 a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}",1,"-1/3*((a + b*x^3)*(a + 3*b*x^3*Log[x] - b*x^3*Log[a + b*x^3]))/(a^2*x^3*Sqrt[(a + b*x^3)^2])","A",1
98,1,235,280,0.0753169,"\int \frac{x^4}{\left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}} \, dx","Integrate[x^4/(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2),x]","\frac{-3 a^{4/3} b^{2/3} x^2+2 a b x^3 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+b^2 x^6 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+a^2 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+6 \sqrt[3]{a} b^{5/3} x^5-2 \left(a+b x^3\right)^2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-2 \sqrt{3} \left(a+b x^3\right)^2 \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{54 a^{4/3} b^{5/3} \left(a+b x^3\right) \sqrt{\left(a+b x^3\right)^2}}","\frac{x^2}{9 a b \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{x^2}{6 b \left(a+b x^3\right) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left(a+b x^3\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{27 a^{4/3} b^{5/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left(a+b x^3\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} a^{4/3} b^{5/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left(a+b x^3\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{54 a^{4/3} b^{5/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}",1,"(-3*a^(4/3)*b^(2/3)*x^2 + 6*a^(1/3)*b^(5/3)*x^5 - 2*Sqrt[3]*(a + b*x^3)^2*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 2*(a + b*x^3)^2*Log[a^(1/3) + b^(1/3)*x] + a^2*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + 2*a*b*x^3*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + b^2*x^6*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(4/3)*b^(5/3)*(a + b*x^3)*Sqrt[(a + b*x^3)^2])","A",1
99,1,235,276,0.0655695,"\int \frac{x^3}{\left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}} \, dx","Integrate[x^3/(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2),x]","\frac{3 a^{2/3} b^{4/3} x^4-2 a b x^3 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-b^2 x^6 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-6 a^{5/3} \sqrt[3]{b} x-a^2 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+2 \left(a+b x^3\right)^2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-2 \sqrt{3} \left(a+b x^3\right)^2 \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{54 a^{5/3} b^{4/3} \left(a+b x^3\right) \sqrt{\left(a+b x^3\right)^2}}","\frac{x}{18 a b \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{x}{6 b \left(a+b x^3\right) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left(a+b x^3\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{27 a^{5/3} b^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left(a+b x^3\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} a^{5/3} b^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left(a+b x^3\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{54 a^{5/3} b^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}",1,"(-6*a^(5/3)*b^(1/3)*x + 3*a^(2/3)*b^(4/3)*x^4 - 2*Sqrt[3]*(a + b*x^3)^2*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*(a + b*x^3)^2*Log[a^(1/3) + b^(1/3)*x] - a^2*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 2*a*b*x^3*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - b^2*x^6*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(5/3)*b^(4/3)*(a + b*x^3)*Sqrt[(a + b*x^3)^2])","A",1
100,1,27,38,0.0107895,"\int \frac{x^2}{\left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}} \, dx","Integrate[x^2/(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2),x]","-\frac{a+b x^3}{6 b \left(\left(a+b x^3\right)^2\right)^{3/2}}","-\frac{1}{6 b \left(a+b x^3\right) \sqrt{a^2+2 a b x^3+b^2 x^6}}",1,"-1/6*(a + b*x^3)/(b*((a + b*x^3)^2)^(3/2))","A",1
101,1,237,277,0.0689866,"\int \frac{x}{\left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}} \, dx","Integrate[x/(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2),x]","\frac{21 a^{4/3} b^{2/3} x^2+4 a b x^3 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+2 b^2 x^6 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+2 a^2 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+12 \sqrt[3]{a} b^{5/3} x^5-4 \left(a+b x^3\right)^2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-4 \sqrt{3} \left(a+b x^3\right)^2 \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{54 a^{7/3} b^{2/3} \left(a+b x^3\right) \sqrt{\left(a+b x^3\right)^2}}","\frac{x^2}{6 a \left(a+b x^3\right) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{2 x^2}{9 a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{2 \left(a+b x^3\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{27 a^{7/3} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{2 \left(a+b x^3\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} a^{7/3} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left(a+b x^3\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{27 a^{7/3} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}",1,"(21*a^(4/3)*b^(2/3)*x^2 + 12*a^(1/3)*b^(5/3)*x^5 - 4*Sqrt[3]*(a + b*x^3)^2*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 4*(a + b*x^3)^2*Log[a^(1/3) + b^(1/3)*x] + 2*a^2*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + 4*a*b*x^3*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + 2*b^2*x^6*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(7/3)*b^(2/3)*(a + b*x^3)*Sqrt[(a + b*x^3)^2])","A",1
102,1,235,286,0.0675754,"\int \frac{1}{\left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(-3/2),x]","\frac{15 a^{2/3} b^{4/3} x^4-10 a b x^3 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-5 b^2 x^6 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+24 a^{5/3} \sqrt[3]{b} x-5 a^2 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+10 \left(a+b x^3\right)^2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-10 \sqrt{3} \left(a+b x^3\right)^2 \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{54 a^{8/3} \sqrt[3]{b} \left(a+b x^3\right) \sqrt{\left(a+b x^3\right)^2}}","\frac{5 x \left(a+b x^3\right)^2}{18 a^2 \left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}}+\frac{x \left(a+b x^3\right)}{6 a \left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}}+\frac{5 \left(a+b x^3\right)^3 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{27 a^{8/3} \sqrt[3]{b} \left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}}-\frac{5 \left(a+b x^3\right)^3 \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} a^{8/3} \sqrt[3]{b} \left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}}-\frac{5 \left(a+b x^3\right)^3 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{54 a^{8/3} \sqrt[3]{b} \left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}}",1,"(24*a^(5/3)*b^(1/3)*x + 15*a^(2/3)*b^(4/3)*x^4 - 10*Sqrt[3]*(a + b*x^3)^2*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 10*(a + b*x^3)^2*Log[a^(1/3) + b^(1/3)*x] - 5*a^2*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 10*a*b*x^3*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 5*b^2*x^6*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(8/3)*b^(1/3)*(a + b*x^3)*Sqrt[(a + b*x^3)^2])","A",1
103,1,74,147,0.0299187,"\int \frac{1}{x \left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}} \, dx","Integrate[1/(x*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)),x]","\frac{a \left(3 a+2 b x^3\right)+6 \log (x) \left(a+b x^3\right)^2-2 \left(a+b x^3\right)^2 \log \left(a+b x^3\right)}{6 a^3 \left(a+b x^3\right) \sqrt{\left(a+b x^3\right)^2}}","\frac{1}{6 a \left(a+b x^3\right) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{3 a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\log (x) \left(a+b x^3\right)}{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left(a+b x^3\right) \log \left(a+b x^3\right)}{3 a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}",1,"(a*(3*a + 2*b*x^3) + 6*(a + b*x^3)^2*Log[x] - 2*(a + b*x^3)^2*Log[a + b*x^3])/(6*a^3*(a + b*x^3)*Sqrt[(a + b*x^3)^2])","A",1
104,1,260,316,0.0844383,"\int \frac{1}{x^2 \left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}} \, dx","Integrate[1/(x^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)),x]","\frac{-14 b^{7/3} x^7 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-28 a b^{4/3} x^4 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-147 a^{4/3} b x^3-54 a^{7/3}-14 a^2 \sqrt[3]{b} x \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-84 \sqrt[3]{a} b^2 x^6+28 \sqrt[3]{b} x \left(a+b x^3\right)^2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)+28 \sqrt{3} \sqrt[3]{b} x \left(a+b x^3\right)^2 \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{54 a^{10/3} x \left(a+b x^3\right) \sqrt{\left(a+b x^3\right)^2}}","\frac{7}{18 a^2 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{6 a x \sqrt{a^2+2 a b x^3+b^2 x^6} \left(a+b x^3\right)}+\frac{14 \sqrt[3]{b} \left(a+b x^3\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{27 a^{10/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{14 \sqrt[3]{b} \left(a+b x^3\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} a^{10/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{7 \sqrt[3]{b} \left(a+b x^3\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{27 a^{10/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{14 \left(a+b x^3\right)}{9 a^3 x \sqrt{a^2+2 a b x^3+b^2 x^6}}",1,"(-54*a^(7/3) - 147*a^(4/3)*b*x^3 - 84*a^(1/3)*b^2*x^6 + 28*Sqrt[3]*b^(1/3)*x*(a + b*x^3)^2*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 28*b^(1/3)*x*(a + b*x^3)^2*Log[a^(1/3) + b^(1/3)*x] - 14*a^2*b^(1/3)*x*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 28*a*b^(4/3)*x^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 14*b^(7/3)*x^7*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(10/3)*x*(a + b*x^3)*Sqrt[(a + b*x^3)^2])","A",1
105,1,266,316,0.0883536,"\int \frac{1}{x^3 \left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}} \, dx","Integrate[1/(x^3*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)),x]","\frac{20 b^{8/3} x^8 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+40 a b^{5/3} x^5 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-60 a^{2/3} b^2 x^6-96 a^{5/3} b x^3-27 a^{8/3}+20 a^2 b^{2/3} x^2 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-40 b^{2/3} x^2 \left(a+b x^3\right)^2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)+40 \sqrt{3} b^{2/3} x^2 \left(a+b x^3\right)^2 \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a}}}{\sqrt{3}}\right)}{54 a^{11/3} x^2 \left(a+b x^3\right) \sqrt{\left(a+b x^3\right)^2}}","\frac{4}{9 a^2 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{6 a x^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left(a+b x^3\right)}-\frac{20 b^{2/3} \left(a+b x^3\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{27 a^{11/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{20 b^{2/3} \left(a+b x^3\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{9 \sqrt{3} a^{11/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{10 b^{2/3} \left(a+b x^3\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{27 a^{11/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{10 \left(a+b x^3\right)}{9 a^3 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}",1,"(-27*a^(8/3) - 96*a^(5/3)*b*x^3 - 60*a^(2/3)*b^2*x^6 + 40*Sqrt[3]*b^(2/3)*x^2*(a + b*x^3)^2*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 40*b^(2/3)*x^2*(a + b*x^3)^2*Log[a^(1/3) + b^(1/3)*x] + 20*a^2*b^(2/3)*x^2*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + 40*a*b^(5/3)*x^5*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + 20*b^(8/3)*x^8*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(11/3)*x^2*(a + b*x^3)*Sqrt[(a + b*x^3)^2])","A",1
106,1,97,188,0.0329094,"\int \frac{1}{x^4 \left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}} \, dx","Integrate[1/(x^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)),x]","\frac{-a \left(2 a^2+9 a b x^3+6 b^2 x^6\right)-18 b x^3 \log (x) \left(a+b x^3\right)^2+6 b x^3 \left(a+b x^3\right)^2 \log \left(a+b x^3\right)}{6 a^4 x^3 \left(a+b x^3\right) \sqrt{\left(a+b x^3\right)^2}}","-\frac{b}{6 a^2 \left(a+b x^3\right) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{3 b \log (x) \left(a+b x^3\right)}{a^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{b \left(a+b x^3\right) \log \left(a+b x^3\right)}{a^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{2 b}{3 a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{a+b x^3}{3 a^3 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}",1,"(-(a*(2*a^2 + 9*a*b*x^3 + 6*b^2*x^6)) - 18*b*x^3*(a + b*x^3)^2*Log[x] + 6*b*x^3*(a + b*x^3)^2*Log[a + b*x^3])/(6*a^4*x^3*(a + b*x^3)*Sqrt[(a + b*x^3)^2])","A",1
107,1,218,359,0.1279132,"\int \frac{x^6}{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}} \, dx","Integrate[x^6/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]","\frac{\left(a+b x^3\right) \left(-\frac{10 \left(a+b x^3\right)^4 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{a^{8/3}}+\frac{20 \left(a+b x^3\right)^4 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{a^{8/3}}+\frac{20 \sqrt{3} \left(a+b x^3\right)^4 \tan ^{-1}\left(\frac{2 \sqrt[3]{b} x-\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{a^{8/3}}+\frac{30 \sqrt[3]{b} x \left(a+b x^3\right)^3}{a^2}+\frac{18 \sqrt[3]{b} x \left(a+b x^3\right)^2}{a}-351 \sqrt[3]{b} x \left(a+b x^3\right)+243 a \sqrt[3]{b} x\right)}{2916 b^{7/3} \left(\left(a+b x^3\right)^2\right)^{5/2}}","\frac{x}{162 a b^2 \left(a+b x^3\right) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{5 x}{486 a^2 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{x}{27 b^2 \left(a+b x^3\right)^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{x^4}{12 b \left(a+b x^3\right)^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{5 \left(a+b x^3\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{729 a^{8/3} b^{7/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{5 \left(a+b x^3\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{243 \sqrt{3} a^{8/3} b^{7/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{5 \left(a+b x^3\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{1458 a^{8/3} b^{7/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}",1,"((a + b*x^3)*(243*a*b^(1/3)*x - 351*b^(1/3)*x*(a + b*x^3) + (18*b^(1/3)*x*(a + b*x^3)^2)/a + (30*b^(1/3)*x*(a + b*x^3)^3)/a^2 + (20*Sqrt[3]*(a + b*x^3)^4*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/a^(8/3) + (20*(a + b*x^3)^4*Log[a^(1/3) + b^(1/3)*x])/a^(8/3) - (10*(a + b*x^3)^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(8/3)))/(2916*b^(7/3)*((a + b*x^3)^2)^(5/2))","A",1
108,1,39,78,0.0156485,"\int \frac{x^5}{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}} \, dx","Integrate[x^5/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]","\frac{-a-4 b x^3}{36 b^2 \left(a+b x^3\right)^3 \sqrt{\left(a+b x^3\right)^2}}","\frac{a}{12 b^2 \left(a+b x^3\right)^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{1}{9 b^2 \left(a+b x^3\right)^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}",1,"(-a - 4*b*x^3)/(36*b^2*(a + b*x^3)^3*Sqrt[(a + b*x^3)^2])","A",1
109,1,229,368,0.1352947,"\int \frac{x^4}{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}} \, dx","Integrate[x^4/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]","\frac{\left(a+b x^3\right) \left(-243 a^{10/3} b^{2/3} x^2+63 a^{4/3} b^{2/3} x^2 \left(a+b x^3\right)^2+54 a^{7/3} b^{2/3} x^2 \left(a+b x^3\right)+14 \left(a+b x^3\right)^4 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+84 \sqrt[3]{a} b^{2/3} x^2 \left(a+b x^3\right)^3-28 \left(a+b x^3\right)^4 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)+28 \sqrt{3} \left(a+b x^3\right)^4 \tan ^{-1}\left(\frac{2 \sqrt[3]{b} x-\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)\right)}{2916 a^{10/3} b^{5/3} \left(\left(a+b x^3\right)^2\right)^{5/2}}","\frac{7 x^2}{324 a^2 b \left(a+b x^3\right) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{x^2}{54 a b \left(a+b x^3\right)^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{x^2}{12 b \left(a+b x^3\right)^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{7 \left(a+b x^3\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{729 a^{10/3} b^{5/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{7 \left(a+b x^3\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{243 \sqrt{3} a^{10/3} b^{5/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{7 \left(a+b x^3\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{1458 a^{10/3} b^{5/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{7 x^2}{243 a^3 b \sqrt{a^2+2 a b x^3+b^2 x^6}}",1,"((a + b*x^3)*(-243*a^(10/3)*b^(2/3)*x^2 + 54*a^(7/3)*b^(2/3)*x^2*(a + b*x^3) + 63*a^(4/3)*b^(2/3)*x^2*(a + b*x^3)^2 + 84*a^(1/3)*b^(2/3)*x^2*(a + b*x^3)^3 + 28*Sqrt[3]*(a + b*x^3)^4*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))] - 28*(a + b*x^3)^4*Log[a^(1/3) + b^(1/3)*x] + 14*(a + b*x^3)^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]))/(2916*a^(10/3)*b^(5/3)*((a + b*x^3)^2)^(5/2))","A",1
110,1,221,360,0.1315032,"\int \frac{x^3}{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}} \, dx","Integrate[x^3/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]","\frac{\left(a+b x^3\right) \left(-20 \left(a+b x^3\right)^4 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)+60 a^{2/3} \sqrt[3]{b} x \left(a+b x^3\right)^3+36 a^{5/3} \sqrt[3]{b} x \left(a+b x^3\right)^2+27 a^{8/3} \sqrt[3]{b} x \left(a+b x^3\right)-243 a^{11/3} \sqrt[3]{b} x+40 \left(a+b x^3\right)^4 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)+40 \sqrt{3} \left(a+b x^3\right)^4 \tan ^{-1}\left(\frac{2 \sqrt[3]{b} x-\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)\right)}{2916 a^{11/3} b^{4/3} \left(\left(a+b x^3\right)^2\right)^{5/2}}","\frac{x}{81 a^2 b \left(a+b x^3\right) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{x}{108 a b \left(a+b x^3\right)^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{x}{12 b \left(a+b x^3\right)^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{10 \left(a+b x^3\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{729 a^{11/3} b^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{10 \left(a+b x^3\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{243 \sqrt{3} a^{11/3} b^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{5 \left(a+b x^3\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{729 a^{11/3} b^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{5 x}{243 a^3 b \sqrt{a^2+2 a b x^3+b^2 x^6}}",1,"((a + b*x^3)*(-243*a^(11/3)*b^(1/3)*x + 27*a^(8/3)*b^(1/3)*x*(a + b*x^3) + 36*a^(5/3)*b^(1/3)*x*(a + b*x^3)^2 + 60*a^(2/3)*b^(1/3)*x*(a + b*x^3)^3 + 40*Sqrt[3]*(a + b*x^3)^4*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))] + 40*(a + b*x^3)^4*Log[a^(1/3) + b^(1/3)*x] - 20*(a + b*x^3)^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]))/(2916*a^(11/3)*b^(4/3)*((a + b*x^3)^2)^(5/2))","A",1
111,1,27,38,0.0099919,"\int \frac{x^2}{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}} \, dx","Integrate[x^2/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]","-\frac{a+b x^3}{12 b \left(\left(a+b x^3\right)^2\right)^{5/2}}","-\frac{1}{12 b \left(a+b x^3\right) \left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}}",1,"-1/12*(a + b*x^3)/(b*((a + b*x^3)^2)^(5/2))","A",1
112,1,219,359,0.1203149,"\int \frac{x}{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}} \, dx","Integrate[x/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]","\frac{\left(a+b x^3\right) \left(\frac{70 \left(a+b x^3\right)^4 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{b^{2/3}}+315 a^{4/3} x^2 \left(a+b x^3\right)^2+270 a^{7/3} x^2 \left(a+b x^3\right)+243 a^{10/3} x^2-\frac{140 \left(a+b x^3\right)^4 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{b^{2/3}}+\frac{140 \sqrt{3} \left(a+b x^3\right)^4 \tan ^{-1}\left(\frac{2 \sqrt[3]{b} x-\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{b^{2/3}}+420 \sqrt[3]{a} x^2 \left(a+b x^3\right)^3\right)}{2916 a^{13/3} \left(\left(a+b x^3\right)^2\right)^{5/2}}","\frac{5 x^2}{54 a^2 \left(a+b x^3\right)^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{x^2}{12 a \left(a+b x^3\right)^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{35 \left(a+b x^3\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{729 a^{13/3} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{35 \left(a+b x^3\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{243 \sqrt{3} a^{13/3} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{35 \left(a+b x^3\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{1458 a^{13/3} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{35 x^2}{243 a^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{35 x^2}{324 a^3 \left(a+b x^3\right) \sqrt{a^2+2 a b x^3+b^2 x^6}}",1,"((a + b*x^3)*(243*a^(10/3)*x^2 + 270*a^(7/3)*x^2*(a + b*x^3) + 315*a^(4/3)*x^2*(a + b*x^3)^2 + 420*a^(1/3)*x^2*(a + b*x^3)^3 + (140*Sqrt[3]*(a + b*x^3)^4*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/b^(2/3) - (140*(a + b*x^3)^4*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + (70*(a + b*x^3)^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3)))/(2916*a^(13/3)*((a + b*x^3)^2)^(5/2))","A",1
113,1,211,364,0.1114749,"\int \frac{1}{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(-5/2),x]","\frac{\left(a+b x^3\right) \left(-\frac{220 \left(a+b x^3\right)^4 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{\sqrt[3]{b}}+660 a^{2/3} x \left(a+b x^3\right)^3+396 a^{5/3} x \left(a+b x^3\right)^2+297 a^{8/3} x \left(a+b x^3\right)+243 a^{11/3} x+\frac{440 \left(a+b x^3\right)^4 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt[3]{b}}+\frac{440 \sqrt{3} \left(a+b x^3\right)^4 \tan ^{-1}\left(\frac{2 \sqrt[3]{b} x-\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt[3]{b}}\right)}{2916 a^{14/3} \left(\left(a+b x^3\right)^2\right)^{5/2}}","\frac{11 x \left(a+b x^3\right)^2}{108 a^2 \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}+\frac{x \left(a+b x^3\right)}{12 a \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}+\frac{110 \left(a+b x^3\right)^5 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{729 a^{14/3} \sqrt[3]{b} \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}-\frac{110 \left(a+b x^3\right)^5 \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{243 \sqrt{3} a^{14/3} \sqrt[3]{b} \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}-\frac{55 \left(a+b x^3\right)^5 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{729 a^{14/3} \sqrt[3]{b} \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}+\frac{55 x \left(a+b x^3\right)^4}{243 a^4 \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}+\frac{11 x \left(a+b x^3\right)^3}{81 a^3 \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}}",1,"((a + b*x^3)*(243*a^(11/3)*x + 297*a^(8/3)*x*(a + b*x^3) + 396*a^(5/3)*x*(a + b*x^3)^2 + 660*a^(2/3)*x*(a + b*x^3)^3 + (440*Sqrt[3]*(a + b*x^3)^4*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/b^(1/3) + (440*(a + b*x^3)^4*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) - (220*(a + b*x^3)^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(1/3)))/(2916*a^(14/3)*((a + b*x^3)^2)^(5/2))","A",1
114,1,96,223,0.0444835,"\int \frac{1}{x \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}} \, dx","Integrate[1/(x*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)),x]","\frac{a \left(25 a^3+52 a^2 b x^3+42 a b^2 x^6+12 b^3 x^9\right)+36 \log (x) \left(a+b x^3\right)^4-12 \left(a+b x^3\right)^4 \log \left(a+b x^3\right)}{36 a^5 \left(a+b x^3\right)^3 \sqrt{\left(a+b x^3\right)^2}}","\frac{1}{9 a^2 \left(a+b x^3\right)^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{12 a \left(a+b x^3\right)^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\log (x) \left(a+b x^3\right)}{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left(a+b x^3\right) \log \left(a+b x^3\right)}{3 a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{3 a^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{6 a^3 \left(a+b x^3\right) \sqrt{a^2+2 a b x^3+b^2 x^6}}",1,"(a*(25*a^3 + 52*a^2*b*x^3 + 42*a*b^2*x^6 + 12*b^3*x^9) + 36*(a + b*x^3)^4*Log[x] - 12*(a + b*x^3)^4*Log[a + b*x^3])/(36*a^5*(a + b*x^3)^3*Sqrt[(a + b*x^3)^2])","A",1
115,1,242,398,0.1288197,"\int \frac{1}{x^2 \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}} \, dx","Integrate[1/(x^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)),x]","\frac{\left(a+b x^3\right) \left(-910 \sqrt[3]{b} \left(a+b x^3\right)^4 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-243 a^{10/3} b x^2-1179 a^{4/3} b x^2 \left(a+b x^3\right)^2-594 a^{7/3} b x^2 \left(a+b x^3\right)-\frac{2916 \sqrt[3]{a} \left(a+b x^3\right)^4}{x}+1820 \sqrt[3]{b} \left(a+b x^3\right)^4 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-1820 \sqrt{3} \sqrt[3]{b} \left(a+b x^3\right)^4 \tan ^{-1}\left(\frac{2 \sqrt[3]{b} x-\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)-2544 \sqrt[3]{a} b x^2 \left(a+b x^3\right)^3\right)}{2916 a^{16/3} \left(\left(a+b x^3\right)^2\right)^{5/2}}","\frac{13}{108 a^2 x \sqrt{a^2+2 a b x^3+b^2 x^6} \left(a+b x^3\right)^2}+\frac{1}{12 a x \sqrt{a^2+2 a b x^3+b^2 x^6} \left(a+b x^3\right)^3}+\frac{455 \sqrt[3]{b} \left(a+b x^3\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{729 a^{16/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{455 \sqrt[3]{b} \left(a+b x^3\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{243 \sqrt{3} a^{16/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{455 \sqrt[3]{b} \left(a+b x^3\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{1458 a^{16/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{455 \left(a+b x^3\right)}{243 a^5 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{455}{972 a^4 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{65}{324 a^3 x \sqrt{a^2+2 a b x^3+b^2 x^6} \left(a+b x^3\right)}",1,"((a + b*x^3)*(-243*a^(10/3)*b*x^2 - 594*a^(7/3)*b*x^2*(a + b*x^3) - 1179*a^(4/3)*b*x^2*(a + b*x^3)^2 - 2544*a^(1/3)*b*x^2*(a + b*x^3)^3 - (2916*a^(1/3)*(a + b*x^3)^4)/x - 1820*Sqrt[3]*b^(1/3)*(a + b*x^3)^4*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))] + 1820*b^(1/3)*(a + b*x^3)^4*Log[a^(1/3) + b^(1/3)*x] - 910*b^(1/3)*(a + b*x^3)^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]))/(2916*a^(16/3)*((a + b*x^3)^2)^(5/2))","A",1
116,1,234,398,0.135437,"\int \frac{1}{x^3 \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}} \, dx","Integrate[1/(x^3*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)),x]","\frac{\left(a+b x^3\right) \left(1540 b^{2/3} \left(a+b x^3\right)^4 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)-3162 a^{2/3} b x \left(a+b x^3\right)^3-1314 a^{5/3} b x \left(a+b x^3\right)^2-621 a^{8/3} b x \left(a+b x^3\right)-\frac{1458 a^{2/3} \left(a+b x^3\right)^4}{x^2}-243 a^{11/3} b x-3080 b^{2/3} \left(a+b x^3\right)^4 \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)-3080 \sqrt{3} b^{2/3} \left(a+b x^3\right)^4 \tan ^{-1}\left(\frac{2 \sqrt[3]{b} x-\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)\right)}{2916 a^{17/3} \left(\left(a+b x^3\right)^2\right)^{5/2}}","\frac{7}{54 a^2 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left(a+b x^3\right)^2}+\frac{1}{12 a x^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left(a+b x^3\right)^3}-\frac{770 b^{2/3} \left(a+b x^3\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{729 a^{17/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{770 b^{2/3} \left(a+b x^3\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} x}{\sqrt{3} \sqrt[3]{a}}\right)}{243 \sqrt{3} a^{17/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{385 b^{2/3} \left(a+b x^3\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2\right)}{729 a^{17/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{385 \left(a+b x^3\right)}{243 a^5 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{154}{243 a^4 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{77}{324 a^3 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left(a+b x^3\right)}",1,"((a + b*x^3)*(-243*a^(11/3)*b*x - 621*a^(8/3)*b*x*(a + b*x^3) - 1314*a^(5/3)*b*x*(a + b*x^3)^2 - 3162*a^(2/3)*b*x*(a + b*x^3)^3 - (1458*a^(2/3)*(a + b*x^3)^4)/x^2 - 3080*Sqrt[3]*b^(2/3)*(a + b*x^3)^4*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))] - 3080*b^(2/3)*(a + b*x^3)^4*Log[a^(1/3) + b^(1/3)*x] + 1540*b^(2/3)*(a + b*x^3)^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]))/(2916*a^(17/3)*((a + b*x^3)^2)^(5/2))","A",1
117,1,119,269,0.0506198,"\int \frac{1}{x^4 \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}} \, dx","Integrate[1/(x^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)),x]","\frac{-a \left(12 a^4+125 a^3 b x^3+260 a^2 b^2 x^6+210 a b^3 x^9+60 b^4 x^{12}\right)-180 b x^3 \log (x) \left(a+b x^3\right)^4+60 b x^3 \left(a+b x^3\right)^4 \log \left(a+b x^3\right)}{36 a^6 x^3 \left(a+b x^3\right)^3 \sqrt{\left(a+b x^3\right)^2}}","-\frac{b}{12 a^2 \left(a+b x^3\right)^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{5 b \log (x) \left(a+b x^3\right)}{a^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{5 b \left(a+b x^3\right) \log \left(a+b x^3\right)}{3 a^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{4 b}{3 a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{a+b x^3}{3 a^5 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{b}{2 a^4 \left(a+b x^3\right) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{2 b}{9 a^3 \left(a+b x^3\right)^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}",1,"(-(a*(12*a^4 + 125*a^3*b*x^3 + 260*a^2*b^2*x^6 + 210*a*b^3*x^9 + 60*b^4*x^12)) - 180*b*x^3*(a + b*x^3)^4*Log[x] + 60*b*x^3*(a + b*x^3)^4*Log[a + b*x^3])/(36*a^6*x^3*(a + b*x^3)^3*Sqrt[(a + b*x^3)^2])","A",1
118,1,111,313,0.0913967,"\int (d x)^m \left(a^2+2 a b x^3+b^2 x^6\right)^{5/2} \, dx","Integrate[(d*x)^m*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]","\frac{x \left(\left(a+b x^3\right)^2\right)^{5/2} (d x)^m \left(\frac{a^5}{m+1}+\frac{5 a^4 b x^3}{m+4}+\frac{10 a^3 b^2 x^6}{m+7}+\frac{10 a^2 b^3 x^9}{m+10}+\frac{5 a b^4 x^{12}}{m+13}+\frac{b^5 x^{15}}{m+16}\right)}{\left(a+b x^3\right)^5}","\frac{b^5 \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+16}}{d^{16} (m+16) \left(a+b x^3\right)}+\frac{5 a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+13}}{d^{13} (m+13) \left(a+b x^3\right)}+\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+10}}{d^{10} (m+10) \left(a+b x^3\right)}+\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+1}}{d (m+1) \left(a+b x^3\right)}+\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+4}}{d^4 (m+4) \left(a+b x^3\right)}+\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+7}}{d^7 (m+7) \left(a+b x^3\right)}",1,"(x*(d*x)^m*((a + b*x^3)^2)^(5/2)*(a^5/(1 + m) + (5*a^4*b*x^3)/(4 + m) + (10*a^3*b^2*x^6)/(7 + m) + (10*a^2*b^3*x^9)/(10 + m) + (5*a*b^4*x^12)/(13 + m) + (b^5*x^15)/(16 + m)))/(a + b*x^3)^5","A",1
119,1,131,205,0.064599,"\int (d x)^m \left(a^2+2 a b x^3+b^2 x^6\right)^{3/2} \, dx","Integrate[(d*x)^m*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2),x]","\frac{x \sqrt{\left(a+b x^3\right)^2} (d x)^m \left(a^3 \left(m^3+21 m^2+138 m+280\right)+3 a^2 b \left(m^3+18 m^2+87 m+70\right) x^3+3 a b^2 \left(m^3+15 m^2+54 m+40\right) x^6+b^3 \left(m^3+12 m^2+39 m+28\right) x^9\right)}{(m+1) (m+4) (m+7) (m+10) \left(a+b x^3\right)}","\frac{3 a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+7}}{d^7 (m+7) \left(a+b x^3\right)}+\frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+4}}{d^4 (m+4) \left(a+b x^3\right)}+\frac{b^3 \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+10}}{d^{10} (m+10) \left(a+b x^3\right)}+\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+1}}{d (m+1) \left(a+b x^3\right)}",1,"(x*(d*x)^m*Sqrt[(a + b*x^3)^2]*(a^3*(280 + 138*m + 21*m^2 + m^3) + 3*a^2*b*(70 + 87*m + 18*m^2 + m^3)*x^3 + 3*a*b^2*(40 + 54*m + 15*m^2 + m^3)*x^6 + b^3*(28 + 39*m + 12*m^2 + m^3)*x^9))/((1 + m)*(4 + m)*(7 + m)*(10 + m)*(a + b*x^3))","A",1
120,1,53,97,0.0231659,"\int (d x)^m \sqrt{a^2+2 a b x^3+b^2 x^6} \, dx","Integrate[(d*x)^m*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]","\frac{x \sqrt{\left(a+b x^3\right)^2} (d x)^m \left(a (m+4)+b (m+1) x^3\right)}{(m+1) (m+4) \left(a+b x^3\right)}","\frac{b \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+4}}{d^4 (m+4) \left(a+b x^3\right)}+\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+1}}{d (m+1) \left(a+b x^3\right)}",1,"(x*(d*x)^m*Sqrt[(a + b*x^3)^2]*(a*(4 + m) + b*(1 + m)*x^3))/((1 + m)*(4 + m)*(a + b*x^3))","A",1
121,1,62,73,0.0181521,"\int \frac{(d x)^m}{\sqrt{a^2+2 a b x^3+b^2 x^6}} \, dx","Integrate[(d*x)^m/Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]","\frac{x \left(a+b x^3\right) (d x)^m \, _2F_1\left(1,\frac{m+1}{3};\frac{m+1}{3}+1;-\frac{b x^3}{a}\right)}{a (m+1) \sqrt{\left(a+b x^3\right)^2}}","\frac{\left(a+b x^3\right) (d x)^{m+1} \, _2F_1\left(1,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right)}{a d (m+1) \sqrt{a^2+2 a b x^3+b^2 x^6}}",1,"(x*(d*x)^m*(a + b*x^3)*Hypergeometric2F1[1, (1 + m)/3, 1 + (1 + m)/3, -((b*x^3)/a)])/(a*(1 + m)*Sqrt[(a + b*x^3)^2])","A",1
122,1,60,73,0.0198201,"\int \frac{(d x)^m}{\left(a^2+2 a b x^3+b^2 x^6\right)^{3/2}} \, dx","Integrate[(d*x)^m/(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2),x]","\frac{x \left(a+b x^3\right) (d x)^m \, _2F_1\left(3,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right)}{a^3 (m+1) \sqrt{\left(a+b x^3\right)^2}}","\frac{\left(a+b x^3\right) (d x)^{m+1} \, _2F_1\left(3,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right)}{a^3 d (m+1) \sqrt{a^2+2 a b x^3+b^2 x^6}}",1,"(x*(d*x)^m*(a + b*x^3)*Hypergeometric2F1[3, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)])/(a^3*(1 + m)*Sqrt[(a + b*x^3)^2])","A",1
123,1,60,73,0.0184297,"\int \frac{(d x)^m}{\left(a^2+2 a b x^3+b^2 x^6\right)^{5/2}} \, dx","Integrate[(d*x)^m/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]","\frac{x \left(a+b x^3\right) (d x)^m \, _2F_1\left(5,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right)}{a^5 (m+1) \sqrt{\left(a+b x^3\right)^2}}","\frac{\left(a+b x^3\right) (d x)^{m+1} \, _2F_1\left(5,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right)}{a^5 d (m+1) \sqrt{a^2+2 a b x^3+b^2 x^6}}",1,"(x*(d*x)^m*(a + b*x^3)*Hypergeometric2F1[5, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)])/(a^5*(1 + m)*Sqrt[(a + b*x^3)^2])","A",1
124,1,66,77,0.0189035,"\int (d x)^m \left(a^2+2 a b x^3+b^2 x^6\right)^p \, dx","Integrate[(d*x)^m*(a^2 + 2*a*b*x^3 + b^2*x^6)^p,x]","\frac{x (d x)^m \left(\left(a+b x^3\right)^2\right)^p \left(\frac{b x^3}{a}+1\right)^{-2 p} \, _2F_1\left(\frac{m+1}{3},-2 p;\frac{m+1}{3}+1;-\frac{b x^3}{a}\right)}{m+1}","\frac{(d x)^{m+1} \left(\frac{b x^3}{a}+1\right)^{-2 p} \left(a^2+2 a b x^3+b^2 x^6\right)^p \, _2F_1\left(\frac{m+1}{3},-2 p;\frac{m+4}{3};-\frac{b x^3}{a}\right)}{d (m+1)}",1,"(x*(d*x)^m*((a + b*x^3)^2)^p*Hypergeometric2F1[(1 + m)/3, -2*p, 1 + (1 + m)/3, -((b*x^3)/a)])/((1 + m)*(1 + (b*x^3)/a)^(2*p))","A",1
125,1,110,172,0.0563919,"\int x^{11} \left(a^2+2 a b x^3+b^2 x^6\right)^p \, dx","Integrate[x^11*(a^2 + 2*a*b*x^3 + b^2*x^6)^p,x]","\frac{\left(a+b x^3\right) \left(\left(a+b x^3\right)^2\right)^p \left(-3 a^3+3 a^2 b (2 p+1) x^3-3 a b^2 \left(2 p^2+3 p+1\right) x^6+b^3 \left(4 p^3+12 p^2+11 p+3\right) x^9\right)}{6 b^4 (p+1) (p+2) (2 p+1) (2 p+3)}","\frac{\left(a+b x^3\right)^4 \left(a^2+2 a b x^3+b^2 x^6\right)^p}{6 b^4 (p+2)}-\frac{a \left(a+b x^3\right)^3 \left(a^2+2 a b x^3+b^2 x^6\right)^p}{b^4 (2 p+3)}+\frac{a^2 \left(a+b x^3\right)^2 \left(a^2+2 a b x^3+b^2 x^6\right)^p}{2 b^4 (p+1)}-\frac{a^3 \left(a+b x^3\right) \left(a^2+2 a b x^3+b^2 x^6\right)^p}{3 b^4 (2 p+1)}",1,"((a + b*x^3)*((a + b*x^3)^2)^p*(-3*a^3 + 3*a^2*b*(1 + 2*p)*x^3 - 3*a*b^2*(1 + 3*p + 2*p^2)*x^6 + b^3*(3 + 11*p + 12*p^2 + 4*p^3)*x^9))/(6*b^4*(1 + p)*(2 + p)*(1 + 2*p)*(3 + 2*p))","A",1
126,1,77,130,0.0326086,"\int x^8 \left(a^2+2 a b x^3+b^2 x^6\right)^p \, dx","Integrate[x^8*(a^2 + 2*a*b*x^3 + b^2*x^6)^p,x]","\frac{\left(a+b x^3\right) \left(\left(a+b x^3\right)^2\right)^p \left(a^2-a b (2 p+1) x^3+b^2 \left(2 p^2+3 p+1\right) x^6\right)}{3 b^3 (p+1) (2 p+1) (2 p+3)}","\frac{\left(a+b x^3\right)^3 \left(a^2+2 a b x^3+b^2 x^6\right)^p}{3 b^3 (2 p+3)}-\frac{a \left(a+b x^3\right)^2 \left(a^2+2 a b x^3+b^2 x^6\right)^p}{3 b^3 (p+1)}+\frac{a^2 \left(a+b x^3\right) \left(a^2+2 a b x^3+b^2 x^6\right)^p}{3 b^3 (2 p+1)}",1,"((a + b*x^3)*((a + b*x^3)^2)^p*(a^2 - a*b*(1 + 2*p)*x^3 + b^2*(1 + 3*p + 2*p^2)*x^6))/(3*b^3*(1 + p)*(1 + 2*p)*(3 + 2*p))","A",1
127,1,51,84,0.0203562,"\int x^5 \left(a^2+2 a b x^3+b^2 x^6\right)^p \, dx","Integrate[x^5*(a^2 + 2*a*b*x^3 + b^2*x^6)^p,x]","\frac{\left(a+b x^3\right) \left(\left(a+b x^3\right)^2\right)^p \left(b (2 p+1) x^3-a\right)}{6 b^2 (p+1) (2 p+1)}","\frac{\left(a+b x^3\right)^2 \left(a^2+2 a b x^3+b^2 x^6\right)^p}{6 b^2 (p+1)}-\frac{a \left(a+b x^3\right) \left(a^2+2 a b x^3+b^2 x^6\right)^p}{3 b^2 (2 p+1)}",1,"((a + b*x^3)*((a + b*x^3)^2)^p*(-a + b*(1 + 2*p)*x^3))/(6*b^2*(1 + p)*(1 + 2*p))","A",1
128,1,51,60,0.0064841,"\int x^4 \left(a^2+2 a b x^3+b^2 x^6\right)^p \, dx","Integrate[x^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^p,x]","\frac{1}{5} x^5 \left(\left(a+b x^3\right)^2\right)^p \left(\frac{b x^3}{a}+1\right)^{-2 p} \, _2F_1\left(\frac{5}{3},-2 p;\frac{8}{3};-\frac{b x^3}{a}\right)","\frac{1}{5} x^5 \left(\frac{b x^3}{a}+1\right)^{-2 p} \left(a^2+2 a b x^3+b^2 x^6\right)^p \, _2F_1\left(\frac{5}{3},-2 p;\frac{8}{3};-\frac{b x^3}{a}\right)",1,"(x^5*((a + b*x^3)^2)^p*Hypergeometric2F1[5/3, -2*p, 8/3, -((b*x^3)/a)])/(5*(1 + (b*x^3)/a)^(2*p))","A",1
129,1,51,60,0.0061597,"\int x^3 \left(a^2+2 a b x^3+b^2 x^6\right)^p \, dx","Integrate[x^3*(a^2 + 2*a*b*x^3 + b^2*x^6)^p,x]","\frac{1}{4} x^4 \left(\left(a+b x^3\right)^2\right)^p \left(\frac{b x^3}{a}+1\right)^{-2 p} \, _2F_1\left(\frac{4}{3},-2 p;\frac{7}{3};-\frac{b x^3}{a}\right)","\frac{1}{4} x^4 \left(\frac{b x^3}{a}+1\right)^{-2 p} \left(a^2+2 a b x^3+b^2 x^6\right)^p \, _2F_1\left(\frac{4}{3},-2 p;\frac{7}{3};-\frac{b x^3}{a}\right)",1,"(x^4*((a + b*x^3)^2)^p*Hypergeometric2F1[4/3, -2*p, 7/3, -((b*x^3)/a)])/(4*(1 + (b*x^3)/a)^(2*p))","A",1
130,1,32,41,0.0061317,"\int x^2 \left(a^2+2 a b x^3+b^2 x^6\right)^p \, dx","Integrate[x^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^p,x]","\frac{\left(a+b x^3\right) \left(\left(a+b x^3\right)^2\right)^p}{3 b (2 p+1)}","\frac{\left(a+b x^3\right) \left(a^2+2 a b x^3+b^2 x^6\right)^p}{3 b (2 p+1)}",1,"((a + b*x^3)*((a + b*x^3)^2)^p)/(3*b*(1 + 2*p))","A",1
131,1,51,58,0.0065816,"\int x \left(a^2+2 a b x^3+b^2 x^6\right)^p \, dx","Integrate[x*(a^2 + 2*a*b*x^3 + b^2*x^6)^p,x]","\frac{1}{2} x^2 \left(\left(a+b x^3\right)^2\right)^p \left(\frac{b x^3}{a}+1\right)^{-2 p} \, _2F_1\left(\frac{2}{3},-2 p;\frac{5}{3};-\frac{b x^3}{a}\right)","\frac{x^2 \left(a+b x^3\right) \left(a^2+2 a b x^3+b^2 x^6\right)^p \, _2F_1\left(1,2 p+\frac{5}{3};\frac{5}{3};-\frac{b x^3}{a}\right)}{2 a}",1,"(x^2*((a + b*x^3)^2)^p*Hypergeometric2F1[2/3, -2*p, 5/3, -((b*x^3)/a)])/(2*(1 + (b*x^3)/a)^(2*p))","A",1
132,1,204,53,0.1729395,"\int \left(a^2+2 a b x^3+b^2 x^6\right)^p \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^p,x]","\frac{4^{-p} \left((-1)^{2/3} \sqrt[3]{a}+\sqrt[3]{b} x\right) \left(\frac{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}\right)^{-2 p} \left(\frac{i \left(\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1\right)}{\sqrt{3}+3 i}\right)^{-2 p} \left(\left(a+b x^3\right)^2\right)^p F_1\left(2 p+1;-2 p,-2 p;2 (p+1);-\frac{i \left(\sqrt[3]{b} x+(-1)^{2/3} \sqrt[3]{a}\right)}{\sqrt{3} \sqrt[3]{a}},\frac{-\frac{2 i \sqrt[3]{b} x}{\sqrt[3]{a}}+\sqrt{3}+i}{3 i+\sqrt{3}}\right)}{\sqrt[3]{b} (2 p+1)}","\frac{x \left(a+b x^3\right) \left(a^2+2 a b x^3+b^2 x^6\right)^p \, _2F_1\left(1,2 p+\frac{4}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{a}",1,"(((-1)^(2/3)*a^(1/3) + b^(1/3)*x)*((a + b*x^3)^2)^p*AppellF1[1 + 2*p, -2*p, -2*p, 2*(1 + p), ((-I)*((-1)^(2/3)*a^(1/3) + b^(1/3)*x))/(Sqrt[3]*a^(1/3)), (I + Sqrt[3] - ((2*I)*b^(1/3)*x)/a^(1/3))/(3*I + Sqrt[3])])/(4^p*b^(1/3)*(1 + 2*p)*((a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3)))^(2*p)*((I*(1 + (b^(1/3)*x)/a^(1/3)))/(3*I + Sqrt[3]))^(2*p))","C",0
133,1,54,63,0.0108835,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^p}{x} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^p/x,x]","-\frac{\left(a+b x^3\right) \left(\left(a+b x^3\right)^2\right)^p \, _2F_1\left(1,2 p+1;2 p+2;\frac{b x^3}{a}+1\right)}{3 a (2 p+1)}","-\frac{\left(a+b x^3\right) \left(a^2+2 a b x^3+b^2 x^6\right)^p \, _2F_1\left(1,2 p+1;2 (p+1);\frac{b x^3}{a}+1\right)}{3 a (2 p+1)}",1,"-1/3*((a + b*x^3)*((a + b*x^3)^2)^p*Hypergeometric2F1[1, 1 + 2*p, 2 + 2*p, 1 + (b*x^3)/a])/(a*(1 + 2*p))","A",1
134,1,49,58,0.0068286,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^p}{x^2} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^p/x^2,x]","-\frac{\left(\left(a+b x^3\right)^2\right)^p \left(\frac{b x^3}{a}+1\right)^{-2 p} \, _2F_1\left(-\frac{1}{3},-2 p;\frac{2}{3};-\frac{b x^3}{a}\right)}{x}","-\frac{\left(\frac{b x^3}{a}+1\right)^{-2 p} \left(a^2+2 a b x^3+b^2 x^6\right)^p \, _2F_1\left(-\frac{1}{3},-2 p;\frac{2}{3};-\frac{b x^3}{a}\right)}{x}",1,"-((((a + b*x^3)^2)^p*Hypergeometric2F1[-1/3, -2*p, 2/3, -((b*x^3)/a)])/(x*(1 + (b*x^3)/a)^(2*p)))","A",1
135,1,51,60,0.006134,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^p}{x^3} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^p/x^3,x]","-\frac{\left(\left(a+b x^3\right)^2\right)^p \left(\frac{b x^3}{a}+1\right)^{-2 p} \, _2F_1\left(-\frac{2}{3},-2 p;\frac{1}{3};-\frac{b x^3}{a}\right)}{2 x^2}","-\frac{\left(\frac{b x^3}{a}+1\right)^{-2 p} \left(a^2+2 a b x^3+b^2 x^6\right)^p \, _2F_1\left(-\frac{2}{3},-2 p;\frac{1}{3};-\frac{b x^3}{a}\right)}{2 x^2}",1,"-1/2*(((a + b*x^3)^2)^p*Hypergeometric2F1[-2/3, -2*p, 1/3, -((b*x^3)/a)])/(x^2*(1 + (b*x^3)/a)^(2*p))","A",1
136,1,55,64,0.0115586,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^p}{x^4} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^p/x^4,x]","\frac{b \left(a+b x^3\right) \left(\left(a+b x^3\right)^2\right)^p \, _2F_1\left(2,2 p+1;2 p+2;\frac{b x^3}{a}+1\right)}{3 a^2 (2 p+1)}","\frac{b \left(a+b x^3\right) \left(a^2+2 a b x^3+b^2 x^6\right)^p \, _2F_1\left(2,2 p+1;2 (p+1);\frac{b x^3}{a}+1\right)}{3 a^2 (2 p+1)}",1,"(b*(a + b*x^3)*((a + b*x^3)^2)^p*Hypergeometric2F1[2, 1 + 2*p, 2 + 2*p, 1 + (b*x^3)/a])/(3*a^2*(1 + 2*p))","A",1
137,1,51,60,0.0063701,"\int \frac{\left(a^2+2 a b x^3+b^2 x^6\right)^p}{x^5} \, dx","Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^p/x^5,x]","-\frac{\left(\left(a+b x^3\right)^2\right)^p \left(\frac{b x^3}{a}+1\right)^{-2 p} \, _2F_1\left(-\frac{4}{3},-2 p;-\frac{1}{3};-\frac{b x^3}{a}\right)}{4 x^4}","-\frac{\left(\frac{b x^3}{a}+1\right)^{-2 p} \left(a^2+2 a b x^3+b^2 x^6\right)^p \, _2F_1\left(-\frac{4}{3},-2 p;-\frac{1}{3};-\frac{b x^3}{a}\right)}{4 x^4}",1,"-1/4*(((a + b*x^3)^2)^p*Hypergeometric2F1[-4/3, -2*p, -1/3, -((b*x^3)/a)])/(x^4*(1 + (b*x^3)/a)^(2*p))","A",1
138,1,78,81,0.0523753,"\int \frac{x^8}{a+b x^3+c x^6} \, dx","Integrate[x^8/(a + b*x^3 + c*x^6),x]","\frac{\frac{2 \left(b^2-2 a c\right) \tan ^{-1}\left(\frac{b+2 c x^3}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}-b \log \left(a+b x^3+c x^6\right)+2 c x^3}{6 c^2}","-\frac{\left(b^2-2 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right)}{3 c^2 \sqrt{b^2-4 a c}}-\frac{b \log \left(a+b x^3+c x^6\right)}{6 c^2}+\frac{x^3}{3 c}",1,"(2*c*x^3 + (2*(b^2 - 2*a*c)*ArcTan[(b + 2*c*x^3)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] - b*Log[a + b*x^3 + c*x^6])/(6*c^2)","A",1
139,1,62,63,0.0226781,"\int \frac{x^5}{a+b x^3+c x^6} \, dx","Integrate[x^5/(a + b*x^3 + c*x^6),x]","\frac{\log \left(a+b x^3+c x^6\right)-\frac{2 b \tan ^{-1}\left(\frac{b+2 c x^3}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}}{6 c}","\frac{b \tanh ^{-1}\left(\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right)}{3 c \sqrt{b^2-4 a c}}+\frac{\log \left(a+b x^3+c x^6\right)}{6 c}",1,"((-2*b*ArcTan[(b + 2*c*x^3)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + Log[a + b*x^3 + c*x^6])/(6*c)","A",1
140,1,42,38,0.0099631,"\int \frac{x^2}{a+b x^3+c x^6} \, dx","Integrate[x^2/(a + b*x^3 + c*x^6),x]","\frac{2 \tan ^{-1}\left(\frac{b+2 c x^3}{\sqrt{4 a c-b^2}}\right)}{3 \sqrt{4 a c-b^2}}","-\frac{2 \tanh ^{-1}\left(\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right)}{3 \sqrt{b^2-4 a c}}",1,"(2*ArcTan[(b + 2*c*x^3)/Sqrt[-b^2 + 4*a*c]])/(3*Sqrt[-b^2 + 4*a*c])","A",1
141,1,66,69,0.0233192,"\int \frac{1}{x \left(a+b x^3+c x^6\right)} \, dx","Integrate[1/(x*(a + b*x^3 + c*x^6)),x]","\frac{\log (x)}{a}-\frac{\text{RootSum}\left[\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{\text{$\#$1}^3 c \log (x-\text{$\#$1})+b \log (x-\text{$\#$1})}{2 \text{$\#$1}^3 c+b}\&\right]}{3 a}","\frac{b \tanh ^{-1}\left(\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right)}{3 a \sqrt{b^2-4 a c}}-\frac{\log \left(a+b x^3+c x^6\right)}{6 a}+\frac{\log (x)}{a}",1,"Log[x]/a - RootSum[a + b*#1^3 + c*#1^6 & , (b*Log[x - #1] + c*Log[x - #1]*#1^3)/(b + 2*c*#1^3) & ]/(3*a)","C",1
142,1,92,89,0.0294779,"\int \frac{1}{x^4 \left(a+b x^3+c x^6\right)} \, dx","Integrate[1/(x^4*(a + b*x^3 + c*x^6)),x]","\frac{\text{RootSum}\left[\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{\text{$\#$1}^3 b c \log (x-\text{$\#$1})-a c \log (x-\text{$\#$1})+b^2 \log (x-\text{$\#$1})}{2 \text{$\#$1}^3 c+b}\&\right]}{3 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{3 a x^3}","-\frac{\left(b^2-2 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right)}{3 a^2 \sqrt{b^2-4 a c}}+\frac{b \log \left(a+b x^3+c x^6\right)}{6 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{3 a x^3}",1,"-1/3*1/(a*x^3) - (b*Log[x])/a^2 + RootSum[a + b*#1^3 + c*#1^6 & , (b^2*Log[x - #1] - a*c*Log[x - #1] + b*c*Log[x - #1]*#1^3)/(b + 2*c*#1^3) & ]/(3*a^2)","C",1
143,1,70,636,0.0304591,"\int \frac{x^7}{a+b x^3+c x^6} \, dx","Integrate[x^7/(a + b*x^3 + c*x^6),x]","\frac{3 x^2-2 \text{RootSum}\left[\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{\text{$\#$1}^3 b \log (x-\text{$\#$1})+a \log (x-\text{$\#$1})}{2 \text{$\#$1}^4 c+\text{$\#$1} b}\&\right]}{6 c}","-\frac{\left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{b-\sqrt{b^2-4 a c}}+\left(b-\sqrt{b^2-4 a c}\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6\ 2^{2/3} c^{5/3} \sqrt[3]{b-\sqrt{b^2-4 a c}}}-\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{\sqrt{b^2-4 a c}+b}+\left(\sqrt{b^2-4 a c}+b\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6\ 2^{2/3} c^{5/3} \sqrt[3]{\sqrt{b^2-4 a c}+b}}+\frac{\left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \log \left(\sqrt[3]{b-\sqrt{b^2-4 a c}}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3\ 2^{2/3} c^{5/3} \sqrt[3]{b-\sqrt{b^2-4 a c}}}+\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \log \left(\sqrt[3]{\sqrt{b^2-4 a c}+b}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3\ 2^{2/3} c^{5/3} \sqrt[3]{\sqrt{b^2-4 a c}+b}}+\frac{\left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3} c^{5/3} \sqrt[3]{b-\sqrt{b^2-4 a c}}}+\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3} c^{5/3} \sqrt[3]{\sqrt{b^2-4 a c}+b}}+\frac{x^2}{2 c}",1,"(3*x^2 - 2*RootSum[a + b*#1^3 + c*#1^6 & , (a*Log[x - #1] + b*Log[x - #1]*#1^3)/(b*#1 + 2*c*#1^4) & ])/(6*c)","C",1
144,1,70,631,0.0303963,"\int \frac{x^6}{a+b x^3+c x^6} \, dx","Integrate[x^6/(a + b*x^3 + c*x^6),x]","\frac{x}{c}-\frac{\text{RootSum}\left[\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{\text{$\#$1}^3 b \log (x-\text{$\#$1})+a \log (x-\text{$\#$1})}{2 \text{$\#$1}^5 c+\text{$\#$1}^2 b}\&\right]}{3 c}","\frac{\left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{b-\sqrt{b^2-4 a c}}+\left(b-\sqrt{b^2-4 a c}\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6 \sqrt[3]{2} c^{4/3} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}+\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{\sqrt{b^2-4 a c}+b}+\left(\sqrt{b^2-4 a c}+b\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6 \sqrt[3]{2} c^{4/3} \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}-\frac{\left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \log \left(\sqrt[3]{b-\sqrt{b^2-4 a c}}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3 \sqrt[3]{2} c^{4/3} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}-\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \log \left(\sqrt[3]{\sqrt{b^2-4 a c}+b}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3 \sqrt[3]{2} c^{4/3} \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}+\frac{\left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3} c^{4/3} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}+\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3} c^{4/3} \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}+\frac{x}{c}",1,"x/c - RootSum[a + b*#1^3 + c*#1^6 & , (a*Log[x - #1] + b*Log[x - #1]*#1^3)/(b*#1^2 + 2*c*#1^5) & ]/(3*c)","C",1
145,1,44,558,0.0171338,"\int \frac{x^4}{a+b x^3+c x^6} \, dx","Integrate[x^4/(a + b*x^3 + c*x^6),x]","\frac{1}{3} \text{RootSum}\left[\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{\text{$\#$1}^2 \log (x-\text{$\#$1})}{2 \text{$\#$1}^3 c+b}\&\right]","-\frac{\left(b-\sqrt{b^2-4 a c}\right)^{2/3} \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{b-\sqrt{b^2-4 a c}}+\left(b-\sqrt{b^2-4 a c}\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}+\frac{\left(\sqrt{b^2-4 a c}+b\right)^{2/3} \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{\sqrt{b^2-4 a c}+b}+\left(\sqrt{b^2-4 a c}+b\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}+\frac{\left(b-\sqrt{b^2-4 a c}\right)^{2/3} \log \left(\sqrt[3]{b-\sqrt{b^2-4 a c}}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}-\frac{\left(\sqrt{b^2-4 a c}+b\right)^{2/3} \log \left(\sqrt[3]{\sqrt{b^2-4 a c}+b}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}+\frac{\left(b-\sqrt{b^2-4 a c}\right)^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3} c^{2/3} \sqrt{b^2-4 a c}}-\frac{\left(\sqrt{b^2-4 a c}+b\right)^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3} c^{2/3} \sqrt{b^2-4 a c}}",1,"RootSum[a + b*#1^3 + c*#1^6 & , (Log[x - #1]*#1^2)/(b + 2*c*#1^3) & ]/3","C",1
146,1,42,558,0.0157384,"\int \frac{x^3}{a+b x^3+c x^6} \, dx","Integrate[x^3/(a + b*x^3 + c*x^6),x]","\frac{1}{3} \text{RootSum}\left[\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{\text{$\#$1} \log (x-\text{$\#$1})}{2 \text{$\#$1}^3 c+b}\&\right]","\frac{\sqrt[3]{b-\sqrt{b^2-4 a c}} \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{b-\sqrt{b^2-4 a c}}+\left(b-\sqrt{b^2-4 a c}\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6 \sqrt[3]{2} \sqrt[3]{c} \sqrt{b^2-4 a c}}-\frac{\sqrt[3]{\sqrt{b^2-4 a c}+b} \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{\sqrt{b^2-4 a c}+b}+\left(\sqrt{b^2-4 a c}+b\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6 \sqrt[3]{2} \sqrt[3]{c} \sqrt{b^2-4 a c}}-\frac{\sqrt[3]{b-\sqrt{b^2-4 a c}} \log \left(\sqrt[3]{b-\sqrt{b^2-4 a c}}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3 \sqrt[3]{2} \sqrt[3]{c} \sqrt{b^2-4 a c}}+\frac{\sqrt[3]{\sqrt{b^2-4 a c}+b} \log \left(\sqrt[3]{\sqrt{b^2-4 a c}+b}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3 \sqrt[3]{2} \sqrt[3]{c} \sqrt{b^2-4 a c}}+\frac{\sqrt[3]{b-\sqrt{b^2-4 a c}} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3} \sqrt[3]{c} \sqrt{b^2-4 a c}}-\frac{\sqrt[3]{\sqrt{b^2-4 a c}+b} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3} \sqrt[3]{c} \sqrt{b^2-4 a c}}",1,"RootSum[a + b*#1^3 + c*#1^6 & , (Log[x - #1]*#1)/(b + 2*c*#1^3) & ]/3","C",1
147,1,43,558,0.0168077,"\int \frac{x}{a+b x^3+c x^6} \, dx","Integrate[x/(a + b*x^3 + c*x^6),x]","\frac{1}{3} \text{RootSum}\left[\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^4 c+\text{$\#$1} b}\&\right]","\frac{\sqrt[3]{c} \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{b-\sqrt{b^2-4 a c}}+\left(b-\sqrt{b^2-4 a c}\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{3\ 2^{2/3} \sqrt{b^2-4 a c} \sqrt[3]{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt[3]{c} \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{\sqrt{b^2-4 a c}+b}+\left(\sqrt{b^2-4 a c}+b\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{3\ 2^{2/3} \sqrt{b^2-4 a c} \sqrt[3]{\sqrt{b^2-4 a c}+b}}-\frac{\sqrt[3]{2} \sqrt[3]{c} \log \left(\sqrt[3]{b-\sqrt{b^2-4 a c}}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3 \sqrt{b^2-4 a c} \sqrt[3]{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt[3]{2} \sqrt[3]{c} \log \left(\sqrt[3]{\sqrt{b^2-4 a c}+b}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3 \sqrt{b^2-4 a c} \sqrt[3]{\sqrt{b^2-4 a c}+b}}-\frac{\sqrt[3]{2} \sqrt[3]{c} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right)}{\sqrt{3} \sqrt{b^2-4 a c} \sqrt[3]{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt[3]{2} \sqrt[3]{c} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right)}{\sqrt{3} \sqrt{b^2-4 a c} \sqrt[3]{\sqrt{b^2-4 a c}+b}}",1,"RootSum[a + b*#1^3 + c*#1^6 & , Log[x - #1]/(b*#1 + 2*c*#1^4) & ]/3","C",1
148,1,45,558,0.0171098,"\int \frac{1}{a+b x^3+c x^6} \, dx","Integrate[(a + b*x^3 + c*x^6)^(-1),x]","\frac{1}{3} \text{RootSum}\left[\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^5 c+\text{$\#$1}^2 b}\&\right]","-\frac{c^{2/3} \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{b-\sqrt{b^2-4 a c}}+\left(b-\sqrt{b^2-4 a c}\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{3 \sqrt[3]{2} \sqrt{b^2-4 a c} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}+\frac{c^{2/3} \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{\sqrt{b^2-4 a c}+b}+\left(\sqrt{b^2-4 a c}+b\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{3 \sqrt[3]{2} \sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}+\frac{2^{2/3} c^{2/3} \log \left(\sqrt[3]{b-\sqrt{b^2-4 a c}}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3 \sqrt{b^2-4 a c} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}-\frac{2^{2/3} c^{2/3} \log \left(\sqrt[3]{\sqrt{b^2-4 a c}+b}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3 \sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}-\frac{2^{2/3} c^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right)}{\sqrt{3} \sqrt{b^2-4 a c} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}+\frac{2^{2/3} c^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right)}{\sqrt{3} \sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}",1,"RootSum[a + b*#1^3 + c*#1^6 & , Log[x - #1]/(b*#1^2 + 2*c*#1^5) & ]/3","C",1
149,1,71,610,0.030846,"\int \frac{1}{x^2 \left(a+b x^3+c x^6\right)} \, dx","Integrate[1/(x^2*(a + b*x^3 + c*x^6)),x]","-\frac{\text{RootSum}\left[\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{\text{$\#$1}^3 c \log (x-\text{$\#$1})+b \log (x-\text{$\#$1})}{2 \text{$\#$1}^4 c+\text{$\#$1} b}\&\right]}{3 a}-\frac{1}{a x}","-\frac{\sqrt[3]{c} \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{b-\sqrt{b^2-4 a c}}+\left(b-\sqrt{b^2-4 a c}\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6\ 2^{2/3} a \sqrt[3]{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt[3]{c} \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{\sqrt{b^2-4 a c}+b}+\left(\sqrt{b^2-4 a c}+b\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6\ 2^{2/3} a \sqrt[3]{\sqrt{b^2-4 a c}+b}}+\frac{\sqrt[3]{c} \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \log \left(\sqrt[3]{b-\sqrt{b^2-4 a c}}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3\ 2^{2/3} a \sqrt[3]{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt[3]{c} \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \log \left(\sqrt[3]{\sqrt{b^2-4 a c}+b}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3\ 2^{2/3} a \sqrt[3]{\sqrt{b^2-4 a c}+b}}+\frac{\sqrt[3]{c} \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3} a \sqrt[3]{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt[3]{c} \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3} a \sqrt[3]{\sqrt{b^2-4 a c}+b}}-\frac{1}{a x}",1,"-(1/(a*x)) - RootSum[a + b*#1^3 + c*#1^6 & , (b*Log[x - #1] + c*Log[x - #1]*#1^3)/(b*#1 + 2*c*#1^4) & ]/(3*a)","C",1
150,1,75,612,0.0342587,"\int \frac{1}{x^3 \left(a+b x^3+c x^6\right)} \, dx","Integrate[1/(x^3*(a + b*x^3 + c*x^6)),x]","-\frac{\text{RootSum}\left[\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{\text{$\#$1}^3 c \log (x-\text{$\#$1})+b \log (x-\text{$\#$1})}{2 \text{$\#$1}^5 c+\text{$\#$1}^2 b}\&\right]}{3 a}-\frac{1}{2 a x^2}","\frac{c^{2/3} \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{b-\sqrt{b^2-4 a c}}+\left(b-\sqrt{b^2-4 a c}\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6 \sqrt[3]{2} a \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}+\frac{c^{2/3} \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{\sqrt{b^2-4 a c}+b}+\left(\sqrt{b^2-4 a c}+b\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6 \sqrt[3]{2} a \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}-\frac{c^{2/3} \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \log \left(\sqrt[3]{b-\sqrt{b^2-4 a c}}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3 \sqrt[3]{2} a \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}-\frac{c^{2/3} \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \log \left(\sqrt[3]{\sqrt{b^2-4 a c}+b}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3 \sqrt[3]{2} a \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}+\frac{c^{2/3} \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3} a \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}+\frac{c^{2/3} \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3} a \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}-\frac{1}{2 a x^2}",1,"-1/2*1/(a*x^2) - RootSum[a + b*#1^3 + c*#1^6 & , (b*Log[x - #1] + c*Log[x - #1]*#1^3)/(b*#1^2 + 2*c*#1^5) & ]/(3*a)","C",1
151,1,35,35,0.0060188,"\int \frac{x^{11}}{3+4 x^3+x^6} \, dx","Integrate[x^11/(3 + 4*x^3 + x^6),x]","\frac{x^6}{6}-\frac{4 x^3}{3}-\frac{1}{6} \log \left(x^3+1\right)+\frac{9}{2} \log \left(x^3+3\right)","\frac{x^6}{6}-\frac{4 x^3}{3}-\frac{1}{6} \log \left(x^3+1\right)+\frac{9}{2} \log \left(x^3+3\right)",1,"(-4*x^3)/3 + x^6/6 - Log[1 + x^3]/6 + (9*Log[3 + x^3])/2","A",1
152,1,28,28,0.0051723,"\int \frac{x^8}{3+4 x^3+x^6} \, dx","Integrate[x^8/(3 + 4*x^3 + x^6),x]","\frac{x^3}{3}+\frac{1}{6} \log \left(x^3+1\right)-\frac{3}{2} \log \left(x^3+3\right)","\frac{x^3}{3}+\frac{1}{6} \log \left(x^3+1\right)-\frac{3}{2} \log \left(x^3+3\right)",1,"x^3/3 + Log[1 + x^3]/6 - (3*Log[3 + x^3])/2","A",1
153,1,21,21,0.0040195,"\int \frac{x^5}{3+4 x^3+x^6} \, dx","Integrate[x^5/(3 + 4*x^3 + x^6),x]","\frac{1}{2} \log \left(x^3+3\right)-\frac{1}{6} \log \left(x^3+1\right)","\frac{1}{2} \log \left(x^3+3\right)-\frac{1}{6} \log \left(x^3+1\right)",1,"-1/6*Log[1 + x^3] + Log[3 + x^3]/2","A",1
154,1,21,10,0.0035321,"\int \frac{x^2}{3+4 x^3+x^6} \, dx","Integrate[x^2/(3 + 4*x^3 + x^6),x]","\frac{1}{6} \log \left(x^3+1\right)-\frac{1}{6} \log \left(x^3+3\right)","-\frac{1}{3} \tanh ^{-1}\left(x^3+2\right)",1,"Log[1 + x^3]/6 - Log[3 + x^3]/6","B",1
155,1,27,27,0.0057056,"\int \frac{1}{x \left(3+4 x^3+x^6\right)} \, dx","Integrate[1/(x*(3 + 4*x^3 + x^6)),x]","-\frac{1}{6} \log \left(x^3+1\right)+\frac{1}{18} \log \left(x^3+3\right)+\frac{\log (x)}{3}","-\frac{1}{6} \log \left(x^3+1\right)+\frac{1}{18} \log \left(x^3+3\right)+\frac{\log (x)}{3}",1,"Log[x]/3 - Log[1 + x^3]/6 + Log[3 + x^3]/18","A",1
156,1,34,34,0.0057892,"\int \frac{1}{x^4 \left(3+4 x^3+x^6\right)} \, dx","Integrate[1/(x^4*(3 + 4*x^3 + x^6)),x]","-\frac{1}{9 x^3}+\frac{1}{6} \log \left(x^3+1\right)-\frac{1}{54} \log \left(x^3+3\right)-\frac{4 \log (x)}{9}","-\frac{1}{9 x^3}+\frac{1}{6} \log \left(x^3+1\right)-\frac{1}{54} \log \left(x^3+3\right)-\frac{4 \log (x)}{9}",1,"-1/9*1/x^3 - (4*Log[x])/9 + Log[1 + x^3]/6 - Log[3 + x^3]/54","A",1
157,1,41,41,0.0054246,"\int \frac{1}{x^7 \left(3+4 x^3+x^6\right)} \, dx","Integrate[1/(x^7*(3 + 4*x^3 + x^6)),x]","-\frac{1}{18 x^6}+\frac{4}{27 x^3}-\frac{1}{6} \log \left(x^3+1\right)+\frac{1}{162} \log \left(x^3+3\right)+\frac{13 \log (x)}{27}","-\frac{1}{18 x^6}+\frac{4}{27 x^3}-\frac{1}{6} \log \left(x^3+1\right)+\frac{1}{162} \log \left(x^3+3\right)+\frac{13 \log (x)}{27}",1,"-1/18*1/x^6 + 4/(27*x^3) + (13*Log[x])/27 - Log[1 + x^3]/6 + Log[3 + x^3]/162","A",1
158,1,118,124,0.0544064,"\int \frac{x^{10}}{3+4 x^3+x^6} \, dx","Integrate[x^10/(3 + 4*x^3 + x^6),x]","\frac{1}{60} \left(12 x^5-120 x^2-5 \log \left(x^2-x+1\right)+45\ 3^{2/3} \log \left(\sqrt[3]{3} x^2-3^{2/3} x+3\right)+10 \log (x+1)-90\ 3^{2/3} \log \left(3^{2/3} x+3\right)-270 \sqrt[6]{3} \tan ^{-1}\left(\frac{\sqrt[3]{3}-2 x}{3^{5/6}}\right)-10 \sqrt{3} \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)\right)","\frac{x^5}{5}-2 x^2-\frac{1}{12} \log \left(x^2-x+1\right)+\frac{3}{4} 3^{2/3} \log \left(x^2-\sqrt[3]{3} x+3^{2/3}\right)+\frac{1}{6} \log (x+1)-\frac{3}{2} 3^{2/3} \log \left(x+\sqrt[3]{3}\right)+\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{2 \sqrt{3}}-\frac{9}{2} \sqrt[6]{3} \tan ^{-1}\left(\frac{\sqrt[3]{3}-2 x}{3^{5/6}}\right)",1,"(-120*x^2 + 12*x^5 - 270*3^(1/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)] - 10*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] + 10*Log[1 + x] - 90*3^(2/3)*Log[3 + 3^(2/3)*x] - 5*Log[1 - x + x^2] + 45*3^(2/3)*Log[3 - 3^(2/3)*x + 3^(1/3)*x^2])/60","A",1
159,1,114,122,0.0281837,"\int \frac{x^9}{3+4 x^3+x^6} \, dx","Integrate[x^9/(3 + 4*x^3 + x^6),x]","\frac{1}{12} \left(3 x^4+\log \left(x^2-x+1\right)-9 \sqrt[3]{3} \log \left(\sqrt[3]{3} x^2-3^{2/3} x+3\right)-48 x-2 \log (x+1)+18 \sqrt[3]{3} \log \left(3^{2/3} x+3\right)-18\ 3^{5/6} \tan ^{-1}\left(\frac{\sqrt[3]{3}-2 x}{3^{5/6}}\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)\right)","\frac{x^4}{4}+\frac{1}{12} \log \left(x^2-x+1\right)-\frac{3}{4} \sqrt[3]{3} \log \left(x^2-\sqrt[3]{3} x+3^{2/3}\right)-4 x-\frac{1}{6} \log (x+1)+\frac{3}{2} \sqrt[3]{3} \log \left(x+\sqrt[3]{3}\right)+\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{2 \sqrt{3}}-\frac{3}{2} 3^{5/6} \tan ^{-1}\left(\frac{\sqrt[3]{3}-2 x}{3^{5/6}}\right)",1,"(-48*x + 3*x^4 - 18*3^(5/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)] - 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] - 2*Log[1 + x] + 18*3^(1/3)*Log[3 + 3^(2/3)*x] + Log[1 - x + x^2] - 9*3^(1/3)*Log[3 - 3^(2/3)*x + 3^(1/3)*x^2])/12","A",1
160,1,111,119,0.0268208,"\int \frac{x^7}{3+4 x^3+x^6} \, dx","Integrate[x^7/(3 + 4*x^3 + x^6),x]","\frac{1}{12} \left(6 x^2+\log \left(x^2-x+1\right)-3\ 3^{2/3} \log \left(\sqrt[3]{3} x^2-3^{2/3} x+3\right)-2 \log (x+1)+6\ 3^{2/3} \log \left(3^{2/3} x+3\right)+18 \sqrt[6]{3} \tan ^{-1}\left(\frac{\sqrt[3]{3}-2 x}{3^{5/6}}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)\right)","\frac{x^2}{2}+\frac{1}{12} \log \left(x^2-x+1\right)-\frac{1}{4} 3^{2/3} \log \left(x^2-\sqrt[3]{3} x+3^{2/3}\right)-\frac{1}{6} \log (x+1)+\frac{1}{2} 3^{2/3} \log \left(x+\sqrt[3]{3}\right)-\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{2 \sqrt{3}}+\frac{3}{2} \sqrt[6]{3} \tan ^{-1}\left(\frac{\sqrt[3]{3}-2 x}{3^{5/6}}\right)",1,"(6*x^2 + 18*3^(1/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)] + 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] - 2*Log[1 + x] + 6*3^(2/3)*Log[3 + 3^(2/3)*x] + Log[1 - x + x^2] - 3*3^(2/3)*Log[3 - 3^(2/3)*x + 3^(1/3)*x^2])/12","A",1
161,1,111,113,0.0256646,"\int \frac{x^6}{3+4 x^3+x^6} \, dx","Integrate[x^6/(3 + 4*x^3 + x^6),x]","\frac{1}{12} \left(-\log \left(x^2-x+1\right)+3 \sqrt[3]{3} \log \left(\sqrt[3]{3} x^2-3^{2/3} x+3\right)+12 x+2 \log (x+1)-6 \sqrt[3]{3} \log \left(3^{2/3} x+3\right)+6\ 3^{5/6} \tan ^{-1}\left(\frac{\sqrt[3]{3}-2 x}{3^{5/6}}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)\right)","-\frac{1}{12} \log \left(x^2-x+1\right)+\frac{1}{4} \sqrt[3]{3} \log \left(x^2-\sqrt[3]{3} x+3^{2/3}\right)+x+\frac{1}{6} \log (x+1)-\frac{1}{2} \sqrt[3]{3} \log \left(x+\sqrt[3]{3}\right)-\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{2 \sqrt{3}}+\frac{1}{2} 3^{5/6} \tan ^{-1}\left(\frac{\sqrt[3]{3}-2 x}{3^{5/6}}\right)",1,"(12*x + 6*3^(5/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)] + 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] + 2*Log[1 + x] - 6*3^(1/3)*Log[3 + 3^(2/3)*x] - Log[1 - x + x^2] + 3*3^(1/3)*Log[3 - 3^(2/3)*x + 3^(1/3)*x^2])/12","A",1
162,1,107,112,0.0242026,"\int \frac{x^4}{3+4 x^3+x^6} \, dx","Integrate[x^4/(3 + 4*x^3 + x^6),x]","\frac{1}{12} \left(-\log \left(x^2-x+1\right)+3^{2/3} \log \left(\sqrt[3]{3} x^2-3^{2/3} x+3\right)+2 \log (x+1)-2\ 3^{2/3} \log \left(3^{2/3} x+3\right)-6 \sqrt[6]{3} \tan ^{-1}\left(\frac{\sqrt[3]{3}-2 x}{3^{5/6}}\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)\right)","-\frac{1}{12} \log \left(x^2-x+1\right)+\frac{\log \left(x^2-\sqrt[3]{3} x+3^{2/3}\right)}{4 \sqrt[3]{3}}+\frac{1}{6} \log (x+1)-\frac{\log \left(x+\sqrt[3]{3}\right)}{2 \sqrt[3]{3}}+\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{2 \sqrt{3}}-\frac{1}{2} \sqrt[6]{3} \tan ^{-1}\left(\frac{\sqrt[3]{3}-2 x}{3^{5/6}}\right)",1,"(-6*3^(1/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)] - 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] + 2*Log[1 + x] - 2*3^(2/3)*Log[3 + 3^(2/3)*x] - Log[1 - x + x^2] + 3^(2/3)*Log[3 - 3^(2/3)*x + 3^(1/3)*x^2])/12","A",1
163,1,106,112,0.0234405,"\int \frac{x^3}{3+4 x^3+x^6} \, dx","Integrate[x^3/(3 + 4*x^3 + x^6),x]","\frac{1}{12} \left(\log \left(x^2-x+1\right)-\sqrt[3]{3} \log \left(\sqrt[3]{3} x^2-3^{2/3} x+3\right)-2 \log (x+1)+2 \sqrt[3]{3} \log \left(3^{2/3} x+3\right)-2\ 3^{5/6} \tan ^{-1}\left(\frac{\sqrt[3]{3}-2 x}{3^{5/6}}\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)\right)","\frac{1}{12} \log \left(x^2-x+1\right)-\frac{\log \left(x^2-\sqrt[3]{3} x+3^{2/3}\right)}{4\ 3^{2/3}}-\frac{1}{6} \log (x+1)+\frac{\log \left(x+\sqrt[3]{3}\right)}{2\ 3^{2/3}}+\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{2 \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{3}-2 x}{3^{5/6}}\right)}{2 \sqrt[6]{3}}",1,"(-2*3^(5/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)] - 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] - 2*Log[1 + x] + 2*3^(1/3)*Log[3 + 3^(2/3)*x] + Log[1 - x + x^2] - 3^(1/3)*Log[3 - 3^(2/3)*x + 3^(1/3)*x^2])/12","A",1
164,1,108,112,0.0252178,"\int \frac{x}{3+4 x^3+x^6} \, dx","Integrate[x/(3 + 4*x^3 + x^6),x]","\frac{1}{36} \left(3 \log \left(x^2-x+1\right)-3^{2/3} \log \left(\sqrt[3]{3} x^2-3^{2/3} x+3\right)-6 \log (x+1)+2\ 3^{2/3} \log \left(3^{2/3} x+3\right)+6 \sqrt[6]{3} \tan ^{-1}\left(\frac{\sqrt[3]{3}-2 x}{3^{5/6}}\right)+6 \sqrt{3} \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)\right)","\frac{1}{12} \log \left(x^2-x+1\right)-\frac{\log \left(x^2-\sqrt[3]{3} x+3^{2/3}\right)}{12 \sqrt[3]{3}}-\frac{1}{6} \log (x+1)+\frac{\log \left(x+\sqrt[3]{3}\right)}{6 \sqrt[3]{3}}-\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{2 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{3}-2 x}{3^{5/6}}\right)}{2\ 3^{5/6}}",1,"(6*3^(1/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)] + 6*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] - 6*Log[1 + x] + 2*3^(2/3)*Log[3 + 3^(2/3)*x] + 3*Log[1 - x + x^2] - 3^(2/3)*Log[3 - 3^(2/3)*x + 3^(1/3)*x^2])/36","A",1
165,1,107,112,0.0226158,"\int \frac{1}{3+4 x^3+x^6} \, dx","Integrate[(3 + 4*x^3 + x^6)^(-1),x]","\frac{1}{36} \left(-3 \log \left(x^2-x+1\right)+\sqrt[3]{3} \log \left(\sqrt[3]{3} x^2-3^{2/3} x+3\right)+6 \log (x+1)-2 \sqrt[3]{3} \log \left(3^{2/3} x+3\right)+2\ 3^{5/6} \tan ^{-1}\left(\frac{\sqrt[3]{3}-2 x}{3^{5/6}}\right)+6 \sqrt{3} \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)\right)","-\frac{1}{12} \log \left(x^2-x+1\right)+\frac{\log \left(x^2-\sqrt[3]{3} x+3^{2/3}\right)}{12\ 3^{2/3}}+\frac{1}{6} \log (x+1)-\frac{\log \left(x+\sqrt[3]{3}\right)}{6\ 3^{2/3}}-\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{2 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{3}-2 x}{3^{5/6}}\right)}{6 \sqrt[6]{3}}",1,"(2*3^(5/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)] + 6*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] + 6*Log[1 + x] - 2*3^(1/3)*Log[3 + 3^(2/3)*x] - 3*Log[1 - x + x^2] + 3^(1/3)*Log[3 - 3^(2/3)*x + 3^(1/3)*x^2])/36","A",1
166,1,118,119,0.0411651,"\int \frac{1}{x^2 \left(3+4 x^3+x^6\right)} \, dx","Integrate[1/(x^2*(3 + 4*x^3 + x^6)),x]","-\frac{9 x \log \left(x^2-x+1\right)-3^{2/3} x \log \left(\sqrt[3]{3} x^2-3^{2/3} x+3\right)-18 x \log (x+1)+2\ 3^{2/3} x \log \left(3^{2/3} x+3\right)+6 \sqrt[6]{3} x \tan ^{-1}\left(\frac{\sqrt[3]{3}-2 x}{3^{5/6}}\right)+18 \sqrt{3} x \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)+36}{108 x}","-\frac{1}{12} \log \left(x^2-x+1\right)+\frac{\log \left(x^2-\sqrt[3]{3} x+3^{2/3}\right)}{36 \sqrt[3]{3}}-\frac{1}{3 x}+\frac{1}{6} \log (x+1)-\frac{\log \left(x+\sqrt[3]{3}\right)}{18 \sqrt[3]{3}}+\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{2 \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{3}-2 x}{3^{5/6}}\right)}{6\ 3^{5/6}}",1,"-1/108*(36 + 6*3^(1/6)*x*ArcTan[(3^(1/3) - 2*x)/3^(5/6)] + 18*Sqrt[3]*x*ArcTan[(-1 + 2*x)/Sqrt[3]] - 18*x*Log[1 + x] + 2*3^(2/3)*x*Log[3 + 3^(2/3)*x] + 9*x*Log[1 - x + x^2] - 3^(2/3)*x*Log[3 - 3^(2/3)*x + 3^(1/3)*x^2])/x","A",1
167,1,113,119,0.048974,"\int \frac{1}{x^3 \left(3+4 x^3+x^6\right)} \, dx","Integrate[1/(x^3*(3 + 4*x^3 + x^6)),x]","\frac{1}{108} \left(-\frac{18}{x^2}+9 \log \left(x^2-x+1\right)-\sqrt[3]{3} \log \left(\sqrt[3]{3} x^2-3^{2/3} x+3\right)-18 \log (x+1)+2 \sqrt[3]{3} \log \left(3^{2/3} x+3\right)-2\ 3^{5/6} \tan ^{-1}\left(\frac{\sqrt[3]{3}-2 x}{3^{5/6}}\right)-18 \sqrt{3} \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)\right)","-\frac{1}{6 x^2}+\frac{1}{12} \log \left(x^2-x+1\right)-\frac{\log \left(x^2-\sqrt[3]{3} x+3^{2/3}\right)}{36\ 3^{2/3}}-\frac{1}{6} \log (x+1)+\frac{\log \left(x+\sqrt[3]{3}\right)}{18\ 3^{2/3}}+\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{2 \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{3}-2 x}{3^{5/6}}\right)}{18 \sqrt[6]{3}}",1,"(-18/x^2 - 2*3^(5/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)] - 18*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] - 18*Log[1 + x] + 2*3^(1/3)*Log[3 + 3^(2/3)*x] + 9*Log[1 - x + x^2] - 3^(1/3)*Log[3 - 3^(2/3)*x + 3^(1/3)*x^2])/108","A",1
168,1,118,126,0.0476808,"\int \frac{1}{x^5 \left(3+4 x^3+x^6\right)} \, dx","Integrate[1/(x^5*(3 + 4*x^3 + x^6)),x]","\frac{1}{324} \left(-\frac{27}{x^4}+27 \log \left(x^2-x+1\right)-3^{2/3} \log \left(\sqrt[3]{3} x^2-3^{2/3} x+3\right)+\frac{144}{x}-54 \log (x+1)+2\ 3^{2/3} \log \left(3^{2/3} x+3\right)+6 \sqrt[6]{3} \tan ^{-1}\left(\frac{\sqrt[3]{3}-2 x}{3^{5/6}}\right)+54 \sqrt{3} \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)\right)","-\frac{1}{12 x^4}+\frac{1}{12} \log \left(x^2-x+1\right)-\frac{\log \left(x^2-\sqrt[3]{3} x+3^{2/3}\right)}{108 \sqrt[3]{3}}+\frac{4}{9 x}-\frac{1}{6} \log (x+1)+\frac{\log \left(x+\sqrt[3]{3}\right)}{54 \sqrt[3]{3}}-\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{2 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{3}-2 x}{3^{5/6}}\right)}{18\ 3^{5/6}}",1,"(-27/x^4 + 144/x + 6*3^(1/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)] + 54*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] - 54*Log[1 + x] + 2*3^(2/3)*Log[3 + 3^(2/3)*x] + 27*Log[1 - x + x^2] - 3^(2/3)*Log[3 - 3^(2/3)*x + 3^(1/3)*x^2])/324","A",1
169,1,118,126,0.0585672,"\int \frac{1}{x^6 \left(3+4 x^3+x^6\right)} \, dx","Integrate[1/(x^6*(3 + 4*x^3 + x^6)),x]","\frac{-\frac{108}{x^5}+\frac{360}{x^2}-135 \log \left(x^2-x+1\right)+5 \sqrt[3]{3} \log \left(\sqrt[3]{3} x^2-3^{2/3} x+3\right)+270 \log (x+1)-10 \sqrt[3]{3} \log \left(3^{2/3} x+3\right)+10\ 3^{5/6} \tan ^{-1}\left(\frac{\sqrt[3]{3}-2 x}{3^{5/6}}\right)+270 \sqrt{3} \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)}{1620}","-\frac{1}{15 x^5}+\frac{2}{9 x^2}-\frac{1}{12} \log \left(x^2-x+1\right)+\frac{\log \left(x^2-\sqrt[3]{3} x+3^{2/3}\right)}{108\ 3^{2/3}}+\frac{1}{6} \log (x+1)-\frac{\log \left(x+\sqrt[3]{3}\right)}{54\ 3^{2/3}}-\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{2 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{3}-2 x}{3^{5/6}}\right)}{54 \sqrt[6]{3}}",1,"(-108/x^5 + 360/x^2 + 10*3^(5/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)] + 270*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] + 270*Log[1 + x] - 10*3^(1/3)*Log[3 + 3^(2/3)*x] - 135*Log[1 - x + x^2] + 5*3^(1/3)*Log[3 - 3^(2/3)*x + 3^(1/3)*x^2])/1620","A",1
170,1,59,412,0.0132267,"\int \frac{x^6}{1-x^3+x^6} \, dx","Integrate[x^6/(1 - x^3 + x^6),x]","\frac{1}{3} \text{RootSum}\left[\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\text{$\#$1}^3 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^5-\text{$\#$1}^2}\&\right]+x","-\frac{\left(3-i \sqrt{3}\right) \log \left(2^{2/3} x^2+\sqrt[3]{2 \left(1-i \sqrt{3}\right)} x+\left(1-i \sqrt{3}\right)^{2/3}\right)}{18 \sqrt[3]{2} \left(1-i \sqrt{3}\right)^{2/3}}-\frac{\left(3+i \sqrt{3}\right) \log \left(2^{2/3} x^2+\sqrt[3]{2 \left(1+i \sqrt{3}\right)} x+\left(1+i \sqrt{3}\right)^{2/3}\right)}{18 \sqrt[3]{2} \left(1+i \sqrt{3}\right)^{2/3}}+x+\frac{\left(3-i \sqrt{3}\right) \log \left(-\sqrt[3]{2} x+\sqrt[3]{1-i \sqrt{3}}\right)}{9 \sqrt[3]{2} \left(1-i \sqrt{3}\right)^{2/3}}+\frac{\left(3+i \sqrt{3}\right) \log \left(-\sqrt[3]{2} x+\sqrt[3]{1+i \sqrt{3}}\right)}{9 \sqrt[3]{2} \left(1+i \sqrt{3}\right)^{2/3}}+\frac{\left(-\sqrt{3}+i\right) \tan ^{-1}\left(\frac{1+\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1-i \sqrt{3}\right)}}}{\sqrt{3}}\right)}{3 \sqrt[3]{2} \left(1-i \sqrt{3}\right)^{2/3}}-\frac{\left(\sqrt{3}+i\right) \tan ^{-1}\left(\frac{1+\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1+i \sqrt{3}\right)}}}{\sqrt{3}}\right)}{3 \sqrt[3]{2} \left(1+i \sqrt{3}\right)^{2/3}}",1,"x + RootSum[1 - #1^3 + #1^6 & , (-Log[x - #1] + Log[x - #1]*#1^3)/(-#1^2 + 2*#1^5) & ]/3","C",1
171,1,39,39,0.0112753,"\int \frac{x^5}{1-x^3+x^6} \, dx","Integrate[x^5/(1 - x^3 + x^6),x]","\frac{\tan ^{-1}\left(\frac{2 x^3-1}{\sqrt{3}}\right)}{3 \sqrt{3}}+\frac{1}{6} \log \left(x^6-x^3+1\right)","\frac{1}{6} \log \left(x^6-x^3+1\right)-\frac{\tan ^{-1}\left(\frac{1-2 x^3}{\sqrt{3}}\right)}{3 \sqrt{3}}",1,"ArcTan[(-1 + 2*x^3)/Sqrt[3]]/(3*Sqrt[3]) + Log[1 - x^3 + x^6]/6","A",1
172,1,41,411,0.0089472,"\int \frac{x^4}{1-x^3+x^6} \, dx","Integrate[x^4/(1 - x^3 + x^6),x]","\frac{1}{3} \text{RootSum}\left[\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\text{$\#$1}^2 \log (x-\text{$\#$1})}{2 \text{$\#$1}^3-1}\&\right]","-\frac{\left(3+i \sqrt{3}\right) \log \left(2^{2/3} x^2+\sqrt[3]{2 \left(1-i \sqrt{3}\right)} x+\left(1-i \sqrt{3}\right)^{2/3}\right)}{18\ 2^{2/3} \sqrt[3]{1-i \sqrt{3}}}-\frac{\left(3-i \sqrt{3}\right) \log \left(2^{2/3} x^2+\sqrt[3]{2 \left(1+i \sqrt{3}\right)} x+\left(1+i \sqrt{3}\right)^{2/3}\right)}{18\ 2^{2/3} \sqrt[3]{1+i \sqrt{3}}}+\frac{\left(3+i \sqrt{3}\right) \log \left(-\sqrt[3]{2} x+\sqrt[3]{1-i \sqrt{3}}\right)}{9\ 2^{2/3} \sqrt[3]{1-i \sqrt{3}}}+\frac{\left(3-i \sqrt{3}\right) \log \left(-\sqrt[3]{2} x+\sqrt[3]{1+i \sqrt{3}}\right)}{9\ 2^{2/3} \sqrt[3]{1+i \sqrt{3}}}+\frac{\left(\sqrt{3}+i\right) \tan ^{-1}\left(\frac{1+\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1-i \sqrt{3}\right)}}}{\sqrt{3}}\right)}{3\ 2^{2/3} \sqrt[3]{1-i \sqrt{3}}}-\frac{\left(-\sqrt{3}+i\right) \tan ^{-1}\left(\frac{1+\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1+i \sqrt{3}\right)}}}{\sqrt{3}}\right)}{3\ 2^{2/3} \sqrt[3]{1+i \sqrt{3}}}",1,"RootSum[1 - #1^3 + #1^6 & , (Log[x - #1]*#1^2)/(-1 + 2*#1^3) & ]/3","C",1
173,1,39,411,0.0086371,"\int \frac{x^3}{1-x^3+x^6} \, dx","Integrate[x^3/(1 - x^3 + x^6),x]","\frac{1}{3} \text{RootSum}\left[\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\text{$\#$1} \log (x-\text{$\#$1})}{2 \text{$\#$1}^3-1}\&\right]","-\frac{\left(3+i \sqrt{3}\right) \log \left(2^{2/3} x^2+\sqrt[3]{2 \left(1-i \sqrt{3}\right)} x+\left(1-i \sqrt{3}\right)^{2/3}\right)}{18 \sqrt[3]{2} \left(1-i \sqrt{3}\right)^{2/3}}-\frac{\left(3-i \sqrt{3}\right) \log \left(2^{2/3} x^2+\sqrt[3]{2 \left(1+i \sqrt{3}\right)} x+\left(1+i \sqrt{3}\right)^{2/3}\right)}{18 \sqrt[3]{2} \left(1+i \sqrt{3}\right)^{2/3}}+\frac{\left(3+i \sqrt{3}\right) \log \left(-\sqrt[3]{2} x+\sqrt[3]{1-i \sqrt{3}}\right)}{9 \sqrt[3]{2} \left(1-i \sqrt{3}\right)^{2/3}}+\frac{\left(3-i \sqrt{3}\right) \log \left(-\sqrt[3]{2} x+\sqrt[3]{1+i \sqrt{3}}\right)}{9 \sqrt[3]{2} \left(1+i \sqrt{3}\right)^{2/3}}-\frac{\left(\sqrt{3}+i\right) \tan ^{-1}\left(\frac{1+\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1-i \sqrt{3}\right)}}}{\sqrt{3}}\right)}{3 \sqrt[3]{2} \left(1-i \sqrt{3}\right)^{2/3}}+\frac{\left(-\sqrt{3}+i\right) \tan ^{-1}\left(\frac{1+\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1+i \sqrt{3}\right)}}}{\sqrt{3}}\right)}{3 \sqrt[3]{2} \left(1+i \sqrt{3}\right)^{2/3}}",1,"RootSum[1 - #1^3 + #1^6 & , (Log[x - #1]*#1)/(-1 + 2*#1^3) & ]/3","C",1
174,1,23,23,0.0062704,"\int \frac{x^2}{1-x^3+x^6} \, dx","Integrate[x^2/(1 - x^3 + x^6),x]","\frac{2 \tan ^{-1}\left(\frac{2 x^3-1}{\sqrt{3}}\right)}{3 \sqrt{3}}","-\frac{2 \tan ^{-1}\left(\frac{1-2 x^3}{\sqrt{3}}\right)}{3 \sqrt{3}}",1,"(2*ArcTan[(-1 + 2*x^3)/Sqrt[3]])/(3*Sqrt[3])","A",1
175,1,40,375,0.0098403,"\int \frac{x}{1-x^3+x^6} \, dx","Integrate[x/(1 - x^3 + x^6),x]","\frac{1}{3} \text{RootSum}\left[\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^4-\text{$\#$1}}\&\right]","-\frac{i \log \left(2^{2/3} x^2+\sqrt[3]{2 \left(1-i \sqrt{3}\right)} x+\left(1-i \sqrt{3}\right)^{2/3}\right)}{3\ 2^{2/3} \sqrt{3} \sqrt[3]{1-i \sqrt{3}}}+\frac{i \log \left(2^{2/3} x^2+\sqrt[3]{2 \left(1+i \sqrt{3}\right)} x+\left(1+i \sqrt{3}\right)^{2/3}\right)}{3\ 2^{2/3} \sqrt{3} \sqrt[3]{1+i \sqrt{3}}}+\frac{i \log \left(-\sqrt[3]{2} x+\sqrt[3]{1-i \sqrt{3}}\right)}{3 \sqrt{3} \sqrt[3]{\frac{1}{2} \left(1-i \sqrt{3}\right)}}-\frac{i \log \left(-\sqrt[3]{2} x+\sqrt[3]{1+i \sqrt{3}}\right)}{3 \sqrt{3} \sqrt[3]{\frac{1}{2} \left(1+i \sqrt{3}\right)}}+\frac{i \tan ^{-1}\left(\frac{1+\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1-i \sqrt{3}\right)}}}{\sqrt{3}}\right)}{3 \sqrt[3]{\frac{1}{2} \left(1-i \sqrt{3}\right)}}-\frac{i \tan ^{-1}\left(\frac{1+\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1+i \sqrt{3}\right)}}}{\sqrt{3}}\right)}{3 \sqrt[3]{\frac{1}{2} \left(1+i \sqrt{3}\right)}}",1,"RootSum[1 - #1^3 + #1^6 & , Log[x - #1]/(-#1 + 2*#1^4) & ]/3","C",1
176,1,42,186,0.008437,"\int \frac{1}{1-x^3+x^6} \, dx","Integrate[(1 - x^3 + x^6)^(-1),x]","\frac{1}{3} \text{RootSum}\left[\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^5-\text{$\#$1}^2}\&\right]","-\frac{(-1)^{5/18} \left(3 \log \left(\sqrt[9]{-1}-x\right)+\log (2)\right)}{9 \sqrt{3}}+\frac{(-1)^{13/18} \log \left(-\sqrt[3]{2} \left(x+(-1)^{8/9}\right)\right)}{3 \sqrt{3}}-\frac{(-1)^{13/18} \log \left(-2^{2/3} \left(\left((-1)^{8/9}-x\right) x+(-1)^{7/9}\right)\right)}{6 \sqrt{3}}+\frac{(-1)^{5/18} \log \left(2^{2/3} \left(x \left(x+\sqrt[9]{-1}\right)+(-1)^{2/9}\right)\right)}{6 \sqrt{3}}-\frac{1}{3} (-1)^{13/18} \tan ^{-1}\left(\frac{2 \sqrt[9]{-1} x+1}{\sqrt{3}}\right)+\frac{1}{3} (-1)^{5/18} \tan ^{-1}\left(\frac{1-2 (-1)^{8/9} x}{\sqrt{3}}\right)",1,"RootSum[1 - #1^3 + #1^6 & , Log[x - #1]/(-#1^2 + 2*#1^5) & ]/3","C",1
177,1,55,41,0.0117629,"\int \frac{1}{x \left(1-x^3+x^6\right)} \, dx","Integrate[1/(x*(1 - x^3 + x^6)),x]","\log (x)-\frac{1}{3} \text{RootSum}\left[\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\text{$\#$1}^3 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^3-1}\&\right]","-\frac{\tan ^{-1}\left(\frac{1-2 x^3}{\sqrt{3}}\right)}{3 \sqrt{3}}-\frac{1}{6} \log \left(x^6-x^3+1\right)+\log (x)",1,"Log[x] - RootSum[1 - #1^3 + #1^6 & , (-Log[x - #1] + Log[x - #1]*#1^3)/(-1 + 2*#1^3) & ]/3","C",1
178,1,61,416,0.0126215,"\int \frac{1}{x^2 \left(1-x^3+x^6\right)} \, dx","Integrate[1/(x^2*(1 - x^3 + x^6)),x]","-\frac{1}{3} \text{RootSum}\left[\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\text{$\#$1}^3 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^4-\text{$\#$1}}\&\right]-\frac{1}{x}","\frac{\left(3-i \sqrt{3}\right) \log \left(2^{2/3} x^2+\sqrt[3]{2 \left(1-i \sqrt{3}\right)} x+\left(1-i \sqrt{3}\right)^{2/3}\right)}{18\ 2^{2/3} \sqrt[3]{1-i \sqrt{3}}}+\frac{\left(3+i \sqrt{3}\right) \log \left(2^{2/3} x^2+\sqrt[3]{2 \left(1+i \sqrt{3}\right)} x+\left(1+i \sqrt{3}\right)^{2/3}\right)}{18\ 2^{2/3} \sqrt[3]{1+i \sqrt{3}}}-\frac{1}{x}-\frac{\left(3-i \sqrt{3}\right) \log \left(-\sqrt[3]{2} x+\sqrt[3]{1-i \sqrt{3}}\right)}{9\ 2^{2/3} \sqrt[3]{1-i \sqrt{3}}}-\frac{\left(3+i \sqrt{3}\right) \log \left(-\sqrt[3]{2} x+\sqrt[3]{1+i \sqrt{3}}\right)}{9\ 2^{2/3} \sqrt[3]{1+i \sqrt{3}}}+\frac{\left(-\sqrt{3}+i\right) \tan ^{-1}\left(\frac{1+\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1-i \sqrt{3}\right)}}}{\sqrt{3}}\right)}{3\ 2^{2/3} \sqrt[3]{1-i \sqrt{3}}}-\frac{\left(\sqrt{3}+i\right) \tan ^{-1}\left(\frac{1+\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1+i \sqrt{3}\right)}}}{\sqrt{3}}\right)}{3\ 2^{2/3} \sqrt[3]{1+i \sqrt{3}}}",1,"-x^(-1) - RootSum[1 - #1^3 + #1^6 & , (-Log[x - #1] + Log[x - #1]*#1^3)/(-#1 + 2*#1^4) & ]/3","C",1
179,1,65,418,0.0127452,"\int \frac{1}{x^3 \left(1-x^3+x^6\right)} \, dx","Integrate[1/(x^3*(1 - x^3 + x^6)),x]","-\frac{1}{3} \text{RootSum}\left[\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\text{$\#$1}^3 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^5-\text{$\#$1}^2}\&\right]-\frac{1}{2 x^2}","-\frac{1}{2 x^2}+\frac{\left(3-i \sqrt{3}\right) \log \left(2^{2/3} x^2+\sqrt[3]{2 \left(1-i \sqrt{3}\right)} x+\left(1-i \sqrt{3}\right)^{2/3}\right)}{18 \sqrt[3]{2} \left(1-i \sqrt{3}\right)^{2/3}}+\frac{\left(3+i \sqrt{3}\right) \log \left(2^{2/3} x^2+\sqrt[3]{2 \left(1+i \sqrt{3}\right)} x+\left(1+i \sqrt{3}\right)^{2/3}\right)}{18 \sqrt[3]{2} \left(1+i \sqrt{3}\right)^{2/3}}-\frac{\left(3-i \sqrt{3}\right) \log \left(-\sqrt[3]{2} x+\sqrt[3]{1-i \sqrt{3}}\right)}{9 \sqrt[3]{2} \left(1-i \sqrt{3}\right)^{2/3}}-\frac{\left(3+i \sqrt{3}\right) \log \left(-\sqrt[3]{2} x+\sqrt[3]{1+i \sqrt{3}}\right)}{9 \sqrt[3]{2} \left(1+i \sqrt{3}\right)^{2/3}}-\frac{\left(-\sqrt{3}+i\right) \tan ^{-1}\left(\frac{1+\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1-i \sqrt{3}\right)}}}{\sqrt{3}}\right)}{3 \sqrt[3]{2} \left(1-i \sqrt{3}\right)^{2/3}}+\frac{\left(\sqrt{3}+i\right) \tan ^{-1}\left(\frac{1+\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1+i \sqrt{3}\right)}}}{\sqrt{3}}\right)}{3 \sqrt[3]{2} \left(1+i \sqrt{3}\right)^{2/3}}",1,"-1/2*1/x^2 - RootSum[1 - #1^3 + #1^6 & , (-Log[x - #1] + Log[x - #1]*#1^3)/(-#1^2 + 2*#1^5) & ]/3","C",1
180,1,51,48,0.0133914,"\int \frac{1}{x^4 \left(1-x^3+x^6\right)} \, dx","Integrate[1/(x^4*(1 - x^3 + x^6)),x]","-\frac{1}{3} \text{RootSum}\left[\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\text{$\#$1}^3 \log (x-\text{$\#$1})}{2 \text{$\#$1}^3-1}\&\right]-\frac{1}{3 x^3}+\log (x)","-\frac{1}{3 x^3}+\frac{\tan ^{-1}\left(\frac{1-2 x^3}{\sqrt{3}}\right)}{3 \sqrt{3}}-\frac{1}{6} \log \left(x^6-x^3+1\right)+\log (x)",1,"-1/3*1/x^3 + Log[x] - RootSum[1 - #1^3 + #1^6 & , (Log[x - #1]*#1^3)/(-1 + 2*#1^3) & ]/3","C",1
181,1,54,423,0.0139518,"\int \frac{1}{x^5 \left(1-x^3+x^6\right)} \, dx","Integrate[1/(x^5*(1 - x^3 + x^6)),x]","-\frac{1}{3} \text{RootSum}\left[\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\text{$\#$1}^2 \log (x-\text{$\#$1})}{2 \text{$\#$1}^3-1}\&\right]-\frac{1}{4 x^4}-\frac{1}{x}","-\frac{1}{4 x^4}+\frac{\left(3+i \sqrt{3}\right) \log \left(2^{2/3} x^2+\sqrt[3]{2 \left(1-i \sqrt{3}\right)} x+\left(1-i \sqrt{3}\right)^{2/3}\right)}{18\ 2^{2/3} \sqrt[3]{1-i \sqrt{3}}}+\frac{\left(3-i \sqrt{3}\right) \log \left(2^{2/3} x^2+\sqrt[3]{2 \left(1+i \sqrt{3}\right)} x+\left(1+i \sqrt{3}\right)^{2/3}\right)}{18\ 2^{2/3} \sqrt[3]{1+i \sqrt{3}}}-\frac{1}{x}-\frac{\left(3+i \sqrt{3}\right) \log \left(-\sqrt[3]{2} x+\sqrt[3]{1-i \sqrt{3}}\right)}{9\ 2^{2/3} \sqrt[3]{1-i \sqrt{3}}}-\frac{\left(3-i \sqrt{3}\right) \log \left(-\sqrt[3]{2} x+\sqrt[3]{1+i \sqrt{3}}\right)}{9\ 2^{2/3} \sqrt[3]{1+i \sqrt{3}}}-\frac{\left(\sqrt{3}+i\right) \tan ^{-1}\left(\frac{1+\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1-i \sqrt{3}\right)}}}{\sqrt{3}}\right)}{3\ 2^{2/3} \sqrt[3]{1-i \sqrt{3}}}+\frac{\left(-\sqrt{3}+i\right) \tan ^{-1}\left(\frac{1+\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1+i \sqrt{3}\right)}}}{\sqrt{3}}\right)}{3\ 2^{2/3} \sqrt[3]{1+i \sqrt{3}}}",1,"-1/4*1/x^4 - x^(-1) - RootSum[1 - #1^3 + #1^6 & , (Log[x - #1]*#1^2)/(-1 + 2*#1^3) & ]/3","C",1
182,1,38,381,0.0095799,"\int \frac{1}{2+x^3+x^6} \, dx","Integrate[(2 + x^3 + x^6)^(-1),x]","\frac{1}{3} \text{RootSum}\left[\text{$\#$1}^6+\text{$\#$1}^3+2\&,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^5+\text{$\#$1}^2}\&\right]","\frac{i \log \left(2^{2/3} x^2-\sqrt[3]{2 \left(1-i \sqrt{7}\right)} x+\left(1-i \sqrt{7}\right)^{2/3}\right)}{3 \sqrt[3]{2} \sqrt{7} \left(1-i \sqrt{7}\right)^{2/3}}-\frac{i \log \left(2^{2/3} x^2-\sqrt[3]{2 \left(1+i \sqrt{7}\right)} x+\left(1+i \sqrt{7}\right)^{2/3}\right)}{3 \sqrt[3]{2} \sqrt{7} \left(1+i \sqrt{7}\right)^{2/3}}-\frac{i \log \left(\sqrt[3]{2} x+\sqrt[3]{1-i \sqrt{7}}\right)}{3 \sqrt{7} \left(\frac{1}{2} \left(1-i \sqrt{7}\right)\right)^{2/3}}+\frac{i \log \left(\sqrt[3]{2} x+\sqrt[3]{1+i \sqrt{7}}\right)}{3 \sqrt{7} \left(\frac{1}{2} \left(1+i \sqrt{7}\right)\right)^{2/3}}+\frac{i \tan ^{-1}\left(\frac{1-\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1-i \sqrt{7}\right)}}}{\sqrt{3}}\right)}{\sqrt{21} \left(\frac{1}{2} \left(1-i \sqrt{7}\right)\right)^{2/3}}-\frac{i \tan ^{-1}\left(\frac{1-\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1+i \sqrt{7}\right)}}}{\sqrt{3}}\right)}{\sqrt{21} \left(\frac{1}{2} \left(1+i \sqrt{7}\right)\right)^{2/3}}",1,"RootSum[2 + #1^3 + #1^6 & , Log[x - #1]/(#1^2 + 2*#1^5) & ]/3","C",1
183,1,23,23,0.0083847,"\int \frac{x^2}{2+x^3+x^6} \, dx","Integrate[x^2/(2 + x^3 + x^6),x]","\frac{2 \tan ^{-1}\left(\frac{2 x^3+1}{\sqrt{7}}\right)}{3 \sqrt{7}}","\frac{2 \tan ^{-1}\left(\frac{2 x^3+1}{\sqrt{7}}\right)}{3 \sqrt{7}}",1,"(2*ArcTan[(1 + 2*x^3)/Sqrt[7]])/(3*Sqrt[7])","A",1
184,1,37,399,0.0085276,"\int \frac{x^3}{2+x^3+x^6} \, dx","Integrate[x^3/(2 + x^3 + x^6),x]","\frac{1}{3} \text{RootSum}\left[\text{$\#$1}^6+\text{$\#$1}^3+2\&,\frac{\text{$\#$1} \log (x-\text{$\#$1})}{2 \text{$\#$1}^3+1}\&\right]","-\frac{\left(7+i \sqrt{7}\right) \log \left(2^{2/3} x^2-\sqrt[3]{2 \left(1-i \sqrt{7}\right)} x+\left(1-i \sqrt{7}\right)^{2/3}\right)}{42 \sqrt[3]{2} \left(1-i \sqrt{7}\right)^{2/3}}-\frac{\left(7-i \sqrt{7}\right) \log \left(2^{2/3} x^2-\sqrt[3]{2 \left(1+i \sqrt{7}\right)} x+\left(1+i \sqrt{7}\right)^{2/3}\right)}{42 \sqrt[3]{2} \left(1+i \sqrt{7}\right)^{2/3}}+\frac{\left(7+i \sqrt{7}\right) \log \left(\sqrt[3]{2} x+\sqrt[3]{1-i \sqrt{7}}\right)}{21 \sqrt[3]{2} \left(1-i \sqrt{7}\right)^{2/3}}+\frac{\left(7-i \sqrt{7}\right) \log \left(\sqrt[3]{2} x+\sqrt[3]{1+i \sqrt{7}}\right)}{21 \sqrt[3]{2} \left(1+i \sqrt{7}\right)^{2/3}}-\frac{i \sqrt[3]{\frac{1}{2} \left(1-i \sqrt{7}\right)} \tan ^{-1}\left(\frac{1-\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1-i \sqrt{7}\right)}}}{\sqrt{3}}\right)}{\sqrt{21}}+\frac{i \sqrt[3]{\frac{1}{2} \left(1+i \sqrt{7}\right)} \tan ^{-1}\left(\frac{1-\frac{2 x}{\sqrt[3]{\frac{1}{2} \left(1+i \sqrt{7}\right)}}}{\sqrt{3}}\right)}{\sqrt{21}}",1,"RootSum[2 + #1^3 + #1^6 & , (Log[x - #1]*#1)/(1 + 2*#1^3) & ]/3","C",1
185,1,208,231,0.1601427,"\int x^{14} \sqrt{a+b x^3+c x^6} \, dx","Integrate[x^14*Sqrt[a + b*x^3 + c*x^6],x]","\frac{2 \sqrt{c} \sqrt{a+b x^3+c x^6} \left(16 b c^2 \left(113 a^2-34 a c x^6+8 c^2 x^{12}\right)+160 c^3 x^3 \left(-3 a^2+2 a c x^6+8 c^2 x^{12}\right)+168 b^3 c \left(c x^6-10 a\right)+16 b^2 c^2 x^3 \left(56 a-9 c x^6\right)+315 b^5-210 b^4 c x^3\right)-15 \left(-64 a^3 c^3+240 a^2 b^2 c^2-140 a b^4 c+21 b^6\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{46080 c^{11/2}}","-\frac{\left(b^2-4 a c\right) \left(16 a^2 c^2-56 a b^2 c+21 b^4\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{3072 c^{11/2}}+\frac{\left(16 a^2 c^2-56 a b^2 c+21 b^4\right) \left(b+2 c x^3\right) \sqrt{a+b x^3+c x^6}}{1536 c^5}-\frac{\left(7 b \left(15 b^2-28 a c\right)-6 c x^3 \left(21 b^2-20 a c\right)\right) \left(a+b x^3+c x^6\right)^{3/2}}{2880 c^4}-\frac{b x^6 \left(a+b x^3+c x^6\right)^{3/2}}{20 c^2}+\frac{x^9 \left(a+b x^3+c x^6\right)^{3/2}}{18 c}",1,"(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6]*(315*b^5 - 210*b^4*c*x^3 + 16*b^2*c^2*x^3*(56*a - 9*c*x^6) + 168*b^3*c*(-10*a + c*x^6) + 16*b*c^2*(113*a^2 - 34*a*c*x^6 + 8*c^2*x^12) + 160*c^3*x^3*(-3*a^2 + 2*a*c*x^6 + 8*c^2*x^12)) - 15*(21*b^6 - 140*a*b^4*c + 240*a^2*b^2*c^2 - 64*a^3*c^3)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(46080*c^(11/2))","A",1
186,1,164,171,0.1341935,"\int x^{11} \sqrt{a+b x^3+c x^6} \, dx","Integrate[x^11*Sqrt[a + b*x^3 + c*x^6],x]","\frac{-\frac{\left(32 a c-35 b^2+42 b c x^3\right) \left(a+b x^3+c x^6\right)^{3/2}}{48 c^2}+\frac{5 \left(12 a b c-7 b^3\right) \left(2 \sqrt{c} \left(b+2 c x^3\right) \sqrt{a+b x^3+c x^6}-\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)\right)}{256 c^{7/2}}+x^6 \left(a+b x^3+c x^6\right)^{3/2}}{15 c}","\frac{b \left(7 b^2-12 a c\right) \left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{768 c^{9/2}}-\frac{b \left(7 b^2-12 a c\right) \left(b+2 c x^3\right) \sqrt{a+b x^3+c x^6}}{384 c^4}+\frac{\left(-32 a c+35 b^2-42 b c x^3\right) \left(a+b x^3+c x^6\right)^{3/2}}{720 c^3}+\frac{x^6 \left(a+b x^3+c x^6\right)^{3/2}}{15 c}",1,"(x^6*(a + b*x^3 + c*x^6)^(3/2) - ((-35*b^2 + 32*a*c + 42*b*c*x^3)*(a + b*x^3 + c*x^6)^(3/2))/(48*c^2) + (5*(-7*b^3 + 12*a*b*c)*(2*Sqrt[c]*(b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6] - (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])]))/(256*c^(7/2)))/(15*c)","A",1
187,1,136,153,0.0693468,"\int x^8 \sqrt{a+b x^3+c x^6} \, dx","Integrate[x^8*Sqrt[a + b*x^3 + c*x^6],x]","\frac{2 \sqrt{c} \sqrt{a+b x^3+c x^6} \left(b \left(8 c^2 x^6-52 a c\right)+24 c^2 x^3 \left(a+2 c x^6\right)+15 b^3-10 b^2 c x^3\right)-3 \left(16 a^2 c^2-24 a b^2 c+5 b^4\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{1152 c^{7/2}}","-\frac{\left(b^2-4 a c\right) \left(5 b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{384 c^{7/2}}+\frac{\left(5 b^2-4 a c\right) \left(b+2 c x^3\right) \sqrt{a+b x^3+c x^6}}{192 c^3}-\frac{5 b \left(a+b x^3+c x^6\right)^{3/2}}{72 c^2}+\frac{x^3 \left(a+b x^3+c x^6\right)^{3/2}}{12 c}",1,"(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6]*(15*b^3 - 10*b^2*c*x^3 + 24*c^2*x^3*(a + 2*c*x^6) + b*(-52*a*c + 8*c^2*x^6)) - 3*(5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(1152*c^(7/2))","A",1
188,1,99,108,0.0742972,"\int x^5 \sqrt{a+b x^3+c x^6} \, dx","Integrate[x^5*Sqrt[a + b*x^3 + c*x^6],x]","\frac{\left(b^3-4 a b c\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{48 c^{5/2}}+\frac{\sqrt{a+b x^3+c x^6} \left(8 c \left(a+c x^6\right)-3 b^2+2 b c x^3\right)}{72 c^2}","\frac{b \left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{48 c^{5/2}}-\frac{b \left(b+2 c x^3\right) \sqrt{a+b x^3+c x^6}}{24 c^2}+\frac{\left(a+b x^3+c x^6\right)^{3/2}}{9 c}",1,"(Sqrt[a + b*x^3 + c*x^6]*(-3*b^2 + 2*b*c*x^3 + 8*c*(a + c*x^6)))/(72*c^2) + ((b^3 - 4*a*b*c)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(48*c^(5/2))","A",1
189,1,87,83,0.0102146,"\int x^2 \sqrt{a+b x^3+c x^6} \, dx","Integrate[x^2*Sqrt[a + b*x^3 + c*x^6],x]","\frac{1}{3} \left(\frac{\left(b+2 c x^3\right) \sqrt{a+b x^3+c x^6}}{4 c}-\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{8 c^{3/2}}\right)","\frac{\left(b+2 c x^3\right) \sqrt{a+b x^3+c x^6}}{12 c}-\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{24 c^{3/2}}",1,"(((b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6])/(4*c) - ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(8*c^(3/2)))/3","A",1
190,1,106,109,0.0420132,"\int \frac{\sqrt{a+b x^3+c x^6}}{x} \, dx","Integrate[Sqrt[a + b*x^3 + c*x^6]/x,x]","\frac{1}{6} \left(2 \sqrt{a+b x^3+c x^6}-2 \sqrt{a} \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)+\frac{b \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{\sqrt{c}}\right)","\frac{1}{3} \sqrt{a+b x^3+c x^6}-\frac{1}{3} \sqrt{a} \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)+\frac{b \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{6 \sqrt{c}}",1,"(2*Sqrt[a + b*x^3 + c*x^6] - 2*Sqrt[a]*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])] + (b*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/Sqrt[c])/6","A",1
191,1,112,112,0.0478511,"\int \frac{\sqrt{a+b x^3+c x^6}}{x^4} \, dx","Integrate[Sqrt[a + b*x^3 + c*x^6]/x^4,x]","-\frac{\sqrt{a+b x^3+c x^6}}{3 x^3}-\frac{b \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{6 \sqrt{a}}+\frac{1}{3} \sqrt{c} \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)","-\frac{\sqrt{a+b x^3+c x^6}}{3 x^3}-\frac{b \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{6 \sqrt{a}}+\frac{1}{3} \sqrt{c} \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)",1,"-1/3*Sqrt[a + b*x^3 + c*x^6]/x^3 - (b*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(6*Sqrt[a]) + (Sqrt[c]*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/3","A",1
192,1,89,88,0.0431832,"\int \frac{\sqrt{a+b x^3+c x^6}}{x^7} \, dx","Integrate[Sqrt[a + b*x^3 + c*x^6]/x^7,x]","\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)-\frac{2 \sqrt{a} \left(2 a+b x^3\right) \sqrt{a+b x^3+c x^6}}{x^6}}{24 a^{3/2}}","\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{24 a^{3/2}}-\frac{\left(2 a+b x^3\right) \sqrt{a+b x^3+c x^6}}{12 a x^6}",1,"((-2*Sqrt[a]*(2*a + b*x^3)*Sqrt[a + b*x^3 + c*x^6])/x^6 + (b^2 - 4*a*c)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(24*a^(3/2))","A",1
193,1,108,116,0.0735259,"\int \frac{\sqrt{a+b x^3+c x^6}}{x^{10}} \, dx","Integrate[Sqrt[a + b*x^3 + c*x^6]/x^10,x]","-\frac{b \left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{48 a^{5/2}}-\frac{\sqrt{a+b x^3+c x^6} \left(8 a^2+2 a x^3 \left(b+4 c x^3\right)-3 b^2 x^6\right)}{72 a^2 x^9}","-\frac{b \left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{48 a^{5/2}}+\frac{b \left(2 a+b x^3\right) \sqrt{a+b x^3+c x^6}}{24 a^2 x^6}-\frac{\left(a+b x^3+c x^6\right)^{3/2}}{9 a x^9}",1,"-1/72*(Sqrt[a + b*x^3 + c*x^6]*(8*a^2 - 3*b^2*x^6 + 2*a*x^3*(b + 4*c*x^3)))/(a^2*x^9) - (b*(b^2 - 4*a*c)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(48*a^(5/2))","A",1
194,1,139,161,0.0878171,"\int \frac{\sqrt{a+b x^3+c x^6}}{x^{13}} \, dx","Integrate[Sqrt[a + b*x^3 + c*x^6]/x^13,x]","\frac{\left(16 a^2 c^2-24 a b^2 c+5 b^4\right) \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{384 a^{7/2}}-\frac{\sqrt{a+b x^3+c x^6} \left(48 a^3+8 a^2 x^3 \left(b+3 c x^3\right)-2 a b x^6 \left(5 b+26 c x^3\right)+15 b^3 x^9\right)}{576 a^3 x^{12}}","\frac{\left(b^2-4 a c\right) \left(5 b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{384 a^{7/2}}-\frac{\left(5 b^2-4 a c\right) \left(2 a+b x^3\right) \sqrt{a+b x^3+c x^6}}{192 a^3 x^6}+\frac{5 b \left(a+b x^3+c x^6\right)^{3/2}}{72 a^2 x^9}-\frac{\left(a+b x^3+c x^6\right)^{3/2}}{12 a x^{12}}",1,"-1/576*(Sqrt[a + b*x^3 + c*x^6]*(48*a^3 + 15*b^3*x^9 + 8*a^2*x^3*(b + 3*c*x^3) - 2*a*b*x^6*(5*b + 26*c*x^3)))/(a^3*x^12) + ((5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(384*a^(7/2))","A",1
195,1,173,199,0.116403,"\int \frac{\sqrt{a+b x^3+c x^6}}{x^{16}} \, dx","Integrate[Sqrt[a + b*x^3 + c*x^6]/x^16,x]","-\frac{b \left(48 a^2 c^2-40 a b^2 c+7 b^4\right) \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{768 a^{9/2}}-\frac{\sqrt{a+b x^3+c x^6} \left(384 a^4+16 a^3 \left(3 b x^3+8 c x^6\right)-8 a^2 x^6 \left(7 b^2+29 b c x^3+32 c^2 x^6\right)+10 a b^2 x^9 \left(7 b+46 c x^3\right)-105 b^4 x^{12}\right)}{5760 a^4 x^{15}}","-\frac{b \left(7 b^2-12 a c\right) \left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{768 a^{9/2}}+\frac{b \left(7 b^2-12 a c\right) \left(2 a+b x^3\right) \sqrt{a+b x^3+c x^6}}{384 a^4 x^6}-\frac{\left(35 b^2-32 a c\right) \left(a+b x^3+c x^6\right)^{3/2}}{720 a^3 x^9}+\frac{7 b \left(a+b x^3+c x^6\right)^{3/2}}{120 a^2 x^{12}}-\frac{\left(a+b x^3+c x^6\right)^{3/2}}{15 a x^{15}}",1,"-1/5760*(Sqrt[a + b*x^3 + c*x^6]*(384*a^4 - 105*b^4*x^12 + 10*a*b^2*x^9*(7*b + 46*c*x^3) + 16*a^3*(3*b*x^3 + 8*c*x^6) - 8*a^2*x^6*(7*b^2 + 29*b*c*x^3 + 32*c^2*x^6)))/(a^4*x^15) - (b*(7*b^4 - 40*a*b^2*c + 48*a^2*c^2)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(768*a^(9/2))","A",1
196,1,358,140,0.5610282,"\int x^3 \sqrt{a+b x^3+c x^6} \, dx","Integrate[x^3*Sqrt[a + b*x^3 + c*x^6],x]","\frac{x \left(3 x^3 \left(16 a c-5 b^2\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)-24 a b \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)+8 \left(3 b+8 c x^3\right) \left(a+b x^3+c x^6\right)\right)}{448 c \sqrt{a+b x^3+c x^6}}","\frac{x^4 \sqrt{a+b x^3+c x^6} F_1\left(\frac{4}{3};-\frac{1}{2},-\frac{1}{2};\frac{7}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{4 \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1}}",1,"(x*(8*(3*b + 8*c*x^3)*(a + b*x^3 + c*x^6) - 24*a*b*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[1/3, 1/2, 1/2, 4/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + 3*(-5*b^2 + 16*a*c)*x^3*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[4/3, 1/2, 1/2, 7/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])]))/(448*c*Sqrt[a + b*x^3 + c*x^6])","B",0
197,1,337,140,0.4114375,"\int x \sqrt{a+b x^3+c x^6} \, dx","Integrate[x*Sqrt[a + b*x^3 + c*x^6],x]","\frac{x^2 \left(3 b x^3 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)+15 a \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)+10 \left(a+b x^3+c x^6\right)\right)}{50 \sqrt{a+b x^3+c x^6}}","\frac{x^2 \sqrt{a+b x^3+c x^6} F_1\left(\frac{2}{3};-\frac{1}{2},-\frac{1}{2};\frac{5}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{2 \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1}}",1,"(x^2*(10*(a + b*x^3 + c*x^6) + 15*a*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[2/3, 1/2, 1/2, 5/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + 3*b*x^3*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[5/3, 1/2, 1/2, 8/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])]))/(50*Sqrt[a + b*x^3 + c*x^6])","B",0
198,1,335,135,0.3384544,"\int \sqrt{a+b x^3+c x^6} \, dx","Integrate[Sqrt[a + b*x^3 + c*x^6],x]","\frac{x \left(3 b x^3 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)+24 a \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)+8 \left(a+b x^3+c x^6\right)\right)}{32 \sqrt{a+b x^3+c x^6}}","\frac{x \sqrt{a+b x^3+c x^6} F_1\left(\frac{1}{3};-\frac{1}{2},-\frac{1}{2};\frac{4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{\sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1}}",1,"(x*(8*(a + b*x^3 + c*x^6) + 24*a*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[1/3, 1/2, 1/2, 4/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + 3*b*x^3*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[4/3, 1/2, 1/2, 7/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])]))/(32*Sqrt[a + b*x^3 + c*x^6])","B",0
199,1,340,138,0.3546156,"\int \frac{\sqrt{a+b x^3+c x^6}}{x^2} \, dx","Integrate[Sqrt[a + b*x^3 + c*x^6]/x^2,x]","\frac{15 b x^3 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)+12 c x^6 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)-20 \left(a+b x^3+c x^6\right)}{20 x \sqrt{a+b x^3+c x^6}}","-\frac{\sqrt{a+b x^3+c x^6} F_1\left(-\frac{1}{3};-\frac{1}{2},-\frac{1}{2};\frac{2}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{x \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1}}",1,"(-20*(a + b*x^3 + c*x^6) + 15*b*x^3*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[2/3, 1/2, 1/2, 5/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + 12*c*x^6*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[5/3, 1/2, 1/2, 8/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/(20*x*Sqrt[a + b*x^3 + c*x^6])","B",0
200,1,340,140,0.3179946,"\int \frac{\sqrt{a+b x^3+c x^6}}{x^3} \, dx","Integrate[Sqrt[a + b*x^3 + c*x^6]/x^3,x]","\frac{6 b x^3 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)+3 c x^6 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)-4 \left(a+b x^3+c x^6\right)}{8 x^2 \sqrt{a+b x^3+c x^6}}","-\frac{\sqrt{a+b x^3+c x^6} F_1\left(-\frac{2}{3};-\frac{1}{2},-\frac{1}{2};\frac{1}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{2 x^2 \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1}}",1,"(-4*(a + b*x^3 + c*x^6) + 6*b*x^3*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[1/3, 1/2, 1/2, 4/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + 3*c*x^6*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[4/3, 1/2, 1/2, 7/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/(8*x^2*Sqrt[a + b*x^3 + c*x^6])","B",0
201,1,241,293,0.3487288,"\int x^{14} \left(a+b x^3+c x^6\right)^{3/2} \, dx","Integrate[x^14*(a + b*x^3 + c*x^6)^(3/2),x]","\frac{\frac{\left(16 a^2 c^2-72 a b^2 c+33 b^4\right) \left(2 \sqrt{c} \left(b+2 c x^3\right) \sqrt{a+b x^3+c x^6} \left(4 c \left(5 a+2 c x^6\right)-3 b^2+8 b c x^3\right)+3 \left(b^2-4 a c\right)^2 \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)\right)}{4096 c^{11/2}}+\frac{\left(372 a b c-280 a c^2 x^3-231 b^3+330 b^2 c x^3\right) \left(a+b x^3+c x^6\right)^{5/2}}{560 c^3}-\frac{11 b x^6 \left(a+b x^3+c x^6\right)^{5/2}}{14 c}+x^9 \left(a+b x^3+c x^6\right)^{5/2}}{24 c}","\frac{\left(b^2-4 a c\right)^2 \left(16 a^2 c^2-72 a b^2 c+33 b^4\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{32768 c^{13/2}}-\frac{\left(b^2-4 a c\right) \left(16 a^2 c^2-72 a b^2 c+33 b^4\right) \left(b+2 c x^3\right) \sqrt{a+b x^3+c x^6}}{16384 c^6}+\frac{\left(16 a^2 c^2-72 a b^2 c+33 b^4\right) \left(b+2 c x^3\right) \left(a+b x^3+c x^6\right)^{3/2}}{6144 c^5}-\frac{\left(3 b \left(77 b^2-124 a c\right)-10 c x^3 \left(33 b^2-28 a c\right)\right) \left(a+b x^3+c x^6\right)^{5/2}}{13440 c^4}-\frac{11 b x^6 \left(a+b x^3+c x^6\right)^{5/2}}{336 c^2}+\frac{x^9 \left(a+b x^3+c x^6\right)^{5/2}}{24 c}",1,"((-11*b*x^6*(a + b*x^3 + c*x^6)^(5/2))/(14*c) + x^9*(a + b*x^3 + c*x^6)^(5/2) + ((-231*b^3 + 372*a*b*c + 330*b^2*c*x^3 - 280*a*c^2*x^3)*(a + b*x^3 + c*x^6)^(5/2))/(560*c^3) + ((33*b^4 - 72*a*b^2*c + 16*a^2*c^2)*(2*Sqrt[c]*(b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6]*(-3*b^2 + 8*b*c*x^3 + 4*c*(5*a + 2*c*x^6)) + 3*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])]))/(4096*c^(11/2)))/(24*c)","A",1
202,1,192,223,0.1916932,"\int x^{11} \left(a+b x^3+c x^6\right)^{3/2} \, dx","Integrate[x^11*(a + b*x^3 + c*x^6)^(3/2),x]","\frac{-\frac{\left(16 a c-21 b^2+30 b c x^3\right) \left(a+b x^3+c x^6\right)^{5/2}}{40 c^2}+\frac{7 \left(4 a b c-3 b^3\right) \left(2 \sqrt{c} \left(b+2 c x^3\right) \sqrt{a+b x^3+c x^6} \left(4 c \left(5 a+2 c x^6\right)-3 b^2+8 b c x^3\right)+3 \left(b^2-4 a c\right)^2 \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)\right)}{2048 c^{9/2}}+x^6 \left(a+b x^3+c x^6\right)^{5/2}}{21 c}","-\frac{b \left(b^2-4 a c\right)^2 \left(3 b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{2048 c^{11/2}}+\frac{b \left(b^2-4 a c\right) \left(3 b^2-4 a c\right) \left(b+2 c x^3\right) \sqrt{a+b x^3+c x^6}}{1024 c^5}-\frac{b \left(3 b^2-4 a c\right) \left(b+2 c x^3\right) \left(a+b x^3+c x^6\right)^{3/2}}{384 c^4}+\frac{\left(-16 a c+21 b^2-30 b c x^3\right) \left(a+b x^3+c x^6\right)^{5/2}}{840 c^3}+\frac{x^6 \left(a+b x^3+c x^6\right)^{5/2}}{21 c}",1,"(x^6*(a + b*x^3 + c*x^6)^(5/2) - ((-21*b^2 + 16*a*c + 30*b*c*x^3)*(a + b*x^3 + c*x^6)^(5/2))/(40*c^2) + (7*(-3*b^3 + 4*a*b*c)*(2*Sqrt[c]*(b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6]*(-3*b^2 + 8*b*c*x^3 + 4*c*(5*a + 2*c*x^6)) + 3*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])]))/(2048*c^(9/2)))/(21*c)","A",1
203,1,175,204,0.159255,"\int x^8 \left(a+b x^3+c x^6\right)^{3/2} \, dx","Integrate[x^8*(a + b*x^3 + c*x^6)^(3/2),x]","\frac{\frac{\left(7 b^2-4 a c\right) \left(2 \sqrt{c} \left(b+2 c x^3\right) \sqrt{a+b x^3+c x^6} \left(4 c \left(5 a+2 c x^6\right)-3 b^2+8 b c x^3\right)+3 \left(b^2-4 a c\right)^2 \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)\right)}{512 c^{7/2}}+x^3 \left(a+b x^3+c x^6\right)^{5/2}-\frac{7 b \left(a+b x^3+c x^6\right)^{5/2}}{10 c}}{18 c}","\frac{\left(b^2-4 a c\right)^2 \left(7 b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{3072 c^{9/2}}-\frac{\left(b^2-4 a c\right) \left(7 b^2-4 a c\right) \left(b+2 c x^3\right) \sqrt{a+b x^3+c x^6}}{1536 c^4}+\frac{\left(7 b^2-4 a c\right) \left(b+2 c x^3\right) \left(a+b x^3+c x^6\right)^{3/2}}{576 c^3}-\frac{7 b \left(a+b x^3+c x^6\right)^{5/2}}{180 c^2}+\frac{x^3 \left(a+b x^3+c x^6\right)^{5/2}}{18 c}",1,"((-7*b*(a + b*x^3 + c*x^6)^(5/2))/(10*c) + x^3*(a + b*x^3 + c*x^6)^(5/2) + ((7*b^2 - 4*a*c)*(2*Sqrt[c]*(b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6]*(-3*b^2 + 8*b*c*x^3 + 4*c*(5*a + 2*c*x^6)) + 3*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])]))/(512*c^(7/2)))/(18*c)","A",1
204,1,149,150,0.1462555,"\int x^5 \left(a+b x^3+c x^6\right)^{3/2} \, dx","Integrate[x^5*(a + b*x^3 + c*x^6)^(3/2),x]","-\frac{b \left(b^2-4 a c\right) \left(\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)-2 \sqrt{c} \left(b+2 c x^3\right) \sqrt{a+b x^3+c x^6}\right)}{256 c^{7/2}}-\frac{b \left(b+2 c x^3\right) \left(a+b x^3+c x^6\right)^{3/2}}{48 c^2}+\frac{\left(a+b x^3+c x^6\right)^{5/2}}{15 c}","-\frac{b \left(b^2-4 a c\right)^2 \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{256 c^{7/2}}+\frac{b \left(b^2-4 a c\right) \left(b+2 c x^3\right) \sqrt{a+b x^3+c x^6}}{128 c^3}-\frac{b \left(b+2 c x^3\right) \left(a+b x^3+c x^6\right)^{3/2}}{48 c^2}+\frac{\left(a+b x^3+c x^6\right)^{5/2}}{15 c}",1,"-1/48*(b*(b + 2*c*x^3)*(a + b*x^3 + c*x^6)^(3/2))/c^2 + (a + b*x^3 + c*x^6)^(5/2)/(15*c) - (b*(b^2 - 4*a*c)*(-2*Sqrt[c]*(b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6] + (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])]))/(256*c^(7/2))","A",1
205,1,126,124,0.085438,"\int x^2 \left(a+b x^3+c x^6\right)^{3/2} \, dx","Integrate[x^2*(a + b*x^3 + c*x^6)^(3/2),x]","\frac{\frac{3 \left(b^2-4 a c\right) \left(\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)-2 \sqrt{c} \left(b+2 c x^3\right) \sqrt{a+b x^3+c x^6}\right)}{8 c^{3/2}}+2 \left(b+2 c x^3\right) \left(a+b x^3+c x^6\right)^{3/2}}{48 c}","\frac{\left(b^2-4 a c\right)^2 \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{128 c^{5/2}}-\frac{\left(b^2-4 a c\right) \left(b+2 c x^3\right) \sqrt{a+b x^3+c x^6}}{64 c^2}+\frac{\left(b+2 c x^3\right) \left(a+b x^3+c x^6\right)^{3/2}}{24 c}",1,"(2*(b + 2*c*x^3)*(a + b*x^3 + c*x^6)^(3/2) + (3*(b^2 - 4*a*c)*(-2*Sqrt[c]*(b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6] + (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])]))/(8*c^(3/2)))/(48*c)","A",1
206,1,143,155,0.143401,"\int \frac{\left(a+b x^3+c x^6\right)^{3/2}}{x} \, dx","Integrate[(a + b*x^3 + c*x^6)^(3/2)/x,x]","\frac{1}{144} \left(-48 a^{3/2} \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)-\frac{3 b \left(b^2-12 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{c^{3/2}}+\frac{2 \sqrt{a+b x^3+c x^6} \left(8 c \left(4 a+c x^6\right)+3 b^2+14 b c x^3\right)}{c}\right)","-\frac{1}{3} a^{3/2} \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)-\frac{b \left(b^2-12 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{48 c^{3/2}}+\frac{\left(8 a c+b^2+2 b c x^3\right) \sqrt{a+b x^3+c x^6}}{24 c}+\frac{1}{9} \left(a+b x^3+c x^6\right)^{3/2}",1,"((2*Sqrt[a + b*x^3 + c*x^6]*(3*b^2 + 14*b*c*x^3 + 8*c*(4*a + c*x^6)))/c - 48*a^(3/2)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])] - (3*b*(b^2 - 12*a*c)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/c^(3/2))/144","A",1
207,1,134,150,0.1114509,"\int \frac{\left(a+b x^3+c x^6\right)^{3/2}}{x^4} \, dx","Integrate[(a + b*x^3 + c*x^6)^(3/2)/x^4,x]","\frac{1}{24} \left(\frac{3 \left(4 a c+b^2\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{\sqrt{c}}+\frac{2 \sqrt{a+b x^3+c x^6} \left(-4 a+5 b x^3+2 c x^6\right)}{x^3}-12 \sqrt{a} b \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)\right)","\frac{\left(4 a c+b^2\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{8 \sqrt{c}}-\frac{\left(a+b x^3+c x^6\right)^{3/2}}{3 x^3}+\frac{1}{4} \left(3 b+2 c x^3\right) \sqrt{a+b x^3+c x^6}-\frac{1}{2} \sqrt{a} b \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)",1,"((2*Sqrt[a + b*x^3 + c*x^6]*(-4*a + 5*b*x^3 + 2*c*x^6))/x^3 - 12*Sqrt[a]*b*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])] + (3*(b^2 + 4*a*c)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/Sqrt[c])/24","A",1
208,1,134,151,0.1719068,"\int \frac{\left(a+b x^3+c x^6\right)^{3/2}}{x^7} \, dx","Integrate[(a + b*x^3 + c*x^6)^(3/2)/x^7,x]","\frac{1}{24} \left(-\frac{3 \left(4 a c+b^2\right) \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{\sqrt{a}}-\frac{2 \sqrt{a+b x^3+c x^6} \left(2 a+5 b x^3-4 c x^6\right)}{x^6}+12 b \sqrt{c} \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)\right)","-\frac{\left(4 a c+b^2\right) \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{8 \sqrt{a}}-\frac{\left(a+b x^3+c x^6\right)^{3/2}}{6 x^6}-\frac{\left(b-2 c x^3\right) \sqrt{a+b x^3+c x^6}}{4 x^3}+\frac{1}{2} b \sqrt{c} \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)",1,"((-2*(2*a + 5*b*x^3 - 4*c*x^6)*Sqrt[a + b*x^3 + c*x^6])/x^6 - (3*(b^2 + 4*a*c)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/Sqrt[a] + 12*b*Sqrt[c]*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/24","A",1
209,1,149,163,0.2063031,"\int \frac{\left(a+b x^3+c x^6\right)^{3/2}}{x^{10}} \, dx","Integrate[(a + b*x^3 + c*x^6)^(3/2)/x^10,x]","\frac{1}{144} \left(\frac{3 b \left(b^2-12 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{a^{3/2}}-\frac{2 \sqrt{a+b x^3+c x^6} \left(8 a^2+14 a b x^3+32 a c x^6+3 b^2 x^6\right)}{a x^9}+48 c^{3/2} \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)\right)","\frac{b \left(b^2-12 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{48 a^{3/2}}-\frac{\left(x^3 \left(8 a c+b^2\right)+2 a b\right) \sqrt{a+b x^3+c x^6}}{24 a x^6}+\frac{1}{3} c^{3/2} \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)-\frac{\left(a+b x^3+c x^6\right)^{3/2}}{9 x^9}",1,"((-2*Sqrt[a + b*x^3 + c*x^6]*(8*a^2 + 14*a*b*x^3 + 3*b^2*x^6 + 32*a*c*x^6))/(a*x^9) + (3*b*(b^2 - 12*a*c)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/a^(3/2) + 48*c^(3/2)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/144","A",1
210,1,138,133,0.1635162,"\int \frac{\left(a+b x^3+c x^6\right)^{3/2}}{x^{13}} \, dx","Integrate[(a + b*x^3 + c*x^6)^(3/2)/x^13,x]","-\frac{\frac{3 \left(b^2-4 a c\right) \left(x^6 \left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)-2 \sqrt{a} \left(2 a+b x^3\right) \sqrt{a+b x^3+c x^6}\right)}{8 a^{3/2} x^6}+\frac{2 \left(2 a+b x^3\right) \left(a+b x^3+c x^6\right)^{3/2}}{x^{12}}}{48 a}","-\frac{\left(b^2-4 a c\right)^2 \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{128 a^{5/2}}+\frac{\left(b^2-4 a c\right) \left(2 a+b x^3\right) \sqrt{a+b x^3+c x^6}}{64 a^2 x^6}-\frac{\left(2 a+b x^3\right) \left(a+b x^3+c x^6\right)^{3/2}}{24 a x^{12}}",1,"-1/48*((2*(2*a + b*x^3)*(a + b*x^3 + c*x^6)^(3/2))/x^12 + (3*(b^2 - 4*a*c)*(-2*Sqrt[a]*(2*a + b*x^3)*Sqrt[a + b*x^3 + c*x^6] + (b^2 - 4*a*c)*x^6*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])]))/(8*a^(3/2)*x^6))/a","A",1
211,1,167,162,0.1399236,"\int \frac{\left(a+b x^3+c x^6\right)^{3/2}}{x^{16}} \, dx","Integrate[(a + b*x^3 + c*x^6)^(3/2)/x^16,x]","\frac{b \left(16 a^{3/2} \left(2 a+b x^3\right) \left(a+b x^3+c x^6\right)^{3/2}-3 x^6 \left(b^2-4 a c\right) \left(2 \sqrt{a} \left(2 a+b x^3\right) \sqrt{a+b x^3+c x^6}-x^6 \left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)\right)\right)}{768 a^{7/2} x^{12}}-\frac{\left(a+b x^3+c x^6\right)^{5/2}}{15 a x^{15}}","\frac{b \left(b^2-4 a c\right)^2 \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{256 a^{7/2}}-\frac{b \left(b^2-4 a c\right) \left(2 a+b x^3\right) \sqrt{a+b x^3+c x^6}}{128 a^3 x^6}+\frac{b \left(2 a+b x^3\right) \left(a+b x^3+c x^6\right)^{3/2}}{48 a^2 x^{12}}-\frac{\left(a+b x^3+c x^6\right)^{5/2}}{15 a x^{15}}",1,"-1/15*(a + b*x^3 + c*x^6)^(5/2)/(a*x^15) + (b*(16*a^(3/2)*(2*a + b*x^3)*(a + b*x^3 + c*x^6)^(3/2) - 3*(b^2 - 4*a*c)*x^6*(2*Sqrt[a]*(2*a + b*x^3)*Sqrt[a + b*x^3 + c*x^6] - (b^2 - 4*a*c)*x^6*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])))/(768*a^(7/2)*x^12)","A",1
212,1,206,216,0.2095243,"\int \frac{\left(a+b x^3+c x^6\right)^{3/2}}{x^{19}} \, dx","Integrate[(a + b*x^3 + c*x^6)^(3/2)/x^19,x]","-\frac{\frac{\left(\frac{7 b^2}{2}-2 a c\right) \left(16 a^{3/2} \left(2 a+b x^3\right) \left(a+b x^3+c x^6\right)^{3/2}-3 x^6 \left(b^2-4 a c\right) \left(2 \sqrt{a} \left(2 a+b x^3\right) \sqrt{a+b x^3+c x^6}-x^6 \left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)\right)\right)}{256 a^{7/2} x^{12}}+\frac{\left(a+b x^3+c x^6\right)^{5/2}}{x^{18}}-\frac{7 b \left(a+b x^3+c x^6\right)^{5/2}}{10 a x^{15}}}{18 a}","-\frac{\left(b^2-4 a c\right)^2 \left(7 b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{3072 a^{9/2}}+\frac{\left(b^2-4 a c\right) \left(7 b^2-4 a c\right) \left(2 a+b x^3\right) \sqrt{a+b x^3+c x^6}}{1536 a^4 x^6}-\frac{\left(7 b^2-4 a c\right) \left(2 a+b x^3\right) \left(a+b x^3+c x^6\right)^{3/2}}{576 a^3 x^{12}}+\frac{7 b \left(a+b x^3+c x^6\right)^{5/2}}{180 a^2 x^{15}}-\frac{\left(a+b x^3+c x^6\right)^{5/2}}{18 a x^{18}}",1,"-1/18*((a + b*x^3 + c*x^6)^(5/2)/x^18 - (7*b*(a + b*x^3 + c*x^6)^(5/2))/(10*a*x^15) + (((7*b^2)/2 - 2*a*c)*(16*a^(3/2)*(2*a + b*x^3)*(a + b*x^3 + c*x^6)^(3/2) - 3*(b^2 - 4*a*c)*x^6*(2*Sqrt[a]*(2*a + b*x^3)*Sqrt[a + b*x^3 + c*x^6] - (b^2 - 4*a*c)*x^6*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])))/(256*a^(7/2)*x^12))/a","A",1
213,1,243,255,0.4859778,"\int \frac{\left(a+b x^3+c x^6\right)^{3/2}}{x^{22}} \, dx","Integrate[(a + b*x^3 + c*x^6)^(3/2)/x^22,x]","-\frac{\frac{\left(21 a b c-\frac{63 b^3}{4}\right) \left(16 a^{3/2} \left(2 a+b x^3\right) \left(a+b x^3+c x^6\right)^{3/2}-3 x^6 \left(b^2-4 a c\right) \left(2 \sqrt{a} \left(2 a+b x^3\right) \sqrt{a+b x^3+c x^6}-x^6 \left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)\right)\right)}{1536 a^{9/2} x^{12}}+\frac{\left(21 b^2-16 a c\right) \left(a+b x^3+c x^6\right)^{5/2}}{40 a^2 x^{15}}+\frac{\left(a+b x^3+c x^6\right)^{5/2}}{x^{21}}-\frac{3 b \left(a+b x^3+c x^6\right)^{5/2}}{4 a x^{18}}}{21 a}","\frac{b \left(b^2-4 a c\right)^2 \left(3 b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{2048 a^{11/2}}-\frac{b \left(b^2-4 a c\right) \left(3 b^2-4 a c\right) \left(2 a+b x^3\right) \sqrt{a+b x^3+c x^6}}{1024 a^5 x^6}+\frac{b \left(3 b^2-4 a c\right) \left(2 a+b x^3\right) \left(a+b x^3+c x^6\right)^{3/2}}{384 a^4 x^{12}}-\frac{\left(21 b^2-16 a c\right) \left(a+b x^3+c x^6\right)^{5/2}}{840 a^3 x^{15}}+\frac{b \left(a+b x^3+c x^6\right)^{5/2}}{28 a^2 x^{18}}-\frac{\left(a+b x^3+c x^6\right)^{5/2}}{21 a x^{21}}",1,"-1/21*((a + b*x^3 + c*x^6)^(5/2)/x^21 - (3*b*(a + b*x^3 + c*x^6)^(5/2))/(4*a*x^18) + ((21*b^2 - 16*a*c)*(a + b*x^3 + c*x^6)^(5/2))/(40*a^2*x^15) + (((-63*b^3)/4 + 21*a*b*c)*(16*a^(3/2)*(2*a + b*x^3)*(a + b*x^3 + c*x^6)^(3/2) - 3*(b^2 - 4*a*c)*x^6*(2*Sqrt[a]*(2*a + b*x^3)*Sqrt[a + b*x^3 + c*x^6] - (b^2 - 4*a*c)*x^6*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])))/(1536*a^(9/2)*x^12))/a","A",1
214,1,453,141,0.8716558,"\int x^3 \left(a+b x^3+c x^6\right)^{3/2} \, dx","Integrate[x^3*(a + b*x^3 + c*x^6)^(3/2),x]","\frac{x \left(27 x^3 \left(640 a^2 c^2-404 a b^2 c+55 b^4\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)+8 \left(4 a^2 c \left(459 b+1280 c x^3\right)+a \left(-297 b^3+2052 b^2 c x^3+10204 b c^2 x^6+7360 c^3 x^9\right)-297 b^4 x^3-81 b^3 c x^6+3464 b^2 c^2 x^9+5488 b c^3 x^{12}+2240 c^4 x^{15}\right)+216 a b \left(11 b^2-68 a c\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)\right)}{232960 c^2 \sqrt{a+b x^3+c x^6}}","\frac{a x^4 \sqrt{a+b x^3+c x^6} F_1\left(\frac{4}{3};-\frac{3}{2},-\frac{3}{2};\frac{7}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{4 \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1}}",1,"(x*(8*(-297*b^4*x^3 - 81*b^3*c*x^6 + 3464*b^2*c^2*x^9 + 5488*b*c^3*x^12 + 2240*c^4*x^15 + 4*a^2*c*(459*b + 1280*c*x^3) + a*(-297*b^3 + 2052*b^2*c*x^3 + 10204*b*c^2*x^6 + 7360*c^3*x^9)) + 216*a*b*(11*b^2 - 68*a*c)*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[1/3, 1/2, 1/2, 4/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + 27*(55*b^4 - 404*a*b^2*c + 640*a^2*c^2)*x^3*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[4/3, 1/2, 1/2, 7/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])]))/(232960*c^2*Sqrt[a + b*x^3 + c*x^6])","B",0
215,1,410,141,0.6964224,"\int x \left(a+b x^3+c x^6\right)^{3/2} \, dx","Integrate[x*(a + b*x^3 + c*x^6)^(3/2),x]","\frac{x^2 \left(10 \left(448 a^2 c+27 a b^2+698 a b c x^3+608 a c^2 x^6+27 b^3 x^3+277 b^2 c x^6+410 b c^2 x^9+160 c^3 x^{12}\right)-27 b x^3 \left(7 b^2-52 a c\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)-270 a \left(b^2-16 a c\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)\right)}{17600 c \sqrt{a+b x^3+c x^6}}","\frac{a x^2 \sqrt{a+b x^3+c x^6} F_1\left(\frac{2}{3};-\frac{3}{2},-\frac{3}{2};\frac{5}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{2 \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1}}",1,"(x^2*(10*(27*a*b^2 + 448*a^2*c + 27*b^3*x^3 + 698*a*b*c*x^3 + 277*b^2*c*x^6 + 608*a*c^2*x^6 + 410*b*c^2*x^9 + 160*c^3*x^12) - 270*a*(b^2 - 16*a*c)*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[2/3, 1/2, 1/2, 5/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] - 27*b*(7*b^2 - 52*a*c)*x^3*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[5/3, 1/2, 1/2, 8/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])]))/(17600*c*Sqrt[a + b*x^3 + c*x^6])","B",0
216,1,408,136,0.6704689,"\int \left(a+b x^3+c x^6\right)^{3/2} \, dx","Integrate[(a + b*x^3 + c*x^6)^(3/2),x]","\frac{x \left(8 \left(364 a^2 c+27 a b^2+548 a b c x^3+476 a c^2 x^6+27 b^3 x^3+211 b^2 c x^6+296 b c^2 x^9+112 c^3 x^{12}\right)-27 b x^3 \left(5 b^2-44 a c\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)-216 a \left(b^2-28 a c\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)\right)}{8960 c \sqrt{a+b x^3+c x^6}}","\frac{a x \sqrt{a+b x^3+c x^6} F_1\left(\frac{1}{3};-\frac{3}{2},-\frac{3}{2};\frac{4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{\sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1}}",1,"(x*(8*(27*a*b^2 + 364*a^2*c + 27*b^3*x^3 + 548*a*b*c*x^3 + 211*b^2*c*x^6 + 476*a*c^2*x^6 + 296*b*c^2*x^9 + 112*c^3*x^12) - 216*a*(b^2 - 28*a*c)*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[1/3, 1/2, 1/2, 4/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] - 27*b*(5*b^2 - 44*a*c)*x^3*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[4/3, 1/2, 1/2, 7/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])]))/(8960*c*Sqrt[a + b*x^3 + c*x^6])","B",0
217,1,379,139,0.4831227,"\int \frac{\left(a+b x^3+c x^6\right)^{3/2}}{x^2} \, dx","Integrate[(a + b*x^3 + c*x^6)^(3/2)/x^2,x]","\frac{10 \left(-80 a^2-61 a b x^3-70 a c x^6+19 b^2 x^6+29 b c x^9+10 c^2 x^{12}\right)+810 a b x^3 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)+27 x^6 \left(20 a c+b^2\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)}{800 x \sqrt{a+b x^3+c x^6}}","-\frac{a \sqrt{a+b x^3+c x^6} F_1\left(-\frac{1}{3};-\frac{3}{2},-\frac{3}{2};\frac{2}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{x \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1}}",1,"(10*(-80*a^2 - 61*a*b*x^3 + 19*b^2*x^6 - 70*a*c*x^6 + 29*b*c*x^9 + 10*c^2*x^12) + 810*a*b*x^3*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[2/3, 1/2, 1/2, 5/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + 27*(b^2 + 20*a*c)*x^6*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[5/3, 1/2, 1/2, 8/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/(800*x*Sqrt[a + b*x^3 + c*x^6])","B",0
218,1,379,141,0.4822316,"\int \frac{\left(a+b x^3+c x^6\right)^{3/2}}{x^3} \, dx","Integrate[(a + b*x^3 + c*x^6)^(3/2)/x^3,x]","\frac{8 \left(-28 a^2-11 a b x^3-20 a c x^6+17 b^2 x^6+25 b c x^9+8 c^2 x^{12}\right)+648 a b x^3 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)+27 x^6 \left(8 a c+b^2\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)}{448 x^2 \sqrt{a+b x^3+c x^6}}","-\frac{a \sqrt{a+b x^3+c x^6} F_1\left(-\frac{2}{3};-\frac{3}{2},-\frac{3}{2};\frac{1}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{2 x^2 \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1}}",1,"(8*(-28*a^2 - 11*a*b*x^3 + 17*b^2*x^6 - 20*a*c*x^6 + 25*b*c*x^9 + 8*c^2*x^12) + 648*a*b*x^3*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[1/3, 1/2, 1/2, 4/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + 27*(b^2 + 8*a*c)*x^6*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[4/3, 1/2, 1/2, 7/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/(448*x^2*Sqrt[a + b*x^3 + c*x^6])","B",0
219,1,137,171,0.110789,"\int \frac{x^{14}}{\sqrt{a+b x^3+c x^6}} \, dx","Integrate[x^14/Sqrt[a + b*x^3 + c*x^6],x]","\frac{3 \left(48 a^2 c^2-120 a b^2 c+35 b^4\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)+2 \sqrt{c} \sqrt{a+b x^3+c x^6} \left(4 b c \left(55 a-14 c x^6\right)+24 c^2 x^3 \left(2 c x^6-3 a\right)-105 b^3+70 b^2 c x^3\right)}{1152 c^{9/2}}","\frac{\left(48 a^2 c^2-120 a b^2 c+35 b^4\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{384 c^{9/2}}-\frac{\left(5 b \left(21 b^2-44 a c\right)-2 c x^3 \left(35 b^2-36 a c\right)\right) \sqrt{a+b x^3+c x^6}}{576 c^4}-\frac{7 b x^6 \sqrt{a+b x^3+c x^6}}{72 c^2}+\frac{x^9 \sqrt{a+b x^3+c x^6}}{12 c}",1,"(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6]*(-105*b^3 + 70*b^2*c*x^3 + 4*b*c*(55*a - 14*c*x^6) + 24*c^2*x^3*(-3*a + 2*c*x^6)) + 3*(35*b^4 - 120*a*b^2*c + 48*a^2*c^2)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(1152*c^(9/2))","A",1
220,1,104,121,0.050453,"\int \frac{x^{11}}{\sqrt{a+b x^3+c x^6}} \, dx","Integrate[x^11/Sqrt[a + b*x^3 + c*x^6],x]","\frac{\left(36 a b c-15 b^3\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)+2 \sqrt{c} \sqrt{a+b x^3+c x^6} \left(8 c \left(c x^6-2 a\right)+15 b^2-10 b c x^3\right)}{144 c^{7/2}}","-\frac{b \left(5 b^2-12 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{48 c^{7/2}}+\frac{\left(-16 a c+15 b^2-10 b c x^3\right) \sqrt{a+b x^3+c x^6}}{72 c^3}+\frac{x^6 \sqrt{a+b x^3+c x^6}}{9 c}",1,"(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6]*(15*b^2 - 10*b*c*x^3 + 8*c*(-2*a + c*x^6)) + (-15*b^3 + 36*a*b*c)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(144*c^(7/2))","A",1
221,1,88,104,0.0329483,"\int \frac{x^8}{\sqrt{a+b x^3+c x^6}} \, dx","Integrate[x^8/Sqrt[a + b*x^3 + c*x^6],x]","\frac{\left(3 b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)+2 \sqrt{c} \left(2 c x^3-3 b\right) \sqrt{a+b x^3+c x^6}}{24 c^{5/2}}","\frac{\left(3 b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{24 c^{5/2}}-\frac{b \sqrt{a+b x^3+c x^6}}{4 c^2}+\frac{x^3 \sqrt{a+b x^3+c x^6}}{6 c}",1,"(2*Sqrt[c]*(-3*b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6] + (3*b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(24*c^(5/2))","A",1
222,1,68,68,0.0156873,"\int \frac{x^5}{\sqrt{a+b x^3+c x^6}} \, dx","Integrate[x^5/Sqrt[a + b*x^3 + c*x^6],x]","\frac{\sqrt{a+b x^3+c x^6}}{3 c}-\frac{b \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{6 c^{3/2}}","\frac{\sqrt{a+b x^3+c x^6}}{3 c}-\frac{b \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{6 c^{3/2}}",1,"Sqrt[a + b*x^3 + c*x^6]/(3*c) - (b*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(6*c^(3/2))","A",1
223,1,43,43,0.0055582,"\int \frac{x^2}{\sqrt{a+b x^3+c x^6}} \, dx","Integrate[x^2/Sqrt[a + b*x^3 + c*x^6],x]","\frac{\tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{3 \sqrt{c}}","\frac{\tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{3 \sqrt{c}}",1,"ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])]/(3*Sqrt[c])","A",1
224,1,44,44,0.0059867,"\int \frac{1}{x \sqrt{a+b x^3+c x^6}} \, dx","Integrate[1/(x*Sqrt[a + b*x^3 + c*x^6]),x]","-\frac{\tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{3 \sqrt{a}}","-\frac{\tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{3 \sqrt{a}}",1,"-1/3*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])]/Sqrt[a]","A",1
225,1,72,72,0.0239811,"\int \frac{1}{x^4 \sqrt{a+b x^3+c x^6}} \, dx","Integrate[1/(x^4*Sqrt[a + b*x^3 + c*x^6]),x]","\frac{b \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{6 a^{3/2}}-\frac{\sqrt{a+b x^3+c x^6}}{3 a x^3}","\frac{b \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{6 a^{3/2}}-\frac{\sqrt{a+b x^3+c x^6}}{3 a x^3}",1,"-1/3*Sqrt[a + b*x^3 + c*x^6]/(a*x^3) + (b*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(6*a^(3/2))","A",1
226,1,92,108,0.0593362,"\int \frac{1}{x^7 \sqrt{a+b x^3+c x^6}} \, dx","Integrate[1/(x^7*Sqrt[a + b*x^3 + c*x^6]),x]","\frac{\left(4 a c-3 b^2\right) \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)+\frac{2 \sqrt{a} \left(3 b x^3-2 a\right) \sqrt{a+b x^3+c x^6}}{x^6}}{24 a^{5/2}}","-\frac{\left(3 b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{24 a^{5/2}}+\frac{b \sqrt{a+b x^3+c x^6}}{4 a^2 x^3}-\frac{\sqrt{a+b x^3+c x^6}}{6 a x^6}",1,"((2*Sqrt[a]*(-2*a + 3*b*x^3)*Sqrt[a + b*x^3 + c*x^6])/x^6 + (-3*b^2 + 4*a*c)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(24*a^(5/2))","A",1
227,1,112,145,0.0776626,"\int \frac{1}{x^{10} \sqrt{a+b x^3+c x^6}} \, dx","Integrate[1/(x^10*Sqrt[a + b*x^3 + c*x^6]),x]","\frac{b \left(5 b^2-12 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{48 a^{7/2}}+\frac{\sqrt{a+b x^3+c x^6} \left(-8 a^2+2 a \left(5 b x^3+8 c x^6\right)-15 b^2 x^6\right)}{72 a^3 x^9}","\frac{b \left(5 b^2-12 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{48 a^{7/2}}-\frac{\left(15 b^2-16 a c\right) \sqrt{a+b x^3+c x^6}}{72 a^3 x^3}+\frac{5 b \sqrt{a+b x^3+c x^6}}{36 a^2 x^6}-\frac{\sqrt{a+b x^3+c x^6}}{9 a x^9}",1,"(Sqrt[a + b*x^3 + c*x^6]*(-8*a^2 - 15*b^2*x^6 + 2*a*(5*b*x^3 + 8*c*x^6)))/(72*a^3*x^9) + (b*(5*b^2 - 12*a*c)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(48*a^(7/2))","A",1
228,1,141,192,0.0979258,"\int \frac{1}{x^{13} \sqrt{a+b x^3+c x^6}} \, dx","Integrate[1/(x^13*Sqrt[a + b*x^3 + c*x^6]),x]","\frac{\sqrt{a+b x^3+c x^6} \left(-48 a^3+8 a^2 \left(7 b x^3+9 c x^6\right)-10 a b x^6 \left(7 b+22 c x^3\right)+105 b^3 x^9\right)}{576 a^4 x^{12}}-\frac{\left(48 a^2 c^2-120 a b^2 c+35 b^4\right) \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{384 a^{9/2}}","\frac{5 b \left(21 b^2-44 a c\right) \sqrt{a+b x^3+c x^6}}{576 a^4 x^3}-\frac{\left(35 b^2-36 a c\right) \sqrt{a+b x^3+c x^6}}{288 a^3 x^6}+\frac{7 b \sqrt{a+b x^3+c x^6}}{72 a^2 x^9}-\frac{\left(48 a^2 c^2-120 a b^2 c+35 b^4\right) \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{384 a^{9/2}}-\frac{\sqrt{a+b x^3+c x^6}}{12 a x^{12}}",1,"(Sqrt[a + b*x^3 + c*x^6]*(-48*a^3 + 105*b^3*x^9 - 10*a*b*x^6*(7*b + 22*c*x^3) + 8*a^2*(7*b*x^3 + 9*c*x^6)))/(576*a^4*x^12) - ((35*b^4 - 120*a*b^2*c + 48*a^2*c^2)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(384*a^(9/2))","A",1
229,1,168,140,0.0776952,"\int \frac{x^3}{\sqrt{a+b x^3+c x^6}} \, dx","Integrate[x^3/Sqrt[a + b*x^3 + c*x^6],x]","\frac{x^4 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)}{4 \sqrt{a+b x^3+c x^6}}","\frac{x^4 \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{4 \sqrt{a+b x^3+c x^6}}",1,"(x^4*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[4/3, 1/2, 1/2, 7/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/(4*Sqrt[a + b*x^3 + c*x^6])","A",0
230,1,168,140,0.0730179,"\int \frac{x}{\sqrt{a+b x^3+c x^6}} \, dx","Integrate[x/Sqrt[a + b*x^3 + c*x^6],x]","\frac{x^2 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)}{2 \sqrt{a+b x^3+c x^6}}","\frac{x^2 \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{2 \sqrt{a+b x^3+c x^6}}",1,"(x^2*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[2/3, 1/2, 1/2, 5/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/(2*Sqrt[a + b*x^3 + c*x^6])","A",0
231,1,163,135,0.0730221,"\int \frac{1}{\sqrt{a+b x^3+c x^6}} \, dx","Integrate[1/Sqrt[a + b*x^3 + c*x^6],x]","\frac{x \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)}{\sqrt{a+b x^3+c x^6}}","\frac{x \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{\sqrt{a+b x^3+c x^6}}",1,"(x*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[1/3, 1/2, 1/2, 4/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/Sqrt[a + b*x^3 + c*x^6]","A",0
232,1,343,138,0.3450106,"\int \frac{1}{x^2 \sqrt{a+b x^3+c x^6}} \, dx","Integrate[1/(x^2*Sqrt[a + b*x^3 + c*x^6]),x]","\frac{5 b x^3 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)+8 c x^6 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)-20 \left(a+b x^3+c x^6\right)}{20 a x \sqrt{a+b x^3+c x^6}}","-\frac{\sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1} F_1\left(-\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{2}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{x \sqrt{a+b x^3+c x^6}}",1,"(-20*(a + b*x^3 + c*x^6) + 5*b*x^3*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[2/3, 1/2, 1/2, 5/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + 8*c*x^6*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[5/3, 1/2, 1/2, 8/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/(20*a*x*Sqrt[a + b*x^3 + c*x^6])","B",0
233,1,342,140,0.2992299,"\int \frac{1}{x^3 \sqrt{a+b x^3+c x^6}} \, dx","Integrate[1/(x^3*Sqrt[a + b*x^3 + c*x^6]),x]","\frac{-2 b x^3 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)+c x^6 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)-4 \left(a+b x^3+c x^6\right)}{8 a x^2 \sqrt{a+b x^3+c x^6}}","-\frac{\sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1} F_1\left(-\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{1}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{2 x^2 \sqrt{a+b x^3+c x^6}}",1,"(-4*(a + b*x^3 + c*x^6) - 2*b*x^3*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[1/3, 1/2, 1/2, 4/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + c*x^6*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[4/3, 1/2, 1/2, 7/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/(8*a*x^2*Sqrt[a + b*x^3 + c*x^6])","B",0
234,1,181,195,0.1813909,"\int \frac{x^{14}}{\left(a+b x^3+c x^6\right)^{3/2}} \, dx","Integrate[x^14/(a + b*x^3 + c*x^6)^(3/2),x]","\frac{\frac{2 \sqrt{c} \left(4 a^2 c \left(6 c x^3-13 b\right)+a \left(15 b^3-62 b^2 c x^3-20 b c^2 x^6+8 c^3 x^9\right)+b^2 x^3 \left(15 b^2+5 b c x^3-2 c^2 x^6\right)\right)}{\sqrt{a+b x^3+c x^6}}-3 \left(16 a^2 c^2-24 a b^2 c+5 b^4\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{24 c^{7/2} \left(4 a c-b^2\right)}","\frac{\left(5 b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{8 c^{7/2}}-\frac{\left(b \left(15 b^2-52 a c\right)-2 c x^3 \left(5 b^2-12 a c\right)\right) \sqrt{a+b x^3+c x^6}}{12 c^3 \left(b^2-4 a c\right)}-\frac{2 b x^6 \sqrt{a+b x^3+c x^6}}{3 c \left(b^2-4 a c\right)}+\frac{2 x^9 \left(2 a+b x^3\right)}{3 \left(b^2-4 a c\right) \sqrt{a+b x^3+c x^6}}",1,"((2*Sqrt[c]*(4*a^2*c*(-13*b + 6*c*x^3) + b^2*x^3*(15*b^2 + 5*b*c*x^3 - 2*c^2*x^6) + a*(15*b^3 - 62*b^2*c*x^3 - 20*b*c^2*x^6 + 8*c^3*x^9)))/Sqrt[a + b*x^3 + c*x^6] - 3*(5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(24*c^(7/2)*(-b^2 + 4*a*c))","A",1
235,1,137,137,0.1190635,"\int \frac{x^{11}}{\left(a+b x^3+c x^6\right)^{3/2}} \, dx","Integrate[x^11/(a + b*x^3 + c*x^6)^(3/2),x]","\frac{\frac{2 \sqrt{c} \left(8 a^2 c+a \left(-3 b^2+10 b c x^3+4 c^2 x^6\right)-b^2 x^3 \left(3 b+c x^3\right)\right)}{\sqrt{a+b x^3+c x^6}}+3 b \left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{6 c^{5/2} \left(4 a c-b^2\right)}","\frac{\left(-8 a c+3 b^2-2 b c x^3\right) \sqrt{a+b x^3+c x^6}}{3 c^2 \left(b^2-4 a c\right)}+\frac{2 x^6 \left(2 a+b x^3\right)}{3 \left(b^2-4 a c\right) \sqrt{a+b x^3+c x^6}}-\frac{b \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{2 c^{5/2}}",1,"((2*Sqrt[c]*(8*a^2*c - b^2*x^3*(3*b + c*x^3) + a*(-3*b^2 + 10*b*c*x^3 + 4*c^2*x^6)))/Sqrt[a + b*x^3 + c*x^6] + 3*b*(b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(6*c^(5/2)*(-b^2 + 4*a*c))","A",1
236,1,107,120,0.097817,"\int \frac{x^8}{\left(a+b x^3+c x^6\right)^{3/2}} \, dx","Integrate[x^8/(a + b*x^3 + c*x^6)^(3/2),x]","\frac{\frac{2 \sqrt{c} \left(a \left(b-2 c x^3\right)+b^2 x^3\right)}{\sqrt{a+b x^3+c x^6}}-\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{3 c^{3/2} \left(4 a c-b^2\right)}","\frac{2 x^3 \left(2 a+b x^3\right)}{3 \left(b^2-4 a c\right) \sqrt{a+b x^3+c x^6}}-\frac{2 b \sqrt{a+b x^3+c x^6}}{3 c \left(b^2-4 a c\right)}+\frac{\tanh ^{-1}\left(\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right)}{3 c^{3/2}}",1,"((2*Sqrt[c]*(b^2*x^3 + a*(b - 2*c*x^3)))/Sqrt[a + b*x^3 + c*x^6] - (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(3*c^(3/2)*(-b^2 + 4*a*c))","A",1
237,1,41,39,0.0986641,"\int \frac{x^5}{\left(a+b x^3+c x^6\right)^{3/2}} \, dx","Integrate[x^5/(a + b*x^3 + c*x^6)^(3/2),x]","-\frac{2 \left(2 a+b x^3\right)}{3 \left(4 a c-b^2\right) \sqrt{a+b x^3+c x^6}}","\frac{2 \left(2 a+b x^3\right)}{3 \left(b^2-4 a c\right) \sqrt{a+b x^3+c x^6}}",1,"(-2*(2*a + b*x^3))/(3*(-b^2 + 4*a*c)*Sqrt[a + b*x^3 + c*x^6])","A",1
238,1,38,38,0.0235735,"\int \frac{x^2}{\left(a+b x^3+c x^6\right)^{3/2}} \, dx","Integrate[x^2/(a + b*x^3 + c*x^6)^(3/2),x]","-\frac{2 \left(b+2 c x^3\right)}{3 \left(b^2-4 a c\right) \sqrt{a+b x^3+c x^6}}","-\frac{2 \left(b+2 c x^3\right)}{3 \left(b^2-4 a c\right) \sqrt{a+b x^3+c x^6}}",1,"(-2*(b + 2*c*x^3))/(3*(b^2 - 4*a*c)*Sqrt[a + b*x^3 + c*x^6])","A",1
239,1,92,92,0.1231691,"\int \frac{1}{x \left(a+b x^3+c x^6\right)^{3/2}} \, dx","Integrate[1/(x*(a + b*x^3 + c*x^6)^(3/2)),x]","\frac{1}{3} \left(\frac{2 \left(-2 a c+b^2+b c x^3\right)}{a \left(b^2-4 a c\right) \sqrt{a+b x^3+c x^6}}-\frac{\tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{a^{3/2}}\right)","\frac{2 \left(-2 a c+b^2+b c x^3\right)}{3 a \left(b^2-4 a c\right) \sqrt{a+b x^3+c x^6}}-\frac{\tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{3 a^{3/2}}",1,"((2*(b^2 - 2*a*c + b*c*x^3))/(a*(b^2 - 4*a*c)*Sqrt[a + b*x^3 + c*x^6]) - ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])]/a^(3/2))/3","A",1
240,1,137,142,0.0834405,"\int \frac{1}{x^4 \left(a+b x^3+c x^6\right)^{3/2}} \, dx","Integrate[1/(x^4*(a + b*x^3 + c*x^6)^(3/2)),x]","\frac{\frac{2 \sqrt{a} \left(-4 a^2 c+a \left(b^2-10 b c x^3-8 c^2 x^6\right)+3 b^2 x^3 \left(b+c x^3\right)\right)}{x^3 \sqrt{a+b x^3+c x^6}}-3 b \left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{6 a^{5/2} \left(4 a c-b^2\right)}","\frac{b \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{2 a^{5/2}}-\frac{\left(3 b^2-8 a c\right) \sqrt{a+b x^3+c x^6}}{3 a^2 x^3 \left(b^2-4 a c\right)}+\frac{2 \left(-2 a c+b^2+b c x^3\right)}{3 a x^3 \left(b^2-4 a c\right) \sqrt{a+b x^3+c x^6}}",1,"((2*Sqrt[a]*(-4*a^2*c + 3*b^2*x^3*(b + c*x^3) + a*(b^2 - 10*b*c*x^3 - 8*c^2*x^6)))/(x^3*Sqrt[a + b*x^3 + c*x^6]) - 3*b*(b^2 - 4*a*c)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(6*a^(5/2)*(-b^2 + 4*a*c))","A",1
241,1,179,198,0.12636,"\int \frac{1}{x^7 \left(a+b x^3+c x^6\right)^{3/2}} \, dx","Integrate[1/(x^7*(a + b*x^3 + c*x^6)^(3/2)),x]","\frac{3 \left(16 a^2 c^2-24 a b^2 c+5 b^4\right) \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)+\frac{2 \sqrt{a} \left(-8 a^3 c+2 a^2 \left(b^2+10 b c x^3-12 c^2 x^6\right)+a b x^3 \left(-5 b^2+62 b c x^3+52 c^2 x^6\right)-15 b^3 x^6 \left(b+c x^3\right)\right)}{x^6 \sqrt{a+b x^3+c x^6}}}{24 a^{7/2} \left(4 a c-b^2\right)}","-\frac{\left(5 b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{8 a^{7/2}}+\frac{b \left(15 b^2-52 a c\right) \sqrt{a+b x^3+c x^6}}{12 a^3 x^3 \left(b^2-4 a c\right)}-\frac{\left(5 b^2-12 a c\right) \sqrt{a+b x^3+c x^6}}{6 a^2 x^6 \left(b^2-4 a c\right)}+\frac{2 \left(-2 a c+b^2+b c x^3\right)}{3 a x^6 \left(b^2-4 a c\right) \sqrt{a+b x^3+c x^6}}",1,"((2*Sqrt[a]*(-8*a^3*c - 15*b^3*x^6*(b + c*x^3) + 2*a^2*(b^2 + 10*b*c*x^3 - 12*c^2*x^6) + a*b*x^3*(-5*b^2 + 62*b*c*x^3 + 52*c^2*x^6)))/(x^6*Sqrt[a + b*x^3 + c*x^6]) + 3*(5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(24*a^(7/2)*(-b^2 + 4*a*c))","A",1
242,1,223,256,0.1772462,"\int \frac{1}{x^{10} \left(a+b x^3+c x^6\right)^{3/2}} \, dx","Integrate[1/(x^10*(a + b*x^3 + c*x^6)^(3/2)),x]","\frac{\frac{2 \sqrt{a} \left(-32 a^4 c+8 a^3 \left(b^2+7 b c x^3+16 c^2 x^6\right)+2 a^2 x^3 \left(-7 b^3-86 b^2 c x^3+244 b c^2 x^6+128 c^3 x^9\right)+5 a b^2 x^6 \left(7 b^2-106 b c x^3-92 c^2 x^6\right)+105 b^4 x^9 \left(b+c x^3\right)\right)}{x^9 \sqrt{a+b x^3+c x^6}}-15 b \left(48 a^2 c^2-40 a b^2 c+7 b^4\right) \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{144 a^{9/2} \left(4 a c-b^2\right)}","\frac{5 b \left(7 b^2-12 a c\right) \tanh ^{-1}\left(\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right)}{48 a^{9/2}}+\frac{b \left(35 b^2-116 a c\right) \sqrt{a+b x^3+c x^6}}{36 a^3 x^6 \left(b^2-4 a c\right)}-\frac{\left(7 b^2-16 a c\right) \sqrt{a+b x^3+c x^6}}{9 a^2 x^9 \left(b^2-4 a c\right)}-\frac{\left(256 a^2 c^2-460 a b^2 c+105 b^4\right) \sqrt{a+b x^3+c x^6}}{72 a^4 x^3 \left(b^2-4 a c\right)}+\frac{2 \left(-2 a c+b^2+b c x^3\right)}{3 a x^9 \left(b^2-4 a c\right) \sqrt{a+b x^3+c x^6}}",1,"((2*Sqrt[a]*(-32*a^4*c + 105*b^4*x^9*(b + c*x^3) + 5*a*b^2*x^6*(7*b^2 - 106*b*c*x^3 - 92*c^2*x^6) + 8*a^3*(b^2 + 7*b*c*x^3 + 16*c^2*x^6) + 2*a^2*x^3*(-7*b^3 - 86*b^2*c*x^3 + 244*b*c^2*x^6 + 128*c^3*x^9)))/(x^9*Sqrt[a + b*x^3 + c*x^6]) - 15*b*(7*b^4 - 40*a*b^2*c + 48*a^2*c^2)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(144*a^(9/2)*(-b^2 + 4*a*c))","A",1
243,1,340,143,0.3175618,"\int \frac{x^3}{\left(a+b x^3+c x^6\right)^{3/2}} \, dx","Integrate[x^3/(a + b*x^3 + c*x^6)^(3/2),x]","\frac{x \left(c x^3 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)+2 b \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)-2 \left(b+2 c x^3\right)\right)}{3 \left(b^2-4 a c\right) \sqrt{a+b x^3+c x^6}}","\frac{x^4 \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{4}{3};\frac{3}{2},\frac{3}{2};\frac{7}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{4 a \sqrt{a+b x^3+c x^6}}",1,"(x*(-2*(b + 2*c*x^3) + 2*b*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[1/3, 1/2, 1/2, 4/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + c*x^3*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[4/3, 1/2, 1/2, 7/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])]))/(3*(b^2 - 4*a*c)*Sqrt[a + b*x^3 + c*x^6])","B",0
244,1,362,143,0.480924,"\int \frac{x}{\left(a+b x^3+c x^6\right)^{3/2}} \, dx","Integrate[x/(a + b*x^3 + c*x^6)^(3/2),x]","\frac{x^2 \left(8 b c x^3 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)+5 \left(4 a c+b^2\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)-20 \left(-2 a c+b^2+b c x^3\right)\right)}{30 a \left(4 a c-b^2\right) \sqrt{a+b x^3+c x^6}}","\frac{x^2 \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{2}{3};\frac{3}{2},\frac{3}{2};\frac{5}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{2 a \sqrt{a+b x^3+c x^6}}",1,"(x^2*(-20*(b^2 - 2*a*c + b*c*x^3) + 5*(b^2 + 4*a*c)*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[2/3, 1/2, 1/2, 5/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + 8*b*c*x^3*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[5/3, 1/2, 1/2, 8/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])]))/(30*a*(-b^2 + 4*a*c)*Sqrt[a + b*x^3 + c*x^6])","B",0
245,1,359,138,0.4652647,"\int \frac{1}{\left(a+b x^3+c x^6\right)^{3/2}} \, dx","Integrate[(a + b*x^3 + c*x^6)^(-3/2),x]","\frac{x \left(b c x^3 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)-2 \left(b^2-8 a c\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)-4 \left(-2 a c+b^2+b c x^3\right)\right)}{6 a \left(4 a c-b^2\right) \sqrt{a+b x^3+c x^6}}","\frac{x \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{1}{3};\frac{3}{2},\frac{3}{2};\frac{4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{a \sqrt{a+b x^3+c x^6}}",1,"(x*(-4*(b^2 - 2*a*c + b*c*x^3) - 2*(b^2 - 8*a*c)*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[1/3, 1/2, 1/2, 4/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + b*c*x^3*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[4/3, 1/2, 1/2, 7/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])]))/(6*a*(-b^2 + 4*a*c)*Sqrt[a + b*x^3 + c*x^6])","B",0
246,1,407,141,0.7147428,"\int \frac{1}{x^2 \left(a+b x^3+c x^6\right)^{3/2}} \, dx","Integrate[1/(x^2*(a + b*x^3 + c*x^6)^(3/2)),x]","-\frac{5 b x^3 \left(12 a c-5 b^2\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)-4 \left(60 a^2 c+2 c x^6 \left(5 b^2-16 a c\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)+5 a \left(-3 b^2+18 b c x^3+16 c^2 x^6\right)-25 b^2 x^3 \left(b+c x^3\right)\right)}{60 a^2 x \left(b^2-4 a c\right) \sqrt{a+b x^3+c x^6}}","-\frac{\sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1} F_1\left(-\frac{1}{3};\frac{3}{2},\frac{3}{2};\frac{2}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{a x \sqrt{a+b x^3+c x^6}}",1,"-1/60*(5*b*(-5*b^2 + 12*a*c)*x^3*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[2/3, 1/2, 1/2, 5/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] - 4*(60*a^2*c - 25*b^2*x^3*(b + c*x^3) + 5*a*(-3*b^2 + 18*b*c*x^3 + 16*c^2*x^6) + 2*c*(5*b^2 - 16*a*c)*x^6*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[5/3, 1/2, 1/2, 8/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])]))/(a^2*(b^2 - 4*a*c)*x*Sqrt[a + b*x^3 + c*x^6])","B",0
247,1,405,143,0.6418718,"\int \frac{1}{x^3 \left(a+b x^3+c x^6\right)^{3/2}} \, dx","Integrate[1/(x^3*(a + b*x^3 + c*x^6)^(3/2)),x]","\frac{-48 a^2 c+2 b x^3 \left(7 b^2-36 a c\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)+c x^6 \left(20 a c-7 b^2\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)+4 a \left(3 b^2-24 b c x^3-20 c^2 x^6\right)+28 b^2 x^3 \left(b+c x^3\right)}{24 a^2 x^2 \left(4 a c-b^2\right) \sqrt{a+b x^3+c x^6}}","-\frac{\sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1} F_1\left(-\frac{2}{3};\frac{3}{2},\frac{3}{2};\frac{1}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{2 a x^2 \sqrt{a+b x^3+c x^6}}",1,"(-48*a^2*c + 28*b^2*x^3*(b + c*x^3) + 4*a*(3*b^2 - 24*b*c*x^3 - 20*c^2*x^6) + 2*b*(7*b^2 - 36*a*c)*x^3*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[1/3, 1/2, 1/2, 4/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + c*(-7*b^2 + 20*a*c)*x^6*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[4/3, 1/2, 1/2, 7/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/(24*a^2*(-b^2 + 4*a*c)*x^2*Sqrt[a + b*x^3 + c*x^6])","B",0
248,1,70,101,0.0712602,"\int (d x)^m \left(a+b x^3+c x^6\right)^2 \, dx","Integrate[(d*x)^m*(a + b*x^3 + c*x^6)^2,x]","x (d x)^m \left(\frac{a^2}{m+1}+\frac{x^6 \left(2 a c+b^2\right)}{m+7}+\frac{2 a b x^3}{m+4}+\frac{2 b c x^9}{m+10}+\frac{c^2 x^{12}}{m+13}\right)","\frac{a^2 (d x)^{m+1}}{d (m+1)}+\frac{\left(2 a c+b^2\right) (d x)^{m+7}}{d^7 (m+7)}+\frac{2 a b (d x)^{m+4}}{d^4 (m+4)}+\frac{2 b c (d x)^{m+10}}{d^{10} (m+10)}+\frac{c^2 (d x)^{m+13}}{d^{13} (m+13)}",1,"x*(d*x)^m*(a^2/(1 + m) + (2*a*b*x^3)/(4 + m) + ((b^2 + 2*a*c)*x^6)/(7 + m) + (2*b*c*x^9)/(10 + m) + (c^2*x^12)/(13 + m))","A",1
249,1,35,52,0.0287939,"\int (d x)^m \left(a+b x^3+c x^6\right) \, dx","Integrate[(d*x)^m*(a + b*x^3 + c*x^6),x]","x (d x)^m \left(\frac{a}{m+1}+\frac{b x^3}{m+4}+\frac{c x^6}{m+7}\right)","\frac{a (d x)^{m+1}}{d (m+1)}+\frac{b (d x)^{m+4}}{d^4 (m+4)}+\frac{c (d x)^{m+7}}{d^7 (m+7)}",1,"x*(d*x)^m*(a/(1 + m) + (b*x^3)/(4 + m) + (c*x^6)/(7 + m))","A",1
250,1,84,173,0.0608474,"\int \frac{(d x)^m}{a+b x^3+c x^6} \, dx","Integrate[(d*x)^m/(a + b*x^3 + c*x^6),x]","\frac{(d x)^m \text{RootSum}\left[\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{\left(\frac{x}{x-\text{$\#$1}}\right)^{-m} \, _2F_1\left(-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right)}{2 \text{$\#$1}^5 c+\text{$\#$1}^2 b}\&\right]}{3 m}","\frac{2 c (d x)^{m+1} \, _2F_1\left(1,\frac{m+1}{3};\frac{m+4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}\right)}{d (m+1) \sqrt{b^2-4 a c} \left(b-\sqrt{b^2-4 a c}\right)}-\frac{2 c (d x)^{m+1} \, _2F_1\left(1,\frac{m+1}{3};\frac{m+4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{d (m+1) \sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}+b\right)}",1,"((d*x)^m*RootSum[a + b*#1^3 + c*#1^6 & , Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1))]/((x/(x - #1))^m*(b*#1^2 + 2*c*#1^5)) & ])/(3*m)","C",0
251,1,78,315,0.0810345,"\int \frac{(d x)^m}{\left(a+b x^3+c x^6\right)^2} \, dx","Integrate[(d*x)^m/(a + b*x^3 + c*x^6)^2,x]","\frac{x (d x)^m F_1\left(\frac{m+1}{3};2,2;\frac{m+4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)}{a^2 (m+1)}","\frac{c (d x)^{m+1} \left(b (2-m) \sqrt{b^2-4 a c}-4 a c (5-m)+b^2 (2-m)\right) \, _2F_1\left(1,\frac{m+1}{3};\frac{m+4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}\right)}{3 a d (m+1) \left(b^2-4 a c\right)^{3/2} \left(b-\sqrt{b^2-4 a c}\right)}-\frac{c (d x)^{m+1} \left(-b (2-m) \sqrt{b^2-4 a c}-4 a c (5-m)+b^2 (2-m)\right) \, _2F_1\left(1,\frac{m+1}{3};\frac{m+4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{3 a d (m+1) \left(b^2-4 a c\right)^{3/2} \left(\sqrt{b^2-4 a c}+b\right)}+\frac{(d x)^{m+1} \left(-2 a c+b^2+b c x^3\right)}{3 a d \left(b^2-4 a c\right) \left(a+b x^3+c x^6\right)}",1,"(x*(d*x)^m*AppellF1[(1 + m)/3, 2, 2, (4 + m)/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/(a^2*(1 + m))","C",0
252,1,357,158,0.2705786,"\int (d x)^m \left(a+b x^3+c x^6\right)^{3/2} \, dx","Integrate[(d*x)^m*(a + b*x^3 + c*x^6)^(3/2),x]","\frac{x (d x)^m \sqrt{a+b x^3+c x^6} \left(a \left(m^2+11 m+28\right) F_1\left(\frac{m+1}{3};-\frac{1}{2},-\frac{1}{2};\frac{m+4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)+(m+1) x^3 \left(c (m+4) x^3 F_1\left(\frac{m+7}{3};-\frac{1}{2},-\frac{1}{2};\frac{m+10}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)+b (m+7) F_1\left(\frac{m+4}{3};-\frac{1}{2},-\frac{1}{2};\frac{m+7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)\right)\right)}{(m+1) (m+4) (m+7) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}}}","\frac{a (d x)^{m+1} \sqrt{a+b x^3+c x^6} F_1\left(\frac{m+1}{3};-\frac{3}{2},-\frac{3}{2};\frac{m+4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{d (m+1) \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1}}",1,"(x*(d*x)^m*Sqrt[a + b*x^3 + c*x^6]*(a*(28 + 11*m + m^2)*AppellF1[(1 + m)/3, -1/2, -1/2, (4 + m)/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + (1 + m)*x^3*(b*(7 + m)*AppellF1[(4 + m)/3, -1/2, -1/2, (7 + m)/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + c*(4 + m)*x^3*AppellF1[(7 + m)/3, -1/2, -1/2, (10 + m)/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])))/((1 + m)*(4 + m)*(7 + m)*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])])","B",0
253,1,181,157,0.0943512,"\int (d x)^m \sqrt{a+b x^3+c x^6} \, dx","Integrate[(d*x)^m*Sqrt[a + b*x^3 + c*x^6],x]","\frac{x (d x)^m \sqrt{a+b x^3+c x^6} F_1\left(\frac{m+1}{3};-\frac{1}{2},-\frac{1}{2};\frac{m+4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)}{(m+1) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}}}","\frac{(d x)^{m+1} \sqrt{a+b x^3+c x^6} F_1\left(\frac{m+1}{3};-\frac{1}{2},-\frac{1}{2};\frac{m+4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{d (m+1) \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1}}",1,"(x*(d*x)^m*Sqrt[a + b*x^3 + c*x^6]*AppellF1[(1 + m)/3, -1/2, -1/2, (4 + m)/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/((1 + m)*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])])","A",0
254,1,181,157,0.1054128,"\int \frac{(d x)^m}{\sqrt{a+b x^3+c x^6}} \, dx","Integrate[(d*x)^m/Sqrt[a + b*x^3 + c*x^6],x]","\frac{x (d x)^m \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{m+1}{3};\frac{1}{2},\frac{1}{2};\frac{m+4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)}{(m+1) \sqrt{a+b x^3+c x^6}}","\frac{(d x)^{m+1} \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{m+1}{3};\frac{1}{2},\frac{1}{2};\frac{m+4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{d (m+1) \sqrt{a+b x^3+c x^6}}",1,"(x*(d*x)^m*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[(1 + m)/3, 1/2, 1/2, (4 + m)/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/((1 + m)*Sqrt[a + b*x^3 + c*x^6])","A",0
255,1,221,160,0.1703749,"\int \frac{(d x)^m}{\left(a+b x^3+c x^6\right)^{3/2}} \, dx","Integrate[(d*x)^m/(a + b*x^3 + c*x^6)^(3/2),x]","\frac{x (d x)^m \left(\sqrt{b^2-4 a c}-b-2 c x^3\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}\right)^{3/2} F_1\left(\frac{m+1}{3};\frac{3}{2},\frac{3}{2};\frac{m+4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)}{(m+1) \left(\sqrt{b^2-4 a c}-b\right) \left(a+b x^3+c x^6\right)^{3/2}}","\frac{(d x)^{m+1} \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{m+1}{3};\frac{3}{2},\frac{3}{2};\frac{m+4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{a d (m+1) \sqrt{a+b x^3+c x^6}}",1,"(x*(d*x)^m*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^3)*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]))^(3/2)*AppellF1[(1 + m)/3, 3/2, 3/2, (4 + m)/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/((-b + Sqrt[b^2 - 4*a*c])*(1 + m)*(a + b*x^3 + c*x^6)^(3/2))","A",0
256,1,179,155,0.2398765,"\int (d x)^m \left(a+b x^3+c x^6\right)^p \, dx","Integrate[(d*x)^m*(a + b*x^3 + c*x^6)^p,x]","\frac{x (d x)^m \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}\right)^{-p} \left(a+b x^3+c x^6\right)^p F_1\left(\frac{m+1}{3};-p,-p;\frac{m+4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)}{m+1}","\frac{(d x)^{m+1} \left(\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^3+c x^6\right)^p F_1\left(\frac{m+1}{3};-p,-p;\frac{m+4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{d (m+1)}",1,"(x*(d*x)^m*(a + b*x^3 + c*x^6)^p*AppellF1[(1 + m)/3, -p, -p, (4 + m)/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/((1 + m)*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]))^p)","A",0
257,1,162,224,0.2032469,"\int x^8 \left(a+b x^3+c x^6\right)^p \, dx","Integrate[x^8*(a + b*x^3 + c*x^6)^p,x]","\frac{1}{9} x^9 \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}\right)^{-p} \left(a+b x^3+c x^6\right)^p F_1\left(3;-p,-p;4;-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)","\frac{2^p \left(2 a c-b^2 (p+2)\right) \left(a+b x^3+c x^6\right)^{p+1} \left(-\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}}\right)^{-p-1} \, _2F_1\left(-p,p+1;p+2;\frac{2 c x^3+b+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right)}{3 c^2 (p+1) (2 p+3) \sqrt{b^2-4 a c}}-\frac{b (p+2) \left(a+b x^3+c x^6\right)^{p+1}}{6 c^2 (p+1) (2 p+3)}+\frac{x^3 \left(a+b x^3+c x^6\right)^{p+1}}{3 c (2 p+3)}",1,"(x^9*(a + b*x^3 + c*x^6)^p*AppellF1[3, -p, -p, 4, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/(9*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]))^p)","C",0
258,1,162,161,0.2103483,"\int x^5 \left(a+b x^3+c x^6\right)^p \, dx","Integrate[x^5*(a + b*x^3 + c*x^6)^p,x]","\frac{1}{6} x^6 \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}\right)^{-p} \left(a+b x^3+c x^6\right)^p F_1\left(2;-p,-p;3;-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)","\frac{b 2^p \left(a+b x^3+c x^6\right)^{p+1} \left(-\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}}\right)^{-p-1} \, _2F_1\left(-p,p+1;p+2;\frac{2 c x^3+b+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right)}{3 c (p+1) \sqrt{b^2-4 a c}}+\frac{\left(a+b x^3+c x^6\right)^{p+1}}{6 c (p+1)}",1,"(x^6*(a + b*x^3 + c*x^6)^p*AppellF1[2, -p, -p, 3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/(6*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]))^p)","C",0
259,1,138,130,0.1057374,"\int x^2 \left(a+b x^3+c x^6\right)^p \, dx","Integrate[x^2*(a + b*x^3 + c*x^6)^p,x]","\frac{2^{p-1} \left(-\sqrt{b^2-4 a c}+b+2 c x^3\right) \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}}\right)^{-p} \left(a+b x^3+c x^6\right)^p \, _2F_1\left(-p,p+1;p+2;\frac{-2 c x^3-b+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right)}{3 c (p+1)}","-\frac{2^{p+1} \left(-\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}}\right)^{-p-1} \left(a+b x^3+c x^6\right)^{p+1} \, _2F_1\left(-p,p+1;p+2;\frac{2 c x^3+b+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right)}{3 (p+1) \sqrt{b^2-4 a c}}",1,"(2^(-1 + p)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)*(a + b*x^3 + c*x^6)^p*Hypergeometric2F1[-p, 1 + p, 2 + p, (-b + Sqrt[b^2 - 4*a*c] - 2*c*x^3)/(2*Sqrt[b^2 - 4*a*c])])/(3*c*(1 + p)*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/Sqrt[b^2 - 4*a*c])^p)","A",1
260,1,166,138,0.1577892,"\int x^4 \left(a+b x^3+c x^6\right)^p \, dx","Integrate[x^4*(a + b*x^3 + c*x^6)^p,x]","\frac{1}{5} x^5 \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}\right)^{-p} \left(a+b x^3+c x^6\right)^p F_1\left(\frac{5}{3};-p,-p;\frac{8}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)","\frac{1}{5} x^5 \left(\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^3+c x^6\right)^p F_1\left(\frac{5}{3};-p,-p;\frac{8}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)",1,"(x^5*(a + b*x^3 + c*x^6)^p*AppellF1[5/3, -p, -p, 8/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/(5*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]))^p)","A",0
261,1,166,138,0.1776258,"\int x^3 \left(a+b x^3+c x^6\right)^p \, dx","Integrate[x^3*(a + b*x^3 + c*x^6)^p,x]","\frac{1}{4} x^4 \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}\right)^{-p} \left(a+b x^3+c x^6\right)^p F_1\left(\frac{4}{3};-p,-p;\frac{7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)","\frac{1}{4} x^4 \left(\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^3+c x^6\right)^p F_1\left(\frac{4}{3};-p,-p;\frac{7}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)",1,"(x^4*(a + b*x^3 + c*x^6)^p*AppellF1[4/3, -p, -p, 7/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/(4*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]))^p)","A",0
262,1,166,138,0.1831214,"\int x \left(a+b x^3+c x^6\right)^p \, dx","Integrate[x*(a + b*x^3 + c*x^6)^p,x]","\frac{1}{2} x^2 \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}\right)^{-p} \left(a+b x^3+c x^6\right)^p F_1\left(\frac{2}{3};-p,-p;\frac{5}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)","\frac{1}{2} x^2 \left(\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^3+c x^6\right)^p F_1\left(\frac{2}{3};-p,-p;\frac{5}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)",1,"(x^2*(a + b*x^3 + c*x^6)^p*AppellF1[2/3, -p, -p, 5/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/(2*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]))^p)","A",0
263,1,161,133,0.1469214,"\int \left(a+b x^3+c x^6\right)^p \, dx","Integrate[(a + b*x^3 + c*x^6)^p,x]","x \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}\right)^{-p} \left(a+b x^3+c x^6\right)^p F_1\left(\frac{1}{3};-p,-p;\frac{4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)","x \left(\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^3+c x^6\right)^p F_1\left(\frac{1}{3};-p,-p;\frac{4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)",1,"(x*(a + b*x^3 + c*x^6)^p*AppellF1[1/3, -p, -p, 4/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/(((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]))^p)","A",0
264,1,157,157,0.1995736,"\int \frac{\left(a+b x^3+c x^6\right)^p}{x} \, dx","Integrate[(a + b*x^3 + c*x^6)^p/x,x]","\frac{2^{2 p-1} \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{c x^3}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{c x^3}\right)^{-p} \left(a+b x^3+c x^6\right)^p F_1\left(-2 p;-p,-p;1-2 p;-\frac{b+\sqrt{b^2-4 a c}}{2 c x^3},\frac{\sqrt{b^2-4 a c}-b}{2 c x^3}\right)}{3 p}","\frac{2^{2 p-1} \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{c x^3}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{c x^3}\right)^{-p} \left(a+b x^3+c x^6\right)^p F_1\left(-2 p;-p,-p;1-2 p;-\frac{b-\sqrt{b^2-4 a c}}{2 c x^3},-\frac{b+\sqrt{b^2-4 a c}}{2 c x^3}\right)}{3 p}",1,"(2^(-1 + 2*p)*(a + b*x^3 + c*x^6)^p*AppellF1[-2*p, -p, -p, 1 - 2*p, -1/2*(b + Sqrt[b^2 - 4*a*c])/(c*x^3), (-b + Sqrt[b^2 - 4*a*c])/(2*c*x^3)])/(3*p*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(c*x^3))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(c*x^3))^p)","A",0
265,1,164,136,0.1600661,"\int \frac{\left(a+b x^3+c x^6\right)^p}{x^2} \, dx","Integrate[(a + b*x^3 + c*x^6)^p/x^2,x]","-\frac{\left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}\right)^{-p} \left(a+b x^3+c x^6\right)^p F_1\left(-\frac{1}{3};-p,-p;\frac{2}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)}{x}","-\frac{\left(\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^3+c x^6\right)^p F_1\left(-\frac{1}{3};-p,-p;\frac{2}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{x}",1,"-(((a + b*x^3 + c*x^6)^p*AppellF1[-1/3, -p, -p, 2/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/(x*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]))^p))","A",0
266,1,166,138,0.1772728,"\int \frac{\left(a+b x^3+c x^6\right)^p}{x^3} \, dx","Integrate[(a + b*x^3 + c*x^6)^p/x^3,x]","-\frac{\left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}\right)^{-p} \left(a+b x^3+c x^6\right)^p F_1\left(-\frac{2}{3};-p,-p;\frac{1}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)}{2 x^2}","-\frac{\left(\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^3+c x^6\right)^p F_1\left(-\frac{2}{3};-p,-p;\frac{1}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{2 x^2}",1,"-1/2*((a + b*x^3 + c*x^6)^p*AppellF1[-2/3, -p, -p, 1/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/(x^2*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]))^p)","A",0
267,1,162,164,0.2012262,"\int \frac{\left(a+b x^3+c x^6\right)^p}{x^4} \, dx","Integrate[(a + b*x^3 + c*x^6)^p/x^4,x]","\frac{4^p \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{c x^3}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{c x^3}\right)^{-p} \left(a+b x^3+c x^6\right)^p F_1\left(1-2 p;-p,-p;2-2 p;-\frac{b+\sqrt{b^2-4 a c}}{2 c x^3},\frac{\sqrt{b^2-4 a c}-b}{2 c x^3}\right)}{3 (2 p-1) x^3}","-\frac{4^p \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{c x^3}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{c x^3}\right)^{-p} \left(a+b x^3+c x^6\right)^p F_1\left(1-2 p;-p,-p;2 (1-p);-\frac{b-\sqrt{b^2-4 a c}}{2 c x^3},-\frac{b+\sqrt{b^2-4 a c}}{2 c x^3}\right)}{3 (1-2 p) x^3}",1,"(4^p*(a + b*x^3 + c*x^6)^p*AppellF1[1 - 2*p, -p, -p, 2 - 2*p, -1/2*(b + Sqrt[b^2 - 4*a*c])/(c*x^3), (-b + Sqrt[b^2 - 4*a*c])/(2*c*x^3)])/(3*(-1 + 2*p)*x^3*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(c*x^3))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(c*x^3))^p)","A",0
268,1,166,138,0.1976785,"\int \frac{\left(a+b x^3+c x^6\right)^p}{x^5} \, dx","Integrate[(a + b*x^3 + c*x^6)^p/x^5,x]","-\frac{\left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}\right)^{-p} \left(a+b x^3+c x^6\right)^p F_1\left(-\frac{4}{3};-p,-p;-\frac{1}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)}{4 x^4}","-\frac{\left(\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^3+c x^6\right)^p F_1\left(-\frac{4}{3};-p,-p;-\frac{1}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{4 x^4}",1,"-1/4*((a + b*x^3 + c*x^6)^p*AppellF1[-4/3, -p, -p, -1/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/(x^4*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]))^p)","A",0
269,1,166,138,0.1628854,"\int \frac{\left(a+b x^3+c x^6\right)^p}{x^6} \, dx","Integrate[(a + b*x^3 + c*x^6)^p/x^6,x]","-\frac{\left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}+b}\right)^{-p} \left(a+b x^3+c x^6\right)^p F_1\left(-\frac{5}{3};-p,-p;-\frac{2}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right)}{5 x^5}","-\frac{\left(\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^3+c x^6\right)^p F_1\left(-\frac{5}{3};-p,-p;-\frac{2}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right)}{5 x^5}",1,"-1/5*((a + b*x^3 + c*x^6)^p*AppellF1[-5/3, -p, -p, -2/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/(x^5*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]))^p)","A",0
270,1,164,168,0.2209289,"\int \frac{\left(a+b x^3+c x^6\right)^p}{x^7} \, dx","Integrate[(a + b*x^3 + c*x^6)^p/x^7,x]","\frac{2^{2 p-1} \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{c x^3}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{c x^3}\right)^{-p} \left(a+b x^3+c x^6\right)^p F_1\left(2-2 p;-p,-p;3-2 p;-\frac{b+\sqrt{b^2-4 a c}}{2 c x^3},\frac{\sqrt{b^2-4 a c}-b}{2 c x^3}\right)}{3 (p-1) x^6}","-\frac{2^{2 p-1} \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{c x^3}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{c x^3}\right)^{-p} \left(a+b x^3+c x^6\right)^p F_1\left(2 (1-p);-p,-p;3-2 p;-\frac{b-\sqrt{b^2-4 a c}}{2 c x^3},-\frac{b+\sqrt{b^2-4 a c}}{2 c x^3}\right)}{3 (1-p) x^6}",1,"(2^(-1 + 2*p)*(a + b*x^3 + c*x^6)^p*AppellF1[2 - 2*p, -p, -p, 3 - 2*p, -1/2*(b + Sqrt[b^2 - 4*a*c])/(c*x^3), (-b + Sqrt[b^2 - 4*a*c])/(2*c*x^3)])/(3*(-1 + p)*x^6*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(c*x^3))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(c*x^3))^p)","A",0
271,1,34,32,0.0076944,"\int \frac{x^m}{1+2 x^4+x^8} \, dx","Integrate[x^m/(1 + 2*x^4 + x^8),x]","\frac{x^{m+1} \, _2F_1\left(2,\frac{m+1}{4};\frac{m+1}{4}+1;-x^4\right)}{m+1}","\frac{x^{m+1} \, _2F_1\left(2,\frac{m+1}{4};\frac{m+5}{4};-x^4\right)}{m+1}",1,"(x^(1 + m)*Hypergeometric2F1[2, (1 + m)/4, 1 + (1 + m)/4, -x^4])/(1 + m)","A",1
272,1,24,30,0.014305,"\int \frac{x^9}{1+2 x^4+x^8} \, dx","Integrate[x^9/(1 + 2*x^4 + x^8),x]","\frac{1}{4} \left(x^2 \left(\frac{1}{x^4+1}+2\right)-3 \tan ^{-1}\left(x^2\right)\right)","\frac{3 x^2}{4}-\frac{3}{4} \tan ^{-1}\left(x^2\right)-\frac{x^6}{4 \left(x^4+1\right)}",1,"(x^2*(2 + (1 + x^4)^(-1)) - 3*ArcTan[x^2])/4","A",1
273,1,18,22,0.0056782,"\int \frac{x^7}{1+2 x^4+x^8} \, dx","Integrate[x^7/(1 + 2*x^4 + x^8),x]","\frac{1}{4} \left(\frac{1}{x^4+1}+\log \left(x^4+1\right)\right)","\frac{1}{4 \left(x^4+1\right)}+\frac{1}{4} \log \left(x^4+1\right)",1,"((1 + x^4)^(-1) + Log[1 + x^4])/4","A",1
274,1,23,23,0.0093323,"\int \frac{x^5}{1+2 x^4+x^8} \, dx","Integrate[x^5/(1 + 2*x^4 + x^8),x]","\frac{1}{4} \tan ^{-1}\left(x^2\right)-\frac{x^2}{4 \left(x^4+1\right)}","\frac{1}{4} \tan ^{-1}\left(x^2\right)-\frac{x^2}{4 \left(x^4+1\right)}",1,"-1/4*x^2/(1 + x^4) + ArcTan[x^2]/4","A",1
275,1,11,11,0.0022579,"\int \frac{x^3}{1+2 x^4+x^8} \, dx","Integrate[x^3/(1 + 2*x^4 + x^8),x]","-\frac{1}{4 \left(x^4+1\right)}","-\frac{1}{4 \left(x^4+1\right)}",1,"-1/4*1/(1 + x^4)","A",1
276,1,20,23,0.0054374,"\int \frac{x}{1+2 x^4+x^8} \, dx","Integrate[x/(1 + 2*x^4 + x^8),x]","\frac{1}{4} \left(\tan ^{-1}\left(x^2\right)+\frac{x^2}{x^4+1}\right)","\frac{1}{4} \tan ^{-1}\left(x^2\right)+\frac{x^2}{4 \left(x^4+1\right)}",1,"(x^2/(1 + x^4) + ArcTan[x^2])/4","A",1
277,1,24,24,0.0099483,"\int \frac{1}{x \left(1+2 x^4+x^8\right)} \, dx","Integrate[1/(x*(1 + 2*x^4 + x^8)),x]","\frac{1}{4 \left(x^4+1\right)}-\frac{1}{4} \log \left(x^4+1\right)+\log (x)","\frac{1}{4 \left(x^4+1\right)}-\frac{1}{4} \log \left(x^4+1\right)+\log (x)",1,"1/(4*(1 + x^4)) + Log[x] - Log[1 + x^4]/4","A",1
278,1,30,30,0.0110507,"\int \frac{1}{x^3 \left(1+2 x^4+x^8\right)} \, dx","Integrate[1/(x^3*(1 + 2*x^4 + x^8)),x]","-\frac{1}{2 x^2}+\frac{3}{4} \tan ^{-1}\left(\frac{1}{x^2}\right)-\frac{x^2}{4 \left(x^4+1\right)}","-\frac{3}{4 x^2}-\frac{3}{4} \tan ^{-1}\left(x^2\right)+\frac{1}{4 x^2 \left(x^4+1\right)}",1,"-1/2*1/x^2 - x^2/(4*(1 + x^4)) + (3*ArcTan[x^(-2)])/4","A",1
279,1,33,33,0.011554,"\int \frac{1}{x^5 \left(1+2 x^4+x^8\right)} \, dx","Integrate[1/(x^5*(1 + 2*x^4 + x^8)),x]","-\frac{1}{4 \left(x^4+1\right)}-\frac{1}{4 x^4}+\frac{1}{2} \log \left(x^4+1\right)-2 \log (x)","-\frac{1}{4 \left(x^4+1\right)}-\frac{1}{4 x^4}+\frac{1}{2} \log \left(x^4+1\right)-2 \log (x)",1,"-1/4*1/x^4 - 1/(4*(1 + x^4)) - 2*Log[x] + Log[1 + x^4]/2","A",1
280,1,33,37,0.0106675,"\int \frac{1}{x^7 \left(1+2 x^4+x^8\right)} \, dx","Integrate[1/(x^7*(1 + 2*x^4 + x^8)),x]","-\frac{1}{6 x^6}+\frac{1}{x^2}-\frac{5}{4} \tan ^{-1}\left(\frac{1}{x^2}\right)+\frac{x^2}{4 \left(x^4+1\right)}","-\frac{5}{12 x^6}+\frac{5}{4 x^2}+\frac{5}{4} \tan ^{-1}\left(x^2\right)+\frac{1}{4 x^6 \left(x^4+1\right)}",1,"-1/6*1/x^6 + x^(-2) + x^2/(4*(1 + x^4)) - (5*ArcTan[x^(-2)])/4","A",1
281,1,94,104,0.0681393,"\int \frac{x^8}{1+2 x^4+x^8} \, dx","Integrate[x^8/(1 + 2*x^4 + x^8),x]","\frac{1}{32} \left(\frac{8 x}{x^4+1}+5 \sqrt{2} \log \left(x^2-\sqrt{2} x+1\right)-5 \sqrt{2} \log \left(x^2+\sqrt{2} x+1\right)+32 x+10 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} x\right)-10 \sqrt{2} \tan ^{-1}\left(\sqrt{2} x+1\right)\right)","\frac{5 \log \left(x^2-\sqrt{2} x+1\right)}{16 \sqrt{2}}-\frac{5 \log \left(x^2+\sqrt{2} x+1\right)}{16 \sqrt{2}}-\frac{x^5}{4 \left(x^4+1\right)}+\frac{5 x}{4}+\frac{5 \tan ^{-1}\left(1-\sqrt{2} x\right)}{8 \sqrt{2}}-\frac{5 \tan ^{-1}\left(\sqrt{2} x+1\right)}{8 \sqrt{2}}",1,"(32*x + (8*x)/(1 + x^4) + 10*Sqrt[2]*ArcTan[1 - Sqrt[2]*x] - 10*Sqrt[2]*ArcTan[1 + Sqrt[2]*x] + 5*Sqrt[2]*Log[1 - Sqrt[2]*x + x^2] - 5*Sqrt[2]*Log[1 + Sqrt[2]*x + x^2])/32","A",1
282,1,93,99,0.0607944,"\int \frac{x^6}{1+2 x^4+x^8} \, dx","Integrate[x^6/(1 + 2*x^4 + x^8),x]","\frac{1}{32} \left(3 \sqrt{2} \log \left(x^2-\sqrt{2} x+1\right)-3 \sqrt{2} \log \left(x^2+\sqrt{2} x+1\right)-\frac{8 x^3}{x^4+1}-6 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} x\right)+6 \sqrt{2} \tan ^{-1}\left(\sqrt{2} x+1\right)\right)","\frac{3 \log \left(x^2-\sqrt{2} x+1\right)}{16 \sqrt{2}}-\frac{3 \log \left(x^2+\sqrt{2} x+1\right)}{16 \sqrt{2}}-\frac{x^3}{4 \left(x^4+1\right)}-\frac{3 \tan ^{-1}\left(1-\sqrt{2} x\right)}{8 \sqrt{2}}+\frac{3 \tan ^{-1}\left(\sqrt{2} x+1\right)}{8 \sqrt{2}}",1,"((-8*x^3)/(1 + x^4) - 6*Sqrt[2]*ArcTan[1 - Sqrt[2]*x] + 6*Sqrt[2]*ArcTan[1 + Sqrt[2]*x] + 3*Sqrt[2]*Log[1 - Sqrt[2]*x + x^2] - 3*Sqrt[2]*Log[1 + Sqrt[2]*x + x^2])/32","A",1
283,1,90,97,0.0646521,"\int \frac{x^4}{1+2 x^4+x^8} \, dx","Integrate[x^4/(1 + 2*x^4 + x^8),x]","\frac{1}{32} \left(-\frac{8 x}{x^4+1}-\sqrt{2} \log \left(x^2-\sqrt{2} x+1\right)+\sqrt{2} \log \left(x^2+\sqrt{2} x+1\right)-2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} x\right)+2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} x+1\right)\right)","-\frac{x}{4 \left(x^4+1\right)}-\frac{\log \left(x^2-\sqrt{2} x+1\right)}{16 \sqrt{2}}+\frac{\log \left(x^2+\sqrt{2} x+1\right)}{16 \sqrt{2}}-\frac{\tan ^{-1}\left(1-\sqrt{2} x\right)}{8 \sqrt{2}}+\frac{\tan ^{-1}\left(\sqrt{2} x+1\right)}{8 \sqrt{2}}",1,"((-8*x)/(1 + x^4) - 2*Sqrt[2]*ArcTan[1 - Sqrt[2]*x] + 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*x] - Sqrt[2]*Log[1 - Sqrt[2]*x + x^2] + Sqrt[2]*Log[1 + Sqrt[2]*x + x^2])/32","A",1
284,1,92,99,0.0486081,"\int \frac{x^2}{1+2 x^4+x^8} \, dx","Integrate[x^2/(1 + 2*x^4 + x^8),x]","\frac{1}{32} \left(\sqrt{2} \log \left(x^2-\sqrt{2} x+1\right)-\sqrt{2} \log \left(x^2+\sqrt{2} x+1\right)+\frac{8 x^3}{x^4+1}-2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} x\right)+2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} x+1\right)\right)","\frac{\log \left(x^2-\sqrt{2} x+1\right)}{16 \sqrt{2}}-\frac{\log \left(x^2+\sqrt{2} x+1\right)}{16 \sqrt{2}}+\frac{x^3}{4 \left(x^4+1\right)}-\frac{\tan ^{-1}\left(1-\sqrt{2} x\right)}{8 \sqrt{2}}+\frac{\tan ^{-1}\left(\sqrt{2} x+1\right)}{8 \sqrt{2}}",1,"((8*x^3)/(1 + x^4) - 2*Sqrt[2]*ArcTan[1 - Sqrt[2]*x] + 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*x] + Sqrt[2]*Log[1 - Sqrt[2]*x + x^2] - Sqrt[2]*Log[1 + Sqrt[2]*x + x^2])/32","A",1
285,1,91,97,0.0461038,"\int \frac{1}{1+2 x^4+x^8} \, dx","Integrate[(1 + 2*x^4 + x^8)^(-1),x]","\frac{1}{32} \left(\frac{8 x}{x^4+1}-3 \sqrt{2} \log \left(x^2-\sqrt{2} x+1\right)+3 \sqrt{2} \log \left(x^2+\sqrt{2} x+1\right)-6 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} x\right)+6 \sqrt{2} \tan ^{-1}\left(\sqrt{2} x+1\right)\right)","\frac{x}{4 \left(x^4+1\right)}-\frac{3 \log \left(x^2-\sqrt{2} x+1\right)}{16 \sqrt{2}}+\frac{3 \log \left(x^2+\sqrt{2} x+1\right)}{16 \sqrt{2}}-\frac{3 \tan ^{-1}\left(1-\sqrt{2} x\right)}{8 \sqrt{2}}+\frac{3 \tan ^{-1}\left(\sqrt{2} x+1\right)}{8 \sqrt{2}}",1,"((8*x)/(1 + x^4) - 6*Sqrt[2]*ArcTan[1 - Sqrt[2]*x] + 6*Sqrt[2]*ArcTan[1 + Sqrt[2]*x] - 3*Sqrt[2]*Log[1 - Sqrt[2]*x + x^2] + 3*Sqrt[2]*Log[1 + Sqrt[2]*x + x^2])/32","A",1
286,1,98,106,0.0685563,"\int \frac{1}{x^2 \left(1+2 x^4+x^8\right)} \, dx","Integrate[1/(x^2*(1 + 2*x^4 + x^8)),x]","\frac{1}{32} \left(-5 \sqrt{2} \log \left(x^2-\sqrt{2} x+1\right)+5 \sqrt{2} \log \left(x^2+\sqrt{2} x+1\right)-\frac{8 x^3}{x^4+1}-\frac{32}{x}+10 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} x\right)-10 \sqrt{2} \tan ^{-1}\left(\sqrt{2} x+1\right)\right)","\frac{1}{4 x \left(x^4+1\right)}-\frac{5 \log \left(x^2-\sqrt{2} x+1\right)}{16 \sqrt{2}}+\frac{5 \log \left(x^2+\sqrt{2} x+1\right)}{16 \sqrt{2}}-\frac{5}{4 x}+\frac{5 \tan ^{-1}\left(1-\sqrt{2} x\right)}{8 \sqrt{2}}-\frac{5 \tan ^{-1}\left(\sqrt{2} x+1\right)}{8 \sqrt{2}}",1,"(-32/x - (8*x^3)/(1 + x^4) + 10*Sqrt[2]*ArcTan[1 - Sqrt[2]*x] - 10*Sqrt[2]*ArcTan[1 + Sqrt[2]*x] - 5*Sqrt[2]*Log[1 - Sqrt[2]*x + x^2] + 5*Sqrt[2]*Log[1 + Sqrt[2]*x + x^2])/32","A",1
287,1,96,106,0.0714279,"\int \frac{1}{x^4 \left(1+2 x^4+x^8\right)} \, dx","Integrate[1/(x^4*(1 + 2*x^4 + x^8)),x]","\frac{1}{96} \left(-\frac{24 x}{x^4+1}-\frac{32}{x^3}+21 \sqrt{2} \log \left(x^2-\sqrt{2} x+1\right)-21 \sqrt{2} \log \left(x^2+\sqrt{2} x+1\right)+42 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} x\right)-42 \sqrt{2} \tan ^{-1}\left(\sqrt{2} x+1\right)\right)","-\frac{7}{12 x^3}+\frac{7 \log \left(x^2-\sqrt{2} x+1\right)}{16 \sqrt{2}}-\frac{7 \log \left(x^2+\sqrt{2} x+1\right)}{16 \sqrt{2}}+\frac{1}{4 x^3 \left(x^4+1\right)}+\frac{7 \tan ^{-1}\left(1-\sqrt{2} x\right)}{8 \sqrt{2}}-\frac{7 \tan ^{-1}\left(\sqrt{2} x+1\right)}{8 \sqrt{2}}",1,"(-32/x^3 - (24*x)/(1 + x^4) + 42*Sqrt[2]*ArcTan[1 - Sqrt[2]*x] - 42*Sqrt[2]*ArcTan[1 + Sqrt[2]*x] + 21*Sqrt[2]*Log[1 - Sqrt[2]*x + x^2] - 21*Sqrt[2]*Log[1 + Sqrt[2]*x + x^2])/96","A",1
288,1,103,113,0.0755597,"\int \frac{1}{x^6 \left(1+2 x^4+x^8\right)} \, dx","Integrate[1/(x^6*(1 + 2*x^4 + x^8)),x]","\frac{1}{160} \left(-\frac{32}{x^5}+45 \sqrt{2} \log \left(x^2-\sqrt{2} x+1\right)-45 \sqrt{2} \log \left(x^2+\sqrt{2} x+1\right)+\frac{40 x^3}{x^4+1}+\frac{320}{x}-90 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} x\right)+90 \sqrt{2} \tan ^{-1}\left(\sqrt{2} x+1\right)\right)","-\frac{9}{20 x^5}+\frac{9 \log \left(x^2-\sqrt{2} x+1\right)}{16 \sqrt{2}}-\frac{9 \log \left(x^2+\sqrt{2} x+1\right)}{16 \sqrt{2}}+\frac{1}{4 x^5 \left(x^4+1\right)}+\frac{9}{4 x}-\frac{9 \tan ^{-1}\left(1-\sqrt{2} x\right)}{8 \sqrt{2}}+\frac{9 \tan ^{-1}\left(\sqrt{2} x+1\right)}{8 \sqrt{2}}",1,"(-32/x^5 + 320/x + (40*x^3)/(1 + x^4) - 90*Sqrt[2]*ArcTan[1 - Sqrt[2]*x] + 90*Sqrt[2]*ArcTan[1 + Sqrt[2]*x] + 45*Sqrt[2]*Log[1 - Sqrt[2]*x + x^2] - 45*Sqrt[2]*Log[1 + Sqrt[2]*x + x^2])/160","A",1
289,1,101,113,0.0760744,"\int \frac{1}{x^8 \left(1+2 x^4+x^8\right)} \, dx","Integrate[1/(x^8*(1 + 2*x^4 + x^8)),x]","\frac{1}{672} \left(-\frac{96}{x^7}+\frac{168 x}{x^4+1}+\frac{448}{x^3}-231 \sqrt{2} \log \left(x^2-\sqrt{2} x+1\right)+231 \sqrt{2} \log \left(x^2+\sqrt{2} x+1\right)-462 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} x\right)+462 \sqrt{2} \tan ^{-1}\left(\sqrt{2} x+1\right)\right)","-\frac{11}{28 x^7}+\frac{11}{12 x^3}-\frac{11 \log \left(x^2-\sqrt{2} x+1\right)}{16 \sqrt{2}}+\frac{11 \log \left(x^2+\sqrt{2} x+1\right)}{16 \sqrt{2}}+\frac{1}{4 x^7 \left(x^4+1\right)}-\frac{11 \tan ^{-1}\left(1-\sqrt{2} x\right)}{8 \sqrt{2}}+\frac{11 \tan ^{-1}\left(\sqrt{2} x+1\right)}{8 \sqrt{2}}",1,"(-96/x^7 + 448/x^3 + (168*x)/(1 + x^4) - 462*Sqrt[2]*ArcTan[1 - Sqrt[2]*x] + 462*Sqrt[2]*ArcTan[1 + Sqrt[2]*x] - 231*Sqrt[2]*Log[1 - Sqrt[2]*x + x^2] + 231*Sqrt[2]*Log[1 + Sqrt[2]*x + x^2])/672","A",1
290,1,32,30,0.0060983,"\int \frac{x^m}{1-2 x^4+x^8} \, dx","Integrate[x^m/(1 - 2*x^4 + x^8),x]","\frac{x^{m+1} \, _2F_1\left(2,\frac{m+1}{4};\frac{m+1}{4}+1;x^4\right)}{m+1}","\frac{x^{m+1} \, _2F_1\left(2,\frac{m+1}{4};\frac{m+5}{4};x^4\right)}{m+1}",1,"(x^(1 + m)*Hypergeometric2F1[2, (1 + m)/4, 1 + (1 + m)/4, x^4])/(1 + m)","A",1
291,1,39,32,0.0221712,"\int \frac{x^9}{1-2 x^4+x^8} \, dx","Integrate[x^9/(1 - 2*x^4 + x^8),x]","\frac{1}{8} \left(3 \log \left(1-x^2\right)-3 \log \left(x^2+1\right)+2 \left(\frac{1}{1-x^4}+2\right) x^2\right)","\frac{3 x^2}{4}-\frac{3}{4} \tanh ^{-1}\left(x^2\right)+\frac{x^6}{4 \left(1-x^4\right)}",1,"(2*x^2*(2 + (1 - x^4)^(-1)) + 3*Log[1 - x^2] - 3*Log[1 + x^2])/8","A",1
292,1,22,26,0.0066377,"\int \frac{x^7}{1-2 x^4+x^8} \, dx","Integrate[x^7/(1 - 2*x^4 + x^8),x]","\frac{1}{4} \log \left(x^4-1\right)-\frac{1}{4 \left(x^4-1\right)}","\frac{1}{4 \left(1-x^4\right)}+\frac{1}{4} \log \left(1-x^4\right)",1,"-1/4*1/(-1 + x^4) + Log[-1 + x^4]/4","A",1
293,1,33,25,0.0110393,"\int \frac{x^5}{1-2 x^4+x^8} \, dx","Integrate[x^5/(1 - 2*x^4 + x^8),x]","\frac{1}{8} \left(\log \left(1-x^2\right)-\log \left(x^2+1\right)-\frac{2 x^2}{x^4-1}\right)","\frac{x^2}{4 \left(1-x^4\right)}-\frac{1}{4} \tanh ^{-1}\left(x^2\right)",1,"((-2*x^2)/(-1 + x^4) + Log[1 - x^2] - Log[1 + x^2])/8","A",1
294,1,11,13,0.0024013,"\int \frac{x^3}{1-2 x^4+x^8} \, dx","Integrate[x^3/(1 - 2*x^4 + x^8),x]","-\frac{1}{4 \left(x^4-1\right)}","\frac{1}{4 \left(1-x^4\right)}",1,"-1/4*1/(-1 + x^4)","A",1
295,1,33,25,0.007364,"\int \frac{x}{1-2 x^4+x^8} \, dx","Integrate[x/(1 - 2*x^4 + x^8),x]","\frac{1}{8} \left(-\log \left(1-x^2\right)+\log \left(x^2+1\right)-\frac{2 x^2}{x^4-1}\right)","\frac{1}{4} \tanh ^{-1}\left(x^2\right)+\frac{x^2}{4 \left(1-x^4\right)}",1,"((-2*x^2)/(-1 + x^4) - Log[1 - x^2] + Log[1 + x^2])/8","A",1
296,1,26,28,0.0096974,"\int \frac{1}{x \left(1-2 x^4+x^8\right)} \, dx","Integrate[1/(x*(1 - 2*x^4 + x^8)),x]","-\frac{1}{4 \left(x^4-1\right)}-\frac{1}{4} \log \left(1-x^4\right)+\log (x)","\frac{1}{4 \left(1-x^4\right)}-\frac{1}{4} \log \left(1-x^4\right)+\log (x)",1,"-1/4*1/(-1 + x^4) + Log[x] - Log[1 - x^4]/4","A",1
297,1,41,32,0.0171664,"\int \frac{1}{x^3 \left(1-2 x^4+x^8\right)} \, dx","Integrate[1/(x^3*(1 - 2*x^4 + x^8)),x]","\frac{1}{8} \left(-3 \log \left(1-x^2\right)+3 \log \left(x^2+1\right)+\frac{4-6 x^4}{x^2 \left(x^4-1\right)}\right)","-\frac{3}{4 x^2}+\frac{3}{4} \tanh ^{-1}\left(x^2\right)+\frac{1}{4 x^2 \left(1-x^4\right)}",1,"((4 - 6*x^4)/(x^2*(-1 + x^4)) - 3*Log[1 - x^2] + 3*Log[1 + x^2])/8","A",1
298,1,35,37,0.0120184,"\int \frac{1}{x^5 \left(1-2 x^4+x^8\right)} \, dx","Integrate[1/(x^5*(1 - 2*x^4 + x^8)),x]","-\frac{1}{4 \left(x^4-1\right)}-\frac{1}{4 x^4}-\frac{1}{2} \log \left(1-x^4\right)+2 \log (x)","\frac{1}{4 \left(1-x^4\right)}-\frac{1}{4 x^4}-\frac{1}{2} \log \left(1-x^4\right)+2 \log (x)",1,"-1/4*1/x^4 - 1/(4*(-1 + x^4)) + 2*Log[x] - Log[1 - x^4]/2","A",1
299,1,49,39,0.0142252,"\int \frac{1}{x^7 \left(1-2 x^4+x^8\right)} \, dx","Integrate[1/(x^7*(1 - 2*x^4 + x^8)),x]","-\frac{1}{6 x^6}-\frac{1}{x^2}-\frac{5}{8} \log \left(1-x^2\right)+\frac{5}{8} \log \left(x^2+1\right)-\frac{x^2}{4 \left(x^4-1\right)}","-\frac{5}{12 x^6}-\frac{5}{4 x^2}+\frac{5}{4} \tanh ^{-1}\left(x^2\right)+\frac{1}{4 x^6 \left(1-x^4\right)}",1,"-1/6*1/x^6 - x^(-2) - x^2/(4*(-1 + x^4)) - (5*Log[1 - x^2])/8 + (5*Log[1 + x^2])/8","A",1
300,1,38,34,0.0156802,"\int \frac{x^8}{1-2 x^4+x^8} \, dx","Integrate[x^8/(1 - 2*x^4 + x^8),x]","-\frac{x}{4 \left(x^4-1\right)}+x+\frac{5}{16} \log (1-x)-\frac{5}{16} \log (x+1)-\frac{5}{8} \tan ^{-1}(x)","\frac{x^5}{4 \left(1-x^4\right)}+\frac{5 x}{4}-\frac{5}{8} \tan ^{-1}(x)-\frac{5}{8} \tanh ^{-1}(x)",1,"x - x/(4*(-1 + x^4)) - (5*ArcTan[x])/8 + (5*Log[1 - x])/16 - (5*Log[1 + x])/16","A",1
301,1,35,29,0.0144554,"\int \frac{x^6}{1-2 x^4+x^8} \, dx","Integrate[x^6/(1 - 2*x^4 + x^8),x]","\frac{1}{16} \left(-\frac{4 x^3}{x^4-1}+3 \log (1-x)-3 \log (x+1)+6 \tan ^{-1}(x)\right)","\frac{x^3}{4 \left(1-x^4\right)}+\frac{3}{8} \tan ^{-1}(x)-\frac{3}{8} \tanh ^{-1}(x)",1,"((-4*x^3)/(-1 + x^4) + 6*ArcTan[x] + 3*Log[1 - x] - 3*Log[1 + x])/16","A",1
302,1,31,27,0.0132818,"\int \frac{x^4}{1-2 x^4+x^8} \, dx","Integrate[x^4/(1 - 2*x^4 + x^8),x]","\frac{1}{16} \left(-\frac{4 x}{x^4-1}+\log (1-x)-\log (x+1)-2 \tan ^{-1}(x)\right)","\frac{x}{4 \left(1-x^4\right)}-\frac{1}{8} \tan ^{-1}(x)-\frac{1}{8} \tanh ^{-1}(x)",1,"((-4*x)/(-1 + x^4) - 2*ArcTan[x] + Log[1 - x] - Log[1 + x])/16","A",1
303,1,33,29,0.0116216,"\int \frac{x^2}{1-2 x^4+x^8} \, dx","Integrate[x^2/(1 - 2*x^4 + x^8),x]","\frac{1}{16} \left(-\frac{4 x^3}{x^4-1}-\log (1-x)+\log (x+1)-2 \tan ^{-1}(x)\right)","\frac{x^3}{4 \left(1-x^4\right)}-\frac{1}{8} \tan ^{-1}(x)+\frac{1}{8} \tanh ^{-1}(x)",1,"((-4*x^3)/(-1 + x^4) - 2*ArcTan[x] - Log[1 - x] + Log[1 + x])/16","A",1
304,1,33,27,0.0092737,"\int \frac{1}{1-2 x^4+x^8} \, dx","Integrate[(1 - 2*x^4 + x^8)^(-1),x]","\frac{1}{16} \left(-\frac{4 x}{x^4-1}-3 \log (1-x)+3 \log (x+1)+6 \tan ^{-1}(x)\right)","\frac{x}{4 \left(1-x^4\right)}+\frac{3}{8} \tan ^{-1}(x)+\frac{3}{8} \tanh ^{-1}(x)",1,"((-4*x)/(-1 + x^4) + 6*ArcTan[x] - 3*Log[1 - x] + 3*Log[1 + x])/16","A",1
305,1,40,36,0.0161396,"\int \frac{1}{x^2 \left(1-2 x^4+x^8\right)} \, dx","Integrate[1/(x^2*(1 - 2*x^4 + x^8)),x]","\frac{1}{16} \left(-\frac{4 x^3}{x^4-1}-\frac{16}{x}-5 \log (1-x)+5 \log (x+1)-10 \tan ^{-1}(x)\right)","\frac{1}{4 x \left(1-x^4\right)}-\frac{5}{4 x}-\frac{5}{8} \tan ^{-1}(x)+\frac{5}{8} \tanh ^{-1}(x)",1,"(-16/x - (4*x^3)/(-1 + x^4) - 10*ArcTan[x] - 5*Log[1 - x] + 5*Log[1 + x])/16","A",1
306,1,38,36,0.017909,"\int \frac{1}{x^4 \left(1-2 x^4+x^8\right)} \, dx","Integrate[1/(x^4*(1 - 2*x^4 + x^8)),x]","\frac{1}{48} \left(-\frac{12 x}{x^4-1}-\frac{16}{x^3}-21 \log (1-x)+21 \log (x+1)+42 \tan ^{-1}(x)\right)","-\frac{7}{12 x^3}+\frac{1}{4 x^3 \left(1-x^4\right)}+\frac{7}{8} \tan ^{-1}(x)+\frac{7}{8} \tanh ^{-1}(x)",1,"(-16/x^3 - (12*x)/(-1 + x^4) + 42*ArcTan[x] - 21*Log[1 - x] + 21*Log[1 + x])/48","A",1
307,1,51,43,0.02031,"\int \frac{1}{x^6 \left(1-2 x^4+x^8\right)} \, dx","Integrate[1/(x^6*(1 - 2*x^4 + x^8)),x]","-\frac{1}{5 x^5}-\frac{x^3}{4 \left(x^4-1\right)}-\frac{2}{x}-\frac{9}{16} \log (1-x)+\frac{9}{16} \log (x+1)-\frac{9}{8} \tan ^{-1}(x)","-\frac{9}{20 x^5}+\frac{1}{4 x^5 \left(1-x^4\right)}-\frac{9}{4 x}-\frac{9}{8} \tan ^{-1}(x)+\frac{9}{8} \tanh ^{-1}(x)",1,"-1/5*1/x^5 - 2/x - x^3/(4*(-1 + x^4)) - (9*ArcTan[x])/8 - (9*Log[1 - x])/16 + (9*Log[1 + x])/16","A",1
308,1,43,43,0.0190596,"\int \frac{1}{x^8 \left(1-2 x^4+x^8\right)} \, dx","Integrate[1/(x^8*(1 - 2*x^4 + x^8)),x]","\frac{1}{336} \left(-\frac{48}{x^7}-\frac{84 x}{x^4-1}-\frac{224}{x^3}-231 \log (1-x)+231 \log (x+1)+462 \tan ^{-1}(x)\right)","-\frac{11}{28 x^7}-\frac{11}{12 x^3}+\frac{1}{4 x^7 \left(1-x^4\right)}+\frac{11}{8} \tan ^{-1}(x)+\frac{11}{8} \tanh ^{-1}(x)",1,"(-48/x^7 - 224/x^3 - (84*x)/(-1 + x^4) + 462*ArcTan[x] - 231*Log[1 - x] + 231*Log[1 + x])/336","A",1
309,1,82,163,0.058858,"\int \frac{x^m}{a+b x^4+c x^8} \, dx","Integrate[x^m/(a + b*x^4 + c*x^8),x]","\frac{x^m \text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\left(\frac{x}{x-\text{$\#$1}}\right)^{-m} \, _2F_1\left(-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right)}{2 \text{$\#$1}^7 c+\text{$\#$1}^3 b}\&\right]}{4 m}","\frac{2 c x^{m+1} \, _2F_1\left(1,\frac{m+1}{4};\frac{m+5}{4};-\frac{2 c x^4}{b-\sqrt{b^2-4 a c}}\right)}{(m+1) \sqrt{b^2-4 a c} \left(b-\sqrt{b^2-4 a c}\right)}-\frac{2 c x^{m+1} \, _2F_1\left(1,\frac{m+1}{4};\frac{m+5}{4};-\frac{2 c x^4}{b+\sqrt{b^2-4 a c}}\right)}{(m+1) \sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}+b\right)}",1,"(x^m*RootSum[a + b*#1^4 + c*#1^8 & , Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1))]/((x/(x - #1))^m*(b*#1^3 + 2*c*#1^7)) & ])/(4*m)","C",0
310,1,78,81,0.050666,"\int \frac{x^{11}}{a+b x^4+c x^8} \, dx","Integrate[x^11/(a + b*x^4 + c*x^8),x]","\frac{\frac{2 \left(b^2-2 a c\right) \tan ^{-1}\left(\frac{b+2 c x^4}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}-b \log \left(a+b x^4+c x^8\right)+2 c x^4}{8 c^2}","-\frac{\left(b^2-2 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^4}{\sqrt{b^2-4 a c}}\right)}{4 c^2 \sqrt{b^2-4 a c}}-\frac{b \log \left(a+b x^4+c x^8\right)}{8 c^2}+\frac{x^4}{4 c}",1,"(2*c*x^4 + (2*(b^2 - 2*a*c)*ArcTan[(b + 2*c*x^4)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] - b*Log[a + b*x^4 + c*x^8])/(8*c^2)","A",1
311,1,210,192,0.1234359,"\int \frac{x^9}{a+b x^4+c x^8} \, dx","Integrate[x^9/(a + b*x^4 + c*x^8),x]","\frac{-\frac{\sqrt{2} \left(b \sqrt{b^2-4 a c}+2 a c-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x^2}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \left(b \sqrt{b^2-4 a c}-2 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x^2}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+2 \sqrt{c} x^2}{4 c^{3/2}}","-\frac{\left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x^2}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} c^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x^2}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} c^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x^2}{2 c}",1,"(2*Sqrt[c]*x^2 - (Sqrt[2]*(-b^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x^2)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*(b^2 - 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x^2)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(4*c^(3/2))","A",1
312,1,62,63,0.0239046,"\int \frac{x^7}{a+b x^4+c x^8} \, dx","Integrate[x^7/(a + b*x^4 + c*x^8),x]","\frac{\log \left(a+b x^4+c x^8\right)-\frac{2 b \tan ^{-1}\left(\frac{b+2 c x^4}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}}{8 c}","\frac{b \tanh ^{-1}\left(\frac{b+2 c x^4}{\sqrt{b^2-4 a c}}\right)}{4 c \sqrt{b^2-4 a c}}+\frac{\log \left(a+b x^4+c x^8\right)}{8 c}",1,"((-2*b*ArcTan[(b + 2*c*x^4)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + Log[a + b*x^4 + c*x^8])/(8*c)","A",1
313,1,171,159,0.0861632,"\int \frac{x^5}{a+b x^4+c x^8} \, dx","Integrate[x^5/(a + b*x^4 + c*x^8),x]","\frac{\left(\sqrt{b^2-4 a c}-b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x^2}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)+\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{\sqrt{b^2-4 a c}+b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x^2}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} \sqrt{c} \sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}","\frac{\sqrt{\sqrt{b^2-4 a c}+b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x^2}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}-\frac{\sqrt{b-\sqrt{b^2-4 a c}} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x^2}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}",1,"((-b + Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x^2)/Sqrt[b - Sqrt[b^2 - 4*a*c]]] + Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[b + Sqrt[b^2 - 4*a*c]]*ArcTan[(Sqrt[2]*Sqrt[c]*x^2)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*Sqrt[c]*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]])","A",1
314,1,42,38,0.0094912,"\int \frac{x^3}{a+b x^4+c x^8} \, dx","Integrate[x^3/(a + b*x^4 + c*x^8),x]","\frac{\tan ^{-1}\left(\frac{b+2 c x^4}{\sqrt{4 a c-b^2}}\right)}{2 \sqrt{4 a c-b^2}}","-\frac{\tanh ^{-1}\left(\frac{b+2 c x^4}{\sqrt{b^2-4 a c}}\right)}{2 \sqrt{b^2-4 a c}}",1,"ArcTan[(b + 2*c*x^4)/Sqrt[-b^2 + 4*a*c]]/(2*Sqrt[-b^2 + 4*a*c])","A",1
315,1,133,154,0.0812504,"\int \frac{x}{a+b x^4+c x^8} \, dx","Integrate[x/(a + b*x^4 + c*x^8),x]","\frac{\sqrt{c} \left(\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x^2}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x^2}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} \sqrt{b^2-4 a c}}","\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x^2}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x^2}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}",1,"(Sqrt[c]*(ArcTan[(Sqrt[2]*Sqrt[c]*x^2)/Sqrt[b - Sqrt[b^2 - 4*a*c]]]/Sqrt[b - Sqrt[b^2 - 4*a*c]] - ArcTan[(Sqrt[2]*Sqrt[c]*x^2)/Sqrt[b + Sqrt[b^2 - 4*a*c]]]/Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(Sqrt[2]*Sqrt[b^2 - 4*a*c])","A",1
316,1,66,69,0.0226932,"\int \frac{1}{x \left(a+b x^4+c x^8\right)} \, dx","Integrate[1/(x*(a + b*x^4 + c*x^8)),x]","\frac{\log (x)}{a}-\frac{\text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 c \log (x-\text{$\#$1})+b \log (x-\text{$\#$1})}{2 \text{$\#$1}^4 c+b}\&\right]}{4 a}","\frac{b \tanh ^{-1}\left(\frac{b+2 c x^4}{\sqrt{b^2-4 a c}}\right)}{4 a \sqrt{b^2-4 a c}}-\frac{\log \left(a+b x^4+c x^8\right)}{8 a}+\frac{\log (x)}{a}",1,"Log[x]/a - RootSum[a + b*#1^4 + c*#1^8 & , (b*Log[x - #1] + c*Log[x - #1]*#1^4)/(b + 2*c*#1^4) & ]/(4*a)","C",1
317,1,75,184,0.0302532,"\int \frac{1}{x^3 \left(a+b x^4+c x^8\right)} \, dx","Integrate[1/(x^3*(a + b*x^4 + c*x^8)),x]","-\frac{\text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 c \log (x-\text{$\#$1})+b \log (x-\text{$\#$1})}{2 \text{$\#$1}^6 c+\text{$\#$1}^2 b}\&\right]}{4 a}-\frac{1}{2 a x^2}","-\frac{\sqrt{c} \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x^2}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} a \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x^2}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} a \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{1}{2 a x^2}",1,"-1/2*1/(a*x^2) - RootSum[a + b*#1^4 + c*#1^8 & , (b*Log[x - #1] + c*Log[x - #1]*#1^4)/(b*#1^2 + 2*c*#1^6) & ]/(4*a)","C",1
318,1,92,89,0.0320401,"\int \frac{1}{x^5 \left(a+b x^4+c x^8\right)} \, dx","Integrate[1/(x^5*(a + b*x^4 + c*x^8)),x]","\frac{\text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 b c \log (x-\text{$\#$1})-a c \log (x-\text{$\#$1})+b^2 \log (x-\text{$\#$1})}{2 \text{$\#$1}^4 c+b}\&\right]}{4 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{4 a x^4}","-\frac{\left(b^2-2 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^4}{\sqrt{b^2-4 a c}}\right)}{4 a^2 \sqrt{b^2-4 a c}}+\frac{b \log \left(a+b x^4+c x^8\right)}{8 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{4 a x^4}",1,"-1/4*1/(a*x^4) - (b*Log[x])/a^2 + RootSum[a + b*#1^4 + c*#1^8 & , (b^2*Log[x - #1] - a*c*Log[x - #1] + b*c*Log[x - #1]*#1^4)/(b + 2*c*#1^4) & ]/(4*a^2)","C",1
319,1,70,381,0.0400647,"\int \frac{x^{10}}{a+b x^4+c x^8} \, dx","Integrate[x^10/(a + b*x^4 + c*x^8),x]","\frac{4 x^3-3 \text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 b \log (x-\text{$\#$1})+a \log (x-\text{$\#$1})}{2 \text{$\#$1}^5 c+\text{$\#$1} b}\&\right]}{12 c}","-\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2\ 2^{3/4} c^{7/4} \sqrt[4]{-\sqrt{b^2-4 a c}-b}}-\frac{\left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2\ 2^{3/4} c^{7/4} \sqrt[4]{\sqrt{b^2-4 a c}-b}}+\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2\ 2^{3/4} c^{7/4} \sqrt[4]{-\sqrt{b^2-4 a c}-b}}+\frac{\left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2\ 2^{3/4} c^{7/4} \sqrt[4]{\sqrt{b^2-4 a c}-b}}+\frac{x^3}{3 c}",1,"(4*x^3 - 3*RootSum[a + b*#1^4 + c*#1^8 & , (a*Log[x - #1] + b*Log[x - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ])/(12*c)","C",1
320,1,70,376,0.0373616,"\int \frac{x^8}{a+b x^4+c x^8} \, dx","Integrate[x^8/(a + b*x^4 + c*x^8),x]","\frac{x}{c}-\frac{\text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 b \log (x-\text{$\#$1})+a \log (x-\text{$\#$1})}{2 \text{$\#$1}^7 c+\text{$\#$1}^3 b}\&\right]}{4 c}","\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} c^{5/4} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{\left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} c^{5/4} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} c^{5/4} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{\left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} c^{5/4} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{x}{c}",1,"x/c - RootSum[a + b*#1^4 + c*#1^8 & , (a*Log[x - #1] + b*Log[x - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ]/(4*c)","C",1
321,1,44,325,0.0243097,"\int \frac{x^6}{a+b x^4+c x^8} \, dx","Integrate[x^6/(a + b*x^4 + c*x^8),x]","\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^3 \log (x-\text{$\#$1})}{2 \text{$\#$1}^4 c+b}\&\right]","-\frac{\left(-\sqrt{b^2-4 a c}-b\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2\ 2^{3/4} c^{3/4} \sqrt{b^2-4 a c}}+\frac{\left(\sqrt{b^2-4 a c}-b\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2\ 2^{3/4} c^{3/4} \sqrt{b^2-4 a c}}+\frac{\left(-\sqrt{b^2-4 a c}-b\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2\ 2^{3/4} c^{3/4} \sqrt{b^2-4 a c}}-\frac{\left(\sqrt{b^2-4 a c}-b\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2\ 2^{3/4} c^{3/4} \sqrt{b^2-4 a c}}",1,"RootSum[a + b*#1^4 + c*#1^8 & , (Log[x - #1]*#1^3)/(b + 2*c*#1^4) & ]/4","C",1
322,1,42,325,0.0214843,"\int \frac{x^4}{a+b x^4+c x^8} \, dx","Integrate[x^4/(a + b*x^4 + c*x^8),x]","\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1} \log (x-\text{$\#$1})}{2 \text{$\#$1}^4 c+b}\&\right]","\frac{\sqrt[4]{-\sqrt{b^2-4 a c}-b} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} \sqrt[4]{c} \sqrt{b^2-4 a c}}-\frac{\sqrt[4]{\sqrt{b^2-4 a c}-b} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} \sqrt[4]{c} \sqrt{b^2-4 a c}}+\frac{\sqrt[4]{-\sqrt{b^2-4 a c}-b} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} \sqrt[4]{c} \sqrt{b^2-4 a c}}-\frac{\sqrt[4]{\sqrt{b^2-4 a c}-b} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} \sqrt[4]{c} \sqrt{b^2-4 a c}}",1,"RootSum[a + b*#1^4 + c*#1^8 & , (Log[x - #1]*#1)/(b + 2*c*#1^4) & ]/4","C",1
323,1,43,315,0.0225401,"\int \frac{x^2}{a+b x^4+c x^8} \, dx","Integrate[x^2/(a + b*x^4 + c*x^8),x]","\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^5 c+\text{$\#$1} b}\&\right]","-\frac{\sqrt[4]{c} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2^{3/4} \sqrt{b^2-4 a c} \sqrt[4]{-\sqrt{b^2-4 a c}-b}}+\frac{\sqrt[4]{c} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2^{3/4} \sqrt{b^2-4 a c} \sqrt[4]{\sqrt{b^2-4 a c}-b}}+\frac{\sqrt[4]{c} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2^{3/4} \sqrt{b^2-4 a c} \sqrt[4]{-\sqrt{b^2-4 a c}-b}}-\frac{\sqrt[4]{c} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2^{3/4} \sqrt{b^2-4 a c} \sqrt[4]{\sqrt{b^2-4 a c}-b}}",1,"RootSum[a + b*#1^4 + c*#1^8 & , Log[x - #1]/(b*#1 + 2*c*#1^5) & ]/4","C",1
324,1,45,315,0.0249952,"\int \frac{1}{a+b x^4+c x^8} \, dx","Integrate[(a + b*x^4 + c*x^8)^(-1),x]","\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^7 c+\text{$\#$1}^3 b}\&\right]","\frac{c^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{\sqrt[4]{2} \sqrt{b^2-4 a c} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{c^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{\sqrt[4]{2} \sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{c^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{\sqrt[4]{2} \sqrt{b^2-4 a c} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{c^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{\sqrt[4]{2} \sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}",1,"RootSum[a + b*#1^4 + c*#1^8 & , Log[x - #1]/(b*#1^3 + 2*c*#1^7) & ]/4","C",1
325,1,71,363,0.036735,"\int \frac{1}{x^2 \left(a+b x^4+c x^8\right)} \, dx","Integrate[1/(x^2*(a + b*x^4 + c*x^8)),x]","-\frac{\text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 c \log (x-\text{$\#$1})+b \log (x-\text{$\#$1})}{2 \text{$\#$1}^5 c+\text{$\#$1} b}\&\right]}{4 a}-\frac{1}{a x}","-\frac{\sqrt[4]{c} \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2\ 2^{3/4} a \sqrt[4]{-\sqrt{b^2-4 a c}-b}}-\frac{\sqrt[4]{c} \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2\ 2^{3/4} a \sqrt[4]{\sqrt{b^2-4 a c}-b}}+\frac{\sqrt[4]{c} \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2\ 2^{3/4} a \sqrt[4]{-\sqrt{b^2-4 a c}-b}}+\frac{\sqrt[4]{c} \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2\ 2^{3/4} a \sqrt[4]{\sqrt{b^2-4 a c}-b}}-\frac{1}{a x}",1,"-(1/(a*x)) - RootSum[a + b*#1^4 + c*#1^8 & , (b*Log[x - #1] + c*Log[x - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ]/(4*a)","C",1
326,1,75,365,0.0419102,"\int \frac{1}{x^4 \left(a+b x^4+c x^8\right)} \, dx","Integrate[1/(x^4*(a + b*x^4 + c*x^8)),x]","-\frac{\text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 c \log (x-\text{$\#$1})+b \log (x-\text{$\#$1})}{2 \text{$\#$1}^7 c+\text{$\#$1}^3 b}\&\right]}{4 a}-\frac{1}{3 a x^3}","\frac{c^{3/4} \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} a \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{c^{3/4} \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} a \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{c^{3/4} \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} a \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{c^{3/4} \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} a \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{1}{3 a x^3}",1,"-1/3*1/(a*x^3) - RootSum[a + b*#1^4 + c*#1^8 & , (b*Log[x - #1] + c*Log[x - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ]/(4*a)","C",1
327,1,488,127,1.1298071,"\int \frac{x^m}{1+x^4+x^8} \, dx","Integrate[x^m/(1 + x^4 + x^8),x]","\frac{x^m \left(-\frac{\text{RootSum}\left[\text{$\#$1}^4-\text{$\#$1}^2+1\&,\frac{\text{$\#$1}^2 m^2 \left(\frac{x}{x-\text{$\#$1}}\right)^{-m} \, _2F_1\left(-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right)+3 \text{$\#$1}^2 m \left(\frac{x}{x-\text{$\#$1}}\right)^{-m} \, _2F_1\left(-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right)+2 \text{$\#$1}^2 \left(\frac{x}{x-\text{$\#$1}}\right)^{-m} \, _2F_1\left(-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right)+\text{$\#$1}^2 m \left(\frac{x}{\text{$\#$1}}\right)^{-m}+\text{$\#$1} m^2 x+2 \text{$\#$1} m x+m^2 x^2+m x^2}{2 \text{$\#$1}^3-\text{$\#$1}}\&\right]}{m^2+3 m+2}+\text{RootSum}\left[\text{$\#$1}^4-\text{$\#$1}^2+1\&,\frac{\left(\frac{x}{x-\text{$\#$1}}\right)^{-m} \, _2F_1\left(-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right)}{2 \text{$\#$1}^3-\text{$\#$1}}\&\right]-\frac{i \left(\left(\frac{x}{x-\sqrt[3]{-1}}\right)^{-m} \, _2F_1\left(-m,-m;1-m;\frac{\sqrt[3]{-1}}{\sqrt[3]{-1}-x}\right)+\left(\frac{x}{x-(-1)^{2/3}}\right)^{-m} \, _2F_1\left(-m,-m;1-m;\frac{(-1)^{2/3}}{(-1)^{2/3}-x}\right)-\left(\frac{x}{x+\sqrt[3]{-1}}\right)^{-m} \, _2F_1\left(-m,-m;1-m;\frac{\sqrt[3]{-1}}{x+\sqrt[3]{-1}}\right)-\left(\frac{x}{x+(-1)^{2/3}}\right)^{-m} \, _2F_1\left(-m,-m;1-m;\frac{(-1)^{2/3}}{x+(-1)^{2/3}}\right)\right)}{\sqrt{3}}\right)}{4 m}","\frac{2 x^{m+1} \, _2F_1\left(1,\frac{m+1}{4};\frac{m+5}{4};-\frac{2 x^4}{1-i \sqrt{3}}\right)}{\sqrt{3} \left(\sqrt{3}+i\right) (m+1)}-\frac{2 x^{m+1} \, _2F_1\left(1,\frac{m+1}{4};\frac{m+5}{4};-\frac{2 x^4}{1+i \sqrt{3}}\right)}{\sqrt{3} \left(-\sqrt{3}+i\right) (m+1)}",1,"(x^m*(((-I)*(Hypergeometric2F1[-m, -m, 1 - m, (-1)^(1/3)/((-1)^(1/3) - x)]/(x/(-(-1)^(1/3) + x))^m + Hypergeometric2F1[-m, -m, 1 - m, (-1)^(2/3)/((-1)^(2/3) - x)]/(x/(-(-1)^(2/3) + x))^m - Hypergeometric2F1[-m, -m, 1 - m, (-1)^(1/3)/((-1)^(1/3) + x)]/(x/((-1)^(1/3) + x))^m - Hypergeometric2F1[-m, -m, 1 - m, (-1)^(2/3)/((-1)^(2/3) + x)]/(x/((-1)^(2/3) + x))^m))/Sqrt[3] + RootSum[1 - #1^2 + #1^4 & , Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1))]/((x/(x - #1))^m*(-#1 + 2*#1^3)) & ] - RootSum[1 - #1^2 + #1^4 & , (m*x^2 + m^2*x^2 + 2*m*x*#1 + m^2*x*#1 + (2*Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1))]*#1^2)/(x/(x - #1))^m + (3*m*Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1))]*#1^2)/(x/(x - #1))^m + (m^2*Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1))]*#1^2)/(x/(x - #1))^m + (m*#1^2)/(x/#1)^m)/(-#1 + 2*#1^3) & ]/(2 + 3*m + m^2)))/(4*m)","C",0
328,1,44,44,0.0110769,"\int \frac{x^{11}}{1+x^4+x^8} \, dx","Integrate[x^11/(1 + x^4 + x^8),x]","\frac{x^4}{4}-\frac{\tan ^{-1}\left(\frac{2 x^4+1}{\sqrt{3}}\right)}{4 \sqrt{3}}-\frac{1}{8} \log \left(x^8+x^4+1\right)","\frac{x^4}{4}-\frac{\tan ^{-1}\left(\frac{2 x^4+1}{\sqrt{3}}\right)}{4 \sqrt{3}}-\frac{1}{8} \log \left(x^8+x^4+1\right)",1,"x^4/4 - ArcTan[(1 + 2*x^4)/Sqrt[3]]/(4*Sqrt[3]) - Log[1 + x^4 + x^8]/8","A",1
329,1,98,54,0.1726708,"\int \frac{x^9}{1+x^4+x^8} \, dx","Integrate[x^9/(1 + x^4 + x^8),x]","\frac{x^2}{2}-\frac{\left(\sqrt{3}+i\right) \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}-i\right) x^2\right)}{2 \sqrt{6+6 i \sqrt{3}}}-\frac{\left(\sqrt{3}-i\right) \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}+i\right) x^2\right)}{2 \sqrt{6-6 i \sqrt{3}}}","\frac{x^2}{2}+\frac{\tan ^{-1}\left(\frac{1-2 x^2}{\sqrt{3}}\right)}{2 \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{2 x^2+1}{\sqrt{3}}\right)}{2 \sqrt{3}}",1,"x^2/2 - ((I + Sqrt[3])*ArcTan[((-I + Sqrt[3])*x^2)/2])/(2*Sqrt[6 + (6*I)*Sqrt[3]]) - ((-I + Sqrt[3])*ArcTan[((I + Sqrt[3])*x^2)/2])/(2*Sqrt[6 - (6*I)*Sqrt[3]])","C",0
330,1,37,37,0.0079794,"\int \frac{x^7}{1+x^4+x^8} \, dx","Integrate[x^7/(1 + x^4 + x^8),x]","\frac{1}{8} \log \left(x^8+x^4+1\right)-\frac{\tan ^{-1}\left(\frac{2 x^4+1}{\sqrt{3}}\right)}{4 \sqrt{3}}","\frac{1}{8} \log \left(x^8+x^4+1\right)-\frac{\tan ^{-1}\left(\frac{2 x^4+1}{\sqrt{3}}\right)}{4 \sqrt{3}}",1,"-1/4*ArcTan[(1 + 2*x^4)/Sqrt[3]]/Sqrt[3] + Log[1 + x^4 + x^8]/8","A",1
331,1,94,75,0.1191408,"\int \frac{x^5}{1+x^4+x^8} \, dx","Integrate[x^5/(1 + x^4 + x^8),x]","\frac{\sqrt{1-i \sqrt{3}} \left(\sqrt{3}-i\right) \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}-i\right) x^2\right)+\sqrt{1+i \sqrt{3}} \left(\sqrt{3}+i\right) \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}+i\right) x^2\right)}{4 \sqrt{6}}","-\frac{\tan ^{-1}\left(\frac{1-2 x^2}{\sqrt{3}}\right)}{4 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{2 x^2+1}{\sqrt{3}}\right)}{4 \sqrt{3}}+\frac{1}{8} \log \left(x^4-x^2+1\right)-\frac{1}{8} \log \left(x^4+x^2+1\right)",1,"(Sqrt[1 - I*Sqrt[3]]*(-I + Sqrt[3])*ArcTan[((-I + Sqrt[3])*x^2)/2] + Sqrt[1 + I*Sqrt[3]]*(I + Sqrt[3])*ArcTan[((I + Sqrt[3])*x^2)/2])/(4*Sqrt[6])","C",0
332,1,23,23,0.0056587,"\int \frac{x^3}{1+x^4+x^8} \, dx","Integrate[x^3/(1 + x^4 + x^8),x]","\frac{\tan ^{-1}\left(\frac{2 x^4+1}{\sqrt{3}}\right)}{2 \sqrt{3}}","\frac{\tan ^{-1}\left(\frac{2 x^4+1}{\sqrt{3}}\right)}{2 \sqrt{3}}",1,"ArcTan[(1 + 2*x^4)/Sqrt[3]]/(2*Sqrt[3])","A",1
333,1,79,75,0.0482362,"\int \frac{x}{1+x^4+x^8} \, dx","Integrate[x/(1 + x^4 + x^8),x]","\frac{i \left(\sqrt{1-i \sqrt{3}} \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}-i\right) x^2\right)-\sqrt{1+i \sqrt{3}} \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{3}+i\right) x^2\right)\right)}{2 \sqrt{6}}","-\frac{\tan ^{-1}\left(\frac{1-2 x^2}{\sqrt{3}}\right)}{4 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{2 x^2+1}{\sqrt{3}}\right)}{4 \sqrt{3}}-\frac{1}{8} \log \left(x^4-x^2+1\right)+\frac{1}{8} \log \left(x^4+x^2+1\right)",1,"((I/2)*(Sqrt[1 - I*Sqrt[3]]*ArcTan[((-I + Sqrt[3])*x^2)/2] - Sqrt[1 + I*Sqrt[3]]*ArcTan[((I + Sqrt[3])*x^2)/2]))/Sqrt[6]","C",0
334,1,138,39,0.0836826,"\int \frac{1}{x \left(1+x^4+x^8\right)} \, dx","Integrate[1/(x*(1 + x^4 + x^8)),x]","\frac{1}{24} \left(-\sqrt{3} \left(\sqrt{3}-i\right) \log \left(x^2-\frac{i \sqrt{3}}{2}-\frac{1}{2}\right)-\sqrt{3} \left(\sqrt{3}+i\right) \log \left(x^2+\frac{1}{2} i \left(\sqrt{3}+i\right)\right)-3 \log \left(x^2-x+1\right)-3 \log \left(x^2+x+1\right)+24 \log (x)+2 \sqrt{3} \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)\right)","-\frac{\tan ^{-1}\left(\frac{2 x^4+1}{\sqrt{3}}\right)}{4 \sqrt{3}}-\frac{1}{8} \log \left(x^8+x^4+1\right)+\log (x)",1,"(2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] - 2*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] + 24*Log[x] - Sqrt[3]*(-I + Sqrt[3])*Log[-1/2 - (I/2)*Sqrt[3] + x^2] - Sqrt[3]*(I + Sqrt[3])*Log[(I/2)*(I + Sqrt[3]) + x^2] - 3*Log[1 - x + x^2] - 3*Log[1 + x + x^2])/24","C",1
335,1,100,54,0.0487784,"\int \frac{1}{x^3 \left(1+x^4+x^8\right)} \, dx","Integrate[1/(x^3*(1 + x^4 + x^8)),x]","\frac{1}{12} \left(-\frac{6}{x^2}+i \sqrt{3} \log \left(2 x^2-i \sqrt{3}-1\right)-i \sqrt{3} \log \left(2 x^2+i \sqrt{3}-1\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)\right)","-\frac{1}{2 x^2}+\frac{\tan ^{-1}\left(\frac{1-2 x^2}{\sqrt{3}}\right)}{2 \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{2 x^2+1}{\sqrt{3}}\right)}{2 \sqrt{3}}",1,"(-6/x^2 - 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] + 2*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] + I*Sqrt[3]*Log[-1 - I*Sqrt[3] + 2*x^2] - I*Sqrt[3]*Log[-1 + I*Sqrt[3] + 2*x^2])/12","C",1
336,1,141,48,0.1020265,"\int \frac{1}{x^5 \left(1+x^4+x^8\right)} \, dx","Integrate[1/(x^5*(1 + x^4 + x^8)),x]","\frac{1}{24} \left(-\frac{6}{x^4}+\sqrt{3} \left(\sqrt{3}+i\right) \log \left(x^2-\frac{i \sqrt{3}}{2}-\frac{1}{2}\right)+\sqrt{3} \left(\sqrt{3}-i\right) \log \left(x^2+\frac{1}{2} i \left(\sqrt{3}+i\right)\right)+3 \log \left(x^2-x+1\right)+3 \log \left(x^2+x+1\right)-24 \log (x)+2 \sqrt{3} \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)\right)","-\frac{1}{4 x^4}-\frac{\tan ^{-1}\left(\frac{2 x^4+1}{\sqrt{3}}\right)}{4 \sqrt{3}}+\frac{1}{8} \log \left(x^8+x^4+1\right)-\log (x)",1,"(-6/x^4 + 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] - 2*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] - 24*Log[x] + Sqrt[3]*(I + Sqrt[3])*Log[-1/2 - (I/2)*Sqrt[3] + x^2] + Sqrt[3]*(-I + Sqrt[3])*Log[(I/2)*(I + Sqrt[3]) + x^2] + 3*Log[1 - x + x^2] + 3*Log[1 + x + x^2])/24","C",1
337,1,142,89,0.1100339,"\int \frac{1}{x^7 \left(1+x^4+x^8\right)} \, dx","Integrate[1/(x^7*(1 + x^4 + x^8)),x]","\frac{1}{24} \left(-\frac{4}{x^6}+\frac{12}{x^2}+\sqrt{3} \left(\sqrt{3}-i\right) \log \left(x^2-\frac{i \sqrt{3}}{2}-\frac{1}{2}\right)+\sqrt{3} \left(\sqrt{3}+i\right) \log \left(x^2+\frac{1}{2} i \left(\sqrt{3}+i\right)\right)-3 \log \left(x^2-x+1\right)-3 \log \left(x^2+x+1\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)\right)","-\frac{1}{6 x^6}+\frac{1}{2 x^2}-\frac{\tan ^{-1}\left(\frac{1-2 x^2}{\sqrt{3}}\right)}{4 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{2 x^2+1}{\sqrt{3}}\right)}{4 \sqrt{3}}+\frac{1}{8} \log \left(x^4-x^2+1\right)-\frac{1}{8} \log \left(x^4+x^2+1\right)",1,"(-4/x^6 + 12/x^2 + 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] - 2*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] + Sqrt[3]*(-I + Sqrt[3])*Log[-1/2 - (I/2)*Sqrt[3] + x^2] + Sqrt[3]*(I + Sqrt[3])*Log[(I/2)*(I + Sqrt[3]) + x^2] - 3*Log[1 - x + x^2] - 3*Log[1 + x + x^2])/24","C",1
338,1,139,141,0.2754803,"\int \frac{x^8}{1+x^4+x^8} \, dx","Integrate[x^8/(1 + x^4 + x^8),x]","\frac{1}{24} \left(3 \log \left(x^2-x+1\right)-3 \log \left(x^2+x+1\right)+24 x-2 \sqrt{3} \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)\right)-\frac{i \tan ^{-1}\left(\frac{1}{2} \left(1-i \sqrt{3}\right) x\right)}{\sqrt{-6+6 i \sqrt{3}}}+\frac{i \tan ^{-1}\left(\frac{1}{2} \left(1+i \sqrt{3}\right) x\right)}{\sqrt{-6-6 i \sqrt{3}}}","\frac{1}{8} \log \left(x^2-x+1\right)-\frac{1}{8} \log \left(x^2+x+1\right)+\frac{\log \left(x^2-\sqrt{3} x+1\right)}{8 \sqrt{3}}-\frac{\log \left(x^2+\sqrt{3} x+1\right)}{8 \sqrt{3}}+x+\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{4 \sqrt{3}}+\frac{1}{4} \tan ^{-1}\left(\sqrt{3}-2 x\right)-\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{4 \sqrt{3}}-\frac{1}{4} \tan ^{-1}\left(2 x+\sqrt{3}\right)",1,"((-I)*ArcTan[((1 - I*Sqrt[3])*x)/2])/Sqrt[-6 + (6*I)*Sqrt[3]] + (I*ArcTan[((1 + I*Sqrt[3])*x)/2])/Sqrt[-6 - (6*I)*Sqrt[3]] + (24*x - 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] - 2*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] + 3*Log[1 - x + x^2] - 3*Log[1 + x + x^2])/24","C",0
339,1,68,88,0.0201256,"\int \frac{x^6}{1+x^4+x^8} \, dx","Integrate[x^6/(1 + x^4 + x^8),x]","\frac{\log \left(-x^2+\sqrt{3} x-1\right)-\log \left(x^2+\sqrt{3} x+1\right)+2 \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)+2 \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{4 \sqrt{3}}","\frac{\log \left(x^2-\sqrt{3} x+1\right)}{4 \sqrt{3}}-\frac{\log \left(x^2+\sqrt{3} x+1\right)}{4 \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{2 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{2 \sqrt{3}}",1,"(2*ArcTan[(-1 + 2*x)/Sqrt[3]] + 2*ArcTan[(1 + 2*x)/Sqrt[3]] + Log[-1 + Sqrt[3]*x - x^2] - Log[1 + Sqrt[3]*x + x^2])/(4*Sqrt[3])","A",1
340,1,135,140,0.1648541,"\int \frac{x^4}{1+x^4+x^8} \, dx","Integrate[x^4/(1 + x^4 + x^8),x]","\frac{1}{24} \left(-3 \log \left(x^2-x+1\right)+3 \log \left(x^2+x+1\right)-2 i \sqrt{-6+6 i \sqrt{3}} \tan ^{-1}\left(\frac{1}{2} \left(1-i \sqrt{3}\right) x\right)+2 i \sqrt{-6-6 i \sqrt{3}} \tan ^{-1}\left(\frac{1}{2} \left(1+i \sqrt{3}\right) x\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)\right)","-\frac{1}{8} \log \left(x^2-x+1\right)+\frac{1}{8} \log \left(x^2+x+1\right)+\frac{\log \left(x^2-\sqrt{3} x+1\right)}{8 \sqrt{3}}-\frac{\log \left(x^2+\sqrt{3} x+1\right)}{8 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{4 \sqrt{3}}-\frac{1}{4} \tan ^{-1}\left(\sqrt{3}-2 x\right)-\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{4 \sqrt{3}}+\frac{1}{4} \tan ^{-1}\left(2 x+\sqrt{3}\right)",1,"((-2*I)*Sqrt[-6 + (6*I)*Sqrt[3]]*ArcTan[((1 - I*Sqrt[3])*x)/2] + (2*I)*Sqrt[-6 - (6*I)*Sqrt[3]]*ArcTan[((1 + I*Sqrt[3])*x)/2] - 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] - 2*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] - 3*Log[1 - x + x^2] + 3*Log[1 + x + x^2])/24","C",0
341,1,135,140,0.1509371,"\int \frac{x^2}{1+x^4+x^8} \, dx","Integrate[x^2/(1 + x^4 + x^8),x]","\frac{1}{48} \left(6 \log \left(x^2-x+1\right)-6 \log \left(x^2+x+1\right)+4 i \sqrt{-6-6 i \sqrt{3}} \tan ^{-1}\left(\frac{1}{2} \left(1-i \sqrt{3}\right) x\right)-4 i \sqrt{-6+6 i \sqrt{3}} \tan ^{-1}\left(\frac{1}{2} \left(1+i \sqrt{3}\right) x\right)-4 \sqrt{3} \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)-4 \sqrt{3} \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)\right)","\frac{1}{8} \log \left(x^2-x+1\right)-\frac{1}{8} \log \left(x^2+x+1\right)-\frac{\log \left(x^2-\sqrt{3} x+1\right)}{8 \sqrt{3}}+\frac{\log \left(x^2+\sqrt{3} x+1\right)}{8 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{4 \sqrt{3}}-\frac{1}{4} \tan ^{-1}\left(\sqrt{3}-2 x\right)-\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{4 \sqrt{3}}+\frac{1}{4} \tan ^{-1}\left(2 x+\sqrt{3}\right)",1,"((4*I)*Sqrt[-6 - (6*I)*Sqrt[3]]*ArcTan[((1 - I*Sqrt[3])*x)/2] - (4*I)*Sqrt[-6 + (6*I)*Sqrt[3]]*ArcTan[((1 + I*Sqrt[3])*x)/2] - 4*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] - 4*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] + 6*Log[1 - x + x^2] - 6*Log[1 + x + x^2])/48","C",0
342,1,68,88,0.0171897,"\int \frac{1}{1+x^4+x^8} \, dx","Integrate[(1 + x^4 + x^8)^(-1),x]","\frac{-\log \left(-x^2+\sqrt{3} x-1\right)+\log \left(x^2+\sqrt{3} x+1\right)+2 \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)+2 \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{4 \sqrt{3}}","-\frac{\log \left(x^2-\sqrt{3} x+1\right)}{4 \sqrt{3}}+\frac{\log \left(x^2+\sqrt{3} x+1\right)}{4 \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{2 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{2 \sqrt{3}}",1,"(2*ArcTan[(-1 + 2*x)/Sqrt[3]] + 2*ArcTan[(1 + 2*x)/Sqrt[3]] - Log[-1 + Sqrt[3]*x - x^2] + Log[1 + Sqrt[3]*x + x^2])/(4*Sqrt[3])","A",1
343,1,140,145,0.2084831,"\int \frac{1}{x^2 \left(1+x^4+x^8\right)} \, dx","Integrate[1/(x^2*(1 + x^4 + x^8)),x]","\frac{1}{24} \left(-3 \log \left(x^2-x+1\right)+3 \log \left(x^2+x+1\right)-\frac{24}{x}+2 i \sqrt{-6+6 i \sqrt{3}} \tan ^{-1}\left(\frac{1}{2} \left(1-i \sqrt{3}\right) x\right)-2 i \sqrt{-6-6 i \sqrt{3}} \tan ^{-1}\left(\frac{1}{2} \left(1+i \sqrt{3}\right) x\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)\right)","-\frac{1}{8} \log \left(x^2-x+1\right)+\frac{1}{8} \log \left(x^2+x+1\right)-\frac{\log \left(x^2-\sqrt{3} x+1\right)}{8 \sqrt{3}}+\frac{\log \left(x^2+\sqrt{3} x+1\right)}{8 \sqrt{3}}-\frac{1}{x}+\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{4 \sqrt{3}}+\frac{1}{4} \tan ^{-1}\left(\sqrt{3}-2 x\right)-\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{4 \sqrt{3}}-\frac{1}{4} \tan ^{-1}\left(2 x+\sqrt{3}\right)",1,"(-24/x + (2*I)*Sqrt[-6 + (6*I)*Sqrt[3]]*ArcTan[((1 - I*Sqrt[3])*x)/2] - (2*I)*Sqrt[-6 - (6*I)*Sqrt[3]]*ArcTan[((1 + I*Sqrt[3])*x)/2] - 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] - 2*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] - 3*Log[1 - x + x^2] + 3*Log[1 + x + x^2])/24","C",0
344,1,148,147,0.2939661,"\int \frac{1}{x^4 \left(1+x^4+x^8\right)} \, dx","Integrate[1/(x^4*(1 + x^4 + x^8)),x]","\frac{1}{24} \left(-\frac{8}{x^3}+3 \log \left(x^2-x+1\right)-3 \log \left(x^2+x+1\right)-\frac{4 i \tan ^{-1}\left(\frac{1}{2} \left(1-i \sqrt{3}\right) x\right)}{\sqrt{\frac{1}{6} i \left(\sqrt{3}+i\right)}}+\frac{4 i \tan ^{-1}\left(\frac{1}{2} \left(1+i \sqrt{3}\right) x\right)}{\sqrt{-\frac{1}{6} i \left(\sqrt{3}-i\right)}}-2 \sqrt{3} \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)\right)","-\frac{1}{3 x^3}+\frac{1}{8} \log \left(x^2-x+1\right)-\frac{1}{8} \log \left(x^2+x+1\right)+\frac{\log \left(x^2-\sqrt{3} x+1\right)}{8 \sqrt{3}}-\frac{\log \left(x^2+\sqrt{3} x+1\right)}{8 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{4 \sqrt{3}}+\frac{1}{4} \tan ^{-1}\left(\sqrt{3}-2 x\right)-\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{4 \sqrt{3}}-\frac{1}{4} \tan ^{-1}\left(2 x+\sqrt{3}\right)",1,"(-8/x^3 - ((4*I)*ArcTan[((1 - I*Sqrt[3])*x)/2])/Sqrt[(I/6)*(I + Sqrt[3])] + ((4*I)*ArcTan[((1 + I*Sqrt[3])*x)/2])/Sqrt[(-1/6*I)*(-I + Sqrt[3])] - 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] - 2*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] + 3*Log[1 - x + x^2] - 3*Log[1 + x + x^2])/24","C",0
345,1,95,98,0.0338144,"\int \frac{1}{x^6 \left(1+x^4+x^8\right)} \, dx","Integrate[1/(x^6*(1 + x^4 + x^8)),x]","\frac{1}{60} \left(-\frac{12}{x^5}+5 \sqrt{3} \log \left(-x^2+\sqrt{3} x-1\right)-5 \sqrt{3} \log \left(x^2+\sqrt{3} x+1\right)+\frac{60}{x}+10 \sqrt{3} \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)+10 \sqrt{3} \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)\right)","-\frac{1}{5 x^5}+\frac{\log \left(x^2-\sqrt{3} x+1\right)}{4 \sqrt{3}}-\frac{\log \left(x^2+\sqrt{3} x+1\right)}{4 \sqrt{3}}+\frac{1}{x}-\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{2 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{2 \sqrt{3}}",1,"(-12/x^5 + 60/x + 10*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] + 10*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] + 5*Sqrt[3]*Log[-1 + Sqrt[3]*x - x^2] - 5*Sqrt[3]*Log[1 + Sqrt[3]*x + x^2])/60","A",1
346,1,171,154,0.3438865,"\int \frac{1}{x^8 \left(1+x^4+x^8\right)} \, dx","Integrate[1/(x^8*(1 + x^4 + x^8)),x]","-\frac{1}{7 x^7}+\frac{1}{3 x^3}-\frac{1}{8} \log \left(x^2-x+1\right)+\frac{1}{8} \log \left(x^2+x+1\right)+\frac{\left(\sqrt{3}+i\right) \tan ^{-1}\left(\frac{1}{2} \left(1-i \sqrt{3}\right) x\right)}{2 \sqrt{-6+6 i \sqrt{3}}}+\frac{\left(\sqrt{3}-i\right) \tan ^{-1}\left(\frac{1}{2} \left(1+i \sqrt{3}\right) x\right)}{2 \sqrt{-6-6 i \sqrt{3}}}-\frac{\tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)}{4 \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{4 \sqrt{3}}","-\frac{1}{7 x^7}+\frac{1}{3 x^3}-\frac{1}{8} \log \left(x^2-x+1\right)+\frac{1}{8} \log \left(x^2+x+1\right)+\frac{\log \left(x^2-\sqrt{3} x+1\right)}{8 \sqrt{3}}-\frac{\log \left(x^2+\sqrt{3} x+1\right)}{8 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{4 \sqrt{3}}-\frac{1}{4} \tan ^{-1}\left(\sqrt{3}-2 x\right)-\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{4 \sqrt{3}}+\frac{1}{4} \tan ^{-1}\left(2 x+\sqrt{3}\right)",1,"-1/7*1/x^7 + 1/(3*x^3) + ((I + Sqrt[3])*ArcTan[((1 - I*Sqrt[3])*x)/2])/(2*Sqrt[-6 + (6*I)*Sqrt[3]]) + ((-I + Sqrt[3])*ArcTan[((1 + I*Sqrt[3])*x)/2])/(2*Sqrt[-6 - (6*I)*Sqrt[3]]) - ArcTan[(-1 + 2*x)/Sqrt[3]]/(4*Sqrt[3]) - ArcTan[(1 + 2*x)/Sqrt[3]]/(4*Sqrt[3]) - Log[1 - x + x^2]/8 + Log[1 + x + x^2]/8","C",0
347,1,79,127,0.0556536,"\int \frac{x^m}{1-x^4+x^8} \, dx","Integrate[x^m/(1 - x^4 + x^8),x]","\frac{x^m \text{RootSum}\left[\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\left(\frac{x}{x-\text{$\#$1}}\right)^{-m} \, _2F_1\left(-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right)}{2 \text{$\#$1}^7-\text{$\#$1}^3}\&\right]}{4 m}","\frac{2 x^{m+1} \, _2F_1\left(1,\frac{m+1}{4};\frac{m+5}{4};\frac{2 x^4}{1-i \sqrt{3}}\right)}{\sqrt{3} \left(\sqrt{3}+i\right) (m+1)}-\frac{2 x^{m+1} \, _2F_1\left(1,\frac{m+1}{4};\frac{m+5}{4};\frac{2 x^4}{1+i \sqrt{3}}\right)}{\sqrt{3} \left(-\sqrt{3}+i\right) (m+1)}",1,"(x^m*RootSum[1 - #1^4 + #1^8 & , Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1))]/((x/(x - #1))^m*(-#1^3 + 2*#1^7)) & ])/(4*m)","C",0
348,1,46,46,0.0115659,"\int \frac{x^{11}}{1-x^4+x^8} \, dx","Integrate[x^11/(1 - x^4 + x^8),x]","\frac{x^4}{4}-\frac{\tan ^{-1}\left(\frac{2 x^4-1}{\sqrt{3}}\right)}{4 \sqrt{3}}+\frac{1}{8} \log \left(x^8-x^4+1\right)","\frac{x^4}{4}+\frac{\tan ^{-1}\left(\frac{1-2 x^4}{\sqrt{3}}\right)}{4 \sqrt{3}}+\frac{1}{8} \log \left(x^8-x^4+1\right)",1,"x^4/4 - ArcTan[(-1 + 2*x^4)/Sqrt[3]]/(4*Sqrt[3]) + Log[1 - x^4 + x^8]/8","A",1
349,1,55,57,0.0145449,"\int \frac{x^9}{1-x^4+x^8} \, dx","Integrate[x^9/(1 - x^4 + x^8),x]","\frac{1}{12} \left(6 x^2+\sqrt{3} \log \left(-x^4+\sqrt{3} x^2-1\right)-\sqrt{3} \log \left(x^4+\sqrt{3} x^2+1\right)\right)","\frac{x^2}{2}+\frac{\log \left(x^4-\sqrt{3} x^2+1\right)}{4 \sqrt{3}}-\frac{\log \left(x^4+\sqrt{3} x^2+1\right)}{4 \sqrt{3}}",1,"(6*x^2 + Sqrt[3]*Log[-1 + Sqrt[3]*x^2 - x^4] - Sqrt[3]*Log[1 + Sqrt[3]*x^2 + x^4])/12","A",1
350,1,39,39,0.0080341,"\int \frac{x^7}{1-x^4+x^8} \, dx","Integrate[x^7/(1 - x^4 + x^8),x]","\frac{\tan ^{-1}\left(\frac{2 x^4-1}{\sqrt{3}}\right)}{4 \sqrt{3}}+\frac{1}{8} \log \left(x^8-x^4+1\right)","\frac{1}{8} \log \left(x^8-x^4+1\right)-\frac{\tan ^{-1}\left(\frac{1-2 x^4}{\sqrt{3}}\right)}{4 \sqrt{3}}",1,"ArcTan[(-1 + 2*x^4)/Sqrt[3]]/(4*Sqrt[3]) + Log[1 - x^4 + x^8]/8","A",1
351,1,98,82,0.131903,"\int \frac{x^5}{1-x^4+x^8} \, dx","Integrate[x^5/(1 - x^4 + x^8),x]","\frac{\sqrt{-1-i \sqrt{3}} \left(\sqrt{3}+i\right) \tan ^{-1}\left(\frac{1}{2} \left(1-i \sqrt{3}\right) x^2\right)+\sqrt{-1+i \sqrt{3}} \left(\sqrt{3}-i\right) \tan ^{-1}\left(\frac{1}{2} \left(1+i \sqrt{3}\right) x^2\right)}{4 \sqrt{6}}","-\frac{1}{4} \tan ^{-1}\left(\sqrt{3}-2 x^2\right)+\frac{1}{4} \tan ^{-1}\left(2 x^2+\sqrt{3}\right)+\frac{\log \left(x^4-\sqrt{3} x^2+1\right)}{8 \sqrt{3}}-\frac{\log \left(x^4+\sqrt{3} x^2+1\right)}{8 \sqrt{3}}",1,"(Sqrt[-1 - I*Sqrt[3]]*(I + Sqrt[3])*ArcTan[((1 - I*Sqrt[3])*x^2)/2] + Sqrt[-1 + I*Sqrt[3]]*(-I + Sqrt[3])*ArcTan[((1 + I*Sqrt[3])*x^2)/2])/(4*Sqrt[6])","C",0
352,1,23,23,0.0070765,"\int \frac{x^3}{1-x^4+x^8} \, dx","Integrate[x^3/(1 - x^4 + x^8),x]","\frac{\tan ^{-1}\left(\frac{2 x^4-1}{\sqrt{3}}\right)}{2 \sqrt{3}}","-\frac{\tan ^{-1}\left(\frac{1-2 x^4}{\sqrt{3}}\right)}{2 \sqrt{3}}",1,"ArcTan[(-1 + 2*x^4)/Sqrt[3]]/(2*Sqrt[3])","A",1
353,1,83,82,0.0520885,"\int \frac{x}{1-x^4+x^8} \, dx","Integrate[x/(1 - x^4 + x^8),x]","\frac{i \left(\sqrt{-1-i \sqrt{3}} \tan ^{-1}\left(\frac{1}{2} \left(1-i \sqrt{3}\right) x^2\right)-\sqrt{-1+i \sqrt{3}} \tan ^{-1}\left(\frac{1}{2} \left(1+i \sqrt{3}\right) x^2\right)\right)}{2 \sqrt{6}}","-\frac{1}{4} \tan ^{-1}\left(\sqrt{3}-2 x^2\right)+\frac{1}{4} \tan ^{-1}\left(2 x^2+\sqrt{3}\right)-\frac{\log \left(x^4-\sqrt{3} x^2+1\right)}{8 \sqrt{3}}+\frac{\log \left(x^4+\sqrt{3} x^2+1\right)}{8 \sqrt{3}}",1,"((I/2)*(Sqrt[-1 - I*Sqrt[3]]*ArcTan[((1 - I*Sqrt[3])*x^2)/2] - Sqrt[-1 + I*Sqrt[3]]*ArcTan[((1 + I*Sqrt[3])*x^2)/2]))/Sqrt[6]","C",0
354,1,55,41,0.0133832,"\int \frac{1}{x \left(1-x^4+x^8\right)} \, dx","Integrate[1/(x*(1 - x^4 + x^8)),x]","\log (x)-\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^4-1}\&\right]","-\frac{\tan ^{-1}\left(\frac{1-2 x^4}{\sqrt{3}}\right)}{4 \sqrt{3}}-\frac{1}{8} \log \left(x^8-x^4+1\right)+\log (x)",1,"Log[x] - RootSum[1 - #1^4 + #1^8 & , (-Log[x - #1] + Log[x - #1]*#1^4)/(-1 + 2*#1^4) & ]/4","C",1
355,1,55,57,0.0148887,"\int \frac{1}{x^3 \left(1-x^4+x^8\right)} \, dx","Integrate[1/(x^3*(1 - x^4 + x^8)),x]","\frac{1}{12} \left(-\frac{6}{x^2}-\sqrt{3} \log \left(-x^4+\sqrt{3} x^2-1\right)+\sqrt{3} \log \left(x^4+\sqrt{3} x^2+1\right)\right)","-\frac{1}{2 x^2}-\frac{\log \left(x^4-\sqrt{3} x^2+1\right)}{4 \sqrt{3}}+\frac{\log \left(x^4+\sqrt{3} x^2+1\right)}{4 \sqrt{3}}",1,"(-6/x^2 - Sqrt[3]*Log[-1 + Sqrt[3]*x^2 - x^4] + Sqrt[3]*Log[1 + Sqrt[3]*x^2 + x^4])/12","A",1
356,1,51,48,0.013552,"\int \frac{1}{x^5 \left(1-x^4+x^8\right)} \, dx","Integrate[1/(x^5*(1 - x^4 + x^8)),x]","-\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})}{2 \text{$\#$1}^4-1}\&\right]-\frac{1}{4 x^4}+\log (x)","-\frac{1}{4 x^4}+\frac{\tan ^{-1}\left(\frac{1-2 x^4}{\sqrt{3}}\right)}{4 \sqrt{3}}-\frac{1}{8} \log \left(x^8-x^4+1\right)+\log (x)",1,"-1/4*1/x^4 + Log[x] - RootSum[1 - #1^4 + #1^8 & , (Log[x - #1]*#1^4)/(-1 + 2*#1^4) & ]/4","C",1
357,1,56,96,0.0180523,"\int \frac{1}{x^7 \left(1-x^4+x^8\right)} \, dx","Integrate[1/(x^7*(1 - x^4 + x^8)),x]","-\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\text{$\#$1}^2 \log (x-\text{$\#$1})}{2 \text{$\#$1}^4-1}\&\right]-\frac{1}{6 x^6}-\frac{1}{2 x^2}","-\frac{1}{6 x^6}-\frac{1}{2 x^2}+\frac{1}{4} \tan ^{-1}\left(\sqrt{3}-2 x^2\right)-\frac{1}{4} \tan ^{-1}\left(2 x^2+\sqrt{3}\right)-\frac{\log \left(x^4-\sqrt{3} x^2+1\right)}{8 \sqrt{3}}+\frac{\log \left(x^4+\sqrt{3} x^2+1\right)}{8 \sqrt{3}}",1,"-1/6*1/x^6 - 1/(2*x^2) - RootSum[1 - #1^4 + #1^8 & , (Log[x - #1]*#1^2)/(-1 + 2*#1^4) & ]/4","C",1
358,1,59,356,0.0145131,"\int \frac{x^8}{1-x^4+x^8} \, dx","Integrate[x^8/(1 - x^4 + x^8),x]","\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^7-\text{$\#$1}^3}\&\right]+x","-\frac{1}{8} \sqrt{\frac{1}{3} \left(2-\sqrt{3}\right)} \log \left(x^2-\sqrt{2-\sqrt{3}} x+1\right)+\frac{1}{8} \sqrt{\frac{1}{3} \left(2-\sqrt{3}\right)} \log \left(x^2+\sqrt{2-\sqrt{3}} x+1\right)+\frac{1}{8} \sqrt{\frac{1}{3} \left(2+\sqrt{3}\right)} \log \left(x^2-\sqrt{2+\sqrt{3}} x+1\right)-\frac{1}{8} \sqrt{\frac{1}{3} \left(2+\sqrt{3}\right)} \log \left(x^2+\sqrt{2+\sqrt{3}} x+1\right)+x+\frac{\tan ^{-1}\left(\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right)}{4 \sqrt{3 \left(2-\sqrt{3}\right)}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right)}{4 \sqrt{3 \left(2+\sqrt{3}\right)}}-\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right)}{4 \sqrt{3 \left(2-\sqrt{3}\right)}}+\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right)}{4 \sqrt{3 \left(2+\sqrt{3}\right)}}",1,"x + RootSum[1 - #1^4 + #1^8 & , (-Log[x - #1] + Log[x - #1]*#1^4)/(-#1^3 + 2*#1^7) & ]/4","C",1
359,1,41,275,0.0113915,"\int \frac{x^6}{1-x^4+x^8} \, dx","Integrate[x^6/(1 - x^4 + x^8),x]","\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\text{$\#$1}^3 \log (x-\text{$\#$1})}{2 \text{$\#$1}^4-1}\&\right]","\frac{\log \left(x^2-\sqrt{2-\sqrt{3}} x+1\right)}{4 \sqrt{6}}-\frac{\log \left(x^2+\sqrt{2-\sqrt{3}} x+1\right)}{4 \sqrt{6}}+\frac{\log \left(x^2-\sqrt{2+\sqrt{3}} x+1\right)}{4 \sqrt{6}}-\frac{\log \left(x^2+\sqrt{2+\sqrt{3}} x+1\right)}{4 \sqrt{6}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right)}{2 \sqrt{6}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right)}{2 \sqrt{6}}+\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right)}{2 \sqrt{6}}+\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right)}{2 \sqrt{6}}",1,"RootSum[1 - #1^4 + #1^8 & , (Log[x - #1]*#1^3)/(-1 + 2*#1^4) & ]/4","C",1
360,1,39,347,0.0111774,"\int \frac{x^4}{1-x^4+x^8} \, dx","Integrate[x^4/(1 - x^4 + x^8),x]","\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\text{$\#$1} \log (x-\text{$\#$1})}{2 \text{$\#$1}^4-1}\&\right]","-\frac{\log \left(x^2-\sqrt{2-\sqrt{3}} x+1\right)}{8 \sqrt{3 \left(2-\sqrt{3}\right)}}+\frac{\log \left(x^2+\sqrt{2-\sqrt{3}} x+1\right)}{8 \sqrt{3 \left(2-\sqrt{3}\right)}}+\frac{\log \left(x^2-\sqrt{2+\sqrt{3}} x+1\right)}{8 \sqrt{3 \left(2+\sqrt{3}\right)}}-\frac{\log \left(x^2+\sqrt{2+\sqrt{3}} x+1\right)}{8 \sqrt{3 \left(2+\sqrt{3}\right)}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right)}{4 \sqrt{3 \left(2+\sqrt{3}\right)}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right)}{4 \sqrt{3 \left(2-\sqrt{3}\right)}}-\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right)}{4 \sqrt{3 \left(2+\sqrt{3}\right)}}+\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right)}{4 \sqrt{3 \left(2-\sqrt{3}\right)}}",1,"RootSum[1 - #1^4 + #1^8 & , (Log[x - #1]*#1)/(-1 + 2*#1^4) & ]/4","C",1
361,1,40,355,0.0114071,"\int \frac{x^2}{1-x^4+x^8} \, dx","Integrate[x^2/(1 - x^4 + x^8),x]","\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^5-\text{$\#$1}}\&\right]","\frac{\log \left(x^2-\sqrt{2-\sqrt{3}} x+1\right)}{8 \sqrt{3 \left(2-\sqrt{3}\right)}}-\frac{\log \left(x^2+\sqrt{2-\sqrt{3}} x+1\right)}{8 \sqrt{3 \left(2-\sqrt{3}\right)}}-\frac{\log \left(x^2-\sqrt{2+\sqrt{3}} x+1\right)}{8 \sqrt{3 \left(2+\sqrt{3}\right)}}+\frac{\log \left(x^2+\sqrt{2+\sqrt{3}} x+1\right)}{8 \sqrt{3 \left(2+\sqrt{3}\right)}}+\frac{1}{4} \sqrt{\frac{1}{3} \left(2-\sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right)-\frac{1}{4} \sqrt{\frac{1}{3} \left(2+\sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right)-\frac{1}{4} \sqrt{\frac{1}{3} \left(2-\sqrt{3}\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right)+\frac{1}{4} \sqrt{\frac{1}{3} \left(2+\sqrt{3}\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right)",1,"RootSum[1 - #1^4 + #1^8 & , Log[x - #1]/(-#1 + 2*#1^5) & ]/4","C",1
362,1,42,275,0.0106605,"\int \frac{1}{1-x^4+x^8} \, dx","Integrate[(1 - x^4 + x^8)^(-1),x]","\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^7-\text{$\#$1}^3}\&\right]","-\frac{\log \left(x^2-\sqrt{2-\sqrt{3}} x+1\right)}{4 \sqrt{6}}+\frac{\log \left(x^2+\sqrt{2-\sqrt{3}} x+1\right)}{4 \sqrt{6}}-\frac{\log \left(x^2-\sqrt{2+\sqrt{3}} x+1\right)}{4 \sqrt{6}}+\frac{\log \left(x^2+\sqrt{2+\sqrt{3}} x+1\right)}{4 \sqrt{6}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right)}{2 \sqrt{6}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right)}{2 \sqrt{6}}+\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right)}{2 \sqrt{6}}+\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right)}{2 \sqrt{6}}",1,"RootSum[1 - #1^4 + #1^8 & , Log[x - #1]/(-#1^3 + 2*#1^7) & ]/4","C",1
363,1,61,360,0.0155965,"\int \frac{1}{x^2 \left(1-x^4+x^8\right)} \, dx","Integrate[1/(x^2*(1 - x^4 + x^8)),x]","-\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^5-\text{$\#$1}}\&\right]-\frac{1}{x}","\frac{1}{8} \sqrt{\frac{1}{3} \left(2-\sqrt{3}\right)} \log \left(x^2-\sqrt{2-\sqrt{3}} x+1\right)-\frac{1}{8} \sqrt{\frac{1}{3} \left(2-\sqrt{3}\right)} \log \left(x^2+\sqrt{2-\sqrt{3}} x+1\right)-\frac{1}{8} \sqrt{\frac{1}{3} \left(2+\sqrt{3}\right)} \log \left(x^2-\sqrt{2+\sqrt{3}} x+1\right)+\frac{1}{8} \sqrt{\frac{1}{3} \left(2+\sqrt{3}\right)} \log \left(x^2+\sqrt{2+\sqrt{3}} x+1\right)-\frac{1}{x}+\frac{\tan ^{-1}\left(\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right)}{4 \sqrt{3 \left(2-\sqrt{3}\right)}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right)}{4 \sqrt{3 \left(2+\sqrt{3}\right)}}-\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right)}{4 \sqrt{3 \left(2-\sqrt{3}\right)}}+\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right)}{4 \sqrt{3 \left(2+\sqrt{3}\right)}}",1,"-x^(-1) - RootSum[1 - #1^4 + #1^8 & , (-Log[x - #1] + Log[x - #1]*#1^4)/(-#1 + 2*#1^5) & ]/4","C",1
364,1,65,370,0.0136267,"\int \frac{1}{x^4 \left(1-x^4+x^8\right)} \, dx","Integrate[1/(x^4*(1 - x^4 + x^8)),x]","-\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^7-\text{$\#$1}^3}\&\right]-\frac{1}{3 x^3}","-\frac{1}{3 x^3}+\frac{1}{8} \sqrt{\frac{1}{3} \left(2-\sqrt{3}\right)} \log \left(x^2-\sqrt{2-\sqrt{3}} x+1\right)-\frac{1}{8} \sqrt{\frac{1}{3} \left(2-\sqrt{3}\right)} \log \left(x^2+\sqrt{2-\sqrt{3}} x+1\right)-\frac{1}{8} \sqrt{\frac{1}{3} \left(2+\sqrt{3}\right)} \log \left(x^2-\sqrt{2+\sqrt{3}} x+1\right)+\frac{1}{8} \sqrt{\frac{1}{3} \left(2+\sqrt{3}\right)} \log \left(x^2+\sqrt{2+\sqrt{3}} x+1\right)-\frac{1}{4} \sqrt{\frac{1}{3} \left(2+\sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right)+\frac{1}{4} \sqrt{\frac{1}{3} \left(2-\sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right)+\frac{1}{4} \sqrt{\frac{1}{3} \left(2+\sqrt{3}\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right)-\frac{1}{4} \sqrt{\frac{1}{3} \left(2-\sqrt{3}\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right)",1,"-1/3*1/x^3 - RootSum[1 - #1^4 + #1^8 & , (-Log[x - #1] + Log[x - #1]*#1^4)/(-#1^3 + 2*#1^7) & ]/4","C",1
365,1,54,287,0.0165845,"\int \frac{1}{x^6 \left(1-x^4+x^8\right)} \, dx","Integrate[1/(x^6*(1 - x^4 + x^8)),x]","-\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\text{$\#$1}^3 \log (x-\text{$\#$1})}{2 \text{$\#$1}^4-1}\&\right]-\frac{1}{5 x^5}-\frac{1}{x}","-\frac{1}{5 x^5}-\frac{\log \left(x^2-\sqrt{2-\sqrt{3}} x+1\right)}{4 \sqrt{6}}+\frac{\log \left(x^2+\sqrt{2-\sqrt{3}} x+1\right)}{4 \sqrt{6}}-\frac{\log \left(x^2-\sqrt{2+\sqrt{3}} x+1\right)}{4 \sqrt{6}}+\frac{\log \left(x^2+\sqrt{2+\sqrt{3}} x+1\right)}{4 \sqrt{6}}-\frac{1}{x}+\frac{\tan ^{-1}\left(\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right)}{2 \sqrt{6}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right)}{2 \sqrt{6}}-\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right)}{2 \sqrt{6}}-\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right)}{2 \sqrt{6}}",1,"-1/5*1/x^5 - x^(-1) - RootSum[1 - #1^4 + #1^8 & , (Log[x - #1]*#1^3)/(-1 + 2*#1^4) & ]/4","C",1
366,1,54,377,0.0149116,"\int \frac{1}{x^8 \left(1-x^4+x^8\right)} \, dx","Integrate[1/(x^8*(1 - x^4 + x^8)),x]","-\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\text{$\#$1} \log (x-\text{$\#$1})}{2 \text{$\#$1}^4-1}\&\right]-\frac{1}{7 x^7}-\frac{1}{3 x^3}","-\frac{1}{7 x^7}-\frac{1}{3 x^3}+\frac{1}{8} \sqrt{\frac{1}{3} \left(2+\sqrt{3}\right)} \log \left(x^2-\sqrt{2-\sqrt{3}} x+1\right)-\frac{1}{8} \sqrt{\frac{1}{3} \left(2+\sqrt{3}\right)} \log \left(x^2+\sqrt{2-\sqrt{3}} x+1\right)-\frac{1}{8} \sqrt{\frac{1}{3} \left(2-\sqrt{3}\right)} \log \left(x^2-\sqrt{2+\sqrt{3}} x+1\right)+\frac{1}{8} \sqrt{\frac{1}{3} \left(2-\sqrt{3}\right)} \log \left(x^2+\sqrt{2+\sqrt{3}} x+1\right)-\frac{1}{4} \sqrt{\frac{1}{3} \left(2-\sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right)+\frac{1}{4} \sqrt{\frac{1}{3} \left(2+\sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right)+\frac{1}{4} \sqrt{\frac{1}{3} \left(2-\sqrt{3}\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right)-\frac{1}{4} \sqrt{\frac{1}{3} \left(2+\sqrt{3}\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right)",1,"-1/7*1/x^7 - 1/(3*x^3) - RootSum[1 - #1^4 + #1^8 & , (Log[x - #1]*#1)/(-1 + 2*#1^4) & ]/4","C",1
367,1,79,117,0.0583755,"\int \frac{x^m}{1+3 x^4+x^8} \, dx","Integrate[x^m/(1 + 3*x^4 + x^8),x]","\frac{x^m \text{RootSum}\left[\text{$\#$1}^8+3 \text{$\#$1}^4+1\&,\frac{\left(\frac{x}{x-\text{$\#$1}}\right)^{-m} \, _2F_1\left(-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right)}{2 \text{$\#$1}^7+3 \text{$\#$1}^3}\&\right]}{4 m}","\frac{2 x^{m+1} \, _2F_1\left(1,\frac{m+1}{4};\frac{m+5}{4};-\frac{2 x^4}{3-\sqrt{5}}\right)}{\sqrt{5} \left(3-\sqrt{5}\right) (m+1)}-\frac{2 x^{m+1} \, _2F_1\left(1,\frac{m+1}{4};\frac{m+5}{4};-\frac{2 x^4}{3+\sqrt{5}}\right)}{\sqrt{5} \left(3+\sqrt{5}\right) (m+1)}",1,"(x^m*RootSum[1 + 3*#1^4 + #1^8 & , Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1))]/((x/(x - #1))^m*(3*#1^3 + 2*#1^7)) & ])/(4*m)","C",0
368,1,57,62,0.0331108,"\int \frac{x^{11}}{1+3 x^4+x^8} \, dx","Integrate[x^11/(1 + 3*x^4 + x^8),x]","\frac{1}{40} \left(10 x^4+\left(7 \sqrt{5}-15\right) \log \left(-2 x^4+\sqrt{5}-3\right)-\left(15+7 \sqrt{5}\right) \log \left(2 x^4+\sqrt{5}+3\right)\right)","\frac{x^4}{4}-\frac{1}{40} \left(15-7 \sqrt{5}\right) \log \left(2 x^4-\sqrt{5}+3\right)-\frac{1}{40} \left(15+7 \sqrt{5}\right) \log \left(2 x^4+\sqrt{5}+3\right)",1,"(10*x^4 + (-15 + 7*Sqrt[5])*Log[-3 + Sqrt[5] - 2*x^4] - (15 + 7*Sqrt[5])*Log[3 + Sqrt[5] + 2*x^4])/40","A",1
369,1,97,90,0.1518215,"\int \frac{x^9}{1+3 x^4+x^8} \, dx","Integrate[x^9/(1 + 3*x^4 + x^8),x]","\frac{1}{40} \left(20 x^2-\sqrt{6-2 \sqrt{5}} \left(15+7 \sqrt{5}\right) \tan ^{-1}\left(\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right)+\sqrt{2 \left(3+\sqrt{5}\right)} \left(7 \sqrt{5}-15\right) \tan ^{-1}\left(\sqrt{\frac{1}{2} \left(3+\sqrt{5}\right)} x^2\right)\right)","\frac{x^2}{2}-\frac{1}{2} \sqrt{\frac{1}{5} \left(9+4 \sqrt{5}\right)} \tan ^{-1}\left(\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right)+\frac{1}{2} \sqrt{\frac{1}{5} \left(9-4 \sqrt{5}\right)} \tan ^{-1}\left(\sqrt{\frac{1}{2} \left(3+\sqrt{5}\right)} x^2\right)",1,"(20*x^2 - Sqrt[6 - 2*Sqrt[5]]*(15 + 7*Sqrt[5])*ArcTan[Sqrt[2/(3 + Sqrt[5])]*x^2] + Sqrt[2*(3 + Sqrt[5])]*(-15 + 7*Sqrt[5])*ArcTan[Sqrt[(3 + Sqrt[5])/2]*x^2])/40","A",1
370,1,53,55,0.023356,"\int \frac{x^7}{1+3 x^4+x^8} \, dx","Integrate[x^7/(1 + 3*x^4 + x^8),x]","\frac{1}{40} \left(5-3 \sqrt{5}\right) \log \left(-2 x^4+\sqrt{5}-3\right)+\frac{1}{40} \left(5+3 \sqrt{5}\right) \log \left(2 x^4+\sqrt{5}+3\right)","\frac{1}{40} \left(5-3 \sqrt{5}\right) \log \left(2 x^4-\sqrt{5}+3\right)+\frac{1}{40} \left(5+3 \sqrt{5}\right) \log \left(2 x^4+\sqrt{5}+3\right)",1,"((5 - 3*Sqrt[5])*Log[-3 + Sqrt[5] - 2*x^4])/40 + ((5 + 3*Sqrt[5])*Log[3 + Sqrt[5] + 2*x^4])/40","A",1
371,1,75,81,0.0476232,"\int \frac{x^5}{1+3 x^4+x^8} \, dx","Integrate[x^5/(1 + 3*x^4 + x^8),x]","\frac{2 \sqrt{5} \tan ^{-1}\left(\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right)+\left(5-3 \sqrt{5}\right) \tan ^{-1}\left(\sqrt{\frac{1}{2} \left(3+\sqrt{5}\right)} x^2\right)}{10 \sqrt{6-2 \sqrt{5}}}","\frac{1}{2} \sqrt{\frac{1}{10} \left(3+\sqrt{5}\right)} \tan ^{-1}\left(\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(3-\sqrt{5}\right)} \tan ^{-1}\left(\sqrt{\frac{1}{2} \left(3+\sqrt{5}\right)} x^2\right)",1,"(2*Sqrt[5]*ArcTan[Sqrt[2/(3 + Sqrt[5])]*x^2] + (5 - 3*Sqrt[5])*ArcTan[Sqrt[(3 + Sqrt[5])/2]*x^2])/(10*Sqrt[6 - 2*Sqrt[5]])","A",1
372,1,38,23,0.009795,"\int \frac{x^3}{1+3 x^4+x^8} \, dx","Integrate[x^3/(1 + 3*x^4 + x^8),x]","\frac{\log \left(-2 x^4+\sqrt{5}-3\right)-\log \left(2 x^4+\sqrt{5}+3\right)}{4 \sqrt{5}}","-\frac{\tanh ^{-1}\left(\frac{2 x^4+3}{\sqrt{5}}\right)}{2 \sqrt{5}}",1,"(Log[-3 + Sqrt[5] - 2*x^4] - Log[3 + Sqrt[5] + 2*x^4])/(4*Sqrt[5])","A",1
373,1,74,75,0.0371743,"\int \frac{x}{1+3 x^4+x^8} \, dx","Integrate[x/(1 + 3*x^4 + x^8),x]","\frac{\tan ^{-1}\left(\sqrt{\frac{2}{3-\sqrt{5}}} x^2\right)}{\sqrt{10 \left(3-\sqrt{5}\right)}}-\frac{\tan ^{-1}\left(\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right)}{\sqrt{10 \left(3+\sqrt{5}\right)}}","\frac{1}{2} \sqrt{\frac{1}{10} \left(3+\sqrt{5}\right)} \tan ^{-1}\left(\sqrt{\frac{1}{2} \left(3+\sqrt{5}\right)} x^2\right)-\frac{\tan ^{-1}\left(\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right)}{\sqrt{10 \left(3+\sqrt{5}\right)}}",1,"ArcTan[Sqrt[2/(3 - Sqrt[5])]*x^2]/Sqrt[10*(3 - Sqrt[5])] - ArcTan[Sqrt[2/(3 + Sqrt[5])]*x^2]/Sqrt[10*(3 + Sqrt[5])]","A",1
374,1,55,57,0.0297286,"\int \frac{1}{x \left(1+3 x^4+x^8\right)} \, dx","Integrate[1/(x*(1 + 3*x^4 + x^8)),x]","\frac{1}{40} \left(-5-3 \sqrt{5}\right) \log \left(-2 x^4+\sqrt{5}-3\right)+\frac{1}{40} \left(3 \sqrt{5}-5\right) \log \left(2 x^4+\sqrt{5}+3\right)+\log (x)","-\frac{1}{40} \left(5+3 \sqrt{5}\right) \log \left(2 x^4-\sqrt{5}+3\right)-\frac{1}{40} \left(5-3 \sqrt{5}\right) \log \left(2 x^4+\sqrt{5}+3\right)+\log (x)",1,"Log[x] + ((-5 - 3*Sqrt[5])*Log[-3 + Sqrt[5] - 2*x^4])/40 + ((-5 + 3*Sqrt[5])*Log[3 + Sqrt[5] + 2*x^4])/40","A",1
375,1,65,89,0.0174522,"\int \frac{1}{x^3 \left(1+3 x^4+x^8\right)} \, dx","Integrate[1/(x^3*(1 + 3*x^4 + x^8)),x]","-\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8+3 \text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})+3 \log (x-\text{$\#$1})}{2 \text{$\#$1}^6+3 \text{$\#$1}^2}\&\right]-\frac{1}{2 x^2}","-\frac{1}{2 x^2}+\frac{1}{2} \sqrt{\frac{1}{5} \left(9-4 \sqrt{5}\right)} \tan ^{-1}\left(\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right)-\frac{\left(3+\sqrt{5}\right)^{3/2} \tan ^{-1}\left(\sqrt{\frac{1}{2} \left(3+\sqrt{5}\right)} x^2\right)}{4 \sqrt{10}}",1,"-1/2*1/x^2 - RootSum[1 + 3*#1^4 + #1^8 & , (3*Log[x - #1] + Log[x - #1]*#1^4)/(3*#1^2 + 2*#1^6) & ]/4","C",1
376,1,60,66,0.035448,"\int \frac{1}{x^5 \left(1+3 x^4+x^8\right)} \, dx","Integrate[1/(x^5*(1 + 3*x^4 + x^8)),x]","\frac{1}{40} \left(-\frac{10}{x^4}+\left(15+7 \sqrt{5}\right) \log \left(-2 x^4+\sqrt{5}-3\right)+\left(15-7 \sqrt{5}\right) \log \left(2 x^4+\sqrt{5}+3\right)-120 \log (x)\right)","-\frac{1}{4 x^4}+\frac{1}{40} \left(15+7 \sqrt{5}\right) \log \left(2 x^4-\sqrt{5}+3\right)+\frac{1}{40} \left(15-7 \sqrt{5}\right) \log \left(2 x^4+\sqrt{5}+3\right)-3 \log (x)",1,"(-10/x^4 - 120*Log[x] + (15 + 7*Sqrt[5])*Log[-3 + Sqrt[5] - 2*x^4] + (15 - 7*Sqrt[5])*Log[3 + Sqrt[5] + 2*x^4])/40","A",1
377,1,73,97,0.0182868,"\int \frac{1}{x^7 \left(1+3 x^4+x^8\right)} \, dx","Integrate[1/(x^7*(1 + 3*x^4 + x^8)),x]","\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8+3 \text{$\#$1}^4+1\&,\frac{3 \text{$\#$1}^4 \log (x-\text{$\#$1})+8 \log (x-\text{$\#$1})}{2 \text{$\#$1}^6+3 \text{$\#$1}^2}\&\right]-\frac{1}{6 x^6}+\frac{3}{2 x^2}","-\frac{1}{6 x^6}+\frac{3}{2 x^2}-\frac{1}{2} \sqrt{\frac{1}{10} \left(123-55 \sqrt{5}\right)} \tan ^{-1}\left(\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right)+\frac{1}{2} \sqrt{\frac{1}{10} \left(123+55 \sqrt{5}\right)} \tan ^{-1}\left(\sqrt{\frac{1}{2} \left(3+\sqrt{5}\right)} x^2\right)",1,"-1/6*1/x^6 + 3/(2*x^2) + RootSum[1 + 3*#1^4 + #1^8 & , (8*Log[x - #1] + 3*Log[x - #1]*#1^4)/(3*#1^2 + 2*#1^6) & ]/4","C",1
378,1,58,460,0.0147383,"\int \frac{x^8}{1+3 x^4+x^8} \, dx","Integrate[x^8/(1 + 3*x^4 + x^8),x]","x-\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8+3 \text{$\#$1}^4+1\&,\frac{3 \text{$\#$1}^4 \log (x-\text{$\#$1})+\log (x-\text{$\#$1})}{2 \text{$\#$1}^7+3 \text{$\#$1}^3}\&\right]","-\frac{\sqrt[4]{123-55 \sqrt{5}} \log \left(2 x^2-2 \sqrt[4]{2 \left(3-\sqrt{5}\right)} x+\sqrt{2 \left(3-\sqrt{5}\right)}\right)}{4\ 2^{3/4} \sqrt{5}}+\frac{\sqrt[4]{123-55 \sqrt{5}} \log \left(2 x^2+2 \sqrt[4]{2 \left(3-\sqrt{5}\right)} x+\sqrt{2 \left(3-\sqrt{5}\right)}\right)}{4\ 2^{3/4} \sqrt{5}}+\frac{\sqrt[4]{123+55 \sqrt{5}} \log \left(2 x^2-2 \sqrt[4]{2 \left(3+\sqrt{5}\right)} x+\sqrt{2 \left(3+\sqrt{5}\right)}\right)}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt[4]{123+55 \sqrt{5}} \log \left(2 x^2+2 \sqrt[4]{2 \left(3+\sqrt{5}\right)} x+\sqrt{2 \left(3+\sqrt{5}\right)}\right)}{4\ 2^{3/4} \sqrt{5}}+x-\frac{\sqrt[4]{123-55 \sqrt{5}} \tan ^{-1}\left(1-\frac{2^{3/4} x}{\sqrt[4]{3-\sqrt{5}}}\right)}{2\ 2^{3/4} \sqrt{5}}+\frac{\sqrt[4]{123-55 \sqrt{5}} \tan ^{-1}\left(\frac{2^{3/4} x}{\sqrt[4]{3-\sqrt{5}}}+1\right)}{2\ 2^{3/4} \sqrt{5}}+\frac{\sqrt[4]{123+55 \sqrt{5}} \tan ^{-1}\left(1-\frac{2^{3/4} x}{\sqrt[4]{3+\sqrt{5}}}\right)}{2\ 2^{3/4} \sqrt{5}}-\frac{\sqrt[4]{123+55 \sqrt{5}} \tan ^{-1}\left(\frac{2^{3/4} x}{\sqrt[4]{3+\sqrt{5}}}+1\right)}{2\ 2^{3/4} \sqrt{5}}",1,"x - RootSum[1 + 3*#1^4 + #1^8 & , (Log[x - #1] + 3*Log[x - #1]*#1^4)/(3*#1^3 + 2*#1^7) & ]/4","C",1
379,1,41,431,0.01161,"\int \frac{x^6}{1+3 x^4+x^8} \, dx","Integrate[x^6/(1 + 3*x^4 + x^8),x]","\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8+3 \text{$\#$1}^4+1\&,\frac{\text{$\#$1}^3 \log (x-\text{$\#$1})}{2 \text{$\#$1}^4+3}\&\right]","-\frac{\sqrt[4]{9-4 \sqrt{5}} \log \left(2 x^2-2 \sqrt[4]{2 \left(3-\sqrt{5}\right)} x+\sqrt{2 \left(3-\sqrt{5}\right)}\right)}{4 \sqrt{10}}+\frac{\sqrt[4]{9-4 \sqrt{5}} \log \left(2 x^2+2 \sqrt[4]{2 \left(3-\sqrt{5}\right)} x+\sqrt{2 \left(3-\sqrt{5}\right)}\right)}{4 \sqrt{10}}+\frac{\left(3+\sqrt{5}\right)^{3/4} \log \left(2 x^2-2 \sqrt[4]{2 \left(3+\sqrt{5}\right)} x+\sqrt{2 \left(3+\sqrt{5}\right)}\right)}{8 \sqrt[4]{2} \sqrt{5}}-\frac{\left(3+\sqrt{5}\right)^{3/4} \log \left(2 x^2+2 \sqrt[4]{2 \left(3+\sqrt{5}\right)} x+\sqrt{2 \left(3+\sqrt{5}\right)}\right)}{8 \sqrt[4]{2} \sqrt{5}}+\frac{\sqrt[4]{9-4 \sqrt{5}} \tan ^{-1}\left(1-\frac{2^{3/4} x}{\sqrt[4]{3-\sqrt{5}}}\right)}{2 \sqrt{10}}-\frac{\sqrt[4]{9-4 \sqrt{5}} \tan ^{-1}\left(\frac{2^{3/4} x}{\sqrt[4]{3-\sqrt{5}}}+1\right)}{2 \sqrt{10}}-\frac{\left(3+\sqrt{5}\right)^{3/4} \tan ^{-1}\left(1-\frac{2^{3/4} x}{\sqrt[4]{3+\sqrt{5}}}\right)}{4 \sqrt[4]{2} \sqrt{5}}+\frac{\left(3+\sqrt{5}\right)^{3/4} \tan ^{-1}\left(\frac{2^{3/4} x}{\sqrt[4]{3+\sqrt{5}}}+1\right)}{4 \sqrt[4]{2} \sqrt{5}}",1,"RootSum[1 + 3*#1^4 + #1^8 & , (Log[x - #1]*#1^3)/(3 + 2*#1^4) & ]/4","C",1
380,1,39,451,0.0111621,"\int \frac{x^4}{1+3 x^4+x^8} \, dx","Integrate[x^4/(1 + 3*x^4 + x^8),x]","\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8+3 \text{$\#$1}^4+1\&,\frac{\text{$\#$1} \log (x-\text{$\#$1})}{2 \text{$\#$1}^4+3}\&\right]","\frac{\sqrt[4]{3-\sqrt{5}} \log \left(2 x^2-2 \sqrt[4]{2 \left(3-\sqrt{5}\right)} x+\sqrt{2 \left(3-\sqrt{5}\right)}\right)}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt[4]{3-\sqrt{5}} \log \left(2 x^2+2 \sqrt[4]{2 \left(3-\sqrt{5}\right)} x+\sqrt{2 \left(3-\sqrt{5}\right)}\right)}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt[4]{3+\sqrt{5}} \log \left(2 x^2-2 \sqrt[4]{2 \left(3+\sqrt{5}\right)} x+\sqrt{2 \left(3+\sqrt{5}\right)}\right)}{4\ 2^{3/4} \sqrt{5}}+\frac{\sqrt[4]{3+\sqrt{5}} \log \left(2 x^2+2 \sqrt[4]{2 \left(3+\sqrt{5}\right)} x+\sqrt{2 \left(3+\sqrt{5}\right)}\right)}{4\ 2^{3/4} \sqrt{5}}+\frac{\sqrt[4]{3-\sqrt{5}} \tan ^{-1}\left(1-\frac{2^{3/4} x}{\sqrt[4]{3-\sqrt{5}}}\right)}{2\ 2^{3/4} \sqrt{5}}-\frac{\sqrt[4]{3-\sqrt{5}} \tan ^{-1}\left(\frac{2^{3/4} x}{\sqrt[4]{3-\sqrt{5}}}+1\right)}{2\ 2^{3/4} \sqrt{5}}-\frac{\sqrt[4]{3+\sqrt{5}} \tan ^{-1}\left(1-\frac{2^{3/4} x}{\sqrt[4]{3+\sqrt{5}}}\right)}{2\ 2^{3/4} \sqrt{5}}+\frac{\sqrt[4]{3+\sqrt{5}} \tan ^{-1}\left(\frac{2^{3/4} x}{\sqrt[4]{3+\sqrt{5}}}+1\right)}{2\ 2^{3/4} \sqrt{5}}",1,"RootSum[1 + 3*#1^4 + #1^8 & , (Log[x - #1]*#1)/(3 + 2*#1^4) & ]/4","C",1
381,1,40,427,0.0101661,"\int \frac{x^2}{1+3 x^4+x^8} \, dx","Integrate[x^2/(1 + 3*x^4 + x^8),x]","\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8+3 \text{$\#$1}^4+1\&,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^5+3 \text{$\#$1}}\&\right]","\frac{\log \left(2 x^2-2 \sqrt[4]{2 \left(3-\sqrt{5}\right)} x+\sqrt{2 \left(3-\sqrt{5}\right)}\right)}{4 \sqrt{5} \sqrt[4]{2 \left(3-\sqrt{5}\right)}}-\frac{\log \left(2 x^2+2 \sqrt[4]{2 \left(3-\sqrt{5}\right)} x+\sqrt{2 \left(3-\sqrt{5}\right)}\right)}{4 \sqrt{5} \sqrt[4]{2 \left(3-\sqrt{5}\right)}}-\frac{\log \left(2 x^2-2 \sqrt[4]{2 \left(3+\sqrt{5}\right)} x+\sqrt{2 \left(3+\sqrt{5}\right)}\right)}{4 \sqrt{5} \sqrt[4]{2 \left(3+\sqrt{5}\right)}}+\frac{\log \left(2 x^2+2 \sqrt[4]{2 \left(3+\sqrt{5}\right)} x+\sqrt{2 \left(3+\sqrt{5}\right)}\right)}{4 \sqrt{5} \sqrt[4]{2 \left(3+\sqrt{5}\right)}}-\frac{\tan ^{-1}\left(1-\frac{2^{3/4} x}{\sqrt[4]{3-\sqrt{5}}}\right)}{2 \sqrt{5} \sqrt[4]{2 \left(3-\sqrt{5}\right)}}+\frac{\tan ^{-1}\left(\frac{2^{3/4} x}{\sqrt[4]{3-\sqrt{5}}}+1\right)}{2 \sqrt{5} \sqrt[4]{2 \left(3-\sqrt{5}\right)}}+\frac{\tan ^{-1}\left(1-\frac{2^{3/4} x}{\sqrt[4]{3+\sqrt{5}}}\right)}{2 \sqrt{5} \sqrt[4]{2 \left(3+\sqrt{5}\right)}}-\frac{\tan ^{-1}\left(\frac{2^{3/4} x}{\sqrt[4]{3+\sqrt{5}}}+1\right)}{2 \sqrt{5} \sqrt[4]{2 \left(3+\sqrt{5}\right)}}",1,"RootSum[1 + 3*#1^4 + #1^8 & , Log[x - #1]/(3*#1 + 2*#1^5) & ]/4","C",1
382,1,42,414,0.0095935,"\int \frac{1}{1+3 x^4+x^8} \, dx","Integrate[(1 + 3*x^4 + x^8)^(-1),x]","\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8+3 \text{$\#$1}^4+1\&,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^7+3 \text{$\#$1}^3}\&\right]","-\frac{\sqrt[4]{9+4 \sqrt{5}} \log \left(2 x^2-2 \sqrt[4]{2 \left(3-\sqrt{5}\right)} x+\sqrt{2 \left(3-\sqrt{5}\right)}\right)}{4 \sqrt{10}}+\frac{\sqrt[4]{9+4 \sqrt{5}} \log \left(2 x^2+2 \sqrt[4]{2 \left(3-\sqrt{5}\right)} x+\sqrt{2 \left(3-\sqrt{5}\right)}\right)}{4 \sqrt{10}}+\frac{\log \left(2 x^2-2 \sqrt[4]{2 \left(3+\sqrt{5}\right)} x+\sqrt{2 \left(3+\sqrt{5}\right)}\right)}{2 \sqrt{5} \left(2 \left(3+\sqrt{5}\right)\right)^{3/4}}-\frac{\log \left(2 x^2+2 \sqrt[4]{2 \left(3+\sqrt{5}\right)} x+\sqrt{2 \left(3+\sqrt{5}\right)}\right)}{2 \sqrt{5} \left(2 \left(3+\sqrt{5}\right)\right)^{3/4}}-\frac{\sqrt[4]{9+4 \sqrt{5}} \tan ^{-1}\left(1-\frac{2^{3/4} x}{\sqrt[4]{3-\sqrt{5}}}\right)}{2 \sqrt{10}}+\frac{\sqrt[4]{9+4 \sqrt{5}} \tan ^{-1}\left(\frac{2^{3/4} x}{\sqrt[4]{3-\sqrt{5}}}+1\right)}{2 \sqrt{10}}+\frac{\tan ^{-1}\left(1-\frac{2^{3/4} x}{\sqrt[4]{3+\sqrt{5}}}\right)}{\sqrt{5} \left(2 \left(3+\sqrt{5}\right)\right)^{3/4}}-\frac{\tan ^{-1}\left(\frac{2^{3/4} x}{\sqrt[4]{3+\sqrt{5}}}+1\right)}{\sqrt{5} \left(2 \left(3+\sqrt{5}\right)\right)^{3/4}}",1,"RootSum[1 + 3*#1^4 + #1^8 & , Log[x - #1]/(3*#1^3 + 2*#1^7) & ]/4","C",1
383,1,61,416,0.0157319,"\int \frac{1}{x^2 \left(1+3 x^4+x^8\right)} \, dx","Integrate[1/(x^2*(1 + 3*x^4 + x^8)),x]","-\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8+3 \text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})+3 \log (x-\text{$\#$1})}{2 \text{$\#$1}^5+3 \text{$\#$1}}\&\right]-\frac{1}{x}","-\frac{\left(3+\sqrt{5}\right)^{5/4} \log \left(2 x^2-2 \sqrt[4]{2 \left(3-\sqrt{5}\right)} x+\sqrt{2 \left(3-\sqrt{5}\right)}\right)}{8\ 2^{3/4} \sqrt{5}}+\frac{\left(3+\sqrt{5}\right)^{5/4} \log \left(2 x^2+2 \sqrt[4]{2 \left(3-\sqrt{5}\right)} x+\sqrt{2 \left(3-\sqrt{5}\right)}\right)}{8\ 2^{3/4} \sqrt{5}}+\frac{1}{40} \sqrt[4]{6150-2750 \sqrt{5}} \log \left(2 x^2-2 \sqrt[4]{2 \left(3+\sqrt{5}\right)} x+\sqrt{2 \left(3+\sqrt{5}\right)}\right)-\frac{1}{40} \sqrt[4]{6150-2750 \sqrt{5}} \log \left(2 x^2+2 \sqrt[4]{2 \left(3+\sqrt{5}\right)} x+\sqrt{2 \left(3+\sqrt{5}\right)}\right)-\frac{1}{x}+\frac{\left(3+\sqrt{5}\right)^{5/4} \tan ^{-1}\left(1-\frac{2^{3/4} x}{\sqrt[4]{3-\sqrt{5}}}\right)}{4\ 2^{3/4} \sqrt{5}}-\frac{\left(3+\sqrt{5}\right)^{5/4} \tan ^{-1}\left(\frac{2^{3/4} x}{\sqrt[4]{3-\sqrt{5}}}+1\right)}{4\ 2^{3/4} \sqrt{5}}-\frac{1}{20} \sqrt[4]{6150-2750 \sqrt{5}} \tan ^{-1}\left(1-\frac{2^{3/4} x}{\sqrt[4]{3+\sqrt{5}}}\right)+\frac{1}{20} \sqrt[4]{6150-2750 \sqrt{5}} \tan ^{-1}\left(\frac{2^{3/4} x}{\sqrt[4]{3+\sqrt{5}}}+1\right)",1,"-x^(-1) - RootSum[1 + 3*#1^4 + #1^8 & , (3*Log[x - #1] + Log[x - #1]*#1^4)/(3*#1 + 2*#1^5) & ]/4","C",1
384,1,65,466,0.0160643,"\int \frac{1}{x^4 \left(1+3 x^4+x^8\right)} \, dx","Integrate[1/(x^4*(1 + 3*x^4 + x^8)),x]","-\frac{1}{4} \text{RootSum}\left[\text{$\#$1}^8+3 \text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})+3 \log (x-\text{$\#$1})}{2 \text{$\#$1}^7+3 \text{$\#$1}^3}\&\right]-\frac{1}{3 x^3}","-\frac{1}{3 x^3}+\frac{\sqrt[4]{843+377 \sqrt{5}} \log \left(2 x^2-2 \sqrt[4]{2 \left(3-\sqrt{5}\right)} x+\sqrt{2 \left(3-\sqrt{5}\right)}\right)}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt[4]{843+377 \sqrt{5}} \log \left(2 x^2+2 \sqrt[4]{2 \left(3-\sqrt{5}\right)} x+\sqrt{2 \left(3-\sqrt{5}\right)}\right)}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt[4]{843-377 \sqrt{5}} \log \left(2 x^2-2 \sqrt[4]{2 \left(3+\sqrt{5}\right)} x+\sqrt{2 \left(3+\sqrt{5}\right)}\right)}{4\ 2^{3/4} \sqrt{5}}+\frac{\sqrt[4]{843-377 \sqrt{5}} \log \left(2 x^2+2 \sqrt[4]{2 \left(3+\sqrt{5}\right)} x+\sqrt{2 \left(3+\sqrt{5}\right)}\right)}{4\ 2^{3/4} \sqrt{5}}+\frac{\sqrt[4]{843+377 \sqrt{5}} \tan ^{-1}\left(1-\frac{2^{3/4} x}{\sqrt[4]{3-\sqrt{5}}}\right)}{2\ 2^{3/4} \sqrt{5}}-\frac{\sqrt[4]{843+377 \sqrt{5}} \tan ^{-1}\left(\frac{2^{3/4} x}{\sqrt[4]{3-\sqrt{5}}}+1\right)}{2\ 2^{3/4} \sqrt{5}}-\frac{\sqrt[4]{843-377 \sqrt{5}} \tan ^{-1}\left(1-\frac{2^{3/4} x}{\sqrt[4]{3+\sqrt{5}}}\right)}{2\ 2^{3/4} \sqrt{5}}+\frac{\sqrt[4]{843-377 \sqrt{5}} \tan ^{-1}\left(\frac{2^{3/4} x}{\sqrt[4]{3+\sqrt{5}}}+1\right)}{2\ 2^{3/4} \sqrt{5}}",1,"-1/3*1/x^3 - RootSum[1 + 3*#1^4 + #1^8 & , (3*Log[x - #1] + Log[x - #1]*#1^4)/(3*#1^3 + 2*#1^7) & ]/4","C",1
385,1,575,117,0.5718944,"\int \frac{x^m}{1-3 x^4+x^8} \, dx","Integrate[x^m/(1 - 3*x^4 + x^8),x]","\frac{x^m \left(\frac{\text{RootSum}\left[\text{$\#$1}^4-\text{$\#$1}^2-1\&,\frac{\text{$\#$1}^2 m^2 \left(\frac{x}{x-\text{$\#$1}}\right)^{-m} \, _2F_1\left(-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right)+3 \text{$\#$1}^2 m \left(\frac{x}{x-\text{$\#$1}}\right)^{-m} \, _2F_1\left(-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right)+2 \text{$\#$1}^2 \left(\frac{x}{x-\text{$\#$1}}\right)^{-m} \, _2F_1\left(-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right)+\text{$\#$1}^2 m \left(\frac{x}{\text{$\#$1}}\right)^{-m}+\text{$\#$1} m^2 x+2 \text{$\#$1} m x+m^2 x^2+m x^2}{2 \text{$\#$1}^3-\text{$\#$1}}\&\right]-\text{RootSum}\left[\text{$\#$1}^4+\text{$\#$1}^2-1\&,\frac{\text{$\#$1}^2 m^2 \left(\frac{x}{x-\text{$\#$1}}\right)^{-m} \, _2F_1\left(-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right)+3 \text{$\#$1}^2 m \left(\frac{x}{x-\text{$\#$1}}\right)^{-m} \, _2F_1\left(-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right)+2 \text{$\#$1}^2 \left(\frac{x}{x-\text{$\#$1}}\right)^{-m} \, _2F_1\left(-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right)+\text{$\#$1}^2 m \left(\frac{x}{\text{$\#$1}}\right)^{-m}+\text{$\#$1} m^2 x+2 \text{$\#$1} m x+m^2 x^2+m x^2}{2 \text{$\#$1}^3+\text{$\#$1}}\&\right]-\left(m^2+3 m+2\right) \text{RootSum}\left[\text{$\#$1}^4+\text{$\#$1}^2-1\&,\frac{\left(\frac{x}{x-\text{$\#$1}}\right)^{-m} \, _2F_1\left(-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right)}{2 \text{$\#$1}^3+\text{$\#$1}}\&\right]}{m^2+3 m+2}-\text{RootSum}\left[\text{$\#$1}^4-\text{$\#$1}^2-1\&,\frac{\left(\frac{x}{x-\text{$\#$1}}\right)^{-m} \, _2F_1\left(-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right)}{2 \text{$\#$1}^3-\text{$\#$1}}\&\right]\right)}{4 m}","\frac{2 x^{m+1} \, _2F_1\left(1,\frac{m+1}{4};\frac{m+5}{4};\frac{2 x^4}{3-\sqrt{5}}\right)}{\sqrt{5} \left(3-\sqrt{5}\right) (m+1)}-\frac{2 x^{m+1} \, _2F_1\left(1,\frac{m+1}{4};\frac{m+5}{4};\frac{2 x^4}{3+\sqrt{5}}\right)}{\sqrt{5} \left(3+\sqrt{5}\right) (m+1)}",1,"(x^m*(-RootSum[-1 - #1^2 + #1^4 & , Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1))]/((x/(x - #1))^m*(-#1 + 2*#1^3)) & ] + (RootSum[-1 - #1^2 + #1^4 & , (m*x^2 + m^2*x^2 + 2*m*x*#1 + m^2*x*#1 + (2*Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1))]*#1^2)/(x/(x - #1))^m + (3*m*Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1))]*#1^2)/(x/(x - #1))^m + (m^2*Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1))]*#1^2)/(x/(x - #1))^m + (m*#1^2)/(x/#1)^m)/(-#1 + 2*#1^3) & ] - (2 + 3*m + m^2)*RootSum[-1 + #1^2 + #1^4 & , Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1))]/((x/(x - #1))^m*(#1 + 2*#1^3)) & ] - RootSum[-1 + #1^2 + #1^4 & , (m*x^2 + m^2*x^2 + 2*m*x*#1 + m^2*x*#1 + (2*Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1))]*#1^2)/(x/(x - #1))^m + (3*m*Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1))]*#1^2)/(x/(x - #1))^m + (m^2*Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1))]*#1^2)/(x/(x - #1))^m + (m*#1^2)/(x/#1)^m)/(#1 + 2*#1^3) & ])/(2 + 3*m + m^2)))/(4*m)","C",0
386,1,56,62,0.0324213,"\int \frac{x^{11}}{1-3 x^4+x^8} \, dx","Integrate[x^11/(1 - 3*x^4 + x^8),x]","\frac{1}{40} \left(10 x^4+\left(15+7 \sqrt{5}\right) \log \left(-2 x^4+\sqrt{5}+3\right)+\left(15-7 \sqrt{5}\right) \log \left(2 x^4+\sqrt{5}-3\right)\right)","\frac{x^4}{4}+\frac{1}{40} \left(15-7 \sqrt{5}\right) \log \left(-2 x^4-\sqrt{5}+3\right)+\frac{1}{40} \left(15+7 \sqrt{5}\right) \log \left(-2 x^4+\sqrt{5}+3\right)",1,"(10*x^4 + (15 + 7*Sqrt[5])*Log[3 + Sqrt[5] - 2*x^4] + (15 - 7*Sqrt[5])*Log[-3 + Sqrt[5] + 2*x^4])/40","A",1
387,1,103,90,0.0540208,"\int \frac{x^9}{1-3 x^4+x^8} \, dx","Integrate[x^9/(1 - 3*x^4 + x^8),x]","\frac{1}{20} \left(10 x^2+\left(2 \sqrt{5}-5\right) \log \left(-2 x^2+\sqrt{5}-1\right)+\left(5+2 \sqrt{5}\right) \log \left(-2 x^2+\sqrt{5}+1\right)+\left(5-2 \sqrt{5}\right) \log \left(2 x^2+\sqrt{5}-1\right)-\left(5+2 \sqrt{5}\right) \log \left(2 x^2+\sqrt{5}+1\right)\right)","\frac{x^2}{2}-\frac{1}{2} \sqrt{\frac{1}{5} \left(9+4 \sqrt{5}\right)} \tanh ^{-1}\left(\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right)+\frac{1}{2} \sqrt{\frac{1}{5} \left(9-4 \sqrt{5}\right)} \tanh ^{-1}\left(\sqrt{\frac{1}{2} \left(3+\sqrt{5}\right)} x^2\right)",1,"(10*x^2 + (-5 + 2*Sqrt[5])*Log[-1 + Sqrt[5] - 2*x^2] + (5 + 2*Sqrt[5])*Log[1 + Sqrt[5] - 2*x^2] + (5 - 2*Sqrt[5])*Log[-1 + Sqrt[5] + 2*x^2] - (5 + 2*Sqrt[5])*Log[1 + Sqrt[5] + 2*x^2])/20","A",1
388,1,53,55,0.0216843,"\int \frac{x^7}{1-3 x^4+x^8} \, dx","Integrate[x^7/(1 - 3*x^4 + x^8),x]","\frac{1}{40} \left(5+3 \sqrt{5}\right) \log \left(-2 x^4+\sqrt{5}+3\right)+\frac{1}{40} \left(5-3 \sqrt{5}\right) \log \left(2 x^4+\sqrt{5}-3\right)","\frac{1}{40} \left(5-3 \sqrt{5}\right) \log \left(-2 x^4-\sqrt{5}+3\right)+\frac{1}{40} \left(5+3 \sqrt{5}\right) \log \left(-2 x^4+\sqrt{5}+3\right)",1,"((5 + 3*Sqrt[5])*Log[3 + Sqrt[5] - 2*x^4])/40 + ((5 - 3*Sqrt[5])*Log[-3 + Sqrt[5] + 2*x^4])/40","A",1
389,1,91,81,0.0337714,"\int \frac{x^5}{1-3 x^4+x^8} \, dx","Integrate[x^5/(1 - 3*x^4 + x^8),x]","\frac{1}{40} \left(\left(\sqrt{5}-5\right) \log \left(-2 x^2+\sqrt{5}-1\right)+\left(5+\sqrt{5}\right) \log \left(-2 x^2+\sqrt{5}+1\right)-\left(\sqrt{5}-5\right) \log \left(2 x^2+\sqrt{5}-1\right)-\left(5+\sqrt{5}\right) \log \left(2 x^2+\sqrt{5}+1\right)\right)","\frac{1}{2} \sqrt{\frac{1}{10} \left(3-\sqrt{5}\right)} \tanh ^{-1}\left(\sqrt{\frac{1}{2} \left(3+\sqrt{5}\right)} x^2\right)-\frac{1}{2} \sqrt{\frac{1}{10} \left(3+\sqrt{5}\right)} \tanh ^{-1}\left(\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right)",1,"((-5 + Sqrt[5])*Log[-1 + Sqrt[5] - 2*x^2] + (5 + Sqrt[5])*Log[1 + Sqrt[5] - 2*x^2] - (-5 + Sqrt[5])*Log[-1 + Sqrt[5] + 2*x^2] - (5 + Sqrt[5])*Log[1 + Sqrt[5] + 2*x^2])/40","A",1
390,1,38,23,0.0097651,"\int \frac{x^3}{1-3 x^4+x^8} \, dx","Integrate[x^3/(1 - 3*x^4 + x^8),x]","\frac{\log \left(-2 x^4+\sqrt{5}+3\right)-\log \left(2 x^4+\sqrt{5}-3\right)}{4 \sqrt{5}}","\frac{\tanh ^{-1}\left(\frac{3-2 x^4}{\sqrt{5}}\right)}{2 \sqrt{5}}",1,"(Log[3 + Sqrt[5] - 2*x^4] - Log[-3 + Sqrt[5] + 2*x^4])/(4*Sqrt[5])","A",1
391,1,91,75,0.0286893,"\int \frac{x}{1-3 x^4+x^8} \, dx","Integrate[x/(1 - 3*x^4 + x^8),x]","\frac{1}{40} \left(-\left(\left(5+\sqrt{5}\right) \log \left(-2 x^2+\sqrt{5}-1\right)\right)-\left(\sqrt{5}-5\right) \log \left(-2 x^2+\sqrt{5}+1\right)+\left(5+\sqrt{5}\right) \log \left(2 x^2+\sqrt{5}-1\right)+\left(\sqrt{5}-5\right) \log \left(2 x^2+\sqrt{5}+1\right)\right)","\frac{1}{2} \sqrt{\frac{1}{10} \left(3+\sqrt{5}\right)} \tanh ^{-1}\left(\sqrt{\frac{1}{2} \left(3+\sqrt{5}\right)} x^2\right)-\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right)}{\sqrt{10 \left(3+\sqrt{5}\right)}}",1,"(-((5 + Sqrt[5])*Log[-1 + Sqrt[5] - 2*x^2]) - (-5 + Sqrt[5])*Log[1 + Sqrt[5] - 2*x^2] + (5 + Sqrt[5])*Log[-1 + Sqrt[5] + 2*x^2] + (-5 + Sqrt[5])*Log[1 + Sqrt[5] + 2*x^2])/40","A",1
392,1,55,57,0.0312459,"\int \frac{1}{x \left(1-3 x^4+x^8\right)} \, dx","Integrate[1/(x*(1 - 3*x^4 + x^8)),x]","\frac{1}{40} \left(3 \sqrt{5}-5\right) \log \left(-2 x^4+\sqrt{5}+3\right)+\frac{1}{40} \left(-5-3 \sqrt{5}\right) \log \left(2 x^4+\sqrt{5}-3\right)+\log (x)","-\frac{1}{40} \left(5+3 \sqrt{5}\right) \log \left(-2 x^4-\sqrt{5}+3\right)-\frac{1}{40} \left(5-3 \sqrt{5}\right) \log \left(-2 x^4+\sqrt{5}+3\right)+\log (x)",1,"Log[x] + ((-5 + 3*Sqrt[5])*Log[3 + Sqrt[5] - 2*x^4])/40 + ((-5 - 3*Sqrt[5])*Log[-3 + Sqrt[5] + 2*x^4])/40","A",1
393,1,103,89,0.0565857,"\int \frac{1}{x^3 \left(1-3 x^4+x^8\right)} \, dx","Integrate[1/(x^3*(1 - 3*x^4 + x^8)),x]","\frac{1}{20} \left(-\frac{10}{x^2}-\left(\left(5+2 \sqrt{5}\right) \log \left(-2 x^2+\sqrt{5}-1\right)\right)+\left(5-2 \sqrt{5}\right) \log \left(-2 x^2+\sqrt{5}+1\right)+\left(5+2 \sqrt{5}\right) \log \left(2 x^2+\sqrt{5}-1\right)+\left(2 \sqrt{5}-5\right) \log \left(2 x^2+\sqrt{5}+1\right)\right)","-\frac{1}{2 x^2}-\frac{1}{2} \sqrt{\frac{1}{5} \left(9-4 \sqrt{5}\right)} \tanh ^{-1}\left(\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right)+\frac{\left(3+\sqrt{5}\right)^{3/2} \tanh ^{-1}\left(\sqrt{\frac{1}{2} \left(3+\sqrt{5}\right)} x^2\right)}{4 \sqrt{10}}",1,"(-10/x^2 - (5 + 2*Sqrt[5])*Log[-1 + Sqrt[5] - 2*x^2] + (5 - 2*Sqrt[5])*Log[1 + Sqrt[5] - 2*x^2] + (5 + 2*Sqrt[5])*Log[-1 + Sqrt[5] + 2*x^2] + (-5 + 2*Sqrt[5])*Log[1 + Sqrt[5] + 2*x^2])/20","A",1
394,1,61,66,0.0356227,"\int \frac{1}{x^5 \left(1-3 x^4+x^8\right)} \, dx","Integrate[1/(x^5*(1 - 3*x^4 + x^8)),x]","\frac{1}{40} \left(-\frac{10}{x^4}+\left(7 \sqrt{5}-15\right) \log \left(-2 x^4+\sqrt{5}+3\right)-\left(15+7 \sqrt{5}\right) \log \left(2 x^4+\sqrt{5}-3\right)+120 \log (x)\right)","-\frac{1}{4 x^4}-\frac{1}{40} \left(15+7 \sqrt{5}\right) \log \left(-2 x^4-\sqrt{5}+3\right)-\frac{1}{40} \left(15-7 \sqrt{5}\right) \log \left(-2 x^4+\sqrt{5}+3\right)+3 \log (x)",1,"(-10/x^4 + 120*Log[x] + (-15 + 7*Sqrt[5])*Log[3 + Sqrt[5] - 2*x^4] - (15 + 7*Sqrt[5])*Log[-3 + Sqrt[5] + 2*x^4])/40","A",1
395,1,111,97,0.0707142,"\int \frac{1}{x^7 \left(1-3 x^4+x^8\right)} \, dx","Integrate[1/(x^7*(1 - 3*x^4 + x^8)),x]","\frac{1}{120} \left(-\frac{20}{x^6}-\frac{180}{x^2}-3 \left(25+11 \sqrt{5}\right) \log \left(-2 x^2+\sqrt{5}-1\right)+3 \left(25-11 \sqrt{5}\right) \log \left(-2 x^2+\sqrt{5}+1\right)+3 \left(25+11 \sqrt{5}\right) \log \left(2 x^2+\sqrt{5}-1\right)+3 \left(11 \sqrt{5}-25\right) \log \left(2 x^2+\sqrt{5}+1\right)\right)","-\frac{1}{6 x^6}-\frac{3}{2 x^2}-\frac{1}{2} \sqrt{\frac{1}{10} \left(123-55 \sqrt{5}\right)} \tanh ^{-1}\left(\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right)+\frac{1}{2} \sqrt{\frac{1}{10} \left(123+55 \sqrt{5}\right)} \tanh ^{-1}\left(\sqrt{\frac{1}{2} \left(3+\sqrt{5}\right)} x^2\right)",1,"(-20/x^6 - 180/x^2 - 3*(25 + 11*Sqrt[5])*Log[-1 + Sqrt[5] - 2*x^2] + 3*(25 - 11*Sqrt[5])*Log[1 + Sqrt[5] - 2*x^2] + 3*(25 + 11*Sqrt[5])*Log[-1 + Sqrt[5] + 2*x^2] + 3*(-25 + 11*Sqrt[5])*Log[1 + Sqrt[5] + 2*x^2])/120","A",1
396,1,160,170,0.2659913,"\int \frac{x^8}{1-3 x^4+x^8} \, dx","Integrate[x^8/(1 - 3*x^4 + x^8),x]","x+\frac{\left(\sqrt{5}-2\right) \tan ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{\sqrt{10 \left(\sqrt{5}-1\right)}}-\frac{\left(2+\sqrt{5}\right) \tan ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{\sqrt{10 \left(1+\sqrt{5}\right)}}+\frac{\left(\sqrt{5}-2\right) \tanh ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{\sqrt{10 \left(\sqrt{5}-1\right)}}-\frac{\left(2+\sqrt{5}\right) \tanh ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{\sqrt{10 \left(1+\sqrt{5}\right)}}","x-\frac{\sqrt[4]{\frac{1}{2} \left(123+55 \sqrt{5}\right)} \tan ^{-1}\left(\sqrt[4]{\frac{2}{3+\sqrt{5}}} x\right)}{2 \sqrt{5}}+\frac{\sqrt[4]{984-440 \sqrt{5}} \tan ^{-1}\left(\sqrt[4]{\frac{1}{2} \left(3+\sqrt{5}\right)} x\right)}{4 \sqrt{5}}-\frac{\sqrt[4]{\frac{1}{2} \left(123+55 \sqrt{5}\right)} \tanh ^{-1}\left(\sqrt[4]{\frac{2}{3+\sqrt{5}}} x\right)}{2 \sqrt{5}}+\frac{\sqrt[4]{984-440 \sqrt{5}} \tanh ^{-1}\left(\sqrt[4]{\frac{1}{2} \left(3+\sqrt{5}\right)} x\right)}{4 \sqrt{5}}",1,"x + ((-2 + Sqrt[5])*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x])/Sqrt[10*(-1 + Sqrt[5])] - ((2 + Sqrt[5])*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x])/Sqrt[10*(1 + Sqrt[5])] + ((-2 + Sqrt[5])*ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x])/Sqrt[10*(-1 + Sqrt[5])] - ((2 + Sqrt[5])*ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x])/Sqrt[10*(1 + Sqrt[5])]","A",1
397,1,160,167,0.1503827,"\int \frac{x^6}{1-3 x^4+x^8} \, dx","Integrate[x^6/(1 - 3*x^4 + x^8),x]","\frac{\frac{\left(\sqrt{5}-3\right) \tan ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{\sqrt{\sqrt{5}-1}}+\frac{\left(3+\sqrt{5}\right) \tan ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{\sqrt{1+\sqrt{5}}}-\frac{\left(\sqrt{5}-3\right) \tanh ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{\sqrt{\sqrt{5}-1}}-\frac{\left(3+\sqrt{5}\right) \tanh ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{\sqrt{1+\sqrt{5}}}}{2 \sqrt{10}}","\frac{\left(3+\sqrt{5}\right)^{3/4} \tan ^{-1}\left(\sqrt[4]{\frac{2}{3+\sqrt{5}}} x\right)}{2\ 2^{3/4} \sqrt{5}}-\frac{\sqrt[4]{144-64 \sqrt{5}} \tan ^{-1}\left(\sqrt[4]{\frac{1}{2} \left(3+\sqrt{5}\right)} x\right)}{4 \sqrt{5}}-\frac{\left(3+\sqrt{5}\right)^{3/4} \tanh ^{-1}\left(\sqrt[4]{\frac{2}{3+\sqrt{5}}} x\right)}{2\ 2^{3/4} \sqrt{5}}+\frac{\sqrt[4]{144-64 \sqrt{5}} \tanh ^{-1}\left(\sqrt[4]{\frac{1}{2} \left(3+\sqrt{5}\right)} x\right)}{4 \sqrt{5}}",1,"(((-3 + Sqrt[5])*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x])/Sqrt[-1 + Sqrt[5]] + ((3 + Sqrt[5])*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x])/Sqrt[1 + Sqrt[5]] - ((-3 + Sqrt[5])*ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x])/Sqrt[-1 + Sqrt[5]] - ((3 + Sqrt[5])*ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x])/Sqrt[1 + Sqrt[5]])/(2*Sqrt[10])","A",1
398,1,132,173,0.1907705,"\int \frac{x^4}{1-3 x^4+x^8} \, dx","Integrate[x^4/(1 - 3*x^4 + x^8),x]","\frac{\sqrt{\sqrt{5}-1} \tan ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)-\sqrt{1+\sqrt{5}} \tan ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)+\sqrt{\sqrt{5}-1} \tanh ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)-\sqrt{1+\sqrt{5}} \tanh ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{2 \sqrt{10}}","-\frac{\sqrt[4]{\frac{1}{2} \left(3+\sqrt{5}\right)} \tan ^{-1}\left(\sqrt[4]{\frac{2}{3+\sqrt{5}}} x\right)}{2 \sqrt{5}}+\frac{\sqrt[4]{\frac{1}{2} \left(3-\sqrt{5}\right)} \tan ^{-1}\left(\sqrt[4]{\frac{1}{2} \left(3+\sqrt{5}\right)} x\right)}{2 \sqrt{5}}-\frac{\sqrt[4]{\frac{1}{2} \left(3+\sqrt{5}\right)} \tanh ^{-1}\left(\sqrt[4]{\frac{2}{3+\sqrt{5}}} x\right)}{2 \sqrt{5}}+\frac{\sqrt[4]{\frac{1}{2} \left(3-\sqrt{5}\right)} \tanh ^{-1}\left(\sqrt[4]{\frac{1}{2} \left(3+\sqrt{5}\right)} x\right)}{2 \sqrt{5}}",1,"(Sqrt[-1 + Sqrt[5]]*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x] - Sqrt[1 + Sqrt[5]]*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x] + Sqrt[-1 + Sqrt[5]]*ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x] - Sqrt[1 + Sqrt[5]]*ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x])/(2*Sqrt[10])","A",1
399,1,131,145,0.0443874,"\int \frac{x^2}{1-3 x^4+x^8} \, dx","Integrate[x^2/(1 - 3*x^4 + x^8),x]","-\frac{\tan ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{\sqrt{10 \left(\sqrt{5}-1\right)}}+\frac{\tan ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{\sqrt{10 \left(1+\sqrt{5}\right)}}+\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{\sqrt{10 \left(\sqrt{5}-1\right)}}-\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{\sqrt{10 \left(1+\sqrt{5}\right)}}","\frac{1}{20} \sqrt{10 \sqrt{5}-10} \tan ^{-1}\left(\frac{1}{2} \sqrt{2 \sqrt{5}-2} x\right)-\frac{1}{20} \sqrt{10+10 \sqrt{5}} \tan ^{-1}\left(\frac{1}{2} \sqrt{2+2 \sqrt{5}} x\right)-\frac{1}{20} \sqrt{10 \sqrt{5}-10} \tanh ^{-1}\left(\frac{1}{2} \sqrt{2 \sqrt{5}-2} x\right)+\frac{1}{20} \sqrt{10+10 \sqrt{5}} \tanh ^{-1}\left(\frac{1}{2} \sqrt{2+2 \sqrt{5}} x\right)",1,"-(ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x]/Sqrt[10*(-1 + Sqrt[5])]) + ArcTan[Sqrt[2/(1 + Sqrt[5])]*x]/Sqrt[10*(1 + Sqrt[5])] + ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x]/Sqrt[10*(-1 + Sqrt[5])] - ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x]/Sqrt[10*(1 + Sqrt[5])]","A",1
400,1,160,169,0.1564424,"\int \frac{1}{1-3 x^4+x^8} \, dx","Integrate[(1 - 3*x^4 + x^8)^(-1),x]","\frac{\frac{\left(1+\sqrt{5}\right) \tan ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{\sqrt{\sqrt{5}-1}}-\frac{\left(\sqrt{5}-1\right) \tan ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{\sqrt{1+\sqrt{5}}}+\frac{\left(1+\sqrt{5}\right) \tanh ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{\sqrt{\sqrt{5}-1}}-\frac{\left(\sqrt{5}-1\right) \tanh ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{\sqrt{1+\sqrt{5}}}}{2 \sqrt{10}}","-\frac{\tan ^{-1}\left(\sqrt[4]{\frac{2}{3+\sqrt{5}}} x\right)}{\sqrt[4]{2} \sqrt{5} \left(3+\sqrt{5}\right)^{3/4}}+\frac{\left(3+\sqrt{5}\right)^{3/4} \tan ^{-1}\left(\sqrt[4]{\frac{1}{2} \left(3+\sqrt{5}\right)} x\right)}{2\ 2^{3/4} \sqrt{5}}-\frac{\tanh ^{-1}\left(\sqrt[4]{\frac{2}{3+\sqrt{5}}} x\right)}{\sqrt[4]{2} \sqrt{5} \left(3+\sqrt{5}\right)^{3/4}}+\frac{\left(3+\sqrt{5}\right)^{3/4} \tanh ^{-1}\left(\sqrt[4]{\frac{1}{2} \left(3+\sqrt{5}\right)} x\right)}{2\ 2^{3/4} \sqrt{5}}",1,"(((1 + Sqrt[5])*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x])/Sqrt[-1 + Sqrt[5]] - ((-1 + Sqrt[5])*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x])/Sqrt[1 + Sqrt[5]] + ((1 + Sqrt[5])*ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x])/Sqrt[-1 + Sqrt[5]] - ((-1 + Sqrt[5])*ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x])/Sqrt[1 + Sqrt[5]])/(2*Sqrt[10])","A",1
401,1,174,172,0.2671148,"\int \frac{1}{x^2 \left(1-3 x^4+x^8\right)} \, dx","Integrate[1/(x^2*(1 - 3*x^4 + x^8)),x]","-\frac{1}{x}-\frac{\left(3+\sqrt{5}\right) \tan ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{2 \sqrt{10 \left(\sqrt{5}-1\right)}}-\frac{\left(\sqrt{5}-3\right) \tan ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{2 \sqrt{10 \left(1+\sqrt{5}\right)}}+\frac{\left(3+\sqrt{5}\right) \tanh ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{2 \sqrt{10 \left(\sqrt{5}-1\right)}}+\frac{\left(\sqrt{5}-3\right) \tanh ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{2 \sqrt{10 \left(1+\sqrt{5}\right)}}","-\frac{1}{x}+\frac{\sqrt[4]{984-440 \sqrt{5}} \tan ^{-1}\left(\sqrt[4]{\frac{2}{3+\sqrt{5}}} x\right)}{4 \sqrt{5}}-\frac{\left(3+\sqrt{5}\right)^{5/4} \tan ^{-1}\left(\sqrt[4]{\frac{1}{2} \left(3+\sqrt{5}\right)} x\right)}{4 \sqrt[4]{2} \sqrt{5}}-\frac{\sqrt[4]{984-440 \sqrt{5}} \tanh ^{-1}\left(\sqrt[4]{\frac{2}{3+\sqrt{5}}} x\right)}{4 \sqrt{5}}+\frac{\left(3+\sqrt{5}\right)^{5/4} \tanh ^{-1}\left(\sqrt[4]{\frac{1}{2} \left(3+\sqrt{5}\right)} x\right)}{4 \sqrt[4]{2} \sqrt{5}}",1,"-x^(-1) - ((3 + Sqrt[5])*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x])/(2*Sqrt[10*(-1 + Sqrt[5])]) - ((-3 + Sqrt[5])*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x])/(2*Sqrt[10*(1 + Sqrt[5])]) + ((3 + Sqrt[5])*ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x])/(2*Sqrt[10*(-1 + Sqrt[5])]) + ((-3 + Sqrt[5])*ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x])/(2*Sqrt[10*(1 + Sqrt[5])])","A",1
402,1,166,182,0.2617563,"\int \frac{1}{x^4 \left(1-3 x^4+x^8\right)} \, dx","Integrate[1/(x^4*(1 - 3*x^4 + x^8)),x]","-\frac{1}{3 x^3}+\frac{\left(2+\sqrt{5}\right) \tan ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{\sqrt{10 \left(\sqrt{5}-1\right)}}-\frac{\left(\sqrt{5}-2\right) \tan ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{\sqrt{10 \left(1+\sqrt{5}\right)}}+\frac{\left(2+\sqrt{5}\right) \tanh ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{\sqrt{10 \left(\sqrt{5}-1\right)}}-\frac{\left(\sqrt{5}-2\right) \tanh ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{\sqrt{10 \left(1+\sqrt{5}\right)}}","-\frac{1}{3 x^3}-\frac{\sqrt[4]{\frac{1}{2} \left(843-377 \sqrt{5}\right)} \tan ^{-1}\left(\sqrt[4]{\frac{2}{3+\sqrt{5}}} x\right)}{2 \sqrt{5}}+\frac{\left(3+\sqrt{5}\right)^{7/4} \tan ^{-1}\left(\sqrt[4]{\frac{1}{2} \left(3+\sqrt{5}\right)} x\right)}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt[4]{\frac{1}{2} \left(843-377 \sqrt{5}\right)} \tanh ^{-1}\left(\sqrt[4]{\frac{2}{3+\sqrt{5}}} x\right)}{2 \sqrt{5}}+\frac{\left(3+\sqrt{5}\right)^{7/4} \tanh ^{-1}\left(\sqrt[4]{\frac{1}{2} \left(3+\sqrt{5}\right)} x\right)}{4\ 2^{3/4} \sqrt{5}}",1,"-1/3*1/x^3 + ((2 + Sqrt[5])*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x])/Sqrt[10*(-1 + Sqrt[5])] - ((-2 + Sqrt[5])*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x])/Sqrt[10*(1 + Sqrt[5])] + ((2 + Sqrt[5])*ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x])/Sqrt[10*(-1 + Sqrt[5])] - ((-2 + Sqrt[5])*ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x])/Sqrt[10*(1 + Sqrt[5])]","A",1
403,1,189,173,0.2741941,"\int \frac{1}{x^6 \left(1-3 x^4+x^8\right)} \, dx","Integrate[1/(x^6*(1 - 3*x^4 + x^8)),x]","-\frac{1}{5 x^5}-\frac{3}{x}+\frac{\left(-7-3 \sqrt{5}\right) \tan ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{2 \sqrt{10 \left(\sqrt{5}-1\right)}}+\frac{\left(7-3 \sqrt{5}\right) \tan ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{2 \sqrt{10 \left(1+\sqrt{5}\right)}}-\frac{\left(-7-3 \sqrt{5}\right) \tanh ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{2 \sqrt{10 \left(\sqrt{5}-1\right)}}-\frac{\left(7-3 \sqrt{5}\right) \tanh ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{2 \sqrt{10 \left(1+\sqrt{5}\right)}}","-\frac{1}{5 x^5}-\frac{3}{x}+\frac{\sqrt[4]{2889-1292 \sqrt{5}} \tan ^{-1}\left(\sqrt[4]{\frac{2}{3+\sqrt{5}}} x\right)}{2 \sqrt{5}}-\frac{\sqrt[4]{2889+1292 \sqrt{5}} \tan ^{-1}\left(\sqrt[4]{\frac{1}{2} \left(3+\sqrt{5}\right)} x\right)}{2 \sqrt{5}}-\frac{\sqrt[4]{2889-1292 \sqrt{5}} \tanh ^{-1}\left(\sqrt[4]{\frac{2}{3+\sqrt{5}}} x\right)}{2 \sqrt{5}}+\frac{\sqrt[4]{2889+1292 \sqrt{5}} \tanh ^{-1}\left(\sqrt[4]{\frac{1}{2} \left(3+\sqrt{5}\right)} x\right)}{2 \sqrt{5}}",1,"-1/5*1/x^5 - 3/x + ((-7 - 3*Sqrt[5])*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x])/(2*Sqrt[10*(-1 + Sqrt[5])]) + ((7 - 3*Sqrt[5])*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x])/(2*Sqrt[10*(1 + Sqrt[5])]) - ((-7 - 3*Sqrt[5])*ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x])/(2*Sqrt[10*(-1 + Sqrt[5])]) - ((7 - 3*Sqrt[5])*ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x])/(2*Sqrt[10*(1 + Sqrt[5])])","A",1
404,1,189,189,0.2683333,"\int \frac{1}{x^8 \left(1-3 x^4+x^8\right)} \, dx","Integrate[1/(x^8*(1 - 3*x^4 + x^8)),x]","-\frac{1}{7 x^7}-\frac{1}{x^3}+\frac{\left(11+5 \sqrt{5}\right) \tan ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{2 \sqrt{10 \left(\sqrt{5}-1\right)}}+\frac{\left(11-5 \sqrt{5}\right) \tan ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{2 \sqrt{10 \left(1+\sqrt{5}\right)}}-\frac{\left(-11-5 \sqrt{5}\right) \tanh ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{2 \sqrt{10 \left(\sqrt{5}-1\right)}}-\frac{\left(5 \sqrt{5}-11\right) \tanh ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{2 \sqrt{10 \left(1+\sqrt{5}\right)}}","-\frac{1}{7 x^7}-\frac{1}{x^3}-\frac{\sqrt[4]{\frac{1}{2} \left(39603-17711 \sqrt{5}\right)} \tan ^{-1}\left(\sqrt[4]{\frac{2}{3+\sqrt{5}}} x\right)}{2 \sqrt{5}}+\frac{\sqrt[4]{\frac{1}{2} \left(39603+17711 \sqrt{5}\right)} \tan ^{-1}\left(\sqrt[4]{\frac{1}{2} \left(3+\sqrt{5}\right)} x\right)}{2 \sqrt{5}}-\frac{\sqrt[4]{\frac{1}{2} \left(39603-17711 \sqrt{5}\right)} \tanh ^{-1}\left(\sqrt[4]{\frac{2}{3+\sqrt{5}}} x\right)}{2 \sqrt{5}}+\frac{\sqrt[4]{\frac{1}{2} \left(39603+17711 \sqrt{5}\right)} \tanh ^{-1}\left(\sqrt[4]{\frac{1}{2} \left(3+\sqrt{5}\right)} x\right)}{2 \sqrt{5}}",1,"-1/7*1/x^7 - x^(-3) + ((11 + 5*Sqrt[5])*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x])/(2*Sqrt[10*(-1 + Sqrt[5])]) + ((11 - 5*Sqrt[5])*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x])/(2*Sqrt[10*(1 + Sqrt[5])]) - ((-11 - 5*Sqrt[5])*ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x])/(2*Sqrt[10*(-1 + Sqrt[5])]) - ((-11 + 5*Sqrt[5])*ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x])/(2*Sqrt[10*(1 + Sqrt[5])])","A",1
405,1,21,21,0.0042888,"\int \frac{x^3}{2+3 x^4+x^8} \, dx","Integrate[x^3/(2 + 3*x^4 + x^8),x]","\frac{1}{4} \log \left(x^4+1\right)-\frac{1}{4} \log \left(x^4+2\right)","\frac{1}{4} \log \left(x^4+1\right)-\frac{1}{4} \log \left(x^4+2\right)",1,"Log[1 + x^4]/4 - Log[2 + x^4]/4","A",1
406,1,26,26,0.004719,"\int \frac{x^{11}}{2+3 x^4+x^8} \, dx","Integrate[x^11/(2 + 3*x^4 + x^8),x]","\frac{x^4}{4}+\frac{1}{4} \log \left(x^4+1\right)-\log \left(x^4+2\right)","\frac{x^4}{4}+\frac{1}{4} \log \left(x^4+1\right)-\log \left(x^4+2\right)",1,"x^4/4 + Log[1 + x^4]/4 - Log[2 + x^4]","A",1
407,1,37,37,0.0125863,"\int \frac{x^9}{2+x^5+x^{10}} \, dx","Integrate[x^9/(2 + x^5 + x^10),x]","\frac{1}{10} \log \left(x^{10}+x^5+2\right)-\frac{\tan ^{-1}\left(\frac{2 x^5+1}{\sqrt{7}}\right)}{5 \sqrt{7}}","\frac{1}{10} \log \left(x^{10}+x^5+2\right)-\frac{\tan ^{-1}\left(\frac{2 x^5+1}{\sqrt{7}}\right)}{5 \sqrt{7}}",1,"-1/5*ArcTan[(1 + 2*x^5)/Sqrt[7]]/Sqrt[7] + Log[2 + x^5 + x^10]/10","A",1
408,1,23,23,0.0062003,"\int \frac{x^4}{2+x^5+x^{10}} \, dx","Integrate[x^4/(2 + x^5 + x^10),x]","\frac{2 \tan ^{-1}\left(\frac{2 x^5+1}{\sqrt{7}}\right)}{5 \sqrt{7}}","\frac{2 \tan ^{-1}\left(\frac{2 x^5+1}{\sqrt{7}}\right)}{5 \sqrt{7}}",1,"(2*ArcTan[(1 + 2*x^5)/Sqrt[7]])/(5*Sqrt[7])","A",1
409,1,197,39,0.0367093,"\int \frac{1}{x \left(1+x^5+x^{10}\right)} \, dx","Integrate[1/(x*(1 + x^5 + x^10)),x]","-\frac{1}{5} \text{RootSum}\left[\text{$\#$1}^8-\text{$\#$1}^7+\text{$\#$1}^5-\text{$\#$1}^4+\text{$\#$1}^3-\text{$\#$1}+1\&,\frac{4 \text{$\#$1}^7 \log (x-\text{$\#$1})-3 \text{$\#$1}^6 \log (x-\text{$\#$1})-\text{$\#$1}^5 \log (x-\text{$\#$1})+3 \text{$\#$1}^4 \log (x-\text{$\#$1})-\text{$\#$1}^3 \log (x-\text{$\#$1})+2 \text{$\#$1}^2 \log (x-\text{$\#$1})-\text{$\#$1} \log (x-\text{$\#$1})}{8 \text{$\#$1}^7-7 \text{$\#$1}^6+5 \text{$\#$1}^4-4 \text{$\#$1}^3+3 \text{$\#$1}^2-1}\&\right]-\frac{1}{10} \log \left(x^2+x+1\right)+\log (x)+\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{5 \sqrt{3}}","-\frac{\tan ^{-1}\left(\frac{2 x^5+1}{\sqrt{3}}\right)}{5 \sqrt{3}}-\frac{1}{10} \log \left(x^{10}+x^5+1\right)+\log (x)",1,"ArcTan[(1 + 2*x)/Sqrt[3]]/(5*Sqrt[3]) + Log[x] - Log[1 + x + x^2]/10 - RootSum[1 - #1 + #1^3 - #1^4 + #1^5 - #1^7 + #1^8 & , (-(Log[x - #1]*#1) + 2*Log[x - #1]*#1^2 - Log[x - #1]*#1^3 + 3*Log[x - #1]*#1^4 - Log[x - #1]*#1^5 - 3*Log[x - #1]*#1^6 + 4*Log[x - #1]*#1^7)/(-1 + 3*#1^2 - 4*#1^3 + 5*#1^4 - 7*#1^6 + 8*#1^7) & ]/5","C",1
410,1,208,48,0.0416534,"\int \frac{1}{x^6 \left(1+x^5+x^{10}\right)} \, dx","Integrate[1/(x^6*(1 + x^5 + x^10)),x]","\frac{1}{30} \left(6 \text{RootSum}\left[\text{$\#$1}^8-\text{$\#$1}^7+\text{$\#$1}^5-\text{$\#$1}^4+\text{$\#$1}^3-\text{$\#$1}+1\&,\frac{4 \text{$\#$1}^7 \log (x-\text{$\#$1})-4 \text{$\#$1}^6 \log (x-\text{$\#$1})+\text{$\#$1}^5 \log (x-\text{$\#$1})+2 \text{$\#$1}^4 \log (x-\text{$\#$1})-3 \text{$\#$1}^3 \log (x-\text{$\#$1})+\text{$\#$1}^2 \log (x-\text{$\#$1})+\text{$\#$1} \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{8 \text{$\#$1}^7-7 \text{$\#$1}^6+5 \text{$\#$1}^4-4 \text{$\#$1}^3+3 \text{$\#$1}^2-1}\&\right]-\frac{6}{x^5}+3 \log \left(x^2+x+1\right)-30 \log (x)+2 \sqrt{3} \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)\right)","-\frac{1}{5 x^5}-\frac{\tan ^{-1}\left(\frac{2 x^5+1}{\sqrt{3}}\right)}{5 \sqrt{3}}+\frac{1}{10} \log \left(x^{10}+x^5+1\right)-\log (x)",1,"(-6/x^5 + 2*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] - 30*Log[x] + 3*Log[1 + x + x^2] + 6*RootSum[1 - #1 + #1^3 - #1^4 + #1^5 - #1^7 + #1^8 & , (-Log[x - #1] + Log[x - #1]*#1 + Log[x - #1]*#1^2 - 3*Log[x - #1]*#1^3 + 2*Log[x - #1]*#1^4 + Log[x - #1]*#1^5 - 4*Log[x - #1]*#1^6 + 4*Log[x - #1]*#1^7)/(-1 + 3*#1^2 - 4*#1^3 + 5*#1^4 - 7*#1^6 + 8*#1^7) & ])/30","C",1
411,1,197,39,0.0195542,"\int \frac{1}{x+x^6+x^{11}} \, dx","Integrate[(x + x^6 + x^11)^(-1),x]","-\frac{1}{5} \text{RootSum}\left[\text{$\#$1}^8-\text{$\#$1}^7+\text{$\#$1}^5-\text{$\#$1}^4+\text{$\#$1}^3-\text{$\#$1}+1\&,\frac{4 \text{$\#$1}^7 \log (x-\text{$\#$1})-3 \text{$\#$1}^6 \log (x-\text{$\#$1})-\text{$\#$1}^5 \log (x-\text{$\#$1})+3 \text{$\#$1}^4 \log (x-\text{$\#$1})-\text{$\#$1}^3 \log (x-\text{$\#$1})+2 \text{$\#$1}^2 \log (x-\text{$\#$1})-\text{$\#$1} \log (x-\text{$\#$1})}{8 \text{$\#$1}^7-7 \text{$\#$1}^6+5 \text{$\#$1}^4-4 \text{$\#$1}^3+3 \text{$\#$1}^2-1}\&\right]-\frac{1}{10} \log \left(x^2+x+1\right)+\log (x)+\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{5 \sqrt{3}}","-\frac{\tan ^{-1}\left(\frac{2 x^5+1}{\sqrt{3}}\right)}{5 \sqrt{3}}-\frac{1}{10} \log \left(x^{10}+x^5+1\right)+\log (x)",1,"ArcTan[(1 + 2*x)/Sqrt[3]]/(5*Sqrt[3]) + Log[x] - Log[1 + x + x^2]/10 - RootSum[1 - #1 + #1^3 - #1^4 + #1^5 - #1^7 + #1^8 & , (-(Log[x - #1]*#1) + 2*Log[x - #1]*#1^2 - Log[x - #1]*#1^3 + 3*Log[x - #1]*#1^4 - Log[x - #1]*#1^5 - 3*Log[x - #1]*#1^6 + 4*Log[x - #1]*#1^7)/(-1 + 3*#1^2 - 4*#1^3 + 5*#1^4 - 7*#1^6 + 8*#1^7) & ]/5","C",1
412,1,140,147,0.1220569,"\int \frac{x^3}{c+\frac{a}{x^2}+\frac{b}{x}} \, dx","Integrate[x^3/(c + a/x^2 + b/x),x]","\frac{6 \left(a^2 c^2-3 a b^2 c+b^4\right) \log (a+x (b+c x))-\frac{12 b \left(5 a^2 c^2-5 a b^2 c+b^4\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+c x \left(-4 b c \left(c x^2-6 a\right)+3 c^2 x \left(c x^2-2 a\right)-12 b^3+6 b^2 c x\right)}{12 c^5}","\frac{\left(a^2 c^2-3 a b^2 c+b^4\right) \log \left(a+b x+c x^2\right)}{2 c^5}+\frac{b \left(5 a^2 c^2-5 a b^2 c+b^4\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c^5 \sqrt{b^2-4 a c}}-\frac{b x \left(b^2-2 a c\right)}{c^4}+\frac{x^2 \left(b^2-a c\right)}{2 c^3}-\frac{b x^3}{3 c^2}+\frac{x^4}{4 c}",1,"(c*x*(-12*b^3 + 6*b^2*c*x - 4*b*c*(-6*a + c*x^2) + 3*c^2*x*(-2*a + c*x^2)) - (12*b*(b^4 - 5*a*b^2*c + 5*a^2*c^2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + 6*(b^4 - 3*a*b^2*c + a^2*c^2)*Log[a + x*(b + c*x)])/(12*c^5)","A",1
413,1,112,118,0.0845926,"\int \frac{x^2}{c+\frac{a}{x^2}+\frac{b}{x}} \, dx","Integrate[x^2/(c + a/x^2 + b/x),x]","\frac{\frac{6 \left(2 a^2 c^2-4 a b^2 c+b^4\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}-3 \left(b^3-2 a b c\right) \log (a+x (b+c x))+c x \left(-6 a c+6 b^2-3 b c x+2 c^2 x^2\right)}{6 c^4}","-\frac{\left(2 a^2 c^2-4 a b^2 c+b^4\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c^4 \sqrt{b^2-4 a c}}-\frac{b \left(b^2-2 a c\right) \log \left(a+b x+c x^2\right)}{2 c^4}+\frac{x \left(b^2-a c\right)}{c^3}-\frac{b x^2}{2 c^2}+\frac{x^3}{3 c}",1,"(c*x*(6*b^2 - 6*a*c - 3*b*c*x + 2*c^2*x^2) + (6*(b^4 - 4*a*b^2*c + 2*a^2*c^2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] - 3*(b^3 - 2*a*b*c)*Log[a + x*(b + c*x)])/(6*c^4)","A",1
414,1,84,89,0.1038545,"\int \frac{x}{c+\frac{a}{x^2}+\frac{b}{x}} \, dx","Integrate[x/(c + a/x^2 + b/x),x]","\frac{\left(b^2-a c\right) \log (a+x (b+c x))-\frac{2 b \left(b^2-3 a c\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+c x (c x-2 b)}{2 c^3}","\frac{\left(b^2-a c\right) \log \left(a+b x+c x^2\right)}{2 c^3}+\frac{b \left(b^2-3 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c^3 \sqrt{b^2-4 a c}}-\frac{b x}{c^2}+\frac{x^2}{2 c}",1,"(c*x*(-2*b + c*x) - (2*b*(b^2 - 3*a*c)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + (b^2 - a*c)*Log[a + x*(b + c*x)])/(2*c^3)","A",1
415,1,73,70,0.0644238,"\int \frac{1}{c+\frac{a}{x^2}+\frac{b}{x}} \, dx","Integrate[(c + a/x^2 + b/x)^(-1),x]","\frac{\left(b^2-2 a c\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{c^2 \sqrt{4 a c-b^2}}-\frac{b \log \left(a+b x+c x^2\right)}{2 c^2}+\frac{x}{c}","-\frac{\left(b^2-2 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c^2 \sqrt{b^2-4 a c}}-\frac{b \log \left(a+b x+c x^2\right)}{2 c^2}+\frac{x}{c}",1,"x/c + ((b^2 - 2*a*c)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(c^2*Sqrt[-b^2 + 4*a*c]) - (b*Log[a + b*x + c*x^2])/(2*c^2)","A",1
416,1,57,56,0.0310927,"\int \frac{1}{\left(c+\frac{a}{x^2}+\frac{b}{x}\right) x} \, dx","Integrate[1/((c + a/x^2 + b/x)*x),x]","\frac{\log (a+x (b+c x))-\frac{2 b \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}}{2 c}","\frac{b \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c \sqrt{b^2-4 a c}}+\frac{\log \left(a+b x+c x^2\right)}{2 c}",1,"((-2*b*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + Log[a + x*(b + c*x)])/(2*c)","A",1
417,1,38,36,0.0069335,"\int \frac{1}{\left(c+\frac{a}{x^2}+\frac{b}{x}\right) x^2} \, dx","Integrate[1/((c + a/x^2 + b/x)*x^2),x]","\frac{2 \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}","\frac{2 \tanh ^{-1}\left(\frac{\frac{2 a}{x}+b}{\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c}}",1,"(2*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c]","A",1
418,1,61,62,0.06744,"\int \frac{1}{\left(c+\frac{a}{x^2}+\frac{b}{x}\right) x^3} \, dx","Integrate[1/((c + a/x^2 + b/x)*x^3),x]","-\frac{\frac{2 b \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+\log (a+x (b+c x))-2 \log (x)}{2 a}","\frac{b \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{a \sqrt{b^2-4 a c}}-\frac{\log \left(a+b x+c x^2\right)}{2 a}+\frac{\log (x)}{a}",1,"-1/2*((2*b*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] - 2*Log[x] + Log[a + x*(b + c*x)])/a","A",1
419,1,77,81,0.0793403,"\int \frac{1}{\left(c+\frac{a}{x^2}+\frac{b}{x}\right) x^4} \, dx","Integrate[1/((c + a/x^2 + b/x)*x^4),x]","\frac{\frac{2 \left(b^2-2 a c\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+b \log (a+x (b+c x))-\frac{2 a}{x}-2 b \log (x)}{2 a^2}","-\frac{\left(b^2-2 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{a^2 \sqrt{b^2-4 a c}}+\frac{b \log \left(a+b x+c x^2\right)}{2 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{a x}",1,"((-2*a)/x + (2*(b^2 - 2*a*c)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] - 2*b*Log[x] + b*Log[a + x*(b + c*x)])/(2*a^2)","A",1
420,1,102,104,0.1355705,"\int \frac{1}{\left(c+\frac{a}{x^2}+\frac{b}{x}\right) x^5} \, dx","Integrate[1/((c + a/x^2 + b/x)*x^5),x]","\frac{-\frac{a^2}{x^2}+2 \log (x) \left(b^2-a c\right)+\left(a c-b^2\right) \log (a+x (b+c x))-\frac{2 b \left(b^2-3 a c\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+\frac{2 a b}{x}}{2 a^3}","-\frac{\left(b^2-a c\right) \log \left(a+b x+c x^2\right)}{2 a^3}+\frac{\log (x) \left(b^2-a c\right)}{a^3}+\frac{b \left(b^2-3 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{a^3 \sqrt{b^2-4 a c}}+\frac{b}{a^2 x}-\frac{1}{2 a x^2}",1,"(-(a^2/x^2) + (2*a*b)/x - (2*b*(b^2 - 3*a*c)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + 2*(b^2 - a*c)*Log[x] + (-b^2 + a*c)*Log[a + x*(b + c*x)])/(2*a^3)","A",1
421,1,131,137,0.0987714,"\int \frac{1}{\left(c+\frac{a}{x^2}+\frac{b}{x}\right) x^6} \, dx","Integrate[1/((c + a/x^2 + b/x)*x^6),x]","\frac{-\frac{2 a^3}{x^3}+\frac{6 \left(2 a^2 c^2-4 a b^2 c+b^4\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+\frac{3 a^2 b}{x^2}-6 \log (x) \left(b^3-2 a b c\right)+3 \left(b^3-2 a b c\right) \log (a+x (b+c x))+\frac{6 a \left(a c-b^2\right)}{x}}{6 a^4}","\frac{b \left(b^2-2 a c\right) \log \left(a+b x+c x^2\right)}{2 a^4}-\frac{b \log (x) \left(b^2-2 a c\right)}{a^4}-\frac{b^2-a c}{a^3 x}+\frac{b}{2 a^2 x^2}-\frac{\left(2 a^2 c^2-4 a b^2 c+b^4\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{a^4 \sqrt{b^2-4 a c}}-\frac{1}{3 a x^3}",1,"((-2*a^3)/x^3 + (3*a^2*b)/x^2 + (6*a*(-b^2 + a*c))/x + (6*(b^4 - 4*a*b^2*c + 2*a^2*c^2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] - 6*(b^3 - 2*a*b*c)*Log[x] + 3*(b^3 - 2*a*b*c)*Log[a + x*(b + c*x)])/(6*a^4)","A",1
422,1,163,196,0.2223921,"\int \frac{x}{\left(c+\frac{a}{x^2}+\frac{b}{x}\right)^2} \, dx","Integrate[x/(c + a/x^2 + b/x)^2,x]","\frac{\frac{2 b \left(30 a^2 c^2-20 a b^2 c+3 b^4\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{3/2}}+\frac{2 \left(2 a^3 c^2+a^2 b c (5 c x-4 b)+a b^3 (b-5 c x)+b^5 x\right)}{\left(b^2-4 a c\right) (a+x (b+c x))}+\left(3 b^2-2 a c\right) \log (a+x (b+c x))-4 b c x+c^2 x^2}{2 c^4}","\frac{b \left(30 a^2 c^2-20 a b^2 c+3 b^4\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c^4 \left(b^2-4 a c\right)^{3/2}}+\frac{\left(3 b^2-2 a c\right) \log \left(a+b x+c x^2\right)}{2 c^4}-\frac{b x \left(3 b^2-11 a c\right)}{c^3 \left(b^2-4 a c\right)}+\frac{x^2 \left(3 b^2-8 a c\right)}{2 c^2 \left(b^2-4 a c\right)}-\frac{b x^3}{c \left(b^2-4 a c\right)}+\frac{x^4 (2 a+b x)}{\left(b^2-4 a c\right) \left(a+b x+c x^2\right)}",1,"(-4*b*c*x + c^2*x^2 + (2*(2*a^3*c^2 + b^5*x + a*b^3*(b - 5*c*x) + a^2*b*c*(-4*b + 5*c*x)))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (2*b*(3*b^4 - 20*a*b^2*c + 30*a^2*c^2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + (3*b^2 - 2*a*c)*Log[a + x*(b + c*x)])/(2*c^4)","A",1
423,1,132,150,0.1768823,"\int \frac{1}{\left(c+\frac{a}{x^2}+\frac{b}{x}\right)^2} \, dx","Integrate[(c + a/x^2 + b/x)^(-2),x]","\frac{-\frac{2 \left(6 a^2 c^2-6 a b^2 c+b^4\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{3/2}}+\frac{a^2 c (3 b-2 c x)-a b^2 (b-4 c x)+b^4 (-x)}{\left(b^2-4 a c\right) (a+x (b+c x))}-b \log (a+x (b+c x))+c x}{c^3}","-\frac{2 \left(6 a^2 c^2-6 a b^2 c+b^4\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c^3 \left(b^2-4 a c\right)^{3/2}}+\frac{2 x \left(b^2-3 a c\right)}{c^2 \left(b^2-4 a c\right)}-\frac{b x^2}{c \left(b^2-4 a c\right)}+\frac{x^3 (2 a+b x)}{\left(b^2-4 a c\right) \left(a+b x+c x^2\right)}-\frac{b \log \left(a+b x+c x^2\right)}{c^3}",1,"(c*x + (-(b^4*x) - a*b^2*(b - 4*c*x) + a^2*c*(3*b - 2*c*x))/((b^2 - 4*a*c)*(a + x*(b + c*x))) - (2*(b^4 - 6*a*b^2*c + 6*a^2*c^2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) - b*Log[a + x*(b + c*x)])/c^3","A",1
424,1,109,114,0.1396701,"\int \frac{1}{\left(c+\frac{a}{x^2}+\frac{b}{x}\right)^2 x} \, dx","Integrate[1/((c + a/x^2 + b/x)^2*x),x]","\frac{\frac{2 \left(-2 a^2 c+a b (b-3 c x)+b^3 x\right)}{\left(b^2-4 a c\right) (a+x (b+c x))}+\frac{2 b \left(b^2-6 a c\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{3/2}}+\log (a+x (b+c x))}{2 c^2}","\frac{b \left(b^2-6 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c^2 \left(b^2-4 a c\right)^{3/2}}+\frac{x^2 (2 a+b x)}{\left(b^2-4 a c\right) \left(a+b x+c x^2\right)}-\frac{b x}{c \left(b^2-4 a c\right)}+\frac{\log \left(a+b x+c x^2\right)}{2 c^2}",1,"((2*(-2*a^2*c + b^3*x + a*b*(b - 3*c*x)))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (2*b*(b^2 - 6*a*c)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + Log[a + x*(b + c*x)])/(2*c^2)","A",1
425,1,81,71,0.091807,"\int \frac{1}{\left(c+\frac{a}{x^2}+\frac{b}{x}\right)^2 x^2} \, dx","Integrate[1/((c + a/x^2 + b/x)^2*x^2),x]","\frac{a (b-2 c x)+b^2 x}{c \left(4 a c-b^2\right) (a+x (b+c x))}+\frac{4 a \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{3/2}}","\frac{\frac{2 a}{x}+b}{\left(b^2-4 a c\right) \left(\frac{a}{x^2}+\frac{b}{x}+c\right)}-\frac{4 a \tanh ^{-1}\left(\frac{\frac{2 a}{x}+b}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/2}}",1,"(b^2*x + a*(b - 2*c*x))/(c*(-b^2 + 4*a*c)*(a + x*(b + c*x))) + (4*a*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2)","A",1
426,1,69,66,0.0644179,"\int \frac{1}{\left(c+\frac{a}{x^2}+\frac{b}{x}\right)^2 x^3} \, dx","Integrate[1/((c + a/x^2 + b/x)^2*x^3),x]","\frac{2 a+b x}{\left(b^2-4 a c\right) (a+x (b+c x))}-\frac{2 b \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{3/2}}","\frac{2 a+b x}{\left(b^2-4 a c\right) \left(a+b x+c x^2\right)}-\frac{2 b \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/2}}",1,"(2*a + b*x)/((b^2 - 4*a*c)*(a + x*(b + c*x))) - (2*b*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2)","A",1
427,1,70,66,0.0773737,"\int \frac{1}{\left(c+\frac{a}{x^2}+\frac{b}{x}\right)^2 x^4} \, dx","Integrate[1/((c + a/x^2 + b/x)^2*x^4),x]","-\frac{\frac{4 c \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+\frac{b+2 c x}{a+x (b+c x)}}{b^2-4 a c}","\frac{4 c \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{b+2 c x}{\left(b^2-4 a c\right) \left(a+b x+c x^2\right)}",1,"-(((b + 2*c*x)/(a + x*(b + c*x)) + (4*c*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(b^2 - 4*a*c))","A",1
428,1,107,108,0.1851068,"\int \frac{1}{\left(c+\frac{a}{x^2}+\frac{b}{x}\right)^2 x^5} \, dx","Integrate[1/((c + a/x^2 + b/x)^2*x^5),x]","\frac{\frac{2 a \left(-2 a c+b^2+b c x\right)}{\left(b^2-4 a c\right) (a+x (b+c x))}+\frac{2 b \left(b^2-6 a c\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{3/2}}-\log (a+x (b+c x))+2 \log (x)}{2 a^2}","\frac{b \left(b^2-6 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{a^2 \left(b^2-4 a c\right)^{3/2}}-\frac{\log \left(a+b x+c x^2\right)}{2 a^2}+\frac{\log (x)}{a^2}+\frac{-2 a c+b^2+b c x}{a \left(b^2-4 a c\right) \left(a+b x+c x^2\right)}",1,"((2*a*(b^2 - 2*a*c + b*c*x))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (2*b*(b^2 - 6*a*c)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + 2*Log[x] - Log[a + x*(b + c*x)])/(2*a^2)","A",1
429,1,131,148,0.2662961,"\int \frac{1}{\left(c+\frac{a}{x^2}+\frac{b}{x}\right)^2 x^6} \, dx","Integrate[1/((c + a/x^2 + b/x)^2*x^6),x]","-\frac{\frac{2 \left(6 a^2 c^2-6 a b^2 c+b^4\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{3/2}}+\frac{a \left(-3 a b c-2 a c^2 x+b^3+b^2 c x\right)}{\left(b^2-4 a c\right) (a+x (b+c x))}-b \log (a+x (b+c x))+\frac{a}{x}+2 b \log (x)}{a^3}","\frac{b \log \left(a+b x+c x^2\right)}{a^3}-\frac{2 b \log (x)}{a^3}-\frac{2 \left(b^2-3 a c\right)}{a^2 x \left(b^2-4 a c\right)}-\frac{2 \left(6 a^2 c^2-6 a b^2 c+b^4\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{a^3 \left(b^2-4 a c\right)^{3/2}}+\frac{-2 a c+b^2+b c x}{a x \left(b^2-4 a c\right) \left(a+b x+c x^2\right)}",1,"-((a/x + (a*(b^3 - 3*a*b*c + b^2*c*x - 2*a*c^2*x))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (2*(b^4 - 6*a*b^2*c + 6*a^2*c^2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + 2*b*Log[x] - b*Log[a + x*(b + c*x)])/a^3)","A",1
430,1,175,202,0.3431967,"\int \frac{1}{\left(c+\frac{a}{x^2}+\frac{b}{x}\right)^2 x^7} \, dx","Integrate[1/((c + a/x^2 + b/x)^2*x^7),x]","\frac{\frac{2 b \left(30 a^2 c^2-20 a b^2 c+3 b^4\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{3/2}}+\frac{2 a \left(2 a^2 c^2-4 a b^2 c-3 a b c^2 x+b^4+b^3 c x\right)}{\left(b^2-4 a c\right) (a+x (b+c x))}-\frac{a^2}{x^2}+2 \log (x) \left(3 b^2-2 a c\right)+\left(2 a c-3 b^2\right) \log (a+x (b+c x))+\frac{4 a b}{x}}{2 a^4}","-\frac{\left(3 b^2-2 a c\right) \log \left(a+b x+c x^2\right)}{2 a^4}+\frac{\log (x) \left(3 b^2-2 a c\right)}{a^4}+\frac{b \left(3 b^2-11 a c\right)}{a^3 x \left(b^2-4 a c\right)}-\frac{3 b^2-8 a c}{2 a^2 x^2 \left(b^2-4 a c\right)}+\frac{b \left(30 a^2 c^2-20 a b^2 c+3 b^4\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{a^4 \left(b^2-4 a c\right)^{3/2}}+\frac{-2 a c+b^2+b c x}{a x^2 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)}",1,"(-(a^2/x^2) + (4*a*b)/x + (2*a*(b^4 - 4*a*b^2*c + 2*a^2*c^2 + b^3*c*x - 3*a*b*c^2*x))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (2*b*(3*b^4 - 20*a*b^2*c + 30*a^2*c^2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + 2*(3*b^2 - 2*a*c)*Log[x] + (-3*b^2 + 2*a*c)*Log[a + x*(b + c*x)])/(2*a^4)","A",1
431,1,260,238,0.3695682,"\int \frac{1}{\left(c+\frac{a}{x^2}+\frac{b}{x}\right)^3} \, dx","Integrate[(c + a/x^2 + b/x)^(-3),x]","\frac{\frac{a^3 c^2 (2 c x-5 b)+a^2 b^2 c (5 b-9 c x)-a b^4 (b-6 c x)+b^6 (-x)}{\left(b^2-4 a c\right) (a+x (b+c x))^2}+\frac{6 c \left(-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{5/2}}+\frac{-78 a^3 b c^3+36 a^3 c^4 x+61 a^2 b^3 c^2-102 a^2 b^2 c^3 x-14 a b^5 c+48 a b^4 c^2 x+b^7-6 b^6 c x}{\left(b^2-4 a c\right)^2 (a+x (b+c x))}-3 b c \log (a+x (b+c x))+2 c^2 x}{2 c^5}","\frac{3 x \left(10 a^2 c^2-7 a b^2 c+b^4\right)}{c^3 \left(b^2-4 a c\right)^2}-\frac{3 \left(-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c^4 \left(b^2-4 a c\right)^{5/2}}-\frac{3 b x^2 \left(b^2-6 a c\right)}{2 c^2 \left(b^2-4 a c\right)^2}+\frac{x^5 (2 a+b x)}{2 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}+\frac{x^3 \left(b x \left(b^2-7 a c\right)+a \left(b^2-10 a c\right)\right)}{c \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}-\frac{3 b \log \left(a+b x+c x^2\right)}{2 c^4}",1,"(2*c^2*x + (b^7 - 14*a*b^5*c + 61*a^2*b^3*c^2 - 78*a^3*b*c^3 - 6*b^6*c*x + 48*a*b^4*c^2*x - 102*a^2*b^2*c^3*x + 36*a^3*c^4*x)/((b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (-(b^6*x) + a^2*b^2*c*(5*b - 9*c*x) - a*b^4*(b - 6*c*x) + a^3*c^2*(-5*b + 2*c*x))/((b^2 - 4*a*c)*(a + x*(b + c*x))^2) + (6*c*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2) - 3*b*c*Log[a + x*(b + c*x)])/(2*c^5)","A",1
432,1,221,190,0.3084726,"\int \frac{1}{\left(c+\frac{a}{x^2}+\frac{b}{x}\right)^3 x} \, dx","Integrate[1/((c + a/x^2 + b/x)^3*x),x]","\frac{-\frac{2 b c \left(30 a^2 c^2-10 a b^2 c+b^4\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{5/2}}+\frac{2 a^3 c^2+a^2 b c (5 c x-4 b)+a b^3 (b-5 c x)+b^5 x}{\left(b^2-4 a c\right) (a+x (b+c x))^2}+\frac{32 a^3 c^3-39 a^2 b^2 c^2+50 a^2 b c^3 x+11 a b^4 c-30 a b^3 c^2 x-b^6+4 b^5 c x}{\left(b^2-4 a c\right)^2 (a+x (b+c x))}+c \log (a+x (b+c x))}{2 c^4}","\frac{b \left(30 a^2 c^2-10 a b^2 c+b^4\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c^3 \left(b^2-4 a c\right)^{5/2}}-\frac{b x \left(b^2-7 a c\right)}{c^2 \left(b^2-4 a c\right)^2}+\frac{x^2 \left(b x \left(b^2-10 a c\right)+a \left(b^2-16 a c\right)\right)}{2 c \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}+\frac{x^4 (2 a+b x)}{2 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}+\frac{\log \left(a+b x+c x^2\right)}{2 c^3}",1,"((-b^6 + 11*a*b^4*c - 39*a^2*b^2*c^2 + 32*a^3*c^3 + 4*b^5*c*x - 30*a*b^3*c^2*x + 50*a^2*b*c^3*x)/((b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (2*a^3*c^2 + b^5*x + a*b^3*(b - 5*c*x) + a^2*b*c*(-4*b + 5*c*x))/((b^2 - 4*a*c)*(a + x*(b + c*x))^2) - (2*b*c*(b^4 - 10*a*b^2*c + 30*a^2*c^2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2) + c*Log[a + x*(b + c*x)])/(2*c^4)","A",1
433,1,174,111,0.1743712,"\int \frac{1}{\left(c+\frac{a}{x^2}+\frac{b}{x}\right)^3 x^2} \, dx","Integrate[1/((c + a/x^2 + b/x)^3*x^2),x]","\frac{1}{2} \left(\frac{24 a^2 \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{5/2}}+\frac{a^2 c (2 c x-3 b)+a b^2 (b-4 c x)+b^4 x}{c^3 \left(4 a c-b^2\right) (a+x (b+c x))^2}+\frac{22 a^2 b c^2-20 a^2 c^3 x-8 a b^3 c+16 a b^2 c^2 x+b^5-2 b^4 c x}{c^3 \left(b^2-4 a c\right)^2 (a+x (b+c x))}\right)","\frac{12 a^2 \tanh ^{-1}\left(\frac{\frac{2 a}{x}+b}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{5/2}}-\frac{3 a \left(\frac{2 a}{x}+b\right)}{\left(b^2-4 a c\right)^2 \left(\frac{a}{x^2}+\frac{b}{x}+c\right)}+\frac{\frac{2 a}{x}+b}{2 \left(b^2-4 a c\right) \left(\frac{a}{x^2}+\frac{b}{x}+c\right)^2}",1,"((b^5 - 8*a*b^3*c + 22*a^2*b*c^2 - 2*b^4*c*x + 16*a*b^2*c^2*x - 20*a^2*c^3*x)/(c^3*(b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (b^4*x + a*b^2*(b - 4*c*x) + a^2*c*(-3*b + 2*c*x))/(c^3*(-b^2 + 4*a*c)*(a + x*(b + c*x))^2) + (24*a^2*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2))/2","A",1
434,1,126,107,0.1944348,"\int \frac{1}{\left(c+\frac{a}{x^2}+\frac{b}{x}\right)^3 x^3} \, dx","Integrate[1/((c + a/x^2 + b/x)^3*x^3),x]","-\frac{8 a^3 c+a^2 \left(b^2+10 b c x+16 c^2 x^2\right)+a b x \left(2 b^2+b c x+6 c^2 x^2\right)+b^4 x^2}{2 c \left(b^2-4 a c\right)^2 (a+x (b+c x))^2}-\frac{6 a b \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{5/2}}","\frac{3 b x (2 a+b x)}{2 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}-\frac{x^3 (b+2 c x)}{2 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}+\frac{6 a b \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{5/2}}",1,"-1/2*(8*a^3*c + b^4*x^2 + a*b*x*(2*b^2 + b*c*x + 6*c^2*x^2) + a^2*(b^2 + 10*b*c*x + 16*c^2*x^2))/(c*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) - (6*a*b*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2)","A",1
435,1,131,115,0.139577,"\int \frac{1}{\left(c+\frac{a}{x^2}+\frac{b}{x}\right)^3 x^4} \, dx","Integrate[1/((c + a/x^2 + b/x)^3*x^4),x]","\frac{1}{2} \left(\frac{\left(2 a c+b^2\right) (b+2 c x)}{c \left(b^2-4 a c\right)^2 (a+x (b+c x))}+\frac{a (b-2 c x)+b^2 x}{c \left(4 a c-b^2\right) (a+x (b+c x))^2}+\frac{4 \left(2 a c+b^2\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{5/2}}\right)","\frac{x (2 a+b x)}{2 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}+\frac{x \left(2 a c+b^2\right)+3 a b}{\left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}-\frac{2 \left(2 a c+b^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{5/2}}",1,"((b^2*x + a*(b - 2*c*x))/(c*(-b^2 + 4*a*c)*(a + x*(b + c*x))^2) + ((b^2 + 2*a*c)*(b + 2*c*x))/(c*(b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (4*(b^2 + 2*a*c)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2))/2","A",1
436,1,102,103,0.0961894,"\int \frac{1}{\left(c+\frac{a}{x^2}+\frac{b}{x}\right)^3 x^5} \, dx","Integrate[1/((c + a/x^2 + b/x)^3*x^5),x]","\frac{\frac{\left(b^2-4 a c\right) (2 a+b x)}{(a+x (b+c x))^2}-\frac{12 b c \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}-\frac{3 b (b+2 c x)}{a+x (b+c x)}}{2 \left(b^2-4 a c\right)^2}","\frac{2 a+b x}{2 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}-\frac{3 b (b+2 c x)}{2 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}+\frac{6 b c \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{5/2}}",1,"(((b^2 - 4*a*c)*(2*a + b*x))/(a + x*(b + c*x))^2 - (3*b*(b + 2*c*x))/(a + x*(b + c*x)) - (12*b*c*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(2*(b^2 - 4*a*c)^2)","A",1
437,1,97,103,0.0963934,"\int \frac{1}{\left(c+\frac{a}{x^2}+\frac{b}{x}\right)^3 x^6} \, dx","Integrate[1/((c + a/x^2 + b/x)^3*x^6),x]","\frac{\frac{24 c^2 \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}-\frac{(b+2 c x) \left(-2 c \left(5 a+3 c x^2\right)+b^2-6 b c x\right)}{(a+x (b+c x))^2}}{2 \left(b^2-4 a c\right)^2}","-\frac{12 c^2 \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{5/2}}+\frac{3 c (b+2 c x)}{\left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}+\frac{-b-2 c x}{2 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}",1,"(-(((b + 2*c*x)*(b^2 - 6*b*c*x - 2*c*(5*a + 3*c*x^2)))/(a + x*(b + c*x))^2) + (24*c^2*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(2*(b^2 - 4*a*c)^2)","A",1
438,1,178,185,0.3525285,"\int \frac{1}{\left(c+\frac{a}{x^2}+\frac{b}{x}\right)^3 x^7} \, dx","Integrate[1/((c + a/x^2 + b/x)^3*x^7),x]","\frac{\frac{a^2 \left(-2 a c+b^2+b c x\right)}{\left(b^2-4 a c\right) (a+x (b+c x))^2}-\frac{2 b \left(30 a^2 c^2-10 a b^2 c+b^4\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{5/2}}+\frac{a \left(16 a^2 c^2-15 a b^2 c-14 a b c^2 x+2 b^4+2 b^3 c x\right)}{\left(b^2-4 a c\right)^2 (a+x (b+c x))}-\log (a+x (b+c x))+2 \log (x)}{2 a^3}","-\frac{\log \left(a+b x+c x^2\right)}{2 a^3}+\frac{\log (x)}{a^3}+\frac{16 a^2 c^2+2 b c x \left(b^2-7 a c\right)-15 a b^2 c+2 b^4}{2 a^2 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}+\frac{b \left(30 a^2 c^2-10 a b^2 c+b^4\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{a^3 \left(b^2-4 a c\right)^{5/2}}+\frac{-2 a c+b^2+b c x}{2 a \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}",1,"((a^2*(b^2 - 2*a*c + b*c*x))/((b^2 - 4*a*c)*(a + x*(b + c*x))^2) + (a*(2*b^4 - 15*a*b^2*c + 16*a^2*c^2 + 2*b^3*c*x - 14*a*b*c^2*x))/((b^2 - 4*a*c)^2*(a + x*(b + c*x))) - (2*b*(b^4 - 10*a*b^2*c + 30*a^2*c^2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2) + 2*Log[x] - Log[a + x*(b + c*x)])/(2*a^3)","A",1
439,1,221,239,0.436541,"\int \frac{1}{\left(c+\frac{a}{x^2}+\frac{b}{x}\right)^3 x^8} \, dx","Integrate[1/((c + a/x^2 + b/x)^3*x^8),x]","\frac{\frac{a^2 \left(-3 a b c-2 a c^2 x+b^3+b^2 c x\right)}{\left(4 a c-b^2\right) (a+x (b+c x))^2}-\frac{a \left(46 a^2 b c^2+28 a^2 c^3 x-29 a b^3 c-26 a b^2 c^2 x+4 b^5+4 b^4 c x\right)}{\left(b^2-4 a c\right)^2 (a+x (b+c x))}+\frac{6 \left(-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{5/2}}+3 b \log (a+x (b+c x))-\frac{2 a}{x}-6 b \log (x)}{2 a^4}","\frac{3 b \log \left(a+b x+c x^2\right)}{2 a^4}-\frac{3 b \log (x)}{a^4}-\frac{3 \left(b^2-5 a c\right) \left(b^2-2 a c\right)}{a^3 x \left(b^2-4 a c\right)^2}+\frac{20 a^2 c^2+3 b c x \left(b^2-6 a c\right)-20 a b^2 c+3 b^4}{2 a^2 x \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}-\frac{3 \left(-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{a^4 \left(b^2-4 a c\right)^{5/2}}+\frac{-2 a c+b^2+b c x}{2 a x \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}",1,"((-2*a)/x + (a^2*(b^3 - 3*a*b*c + b^2*c*x - 2*a*c^2*x))/((-b^2 + 4*a*c)*(a + x*(b + c*x))^2) - (a*(4*b^5 - 29*a*b^3*c + 46*a^2*b*c^2 + 4*b^4*c*x - 26*a*b^2*c^2*x + 28*a^2*c^3*x))/((b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (6*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2) - 6*b*Log[x] + 3*b*Log[a + x*(b + c*x)])/(2*a^4)","A",1
440,1,40,40,0.0056126,"\int \frac{x^2}{15+\frac{2}{x^2}+\frac{13}{x}} \, dx","Integrate[x^2/(15 + 2/x^2 + 13/x),x]","\frac{x^3}{45}-\frac{13 x^2}{450}+\frac{139 x}{3375}-\frac{16}{567} \log (3 x+2)+\frac{\log (5 x+1)}{4375}","\frac{x^3}{45}-\frac{13 x^2}{450}+\frac{139 x}{3375}-\frac{16}{567} \log (3 x+2)+\frac{\log (5 x+1)}{4375}",1,"(139*x)/3375 - (13*x^2)/450 + x^3/45 - (16*Log[2 + 3*x])/567 + Log[1 + 5*x]/4375","A",1
441,1,33,33,0.0043457,"\int \frac{x}{15+\frac{2}{x^2}+\frac{13}{x}} \, dx","Integrate[x/(15 + 2/x^2 + 13/x),x]","\frac{x^2}{30}-\frac{13 x}{225}+\frac{8}{189} \log (3 x+2)-\frac{1}{875} \log (5 x+1)","\frac{x^2}{30}-\frac{13 x}{225}+\frac{8}{189} \log (3 x+2)-\frac{1}{875} \log (5 x+1)",1,"(-13*x)/225 + x^2/30 + (8*Log[2 + 3*x])/189 - Log[1 + 5*x]/875","A",1
442,1,26,26,0.0038794,"\int \frac{1}{15+\frac{2}{x^2}+\frac{13}{x}} \, dx","Integrate[(15 + 2/x^2 + 13/x)^(-1),x]","\frac{x}{15}-\frac{4}{63} \log (3 x+2)+\frac{1}{175} \log (5 x+1)","\frac{x}{15}-\frac{4}{63} \log (3 x+2)+\frac{1}{175} \log (5 x+1)",1,"x/15 - (4*Log[2 + 3*x])/63 + Log[1 + 5*x]/175","A",1
443,1,21,21,0.0032944,"\int \frac{1}{\left(15+\frac{2}{x^2}+\frac{13}{x}\right) x} \, dx","Integrate[1/((15 + 2/x^2 + 13/x)*x),x]","\frac{2}{21} \log (3 x+2)-\frac{1}{35} \log (5 x+1)","\frac{2}{21} \log (3 x+2)-\frac{1}{35} \log (5 x+1)",1,"(2*Log[2 + 3*x])/21 - Log[1 + 5*x]/35","A",1
444,1,21,23,0.0028802,"\int \frac{1}{\left(15+\frac{2}{x^2}+\frac{13}{x}\right) x^2} \, dx","Integrate[1/((15 + 2/x^2 + 13/x)*x^2),x]","\frac{1}{7} \log (5 x+1)-\frac{1}{7} \log (3 x+2)","\frac{1}{7} \log \left(\frac{1}{x}+5\right)-\frac{1}{7} \log \left(\frac{2}{x}+3\right)",1,"-1/7*Log[2 + 3*x] + Log[1 + 5*x]/7","A",1
445,1,27,27,0.0044146,"\int \frac{1}{\left(15+\frac{2}{x^2}+\frac{13}{x}\right) x^3} \, dx","Integrate[1/((15 + 2/x^2 + 13/x)*x^3),x]","\frac{\log (x)}{2}+\frac{3}{14} \log (3 x+2)-\frac{5}{7} \log (5 x+1)","\frac{\log (x)}{2}+\frac{3}{14} \log (3 x+2)-\frac{5}{7} \log (5 x+1)",1,"Log[x]/2 + (3*Log[2 + 3*x])/14 - (5*Log[1 + 5*x])/7","A",1
446,1,34,34,0.004255,"\int \frac{1}{\left(15+\frac{2}{x^2}+\frac{13}{x}\right) x^4} \, dx","Integrate[1/((15 + 2/x^2 + 13/x)*x^4),x]","-\frac{1}{2 x}-\frac{13 \log (x)}{4}-\frac{9}{28} \log (3 x+2)+\frac{25}{7} \log (5 x+1)","-\frac{1}{2 x}-\frac{13 \log (x)}{4}-\frac{9}{28} \log (3 x+2)+\frac{25}{7} \log (5 x+1)",1,"-1/2*1/x - (13*Log[x])/4 - (9*Log[2 + 3*x])/28 + (25*Log[1 + 5*x])/7","A",1
447,1,41,41,0.0049529,"\int \frac{1}{\left(15+\frac{2}{x^2}+\frac{13}{x}\right) x^5} \, dx","Integrate[1/((15 + 2/x^2 + 13/x)*x^5),x]","-\frac{1}{4 x^2}+\frac{13}{4 x}+\frac{139 \log (x)}{8}+\frac{27}{56} \log (3 x+2)-\frac{125}{7} \log (5 x+1)","-\frac{1}{4 x^2}+\frac{13}{4 x}+\frac{139 \log (x)}{8}+\frac{27}{56} \log (3 x+2)-\frac{125}{7} \log (5 x+1)",1,"-1/4*1/x^2 + 13/(4*x) + (139*Log[x])/8 + (27*Log[2 + 3*x])/56 - (125*Log[1 + 5*x])/7","A",1
448,1,48,48,0.005335,"\int \frac{1}{\left(15+\frac{2}{x^2}+\frac{13}{x}\right) x^6} \, dx","Integrate[1/((15 + 2/x^2 + 13/x)*x^6),x]","-\frac{1}{6 x^3}+\frac{13}{8 x^2}-\frac{139}{8 x}-\frac{1417 \log (x)}{16}-\frac{81}{112} \log (3 x+2)+\frac{625}{7} \log (5 x+1)","-\frac{1}{6 x^3}+\frac{13}{8 x^2}-\frac{139}{8 x}-\frac{1417 \log (x)}{16}-\frac{81}{112} \log (3 x+2)+\frac{625}{7} \log (5 x+1)",1,"-1/6*1/x^3 + 13/(8*x^2) - 139/(8*x) - (1417*Log[x])/16 - (81*Log[2 + 3*x])/112 + (625*Log[1 + 5*x])/7","A",1
449,1,213,204,0.5222234,"\int \left(a+\frac{c}{x^2}+\frac{b}{x}\right)^{5/2} \, dx","Integrate[(a + c/x^2 + b/x)^(5/2),x]","\frac{\sqrt{a+\frac{b x+c}{x^2}} \left(960 a^{3/2} b c^{3/2} x^4 \tanh ^{-1}\left(\frac{2 a x+b}{2 \sqrt{a} \sqrt{x (a x+b)+c}}\right)+15 x^4 \left(-48 a^2 c^2-24 a b^2 c+b^4\right) \tanh ^{-1}\left(\frac{b x+2 c}{2 \sqrt{c} \sqrt{x (a x+b)+c}}\right)-2 \sqrt{c} \sqrt{x (a x+b)+c} \left(2 c x^2 \left(-96 a^2 x^2+278 a b x+59 b^2\right)+8 c^2 x (27 a x+17 b)+15 b^3 x^3+48 c^3\right)\right)}{384 c^{3/2} x^3 \sqrt{x (a x+b)+c}}","\frac{5}{2} a^{3/2} b \tanh ^{-1}\left(\frac{2 a+\frac{b}{x}}{2 \sqrt{a} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right)+\frac{5 \left(-48 a^2 c^2-24 a b^2 c+b^4\right) \tanh ^{-1}\left(\frac{b+\frac{2 c}{x}}{2 \sqrt{c} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right)}{128 c^{3/2}}-\frac{5 \left(\frac{2 c \left(12 a c+b^2\right)}{x}+b \left(44 a c+b^2\right)\right) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}{64 c}+x \left(a+\frac{b}{x}+\frac{c}{x^2}\right)^{5/2}-\frac{5}{24} \left(7 b+\frac{6 c}{x}\right) \left(a+\frac{b}{x}+\frac{c}{x^2}\right)^{3/2}",1,"(Sqrt[a + (c + b*x)/x^2]*(-2*Sqrt[c]*Sqrt[c + x*(b + a*x)]*(48*c^3 + 15*b^3*x^3 + 8*c^2*x*(17*b + 27*a*x) + 2*c*x^2*(59*b^2 + 278*a*b*x - 96*a^2*x^2)) + 960*a^(3/2)*b*c^(3/2)*x^4*ArcTanh[(b + 2*a*x)/(2*Sqrt[a]*Sqrt[c + x*(b + a*x)])] + 15*(b^4 - 24*a*b^2*c - 48*a^2*c^2)*x^4*ArcTanh[(2*c + b*x)/(2*Sqrt[c]*Sqrt[c + x*(b + a*x)])]))/(384*c^(3/2)*x^3*Sqrt[c + x*(b + a*x)])","A",1
450,1,163,145,0.2498484,"\int \left(a+\frac{c}{x^2}+\frac{b}{x}\right)^{3/2} \, dx","Integrate[(a + c/x^2 + b/x)^(3/2),x]","\frac{\sqrt{a+\frac{b x+c}{x^2}} \left(-3 x^2 \left(4 a c+b^2\right) \tanh ^{-1}\left(\frac{b x+2 c}{2 \sqrt{c} \sqrt{x (a x+b)+c}}\right)+12 \sqrt{a} b \sqrt{c} x^2 \tanh ^{-1}\left(\frac{2 a x+b}{2 \sqrt{a} \sqrt{x (a x+b)+c}}\right)-2 \sqrt{c} (x (5 b-4 a x)+2 c) \sqrt{x (a x+b)+c}\right)}{8 \sqrt{c} x \sqrt{x (a x+b)+c}}","-\frac{3 \left(4 a c+b^2\right) \tanh ^{-1}\left(\frac{b+\frac{2 c}{x}}{2 \sqrt{c} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right)}{8 \sqrt{c}}+x \left(a+\frac{b}{x}+\frac{c}{x^2}\right)^{3/2}-\frac{3}{4} \left(3 b+\frac{2 c}{x}\right) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}+\frac{3}{2} \sqrt{a} b \tanh ^{-1}\left(\frac{2 a+\frac{b}{x}}{2 \sqrt{a} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right)",1,"(Sqrt[a + (c + b*x)/x^2]*(-2*Sqrt[c]*(2*c + x*(5*b - 4*a*x))*Sqrt[c + x*(b + a*x)] + 12*Sqrt[a]*b*Sqrt[c]*x^2*ArcTanh[(b + 2*a*x)/(2*Sqrt[a]*Sqrt[c + x*(b + a*x)])] - 3*(b^2 + 4*a*c)*x^2*ArcTanh[(2*c + b*x)/(2*Sqrt[c]*Sqrt[c + x*(b + a*x)])]))/(8*Sqrt[c]*x*Sqrt[c + x*(b + a*x)])","A",1
451,1,128,105,0.0811979,"\int \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} \, dx","Integrate[Sqrt[a + c/x^2 + b/x],x]","\frac{x \sqrt{a+\frac{b x+c}{x^2}} \left(b \tanh ^{-1}\left(\frac{2 a x+b}{2 \sqrt{a} \sqrt{x (a x+b)+c}}\right)+2 \sqrt{a} \left(\sqrt{x (a x+b)+c}-\sqrt{c} \tanh ^{-1}\left(\frac{b x+2 c}{2 \sqrt{c} \sqrt{x (a x+b)+c}}\right)\right)\right)}{2 \sqrt{a} \sqrt{x (a x+b)+c}}","x \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}+\frac{b \tanh ^{-1}\left(\frac{2 a+\frac{b}{x}}{2 \sqrt{a} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right)}{2 \sqrt{a}}-\sqrt{c} \tanh ^{-1}\left(\frac{b+\frac{2 c}{x}}{2 \sqrt{c} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right)",1,"(x*Sqrt[a + (c + b*x)/x^2]*(b*ArcTanh[(b + 2*a*x)/(2*Sqrt[a]*Sqrt[c + x*(b + a*x)])] + 2*Sqrt[a]*(Sqrt[c + x*(b + a*x)] - Sqrt[c]*ArcTanh[(2*c + b*x)/(2*Sqrt[c]*Sqrt[c + x*(b + a*x)])])))/(2*Sqrt[a]*Sqrt[c + x*(b + a*x)])","A",1
452,1,89,67,0.0577916,"\int \frac{1}{\sqrt{a+\frac{c}{x^2}+\frac{b}{x}}} \, dx","Integrate[1/Sqrt[a + c/x^2 + b/x],x]","\frac{2 \sqrt{a} (x (a x+b)+c)-b \sqrt{x (a x+b)+c} \tanh ^{-1}\left(\frac{2 a x+b}{2 \sqrt{a} \sqrt{x (a x+b)+c}}\right)}{2 a^{3/2} x \sqrt{a+\frac{b x+c}{x^2}}}","\frac{x \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}{a}-\frac{b \tanh ^{-1}\left(\frac{2 a+\frac{b}{x}}{2 \sqrt{a} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right)}{2 a^{3/2}}",1,"(2*Sqrt[a]*(c + x*(b + a*x)) - b*Sqrt[c + x*(b + a*x)]*ArcTanh[(b + 2*a*x)/(2*Sqrt[a]*Sqrt[c + x*(b + a*x)])])/(2*a^(3/2)*x*Sqrt[a + (c + b*x)/x^2])","A",1
453,1,138,133,0.1708246,"\int \frac{1}{\left(a+\frac{c}{x^2}+\frac{b}{x}\right)^{3/2}} \, dx","Integrate[(a + c/x^2 + b/x)^(-3/2),x]","-\frac{3 b \left(b^2-4 a c\right) \sqrt{x (a x+b)+c} \tanh ^{-1}\left(\frac{2 a x+b}{2 \sqrt{a} \sqrt{x (a x+b)+c}}\right)+2 \sqrt{a} \left(-b^2 \left(a x^2+3 c\right)+10 a b c x+4 a c \left(a x^2+2 c\right)-3 b^3 x\right)}{2 a^{5/2} x \left(b^2-4 a c\right) \sqrt{a+\frac{b x+c}{x^2}}}","-\frac{3 b \tanh ^{-1}\left(\frac{2 a+\frac{b}{x}}{2 \sqrt{a} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right)}{2 a^{5/2}}+\frac{x \left(3 b^2-8 a c\right) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}{a^2 \left(b^2-4 a c\right)}-\frac{2 x \left(-2 a c+b^2+\frac{b c}{x}\right)}{a \left(b^2-4 a c\right) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}",1,"-1/2*(2*Sqrt[a]*(-3*b^3*x + 10*a*b*c*x + 4*a*c*(2*c + a*x^2) - b^2*(3*c + a*x^2)) + 3*b*(b^2 - 4*a*c)*Sqrt[c + x*(b + a*x)]*ArcTanh[(b + 2*a*x)/(2*Sqrt[a]*Sqrt[c + x*(b + a*x)])])/(a^(5/2)*(b^2 - 4*a*c)*x*Sqrt[a + (c + b*x)/x^2])","A",1
454,1,256,220,0.378313,"\int \frac{1}{\left(a+\frac{c}{x^2}+\frac{b}{x}\right)^{5/2}} \, dx","Integrate[(a + c/x^2 + b/x)^(-5/2),x]","\frac{2 \sqrt{a} \left(3 b^4 \left(a^2 x^4-30 a c x^2+5 c^2\right)-4 a b^2 c \left(6 a^2 x^4-12 a c x^2+25 c^2\right)+8 a^2 b c^2 x \left(32 a x^2+39 c\right)+16 a^2 c^2 \left(3 a^2 x^4+12 a c x^2+8 c^2\right)+10 b^5 \left(2 a x^3+3 c x\right)-2 a b^3 c x \left(74 a x^2+105 c\right)+15 b^6 x^2\right)-15 b \left(b^2-4 a c\right)^2 (x (a x+b)+c)^{3/2} \tanh ^{-1}\left(\frac{2 a x+b}{2 \sqrt{a} \sqrt{x (a x+b)+c}}\right)}{6 a^{7/2} x \left(b^2-4 a c\right)^2 (x (a x+b)+c) \sqrt{a+\frac{b x+c}{x^2}}}","-\frac{5 b \tanh ^{-1}\left(\frac{2 a+\frac{b}{x}}{2 \sqrt{a} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right)}{2 a^{7/2}}-\frac{2 x \left(32 a^2 c^2+\frac{b c \left(5 b^2-28 a c\right)}{x}-32 a b^2 c+5 b^4\right)}{3 a^2 \left(b^2-4 a c\right)^2 \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}+\frac{x \left(128 a^2 c^2-100 a b^2 c+15 b^4\right) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}{3 a^3 \left(b^2-4 a c\right)^2}-\frac{2 x \left(-2 a c+b^2+\frac{b c}{x}\right)}{3 a \left(b^2-4 a c\right) \left(a+\frac{b}{x}+\frac{c}{x^2}\right)^{3/2}}",1,"(2*Sqrt[a]*(15*b^6*x^2 + 8*a^2*b*c^2*x*(39*c + 32*a*x^2) - 2*a*b^3*c*x*(105*c + 74*a*x^2) + 10*b^5*(3*c*x + 2*a*x^3) + 3*b^4*(5*c^2 - 30*a*c*x^2 + a^2*x^4) + 16*a^2*c^2*(8*c^2 + 12*a*c*x^2 + 3*a^2*x^4) - 4*a*b^2*c*(25*c^2 - 12*a*c*x^2 + 6*a^2*x^4)) - 15*b*(b^2 - 4*a*c)^2*(c + x*(b + a*x))^(3/2)*ArcTanh[(b + 2*a*x)/(2*Sqrt[a]*Sqrt[c + x*(b + a*x)])])/(6*a^(7/2)*(b^2 - 4*a*c)^2*x*(c + x*(b + a*x))*Sqrt[a + (c + b*x)/x^2])","A",1
455,1,32,73,0.0192906,"\int \sqrt{a^2+\frac{b^2}{x^2}+\frac{2 a b}{x}} \, dx","Integrate[Sqrt[a^2 + b^2/x^2 + (2*a*b)/x],x]","\frac{x \sqrt{\frac{(a x+b)^2}{x^2}} (a x+b \log (x))}{a x+b}","\frac{a x \sqrt{a^2+\frac{2 a b}{x}+\frac{b^2}{x^2}}}{a+\frac{b}{x}}-\frac{b \log \left(\frac{1}{x}\right) \sqrt{a^2+\frac{2 a b}{x}+\frac{b^2}{x^2}}}{a+\frac{b}{x}}",1,"(x*Sqrt[(b + a*x)^2/x^2]*(a*x + b*Log[x]))/(b + a*x)","A",1
456,1,202,179,0.1171378,"\int \frac{1}{c+\frac{a}{x^4}+\frac{b}{x^2}} \, dx","Integrate[(c + a/x^4 + b/x^2)^(-1),x]","-\frac{\left(b \sqrt{b^2-4 a c}+2 a c-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} c^{3/2} \sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left(b \sqrt{b^2-4 a c}-2 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} c^{3/2} \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x}{c}","-\frac{\left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} c^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} c^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x}{c}",1,"x/c - ((-b^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - ((b^2 - 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]])","A",1
457,1,70,631,0.0355074,"\int \frac{1}{c+\frac{a}{x^6}+\frac{b}{x^3}} \, dx","Integrate[(c + a/x^6 + b/x^3)^(-1),x]","\frac{x}{c}-\frac{\text{RootSum}\left[\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{\text{$\#$1}^3 b \log (x-\text{$\#$1})+a \log (x-\text{$\#$1})}{2 \text{$\#$1}^5 c+\text{$\#$1}^2 b}\&\right]}{3 c}","\frac{\left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{b-\sqrt{b^2-4 a c}}+\left(b-\sqrt{b^2-4 a c}\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6 \sqrt[3]{2} c^{4/3} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}+\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{\sqrt{b^2-4 a c}+b}+\left(\sqrt{b^2-4 a c}+b\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6 \sqrt[3]{2} c^{4/3} \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}-\frac{\left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \log \left(\sqrt[3]{b-\sqrt{b^2-4 a c}}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3 \sqrt[3]{2} c^{4/3} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}-\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \log \left(\sqrt[3]{\sqrt{b^2-4 a c}+b}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3 \sqrt[3]{2} c^{4/3} \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}+\frac{\left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3} c^{4/3} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}+\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3} c^{4/3} \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}+\frac{x}{c}",1,"x/c - RootSum[a + b*#1^3 + c*#1^6 & , (a*Log[x - #1] + b*Log[x - #1]*#1^3)/(b*#1^2 + 2*c*#1^5) & ]/(3*c)","C",1
458,1,70,376,0.0433572,"\int \frac{1}{c+\frac{a}{x^8}+\frac{b}{x^4}} \, dx","Integrate[(c + a/x^8 + b/x^4)^(-1),x]","\frac{x}{c}-\frac{\text{RootSum}\left[\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 b \log (x-\text{$\#$1})+a \log (x-\text{$\#$1})}{2 \text{$\#$1}^7 c+\text{$\#$1}^3 b}\&\right]}{4 c}","\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} c^{5/4} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{\left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} c^{5/4} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} c^{5/4} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{\left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} c^{5/4} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{x}{c}",1,"x/c - RootSum[a + b*#1^4 + c*#1^8 & , (a*Log[x - #1] + b*Log[x - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ]/(4*c)","C",1
459,1,106,106,0.0561042,"\int \frac{\sqrt{a+b \sqrt{x}+c x}}{x} \, dx","Integrate[Sqrt[a + b*Sqrt[x] + c*x]/x,x]","2 \sqrt{a+b \sqrt{x}+c x}-2 \sqrt{a} \tanh ^{-1}\left(\frac{2 a+b \sqrt{x}}{2 \sqrt{a} \sqrt{a+b \sqrt{x}+c x}}\right)+\frac{b \tanh ^{-1}\left(\frac{b+2 c \sqrt{x}}{2 \sqrt{c} \sqrt{a+b \sqrt{x}+c x}}\right)}{\sqrt{c}}","2 \sqrt{a+b \sqrt{x}+c x}-2 \sqrt{a} \tanh ^{-1}\left(\frac{2 a+b \sqrt{x}}{2 \sqrt{a} \sqrt{a+b \sqrt{x}+c x}}\right)+\frac{b \tanh ^{-1}\left(\frac{b+2 c \sqrt{x}}{2 \sqrt{c} \sqrt{a+b \sqrt{x}+c x}}\right)}{\sqrt{c}}",1,"2*Sqrt[a + b*Sqrt[x] + c*x] - 2*Sqrt[a]*ArcTanh[(2*a + b*Sqrt[x])/(2*Sqrt[a]*Sqrt[a + b*Sqrt[x] + c*x])] + (b*ArcTanh[(b + 2*c*Sqrt[x])/(2*Sqrt[c]*Sqrt[a + b*Sqrt[x] + c*x])])/Sqrt[c]","A",1
460,1,29,40,0.0252208,"\int \left(\frac{b^2}{4 c}+b \sqrt{x}+c x\right)^2 \, dx","Integrate[(b^2/(4*c) + b*Sqrt[x] + c*x)^2,x]","-\frac{\left(b-10 c \sqrt{x}\right) \left(b+2 c \sqrt{x}\right)^5}{960 c^4}","\frac{\left(b+2 c \sqrt{x}\right)^6}{192 c^4}-\frac{b \left(b+2 c \sqrt{x}\right)^5}{160 c^4}",1,"-1/960*((b - 10*c*Sqrt[x])*(b + 2*c*Sqrt[x])^5)/c^4","A",1
461,1,50,75,0.0300197,"\int \frac{1}{\sqrt{a^2+2 a b \sqrt{x}+b^2 x}} \, dx","Integrate[1/Sqrt[a^2 + 2*a*b*Sqrt[x] + b^2*x],x]","\frac{2 \left(a+b \sqrt{x}\right) \left(b \sqrt{x}-a \log \left(a+b \sqrt{x}\right)\right)}{b^2 \sqrt{\left(a+b \sqrt{x}\right)^2}}","\frac{2 \sqrt{a^2+2 a b \sqrt{x}+b^2 x}}{b^2}-\frac{2 a \left(a+b \sqrt{x}\right) \log \left(a+b \sqrt{x}\right)}{b^2 \sqrt{a^2+2 a b \sqrt{x}+b^2 x}}",1,"(2*(a + b*Sqrt[x])*(b*Sqrt[x] - a*Log[a + b*Sqrt[x]]))/(b^2*Sqrt[(a + b*Sqrt[x])^2])","A",1
462,1,56,137,0.0491317,"\int \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^{7/2} \, dx","Integrate[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^(7/2),x]","\frac{\left(a+b \sqrt[3]{x}\right)^7 \sqrt{\left(a+b \sqrt[3]{x}\right)^2} \left(a^2-8 a b \sqrt[3]{x}+36 b^2 x^{2/3}\right)}{120 b^3}","\frac{3 \sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}} \left(a+b \sqrt[3]{x}\right)^9}{10 b^3}-\frac{2 a \sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}} \left(a+b \sqrt[3]{x}\right)^8}{3 b^3}+\frac{3 a^2 \sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}} \left(a+b \sqrt[3]{x}\right)^7}{8 b^3}",1,"((a + b*x^(1/3))^7*Sqrt[(a + b*x^(1/3))^2]*(a^2 - 8*a*b*x^(1/3) + 36*b^2*x^(2/3)))/(120*b^3)","A",1
463,1,56,137,0.0352915,"\int \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^{5/2} \, dx","Integrate[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^(5/2),x]","\frac{\left(a+b \sqrt[3]{x}\right)^5 \sqrt{\left(a+b \sqrt[3]{x}\right)^2} \left(a^2-6 a b \sqrt[3]{x}+21 b^2 x^{2/3}\right)}{56 b^3}","\frac{3 \sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}} \left(a+b \sqrt[3]{x}\right)^7}{8 b^3}-\frac{6 a \sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}} \left(a+b \sqrt[3]{x}\right)^6}{7 b^3}+\frac{a^2 \sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}} \left(a+b \sqrt[3]{x}\right)^5}{2 b^3}",1,"((a + b*x^(1/3))^5*Sqrt[(a + b*x^(1/3))^2]*(a^2 - 6*a*b*x^(1/3) + 21*b^2*x^(2/3)))/(56*b^3)","A",1
464,1,65,137,0.0322665,"\int \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^{3/2} \, dx","Integrate[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^(3/2),x]","\frac{x \sqrt{\left(a+b \sqrt[3]{x}\right)^2} \left(20 a^3+45 a^2 b \sqrt[3]{x}+36 a b^2 x^{2/3}+10 b^3 x\right)}{20 \left(a+b \sqrt[3]{x}\right)}","\frac{\sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}} \left(a+b \sqrt[3]{x}\right)^5}{2 b^3}-\frac{6 a \sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}} \left(a+b \sqrt[3]{x}\right)^4}{5 b^3}+\frac{3 a^2 \sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}} \left(a+b \sqrt[3]{x}\right)^3}{4 b^3}",1,"(Sqrt[(a + b*x^(1/3))^2]*x*(20*a^3 + 45*a^2*b*x^(1/3) + 36*a*b^2*x^(2/3) + 10*b^3*x))/(20*(a + b*x^(1/3)))","A",1
465,1,43,88,0.0094698,"\int \sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)],x]","\frac{\sqrt{\left(a+b \sqrt[3]{x}\right)^2} \left(4 a x+3 b x^{4/3}\right)}{4 \left(a+b \sqrt[3]{x}\right)}","\frac{3 b x^{4/3} \sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}}}{4 \left(a+b \sqrt[3]{x}\right)}+\frac{a x \sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}}}{a+b \sqrt[3]{x}}",1,"(Sqrt[(a + b*x^(1/3))^2]*(4*a*x + 3*b*x^(4/3)))/(4*(a + b*x^(1/3)))","A",1
466,1,65,147,0.0362385,"\int \frac{1}{\sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}}} \, dx","Integrate[1/Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)],x]","\frac{3 \left(a+b \sqrt[3]{x}\right) \left(2 a^2 \log \left(a+b \sqrt[3]{x}\right)+b \sqrt[3]{x} \left(b \sqrt[3]{x}-2 a\right)\right)}{2 b^3 \sqrt{\left(a+b \sqrt[3]{x}\right)^2}}","-\frac{3 a \sqrt[3]{x} \left(a+b \sqrt[3]{x}\right)}{b^2 \sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}}}+\frac{3 x^{2/3} \left(a+b \sqrt[3]{x}\right)}{2 b \sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}}}+\frac{3 a^2 \left(a+b \sqrt[3]{x}\right) \log \left(a+b \sqrt[3]{x}\right)}{b^3 \sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}}}",1,"(3*(a + b*x^(1/3))*(b*(-2*a + b*x^(1/3))*x^(1/3) + 2*a^2*Log[a + b*x^(1/3)]))/(2*b^3*Sqrt[(a + b*x^(1/3))^2])","A",1
467,1,72,130,0.0454057,"\int \frac{1}{\left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^{3/2}} \, dx","Integrate[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^(-3/2),x]","\frac{3 a \left(3 a+4 b \sqrt[3]{x}\right)+6 \left(a+b \sqrt[3]{x}\right)^2 \log \left(a+b \sqrt[3]{x}\right)}{2 b^3 \left(a+b \sqrt[3]{x}\right) \sqrt{\left(a+b \sqrt[3]{x}\right)^2}}","-\frac{3 a^2}{2 b^3 \left(a+b \sqrt[3]{x}\right) \sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}}}+\frac{6 a}{b^3 \sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}}}+\frac{3 \left(a+b \sqrt[3]{x}\right) \log \left(a+b \sqrt[3]{x}\right)}{b^3 \sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}}}",1,"(3*a*(3*a + 4*b*x^(1/3)) + 6*(a + b*x^(1/3))^2*Log[a + b*x^(1/3)])/(2*b^3*(a + b*x^(1/3))*Sqrt[(a + b*x^(1/3))^2])","A",1
468,1,58,135,0.036017,"\int \frac{1}{\left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^{5/2}} \, dx","Integrate[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^(-5/2),x]","\frac{-a^2-4 a b \sqrt[3]{x}-6 b^2 x^{2/3}}{4 b^3 \left(a+b \sqrt[3]{x}\right)^3 \sqrt{\left(a+b \sqrt[3]{x}\right)^2}}","-\frac{3 a^2}{4 b^3 \left(a+b \sqrt[3]{x}\right)^3 \sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}}}+\frac{2 a}{b^3 \left(a+b \sqrt[3]{x}\right)^2 \sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}}}-\frac{3}{2 b^3 \left(a+b \sqrt[3]{x}\right) \sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}}}",1,"(-a^2 - 4*a*b*x^(1/3) - 6*b^2*x^(2/3))/(4*b^3*(a + b*x^(1/3))^3*Sqrt[(a + b*x^(1/3))^2])","A",1
469,1,58,137,0.0347593,"\int \frac{1}{\left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^{7/2}} \, dx","Integrate[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^(-7/2),x]","\frac{-a^2-6 a b \sqrt[3]{x}-15 b^2 x^{2/3}}{20 b^3 \left(a+b \sqrt[3]{x}\right)^5 \sqrt{\left(a+b \sqrt[3]{x}\right)^2}}","-\frac{a^2}{2 b^3 \left(a+b \sqrt[3]{x}\right)^5 \sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}}}+\frac{6 a}{5 b^3 \left(a+b \sqrt[3]{x}\right)^4 \sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}}}-\frac{3}{4 b^3 \left(a+b \sqrt[3]{x}\right)^3 \sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}}}",1,"(-a^2 - 6*a*b*x^(1/3) - 15*b^2*x^(2/3))/(20*b^3*(a + b*x^(1/3))^5*Sqrt[(a + b*x^(1/3))^2])","A",1
470,1,58,137,0.0350077,"\int \frac{1}{\left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^{9/2}} \, dx","Integrate[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^(-9/2),x]","\frac{-a^2-8 a b \sqrt[3]{x}-28 b^2 x^{2/3}}{56 b^3 \left(a+b \sqrt[3]{x}\right)^7 \sqrt{\left(a+b \sqrt[3]{x}\right)^2}}","-\frac{3 a^2}{8 b^3 \left(a+b \sqrt[3]{x}\right)^7 \sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}}}+\frac{6 a}{7 b^3 \left(a+b \sqrt[3]{x}\right)^6 \sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}}}-\frac{1}{2 b^3 \left(a+b \sqrt[3]{x}\right)^5 \sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}}}",1,"(-a^2 - 8*a*b*x^(1/3) - 28*b^2*x^(2/3))/(56*b^3*(a + b*x^(1/3))^7*Sqrt[(a + b*x^(1/3))^2])","A",1
471,1,58,137,0.0365837,"\int \frac{1}{\left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^{11/2}} \, dx","Integrate[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^(-11/2),x]","\frac{-a^2-10 a b \sqrt[3]{x}-45 b^2 x^{2/3}}{120 b^3 \left(a+b \sqrt[3]{x}\right)^9 \sqrt{\left(a+b \sqrt[3]{x}\right)^2}}","-\frac{3 a^2}{10 b^3 \left(a+b \sqrt[3]{x}\right)^9 \sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}}}+\frac{2 a}{3 b^3 \left(a+b \sqrt[3]{x}\right)^8 \sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}}}-\frac{3}{8 b^3 \left(a+b \sqrt[3]{x}\right)^7 \sqrt{a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}}}",1,"(-a^2 - 10*a*b*x^(1/3) - 45*b^2*x^(2/3))/(120*b^3*(a + b*x^(1/3))^9*Sqrt[(a + b*x^(1/3))^2])","A",1
472,1,68,77,0.0297652,"\int \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p (d x)^m \, dx","Integrate[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p*(d*x)^m,x]","\frac{x (d x)^m \left(\left(a+b \sqrt[3]{x}\right)^2\right)^p \left(\frac{b \sqrt[3]{x}}{a}+1\right)^{-2 p} \, _2F_1\left(3 (m+1),-2 p;3 (m+1)+1;-\frac{b \sqrt[3]{x}}{a}\right)}{m+1}","\frac{x (d x)^m \left(\frac{b \sqrt[3]{x}}{a}+1\right)^{-2 p} \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p \, _2F_1\left(3 (m+1),-2 p;3 m+4;-\frac{b \sqrt[3]{x}}{a}\right)}{m+1}",1,"(((a + b*x^(1/3))^2)^p*x*(d*x)^m*Hypergeometric2F1[3*(1 + m), -2*p, 1 + 3*(1 + m), -((b*x^(1/3))/a)])/((1 + m)*(1 + (b*x^(1/3))/a)^(2*p))","A",1
473,1,207,468,0.2163145,"\int \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p x^2 \, dx","Integrate[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p*x^2,x]","\frac{3 \left(a+b \sqrt[3]{x}\right) \left(\frac{a^8}{2 p+1}-\frac{4 a^7 \left(a+b \sqrt[3]{x}\right)}{p+1}+\frac{28 a^6 \left(a+b \sqrt[3]{x}\right)^2}{2 p+3}-\frac{28 a^5 \left(a+b \sqrt[3]{x}\right)^3}{p+2}+\frac{70 a^4 \left(a+b \sqrt[3]{x}\right)^4}{2 p+5}-\frac{28 a^3 \left(a+b \sqrt[3]{x}\right)^5}{p+3}+\frac{28 a^2 \left(a+b \sqrt[3]{x}\right)^6}{2 p+7}-\frac{4 a \left(a+b \sqrt[3]{x}\right)^7}{p+4}+\frac{\left(a+b \sqrt[3]{x}\right)^8}{2 p+9}\right) \left(\left(a+b \sqrt[3]{x}\right)^2\right)^p}{b^9}","\frac{3 a^9 \left(\frac{b \sqrt[3]{x}}{a}+1\right)^9 \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p}{b^9 (2 p+9)}-\frac{12 a^9 \left(\frac{b \sqrt[3]{x}}{a}+1\right)^8 \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p}{b^9 (p+4)}+\frac{84 a^9 \left(\frac{b \sqrt[3]{x}}{a}+1\right)^7 \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p}{b^9 (2 p+7)}-\frac{84 a^9 \left(\frac{b \sqrt[3]{x}}{a}+1\right)^6 \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p}{b^9 (p+3)}+\frac{210 a^9 \left(\frac{b \sqrt[3]{x}}{a}+1\right)^5 \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p}{b^9 (2 p+5)}-\frac{84 a^9 \left(\frac{b \sqrt[3]{x}}{a}+1\right)^4 \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p}{b^9 (p+2)}+\frac{84 a^9 \left(\frac{b \sqrt[3]{x}}{a}+1\right)^3 \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p}{b^9 (2 p+3)}-\frac{12 a^9 \left(\frac{b \sqrt[3]{x}}{a}+1\right)^2 \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p}{b^9 (p+1)}+\frac{3 a^9 \left(\frac{b \sqrt[3]{x}}{a}+1\right) \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p}{b^9 (2 p+1)}",1,"(3*(a^8/(1 + 2*p) - (4*a^7*(a + b*x^(1/3)))/(1 + p) + (28*a^6*(a + b*x^(1/3))^2)/(3 + 2*p) - (28*a^5*(a + b*x^(1/3))^3)/(2 + p) + (70*a^4*(a + b*x^(1/3))^4)/(5 + 2*p) - (28*a^3*(a + b*x^(1/3))^5)/(3 + p) + (28*a^2*(a + b*x^(1/3))^6)/(7 + 2*p) - (4*a*(a + b*x^(1/3))^7)/(4 + p) + (a + b*x^(1/3))^8/(9 + 2*p))*(a + b*x^(1/3))*((a + b*x^(1/3))^2)^p)/b^9","A",1
474,1,143,315,0.1736291,"\int \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p x \, dx","Integrate[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p*x,x]","\frac{3 \left(a+b \sqrt[3]{x}\right) \left(-\frac{2 a^5}{2 p+1}+\frac{5 a^4 \left(a+b \sqrt[3]{x}\right)}{p+1}-\frac{20 a^3 \left(a+b \sqrt[3]{x}\right)^2}{2 p+3}+\frac{10 a^2 \left(a+b \sqrt[3]{x}\right)^3}{p+2}-\frac{10 a \left(a+b \sqrt[3]{x}\right)^4}{2 p+5}+\frac{\left(a+b \sqrt[3]{x}\right)^5}{p+3}\right) \left(\left(a+b \sqrt[3]{x}\right)^2\right)^p}{2 b^6}","\frac{3 a^6 \left(\frac{b \sqrt[3]{x}}{a}+1\right)^6 \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p}{2 b^6 (p+3)}-\frac{15 a^6 \left(\frac{b \sqrt[3]{x}}{a}+1\right)^5 \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p}{b^6 (2 p+5)}+\frac{15 a^6 \left(\frac{b \sqrt[3]{x}}{a}+1\right)^4 \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p}{b^6 (p+2)}-\frac{30 a^6 \left(\frac{b \sqrt[3]{x}}{a}+1\right)^3 \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p}{b^6 (2 p+3)}+\frac{15 a^6 \left(\frac{b \sqrt[3]{x}}{a}+1\right)^2 \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p}{2 b^6 (p+1)}-\frac{3 a^6 \left(\frac{b \sqrt[3]{x}}{a}+1\right) \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p}{b^6 (2 p+1)}",1,"(3*((-2*a^5)/(1 + 2*p) + (5*a^4*(a + b*x^(1/3)))/(1 + p) - (20*a^3*(a + b*x^(1/3))^2)/(3 + 2*p) + (10*a^2*(a + b*x^(1/3))^3)/(2 + p) - (10*a*(a + b*x^(1/3))^4)/(5 + 2*p) + (a + b*x^(1/3))^5/(3 + p))*(a + b*x^(1/3))*((a + b*x^(1/3))^2)^p)/(2*b^6)","A",1
475,1,83,142,0.0486024,"\int \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p \, dx","Integrate[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p,x]","\frac{3 \left(a+b \sqrt[3]{x}\right) \left(\left(a+b \sqrt[3]{x}\right)^2\right)^p \left(a^2-a b (2 p+1) \sqrt[3]{x}+b^2 \left(2 p^2+3 p+1\right) x^{2/3}\right)}{b^3 (p+1) (2 p+1) (2 p+3)}","\frac{3 \left(a+b \sqrt[3]{x}\right)^3 \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p}{b^3 (2 p+3)}-\frac{3 a \left(a+b \sqrt[3]{x}\right)^2 \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p}{b^3 (p+1)}+\frac{3 a^2 \left(a+b \sqrt[3]{x}\right) \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p}{b^3 (2 p+1)}",1,"(3*(a + b*x^(1/3))*((a + b*x^(1/3))^2)^p*(a^2 - a*b*(1 + 2*p)*x^(1/3) + b^2*(1 + 3*p + 2*p^2)*x^(2/3)))/(b^3*(1 + p)*(1 + 2*p)*(3 + 2*p))","A",1
476,1,58,69,0.0133734,"\int \frac{\left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p}{x} \, dx","Integrate[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p/x,x]","-\frac{3 \left(a+b \sqrt[3]{x}\right) \left(\left(a+b \sqrt[3]{x}\right)^2\right)^p \, _2F_1\left(1,2 p+1;2 p+2;\frac{\sqrt[3]{x} b}{a}+1\right)}{a (2 p+1)}","-\frac{3 \left(\frac{b \sqrt[3]{x}}{a}+1\right) \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p \, _2F_1\left(1,2 p+1;2 (p+1);\frac{\sqrt[3]{x} b}{a}+1\right)}{2 p+1}",1,"(-3*(a + b*x^(1/3))*((a + b*x^(1/3))^2)^p*Hypergeometric2F1[1, 1 + 2*p, 2 + 2*p, 1 + (b*x^(1/3))/a])/(a*(1 + 2*p))","A",1
477,1,61,75,0.0146916,"\int \frac{\left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p}{x^2} \, dx","Integrate[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p/x^2,x]","\frac{3 b^3 \left(a+b \sqrt[3]{x}\right) \left(\left(a+b \sqrt[3]{x}\right)^2\right)^p \, _2F_1\left(4,2 p+1;2 p+2;\frac{\sqrt[3]{x} b}{a}+1\right)}{a^4 (2 p+1)}","\frac{3 b^3 \left(\frac{b \sqrt[3]{x}}{a}+1\right) \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p \, _2F_1\left(4,2 p+1;2 (p+1);\frac{\sqrt[3]{x} b}{a}+1\right)}{a^3 (2 p+1)}",1,"(3*b^3*(a + b*x^(1/3))*((a + b*x^(1/3))^2)^p*Hypergeometric2F1[4, 1 + 2*p, 2 + 2*p, 1 + (b*x^(1/3))/a])/(a^4*(1 + 2*p))","A",1
478,1,101,146,0.0821772,"\int \left(\frac{\left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p}{x^2}-\frac{2 b^3 (1-2 p) (1-p) p \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p}{3 a^3 x}\right) \, dx","Integrate[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p/x^2 - (2*b^3*(1 - 2*p)*(1 - p)*p*(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^p)/(3*a^3*x),x]","\frac{b^3 \left(a+b \sqrt[3]{x}\right) \left(\left(a+b \sqrt[3]{x}\right)^2\right)^p \left(2 p \left(2 p^2-3 p+1\right) \, _2F_1\left(1,2 p+1;2 (p+1);\frac{\sqrt[3]{x} b}{a}+1\right)+3 \, _2F_1\left(4,2 p+1;2 (p+1);\frac{\sqrt[3]{x} b}{a}+1\right)\right)}{a^3 (2 a p+a)}","\frac{b (1-p) \left(a+b \sqrt[3]{x}\right) \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p}{a^2 x^{2/3}}-\frac{\left(a+b \sqrt[3]{x}\right) \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p}{a x}-\frac{b^2 (1-2 p) (1-p) \left(a+b \sqrt[3]{x}\right) \left(a^2+2 a b \sqrt[3]{x}+b^2 x^{2/3}\right)^p}{a^3 \sqrt[3]{x}}",1,"(b^3*(a + b*x^(1/3))*((a + b*x^(1/3))^2)^p*(2*p*(1 - 3*p + 2*p^2)*Hypergeometric2F1[1, 1 + 2*p, 2*(1 + p), 1 + (b*x^(1/3))/a] + 3*Hypergeometric2F1[4, 1 + 2*p, 2*(1 + p), 1 + (b*x^(1/3))/a]))/(a^3*(a + 2*a*p))","C",1
479,1,93,176,0.0693092,"\int \frac{1}{\left(a^2+2 a b \sqrt[4]{x}+b^2 \sqrt{x}\right)^{3/2}} \, dx","Integrate[(a^2 + 2*a*b*x^(1/4) + b^2*Sqrt[x])^(-3/2),x]","\frac{2 \left(-5 a^3-4 a^2 b \sqrt[4]{x}+4 a b^2 \sqrt{x}-6 a \left(a+b \sqrt[4]{x}\right)^2 \log \left(a+b \sqrt[4]{x}\right)+2 b^3 x^{3/4}\right)}{b^4 \left(a+b \sqrt[4]{x}\right) \sqrt{\left(a+b \sqrt[4]{x}\right)^2}}","-\frac{12 a^2}{b^4 \sqrt{a^2+2 a b \sqrt[4]{x}+b^2 \sqrt{x}}}-\frac{12 a \left(a+b \sqrt[4]{x}\right) \log \left(a+b \sqrt[4]{x}\right)}{b^4 \sqrt{a^2+2 a b \sqrt[4]{x}+b^2 \sqrt{x}}}+\frac{4 \sqrt[4]{x} \left(a+b \sqrt[4]{x}\right)}{b^3 \sqrt{a^2+2 a b \sqrt[4]{x}+b^2 \sqrt{x}}}+\frac{2 a^3}{b^4 \left(a+b \sqrt[4]{x}\right) \sqrt{a^2+2 a b \sqrt[4]{x}+b^2 \sqrt{x}}}",1,"(2*(-5*a^3 - 4*a^2*b*x^(1/4) + 4*a*b^2*Sqrt[x] + 2*b^3*x^(3/4) - 6*a*(a + b*x^(1/4))^2*Log[a + b*x^(1/4)]))/(b^4*(a + b*x^(1/4))*Sqrt[(a + b*x^(1/4))^2])","A",1
480,1,121,268,0.0962586,"\int \frac{1}{\left(a^2+2 a b \sqrt[6]{x}+b^2 \sqrt[3]{x}\right)^{5/2}} \, dx","Integrate[(a^2 + 2*a*b*x^(1/6) + b^2*x^(1/3))^(-5/2),x]","\frac{-77 a^5-248 a^4 b \sqrt[6]{x}-252 a^3 b^2 \sqrt[3]{x}-48 a^2 b^3 \sqrt{x}+48 a b^4 x^{2/3}-60 a \left(a+b \sqrt[6]{x}\right)^4 \log \left(a+b \sqrt[6]{x}\right)+12 b^5 x^{5/6}}{2 b^6 \left(a+b \sqrt[6]{x}\right)^3 \sqrt{\left(a+b \sqrt[6]{x}\right)^2}}","-\frac{60 a^2}{b^6 \sqrt{a^2+2 a b \sqrt[6]{x}+b^2 \sqrt[3]{x}}}-\frac{30 a \left(a+b \sqrt[6]{x}\right) \log \left(a+b \sqrt[6]{x}\right)}{b^6 \sqrt{a^2+2 a b \sqrt[6]{x}+b^2 \sqrt[3]{x}}}+\frac{6 \sqrt[6]{x} \left(a+b \sqrt[6]{x}\right)}{b^5 \sqrt{a^2+2 a b \sqrt[6]{x}+b^2 \sqrt[3]{x}}}+\frac{3 a^5}{2 b^6 \left(a+b \sqrt[6]{x}\right)^3 \sqrt{a^2+2 a b \sqrt[6]{x}+b^2 \sqrt[3]{x}}}-\frac{10 a^4}{b^6 \left(a+b \sqrt[6]{x}\right)^2 \sqrt{a^2+2 a b \sqrt[6]{x}+b^2 \sqrt[3]{x}}}+\frac{30 a^3}{b^6 \left(a+b \sqrt[6]{x}\right) \sqrt{a^2+2 a b \sqrt[6]{x}+b^2 \sqrt[3]{x}}}",1,"(-77*a^5 - 248*a^4*b*x^(1/6) - 252*a^3*b^2*x^(1/3) - 48*a^2*b^3*Sqrt[x] + 48*a*b^4*x^(2/3) + 12*b^5*x^(5/6) - 60*a*(a + b*x^(1/6))^4*Log[a + b*x^(1/6)])/(2*b^6*(a + b*x^(1/6))^3*Sqrt[(a + b*x^(1/6))^2])","A",1
481,1,66,179,0.0340978,"\int \left(a^2+\frac{b^2}{x}+\frac{2 a b}{\sqrt{x}}\right)^{3/2} \, dx","Integrate[(a^2 + b^2/x + (2*a*b)/Sqrt[x])^(3/2),x]","\frac{\sqrt{\frac{\left(a \sqrt{x}+b\right)^2}{x}} \left(a^3 x^{3/2}+6 a^2 b x+3 a b^2 \sqrt{x} \log (x)-2 b^3\right)}{a \sqrt{x}+b}","\frac{6 a^2 b \sqrt{x} \sqrt{a^2+\frac{2 a b}{\sqrt{x}}+\frac{b^2}{x}}}{a+\frac{b}{\sqrt{x}}}+\frac{6 a b^2 \log \left(\sqrt{x}\right) \sqrt{a^2+\frac{2 a b}{\sqrt{x}}+\frac{b^2}{x}}}{a+\frac{b}{\sqrt{x}}}-\frac{2 b^3 \sqrt{a^2+\frac{2 a b}{\sqrt{x}}+\frac{b^2}{x}}}{\sqrt{x} \left(a+\frac{b}{\sqrt{x}}\right)}+\frac{a^3 x \sqrt{a^2+\frac{2 a b}{\sqrt{x}}+\frac{b^2}{x}}}{a+\frac{b}{\sqrt{x}}}",1,"(Sqrt[(b + a*Sqrt[x])^2/x]*(-2*b^3 + 6*a^2*b*x + a^3*x^(3/2) + 3*a*b^2*Sqrt[x]*Log[x]))/(b + a*Sqrt[x])","A",1
482,1,125,391,0.0681008,"\int \left(a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt[3]{x}}\right)^{7/2} \, dx","Integrate[(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))^(7/2),x]","\frac{\sqrt{\frac{\left(a \sqrt[3]{x}+b\right)^2}{x^{2/3}}} \left(4 a^7 x^{7/3}+42 a^6 b x^2+252 a^5 b^2 x^{5/3}+140 a^4 b^3 x^{4/3} \log (x)-420 a^3 b^4 x-126 a^2 b^5 x^{2/3}-28 a b^6 \sqrt[3]{x}-3 b^7\right)}{4 x \left(a \sqrt[3]{x}+b\right)}","-\frac{3 b^7 \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}{4 x^{4/3} \left(a+\frac{b}{\sqrt[3]{x}}\right)}-\frac{7 a b^6 \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}{x \left(a+\frac{b}{\sqrt[3]{x}}\right)}-\frac{63 a^2 b^5 \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}{2 x^{2/3} \left(a+\frac{b}{\sqrt[3]{x}}\right)}+\frac{a^7 x \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}{a+\frac{b}{\sqrt[3]{x}}}+\frac{21 a^6 b x^{2/3} \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}{2 \left(a+\frac{b}{\sqrt[3]{x}}\right)}+\frac{63 a^5 b^2 \sqrt[3]{x} \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}{a+\frac{b}{\sqrt[3]{x}}}+\frac{105 a^4 b^3 \log \left(\sqrt[3]{x}\right) \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}{a+\frac{b}{\sqrt[3]{x}}}-\frac{105 a^3 b^4 \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}{\sqrt[3]{x} \left(a+\frac{b}{\sqrt[3]{x}}\right)}",1,"(Sqrt[(b + a*x^(1/3))^2/x^(2/3)]*(-3*b^7 - 28*a*b^6*x^(1/3) - 126*a^2*b^5*x^(2/3) - 420*a^3*b^4*x + 252*a^5*b^2*x^(5/3) + 42*a^6*b*x^2 + 4*a^7*x^(7/3) + 140*a^4*b^3*x^(4/3)*Log[x]))/(4*(b + a*x^(1/3))*x)","A",1
483,1,99,291,0.0568543,"\int \left(a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt[3]{x}}\right)^{5/2} \, dx","Integrate[(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))^(5/2),x]","\frac{\left(a \sqrt[3]{x}+b\right) \left(2 a^5 x^{5/3}+15 a^4 b x^{4/3}+60 a^3 b^2 x+20 a^2 b^3 x^{2/3} \log (x)-30 a b^4 \sqrt[3]{x}-3 b^5\right)}{2 x \sqrt{\frac{\left(a \sqrt[3]{x}+b\right)^2}{x^{2/3}}}}","-\frac{3 b^5 \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}{2 x^{2/3} \left(a+\frac{b}{\sqrt[3]{x}}\right)}-\frac{15 a b^4 \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}{\sqrt[3]{x} \left(a+\frac{b}{\sqrt[3]{x}}\right)}+\frac{30 a^2 b^3 \log \left(\sqrt[3]{x}\right) \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}{a+\frac{b}{\sqrt[3]{x}}}+\frac{a^5 x \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}{a+\frac{b}{\sqrt[3]{x}}}+\frac{15 a^4 b x^{2/3} \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}{2 \left(a+\frac{b}{\sqrt[3]{x}}\right)}+\frac{30 a^3 b^2 \sqrt[3]{x} \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}{a+\frac{b}{\sqrt[3]{x}}}",1,"((b + a*x^(1/3))*(-3*b^5 - 30*a*b^4*x^(1/3) + 60*a^3*b^2*x + 15*a^4*b*x^(4/3) + 2*a^5*x^(5/3) + 20*a^2*b^3*x^(2/3)*Log[x]))/(2*Sqrt[(b + a*x^(1/3))^2/x^(2/3)]*x)","A",1
484,1,77,189,0.0285437,"\int \left(a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt[3]{x}}\right)^{3/2} \, dx","Integrate[(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))^(3/2),x]","\frac{\sqrt{\frac{\left(a \sqrt[3]{x}+b\right)^2}{x^{2/3}}} \left(2 a^3 x^{4/3}+9 a^2 b x+18 a b^2 x^{2/3}+2 b^3 \sqrt[3]{x} \log (x)\right)}{2 \left(a \sqrt[3]{x}+b\right)}","\frac{9 a^2 b x^{2/3} \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}{2 \left(a+\frac{b}{\sqrt[3]{x}}\right)}+\frac{9 a b^2 \sqrt[3]{x} \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}{a+\frac{b}{\sqrt[3]{x}}}+\frac{3 b^3 \log \left(\sqrt[3]{x}\right) \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}{a+\frac{b}{\sqrt[3]{x}}}+\frac{a^3 x \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}{a+\frac{b}{\sqrt[3]{x}}}",1,"(Sqrt[(b + a*x^(1/3))^2/x^(2/3)]*(18*a*b^2*x^(2/3) + 9*a^2*b*x + 2*a^3*x^(4/3) + 2*b^3*x^(1/3)*Log[x]))/(2*(b + a*x^(1/3)))","A",1
485,1,49,88,0.0143268,"\int \sqrt{a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt[3]{x}}} \, dx","Integrate[Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)],x]","\frac{\sqrt{\frac{\left(a \sqrt[3]{x}+b\right)^2}{x^{2/3}}} \left(2 a x^{4/3}+3 b x\right)}{2 \left(a \sqrt[3]{x}+b\right)}","\frac{3 b x^{2/3} \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}{2 \left(a+\frac{b}{\sqrt[3]{x}}\right)}+\frac{a x \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}{a+\frac{b}{\sqrt[3]{x}}}",1,"(Sqrt[(b + a*x^(1/3))^2/x^(2/3)]*(3*b*x + 2*a*x^(4/3)))/(2*(b + a*x^(1/3)))","A",1
486,1,86,190,0.0399263,"\int \frac{1}{\sqrt{a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt[3]{x}}}} \, dx","Integrate[1/Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)],x]","\frac{\left(a \sqrt[3]{x}+b\right) \left(2 a^3 x-3 a^2 b x^{2/3}-6 b^3 \log \left(a \sqrt[3]{x}+b\right)+6 a b^2 \sqrt[3]{x}\right)}{2 a^4 \sqrt[3]{x} \sqrt{\frac{\left(a \sqrt[3]{x}+b\right)^2}{x^{2/3}}}}","-\frac{3 b x^{2/3} \left(a+\frac{b}{\sqrt[3]{x}}\right)}{2 a^2 \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}+\frac{x \left(a+\frac{b}{\sqrt[3]{x}}\right)}{a \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}-\frac{3 b^3 \left(a+\frac{b}{\sqrt[3]{x}}\right) \log \left(a \sqrt[3]{x}+b\right)}{a^4 \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}+\frac{3 b^2 \sqrt[3]{x} \left(a+\frac{b}{\sqrt[3]{x}}\right)}{a^3 \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}",1,"((b + a*x^(1/3))*(6*a*b^2*x^(1/3) - 3*a^2*b*x^(2/3) + 2*a^3*x - 6*b^3*Log[b + a*x^(1/3)]))/(2*a^4*Sqrt[(b + a*x^(1/3))^2/x^(2/3)]*x^(1/3))","A",1
487,1,126,300,0.0859126,"\int \frac{1}{\left(a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt[3]{x}}\right)^{3/2}} \, dx","Integrate[(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))^(-3/2),x]","\frac{\left(a \sqrt[3]{x}+b\right) \left(2 a^5 x^{5/3}-5 a^4 b x^{4/3}+20 a^3 b^2 x+63 a^2 b^3 x^{2/3}+6 a b^4 \sqrt[3]{x}-60 b^3 \left(a \sqrt[3]{x}+b\right)^2 \log \left(a \sqrt[3]{x}+b\right)-27 b^5\right)}{2 a^6 x \left(\frac{\left(a \sqrt[3]{x}+b\right)^2}{x^{2/3}}\right)^{3/2}}","\frac{3 b^5 \left(a+\frac{b}{\sqrt[3]{x}}\right)}{2 a^6 \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}} \left(a \sqrt[3]{x}+b\right)^2}-\frac{15 b^4 \left(a+\frac{b}{\sqrt[3]{x}}\right)}{a^6 \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}} \left(a \sqrt[3]{x}+b\right)}-\frac{30 b^3 \left(a+\frac{b}{\sqrt[3]{x}}\right) \log \left(a \sqrt[3]{x}+b\right)}{a^6 \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}+\frac{18 b^2 \sqrt[3]{x} \left(a+\frac{b}{\sqrt[3]{x}}\right)}{a^5 \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}-\frac{9 b x^{2/3} \left(a+\frac{b}{\sqrt[3]{x}}\right)}{2 a^4 \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}+\frac{x \left(a+\frac{b}{\sqrt[3]{x}}\right)}{a^3 \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}",1,"((b + a*x^(1/3))*(-27*b^5 + 6*a*b^4*x^(1/3) + 63*a^2*b^3*x^(2/3) + 20*a^3*b^2*x - 5*a^4*b*x^(4/3) + 2*a^5*x^(5/3) - 60*b^3*(b + a*x^(1/3))^2*Log[b + a*x^(1/3)]))/(2*a^6*((b + a*x^(1/3))^2/x^(2/3))^(3/2)*x)","A",1
488,1,152,410,0.1298359,"\int \frac{1}{\left(a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt[3]{x}}\right)^{5/2}} \, dx","Integrate[(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))^(-5/2),x]","\frac{\left(a \sqrt[3]{x}+b\right) \left(4 a^7 x^{7/3}-14 a^6 b x^2+84 a^5 b^2 x^{5/3}+556 a^4 b^3 x^{4/3}+544 a^3 b^4 x-444 a^2 b^5 x^{2/3}-856 a b^6 \sqrt[3]{x}-420 b^3 \left(a \sqrt[3]{x}+b\right)^4 \log \left(a \sqrt[3]{x}+b\right)-319 b^7\right)}{4 a^8 x^{5/3} \left(\frac{\left(a \sqrt[3]{x}+b\right)^2}{x^{2/3}}\right)^{5/2}}","\frac{3 b^7 \left(a+\frac{b}{\sqrt[3]{x}}\right)}{4 a^8 \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}} \left(a \sqrt[3]{x}+b\right)^4}-\frac{7 b^6 \left(a+\frac{b}{\sqrt[3]{x}}\right)}{a^8 \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}} \left(a \sqrt[3]{x}+b\right)^3}+\frac{63 b^5 \left(a+\frac{b}{\sqrt[3]{x}}\right)}{2 a^8 \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}} \left(a \sqrt[3]{x}+b\right)^2}-\frac{105 b^4 \left(a+\frac{b}{\sqrt[3]{x}}\right)}{a^8 \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}} \left(a \sqrt[3]{x}+b\right)}-\frac{105 b^3 \left(a+\frac{b}{\sqrt[3]{x}}\right) \log \left(a \sqrt[3]{x}+b\right)}{a^8 \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}+\frac{45 b^2 \sqrt[3]{x} \left(a+\frac{b}{\sqrt[3]{x}}\right)}{a^7 \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}-\frac{15 b x^{2/3} \left(a+\frac{b}{\sqrt[3]{x}}\right)}{2 a^6 \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}+\frac{x \left(a+\frac{b}{\sqrt[3]{x}}\right)}{a^5 \sqrt{a^2+\frac{2 a b}{\sqrt[3]{x}}+\frac{b^2}{x^{2/3}}}}",1,"((b + a*x^(1/3))*(-319*b^7 - 856*a*b^6*x^(1/3) - 444*a^2*b^5*x^(2/3) + 544*a^3*b^4*x + 556*a^4*b^3*x^(4/3) + 84*a^5*b^2*x^(5/3) - 14*a^6*b*x^2 + 4*a^7*x^(7/3) - 420*b^3*(b + a*x^(1/3))^4*Log[b + a*x^(1/3)]))/(4*a^8*((b + a*x^(1/3))^2/x^(2/3))^(5/2)*x^(5/3))","A",1
489,1,98,289,0.0516741,"\int \left(a^2+\frac{b^2}{\sqrt{x}}+\frac{2 a b}{\sqrt[4]{x}}\right)^{5/2} \, dx","Integrate[(a^2 + b^2/Sqrt[x] + (2*a*b)/x^(1/4))^(5/2),x]","\frac{\sqrt{\frac{\left(a \sqrt[4]{x}+b\right)^2}{\sqrt{x}}} \left(3 a^5 x^{5/4}+20 a^4 b x+60 a^3 b^2 x^{3/4}+120 a^2 b^3 \sqrt{x}+15 a b^4 \sqrt[4]{x} \log (x)-12 b^5\right)}{3 \left(a \sqrt[4]{x}+b\right)}","-\frac{4 b^5 \sqrt{a^2+\frac{2 a b}{\sqrt[4]{x}}+\frac{b^2}{\sqrt{x}}}}{\sqrt[4]{x} \left(a+\frac{b}{\sqrt[4]{x}}\right)}+\frac{20 a b^4 \log \left(\sqrt[4]{x}\right) \sqrt{a^2+\frac{2 a b}{\sqrt[4]{x}}+\frac{b^2}{\sqrt{x}}}}{a+\frac{b}{\sqrt[4]{x}}}+\frac{40 a^2 b^3 \sqrt[4]{x} \sqrt{a^2+\frac{2 a b}{\sqrt[4]{x}}+\frac{b^2}{\sqrt{x}}}}{a+\frac{b}{\sqrt[4]{x}}}+\frac{a^5 x \sqrt{a^2+\frac{2 a b}{\sqrt[4]{x}}+\frac{b^2}{\sqrt{x}}}}{a+\frac{b}{\sqrt[4]{x}}}+\frac{20 a^4 b x^{3/4} \sqrt{a^2+\frac{2 a b}{\sqrt[4]{x}}+\frac{b^2}{\sqrt{x}}}}{3 \left(a+\frac{b}{\sqrt[4]{x}}\right)}+\frac{20 a^3 b^2 \sqrt{x} \sqrt{a^2+\frac{2 a b}{\sqrt[4]{x}}+\frac{b^2}{\sqrt{x}}}}{a+\frac{b}{\sqrt[4]{x}}}",1,"(Sqrt[(b + a*x^(1/4))^2/Sqrt[x]]*(-12*b^5 + 120*a^2*b^3*Sqrt[x] + 60*a^3*b^2*x^(3/4) + 20*a^4*b*x + 3*a^5*x^(5/4) + 15*a*b^4*x^(1/4)*Log[x]))/(3*(b + a*x^(1/4)))","A",1
490,1,103,291,0.0483347,"\int \left(a^2+\frac{b^2}{x^{2/5}}+\frac{2 a b}{\sqrt[5]{x}}\right)^{5/2} \, dx","Integrate[(a^2 + b^2/x^(2/5) + (2*a*b)/x^(1/5))^(5/2),x]","\frac{\sqrt{\frac{\left(a \sqrt[5]{x}+b\right)^2}{x^{2/5}}} \left(12 a^5 x^{6/5}+75 a^4 b x+200 a^3 b^2 x^{4/5}+300 a^2 b^3 x^{3/5}+300 a b^4 x^{2/5}+12 b^5 \sqrt[5]{x} \log (x)\right)}{12 \left(a \sqrt[5]{x}+b\right)}","\frac{5 b^5 \log \left(\sqrt[5]{x}\right) \sqrt{a^2+\frac{2 a b}{\sqrt[5]{x}}+\frac{b^2}{x^{2/5}}}}{a+\frac{b}{\sqrt[5]{x}}}+\frac{25 a b^4 \sqrt[5]{x} \sqrt{a^2+\frac{2 a b}{\sqrt[5]{x}}+\frac{b^2}{x^{2/5}}}}{a+\frac{b}{\sqrt[5]{x}}}+\frac{25 a^2 b^3 x^{2/5} \sqrt{a^2+\frac{2 a b}{\sqrt[5]{x}}+\frac{b^2}{x^{2/5}}}}{a+\frac{b}{\sqrt[5]{x}}}+\frac{a^5 x \sqrt{a^2+\frac{2 a b}{\sqrt[5]{x}}+\frac{b^2}{x^{2/5}}}}{a+\frac{b}{\sqrt[5]{x}}}+\frac{25 a^4 b x^{4/5} \sqrt{a^2+\frac{2 a b}{\sqrt[5]{x}}+\frac{b^2}{x^{2/5}}}}{4 \left(a+\frac{b}{\sqrt[5]{x}}\right)}+\frac{50 a^3 b^2 x^{3/5} \sqrt{a^2+\frac{2 a b}{\sqrt[5]{x}}+\frac{b^2}{x^{2/5}}}}{3 \left(a+\frac{b}{\sqrt[5]{x}}\right)}",1,"(Sqrt[(b + a*x^(1/5))^2/x^(2/5)]*(300*a*b^4*x^(2/5) + 300*a^2*b^3*x^(3/5) + 200*a^3*b^2*x^(4/5) + 75*a^4*b*x + 12*a^5*x^(6/5) + 12*b^5*x^(1/5)*Log[x]))/(12*(b + a*x^(1/5)))","A",1
491,1,98,222,0.0874352,"\int \frac{1}{\left(a^2+2 a b \sqrt[5]{x}+b^2 x^{2/5}\right)^{5/2}} \, dx","Integrate[(a^2 + 2*a*b*x^(1/5) + b^2*x^(2/5))^(-5/2),x]","\frac{5 a \left(25 a^3+88 a^2 b \sqrt[5]{x}+108 a b^2 x^{2/5}+48 b^3 x^{3/5}\right)+60 \left(a+b \sqrt[5]{x}\right)^4 \log \left(a+b \sqrt[5]{x}\right)}{12 b^5 \left(a+b \sqrt[5]{x}\right)^3 \sqrt{\left(a+b \sqrt[5]{x}\right)^2}}","-\frac{15 a^2}{b^5 \left(a+b \sqrt[5]{x}\right) \sqrt{a^2+2 a b \sqrt[5]{x}+b^2 x^{2/5}}}+\frac{20 a}{b^5 \sqrt{a^2+2 a b \sqrt[5]{x}+b^2 x^{2/5}}}+\frac{5 \left(a+b \sqrt[5]{x}\right) \log \left(a+b \sqrt[5]{x}\right)}{b^5 \sqrt{a^2+2 a b \sqrt[5]{x}+b^2 x^{2/5}}}-\frac{5 a^4}{4 b^5 \left(a+b \sqrt[5]{x}\right)^3 \sqrt{a^2+2 a b \sqrt[5]{x}+b^2 x^{2/5}}}+\frac{20 a^3}{3 b^5 \left(a+b \sqrt[5]{x}\right)^2 \sqrt{a^2+2 a b \sqrt[5]{x}+b^2 x^{2/5}}}",1,"(5*a*(25*a^3 + 88*a^2*b*x^(1/5) + 108*a*b^2*x^(2/5) + 48*b^3*x^(3/5)) + 60*(a + b*x^(1/5))^4*Log[a + b*x^(1/5)])/(12*b^5*(a + b*x^(1/5))^3*Sqrt[(a + b*x^(1/5))^2])","A",1
492,1,124,391,0.0660746,"\int \left(a^2+\frac{b^2}{\sqrt[3]{x}}+\frac{2 a b}{\sqrt[6]{x}}\right)^{7/2} \, dx","Integrate[(a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6))^(7/2),x]","\frac{\sqrt{\frac{\left(a \sqrt[6]{x}+b\right)^2}{\sqrt[3]{x}}} \left(10 a^7 x^{7/6}+84 a^6 b x+315 a^5 b^2 x^{5/6}+700 a^4 b^3 x^{2/3}+1050 a^3 b^4 \sqrt{x}+1260 a^2 b^5 \sqrt[3]{x}+70 a b^6 \sqrt[6]{x} \log (x)-60 b^7\right)}{10 \left(a \sqrt[6]{x}+b\right)}","-\frac{6 b^7 \sqrt{a^2+\frac{2 a b}{\sqrt[6]{x}}+\frac{b^2}{\sqrt[3]{x}}}}{\sqrt[6]{x} \left(a+\frac{b}{\sqrt[6]{x}}\right)}+\frac{42 a b^6 \log \left(\sqrt[6]{x}\right) \sqrt{a^2+\frac{2 a b}{\sqrt[6]{x}}+\frac{b^2}{\sqrt[3]{x}}}}{a+\frac{b}{\sqrt[6]{x}}}+\frac{126 a^2 b^5 \sqrt[6]{x} \sqrt{a^2+\frac{2 a b}{\sqrt[6]{x}}+\frac{b^2}{\sqrt[3]{x}}}}{a+\frac{b}{\sqrt[6]{x}}}+\frac{a^7 x \sqrt{a^2+\frac{2 a b}{\sqrt[6]{x}}+\frac{b^2}{\sqrt[3]{x}}}}{a+\frac{b}{\sqrt[6]{x}}}+\frac{42 a^6 b x^{5/6} \sqrt{a^2+\frac{2 a b}{\sqrt[6]{x}}+\frac{b^2}{\sqrt[3]{x}}}}{5 \left(a+\frac{b}{\sqrt[6]{x}}\right)}+\frac{63 a^5 b^2 x^{2/3} \sqrt{a^2+\frac{2 a b}{\sqrt[6]{x}}+\frac{b^2}{\sqrt[3]{x}}}}{2 \left(a+\frac{b}{\sqrt[6]{x}}\right)}+\frac{70 a^4 b^3 \sqrt{x} \sqrt{a^2+\frac{2 a b}{\sqrt[6]{x}}+\frac{b^2}{\sqrt[3]{x}}}}{a+\frac{b}{\sqrt[6]{x}}}+\frac{105 a^3 b^4 \sqrt[3]{x} \sqrt{a^2+\frac{2 a b}{\sqrt[6]{x}}+\frac{b^2}{\sqrt[3]{x}}}}{a+\frac{b}{\sqrt[6]{x}}}",1,"(Sqrt[(b + a*x^(1/6))^2/x^(1/3)]*(-60*b^7 + 1260*a^2*b^5*x^(1/3) + 1050*a^3*b^4*Sqrt[x] + 700*a^4*b^3*x^(2/3) + 315*a^5*b^2*x^(5/6) + 84*a^6*b*x + 10*a^7*x^(7/6) + 70*a*b^6*x^(1/6)*Log[x]))/(10*(b + a*x^(1/6)))","A",1
493,1,38,46,0.0280877,"\int \frac{x^{-1+4 n}}{b x^n+c x^{2 n}} \, dx","Integrate[x^(-1 + 4*n)/(b*x^n + c*x^(2*n)),x]","\frac{2 b^2 \log \left(b+c x^n\right)+c x^n \left(c x^n-2 b\right)}{2 c^3 n}","\frac{b^2 \log \left(b+c x^n\right)}{c^3 n}-\frac{b x^n}{c^2 n}+\frac{x^{2 n}}{2 c n}",1,"(c*x^n*(-2*b + c*x^n) + 2*b^2*Log[b + c*x^n])/(2*c^3*n)","A",1
494,1,26,28,0.013586,"\int \frac{x^{-1+3 n}}{b x^n+c x^{2 n}} \, dx","Integrate[x^(-1 + 3*n)/(b*x^n + c*x^(2*n)),x]","\frac{\frac{x^n}{c}-\frac{b \log \left(b+c x^n\right)}{c^2}}{n}","\frac{x^n}{c n}-\frac{b \log \left(b+c x^n\right)}{c^2 n}",1,"(x^n/c - (b*Log[b + c*x^n])/c^2)/n","A",1
495,1,15,15,0.0032773,"\int \frac{x^{-1+2 n}}{b x^n+c x^{2 n}} \, dx","Integrate[x^(-1 + 2*n)/(b*x^n + c*x^(2*n)),x]","\frac{\log \left(b+c x^n\right)}{c n}","\frac{\log \left(b+c x^n\right)}{c n}",1,"Log[b + c*x^n]/(c*n)","A",1
496,1,22,23,0.0066028,"\int \frac{x^{-1+n}}{b x^n+c x^{2 n}} \, dx","Integrate[x^(-1 + n)/(b*x^n + c*x^(2*n)),x]","\frac{n \log (x)-\log \left(b+c x^n\right)}{b n}","\frac{\log (x)}{b}-\frac{\log \left(b+c x^n\right)}{b n}",1,"(n*Log[x] - Log[b + c*x^n])/(b*n)","A",1
497,1,49,57,0.0659977,"\int \frac{x^{-1-n}}{b x^n+c x^{2 n}} \, dx","Integrate[x^(-1 - n)/(b*x^n + c*x^(2*n)),x]","\frac{-2 c^2 \log \left(b+c x^n\right)+b x^{-2 n} \left(2 c x^n-b\right)+2 c^2 n \log (x)}{2 b^3 n}","-\frac{c^2 \log \left(b+c x^n\right)}{b^3 n}+\frac{c^2 \log (x)}{b^3}+\frac{c x^{-n}}{b^2 n}-\frac{x^{-2 n}}{2 b n}",1,"((b*(-b + 2*c*x^n))/x^(2*n) + 2*c^2*n*Log[x] - 2*c^2*Log[b + c*x^n])/(2*b^3*n)","A",1
498,1,62,76,0.0784023,"\int \frac{x^{-1-2 n}}{b x^n+c x^{2 n}} \, dx","Integrate[x^(-1 - 2*n)/(b*x^n + c*x^(2*n)),x]","-\frac{b x^{-3 n} \left(2 b^2-3 b c x^n+6 c^2 x^{2 n}\right)-6 c^3 \log \left(b+c x^n\right)+6 c^3 n \log (x)}{6 b^4 n}","\frac{c^3 \log \left(b+c x^n\right)}{b^4 n}-\frac{c^3 \log (x)}{b^4}-\frac{c^2 x^{-n}}{b^3 n}+\frac{c x^{-2 n}}{2 b^2 n}-\frac{x^{-3 n}}{3 b n}",1,"-1/6*((b*(2*b^2 - 3*b*c*x^n + 6*c^2*x^(2*n)))/x^(3*n) + 6*c^3*n*Log[x] - 6*c^3*Log[b + c*x^n])/(b^4*n)","A",1
499,1,75,93,0.0945184,"\int \frac{x^{-1-3 n}}{b x^n+c x^{2 n}} \, dx","Integrate[x^(-1 - 3*n)/(b*x^n + c*x^(2*n)),x]","-\frac{b x^{-4 n} \left(3 b^3-4 b^2 c x^n+6 b c^2 x^{2 n}-12 c^3 x^{3 n}\right)+12 c^4 \log \left(b+c x^n\right)-12 c^4 n \log (x)}{12 b^5 n}","-\frac{c^4 \log \left(b+c x^n\right)}{b^5 n}+\frac{c^4 \log (x)}{b^5}+\frac{c^3 x^{-n}}{b^4 n}-\frac{c^2 x^{-2 n}}{2 b^3 n}+\frac{c x^{-3 n}}{3 b^2 n}-\frac{x^{-4 n}}{4 b n}",1,"-1/12*((b*(3*b^3 - 4*b^2*c*x^n + 6*b*c^2*x^(2*n) - 12*c^3*x^(3*n)))/x^(4*n) - 12*c^4*n*Log[x] + 12*c^4*Log[b + c*x^n])/(b^5*n)","A",1
500,1,34,236,0.0087422,"\int \frac{x^{-1+\frac{n}{4}}}{b x^n+c x^{2 n}} \, dx","Integrate[x^(-1 + n/4)/(b*x^n + c*x^(2*n)),x]","-\frac{4 x^{-3 n/4} \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\frac{c x^n}{b}\right)}{3 b n}","\frac{c^{3/4} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} x^{n/4}+\sqrt{b}+\sqrt{c} x^{n/2}\right)}{\sqrt{2} b^{7/4} n}-\frac{c^{3/4} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} x^{n/4}+\sqrt{b}+\sqrt{c} x^{n/2}\right)}{\sqrt{2} b^{7/4} n}+\frac{\sqrt{2} c^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x^{n/4}}{\sqrt[4]{b}}\right)}{b^{7/4} n}-\frac{\sqrt{2} c^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x^{n/4}}{\sqrt[4]{b}}+1\right)}{b^{7/4} n}-\frac{4 x^{-3 n/4}}{3 b n}",1,"(-4*Hypergeometric2F1[-3/4, 1, 1/4, -((c*x^n)/b)])/(3*b*n*x^((3*n)/4))","C",1
501,1,34,160,0.0087902,"\int \frac{x^{-1+\frac{n}{3}}}{b x^n+c x^{2 n}} \, dx","Integrate[x^(-1 + n/3)/(b*x^n + c*x^(2*n)),x]","-\frac{3 x^{-2 n/3} \, _2F_1\left(-\frac{2}{3},1;\frac{1}{3};-\frac{c x^n}{b}\right)}{2 b n}","-\frac{c^{2/3} \log \left(\sqrt[3]{b}+\sqrt[3]{c} x^{n/3}\right)}{b^{5/3} n}+\frac{c^{2/3} \log \left(b^{2/3}-\sqrt[3]{b} \sqrt[3]{c} x^{n/3}+c^{2/3} x^{2 n/3}\right)}{2 b^{5/3} n}+\frac{\sqrt{3} c^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{b}-2 \sqrt[3]{c} x^{n/3}}{\sqrt{3} \sqrt[3]{b}}\right)}{b^{5/3} n}-\frac{3 x^{-2 n/3}}{2 b n}",1,"(-3*Hypergeometric2F1[-2/3, 1, 1/3, -((c*x^n)/b)])/(2*b*n*x^((2*n)/3))","C",1
502,1,32,50,0.0072871,"\int \frac{x^{-1+\frac{n}{2}}}{b x^n+c x^{2 n}} \, dx","Integrate[x^(-1 + n/2)/(b*x^n + c*x^(2*n)),x]","-\frac{2 x^{-n/2} \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\frac{c x^n}{b}\right)}{b n}","\frac{2 \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{b} x^{-n/2}}{\sqrt{c}}\right)}{b^{3/2} n}-\frac{2 x^{-n/2}}{b n}",1,"(-2*Hypergeometric2F1[-1/2, 1, 1/2, -((c*x^n)/b)])/(b*n*x^(n/2))","C",1
503,1,34,68,0.0078936,"\int \frac{x^{-1-\frac{n}{2}}}{b x^n+c x^{2 n}} \, dx","Integrate[x^(-1 - n/2)/(b*x^n + c*x^(2*n)),x]","-\frac{2 x^{-3 n/2} \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\frac{c x^n}{b}\right)}{3 b n}","-\frac{2 c^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x^{-n/2}}{\sqrt{c}}\right)}{b^{5/2} n}+\frac{2 c x^{-n/2}}{b^2 n}-\frac{2 x^{-3 n/2}}{3 b n}",1,"(-2*Hypergeometric2F1[-3/2, 1, -1/2, -((c*x^n)/b)])/(3*b*n*x^((3*n)/2))","C",1
504,1,34,176,0.0078621,"\int \frac{x^{-1-\frac{n}{3}}}{b x^n+c x^{2 n}} \, dx","Integrate[x^(-1 - n/3)/(b*x^n + c*x^(2*n)),x]","-\frac{3 x^{-4 n/3} \, _2F_1\left(-\frac{4}{3},1;-\frac{1}{3};-\frac{c x^n}{b}\right)}{4 b n}","-\frac{c^{4/3} \log \left(\sqrt[3]{b} x^{-n/3}+\sqrt[3]{c}\right)}{b^{7/3} n}+\frac{c^{4/3} \log \left(b^{2/3} x^{-2 n/3}-\sqrt[3]{b} \sqrt[3]{c} x^{-n/3}+c^{2/3}\right)}{2 b^{7/3} n}+\frac{\sqrt{3} c^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{c}-2 \sqrt[3]{b} x^{-n/3}}{\sqrt{3} \sqrt[3]{c}}\right)}{b^{7/3} n}+\frac{3 c x^{-n/3}}{b^2 n}-\frac{3 x^{-4 n/3}}{4 b n}",1,"(-3*Hypergeometric2F1[-4/3, 1, -1/3, -((c*x^n)/b)])/(4*b*n*x^((4*n)/3))","C",1
505,1,34,252,0.0080729,"\int \frac{x^{-1-\frac{n}{4}}}{b x^n+c x^{2 n}} \, dx","Integrate[x^(-1 - n/4)/(b*x^n + c*x^(2*n)),x]","-\frac{4 x^{-5 n/4} \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\frac{c x^n}{b}\right)}{5 b n}","\frac{c^{5/4} \log \left(-\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} x^{-n/4}+\sqrt{b} x^{-n/2}+\sqrt{c}\right)}{\sqrt{2} b^{9/4} n}-\frac{c^{5/4} \log \left(\sqrt{2} \sqrt[4]{b} \sqrt[4]{c} x^{-n/4}+\sqrt{b} x^{-n/2}+\sqrt{c}\right)}{\sqrt{2} b^{9/4} n}+\frac{\sqrt{2} c^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} x^{-n/4}}{\sqrt[4]{c}}\right)}{b^{9/4} n}-\frac{\sqrt{2} c^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} x^{-n/4}}{\sqrt[4]{c}}+1\right)}{b^{9/4} n}+\frac{4 c x^{-n/4}}{b^2 n}-\frac{4 x^{-5 n/4}}{5 b n}",1,"(-4*Hypergeometric2F1[-5/4, 1, -1/4, -((c*x^n)/b)])/(5*b*n*x^((5*n)/4))","C",1
506,1,38,37,0.0170455,"\int x^{-1-n (-1+p)} \left(b x^n+c x^{2 n}\right)^p \, dx","Integrate[x^(-1 - n*(-1 + p))*(b*x^n + c*x^(2*n))^p,x]","\frac{x^{-n p} \left(b+c x^n\right) \left(x^n \left(b+c x^n\right)\right)^p}{c n (p+1)}","\frac{x^{-n (p+1)} \left(b x^n+c x^{2 n}\right)^{p+1}}{c n (p+1)}",1,"((b + c*x^n)*(x^n*(b + c*x^n))^p)/(c*n*(1 + p)*x^(n*p))","A",1
507,1,43,38,0.0194107,"\int x^{-1-n (1+2 p)} \left(b x^n+c x^{2 n}\right)^p \, dx","Integrate[x^(-1 - n*(1 + 2*p))*(b*x^n + c*x^(2*n))^p,x]","-\frac{x^{-n (2 p+1)} \left(b+c x^n\right) \left(x^n \left(b+c x^n\right)\right)^p}{b n (p+1)}","-\frac{x^{-2 n (p+1)} \left(b x^n+c x^{2 n}\right)^{p+1}}{b n (p+1)}",1,"-(((b + c*x^n)*(x^n*(b + c*x^n))^p)/(b*n*(1 + p)*x^(n*(1 + 2*p))))","A",1
508,1,40,112,0.0561999,"\int x^{-1+2 n} \left(a^2+2 a b x^n+b^2 x^{2 n}\right)^{5/2} \, dx","Integrate[x^(-1 + 2*n)*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(5/2),x]","-\frac{\left(a-6 b x^n\right) \left(a+b x^n\right)^5 \sqrt{\left(a+b x^n\right)^2}}{42 b^2 n}","\frac{\left(a+b x^n\right)^7 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{7 n \left(a b^2+b^3 x^n\right)}-\frac{a \left(a+b x^n\right)^6 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{6 n \left(a b^2+b^3 x^n\right)}",1,"-1/42*((a - 6*b*x^n)*(a + b*x^n)^5*Sqrt[(a + b*x^n)^2])/(b^2*n)","A",1
509,1,40,112,0.0373351,"\int x^{-1+2 n} \left(a^2+2 a b x^n+b^2 x^{2 n}\right)^{3/2} \, dx","Integrate[x^(-1 + 2*n)*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2),x]","-\frac{\left(a-4 b x^n\right) \left(a+b x^n\right)^3 \sqrt{\left(a+b x^n\right)^2}}{20 b^2 n}","\frac{\left(a+b x^n\right)^5 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{5 n \left(a b^2+b^3 x^n\right)}-\frac{a \left(a+b x^n\right)^4 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{4 n \left(a b^2+b^3 x^n\right)}",1,"-1/20*((a - 4*b*x^n)*(a + b*x^n)^3*Sqrt[(a + b*x^n)^2])/(b^2*n)","A",1
510,1,44,99,0.0230069,"\int x^{-1+2 n} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \, dx","Integrate[x^(-1 + 2*n)*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)],x]","\frac{x^{2 n} \sqrt{\left(a+b x^n\right)^2} \left(3 a+2 b x^n\right)}{6 n \left(a+b x^n\right)}","\frac{a x^{2 n} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 n \left(a+b x^n\right)}+\frac{b^2 x^{3 n} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{3 n \left(a b+b^2 x^n\right)}",1,"(x^(2*n)*Sqrt[(a + b*x^n)^2]*(3*a + 2*b*x^n))/(6*n*(a + b*x^n))","A",1
511,1,46,90,0.0319319,"\int \frac{x^{-1+2 n}}{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \, dx","Integrate[x^(-1 + 2*n)/Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)],x]","\frac{\left(a+b x^n\right) \left(\frac{x^n}{b}-\frac{a \log \left(a+b x^n\right)}{b^2}\right)}{n \sqrt{\left(a+b x^n\right)^2}}","\frac{x^n \left(a+b x^n\right)}{b n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}-\frac{a \left(a+b x^n\right) \log \left(a+b x^n\right)}{b^2 n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}",1,"((a + b*x^n)*(x^n/b - (a*Log[a + b*x^n])/b^2))/(n*Sqrt[(a + b*x^n)^2])","A",1
512,1,35,48,0.0157535,"\int \frac{x^{-1+2 n}}{\left(a^2+2 a b x^n+b^2 x^{2 n}\right)^{3/2}} \, dx","Integrate[x^(-1 + 2*n)/(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2),x]","\frac{x^{2 n} \left(a+b x^n\right)}{2 a n \left(\left(a+b x^n\right)^2\right)^{3/2}}","\frac{x^{2 n}}{2 a n \left(a+b x^n\right) \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}",1,"(x^(2*n)*(a + b*x^n))/(2*a*n*((a + b*x^n)^2)^(3/2))","A",1
513,1,40,88,0.0323833,"\int \frac{x^{-1+2 n}}{\left(a^2+2 a b x^n+b^2 x^{2 n}\right)^{5/2}} \, dx","Integrate[x^(-1 + 2*n)/(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(5/2),x]","-\frac{a+4 b x^n}{12 b^2 n \left(a+b x^n\right)^3 \sqrt{\left(a+b x^n\right)^2}}","\frac{a}{4 b^2 n \left(a+b x^n\right)^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}-\frac{1}{3 b^2 n \left(a+b x^n\right)^2 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}",1,"-1/12*(a + 4*b*x^n)/(b^2*n*(a + b*x^n)^3*Sqrt[(a + b*x^n)^2])","A",1
514,1,40,88,0.0337863,"\int \frac{x^{-1+2 n}}{\left(a^2+2 a b x^n+b^2 x^{2 n}\right)^{7/2}} \, dx","Integrate[x^(-1 + 2*n)/(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(7/2),x]","-\frac{a+6 b x^n}{30 b^2 n \left(a+b x^n\right)^5 \sqrt{\left(a+b x^n\right)^2}}","\frac{a}{6 b^2 n \left(a+b x^n\right)^5 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}-\frac{1}{5 b^2 n \left(a+b x^n\right)^4 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}",1,"-1/30*(a + 6*b*x^n)/(b^2*n*(a + b*x^n)^5*Sqrt[(a + b*x^n)^2])","A",1
515,1,55,108,0.0334261,"\int (d x)^m \sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \, dx","Integrate[(d*x)^m*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)],x]","\frac{x (d x)^m \sqrt{\left(a+b x^n\right)^2} \left(a (m+n+1)+b (m+1) x^n\right)}{(m+1) (m+n+1) \left(a+b x^n\right)}","\frac{b^2 x^{n+1} (d x)^m \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(m+n+1) \left(a b+b^2 x^n\right)}+\frac{a (d x)^{m+1} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{d (m+1) \left(a+b x^n\right)}",1,"(x*(d*x)^m*Sqrt[(a + b*x^n)^2]*(a*(1 + m + n) + b*(1 + m)*x^n))/((1 + m)*(1 + m + n)*(a + b*x^n))","A",1
516,1,46,93,0.0224983,"\int x^2 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \, dx","Integrate[x^2*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)],x]","\frac{x^3 \sqrt{\left(a+b x^n\right)^2} \left(a (n+3)+3 b x^n\right)}{3 (n+3) \left(a+b x^n\right)}","\frac{b^2 x^{n+3} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(n+3) \left(a b+b^2 x^n\right)}+\frac{a x^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{3 \left(a+b x^n\right)}",1,"(x^3*Sqrt[(a + b*x^n)^2]*(a*(3 + n) + 3*b*x^n))/(3*(3 + n)*(a + b*x^n))","A",1
517,1,46,93,0.0212257,"\int x \sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \, dx","Integrate[x*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)],x]","\frac{x^2 \sqrt{\left(a+b x^n\right)^2} \left(a (n+2)+2 b x^n\right)}{2 (n+2) \left(a+b x^n\right)}","\frac{b^2 x^{n+2} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(n+2) \left(a b+b^2 x^n\right)}+\frac{a x^2 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 \left(a+b x^n\right)}",1,"(x^2*Sqrt[(a + b*x^n)^2]*(a*(2 + n) + 2*b*x^n))/(2*(2 + n)*(a + b*x^n))","A",1
518,1,39,88,0.0137401,"\int \sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)],x]","\frac{x \sqrt{\left(a+b x^n\right)^2} \left(a n+a+b x^n\right)}{(n+1) \left(a+b x^n\right)}","\frac{b^2 x^{n+1} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(n+1) \left(a b+b^2 x^n\right)}+\frac{a x \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{a+b x^n}",1,"(x*Sqrt[(a + b*x^n)^2]*(a + a*n + b*x^n))/((1 + n)*(a + b*x^n))","A",1
519,1,37,85,0.0154105,"\int \frac{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{x} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)]/x,x]","\frac{\sqrt{\left(a+b x^n\right)^2} \left(a n \log (x)+b x^n\right)}{n \left(a+b x^n\right)}","\frac{b^2 x^n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{n \left(a b+b^2 x^n\right)}+\frac{a \log (x) \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{a+b x^n}",1,"(Sqrt[(a + b*x^n)^2]*(b*x^n + a*n*Log[x]))/(n*(a + b*x^n))","A",1
520,1,42,94,0.0245412,"\int \frac{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{x^2} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)]/x^2,x]","\frac{\sqrt{\left(a+b x^n\right)^2} \left(-a n+a+b x^n\right)}{(n-1) x \left(a+b x^n\right)}","-\frac{b^2 x^{n-1} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(1-n) \left(a b+b^2 x^n\right)}-\frac{a \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{x \left(a+b x^n\right)}",1,"(Sqrt[(a + b*x^n)^2]*(a - a*n + b*x^n))/((-1 + n)*x*(a + b*x^n))","A",1
521,1,47,96,0.0255203,"\int \frac{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{x^3} \, dx","Integrate[Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)]/x^3,x]","\frac{\sqrt{\left(a+b x^n\right)^2} \left(2 b x^n-a (n-2)\right)}{2 (n-2) x^2 \left(a+b x^n\right)}","-\frac{b^2 x^{n-2} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(2-n) \left(a b+b^2 x^n\right)}-\frac{a \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 x^2 \left(a+b x^n\right)}",1,"(Sqrt[(a + b*x^n)^2]*(-(a*(-2 + n)) + 2*b*x^n))/(2*(-2 + n)*x^2*(a + b*x^n))","A",1
522,1,90,238,0.1073129,"\int (d x)^m \left(a^2+2 a b x^n+b^2 x^{2 n}\right)^{3/2} \, dx","Integrate[(d*x)^m*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2),x]","\frac{x (d x)^m \left(\left(a+b x^n\right)^2\right)^{3/2} \left(\frac{a^3}{m+1}+\frac{3 a^2 b x^n}{m+n+1}+\frac{3 a b^2 x^{2 n}}{m+2 n+1}+\frac{b^3 x^{3 n}}{m+3 n+1}\right)}{\left(a+b x^n\right)^3}","\frac{3 a^2 b^2 x^{n+1} (d x)^m \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(m+n+1) \left(a b+b^2 x^n\right)}+\frac{b^4 x^{3 n+1} (d x)^m \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(m+3 n+1) \left(a b+b^2 x^n\right)}+\frac{3 a b^3 x^{2 n+1} (d x)^m \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(m+2 n+1) \left(a b+b^2 x^n\right)}+\frac{a^3 (d x)^{m+1} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{d (m+1) \left(a+b x^n\right)}",1,"(x*(d*x)^m*((a + b*x^n)^2)^(3/2)*(a^3/(1 + m) + (3*a^2*b*x^n)/(1 + m + n) + (3*a*b^2*x^(2*n))/(1 + m + 2*n) + (b^3*x^(3*n))/(1 + m + 3*n)))/(a + b*x^n)^3","A",1
523,1,123,212,0.0783595,"\int x^2 \left(a^2+2 a b x^n+b^2 x^{2 n}\right)^{3/2} \, dx","Integrate[x^2*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2),x]","\frac{x^3 \sqrt{\left(a+b x^n\right)^2} \left(a^3 \left(2 n^3+11 n^2+18 n+9\right)+9 a^2 b \left(2 n^2+5 n+3\right) x^n+9 a b^2 \left(n^2+4 n+3\right) x^{2 n}+b^3 \left(2 n^2+9 n+9\right) x^{3 n}\right)}{3 (n+1) (n+3) (2 n+3) \left(a+b x^n\right)}","\frac{3 a^2 b^2 x^{n+3} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(n+3) \left(a b+b^2 x^n\right)}+\frac{b^4 x^{3 (n+1)} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{3 (n+1) \left(a b+b^2 x^n\right)}+\frac{3 a b^3 x^{2 n+3} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(2 n+3) \left(a b+b^2 x^n\right)}+\frac{a^3 x^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{3 \left(a+b x^n\right)}",1,"(x^3*Sqrt[(a + b*x^n)^2]*(a^3*(9 + 18*n + 11*n^2 + 2*n^3) + 9*a^2*b*(3 + 5*n + 2*n^2)*x^n + 9*a*b^2*(3 + 4*n + n^2)*x^(2*n) + b^3*(9 + 9*n + 2*n^2)*x^(3*n)))/(3*(1 + n)*(3 + n)*(3 + 2*n)*(a + b*x^n))","A",1
524,1,124,211,0.0697624,"\int x \left(a^2+2 a b x^n+b^2 x^{2 n}\right)^{3/2} \, dx","Integrate[x*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2),x]","\frac{x^2 \sqrt{\left(a+b x^n\right)^2} \left(a^3 \left(3 n^3+11 n^2+12 n+4\right)+6 a^2 b \left(3 n^2+5 n+2\right) x^n+3 a b^2 \left(3 n^2+8 n+4\right) x^{2 n}+2 b^3 \left(n^2+3 n+2\right) x^{3 n}\right)}{2 (n+1) (n+2) (3 n+2) \left(a+b x^n\right)}","\frac{3 a^2 b^2 x^{n+2} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(n+2) \left(a b+b^2 x^n\right)}+\frac{b^4 x^{3 n+2} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(3 n+2) \left(a b+b^2 x^n\right)}+\frac{3 a b^3 x^{2 (n+1)} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 (n+1) \left(a b+b^2 x^n\right)}+\frac{a^3 x^2 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 \left(a+b x^n\right)}",1,"(x^2*Sqrt[(a + b*x^n)^2]*(a^3*(4 + 12*n + 11*n^2 + 3*n^3) + 6*a^2*b*(2 + 5*n + 3*n^2)*x^n + 3*a*b^2*(4 + 8*n + 3*n^2)*x^(2*n) + 2*b^3*(2 + 3*n + n^2)*x^(3*n)))/(2*(1 + n)*(2 + n)*(2 + 3*n)*(a + b*x^n))","A",1
525,1,122,206,0.0754971,"\int \left(a^2+2 a b x^n+b^2 x^{2 n}\right)^{3/2} \, dx","Integrate[(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2),x]","\frac{x \sqrt{\left(a+b x^n\right)^2} \left(a^3 \left(6 n^3+11 n^2+6 n+1\right)+3 a^2 b \left(6 n^2+5 n+1\right) x^n+3 a b^2 \left(3 n^2+4 n+1\right) x^{2 n}+b^3 \left(2 n^2+3 n+1\right) x^{3 n}\right)}{(n+1) (2 n+1) (3 n+1) \left(a+b x^n\right)}","\frac{b^6 x^{3 n+1} \left(a^2+2 a b x^n+b^2 x^{2 n}\right)^{3/2}}{(3 n+1) \left(a b+b^2 x^n\right)^3}+\frac{3 a b^5 x^{2 n+1} \left(a^2+2 a b x^n+b^2 x^{2 n}\right)^{3/2}}{(2 n+1) \left(a b+b^2 x^n\right)^3}+\frac{3 a^2 b^4 x^{n+1} \left(a^2+2 a b x^n+b^2 x^{2 n}\right)^{3/2}}{(n+1) \left(a b+b^2 x^n\right)^3}+\frac{a^3 x \left(a^2+2 a b x^n+b^2 x^{2 n}\right)^{3/2}}{\left(a+b x^n\right)^3}",1,"(x*Sqrt[(a + b*x^n)^2]*(a^3*(1 + 6*n + 11*n^2 + 6*n^3) + 3*a^2*b*(1 + 5*n + 6*n^2)*x^n + 3*a*b^2*(1 + 4*n + 3*n^2)*x^(2*n) + b^3*(1 + 3*n + 2*n^2)*x^(3*n)))/((1 + n)*(1 + 2*n)*(1 + 3*n)*(a + b*x^n))","A",1
526,1,68,196,0.0371045,"\int \frac{\left(a^2+2 a b x^n+b^2 x^{2 n}\right)^{3/2}}{x} \, dx","Integrate[(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2)/x,x]","\frac{\left(\left(a+b x^n\right)^2\right)^{3/2} \left(a^3 n \log (x)+3 a^2 b x^n+\frac{3}{2} a b^2 x^{2 n}+\frac{1}{3} b^3 x^{3 n}\right)}{n \left(a+b x^n\right)^3}","\frac{3 a^2 b^2 x^n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{n \left(a b+b^2 x^n\right)}+\frac{b^4 x^{3 n} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{3 n \left(a b+b^2 x^n\right)}+\frac{3 a b^3 x^{2 n} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 n \left(a b+b^2 x^n\right)}+\frac{a^3 \log (x) \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{a+b x^n}",1,"(((a + b*x^n)^2)^(3/2)*(3*a^2*b*x^n + (3*a*b^2*x^(2*n))/2 + (b^3*x^(3*n))/3 + a^3*n*Log[x]))/(n*(a + b*x^n)^3)","A",1
527,1,124,212,0.0910082,"\int \frac{\left(a^2+2 a b x^n+b^2 x^{2 n}\right)^{3/2}}{x^2} \, dx","Integrate[(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2)/x^2,x]","\frac{\sqrt{\left(a+b x^n\right)^2} \left(a^3 \left(-6 n^3+11 n^2-6 n+1\right)+3 a^2 b \left(6 n^2-5 n+1\right) x^n+3 a b^2 \left(3 n^2-4 n+1\right) x^{2 n}+b^3 \left(2 n^2-3 n+1\right) x^{3 n}\right)}{(n-1) (2 n-1) (3 n-1) x \left(a+b x^n\right)}","-\frac{3 a^2 b^2 x^{n-1} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(1-n) \left(a b+b^2 x^n\right)}-\frac{b^4 x^{3 n-1} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(1-3 n) \left(a b+b^2 x^n\right)}-\frac{3 a b^3 x^{2 n-1} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(1-2 n) \left(a b+b^2 x^n\right)}-\frac{a^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{x \left(a+b x^n\right)}",1,"(Sqrt[(a + b*x^n)^2]*(a^3*(1 - 6*n + 11*n^2 - 6*n^3) + 3*a^2*b*(1 - 5*n + 6*n^2)*x^n + 3*a*b^2*(1 - 4*n + 3*n^2)*x^(2*n) + b^3*(1 - 3*n + 2*n^2)*x^(3*n)))/((-1 + n)*(-1 + 2*n)*(-1 + 3*n)*x*(a + b*x^n))","A",1
528,1,124,218,0.0871966,"\int \frac{\left(a^2+2 a b x^n+b^2 x^{2 n}\right)^{3/2}}{x^3} \, dx","Integrate[(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2)/x^3,x]","\frac{\sqrt{\left(a+b x^n\right)^2} \left(a^3 \left(-3 n^3+11 n^2-12 n+4\right)+6 a^2 b \left(3 n^2-5 n+2\right) x^n+3 a b^2 \left(3 n^2-8 n+4\right) x^{2 n}+2 b^3 \left(n^2-3 n+2\right) x^{3 n}\right)}{2 (n-2) (n-1) (3 n-2) x^2 \left(a+b x^n\right)}","-\frac{3 a^2 b^2 x^{n-2} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(2-n) \left(a b+b^2 x^n\right)}-\frac{b^4 x^{3 n-2} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(2-3 n) \left(a b+b^2 x^n\right)}-\frac{3 a b^3 x^{-2 (1-n)} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 (1-n) \left(a b+b^2 x^n\right)}-\frac{a^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 x^2 \left(a+b x^n\right)}",1,"(Sqrt[(a + b*x^n)^2]*(a^3*(4 - 12*n + 11*n^2 - 3*n^3) + 6*a^2*b*(2 - 5*n + 3*n^2)*x^n + 3*a*b^2*(4 - 8*n + 3*n^2)*x^(2*n) + 2*b^3*(2 - 3*n + n^2)*x^(3*n)))/(2*(-2 + n)*(-1 + n)*(-2 + 3*n)*x^2*(a + b*x^n))","A",1
529,1,62,76,0.0226174,"\int \frac{(d x)^m}{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \, dx","Integrate[(d*x)^m/Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)],x]","\frac{x (d x)^m \left(a+b x^n\right) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+1}{n}+1;-\frac{b x^n}{a}\right)}{a (m+1) \sqrt{\left(a+b x^n\right)^2}}","\frac{(d x)^{m+1} \left(a+b x^n\right) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a d (m+1) \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}",1,"(x*(d*x)^m*(a + b*x^n)*Hypergeometric2F1[1, (1 + m)/n, 1 + (1 + m)/n, -((b*x^n)/a)])/(a*(1 + m)*Sqrt[(a + b*x^n)^2])","A",1
530,1,53,64,0.0143224,"\int \frac{x^2}{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \, dx","Integrate[x^2/Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)],x]","\frac{x^3 \left(a+b x^n\right) \, _2F_1\left(1,\frac{3}{n};1+\frac{3}{n};-\frac{b x^n}{a}\right)}{3 a \sqrt{\left(a+b x^n\right)^2}}","\frac{x^3 \left(a+b x^n\right) \, _2F_1\left(1,\frac{3}{n};\frac{n+3}{n};-\frac{b x^n}{a}\right)}{3 a \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}",1,"(x^3*(a + b*x^n)*Hypergeometric2F1[1, 3/n, 1 + 3/n, -((b*x^n)/a)])/(3*a*Sqrt[(a + b*x^n)^2])","A",1
531,1,53,64,0.0137537,"\int \frac{x}{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \, dx","Integrate[x/Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)],x]","\frac{x^2 \left(a+b x^n\right) \, _2F_1\left(1,\frac{2}{n};1+\frac{2}{n};-\frac{b x^n}{a}\right)}{2 a \sqrt{\left(a+b x^n\right)^2}}","\frac{x^2 \left(a+b x^n\right) \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right)}{2 a \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}",1,"(x^2*(a + b*x^n)*Hypergeometric2F1[1, 2/n, 1 + 2/n, -((b*x^n)/a)])/(2*a*Sqrt[(a + b*x^n)^2])","A",1
532,1,44,55,0.011052,"\int \frac{1}{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \, dx","Integrate[1/Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)],x]","\frac{x \left(a+b x^n\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)}{a \sqrt{\left(a+b x^n\right)^2}}","\frac{x \left(a+b x^n\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)}{a \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}",1,"(x*(a + b*x^n)*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/(a*Sqrt[(a + b*x^n)^2])","A",1
533,1,42,85,0.0148626,"\int \frac{1}{x \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \, dx","Integrate[1/(x*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)]),x]","\frac{\left(a+b x^n\right) \left(n \log (x)-\log \left(a+b x^n\right)\right)}{a n \sqrt{\left(a+b x^n\right)^2}}","\frac{\log (x) \left(a+b x^n\right)}{a \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}-\frac{\left(a+b x^n\right) \log \left(a+b x^n\right)}{a n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}",1,"((a + b*x^n)*(n*Log[x] - Log[a + b*x^n]))/(a*n*Sqrt[(a + b*x^n)^2])","A",1
534,1,51,65,0.0139587,"\int \frac{1}{x^2 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \, dx","Integrate[1/(x^2*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)]),x]","-\frac{\left(a+b x^n\right) \, _2F_1\left(1,-\frac{1}{n};1-\frac{1}{n};-\frac{b x^n}{a}\right)}{a x \sqrt{\left(a+b x^n\right)^2}}","-\frac{\left(a+b x^n\right) \, _2F_1\left(1,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right)}{a x \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}",1,"-(((a + b*x^n)*Hypergeometric2F1[1, -n^(-1), 1 - n^(-1), -((b*x^n)/a)])/(a*x*Sqrt[(a + b*x^n)^2]))","A",1
535,1,53,67,0.0135748,"\int \frac{1}{x^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \, dx","Integrate[1/(x^3*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)]),x]","-\frac{\left(a+b x^n\right) \, _2F_1\left(1,-\frac{2}{n};1-\frac{2}{n};-\frac{b x^n}{a}\right)}{2 a x^2 \sqrt{\left(a+b x^n\right)^2}}","-\frac{\left(a+b x^n\right) \, _2F_1\left(1,-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right)}{2 a x^2 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}",1,"-1/2*((a + b*x^n)*Hypergeometric2F1[1, -2/n, 1 - 2/n, -((b*x^n)/a)])/(a*x^2*Sqrt[(a + b*x^n)^2])","A",1
536,1,61,76,0.0216747,"\int \frac{(d x)^m}{\left(a^2+2 a b x^n+b^2 x^{2 n}\right)^{3/2}} \, dx","Integrate[(d*x)^m/(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2),x]","\frac{x (d x)^m \left(a+b x^n\right) \, _2F_1\left(3,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a^3 (m+1) \sqrt{\left(a+b x^n\right)^2}}","\frac{(d x)^{m+1} \left(a+b x^n\right) \, _2F_1\left(3,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right)}{a^3 d (m+1) \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}",1,"(x*(d*x)^m*(a + b*x^n)*Hypergeometric2F1[3, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(a^3*(1 + m)*Sqrt[(a + b*x^n)^2])","A",1
537,1,55,64,0.0177162,"\int \frac{x^2}{\left(a^2+2 a b x^n+b^2 x^{2 n}\right)^{3/2}} \, dx","Integrate[x^2/(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2),x]","\frac{x^3 \left(a+b x^n\right)^3 \, _2F_1\left(3,\frac{3}{n};1+\frac{3}{n};-\frac{b x^n}{a}\right)}{3 a^3 \left(\left(a+b x^n\right)^2\right)^{3/2}}","\frac{x^3 \left(a+b x^n\right) \, _2F_1\left(3,\frac{3}{n};\frac{n+3}{n};-\frac{b x^n}{a}\right)}{3 a^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}",1,"(x^3*(a + b*x^n)^3*Hypergeometric2F1[3, 3/n, 1 + 3/n, -((b*x^n)/a)])/(3*a^3*((a + b*x^n)^2)^(3/2))","A",1
538,1,55,64,0.0171137,"\int \frac{x}{\left(a^2+2 a b x^n+b^2 x^{2 n}\right)^{3/2}} \, dx","Integrate[x/(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2),x]","\frac{x^2 \left(a+b x^n\right)^3 \, _2F_1\left(3,\frac{2}{n};1+\frac{2}{n};-\frac{b x^n}{a}\right)}{2 a^3 \left(\left(a+b x^n\right)^2\right)^{3/2}}","\frac{x^2 \left(a+b x^n\right) \, _2F_1\left(3,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right)}{2 a^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}",1,"(x^2*(a + b*x^n)^3*Hypergeometric2F1[3, 2/n, 1 + 2/n, -((b*x^n)/a)])/(2*a^3*((a + b*x^n)^2)^(3/2))","A",1
539,1,46,57,0.0118035,"\int \frac{1}{\left(a^2+2 a b x^n+b^2 x^{2 n}\right)^{3/2}} \, dx","Integrate[(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(-3/2),x]","\frac{x \left(a+b x^n\right)^3 \, _2F_1\left(3,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)}{a^3 \left(\left(a+b x^n\right)^2\right)^{3/2}}","\frac{x \left(a+b x^n\right)^3 \, _2F_1\left(3,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)}{a^3 \left(a^2+2 a b x^n+b^2 x^{2 n}\right)^{3/2}}",1,"(x*(a + b*x^n)^3*Hypergeometric2F1[3, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/(a^3*((a + b*x^n)^2)^(3/2))","A",1
540,1,76,159,0.0634925,"\int \frac{1}{x \left(a^2+2 a b x^n+b^2 x^{2 n}\right)^{3/2}} \, dx","Integrate[1/(x*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2)),x]","\frac{\left(a+b x^n\right)^3 \left(-\frac{\log \left(a+b x^n\right)}{a^3}+\frac{n \log (x)}{a^3}+\frac{1}{a^2 \left(a+b x^n\right)}+\frac{1}{2 a \left(a+b x^n\right)^2}\right)}{n \left(\left(a+b x^n\right)^2\right)^{3/2}}","\frac{1}{a^2 n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}+\frac{1}{2 a n \left(a+b x^n\right) \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}+\frac{\log (x) \left(a+b x^n\right)}{a^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}-\frac{\left(a+b x^n\right) \log \left(a+b x^n\right)}{a^3 n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}",1,"((a + b*x^n)^3*(1/(2*a*(a + b*x^n)^2) + 1/(a^2*(a + b*x^n)) + (n*Log[x])/a^3 - Log[a + b*x^n]/a^3))/(n*((a + b*x^n)^2)^(3/2))","A",1
541,1,53,65,0.015155,"\int \frac{1}{x^2 \left(a^2+2 a b x^n+b^2 x^{2 n}\right)^{3/2}} \, dx","Integrate[1/(x^2*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2)),x]","-\frac{\left(a+b x^n\right)^3 \, _2F_1\left(3,-\frac{1}{n};1-\frac{1}{n};-\frac{b x^n}{a}\right)}{a^3 x \left(\left(a+b x^n\right)^2\right)^{3/2}}","-\frac{\left(a+b x^n\right) \, _2F_1\left(3,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right)}{a^3 x \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}",1,"-(((a + b*x^n)^3*Hypergeometric2F1[3, -n^(-1), 1 - n^(-1), -((b*x^n)/a)])/(a^3*x*((a + b*x^n)^2)^(3/2)))","A",1
542,1,55,67,0.0155225,"\int \frac{1}{x^3 \left(a^2+2 a b x^n+b^2 x^{2 n}\right)^{3/2}} \, dx","Integrate[1/(x^3*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2)),x]","-\frac{\left(a+b x^n\right)^3 \, _2F_1\left(3,-\frac{2}{n};1-\frac{2}{n};-\frac{b x^n}{a}\right)}{2 a^3 x^2 \left(\left(a+b x^n\right)^2\right)^{3/2}}","-\frac{\left(a+b x^n\right) \, _2F_1\left(3,-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right)}{2 a^3 x^2 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}",1,"-1/2*((a + b*x^n)^3*Hypergeometric2F1[3, -2/n, 1 - 2/n, -((b*x^n)/a)])/(a^3*x^2*((a + b*x^n)^2)^(3/2))","A",1
543,1,58,52,0.0234469,"\int \left(a^2+b^2 x^{-\frac{2}{1+2 p}}+2 a b x^{-\frac{1}{1+2 p}}\right)^p \, dx","Integrate[(a^2 + b^2/x^(2/(1 + 2*p)) + (2*a*b)/x^(1 + 2*p)^(-1))^p,x]","\frac{x^{\frac{2 p}{2 p+1}} \left(a x^{\frac{1}{2 p+1}}+b\right) \left(x^{-\frac{2}{2 p+1}} \left(a x^{\frac{1}{2 p+1}}+b\right)^2\right)^p}{a}","\frac{x \left(a+b x^{\frac{1}{-2 p-1}}\right) \left(a^2+2 a b x^{\frac{1}{-2 p-1}}+b^2 x^{-\frac{2}{2 p+1}}\right)^p}{a}",1,"(x^((2*p)/(1 + 2*p))*(b + a*x^(1 + 2*p)^(-1))*((b + a*x^(1 + 2*p)^(-1))^2/x^(2/(1 + 2*p)))^p)/a","A",1
544,1,32,43,0.0621663,"\int \left(a^2+2 a b x^n+b^2 x^{2 n}\right)^{\frac{-1-n}{2 n}} \, dx","Integrate[(a^2 + 2*a*b*x^n + b^2*x^(2*n))^((-1 - n)/(2*n)),x]","\frac{x \left(a+b x^n\right) \left(\left(a+b x^n\right)^2\right)^{-\frac{n+1}{2 n}}}{a}","\frac{x \left(a+b x^n\right) \left(a^2+2 a b x^n+b^2 x^{2 n}\right)^{-\frac{n+1}{2 n}}}{a}",1,"(x*(a + b*x^n))/(a*((a + b*x^n)^2)^((1 + n)/(2*n)))","A",1
545,1,80,130,0.0619507,"\int \left(a^2+b^2 x^{-\frac{1}{1+p}}+2 a b x^{-\frac{1}{2 (1+p)}}\right)^p \, dx","Integrate[(a^2 + b^2/x^(1 + p)^(-1) + (2*a*b)/x^(1/(2*(1 + p))))^p,x]","\frac{x^{\frac{p}{p+1}} \left(a x^{\frac{1}{2 p+2}}+b\right) \left(x^{-\frac{1}{p+1}} \left(a x^{\frac{1}{2 p+2}}+b\right)^2\right)^p \left(a (2 p+1) x^{\frac{1}{2 p+2}}-b\right)}{a^2 (2 p+1)}","\frac{2 (p+1) x \left(a+b x^{-\frac{1}{2 (p+1)}}\right) \left(a^2+2 a b x^{-\frac{1}{2 (p+1)}}+b^2 x^{-\frac{1}{p+1}}\right)^p}{a (2 p+1)}-\frac{x \left(a+b x^{-\frac{1}{2 (p+1)}}\right)^2 \left(a^2+2 a b x^{-\frac{1}{2 (p+1)}}+b^2 x^{-\frac{1}{p+1}}\right)^p}{a^2 (2 p+1)}",1,"(x^(p/(1 + p))*(b + a*x^(2 + 2*p)^(-1))*((b + a*x^(2 + 2*p)^(-1))^2/x^(1 + p)^(-1))^p*(-b + a*(1 + 2*p)*x^(2 + 2*p)^(-1)))/(a^2*(1 + 2*p))","A",1
546,1,59,102,0.0446432,"\int \left(a^2+2 a b x^n+b^2 x^{2 n}\right)^{\frac{-1-2 n}{2 n}} \, dx","Integrate[(a^2 + 2*a*b*x^n + b^2*x^(2*n))^((-1 - 2*n)/(2*n)),x]","\frac{x \left(\left(a+b x^n\right)^2\right)^{\left.-\frac{1}{2}\right/n} \left(\frac{b x^n}{a}+1\right)^{\frac{1}{n}} \, _2F_1\left(2+\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right)}{a^2}","\frac{n x \left(a+b x^n\right)^2 \left(a^2+2 a b x^n+b^2 x^{2 n}\right)^{\frac{1}{2} \left(-\frac{1}{n}-2\right)}}{a^2 (n+1)}+\frac{x \left(a+b x^n\right) \left(a^2+2 a b x^n+b^2 x^{2 n}\right)^{\frac{1}{2} \left(-\frac{1}{n}-2\right)}}{a (n+1)}",1,"(x*(1 + (b*x^n)/a)^n^(-1)*Hypergeometric2F1[2 + n^(-1), n^(-1), 1 + n^(-1), -((b*x^n)/a)])/(a^2*((a + b*x^n)^2)^(1/(2*n)))","C",1
547,1,75,117,0.0347961,"\int (d x)^{-1-2 n (1+p)} \left(a^2+2 a b x^n+b^2 x^{2 n}\right)^p \, dx","Integrate[(d*x)^(-1 - 2*n*(1 + p))*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^p,x]","-\frac{x (d x)^{-2 n (p+1)-1} \left(\left(a+b x^n\right)^2\right)^p \left(\frac{b x^n}{a}+1\right)^{-2 p} \, _2F_1\left(-2 p,-2 (p+1);1-2 (p+1);-\frac{b x^n}{a}\right)}{2 n (p+1)}","\frac{(d x)^{-2 n (p+1)} \left(a^2+2 a b x^n+b^2 x^{2 n}\right)^{p+1}}{2 a^2 d n (p+1) (2 p+1)}-\frac{\left(a+b x^n\right) (d x)^{-2 n (p+1)} \left(a^2+2 a b x^n+b^2 x^{2 n}\right)^p}{a d n (2 p+1)}",1,"-1/2*(x*(d*x)^(-1 - 2*n*(1 + p))*((a + b*x^n)^2)^p*Hypergeometric2F1[-2*p, -2*(1 + p), 1 - 2*(1 + p), -((b*x^n)/a)])/(n*(1 + p)*(1 + (b*x^n)/a)^(2*p))","C",1
548,1,54,103,0.0291024,"\int x^{-1+2 n} \left(a^2+2 a b x^n+b^2 x^{2 n}\right)^p \, dx","Integrate[x^(-1 + 2*n)*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^p,x]","\frac{\left(a+b x^n\right) \left(\left(a+b x^n\right)^2\right)^p \left(b (2 p+1) x^n-a\right)}{2 b^2 n (p+1) (2 p+1)}","\frac{a^2 \left(\frac{b x^n}{a}+1\right)^2 \left(a^2+2 a b x^n+b^2 x^{2 n}\right)^p}{2 b^2 n (p+1)}-\frac{a^2 \left(\frac{b x^n}{a}+1\right) \left(a^2+2 a b x^n+b^2 x^{2 n}\right)^p}{b^2 n (2 p+1)}",1,"((a + b*x^n)*((a + b*x^n)^2)^p*(-a + b*(1 + 2*p)*x^n))/(2*b^2*n*(1 + p)*(1 + 2*p))","A",1
549,1,93,111,0.1936645,"\int \frac{x^{-1+4 n}}{a+b x^n+c x^{2 n}} \, dx","Integrate[x^(-1 + 4*n)/(a + b*x^n + c*x^(2*n)),x]","\frac{\left(b^2-a c\right) \log \left(a+x^n \left(b+c x^n\right)\right)+\frac{2 b \left(b^2-3 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c}}+c x^n \left(c x^n-2 b\right)}{2 c^3 n}","\frac{b \left(b^2-3 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right)}{c^3 n \sqrt{b^2-4 a c}}+\frac{\left(b^2-a c\right) \log \left(a+b x^n+c x^{2 n}\right)}{2 c^3 n}-\frac{b x^n}{c^2 n}+\frac{x^{2 n}}{2 c n}",1,"(c*x^n*(-2*b + c*x^n) + (2*b*(b^2 - 3*a*c)*ArcTanh[(b + 2*c*x^n)/Sqrt[b^2 - 4*a*c]])/Sqrt[b^2 - 4*a*c] + (b^2 - a*c)*Log[a + x^n*(b + c*x^n)])/(2*c^3*n)","A",1
550,1,80,87,0.1266785,"\int \frac{x^{-1+3 n}}{a+b x^n+c x^{2 n}} \, dx","Integrate[x^(-1 + 3*n)/(a + b*x^n + c*x^(2*n)),x]","\frac{-\frac{\left(b^2-2 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right)}{c \sqrt{b^2-4 a c}}-\frac{b \log \left(a+x^n \left(b+c x^n\right)\right)}{2 c}+x^n}{c n}","-\frac{\left(b^2-2 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right)}{c^2 n \sqrt{b^2-4 a c}}-\frac{b \log \left(a+b x^n+c x^{2 n}\right)}{2 c^2 n}+\frac{x^n}{c n}",1,"(x^n - ((b^2 - 2*a*c)*ArcTanh[(b + 2*c*x^n)/Sqrt[b^2 - 4*a*c]])/(c*Sqrt[b^2 - 4*a*c]) - (b*Log[a + x^n*(b + c*x^n)])/(2*c))/(c*n)","A",1
551,1,62,68,0.0660378,"\int \frac{x^{-1+2 n}}{a+b x^n+c x^{2 n}} \, dx","Integrate[x^(-1 + 2*n)/(a + b*x^n + c*x^(2*n)),x]","\frac{\frac{2 b \tanh ^{-1}\left(\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c}}+\log \left(a+x^n \left(b+c x^n\right)\right)}{2 c n}","\frac{b \tanh ^{-1}\left(\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right)}{c n \sqrt{b^2-4 a c}}+\frac{\log \left(a+b x^n+c x^{2 n}\right)}{2 c n}",1,"((2*b*ArcTanh[(b + 2*c*x^n)/Sqrt[b^2 - 4*a*c]])/Sqrt[b^2 - 4*a*c] + Log[a + x^n*(b + c*x^n)])/(2*c*n)","A",1
552,1,39,39,0.0592097,"\int \frac{x^{-1+n}}{a+b x^n+c x^{2 n}} \, dx","Integrate[x^(-1 + n)/(a + b*x^n + c*x^(2*n)),x]","-\frac{2 \tanh ^{-1}\left(\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right)}{n \sqrt{b^2-4 a c}}","-\frac{2 \tanh ^{-1}\left(\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right)}{n \sqrt{b^2-4 a c}}",1,"(-2*ArcTanh[(b + 2*c*x^n)/Sqrt[b^2 - 4*a*c]])/(Sqrt[b^2 - 4*a*c]*n)","A",1
553,1,135,98,0.6653779,"\int \frac{x^{-1-n}}{a+b x^n+c x^{2 n}} \, dx","Integrate[x^(-1 - n)/(a + b*x^n + c*x^(2*n)),x]","-\frac{-\frac{4 c^2 \log \left(x^{-n} \left(b-\sqrt{b^2-4 a c}\right)+2 c\right)}{\sqrt{b^2-4 a c} \left(b-\sqrt{b^2-4 a c}\right)^2}+\frac{4 c^2 \log \left(x^{-n} \left(\sqrt{b^2-4 a c}+b\right)+2 c\right)}{\sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}+b\right)^2}+\frac{x^{-n}}{a}}{n}","-\frac{\left(b^2-2 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right)}{a^2 n \sqrt{b^2-4 a c}}+\frac{b \log \left(a+b x^n+c x^{2 n}\right)}{2 a^2 n}-\frac{b \log (x)}{a^2}-\frac{x^{-n}}{a n}",1,"-((1/(a*x^n) - (4*c^2*Log[2*c + (b - Sqrt[b^2 - 4*a*c])/x^n])/(Sqrt[b^2 - 4*a*c]*(b - Sqrt[b^2 - 4*a*c])^2) + (4*c^2*Log[2*c + (b + Sqrt[b^2 - 4*a*c])/x^n])/(Sqrt[b^2 - 4*a*c]*(b + Sqrt[b^2 - 4*a*c])^2))/n)","A",1
554,1,112,126,0.3325062,"\int \frac{x^{-1-2 n}}{a+b x^n+c x^{2 n}} \, dx","Integrate[x^(-1 - 2*n)/(a + b*x^n + c*x^(2*n)),x]","\frac{-a^2 x^{-2 n}-\left(b^2-a c\right) \log \left(a+x^n \left(b+c x^n\right)\right)+\frac{2 b \left(b^2-3 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c}}+2 n \log (x) \left(b^2-a c\right)+2 a b x^{-n}}{2 a^3 n}","\frac{b \left(b^2-3 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right)}{a^3 n \sqrt{b^2-4 a c}}-\frac{\left(b^2-a c\right) \log \left(a+b x^n+c x^{2 n}\right)}{2 a^3 n}+\frac{\log (x) \left(b^2-a c\right)}{a^3}+\frac{b x^{-n}}{a^2 n}-\frac{x^{-2 n}}{2 a n}",1,"(-(a^2/x^(2*n)) + (2*a*b)/x^n + (2*b*(b^2 - 3*a*c)*ArcTanh[(b + 2*c*x^n)/Sqrt[b^2 - 4*a*c]])/Sqrt[b^2 - 4*a*c] + 2*(b^2 - a*c)*n*Log[x] - (b^2 - a*c)*Log[a + x^n*(b + c*x^n)])/(2*a^3*n)","A",1
555,1,143,164,0.4431767,"\int \frac{x^{-1-3 n}}{a+b x^n+c x^{2 n}} \, dx","Integrate[x^(-1 - 3*n)/(a + b*x^n + c*x^(2*n)),x]","\frac{-2 a^3 x^{-3 n}-\frac{6 \left(2 a^2 c^2-4 a b^2 c+b^4\right) \tanh ^{-1}\left(\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c}}+3 a^2 b x^{-2 n}+6 a x^{-n} \left(a c-b^2\right)+3 b \left(b^2-2 a c\right) \log \left(a+x^n \left(b+c x^n\right)\right)-6 b n \log (x) \left(b^2-2 a c\right)}{6 a^4 n}","\frac{b \left(b^2-2 a c\right) \log \left(a+b x^n+c x^{2 n}\right)}{2 a^4 n}-\frac{b \log (x) \left(b^2-2 a c\right)}{a^4}-\frac{x^{-n} \left(b^2-a c\right)}{a^3 n}+\frac{b x^{-2 n}}{2 a^2 n}-\frac{\left(2 a^2 c^2-4 a b^2 c+b^4\right) \tanh ^{-1}\left(\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right)}{a^4 n \sqrt{b^2-4 a c}}-\frac{x^{-3 n}}{3 a n}",1,"((-2*a^3)/x^(3*n) + (3*a^2*b)/x^(2*n) + (6*a*(-b^2 + a*c))/x^n - (6*(b^4 - 4*a*b^2*c + 2*a^2*c^2)*ArcTanh[(b + 2*c*x^n)/Sqrt[b^2 - 4*a*c]])/Sqrt[b^2 - 4*a*c] - 6*b*(b^2 - 2*a*c)*n*Log[x] + 3*b*(b^2 - 2*a*c)*Log[a + x^n*(b + c*x^n)])/(6*a^4*n)","A",1
556,1,340,353,0.8984129,"\int \frac{x^{-1+\frac{n}{4}}}{a+b x^n+c x^{2 n}} \, dx","Integrate[x^(-1 + n/4)/(a + b*x^n + c*x^(2*n)),x]","\frac{2\ 2^{3/4} c^{3/4} \left(-\frac{\sqrt[4]{-\sqrt{b^2-4 a c}-b} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x^{n/4}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x^{n/4}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{\sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{\sqrt[4]{-\sqrt{b^2-4 a c}-b} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x^{n/4}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x^{n/4}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{\sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}\right)}{n}","\frac{2\ 2^{3/4} c^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x^{n/4}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{n \sqrt{b^2-4 a c} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{2\ 2^{3/4} c^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x^{n/4}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{n \sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{2\ 2^{3/4} c^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x^{n/4}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{n \sqrt{b^2-4 a c} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{2\ 2^{3/4} c^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x^{n/4}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{n \sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}",1,"(2*2^(3/4)*c^(3/4)*(-(((-b - Sqrt[b^2 - 4*a*c])^(1/4)*ArcTan[(2^(1/4)*c^(1/4)*x^(n/4))/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])) - ArcTan[(2^(1/4)*c^(1/4)*x^(n/4))/(-b + Sqrt[b^2 - 4*a*c])^(1/4)]/(Sqrt[b^2 - 4*a*c]*(-b + Sqrt[b^2 - 4*a*c])^(3/4)) - ((-b - Sqrt[b^2 - 4*a*c])^(1/4)*ArcTanh[(2^(1/4)*c^(1/4)*x^(n/4))/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c]) - ArcTanh[(2^(1/4)*c^(1/4)*x^(n/4))/(-b + Sqrt[b^2 - 4*a*c])^(1/4)]/(Sqrt[b^2 - 4*a*c]*(-b + Sqrt[b^2 - 4*a*c])^(3/4))))/n","A",1
557,1,526,610,0.7634277,"\int \frac{x^{-1+\frac{n}{3}}}{a+b x^n+c x^{2 n}} \, dx","Integrate[x^(-1 + n/3)/(a + b*x^n + c*x^(2*n)),x]","\frac{c^{2/3} \left(-\left(\sqrt{b^2-4 a c}+b\right)^{2/3} \log \left(-\sqrt[3]{2} \sqrt[3]{c} x^{n/3} \sqrt[3]{b-\sqrt{b^2-4 a c}}+\left(b-\sqrt{b^2-4 a c}\right)^{2/3}+2^{2/3} c^{2/3} x^{2 n/3}\right)+\left(b-\sqrt{b^2-4 a c}\right)^{2/3} \log \left(-\sqrt[3]{2} \sqrt[3]{c} x^{n/3} \sqrt[3]{\sqrt{b^2-4 a c}+b}+\left(\sqrt{b^2-4 a c}+b\right)^{2/3}+2^{2/3} c^{2/3} x^{2 n/3}\right)+2 \left(\sqrt{b^2-4 a c}+b\right)^{2/3} \log \left(\sqrt[3]{b-\sqrt{b^2-4 a c}}+\sqrt[3]{2} \sqrt[3]{c} x^{n/3}\right)-2 \left(b-\sqrt{b^2-4 a c}\right)^{2/3} \log \left(\sqrt[3]{\sqrt{b^2-4 a c}+b}+\sqrt[3]{2} \sqrt[3]{c} x^{n/3}\right)-2 \sqrt{3} \left(\sqrt{b^2-4 a c}+b\right)^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x^{n/3}}{\sqrt[3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right)+2 \sqrt{3} \left(b-\sqrt{b^2-4 a c}\right)^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x^{n/3}}{\sqrt[3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right)\right)}{\sqrt[3]{2} n \sqrt{b^2-4 a c} \left(b-\sqrt{b^2-4 a c}\right)^{2/3} \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}","\frac{2^{2/3} c^{2/3} \log \left(\sqrt[3]{b-\sqrt{b^2-4 a c}}+\sqrt[3]{2} \sqrt[3]{c} x^{n/3}\right)}{n \sqrt{b^2-4 a c} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}-\frac{2^{2/3} c^{2/3} \log \left(\sqrt[3]{\sqrt{b^2-4 a c}+b}+\sqrt[3]{2} \sqrt[3]{c} x^{n/3}\right)}{n \sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}-\frac{c^{2/3} \log \left(-\sqrt[3]{2} \sqrt[3]{c} x^{n/3} \sqrt[3]{b-\sqrt{b^2-4 a c}}+\left(b-\sqrt{b^2-4 a c}\right)^{2/3}+2^{2/3} c^{2/3} x^{2 n/3}\right)}{\sqrt[3]{2} n \sqrt{b^2-4 a c} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}+\frac{c^{2/3} \log \left(-\sqrt[3]{2} \sqrt[3]{c} x^{n/3} \sqrt[3]{\sqrt{b^2-4 a c}+b}+\left(\sqrt{b^2-4 a c}+b\right)^{2/3}+2^{2/3} c^{2/3} x^{2 n/3}\right)}{\sqrt[3]{2} n \sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}-\frac{2^{2/3} \sqrt{3} c^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x^{n/3}}{\sqrt[3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right)}{n \sqrt{b^2-4 a c} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}+\frac{2^{2/3} \sqrt{3} c^{2/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x^{n/3}}{\sqrt[3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right)}{n \sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}",1,"(c^(2/3)*(-2*Sqrt[3]*(b + Sqrt[b^2 - 4*a*c])^(2/3)*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x^(n/3))/(b - Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[3]] + 2*Sqrt[3]*(b - Sqrt[b^2 - 4*a*c])^(2/3)*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x^(n/3))/(b + Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[3]] + 2*(b + Sqrt[b^2 - 4*a*c])^(2/3)*Log[(b - Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x^(n/3)] - 2*(b - Sqrt[b^2 - 4*a*c])^(2/3)*Log[(b + Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x^(n/3)] - (b + Sqrt[b^2 - 4*a*c])^(2/3)*Log[(b - Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*(b - Sqrt[b^2 - 4*a*c])^(1/3)*x^(n/3) + 2^(2/3)*c^(2/3)*x^((2*n)/3)] + (b - Sqrt[b^2 - 4*a*c])^(2/3)*Log[(b + Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*(b + Sqrt[b^2 - 4*a*c])^(1/3)*x^(n/3) + 2^(2/3)*c^(2/3)*x^((2*n)/3)]))/(2^(1/3)*Sqrt[b^2 - 4*a*c]*(b - Sqrt[b^2 - 4*a*c])^(2/3)*(b + Sqrt[b^2 - 4*a*c])^(2/3)*n)","A",1
558,1,145,169,0.2575701,"\int \frac{x^{-1+\frac{n}{2}}}{a+b x^n+c x^{2 n}} \, dx","Integrate[x^(-1 + n/2)/(a + b*x^n + c*x^(2*n)),x]","\frac{2 \sqrt{2} \sqrt{c} \left(\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x^{n/2}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x^{n/2}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{n \sqrt{b^2-4 a c}}","\frac{2 \sqrt{2} \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x^{n/2}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{n \sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{2 \sqrt{2} \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x^{n/2}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{n \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}",1,"(2*Sqrt[2]*Sqrt[c]*(ArcTan[(Sqrt[2]*Sqrt[c]*x^(n/2))/Sqrt[b - Sqrt[b^2 - 4*a*c]]]/Sqrt[b - Sqrt[b^2 - 4*a*c]] - ArcTan[(Sqrt[2]*Sqrt[c]*x^(n/2))/Sqrt[b + Sqrt[b^2 - 4*a*c]]]/Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(Sqrt[b^2 - 4*a*c]*n)","A",1
559,1,127,205,0.1786758,"\int \frac{x^{-1-\frac{n}{2}}}{a+b x^n+c x^{2 n}} \, dx","Integrate[x^(-1 - n/2)/(a + b*x^n + c*x^(2*n)),x]","\frac{4 c x^{-n/2} \left(\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)}{-b \sqrt{b^2-4 a c}-4 a c+b^2}+\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{b \sqrt{b^2-4 a c}-4 a c+b^2}\right)}{n}","\frac{\sqrt{2} \left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} x^{-n/2}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{a^{3/2} n \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} x^{-n/2}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{a^{3/2} n \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{2 x^{-n/2}}{a n}",1,"(4*c*(Hypergeometric2F1[-1/2, 1, 1/2, (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]) + Hypergeometric2F1[-1/2, 1, 1/2, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]/(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])))/(n*x^(n/2))","C",1
560,1,127,699,0.1283056,"\int \frac{x^{-1-\frac{n}{3}}}{a+b x^n+c x^{2 n}} \, dx","Integrate[x^(-1 - n/3)/(a + b*x^n + c*x^(2*n)),x]","\frac{6 c x^{-n/3} \left(\frac{\, _2F_1\left(-\frac{1}{3},1;\frac{2}{3};\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)}{-b \sqrt{b^2-4 a c}-4 a c+b^2}+\frac{\, _2F_1\left(-\frac{1}{3},1;\frac{2}{3};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{b \sqrt{b^2-4 a c}-4 a c+b^2}\right)}{n}","\frac{\left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \log \left(\sqrt[3]{b-\sqrt{b^2-4 a c}}+\sqrt[3]{2} \sqrt[3]{a} x^{-n/3}\right)}{\sqrt[3]{2} a^{4/3} n \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}+\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \log \left(\sqrt[3]{\sqrt{b^2-4 a c}+b}+\sqrt[3]{2} \sqrt[3]{a} x^{-n/3}\right)}{\sqrt[3]{2} a^{4/3} n \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}-\frac{\left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \log \left(2^{2/3} a^{2/3} x^{-2 n/3}-\sqrt[3]{2} \sqrt[3]{a} x^{-n/3} \sqrt[3]{b-\sqrt{b^2-4 a c}}+\left(b-\sqrt{b^2-4 a c}\right)^{2/3}\right)}{2 \sqrt[3]{2} a^{4/3} n \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}-\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \log \left(2^{2/3} a^{2/3} x^{-2 n/3}-\sqrt[3]{2} \sqrt[3]{a} x^{-n/3} \sqrt[3]{\sqrt{b^2-4 a c}+b}+\left(\sqrt{b^2-4 a c}+b\right)^{2/3}\right)}{2 \sqrt[3]{2} a^{4/3} n \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}-\frac{\sqrt{3} \left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{a} x^{-n/3}}{\sqrt[3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right)}{\sqrt[3]{2} a^{4/3} n \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}-\frac{\sqrt{3} \left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{a} x^{-n/3}}{\sqrt[3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right)}{\sqrt[3]{2} a^{4/3} n \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}-\frac{3 x^{-n/3}}{a n}",1,"(6*c*(Hypergeometric2F1[-1/3, 1, 2/3, (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]) + Hypergeometric2F1[-1/3, 1, 2/3, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]/(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])))/(n*x^(n/3))","C",1
561,1,127,414,0.1248835,"\int \frac{x^{-1-\frac{n}{4}}}{a+b x^n+c x^{2 n}} \, dx","Integrate[x^(-1 - n/4)/(a + b*x^n + c*x^(2*n)),x]","\frac{8 c x^{-n/4} \left(\frac{\, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)}{-b \sqrt{b^2-4 a c}-4 a c+b^2}+\frac{\, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{b \sqrt{b^2-4 a c}-4 a c+b^2}\right)}{n}","-\frac{2^{3/4} \left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x^{-n/4}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{a^{5/4} n \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{2^{3/4} \left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x^{-n/4}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{a^{5/4} n \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{2^{3/4} \left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x^{-n/4}}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{a^{5/4} n \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{2^{3/4} \left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{a} x^{-n/4}}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{a^{5/4} n \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}-\frac{4 x^{-n/4}}{a n}",1,"(8*c*(Hypergeometric2F1[-1/4, 1, 3/4, (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]) + Hypergeometric2F1[-1/4, 1, 3/4, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]/(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])))/(n*x^(n/4))","C",1
562,1,265,140,0.6540012,"\int \frac{x^2}{a+b x^n+c x^{2 n}} \, dx","Integrate[x^2/(a + b*x^n + c*x^(2*n)),x]","-\frac{2}{3} c x^3 \left(\frac{1-\left(\frac{x^n}{x^n-\frac{\sqrt{b^2-4 a c}-b}{2 c}}\right)^{-3/n} \, _2F_1\left(-\frac{3}{n},-\frac{3}{n};\frac{n-3}{n};\frac{b-\sqrt{b^2-4 a c}}{2 c x^n+b-\sqrt{b^2-4 a c}}\right)}{-b \sqrt{b^2-4 a c}-4 a c+b^2}+\frac{1-8^{-1/n} \left(\frac{c x^n}{\sqrt{b^2-4 a c}+b+2 c x^n}\right)^{-3/n} \, _2F_1\left(-\frac{3}{n},-\frac{3}{n};\frac{n-3}{n};\frac{b+\sqrt{b^2-4 a c}}{2 c x^n+b+\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}+b\right)}\right)","-\frac{2 c x^3 \, _2F_1\left(1,\frac{3}{n};\frac{n+3}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{3 \left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right)}-\frac{2 c x^3 \, _2F_1\left(1,\frac{3}{n};\frac{n+3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{3 \left(b \sqrt{b^2-4 a c}-4 a c+b^2\right)}",1,"(-2*c*x^3*((1 - Hypergeometric2F1[-3/n, -3/n, (-3 + n)/n, (b - Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^(3/n))/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]) + (1 - Hypergeometric2F1[-3/n, -3/n, (-3 + n)/n, (b + Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(8^n^(-1)*((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^(3/n)))/(Sqrt[b^2 - 4*a*c]*(b + Sqrt[b^2 - 4*a*c]))))/3","A",1
563,1,263,136,0.5752933,"\int \frac{x}{a+b x^n+c x^{2 n}} \, dx","Integrate[x/(a + b*x^n + c*x^(2*n)),x]","-c x^2 \left(\frac{1-\left(\frac{x^n}{x^n-\frac{\sqrt{b^2-4 a c}-b}{2 c}}\right)^{-2/n} \, _2F_1\left(-\frac{2}{n},-\frac{2}{n};\frac{n-2}{n};\frac{b-\sqrt{b^2-4 a c}}{2 c x^n+b-\sqrt{b^2-4 a c}}\right)}{-b \sqrt{b^2-4 a c}-4 a c+b^2}+\frac{1-4^{-1/n} \left(\frac{c x^n}{\sqrt{b^2-4 a c}+b+2 c x^n}\right)^{-2/n} \, _2F_1\left(-\frac{2}{n},-\frac{2}{n};\frac{n-2}{n};\frac{b+\sqrt{b^2-4 a c}}{2 c x^n+b+\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}+b\right)}\right)","-\frac{c x^2 \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{c x^2 \, _2F_1\left(1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{b \sqrt{b^2-4 a c}-4 a c+b^2}",1,"-(c*x^2*((1 - Hypergeometric2F1[-2/n, -2/n, (-2 + n)/n, (b - Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^(2/n))/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]) + (1 - Hypergeometric2F1[-2/n, -2/n, (-2 + n)/n, (b + Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(4^n^(-1)*((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^(2/n)))/(Sqrt[b^2 - 4*a*c]*(b + Sqrt[b^2 - 4*a*c]))))","A",1
564,1,261,124,0.5324135,"\int \frac{1}{a+b x^n+c x^{2 n}} \, dx","Integrate[(a + b*x^n + c*x^(2*n))^(-1),x]","-2 c x \left(\frac{1-\left(\frac{x^n}{x^n-\frac{\sqrt{b^2-4 a c}-b}{2 c}}\right)^{-1/n} \, _2F_1\left(-\frac{1}{n},-\frac{1}{n};\frac{n-1}{n};\frac{b-\sqrt{b^2-4 a c}}{2 c x^n+b-\sqrt{b^2-4 a c}}\right)}{-b \sqrt{b^2-4 a c}-4 a c+b^2}+\frac{1-2^{-1/n} \left(\frac{c x^n}{\sqrt{b^2-4 a c}+b+2 c x^n}\right)^{-1/n} \, _2F_1\left(-\frac{1}{n},-\frac{1}{n};\frac{n-1}{n};\frac{b+\sqrt{b^2-4 a c}}{2 c x^n+b+\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}+b\right)}\right)","-\frac{2 c x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 c x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{b \sqrt{b^2-4 a c}-4 a c+b^2}",1,"-2*c*x*((1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, (b - Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^n^(-1))/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]) + (1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, (b + Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(2^n^(-1)*((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1)))/(Sqrt[b^2 - 4*a*c]*(b + Sqrt[b^2 - 4*a*c])))","B",1
565,1,74,74,0.1388575,"\int \frac{1}{x \left(a+b x^n+c x^{2 n}\right)} \, dx","Integrate[1/(x*(a + b*x^n + c*x^(2*n))),x]","-\frac{\frac{2 b \tan ^{-1}\left(\frac{b+2 c x^n}{\sqrt{4 a c-b^2}}\right)}{n \sqrt{4 a c-b^2}}+\frac{\log \left(a+x^n \left(b+c x^n\right)\right)}{n}-2 \log (x)}{2 a}","\frac{b \tanh ^{-1}\left(\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right)}{a n \sqrt{b^2-4 a c}}-\frac{\log \left(a+b x^n+c x^{2 n}\right)}{2 a n}+\frac{\log (x)}{a}",1,"-1/2*((2*b*ArcTan[(b + 2*c*x^n)/Sqrt[-b^2 + 4*a*c]])/(Sqrt[-b^2 + 4*a*c]*n) - 2*Log[x] + Log[a + x^n*(b + c*x^n)]/n)/a","A",1
566,1,240,142,0.383719,"\int \frac{1}{x^2 \left(a+b x^n+c x^{2 n}\right)} \, dx","Integrate[1/(x^2*(a + b*x^n + c*x^(2*n))),x]","\frac{c 2^{\frac{1}{n}+1} \left(\frac{\left(\frac{c x^n}{-\sqrt{b^2-4 a c}+b+2 c x^n}\right)^{\frac{1}{n}} \, _2F_1\left(1+\frac{1}{n},1+\frac{1}{n};2+\frac{1}{n};\frac{b-\sqrt{b^2-4 a c}}{2 c x^n+b-\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c}-b-2 c x^n}+\frac{x^{-n} \left(\frac{c x^n}{\sqrt{b^2-4 a c}+b+2 c x^n}\right)^{\frac{1}{n}+1} \, _2F_1\left(1+\frac{1}{n},1+\frac{1}{n};2+\frac{1}{n};\frac{b+\sqrt{b^2-4 a c}}{2 c x^n+b+\sqrt{b^2-4 a c}}\right)}{c}\right)}{(n+1) x \sqrt{b^2-4 a c}}","\frac{2 c \, _2F_1\left(1,-\frac{1}{n};-\frac{1-n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{x \left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right)}+\frac{2 c \, _2F_1\left(1,-\frac{1}{n};-\frac{1-n}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{x \left(b \sqrt{b^2-4 a c}-4 a c+b^2\right)}",1,"(2^(1 + n^(-1))*c*((((c*x^n)/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1)*Hypergeometric2F1[1 + n^(-1), 1 + n^(-1), 2 + n^(-1), (b - Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)])/(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^n) + (((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^(1 + n^(-1))*Hypergeometric2F1[1 + n^(-1), 1 + n^(-1), 2 + n^(-1), (b + Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)])/(c*x^n)))/(Sqrt[b^2 - 4*a*c]*(1 + n)*x)","A",1
567,1,258,140,0.4242399,"\int \frac{1}{x^3 \left(a+b x^n+c x^{2 n}\right)} \, dx","Integrate[1/(x^3*(a + b*x^n + c*x^(2*n))),x]","\frac{c 2^{\frac{n+2}{n}} \left(\frac{\left(\frac{c x^n}{-\sqrt{b^2-4 a c}+b+2 c x^n}\right)^{2/n} \, _2F_1\left(\frac{n+2}{n},\frac{n+2}{n};2+\frac{2}{n};\frac{b-\sqrt{b^2-4 a c}}{2 c x^n+b-\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c}-b-2 c x^n}+\frac{x^{-n} \left(\frac{c x^n}{\sqrt{b^2-4 a c}+b+2 c x^n}\right)^{\frac{n+2}{n}} \, _2F_1\left(\frac{n+2}{n},\frac{n+2}{n};2+\frac{2}{n};\frac{b+\sqrt{b^2-4 a c}}{2 c x^n+b+\sqrt{b^2-4 a c}}\right)}{c}\right)}{(n+2) x^2 \sqrt{b^2-4 a c}}","\frac{c \, _2F_1\left(1,-\frac{2}{n};-\frac{2-n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{x^2 \left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right)}+\frac{c \, _2F_1\left(1,-\frac{2}{n};-\frac{2-n}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{x^2 \left(b \sqrt{b^2-4 a c}-4 a c+b^2\right)}",1,"(2^((2 + n)/n)*c*((((c*x^n)/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n))^(2/n)*Hypergeometric2F1[(2 + n)/n, (2 + n)/n, 2 + 2/n, (b - Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)])/(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^n) + (((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^((2 + n)/n)*Hypergeometric2F1[(2 + n)/n, (2 + n)/n, 2 + 2/n, (b + Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)])/(c*x^n)))/(Sqrt[b^2 - 4*a*c]*(2 + n)*x^2)","A",1
568,1,365,148,0.7542746,"\int x^3 \sqrt{a+b x^n+c x^{2 n}} \, dx","Integrate[x^3*Sqrt[a + b*x^n + c*x^(2*n)],x]","\frac{x^4 \left(2 b n x^n \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{n+4}{n};\frac{1}{2},\frac{1}{2};2+\frac{4}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)+a n (n+4) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{4}{n};\frac{1}{2},\frac{1}{2};\frac{n+4}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)+4 (n+4) \left(a+x^n \left(b+c x^n\right)\right)\right)}{4 (n+4)^2 \sqrt{a+x^n \left(b+c x^n\right)}}","\frac{x^4 \sqrt{a+b x^n+c x^{2 n}} F_1\left(\frac{4}{n};-\frac{1}{2},-\frac{1}{2};\frac{n+4}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{4 \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}}",1,"(x^4*(4*(4 + n)*(a + x^n*(b + c*x^n)) + a*n*(4 + n)*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[4/n, 1/2, 1/2, (4 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + 2*b*n*x^n*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[(4 + n)/n, 1/2, 1/2, 2 + 4/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]))/(4*(4 + n)^2*Sqrt[a + x^n*(b + c*x^n)])","B",0
569,1,366,148,0.7150801,"\int x^2 \sqrt{a+b x^n+c x^{2 n}} \, dx","Integrate[x^2*Sqrt[a + b*x^n + c*x^(2*n)],x]","\frac{x^3 \left(3 b n x^n \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{n+3}{n};\frac{1}{2},\frac{1}{2};2+\frac{3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)+2 a n (n+3) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{3}{n};\frac{1}{2},\frac{1}{2};\frac{n+3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)+6 (n+3) \left(a+x^n \left(b+c x^n\right)\right)\right)}{6 (n+3)^2 \sqrt{a+x^n \left(b+c x^n\right)}}","\frac{x^3 \sqrt{a+b x^n+c x^{2 n}} F_1\left(\frac{3}{n};-\frac{1}{2},-\frac{1}{2};\frac{n+3}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{3 \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}}",1,"(x^3*(6*(3 + n)*(a + x^n*(b + c*x^n)) + 2*a*n*(3 + n)*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[3/n, 1/2, 1/2, (3 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + 3*b*n*x^n*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[(3 + n)/n, 1/2, 1/2, 2 + 3/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]))/(6*(3 + n)^2*Sqrt[a + x^n*(b + c*x^n)])","B",0
570,1,364,148,0.6535897,"\int x \sqrt{a+b x^n+c x^{2 n}} \, dx","Integrate[x*Sqrt[a + b*x^n + c*x^(2*n)],x]","\frac{x^2 \left(b n x^n \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{n+2}{n};\frac{1}{2},\frac{1}{2};2+\frac{2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)+a n (n+2) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{2}{n};\frac{1}{2},\frac{1}{2};\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)+2 (n+2) \left(a+x^n \left(b+c x^n\right)\right)\right)}{2 (n+2)^2 \sqrt{a+x^n \left(b+c x^n\right)}}","\frac{x^2 \sqrt{a+b x^n+c x^{2 n}} F_1\left(\frac{2}{n};-\frac{1}{2},-\frac{1}{2};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{2 \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}}",1,"(x^2*(2*(2 + n)*(a + x^n*(b + c*x^n)) + a*n*(2 + n)*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[2/n, 1/2, 1/2, (2 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + b*n*x^n*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[(2 + n)/n, 1/2, 1/2, 2 + 2/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]))/(2*(2 + n)^2*Sqrt[a + x^n*(b + c*x^n)])","B",0
571,1,351,139,0.6486384,"\int \sqrt{a+b x^n+c x^{2 n}} \, dx","Integrate[Sqrt[a + b*x^n + c*x^(2*n)],x]","\frac{x \left(b n x^n \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(1+\frac{1}{n};\frac{1}{2},\frac{1}{2};2+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)+2 (n+1) \left(a n \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{1}{n};\frac{1}{2},\frac{1}{2};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)+a+x^n \left(b+c x^n\right)\right)\right)}{2 (n+1)^2 \sqrt{a+x^n \left(b+c x^n\right)}}","\frac{x \sqrt{a+b x^n+c x^{2 n}} F_1\left(\frac{1}{n};-\frac{1}{2},-\frac{1}{2};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{\sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}}",1,"(x*(b*n*x^n*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[1 + n^(-1), 1/2, 1/2, 2 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + 2*(1 + n)*(a + x^n*(b + c*x^n) + a*n*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[n^(-1), 1/2, 1/2, 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])))/(2*(1 + n)^2*Sqrt[a + x^n*(b + c*x^n)])","B",0
572,1,110,119,0.1482952,"\int \frac{\sqrt{a+b x^n+c x^{2 n}}}{x} \, dx","Integrate[Sqrt[a + b*x^n + c*x^(2*n)]/x,x]","\frac{\sqrt{a+x^n \left(b+c x^n\right)}-\sqrt{a} \tanh ^{-1}\left(\frac{2 a+b x^n}{2 \sqrt{a} \sqrt{a+x^n \left(b+c x^n\right)}}\right)+\frac{b \tanh ^{-1}\left(\frac{b+2 c x^n}{2 \sqrt{c} \sqrt{a+x^n \left(b+c x^n\right)}}\right)}{2 \sqrt{c}}}{n}","\frac{\sqrt{a+b x^n+c x^{2 n}}}{n}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{2 a+b x^n}{2 \sqrt{a} \sqrt{a+b x^n+c x^{2 n}}}\right)}{n}+\frac{b \tanh ^{-1}\left(\frac{b+2 c x^n}{2 \sqrt{c} \sqrt{a+b x^n+c x^{2 n}}}\right)}{2 \sqrt{c} n}",1,"(Sqrt[a + x^n*(b + c*x^n)] - Sqrt[a]*ArcTanh[(2*a + b*x^n)/(2*Sqrt[a]*Sqrt[a + x^n*(b + c*x^n)])] + (b*ArcTanh[(b + 2*c*x^n)/(2*Sqrt[c]*Sqrt[a + x^n*(b + c*x^n)])])/(2*Sqrt[c]))/n","A",1
573,1,365,149,0.635542,"\int \frac{\sqrt{a+b x^n+c x^{2 n}}}{x^2} \, dx","Integrate[Sqrt[a + b*x^n + c*x^(2*n)]/x^2,x]","\frac{b n x^n \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{n-1}{n};\frac{1}{2},\frac{1}{2};2-\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)-2 a (n-1) n \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(-\frac{1}{n};\frac{1}{2},\frac{1}{2};\frac{n-1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)+2 (n-1) \left(a+x^n \left(b+c x^n\right)\right)}{2 (n-1)^2 x \sqrt{a+x^n \left(b+c x^n\right)}}","-\frac{\sqrt{a+b x^n+c x^{2 n}} F_1\left(-\frac{1}{n};-\frac{1}{2},-\frac{1}{2};-\frac{1-n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{x \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}}",1,"(2*(-1 + n)*(a + x^n*(b + c*x^n)) - 2*a*(-1 + n)*n*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[-n^(-1), 1/2, 1/2, (-1 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + b*n*x^n*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[(-1 + n)/n, 1/2, 1/2, 2 - n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/(2*(-1 + n)^2*x*Sqrt[a + x^n*(b + c*x^n)])","B",0
574,1,365,151,0.6108033,"\int \frac{\sqrt{a+b x^n+c x^{2 n}}}{x^3} \, dx","Integrate[Sqrt[a + b*x^n + c*x^(2*n)]/x^3,x]","\frac{b n x^n \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{n-2}{n};\frac{1}{2},\frac{1}{2};2-\frac{2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)-a (n-2) n \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(-\frac{2}{n};\frac{1}{2},\frac{1}{2};\frac{n-2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)+2 (n-2) \left(a+x^n \left(b+c x^n\right)\right)}{2 (n-2)^2 x^2 \sqrt{a+x^n \left(b+c x^n\right)}}","-\frac{\sqrt{a+b x^n+c x^{2 n}} F_1\left(-\frac{2}{n};-\frac{1}{2},-\frac{1}{2};-\frac{2-n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{2 x^2 \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}}",1,"(2*(-2 + n)*(a + x^n*(b + c*x^n)) - a*(-2 + n)*n*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[-2/n, 1/2, 1/2, (-2 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + b*n*x^n*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[(-2 + n)/n, 1/2, 1/2, 2 - 2/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/(2*(-2 + n)^2*x^2*Sqrt[a + x^n*(b + c*x^n)])","B",0
575,1,518,149,1.4896675,"\int x^3 \left(a+b x^n+c x^{2 n}\right)^{3/2} \, dx","Integrate[x^3*(a + b*x^n + c*x^(2*n))^(3/2),x]","\frac{x^4 \left(2 (n+4) \left(32 a^2 c \left(n^2+3 n+2\right)+a \left(3 b^2 n^2+2 b c \left(23 n^2+84 n+64\right) x^n+8 c^2 \left(5 n^2+18 n+16\right) x^{2 n}\right)+x^n \left(b+c x^n\right) \left(3 b^2 n^2+2 b c \left(7 n^2+36 n+32\right) x^n+8 c^2 \left(n^2+6 n+8\right) x^{2 n}\right)\right)-3 b n^2 x^n \left(b^2 (n+8)-4 a c (3 n+8)\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{n+4}{n};\frac{1}{2},\frac{1}{2};2+\frac{4}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)-6 a n^2 (n+4) \left(b^2-2 a c (n+2)\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{4}{n};\frac{1}{2},\frac{1}{2};\frac{n+4}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)\right)}{16 c (n+2) (n+4)^2 (3 n+4) \sqrt{a+x^n \left(b+c x^n\right)}}","\frac{a x^4 \sqrt{a+b x^n+c x^{2 n}} F_1\left(\frac{4}{n};-\frac{3}{2},-\frac{3}{2};\frac{n+4}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{4 \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}}",1,"(x^4*(2*(4 + n)*(32*a^2*c*(2 + 3*n + n^2) + x^n*(b + c*x^n)*(3*b^2*n^2 + 2*b*c*(32 + 36*n + 7*n^2)*x^n + 8*c^2*(8 + 6*n + n^2)*x^(2*n)) + a*(3*b^2*n^2 + 2*b*c*(64 + 84*n + 23*n^2)*x^n + 8*c^2*(16 + 18*n + 5*n^2)*x^(2*n))) - 6*a*n^2*(4 + n)*(b^2 - 2*a*c*(2 + n))*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[4/n, 1/2, 1/2, (4 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] - 3*b*n^2*(b^2*(8 + n) - 4*a*c*(8 + 3*n))*x^n*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[(4 + n)/n, 1/2, 1/2, 2 + 4/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]))/(16*c*(2 + n)*(4 + n)^2*(4 + 3*n)*Sqrt[a + x^n*(b + c*x^n)])","B",0
576,1,524,149,1.4636817,"\int x^2 \left(a+b x^n+c x^{2 n}\right)^{3/2} \, dx","Integrate[x^2*(a + b*x^n + c*x^(2*n))^(3/2),x]","\frac{x^3 \left(2 (n+3) \left(4 a^2 c \left(8 n^2+18 n+9\right)+a \left(3 b^2 n^2+2 b c \left(23 n^2+63 n+36\right) x^n+4 c^2 \left(10 n^2+27 n+18\right) x^{2 n}\right)+x^n \left(b+c x^n\right) \left(3 b^2 n^2+2 b c \left(7 n^2+27 n+18\right) x^n+4 c^2 \left(2 n^2+9 n+9\right) x^{2 n}\right)\right)-3 b n^2 x^n \left(b^2 (n+6)-12 a c (n+2)\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{n+3}{n};\frac{1}{2},\frac{1}{2};2+\frac{3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)+2 a n^2 (n+3) \left(4 a c (2 n+3)-3 b^2\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{3}{n};\frac{1}{2},\frac{1}{2};\frac{n+3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)\right)}{24 c (n+1) (n+3)^2 (2 n+3) \sqrt{a+x^n \left(b+c x^n\right)}}","\frac{a x^3 \sqrt{a+b x^n+c x^{2 n}} F_1\left(\frac{3}{n};-\frac{3}{2},-\frac{3}{2};\frac{n+3}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{3 \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}}",1,"(x^3*(2*(3 + n)*(4*a^2*c*(9 + 18*n + 8*n^2) + x^n*(b + c*x^n)*(3*b^2*n^2 + 2*b*c*(18 + 27*n + 7*n^2)*x^n + 4*c^2*(9 + 9*n + 2*n^2)*x^(2*n)) + a*(3*b^2*n^2 + 2*b*c*(36 + 63*n + 23*n^2)*x^n + 4*c^2*(18 + 27*n + 10*n^2)*x^(2*n))) + 2*a*n^2*(3 + n)*(-3*b^2 + 4*a*c*(3 + 2*n))*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[3/n, 1/2, 1/2, (3 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] - 3*b*n^2*(-12*a*c*(2 + n) + b^2*(6 + n))*x^n*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[(3 + n)/n, 1/2, 1/2, 2 + 3/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]))/(24*c*(1 + n)*(3 + n)^2*(3 + 2*n)*Sqrt[a + x^n*(b + c*x^n)])","B",0
577,1,520,149,1.4906136,"\int x \left(a+b x^n+c x^{2 n}\right)^{3/2} \, dx","Integrate[x*(a + b*x^n + c*x^(2*n))^(3/2),x]","\frac{x^2 \left(2 (n+2) \left(16 a^2 c \left(2 n^2+3 n+1\right)+a \left(3 b^2 n^2+2 b c \left(23 n^2+42 n+16\right) x^n+8 c^2 \left(5 n^2+9 n+4\right) x^{2 n}\right)+x^n \left(b+c x^n\right) \left(3 b^2 n^2+2 b c \left(7 n^2+18 n+8\right) x^n+8 c^2 \left(n^2+3 n+2\right) x^{2 n}\right)\right)-3 b n^2 x^n \left(b^2 (n+4)-4 a c (3 n+4)\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{n+2}{n};\frac{1}{2},\frac{1}{2};2+\frac{2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)-6 a n^2 (n+2) \left(b^2-4 a c (n+1)\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{2}{n};\frac{1}{2},\frac{1}{2};\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)\right)}{16 c (n+1) (n+2)^2 (3 n+2) \sqrt{a+x^n \left(b+c x^n\right)}}","\frac{a x^2 \sqrt{a+b x^n+c x^{2 n}} F_1\left(\frac{2}{n};-\frac{3}{2},-\frac{3}{2};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{2 \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}}",1,"(x^2*(2*(2 + n)*(16*a^2*c*(1 + 3*n + 2*n^2) + x^n*(b + c*x^n)*(3*b^2*n^2 + 2*b*c*(8 + 18*n + 7*n^2)*x^n + 8*c^2*(2 + 3*n + n^2)*x^(2*n)) + a*(3*b^2*n^2 + 2*b*c*(16 + 42*n + 23*n^2)*x^n + 8*c^2*(4 + 9*n + 5*n^2)*x^(2*n))) - 6*a*n^2*(2 + n)*(b^2 - 4*a*c*(1 + n))*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[2/n, 1/2, 1/2, (2 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] - 3*b*n^2*(b^2*(4 + n) - 4*a*c*(4 + 3*n))*x^n*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[(2 + n)/n, 1/2, 1/2, 2 + 2/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]))/(16*c*(1 + n)*(2 + n)^2*(2 + 3*n)*Sqrt[a + x^n*(b + c*x^n)])","B",0
578,1,513,140,1.5238341,"\int \left(a+b x^n+c x^{2 n}\right)^{3/2} \, dx","Integrate[(a + b*x^n + c*x^(2*n))^(3/2),x]","\frac{x \left(2 (n+1) \left(4 a^2 c \left(8 n^2+6 n+1\right)-3 a n^2 \left(b^2-4 a c (2 n+1)\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{1}{n};\frac{1}{2},\frac{1}{2};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)+a \left(3 b^2 n^2+2 b c \left(23 n^2+21 n+4\right) x^n+4 c^2 \left(10 n^2+9 n+2\right) x^{2 n}\right)+x^n \left(b+c x^n\right) \left(3 b^2 n^2+2 b c \left(7 n^2+9 n+2\right) x^n+4 c^2 \left(2 n^2+3 n+1\right) x^{2 n}\right)\right)-3 b n^2 x^n \left(b^2 (n+2)-4 a c (3 n+2)\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(1+\frac{1}{n};\frac{1}{2},\frac{1}{2};2+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)\right)}{8 c (n+1)^2 (2 n+1) (3 n+1) \sqrt{a+x^n \left(b+c x^n\right)}}","\frac{a x \sqrt{a+b x^n+c x^{2 n}} F_1\left(\frac{1}{n};-\frac{3}{2},-\frac{3}{2};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{\sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}}",1,"(x*(-3*b*n^2*(b^2*(2 + n) - 4*a*c*(2 + 3*n))*x^n*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[1 + n^(-1), 1/2, 1/2, 2 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + 2*(1 + n)*(4*a^2*c*(1 + 6*n + 8*n^2) + x^n*(b + c*x^n)*(3*b^2*n^2 + 2*b*c*(2 + 9*n + 7*n^2)*x^n + 4*c^2*(1 + 3*n + 2*n^2)*x^(2*n)) + a*(3*b^2*n^2 + 2*b*c*(4 + 21*n + 23*n^2)*x^n + 4*c^2*(2 + 9*n + 10*n^2)*x^(2*n)) - 3*a*n^2*(b^2 - 4*a*c*(1 + 2*n))*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[n^(-1), 1/2, 1/2, 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])))/(8*c*(1 + n)^2*(1 + 2*n)*(1 + 3*n)*Sqrt[a + x^n*(b + c*x^n)])","B",0
579,1,158,173,0.2824711,"\int \frac{\left(a+b x^n+c x^{2 n}\right)^{3/2}}{x} \, dx","Integrate[(a + b*x^n + c*x^(2*n))^(3/2)/x,x]","\frac{-48 a^{3/2} c^{3/2} \tanh ^{-1}\left(\frac{2 a+b x^n}{2 \sqrt{a} \sqrt{a+x^n \left(b+c x^n\right)}}\right)-3 b \left(b^2-12 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^n}{2 \sqrt{c} \sqrt{a+x^n \left(b+c x^n\right)}}\right)+2 \sqrt{c} \sqrt{a+x^n \left(b+c x^n\right)} \left(8 c \left(4 a+c x^{2 n}\right)+3 b^2+14 b c x^n\right)}{48 c^{3/2} n}","-\frac{a^{3/2} \tanh ^{-1}\left(\frac{2 a+b x^n}{2 \sqrt{a} \sqrt{a+b x^n+c x^{2 n}}}\right)}{n}-\frac{b \left(b^2-12 a c\right) \tanh ^{-1}\left(\frac{b+2 c x^n}{2 \sqrt{c} \sqrt{a+b x^n+c x^{2 n}}}\right)}{16 c^{3/2} n}+\frac{\left(8 a c+b^2+2 b c x^n\right) \sqrt{a+b x^n+c x^{2 n}}}{8 c n}+\frac{\left(a+b x^n+c x^{2 n}\right)^{3/2}}{3 n}",1,"(2*Sqrt[c]*Sqrt[a + x^n*(b + c*x^n)]*(3*b^2 + 14*b*c*x^n + 8*c*(4*a + c*x^(2*n))) - 48*a^(3/2)*c^(3/2)*ArcTanh[(2*a + b*x^n)/(2*Sqrt[a]*Sqrt[a + x^n*(b + c*x^n)])] - 3*b*(b^2 - 12*a*c)*ArcTanh[(b + 2*c*x^n)/(2*Sqrt[c]*Sqrt[a + x^n*(b + c*x^n)])])/(48*c^(3/2)*n)","A",1
580,1,526,150,1.3877276,"\int \frac{\left(a+b x^n+c x^{2 n}\right)^{3/2}}{x^2} \, dx","Integrate[(a + b*x^n + c*x^(2*n))^(3/2)/x^2,x]","\frac{2 (n-1) \left(4 a^2 c \left(8 n^2-6 n+1\right)+a \left(3 b^2 n^2+2 b c \left(23 n^2-21 n+4\right) x^n+4 c^2 \left(10 n^2-9 n+2\right) x^{2 n}\right)+x^n \left(b+c x^n\right) \left(3 b^2 n^2+2 b c \left(7 n^2-9 n+2\right) x^n+4 c^2 \left(2 n^2-3 n+1\right) x^{2 n}\right)\right)-3 b n^2 x^n \left(4 a c (2-3 n)+b^2 (n-2)\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{n-1}{n};\frac{1}{2},\frac{1}{2};2-\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)-6 a (n-1) n^2 \left(4 a c (2 n-1)+b^2\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(-\frac{1}{n};\frac{1}{2},\frac{1}{2};\frac{n-1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)}{8 c (n-1)^2 (2 n-1) (3 n-1) x \sqrt{a+x^n \left(b+c x^n\right)}}","-\frac{a \sqrt{a+b x^n+c x^{2 n}} F_1\left(-\frac{1}{n};-\frac{3}{2},-\frac{3}{2};-\frac{1-n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{x \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}}",1,"(2*(-1 + n)*(4*a^2*c*(1 - 6*n + 8*n^2) + x^n*(b + c*x^n)*(3*b^2*n^2 + 2*b*c*(2 - 9*n + 7*n^2)*x^n + 4*c^2*(1 - 3*n + 2*n^2)*x^(2*n)) + a*(3*b^2*n^2 + 2*b*c*(4 - 21*n + 23*n^2)*x^n + 4*c^2*(2 - 9*n + 10*n^2)*x^(2*n))) - 6*a*(-1 + n)*n^2*(b^2 + 4*a*c*(-1 + 2*n))*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[-n^(-1), 1/2, 1/2, (-1 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] - 3*b*(4*a*c*(2 - 3*n) + b^2*(-2 + n))*n^2*x^n*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[(-1 + n)/n, 1/2, 1/2, 2 - n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/(8*c*(-1 + n)^2*(-1 + 2*n)*(-1 + 3*n)*x*Sqrt[a + x^n*(b + c*x^n)])","B",0
581,1,520,152,1.3718655,"\int \frac{\left(a+b x^n+c x^{2 n}\right)^{3/2}}{x^3} \, dx","Integrate[(a + b*x^n + c*x^(2*n))^(3/2)/x^3,x]","\frac{2 (n-2) \left(16 a^2 c \left(2 n^2-3 n+1\right)+a \left(3 b^2 n^2+2 b c \left(23 n^2-42 n+16\right) x^n+8 c^2 \left(5 n^2-9 n+4\right) x^{2 n}\right)+x^n \left(b+c x^n\right) \left(3 b^2 n^2+2 b c \left(7 n^2-18 n+8\right) x^n+8 c^2 \left(n^2-3 n+2\right) x^{2 n}\right)\right)-3 b n^2 x^n \left(4 a c (4-3 n)+b^2 (n-4)\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{n-2}{n};\frac{1}{2},\frac{1}{2};2-\frac{2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)-6 a (n-2) n^2 \left(4 a c (n-1)+b^2\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(-\frac{2}{n};\frac{1}{2},\frac{1}{2};\frac{n-2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)}{16 c (n-2)^2 (n-1) (3 n-2) x^2 \sqrt{a+x^n \left(b+c x^n\right)}}","-\frac{a \sqrt{a+b x^n+c x^{2 n}} F_1\left(-\frac{2}{n};-\frac{3}{2},-\frac{3}{2};-\frac{2-n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{2 x^2 \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}}",1,"(2*(-2 + n)*(16*a^2*c*(1 - 3*n + 2*n^2) + x^n*(b + c*x^n)*(3*b^2*n^2 + 2*b*c*(8 - 18*n + 7*n^2)*x^n + 8*c^2*(2 - 3*n + n^2)*x^(2*n)) + a*(3*b^2*n^2 + 2*b*c*(16 - 42*n + 23*n^2)*x^n + 8*c^2*(4 - 9*n + 5*n^2)*x^(2*n))) - 6*a*(b^2 + 4*a*c*(-1 + n))*(-2 + n)*n^2*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[-2/n, 1/2, 1/2, (-2 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] - 3*b*(4*a*c*(4 - 3*n) + b^2*(-4 + n))*n^2*x^n*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[(-2 + n)/n, 1/2, 1/2, 2 - 2/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/(16*c*(-2 + n)^2*(-1 + n)*(-2 + 3*n)*x^2*Sqrt[a + x^n*(b + c*x^n)])","B",0
582,1,175,148,0.1585232,"\int \frac{x^3}{\sqrt{a+b x^n+c x^{2 n}}} \, dx","Integrate[x^3/Sqrt[a + b*x^n + c*x^(2*n)],x]","\frac{x^4 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{4}{n};\frac{1}{2},\frac{1}{2};\frac{n+4}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)}{4 \sqrt{a+x^n \left(b+c x^n\right)}}","\frac{x^4 \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{4}{n};\frac{1}{2},\frac{1}{2};\frac{n+4}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{4 \sqrt{a+b x^n+c x^{2 n}}}",1,"(x^4*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[4/n, 1/2, 1/2, (4 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/(4*Sqrt[a + x^n*(b + c*x^n)])","A",0
583,1,175,148,0.1423352,"\int \frac{x^2}{\sqrt{a+b x^n+c x^{2 n}}} \, dx","Integrate[x^2/Sqrt[a + b*x^n + c*x^(2*n)],x]","\frac{x^3 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{3}{n};\frac{1}{2},\frac{1}{2};\frac{n+3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)}{3 \sqrt{a+x^n \left(b+c x^n\right)}}","\frac{x^3 \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{3}{n};\frac{1}{2},\frac{1}{2};\frac{n+3}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{3 \sqrt{a+b x^n+c x^{2 n}}}",1,"(x^3*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[3/n, 1/2, 1/2, (3 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/(3*Sqrt[a + x^n*(b + c*x^n)])","A",0
584,1,175,148,0.1352403,"\int \frac{x}{\sqrt{a+b x^n+c x^{2 n}}} \, dx","Integrate[x/Sqrt[a + b*x^n + c*x^(2*n)],x]","\frac{x^2 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{2}{n};\frac{1}{2},\frac{1}{2};\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)}{2 \sqrt{a+x^n \left(b+c x^n\right)}}","\frac{x^2 \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{2}{n};\frac{1}{2},\frac{1}{2};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{2 \sqrt{a+b x^n+c x^{2 n}}}",1,"(x^2*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[2/n, 1/2, 1/2, (2 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/(2*Sqrt[a + x^n*(b + c*x^n)])","A",0
585,1,166,139,0.1078706,"\int \frac{1}{\sqrt{a+b x^n+c x^{2 n}}} \, dx","Integrate[1/Sqrt[a + b*x^n + c*x^(2*n)],x]","\frac{x \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{1}{n};\frac{1}{2},\frac{1}{2};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)}{\sqrt{a+x^n \left(b+c x^n\right)}}","\frac{x \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{1}{n};\frac{1}{2},\frac{1}{2};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{\sqrt{a+b x^n+c x^{2 n}}}",1,"(x*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[n^(-1), 1/2, 1/2, 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/Sqrt[a + x^n*(b + c*x^n)]","A",0
586,1,47,47,0.0628828,"\int \frac{1}{x \sqrt{a+b x^n+c x^{2 n}}} \, dx","Integrate[1/(x*Sqrt[a + b*x^n + c*x^(2*n)]),x]","-\frac{\tanh ^{-1}\left(\frac{2 a+b x^n}{2 \sqrt{a} \sqrt{a+b x^n+c x^{2 n}}}\right)}{\sqrt{a} n}","-\frac{\tanh ^{-1}\left(\frac{2 a+b x^n}{2 \sqrt{a} \sqrt{a+b x^n+c x^{2 n}}}\right)}{\sqrt{a} n}",1,"-(ArcTanh[(2*a + b*x^n)/(2*Sqrt[a]*Sqrt[a + b*x^n + c*x^(2*n)])]/(Sqrt[a]*n))","A",1
587,1,173,149,0.1439886,"\int \frac{1}{x^2 \sqrt{a+b x^n+c x^{2 n}}} \, dx","Integrate[1/(x^2*Sqrt[a + b*x^n + c*x^(2*n)]),x]","-\frac{\sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(-\frac{1}{n};\frac{1}{2},\frac{1}{2};\frac{n-1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)}{x \sqrt{a+x^n \left(b+c x^n\right)}}","-\frac{\sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(-\frac{1}{n};\frac{1}{2},\frac{1}{2};-\frac{1-n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{x \sqrt{a+b x^n+c x^{2 n}}}",1,"-((Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[-n^(-1), 1/2, 1/2, (-1 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/(x*Sqrt[a + x^n*(b + c*x^n)]))","A",0
588,1,175,151,0.1455302,"\int \frac{1}{x^3 \sqrt{a+b x^n+c x^{2 n}}} \, dx","Integrate[1/(x^3*Sqrt[a + b*x^n + c*x^(2*n)]),x]","-\frac{\sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(-\frac{2}{n};\frac{1}{2},\frac{1}{2};\frac{n-2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)}{2 x^2 \sqrt{a+x^n \left(b+c x^n\right)}}","-\frac{\sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(-\frac{2}{n};\frac{1}{2},\frac{1}{2};-\frac{2-n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{2 x^2 \sqrt{a+b x^n+c x^{2 n}}}",1,"-1/2*(Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[-2/n, 1/2, 1/2, (-2 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/(x^2*Sqrt[a + x^n*(b + c*x^n)])","A",0
589,1,398,151,0.8594475,"\int \frac{x^3}{\left(a+b x^n+c x^{2 n}\right)^{3/2}} \, dx","Integrate[x^3/(a + b*x^n + c*x^(2*n))^(3/2),x]","\frac{x^4 \left(32 b c x^n \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{n+4}{n};\frac{1}{2},\frac{1}{2};2+\frac{4}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)-(n+4) \left(b^2 (n-8)-4 a c (n-4)\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{4}{n};\frac{1}{2},\frac{1}{2};\frac{n+4}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)-8 (n+4) \left(-2 a c+b^2+b c x^n\right)\right)}{4 a n (n+4) \left(4 a c-b^2\right) \sqrt{a+x^n \left(b+c x^n\right)}}","\frac{x^4 \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{4}{n};\frac{3}{2},\frac{3}{2};\frac{n+4}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{4 a \sqrt{a+b x^n+c x^{2 n}}}",1,"(x^4*(-8*(4 + n)*(b^2 - 2*a*c + b*c*x^n) - (b^2*(-8 + n) - 4*a*c*(-4 + n))*(4 + n)*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[4/n, 1/2, 1/2, (4 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + 32*b*c*x^n*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[(4 + n)/n, 1/2, 1/2, 2 + 4/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]))/(4*a*(-b^2 + 4*a*c)*n*(4 + n)*Sqrt[a + x^n*(b + c*x^n)])","B",0
590,1,398,151,0.8399814,"\int \frac{x^2}{\left(a+b x^n+c x^{2 n}\right)^{3/2}} \, dx","Integrate[x^2/(a + b*x^n + c*x^(2*n))^(3/2),x]","\frac{x^3 \left(18 b c x^n \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{n+3}{n};\frac{1}{2},\frac{1}{2};2+\frac{3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)-(n+3) \left(b^2 (n-6)-4 a c (n-3)\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{3}{n};\frac{1}{2},\frac{1}{2};\frac{n+3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)-6 (n+3) \left(-2 a c+b^2+b c x^n\right)\right)}{3 a n (n+3) \left(4 a c-b^2\right) \sqrt{a+x^n \left(b+c x^n\right)}}","\frac{x^3 \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{3}{n};\frac{3}{2},\frac{3}{2};\frac{n+3}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{3 a \sqrt{a+b x^n+c x^{2 n}}}",1,"(x^3*(-6*(3 + n)*(b^2 - 2*a*c + b*c*x^n) - (b^2*(-6 + n) - 4*a*c*(-3 + n))*(3 + n)*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[3/n, 1/2, 1/2, (3 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + 18*b*c*x^n*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[(3 + n)/n, 1/2, 1/2, 2 + 3/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]))/(3*a*(-b^2 + 4*a*c)*n*(3 + n)*Sqrt[a + x^n*(b + c*x^n)])","B",0
591,1,398,151,0.8388022,"\int \frac{x}{\left(a+b x^n+c x^{2 n}\right)^{3/2}} \, dx","Integrate[x/(a + b*x^n + c*x^(2*n))^(3/2),x]","\frac{x^2 \left(8 b c x^n \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{n+2}{n};\frac{1}{2},\frac{1}{2};2+\frac{2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)-(n+2) \left(b^2 (n-4)-4 a c (n-2)\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{2}{n};\frac{1}{2},\frac{1}{2};\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)-4 (n+2) \left(-2 a c+b^2+b c x^n\right)\right)}{2 a n (n+2) \left(4 a c-b^2\right) \sqrt{a+x^n \left(b+c x^n\right)}}","\frac{x^2 \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{2}{n};\frac{3}{2},\frac{3}{2};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{2 a \sqrt{a+b x^n+c x^{2 n}}}",1,"(x^2*(-4*(2 + n)*(b^2 - 2*a*c + b*c*x^n) - (b^2*(-4 + n) - 4*a*c*(-2 + n))*(2 + n)*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[2/n, 1/2, 1/2, (2 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + 8*b*c*x^n*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[(2 + n)/n, 1/2, 1/2, 2 + 2/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]))/(2*a*(-b^2 + 4*a*c)*n*(2 + n)*Sqrt[a + x^n*(b + c*x^n)])","B",0
592,1,384,142,0.9437472,"\int \frac{1}{\left(a+b x^n+c x^{2 n}\right)^{3/2}} \, dx","Integrate[(a + b*x^n + c*x^(2*n))^(-3/2),x]","\frac{x \left(2 b c x^n \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(1+\frac{1}{n};\frac{1}{2},\frac{1}{2};2+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)-(n+1) \left(\left(b^2 (n-2)-4 a c (n-1)\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{1}{n};\frac{1}{2},\frac{1}{2};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)+2 \left(-2 a c+b^2+b c x^n\right)\right)\right)}{a n (n+1) \left(4 a c-b^2\right) \sqrt{a+x^n \left(b+c x^n\right)}}","\frac{x \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{1}{n};\frac{3}{2},\frac{3}{2};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a \sqrt{a+b x^n+c x^{2 n}}}",1,"(x*(2*b*c*x^n*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[1 + n^(-1), 1/2, 1/2, 2 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] - (1 + n)*(2*(b^2 - 2*a*c + b*c*x^n) + (b^2*(-2 + n) - 4*a*c*(-1 + n))*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[n^(-1), 1/2, 1/2, 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])))/(a*(-b^2 + 4*a*c)*n*(1 + n)*Sqrt[a + x^n*(b + c*x^n)])","B",0
593,1,94,98,0.3102457,"\int \frac{1}{x \left(a+b x^n+c x^{2 n}\right)^{3/2}} \, dx","Integrate[1/(x*(a + b*x^n + c*x^(2*n))^(3/2)),x]","\frac{\frac{2 \left(-2 a c+b^2+b c x^n\right)}{a \left(b^2-4 a c\right) \sqrt{a+x^n \left(b+c x^n\right)}}-\frac{\tanh ^{-1}\left(\frac{2 a+b x^n}{2 \sqrt{a} \sqrt{a+x^n \left(b+c x^n\right)}}\right)}{a^{3/2}}}{n}","\frac{2 \left(-2 a c+b^2+b c x^n\right)}{a n \left(b^2-4 a c\right) \sqrt{a+b x^n+c x^{2 n}}}-\frac{\tanh ^{-1}\left(\frac{2 a+b x^n}{2 \sqrt{a} \sqrt{a+b x^n+c x^{2 n}}}\right)}{a^{3/2} n}",1,"((2*(b^2 - 2*a*c + b*c*x^n))/(a*(b^2 - 4*a*c)*Sqrt[a + x^n*(b + c*x^n)]) - ArcTanh[(2*a + b*x^n)/(2*Sqrt[a]*Sqrt[a + x^n*(b + c*x^n)])]/a^(3/2))/n","A",1
594,1,395,152,0.783707,"\int \frac{1}{x^2 \left(a+b x^n+c x^{2 n}\right)^{3/2}} \, dx","Integrate[1/(x^2*(a + b*x^n + c*x^(2*n))^(3/2)),x]","\frac{(n-1) \left(b^2 (n+2)-4 a c (n+1)\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(-\frac{1}{n};\frac{1}{2},\frac{1}{2};\frac{n-1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)-2 \left(b c x^n \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{n-1}{n};\frac{1}{2},\frac{1}{2};2-\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)+(n-1) \left(-2 a c+b^2+b c x^n\right)\right)}{a (n-1) n x \left(4 a c-b^2\right) \sqrt{a+x^n \left(b+c x^n\right)}}","-\frac{\sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(-\frac{1}{n};\frac{3}{2},\frac{3}{2};-\frac{1-n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a x \sqrt{a+b x^n+c x^{2 n}}}",1,"((-1 + n)*(-4*a*c*(1 + n) + b^2*(2 + n))*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[-n^(-1), 1/2, 1/2, (-1 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] - 2*((-1 + n)*(b^2 - 2*a*c + b*c*x^n) + b*c*x^n*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[(-1 + n)/n, 1/2, 1/2, 2 - n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]))/(a*(-b^2 + 4*a*c)*(-1 + n)*n*x*Sqrt[a + x^n*(b + c*x^n)])","B",0
595,1,399,154,0.7977715,"\int \frac{1}{x^3 \left(a+b x^n+c x^{2 n}\right)^{3/2}} \, dx","Integrate[1/(x^3*(a + b*x^n + c*x^(2*n))^(3/2)),x]","\frac{(n-2) \left(b^2 (n+4)-4 a c (n+2)\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(-\frac{2}{n};\frac{1}{2},\frac{1}{2};\frac{n-2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)-4 \left(2 b c x^n \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{n-2}{n};\frac{1}{2},\frac{1}{2};2-\frac{2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)+(n-2) \left(-2 a c+b^2+b c x^n\right)\right)}{2 a (n-2) n x^2 \left(4 a c-b^2\right) \sqrt{a+x^n \left(b+c x^n\right)}}","-\frac{\sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(-\frac{2}{n};\frac{3}{2},\frac{3}{2};-\frac{2-n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{2 a x^2 \sqrt{a+b x^n+c x^{2 n}}}",1,"((-2 + n)*(-4*a*c*(2 + n) + b^2*(4 + n))*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[-2/n, 1/2, 1/2, (-2 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] - 4*((-2 + n)*(b^2 - 2*a*c + b*c*x^n) + 2*b*c*x^n*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[(-2 + n)/n, 1/2, 1/2, 2 - 2/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]))/(2*a*(-b^2 + 4*a*c)*(-2 + n)*n*x^2*Sqrt[a + x^n*(b + c*x^n)])","B",0
596,1,137,182,0.3547049,"\int (d x)^m \left(a+b x^n+c x^{2 n}\right)^3 \, dx","Integrate[(d*x)^m*(a + b*x^n + c*x^(2*n))^3,x]","x (d x)^m \left(\frac{a^3}{m+1}+\frac{3 a^2 b x^n}{m+n+1}+\frac{3 a x^{2 n} \left(a c+b^2\right)}{m+2 n+1}+\frac{b x^{3 n} \left(6 a c+b^2\right)}{m+3 n+1}+\frac{3 c x^{4 n} \left(a c+b^2\right)}{m+4 n+1}+\frac{3 b c^2 x^{5 n}}{m+5 n+1}+\frac{c^3 x^{6 n}}{m+6 n+1}\right)","\frac{a^3 (d x)^{m+1}}{d (m+1)}+\frac{3 a^2 b x^{n+1} (d x)^m}{m+n+1}+\frac{3 a x^{2 n+1} \left(a c+b^2\right) (d x)^m}{m+2 n+1}+\frac{b x^{3 n+1} \left(6 a c+b^2\right) (d x)^m}{m+3 n+1}+\frac{3 c x^{4 n+1} \left(a c+b^2\right) (d x)^m}{m+4 n+1}+\frac{3 b c^2 x^{5 n+1} (d x)^m}{m+5 n+1}+\frac{c^3 x^{6 n+1} (d x)^m}{m+6 n+1}",1,"x*(d*x)^m*(a^3/(1 + m) + (3*a^2*b*x^n)/(1 + m + n) + (3*a*(b^2 + a*c)*x^(2*n))/(1 + m + 2*n) + (b*(b^2 + 6*a*c)*x^(3*n))/(1 + m + 3*n) + (3*c*(b^2 + a*c)*x^(4*n))/(1 + m + 4*n) + (3*b*c^2*x^(5*n))/(1 + m + 5*n) + (c^3*x^(6*n))/(1 + m + 6*n))","A",1
597,1,86,117,0.1507663,"\int (d x)^m \left(a+b x^n+c x^{2 n}\right)^2 \, dx","Integrate[(d*x)^m*(a + b*x^n + c*x^(2*n))^2,x]","x (d x)^m \left(\frac{a^2}{m+1}+\frac{x^{2 n} \left(2 a c+b^2\right)}{m+2 n+1}+\frac{2 a b x^n}{m+n+1}+\frac{2 b c x^{3 n}}{m+3 n+1}+\frac{c^2 x^{4 n}}{m+4 n+1}\right)","\frac{a^2 (d x)^{m+1}}{d (m+1)}+\frac{x^{2 n+1} \left(2 a c+b^2\right) (d x)^m}{m+2 n+1}+\frac{2 a b x^{n+1} (d x)^m}{m+n+1}+\frac{2 b c x^{3 n+1} (d x)^m}{m+3 n+1}+\frac{c^2 x^{4 n+1} (d x)^m}{m+4 n+1}",1,"x*(d*x)^m*(a^2/(1 + m) + (2*a*b*x^n)/(1 + m + n) + ((b^2 + 2*a*c)*x^(2*n))/(1 + m + 2*n) + (2*b*c*x^(3*n))/(1 + m + 3*n) + (c^2*x^(4*n))/(1 + m + 4*n))","A",1
598,1,41,58,0.0653568,"\int (d x)^m \left(a+b x^n+c x^{2 n}\right) \, dx","Integrate[(d*x)^m*(a + b*x^n + c*x^(2*n)),x]","x (d x)^m \left(\frac{a}{m+1}+x^n \left(\frac{b}{m+n+1}+\frac{c x^n}{m+2 n+1}\right)\right)","\frac{a (d x)^{m+1}}{d (m+1)}+\frac{b x^{n+1} (d x)^m}{m+n+1}+\frac{c x^{2 n+1} (d x)^m}{m+2 n+1}",1,"x*(d*x)^m*(a/(1 + m) + x^n*(b/(1 + m + n) + (c*x^n)/(1 + m + 2*n)))","A",1
599,1,307,175,0.8452164,"\int \frac{(d x)^m}{a+b x^n+c x^{2 n}} \, dx","Integrate[(d*x)^m/(a + b*x^n + c*x^(2*n)),x]","-\frac{x (d x)^m \left(\frac{2 c \left(1-2^{-\frac{m+1}{n}} \left(\frac{c x^n}{-\sqrt{b^2-4 a c}+b+2 c x^n}\right)^{-\frac{m+1}{n}} \, _2F_1\left(-\frac{m+1}{n},-\frac{m+1}{n};1-\frac{m+1}{n};\frac{b-\sqrt{b^2-4 a c}}{2 c x^n+b-\sqrt{b^2-4 a c}}\right)\right)}{-b \sqrt{b^2-4 a c}-4 a c+b^2}+\frac{2 c \left(1-2^{-\frac{m+1}{n}} \left(\frac{c x^n}{\sqrt{b^2-4 a c}+b+2 c x^n}\right)^{-\frac{m+1}{n}} \, _2F_1\left(-\frac{m+1}{n},-\frac{m+1}{n};\frac{-m+n-1}{n};\frac{b+\sqrt{b^2-4 a c}}{2 c x^n+b+\sqrt{b^2-4 a c}}\right)\right)}{\sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}+b\right)}\right)}{m+1}","\frac{2 c (d x)^{m+1} \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{d (m+1) \sqrt{b^2-4 a c} \left(b-\sqrt{b^2-4 a c}\right)}-\frac{2 c (d x)^{m+1} \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{d (m+1) \sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}+b\right)}",1,"-((x*(d*x)^m*((2*c*(1 - Hypergeometric2F1[-((1 + m)/n), -((1 + m)/n), 1 - (1 + m)/n, (b - Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(2^((1 + m)/n)*((c*x^n)/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n))^((1 + m)/n))))/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]) + (2*c*(1 - Hypergeometric2F1[-((1 + m)/n), -((1 + m)/n), (-1 - m + n)/n, (b + Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(2^((1 + m)/n)*((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^((1 + m)/n))))/(Sqrt[b^2 - 4*a*c]*(b + Sqrt[b^2 - 4*a*c]))))/(1 + m))","A",1
600,1,3515,328,6.3595742,"\int \frac{(d x)^m}{\left(a+b x^n+c x^{2 n}\right)^2} \, dx","Integrate[(d*x)^m/(a + b*x^n + c*x^(2*n))^2,x]","\text{Result too large to show}","\frac{c (d x)^{m+1} \left(\frac{4 a c (m-2 n+1)-b^2 (m-n+1)}{\sqrt{b^2-4 a c}}-b (m-n+1)\right) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{a d (m+1) n \left(b^2-4 a c\right) \left(b-\sqrt{b^2-4 a c}\right)}-\frac{c (d x)^{m+1} \left(b (m-n+1) \sqrt{b^2-4 a c}+4 a c (m-2 n+1)-\left(b^2 (m-n+1)\right)\right) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a d (m+1) n \left(b^2-4 a c\right)^{3/2} \left(\sqrt{b^2-4 a c}+b\right)}+\frac{(d x)^{m+1} \left(-2 a c+b^2+b c x^n\right)}{a d n \left(b^2-4 a c\right) \left(a+b x^n+c x^{2 n}\right)}",1,"(x*(d*x)^m*(-b^2 + 2*a*c - b*c*x^n))/(a*(-b^2 + 4*a*c)*n*(a + b*x^n + c*x^(2*n))) - (b*c*x^(1 + n)*(d*x)^m*(x^n)^((1 + m)/n - (1 + m + n)/n)*(-(((x^n/(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))^(-n^(-1) - m/n)*Hypergeometric2F1[-((1 + m)/n), -((1 + m)/n), 1 - (1 + m)/n, -1/2*(-b - Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))])/Sqrt[b^2 - 4*a*c]) + ((x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^(-n^(-1) - m/n)*Hypergeometric2F1[-((1 + m)/n), -((1 + m)/n), 1 - (1 + m)/n, -1/2*(-b + Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))])/Sqrt[b^2 - 4*a*c]))/(a*(-b^2 + 4*a*c)*(1 + m)) + (b*c*x^(1 + n)*(d*x)^m*(x^n)^((1 + m)/n - (1 + m + n)/n)*(-(((x^n/(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))^(-n^(-1) - m/n)*Hypergeometric2F1[-((1 + m)/n), -((1 + m)/n), 1 - (1 + m)/n, -1/2*(-b - Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))])/Sqrt[b^2 - 4*a*c]) + ((x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^(-n^(-1) - m/n)*Hypergeometric2F1[-((1 + m)/n), -((1 + m)/n), 1 - (1 + m)/n, -1/2*(-b + Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))])/Sqrt[b^2 - 4*a*c]))/(a*(-b^2 + 4*a*c)*(1 + m)*n) + (b*c*m*x^(1 + n)*(d*x)^m*(x^n)^((1 + m)/n - (1 + m + n)/n)*(-(((x^n/(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))^(-n^(-1) - m/n)*Hypergeometric2F1[-((1 + m)/n), -((1 + m)/n), 1 - (1 + m)/n, -1/2*(-b - Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))])/Sqrt[b^2 - 4*a*c]) + ((x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^(-n^(-1) - m/n)*Hypergeometric2F1[-((1 + m)/n), -((1 + m)/n), 1 - (1 + m)/n, -1/2*(-b + Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))])/Sqrt[b^2 - 4*a*c]))/(a*(-b^2 + 4*a*c)*(1 + m)*n) + (b^2*x*(d*x)^m*((1 - (x^n/(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))^(-n^(-1) - m/n)*Hypergeometric2F1[-((1 + m)/n), -((1 + m)/n), 1 - (1 + m)/n, -1/2*(-b - Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))])/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + (1 - (x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^(-n^(-1) - m/n)*Hypergeometric2F1[-((1 + m)/n), -((1 + m)/n), 1 - (1 + m)/n, -1/2*(-b + Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))])/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/(a*(-b^2 + 4*a*c)*(1 + m)) - (4*c*x*(d*x)^m*((1 - (x^n/(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))^(-n^(-1) - m/n)*Hypergeometric2F1[-((1 + m)/n), -((1 + m)/n), 1 - (1 + m)/n, -1/2*(-b - Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))])/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + (1 - (x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^(-n^(-1) - m/n)*Hypergeometric2F1[-((1 + m)/n), -((1 + m)/n), 1 - (1 + m)/n, -1/2*(-b + Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))])/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/((-b^2 + 4*a*c)*(1 + m)) - (b^2*x*(d*x)^m*((1 - (x^n/(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))^(-n^(-1) - m/n)*Hypergeometric2F1[-((1 + m)/n), -((1 + m)/n), 1 - (1 + m)/n, -1/2*(-b - Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))])/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + (1 - (x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^(-n^(-1) - m/n)*Hypergeometric2F1[-((1 + m)/n), -((1 + m)/n), 1 - (1 + m)/n, -1/2*(-b + Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))])/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/(a*(-b^2 + 4*a*c)*(1 + m)*n) + (2*c*x*(d*x)^m*((1 - (x^n/(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))^(-n^(-1) - m/n)*Hypergeometric2F1[-((1 + m)/n), -((1 + m)/n), 1 - (1 + m)/n, -1/2*(-b - Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))])/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + (1 - (x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^(-n^(-1) - m/n)*Hypergeometric2F1[-((1 + m)/n), -((1 + m)/n), 1 - (1 + m)/n, -1/2*(-b + Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))])/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/((-b^2 + 4*a*c)*(1 + m)*n) - (b^2*m*x*(d*x)^m*((1 - (x^n/(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))^(-n^(-1) - m/n)*Hypergeometric2F1[-((1 + m)/n), -((1 + m)/n), 1 - (1 + m)/n, -1/2*(-b - Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))])/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + (1 - (x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^(-n^(-1) - m/n)*Hypergeometric2F1[-((1 + m)/n), -((1 + m)/n), 1 - (1 + m)/n, -1/2*(-b + Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))])/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/(a*(-b^2 + 4*a*c)*(1 + m)*n) + (2*c*m*x*(d*x)^m*((1 - (x^n/(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))^(-n^(-1) - m/n)*Hypergeometric2F1[-((1 + m)/n), -((1 + m)/n), 1 - (1 + m)/n, -1/2*(-b - Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b - Sqrt[b^2 - 4*a*c])/c + x^n))])/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + (1 - (x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^(-n^(-1) - m/n)*Hypergeometric2F1[-((1 + m)/n), -((1 + m)/n), 1 - (1 + m)/n, -1/2*(-b + Sqrt[b^2 - 4*a*c])/(c*(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))])/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/((-b^2 + 4*a*c)*(1 + m)*n)","B",1
601,1,12289,615,6.8606038,"\int \frac{(d x)^m}{\left(a+b x^n+c x^{2 n}\right)^3} \, dx","Integrate[(d*x)^m/(a + b*x^n + c*x^(2*n))^3,x]","\text{Result too large to show}","-\frac{c (d x)^{m+1} \left(-8 a^2 c^2 \left(m^2+m (2-6 n)+8 n^2-6 n+1\right)+6 a b^2 c \left(m^2+m (2-4 n)+3 n^2-4 n+1\right)+b (m-n+1) \sqrt{b^2-4 a c} \left(2 a c (2 m-7 n+2)-b^2 (m-2 n+1)\right)-\left(b^4 \left(m^2+m (2-3 n)+2 n^2-3 n+1\right)\right)\right) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{2 a^2 d (m+1) n^2 \left(b^2-4 a c\right)^{5/2} \left(b-\sqrt{b^2-4 a c}\right)}-\frac{c (d x)^{m+1} \left(8 a^2 c^2 \left(m^2+m (2-6 n)+8 n^2-6 n+1\right)-6 a b^2 c \left(m^2+m (2-4 n)+3 n^2-4 n+1\right)+b (m-n+1) \sqrt{b^2-4 a c} \left(2 a c (2 m-7 n+2)-b^2 (m-2 n+1)\right)+b^4 \left(m^2+m (2-3 n)+2 n^2-3 n+1\right)\right) \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{2 a^2 d (m+1) n^2 \left(b^2-4 a c\right)^{5/2} \left(\sqrt{b^2-4 a c}+b\right)}-\frac{(d x)^{m+1} \left(4 a^2 c^2 (m-4 n+1)-b c x^n \left(2 a c (2 m-7 n+2)-b^2 (m-2 n+1)\right)-5 a b^2 c (m-3 n+1)+b^4 (m-2 n+1)\right)}{2 a^2 d n^2 \left(b^2-4 a c\right)^2 \left(a+b x^n+c x^{2 n}\right)}+\frac{(d x)^{m+1} \left(-2 a c+b^2+b c x^n\right)}{2 a d n \left(b^2-4 a c\right) \left(a+b x^n+c x^{2 n}\right)^2}",1,"Result too large to show","B",1
602,1,618,161,3.2110035,"\int (d x)^m \left(a+b x^n+c x^{2 n}\right)^{3/2} \, dx","Integrate[(d*x)^m*(a + b*x^n + c*x^(2*n))^(3/2),x]","\frac{x (d x)^m \left((m+1) \left(2 (m+n+1) \left(4 a^2 c \left(m^2+m (6 n+2)+8 n^2+6 n+1\right)+a \left(3 b^2 n^2+2 b c \left(4 m^2+m (21 n+8)+23 n^2+21 n+4\right) x^n+4 c^2 \left(2 m^2+m (9 n+4)+10 n^2+9 n+2\right) x^{2 n}\right)+x^n \left(b+c x^n\right) \left(3 b^2 n^2+2 b c \left(2 m^2+m (9 n+4)+7 n^2+9 n+2\right) x^n+4 c^2 \left(m^2+m (3 n+2)+2 n^2+3 n+1\right) x^{2 n}\right)\right)-3 b n^2 x^n \left(b^2 (2 m+n+2)-4 a c (2 m+3 n+2)\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{m+n+1}{n};\frac{1}{2},\frac{1}{2};\frac{m+2 n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)\right)-6 a n^2 (m+n+1) \left(b^2 (m+1)-4 a c (m+2 n+1)\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{m+1}{n};\frac{1}{2},\frac{1}{2};\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)\right)}{8 c (m+1) (m+n+1)^2 (m+2 n+1) (m+3 n+1) \sqrt{a+x^n \left(b+c x^n\right)}}","\frac{a (d x)^{m+1} \sqrt{a+b x^n+c x^{2 n}} F_1\left(\frac{m+1}{n};-\frac{3}{2},-\frac{3}{2};\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{d (m+1) \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}}",1,"(x*(d*x)^m*(-6*a*n^2*(1 + m + n)*(b^2*(1 + m) - 4*a*c*(1 + m + 2*n))*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[(1 + m)/n, 1/2, 1/2, (1 + m + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + (1 + m)*(2*(1 + m + n)*(4*a^2*c*(1 + m^2 + 6*n + 8*n^2 + m*(2 + 6*n)) + x^n*(b + c*x^n)*(3*b^2*n^2 + 2*b*c*(2 + 2*m^2 + 9*n + 7*n^2 + m*(4 + 9*n))*x^n + 4*c^2*(1 + m^2 + 3*n + 2*n^2 + m*(2 + 3*n))*x^(2*n)) + a*(3*b^2*n^2 + 2*b*c*(4 + 4*m^2 + 21*n + 23*n^2 + m*(8 + 21*n))*x^n + 4*c^2*(2 + 2*m^2 + 9*n + 10*n^2 + m*(4 + 9*n))*x^(2*n))) - 3*b*n^2*(b^2*(2 + 2*m + n) - 4*a*c*(2 + 2*m + 3*n))*x^n*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[(1 + m + n)/n, 1/2, 1/2, (1 + m + 2*n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])))/(8*c*(1 + m)*(1 + m + n)^2*(1 + m + 2*n)*(1 + m + 3*n)*Sqrt[a + x^n*(b + c*x^n)])","B",0
603,1,388,160,0.6908615,"\int (d x)^m \sqrt{a+b x^n+c x^{2 n}} \, dx","Integrate[(d*x)^m*Sqrt[a + b*x^n + c*x^(2*n)],x]","\frac{x (d x)^m \left(2 a n (m+n+1) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{m+1}{n};\frac{1}{2},\frac{1}{2};\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)+(m+1) \left(b n x^n \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{m+n+1}{n};\frac{1}{2},\frac{1}{2};\frac{m+2 n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)+2 (m+n+1) \left(a+x^n \left(b+c x^n\right)\right)\right)\right)}{2 (m+1) (m+n+1)^2 \sqrt{a+x^n \left(b+c x^n\right)}}","\frac{(d x)^{m+1} \sqrt{a+b x^n+c x^{2 n}} F_1\left(\frac{m+1}{n};-\frac{1}{2},-\frac{1}{2};\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{d (m+1) \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}}",1,"(x*(d*x)^m*(2*a*n*(1 + m + n)*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[(1 + m)/n, 1/2, 1/2, (1 + m + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + (1 + m)*(2*(1 + m + n)*(a + x^n*(b + c*x^n)) + b*n*x^n*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[(1 + m + n)/n, 1/2, 1/2, (1 + m + 2*n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])))/(2*(1 + m)*(1 + m + n)^2*Sqrt[a + x^n*(b + c*x^n)])","B",0
604,1,183,160,0.159262,"\int \frac{(d x)^m}{\sqrt{a+b x^n+c x^{2 n}}} \, dx","Integrate[(d*x)^m/Sqrt[a + b*x^n + c*x^(2*n)],x]","\frac{x (d x)^m \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{m+1}{n};\frac{1}{2},\frac{1}{2};\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)}{(m+1) \sqrt{a+x^n \left(b+c x^n\right)}}","\frac{(d x)^{m+1} \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{m+1}{n};\frac{1}{2},\frac{1}{2};\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{d (m+1) \sqrt{a+b x^n+c x^{2 n}}}",1,"(x*(d*x)^m*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[(1 + m)/n, 1/2, 1/2, (1 + m + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/((1 + m)*Sqrt[a + x^n*(b + c*x^n)])","A",0
605,1,428,163,1.4570016,"\int \frac{(d x)^m}{\left(a+b x^n+c x^{2 n}\right)^{3/2}} \, dx","Integrate[(d*x)^m/(a + b*x^n + c*x^(2*n))^(3/2),x]","\frac{x (d x)^m \left((m+n+1) \left(b^2 (2 m-n+2)-4 a c (m-n+1)\right) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{m+1}{n};\frac{1}{2},\frac{1}{2};\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)-2 (m+1) \left((m+n+1) \left(-2 a c+b^2+b c x^n\right)-b c (m+1) x^n \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left(\frac{m+n+1}{n};\frac{1}{2},\frac{1}{2};\frac{m+2 n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)\right)\right)}{a (m+1) n (m+n+1) \left(4 a c-b^2\right) \sqrt{a+x^n \left(b+c x^n\right)}}","\frac{(d x)^{m+1} \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{m+1}{n};\frac{3}{2},\frac{3}{2};\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a d (m+1) \sqrt{a+b x^n+c x^{2 n}}}",1,"(x*(d*x)^m*((-4*a*c*(1 + m - n) + b^2*(2 + 2*m - n))*(1 + m + n)*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[(1 + m)/n, 1/2, 1/2, (1 + m + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] - 2*(1 + m)*((1 + m + n)*(b^2 - 2*a*c + b*c*x^n) - b*c*(1 + m)*x^n*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[(1 + m + n)/n, 1/2, 1/2, (1 + m + 2*n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])))/(a*(-b^2 + 4*a*c)*(1 + m)*n*(1 + m + n)*Sqrt[a + x^n*(b + c*x^n)])","B",0
606,1,181,158,0.3579105,"\int (d x)^m \left(a+b x^n+c x^{2 n}\right)^p \, dx","Integrate[(d*x)^m*(a + b*x^n + c*x^(2*n))^p,x]","\frac{x (d x)^m \left(\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}\right)^{-p} \left(\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}\right)^{-p} \left(a+x^n \left(b+c x^n\right)\right)^p F_1\left(\frac{m+1}{n};-p,-p;\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right)}{m+1}","\frac{(d x)^{m+1} \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(\frac{m+1}{n};-p,-p;\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{d (m+1)}",1,"(x*(d*x)^m*(a + x^n*(b + c*x^n))^p*AppellF1[(1 + m)/n, -p, -p, (1 + m + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/((1 + m)*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]))^p)","A",0
607,1,150,46,0.0389027,"\int (d+e x)^3 \left(a+b (d+e x)^2+c (d+e x)^4\right) \, dx","Integrate[(d + e*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4),x]","\frac{1}{4} e^3 x^4 \left(a+10 b d^2+35 c d^4\right)+\frac{1}{3} d e^2 x^3 \left(3 a+10 b d^2+21 c d^4\right)+\frac{1}{2} d^2 e x^2 \left(3 a+5 b d^2+7 c d^4\right)+d^3 x \left(a+b d^2+c d^4\right)+\frac{1}{6} e^5 x^6 \left(b+21 c d^2\right)+d e^4 x^5 \left(b+7 c d^2\right)+c d e^6 x^7+\frac{1}{8} c e^7 x^8","\frac{a (d+e x)^4}{4 e}+\frac{b (d+e x)^6}{6 e}+\frac{c (d+e x)^8}{8 e}",1,"d^3*(a + b*d^2 + c*d^4)*x + (d^2*(3*a + 5*b*d^2 + 7*c*d^4)*e*x^2)/2 + (d*(3*a + 10*b*d^2 + 21*c*d^4)*e^2*x^3)/3 + ((a + 10*b*d^2 + 35*c*d^4)*e^3*x^4)/4 + d*(b + 7*c*d^2)*e^4*x^5 + ((b + 21*c*d^2)*e^5*x^6)/6 + c*d*e^6*x^7 + (c*e^7*x^8)/8","B",1
608,1,401,89,0.1137896,"\int (d+e x)^3 \left(a+b (d+e x)^2+c (d+e x)^4\right)^2 \, dx","Integrate[(d + e*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2,x]","\frac{1}{4} e^3 x^4 \left(a^2+20 a b d^2+70 a c d^4+35 b^2 d^4+168 b c d^6+165 c^2 d^8\right)+\frac{1}{3} d e^2 x^3 \left(3 a^2+20 a b d^2+42 a c d^4+21 b^2 d^4+72 b c d^6+55 c^2 d^8\right)+\frac{1}{2} d^2 e x^2 \left(3 a^2+10 a b d^2+14 a c d^4+7 b^2 d^4+18 b c d^6+11 c^2 d^8\right)+\frac{1}{8} e^7 x^8 \left(2 a c+b^2+72 b c d^2+330 c^2 d^4\right)+d e^6 x^7 \left(2 a c+b^2+24 b c d^2+66 c^2 d^4\right)+\frac{1}{6} e^5 x^6 \left(2 a b+42 a c d^2+21 b^2 d^2+252 b c d^4+462 c^2 d^6\right)+\frac{1}{5} d e^4 x^5 \left(10 a b+70 a c d^2+35 b^2 d^2+252 b c d^4+330 c^2 d^6\right)+d^3 x \left(a+b d^2+c d^4\right)^2+\frac{1}{10} c e^9 x^{10} \left(2 b+55 c d^2\right)+\frac{1}{3} c d e^8 x^9 \left(6 b+55 c d^2\right)+c^2 d e^{10} x^{11}+\frac{1}{12} c^2 e^{11} x^{12}","\frac{a^2 (d+e x)^4}{4 e}+\frac{\left(2 a c+b^2\right) (d+e x)^8}{8 e}+\frac{a b (d+e x)^6}{3 e}+\frac{b c (d+e x)^{10}}{5 e}+\frac{c^2 (d+e x)^{12}}{12 e}",1,"d^3*(a + b*d^2 + c*d^4)^2*x + (d^2*(3*a^2 + 10*a*b*d^2 + 7*b^2*d^4 + 14*a*c*d^4 + 18*b*c*d^6 + 11*c^2*d^8)*e*x^2)/2 + (d*(3*a^2 + 20*a*b*d^2 + 21*b^2*d^4 + 42*a*c*d^4 + 72*b*c*d^6 + 55*c^2*d^8)*e^2*x^3)/3 + ((a^2 + 20*a*b*d^2 + 35*b^2*d^4 + 70*a*c*d^4 + 168*b*c*d^6 + 165*c^2*d^8)*e^3*x^4)/4 + (d*(10*a*b + 35*b^2*d^2 + 70*a*c*d^2 + 252*b*c*d^4 + 330*c^2*d^6)*e^4*x^5)/5 + ((2*a*b + 21*b^2*d^2 + 42*a*c*d^2 + 252*b*c*d^4 + 462*c^2*d^6)*e^5*x^6)/6 + d*(b^2 + 2*a*c + 24*b*c*d^2 + 66*c^2*d^4)*e^6*x^7 + ((b^2 + 2*a*c + 72*b*c*d^2 + 330*c^2*d^4)*e^7*x^8)/8 + (c*d*(6*b + 55*c*d^2)*e^8*x^9)/3 + (c*(2*b + 55*c*d^2)*e^9*x^10)/10 + c^2*d*e^10*x^11 + (c^2*e^11*x^12)/12","B",1
609,1,797,138,0.2751066,"\int (d+e x)^3 \left(a+b (d+e x)^2+c (d+e x)^4\right)^3 \, dx","Integrate[(d + e*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3,x]","\frac{1}{16} c^3 e^{15} x^{16}+c^3 d e^{14} x^{15}+\frac{3}{14} c^2 \left(35 c d^2+b\right) e^{13} x^{14}+c^2 d \left(35 c d^2+3 b\right) e^{12} x^{13}+\frac{1}{4} c \left(455 c^2 d^4+78 b c d^2+b^2+a c\right) e^{11} x^{12}+3 c d \left(91 c^2 d^4+26 b c d^2+b^2+a c\right) e^{10} x^{11}+\frac{1}{10} \left(5005 c^3 d^6+2145 b c^2 d^4+165 a c^2 d^2+165 b^2 c d^2+b^3+6 a b c\right) e^9 x^{10}+d \left(715 c^3 d^6+429 b c^2 d^4+55 a c^2 d^2+55 b^2 c d^2+b^3+6 a b c\right) e^8 x^9+\frac{3}{8} \left(2145 c^3 d^8+1716 b c^2 d^6+330 a c^2 d^4+330 b^2 c d^4+12 b^3 d^2+72 a b c d^2+a b^2+a^2 c\right) e^7 x^8+\frac{1}{7} d \left(5005 c^3 d^8+5148 b c^2 d^6+1386 a c^2 d^4+1386 b^2 c d^4+84 b^3 d^2+504 a b c d^2+21 a b^2+21 a^2 c\right) e^6 x^7+\frac{1}{2} \left(1001 c^3 d^{10}+1287 b c^2 d^8+462 a c^2 d^6+462 b^2 c d^6+42 b^3 d^4+252 a b c d^4+21 a b^2 d^2+21 a^2 c d^2+a^2 b\right) e^5 x^6+\frac{3}{5} d \left(455 c^3 d^{10}+715 b c^2 d^8+330 a c^2 d^6+330 b^2 c d^6+42 b^3 d^4+252 a b c d^4+35 a b^2 d^2+35 a^2 c d^2+5 a^2 b\right) e^4 x^5+\frac{1}{4} \left(455 c^3 d^{12}+858 b c^2 d^{10}+495 a c^2 d^8+495 b^2 c d^8+84 b^3 d^6+504 a b c d^6+105 a b^2 d^4+105 a^2 c d^4+30 a^2 b d^2+a^3\right) e^3 x^4+d \left(35 c^3 d^{12}+78 b c^2 d^{10}+55 a c^2 d^8+55 b^2 c d^8+12 b^3 d^6+72 a b c d^6+21 a b^2 d^4+21 a^2 c d^4+10 a^2 b d^2+a^3\right) e^2 x^3+\frac{3}{2} d^2 \left(c d^4+b d^2+a\right)^2 \left(5 c d^4+3 b d^2+a\right) e x^2+d^3 \left(c d^4+b d^2+a\right)^3 x","\frac{a^3 (d+e x)^4}{4 e}+\frac{a^2 b (d+e x)^6}{2 e}+\frac{c \left(a c+b^2\right) (d+e x)^{12}}{4 e}+\frac{b \left(6 a c+b^2\right) (d+e x)^{10}}{10 e}+\frac{3 a \left(a c+b^2\right) (d+e x)^8}{8 e}+\frac{3 b c^2 (d+e x)^{14}}{14 e}+\frac{c^3 (d+e x)^{16}}{16 e}",1,"d^3*(a + b*d^2 + c*d^4)^3*x + (3*d^2*(a + b*d^2 + c*d^4)^2*(a + 3*b*d^2 + 5*c*d^4)*e*x^2)/2 + d*(a^3 + 10*a^2*b*d^2 + 21*a*b^2*d^4 + 21*a^2*c*d^4 + 12*b^3*d^6 + 72*a*b*c*d^6 + 55*b^2*c*d^8 + 55*a*c^2*d^8 + 78*b*c^2*d^10 + 35*c^3*d^12)*e^2*x^3 + ((a^3 + 30*a^2*b*d^2 + 105*a*b^2*d^4 + 105*a^2*c*d^4 + 84*b^3*d^6 + 504*a*b*c*d^6 + 495*b^2*c*d^8 + 495*a*c^2*d^8 + 858*b*c^2*d^10 + 455*c^3*d^12)*e^3*x^4)/4 + (3*d*(5*a^2*b + 35*a*b^2*d^2 + 35*a^2*c*d^2 + 42*b^3*d^4 + 252*a*b*c*d^4 + 330*b^2*c*d^6 + 330*a*c^2*d^6 + 715*b*c^2*d^8 + 455*c^3*d^10)*e^4*x^5)/5 + ((a^2*b + 21*a*b^2*d^2 + 21*a^2*c*d^2 + 42*b^3*d^4 + 252*a*b*c*d^4 + 462*b^2*c*d^6 + 462*a*c^2*d^6 + 1287*b*c^2*d^8 + 1001*c^3*d^10)*e^5*x^6)/2 + (d*(21*a*b^2 + 21*a^2*c + 84*b^3*d^2 + 504*a*b*c*d^2 + 1386*b^2*c*d^4 + 1386*a*c^2*d^4 + 5148*b*c^2*d^6 + 5005*c^3*d^8)*e^6*x^7)/7 + (3*(a*b^2 + a^2*c + 12*b^3*d^2 + 72*a*b*c*d^2 + 330*b^2*c*d^4 + 330*a*c^2*d^4 + 1716*b*c^2*d^6 + 2145*c^3*d^8)*e^7*x^8)/8 + d*(b^3 + 6*a*b*c + 55*b^2*c*d^2 + 55*a*c^2*d^2 + 429*b*c^2*d^4 + 715*c^3*d^6)*e^8*x^9 + ((b^3 + 6*a*b*c + 165*b^2*c*d^2 + 165*a*c^2*d^2 + 2145*b*c^2*d^4 + 5005*c^3*d^6)*e^9*x^10)/10 + 3*c*d*(b^2 + a*c + 26*b*c*d^2 + 91*c^2*d^4)*e^10*x^11 + (c*(b^2 + a*c + 78*b*c*d^2 + 455*c^2*d^4)*e^11*x^12)/4 + c^2*d*(3*b + 35*c*d^2)*e^12*x^13 + (3*c^2*(b + 35*c*d^2)*e^13*x^14)/14 + c^3*d*e^14*x^15 + (c^3*e^15*x^16)/16","B",1
610,1,154,55,0.0103423,"\int (d f+e f x)^3 \left(a+b (d+e x)^2+c (d+e x)^4\right) \, dx","Integrate[(d*f + e*f*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4),x]","f^3 \left(\frac{1}{4} e^3 x^4 \left(a+10 b d^2+35 c d^4\right)+\frac{1}{3} d e^2 x^3 \left(3 a+10 b d^2+21 c d^4\right)+\frac{1}{2} d^2 e x^2 \left(3 a+5 b d^2+7 c d^4\right)+d^3 x \left(a+b d^2+c d^4\right)+\frac{1}{6} e^5 x^6 \left(b+21 c d^2\right)+d e^4 x^5 \left(b+7 c d^2\right)+c d e^6 x^7+\frac{1}{8} c e^7 x^8\right)","\frac{a f^3 (d+e x)^4}{4 e}+\frac{b f^3 (d+e x)^6}{6 e}+\frac{c f^3 (d+e x)^8}{8 e}",1,"f^3*(d^3*(a + b*d^2 + c*d^4)*x + (d^2*(3*a + 5*b*d^2 + 7*c*d^4)*e*x^2)/2 + (d*(3*a + 10*b*d^2 + 21*c*d^4)*e^2*x^3)/3 + ((a + 10*b*d^2 + 35*c*d^4)*e^3*x^4)/4 + d*(b + 7*c*d^2)*e^4*x^5 + ((b + 21*c*d^2)*e^5*x^6)/6 + c*d*e^6*x^7 + (c*e^7*x^8)/8)","B",1
611,1,405,104,0.0738036,"\int (d f+e f x)^3 \left(a+b (d+e x)^2+c (d+e x)^4\right)^2 \, dx","Integrate[(d*f + e*f*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2,x]","f^3 \left(\frac{1}{4} e^3 x^4 \left(a^2+20 a b d^2+70 a c d^4+35 b^2 d^4+168 b c d^6+165 c^2 d^8\right)+\frac{1}{3} d e^2 x^3 \left(3 a^2+20 a b d^2+42 a c d^4+21 b^2 d^4+72 b c d^6+55 c^2 d^8\right)+\frac{1}{2} d^2 e x^2 \left(3 a^2+10 a b d^2+14 a c d^4+7 b^2 d^4+18 b c d^6+11 c^2 d^8\right)+\frac{1}{8} e^7 x^8 \left(2 a c+b^2+72 b c d^2+330 c^2 d^4\right)+d e^6 x^7 \left(2 a c+b^2+24 b c d^2+66 c^2 d^4\right)+\frac{1}{6} e^5 x^6 \left(2 a b+42 a c d^2+21 b^2 d^2+252 b c d^4+462 c^2 d^6\right)+\frac{1}{5} d e^4 x^5 \left(10 a b+70 a c d^2+35 b^2 d^2+252 b c d^4+330 c^2 d^6\right)+d^3 x \left(a+b d^2+c d^4\right)^2+\frac{1}{10} c e^9 x^{10} \left(2 b+55 c d^2\right)+\frac{1}{3} c d e^8 x^9 \left(6 b+55 c d^2\right)+c^2 d e^{10} x^{11}+\frac{1}{12} c^2 e^{11} x^{12}\right)","\frac{a^2 f^3 (d+e x)^4}{4 e}+\frac{f^3 \left(2 a c+b^2\right) (d+e x)^8}{8 e}+\frac{a b f^3 (d+e x)^6}{3 e}+\frac{b c f^3 (d+e x)^{10}}{5 e}+\frac{c^2 f^3 (d+e x)^{12}}{12 e}",1,"f^3*(d^3*(a + b*d^2 + c*d^4)^2*x + (d^2*(3*a^2 + 10*a*b*d^2 + 7*b^2*d^4 + 14*a*c*d^4 + 18*b*c*d^6 + 11*c^2*d^8)*e*x^2)/2 + (d*(3*a^2 + 20*a*b*d^2 + 21*b^2*d^4 + 42*a*c*d^4 + 72*b*c*d^6 + 55*c^2*d^8)*e^2*x^3)/3 + ((a^2 + 20*a*b*d^2 + 35*b^2*d^4 + 70*a*c*d^4 + 168*b*c*d^6 + 165*c^2*d^8)*e^3*x^4)/4 + (d*(10*a*b + 35*b^2*d^2 + 70*a*c*d^2 + 252*b*c*d^4 + 330*c^2*d^6)*e^4*x^5)/5 + ((2*a*b + 21*b^2*d^2 + 42*a*c*d^2 + 252*b*c*d^4 + 462*c^2*d^6)*e^5*x^6)/6 + d*(b^2 + 2*a*c + 24*b*c*d^2 + 66*c^2*d^4)*e^6*x^7 + ((b^2 + 2*a*c + 72*b*c*d^2 + 330*c^2*d^4)*e^7*x^8)/8 + (c*d*(6*b + 55*c*d^2)*e^8*x^9)/3 + (c*(2*b + 55*c*d^2)*e^9*x^10)/10 + c^2*d*e^10*x^11 + (c^2*e^11*x^12)/12)","B",1
612,1,801,159,0.0418511,"\int (d f+e f x)^3 \left(a+b (d+e x)^2+c (d+e x)^4\right)^3 \, dx","Integrate[(d*f + e*f*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3,x]","f^3 \left(\frac{1}{16} c^3 e^{15} x^{16}+c^3 d e^{14} x^{15}+\frac{3}{14} c^2 \left(35 c d^2+b\right) e^{13} x^{14}+c^2 d \left(35 c d^2+3 b\right) e^{12} x^{13}+\frac{1}{4} c \left(455 c^2 d^4+78 b c d^2+b^2+a c\right) e^{11} x^{12}+3 c d \left(91 c^2 d^4+26 b c d^2+b^2+a c\right) e^{10} x^{11}+\frac{1}{10} \left(5005 c^3 d^6+2145 b c^2 d^4+165 a c^2 d^2+165 b^2 c d^2+b^3+6 a b c\right) e^9 x^{10}+d \left(715 c^3 d^6+429 b c^2 d^4+55 a c^2 d^2+55 b^2 c d^2+b^3+6 a b c\right) e^8 x^9+\frac{3}{8} \left(2145 c^3 d^8+1716 b c^2 d^6+330 a c^2 d^4+330 b^2 c d^4+12 b^3 d^2+72 a b c d^2+a b^2+a^2 c\right) e^7 x^8+\frac{1}{7} d \left(5005 c^3 d^8+5148 b c^2 d^6+1386 a c^2 d^4+1386 b^2 c d^4+84 b^3 d^2+504 a b c d^2+21 a b^2+21 a^2 c\right) e^6 x^7+\frac{1}{2} \left(1001 c^3 d^{10}+1287 b c^2 d^8+462 a c^2 d^6+462 b^2 c d^6+42 b^3 d^4+252 a b c d^4+21 a b^2 d^2+21 a^2 c d^2+a^2 b\right) e^5 x^6+\frac{3}{5} d \left(455 c^3 d^{10}+715 b c^2 d^8+330 a c^2 d^6+330 b^2 c d^6+42 b^3 d^4+252 a b c d^4+35 a b^2 d^2+35 a^2 c d^2+5 a^2 b\right) e^4 x^5+\frac{1}{4} \left(455 c^3 d^{12}+858 b c^2 d^{10}+495 a c^2 d^8+495 b^2 c d^8+84 b^3 d^6+504 a b c d^6+105 a b^2 d^4+105 a^2 c d^4+30 a^2 b d^2+a^3\right) e^3 x^4+d \left(35 c^3 d^{12}+78 b c^2 d^{10}+55 a c^2 d^8+55 b^2 c d^8+12 b^3 d^6+72 a b c d^6+21 a b^2 d^4+21 a^2 c d^4+10 a^2 b d^2+a^3\right) e^2 x^3+\frac{3}{2} d^2 \left(c d^4+b d^2+a\right)^2 \left(5 c d^4+3 b d^2+a\right) e x^2+d^3 \left(c d^4+b d^2+a\right)^3 x\right)","\frac{a^3 f^3 (d+e x)^4}{4 e}+\frac{a^2 b f^3 (d+e x)^6}{2 e}+\frac{c f^3 \left(a c+b^2\right) (d+e x)^{12}}{4 e}+\frac{b f^3 \left(6 a c+b^2\right) (d+e x)^{10}}{10 e}+\frac{3 a f^3 \left(a c+b^2\right) (d+e x)^8}{8 e}+\frac{3 b c^2 f^3 (d+e x)^{14}}{14 e}+\frac{c^3 f^3 (d+e x)^{16}}{16 e}",1,"f^3*(d^3*(a + b*d^2 + c*d^4)^3*x + (3*d^2*(a + b*d^2 + c*d^4)^2*(a + 3*b*d^2 + 5*c*d^4)*e*x^2)/2 + d*(a^3 + 10*a^2*b*d^2 + 21*a*b^2*d^4 + 21*a^2*c*d^4 + 12*b^3*d^6 + 72*a*b*c*d^6 + 55*b^2*c*d^8 + 55*a*c^2*d^8 + 78*b*c^2*d^10 + 35*c^3*d^12)*e^2*x^3 + ((a^3 + 30*a^2*b*d^2 + 105*a*b^2*d^4 + 105*a^2*c*d^4 + 84*b^3*d^6 + 504*a*b*c*d^6 + 495*b^2*c*d^8 + 495*a*c^2*d^8 + 858*b*c^2*d^10 + 455*c^3*d^12)*e^3*x^4)/4 + (3*d*(5*a^2*b + 35*a*b^2*d^2 + 35*a^2*c*d^2 + 42*b^3*d^4 + 252*a*b*c*d^4 + 330*b^2*c*d^6 + 330*a*c^2*d^6 + 715*b*c^2*d^8 + 455*c^3*d^10)*e^4*x^5)/5 + ((a^2*b + 21*a*b^2*d^2 + 21*a^2*c*d^2 + 42*b^3*d^4 + 252*a*b*c*d^4 + 462*b^2*c*d^6 + 462*a*c^2*d^6 + 1287*b*c^2*d^8 + 1001*c^3*d^10)*e^5*x^6)/2 + (d*(21*a*b^2 + 21*a^2*c + 84*b^3*d^2 + 504*a*b*c*d^2 + 1386*b^2*c*d^4 + 1386*a*c^2*d^4 + 5148*b*c^2*d^6 + 5005*c^3*d^8)*e^6*x^7)/7 + (3*(a*b^2 + a^2*c + 12*b^3*d^2 + 72*a*b*c*d^2 + 330*b^2*c*d^4 + 330*a*c^2*d^4 + 1716*b*c^2*d^6 + 2145*c^3*d^8)*e^7*x^8)/8 + d*(b^3 + 6*a*b*c + 55*b^2*c*d^2 + 55*a*c^2*d^2 + 429*b*c^2*d^4 + 715*c^3*d^6)*e^8*x^9 + ((b^3 + 6*a*b*c + 165*b^2*c*d^2 + 165*a*c^2*d^2 + 2145*b*c^2*d^4 + 5005*c^3*d^6)*e^9*x^10)/10 + 3*c*d*(b^2 + a*c + 26*b*c*d^2 + 91*c^2*d^4)*e^10*x^11 + (c*(b^2 + a*c + 78*b*c*d^2 + 455*c^2*d^4)*e^11*x^12)/4 + c^2*d*(3*b + 35*c*d^2)*e^12*x^13 + (3*c^2*(b + 35*c*d^2)*e^13*x^14)/14 + c^3*d*e^14*x^15 + (c^3*e^15*x^16)/16)","B",1
613,1,219,193,0.1384006,"\int \frac{(d+e x)^4}{a+b (d+e x)^2+c (d+e x)^4} \, dx","Integrate[(d + e*x)^4/(a + b*(d + e*x)^2 + c*(d + e*x)^4),x]","\frac{-\frac{\sqrt{2} \left(b \sqrt{b^2-4 a c}+2 a c-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \left(b \sqrt{b^2-4 a c}-2 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+2 \sqrt{c} (d+e x)}{2 c^{3/2} e}","-\frac{\left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} c^{3/2} e \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} c^{3/2} e \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x}{c}",1,"(2*Sqrt[c]*(d + e*x) - (Sqrt[2]*(-b^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*(b^2 - 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(2*c^(3/2)*e)","A",1
614,1,77,81,0.0436752,"\int \frac{(d+e x)^3}{a+b (d+e x)^2+c (d+e x)^4} \, dx","Integrate[(d + e*x)^3/(a + b*(d + e*x)^2 + c*(d + e*x)^4),x]","\frac{\log \left(a+b (d+e x)^2+c (d+e x)^4\right)-\frac{2 b \tan ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}}{4 c e}","\frac{b \tanh ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right)}{2 c e \sqrt{b^2-4 a c}}+\frac{\log \left(a+b (d+e x)^2+c (d+e x)^4\right)}{4 c e}",1,"((-2*b*ArcTan[(b + 2*c*(d + e*x)^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + Log[a + b*(d + e*x)^2 + c*(d + e*x)^4])/(4*c*e)","A",1
615,1,175,164,0.0913049,"\int \frac{(d+e x)^2}{a+b (d+e x)^2+c (d+e x)^4} \, dx","Integrate[(d + e*x)^2/(a + b*(d + e*x)^2 + c*(d + e*x)^4),x]","\frac{\left(\sqrt{b^2-4 a c}-b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)+\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{\sqrt{b^2-4 a c}+b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} \sqrt{c} e \sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}","\frac{\sqrt{\sqrt{b^2-4 a c}+b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} \sqrt{c} e \sqrt{b^2-4 a c}}-\frac{\sqrt{b-\sqrt{b^2-4 a c}} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} \sqrt{c} e \sqrt{b^2-4 a c}}",1,"((-b + Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]] + Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[b + Sqrt[b^2 - 4*a*c]]*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]*e)","A",1
616,1,46,43,0.0158193,"\int \frac{d+e x}{a+b (d+e x)^2+c (d+e x)^4} \, dx","Integrate[(d + e*x)/(a + b*(d + e*x)^2 + c*(d + e*x)^4),x]","\frac{\tan ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{4 a c-b^2}}\right)}{e \sqrt{4 a c-b^2}}","-\frac{\tanh ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right)}{e \sqrt{b^2-4 a c}}",1,"ArcTan[(b + 2*c*(d + e*x)^2)/Sqrt[-b^2 + 4*a*c]]/(Sqrt[-b^2 + 4*a*c]*e)","A",1
617,1,128,94,0.0775989,"\int \frac{1}{(d+e x) \left(a+b (d+e x)^2+c (d+e x)^4\right)} \, dx","Integrate[1/((d + e*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)),x]","\frac{4 \sqrt{b^2-4 a c} \log (d+e x)-\left(\sqrt{b^2-4 a c}+b\right) \log \left(-\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right)+\left(b-\sqrt{b^2-4 a c}\right) \log \left(\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right)}{4 a e \sqrt{b^2-4 a c}}","\frac{b \tanh ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right)}{2 a e \sqrt{b^2-4 a c}}-\frac{\log \left(a+b (d+e x)^2+c (d+e x)^4\right)}{4 a e}+\frac{\log (d+e x)}{a e}",1,"(4*Sqrt[b^2 - 4*a*c]*Log[d + e*x] - (b + Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2] + (b - Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/(4*a*Sqrt[b^2 - 4*a*c]*e)","A",1
618,1,206,195,0.3578756,"\int \frac{1}{(d+e x)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)} \, dx","Integrate[1/((d + e*x)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)),x]","-\frac{\frac{\sqrt{2} \sqrt{c} \left(\sqrt{b^2-4 a c}+b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left(\sqrt{b^2-4 a c}-b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2}{d+e x}}{2 a e}","-\frac{\sqrt{c} \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} a e \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} a e \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{1}{a e (d+e x)}",1,"-1/2*(2/(d + e*x) + (Sqrt[2]*Sqrt[c]*(b + Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(-b + Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(a*e)","A",1
619,1,154,121,0.1336218,"\int \frac{1}{(d+e x)^3 \left(a+b (d+e x)^2+c (d+e x)^4\right)} \, dx","Integrate[1/((d + e*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)),x]","\frac{\frac{\left(b \sqrt{b^2-4 a c}-2 a c+b^2\right) \log \left(-\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right)}{\sqrt{b^2-4 a c}}+\frac{\left(b \sqrt{b^2-4 a c}+2 a c-b^2\right) \log \left(\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right)}{\sqrt{b^2-4 a c}}-\frac{2 a}{(d+e x)^2}-4 b \log (d+e x)}{4 a^2 e}","-\frac{\left(b^2-2 a c\right) \tanh ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right)}{2 a^2 e \sqrt{b^2-4 a c}}+\frac{b \log \left(a+b (d+e x)^2+c (d+e x)^4\right)}{4 a^2 e}-\frac{b \log (d+e x)}{a^2 e}-\frac{1}{2 a e (d+e x)^2}",1,"((-2*a)/(d + e*x)^2 - 4*b*Log[d + e*x] + ((b^2 - 2*a*c + b*Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/Sqrt[b^2 - 4*a*c] + ((-b^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/Sqrt[b^2 - 4*a*c])/(4*a^2*e)","A",1
620,1,235,224,0.2090407,"\int \frac{1}{(d+e x)^4 \left(a+b (d+e x)^2+c (d+e x)^4\right)} \, dx","Integrate[1/((d + e*x)^4*(a + b*(d + e*x)^2 + c*(d + e*x)^4)),x]","\frac{\frac{3 \sqrt{2} \sqrt{c} \left(b \sqrt{b^2-4 a c}-2 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{2} \sqrt{c} \left(b \sqrt{b^2-4 a c}+2 a c-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{2 a}{(d+e x)^3}+\frac{6 b}{d+e x}}{6 a^2 e}","\frac{\sqrt{c} \left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} a^2 e \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} a^2 e \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{b}{a^2 e (d+e x)}-\frac{1}{3 a e (d+e x)^3}",1,"((-2*a)/(d + e*x)^3 + (6*b)/(d + e*x) + (3*Sqrt[2]*Sqrt[c]*(b^2 - 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[2]*Sqrt[c]*(-b^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(6*a^2*e)","A",1
621,1,263,270,0.4682265,"\int \frac{(d+e x)^4}{\left(a+b (d+e x)^2+c (d+e x)^4\right)^2} \, dx","Integrate[(d + e*x)^4/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2,x]","\frac{-\frac{2 \left(-2 a (d+e x)-b (d+e x)^3\right)}{\left(b^2-4 a c\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)}+\frac{\sqrt{2} \left(b \sqrt{b^2-4 a c}-4 a c-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left(b \sqrt{b^2-4 a c}+4 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}}{4 e}","\frac{(d+e x) \left(2 a+b (d+e x)^2\right)}{2 e \left(b^2-4 a c\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)}+\frac{\left(b-\frac{4 a c+b^2}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} \sqrt{c} e \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left(b \sqrt{b^2-4 a c}+4 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} \sqrt{c} e \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}",1,"((-2*(-2*a*(d + e*x) - b*(d + e*x)^3))/((b^2 - 4*a*c)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (Sqrt[2]*(-b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(b^2 + 4*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(4*e)","A",1
622,1,100,97,0.1309945,"\int \frac{(d+e x)^3}{\left(a+b (d+e x)^2+c (d+e x)^4\right)^2} \, dx","Integrate[(d + e*x)^3/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2,x]","\frac{\frac{2 a+b (d+e x)^2}{\left(b^2-4 a c\right) \left(a+(d+e x)^2 \left(b+c (d+e x)^2\right)\right)}-\frac{2 b \tan ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{3/2}}}{2 e}","\frac{2 a+b (d+e x)^2}{2 e \left(b^2-4 a c\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)}-\frac{b \tanh ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right)}{e \left(b^2-4 a c\right)^{3/2}}",1,"((2*a + b*(d + e*x)^2)/((b^2 - 4*a*c)*(a + (d + e*x)^2*(b + c*(d + e*x)^2))) - (2*b*ArcTan[(b + 2*c*(d + e*x)^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2))/(2*e)","A",1
623,1,247,254,0.9754868,"\int \frac{(d+e x)^2}{\left(a+b (d+e x)^2+c (d+e x)^4\right)^2} \, dx","Integrate[(d + e*x)^2/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2,x]","-\frac{\frac{b (d+e x)+2 c (d+e x)^3}{\left(b^2-4 a c\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)}+\frac{\sqrt{2} \sqrt{c} \left(\sqrt{b^2-4 a c}-2 b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left(\sqrt{b^2-4 a c}+2 b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}}{2 e}","-\frac{(d+e x) \left(b+2 c (d+e x)^2\right)}{2 e \left(b^2-4 a c\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)}+\frac{\sqrt{c} \left(2 b-\sqrt{b^2-4 a c}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} e \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left(\sqrt{b^2-4 a c}+2 b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} e \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}",1,"-1/2*((b*(d + e*x) + 2*c*(d + e*x)^3)/((b^2 - 4*a*c)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (Sqrt[2]*Sqrt[c]*(-2*b + Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(2*b + Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/e","A",1
624,1,98,98,0.1218011,"\int \frac{d+e x}{\left(a+b (d+e x)^2+c (d+e x)^4\right)^2} \, dx","Integrate[(d + e*x)/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2,x]","-\frac{\frac{4 c \tan ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+\frac{b+2 c (d+e x)^2}{a+b (d+e x)^2+c (d+e x)^4}}{2 e \left(b^2-4 a c\right)}","\frac{-b-2 c (d+e x)^2}{2 e \left(b^2-4 a c\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)}+\frac{2 c \tanh ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right)}{e \left(b^2-4 a c\right)^{3/2}}",1,"-1/2*((b + 2*c*(d + e*x)^2)/(a + b*(d + e*x)^2 + c*(d + e*x)^4) + (4*c*ArcTan[(b + 2*c*(d + e*x)^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/((b^2 - 4*a*c)*e)","A",1
625,1,271,299,0.8889106,"\int \frac{1}{\left(a+b (d+e x)^2+c (d+e x)^4\right)^2} \, dx","Integrate[(a + b*(d + e*x)^2 + c*(d + e*x)^4)^(-2),x]","\frac{\frac{2 (d+e x) \left(-2 a c+b^2+b c (d+e x)^2\right)}{\left(b^2-4 a c\right) \left(a+(d+e x)^2 \left(b+c (d+e x)^2\right)\right)}+\frac{\sqrt{2} \sqrt{c} \left(b \sqrt{b^2-4 a c}-12 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left(b \sqrt{b^2-4 a c}+12 a c-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}}{4 a e}","\frac{\left(\frac{d}{e}+x\right) \left(-2 a c+b^2+b c e^2 \left(\frac{d}{e}+x\right)^2\right)}{2 a \left(b^2-4 a c\right) \left(a+b e^2 \left(\frac{d}{e}+x\right)^2+c e^4 \left(\frac{d}{e}+x\right)^4\right)}+\frac{\sqrt{c} \left(b \sqrt{b^2-4 a c}-12 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} a e \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left(-b \sqrt{b^2-4 a c}-12 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} a e \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}",1,"((2*(d + e*x)*(b^2 - 2*a*c + b*c*(d + e*x)^2))/((b^2 - 4*a*c)*(a + (d + e*x)^2*(b + c*(d + e*x)^2))) + (Sqrt[2]*Sqrt[c]*(b^2 - 12*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(-b^2 + 12*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(4*a*e)","A",1
626,1,235,162,0.4730175,"\int \frac{1}{(d+e x) \left(a+b (d+e x)^2+c (d+e x)^4\right)^2} \, dx","Integrate[1/((d + e*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2),x]","\frac{\frac{2 a \left(-2 a c+b^2+b c (d+e x)^2\right)}{\left(b^2-4 a c\right) \left(a+(d+e x)^2 \left(b+c (d+e x)^2\right)\right)}-\frac{\left(b^2 \sqrt{b^2-4 a c}-4 a c \sqrt{b^2-4 a c}-6 a b c+b^3\right) \log \left(-\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{\left(-b^2 \sqrt{b^2-4 a c}+4 a c \sqrt{b^2-4 a c}-6 a b c+b^3\right) \log \left(\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right)}{\left(b^2-4 a c\right)^{3/2}}+4 \log (d+e x)}{4 a^2 e}","\frac{b \left(b^2-6 a c\right) \tanh ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right)}{2 a^2 e \left(b^2-4 a c\right)^{3/2}}-\frac{\log \left(a+b (d+e x)^2+c (d+e x)^4\right)}{4 a^2 e}+\frac{\log (d+e x)}{a^2 e}+\frac{-2 a c+b^2+b c (d+e x)^2}{2 a e \left(b^2-4 a c\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)}",1,"((2*a*(b^2 - 2*a*c + b*c*(d + e*x)^2))/((b^2 - 4*a*c)*(a + (d + e*x)^2*(b + c*(d + e*x)^2))) + 4*Log[d + e*x] - ((b^3 - 6*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - 4*a*c*Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/(b^2 - 4*a*c)^(3/2) + ((b^3 - 6*a*b*c - b^2*Sqrt[b^2 - 4*a*c] + 4*a*c*Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/(b^2 - 4*a*c)^(3/2))/(4*a^2*e)","A",1
627,1,339,348,1.6322149,"\int \frac{1}{(d+e x)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)^2} \, dx","Integrate[1/((d + e*x)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2),x]","\frac{\frac{2 (d+e x) \left(-3 a b c-2 a c^2 (d+e x)^2+b^3+b^2 c (d+e x)^2\right)}{\left(4 a c-b^2\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)}+\frac{\sqrt{2} \sqrt{c} \left(-3 b^2 \sqrt{b^2-4 a c}+10 a c \sqrt{b^2-4 a c}+16 a b c-3 b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left(-3 b^2 \sqrt{b^2-4 a c}+10 a c \sqrt{b^2-4 a c}-16 a b c+3 b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{4}{d+e x}}{4 a^2 e}","-\frac{3 b^2-10 a c}{2 a^2 e \left(b^2-4 a c\right) (d+e x)}-\frac{\sqrt{c} \left(\left(3 b^2-10 a c\right) \sqrt{b^2-4 a c}-16 a b c+3 b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} a^2 e \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left(-\left(3 b^2-10 a c\right) \sqrt{b^2-4 a c}-16 a b c+3 b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} a^2 e \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{-2 a c+b^2+b c (d+e x)^2}{2 a e \left(b^2-4 a c\right) (d+e x) \left(a+b (d+e x)^2+c (d+e x)^4\right)}",1,"(-4/(d + e*x) + (2*(d + e*x)*(b^3 - 3*a*b*c + b^2*c*(d + e*x)^2 - 2*a*c^2*(d + e*x)^2))/((-b^2 + 4*a*c)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (Sqrt[2]*Sqrt[c]*(-3*b^3 + 16*a*b*c - 3*b^2*Sqrt[b^2 - 4*a*c] + 10*a*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(3*b^3 - 16*a*b*c - 3*b^2*Sqrt[b^2 - 4*a*c] + 10*a*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(4*a^2*e)","A",1
628,1,284,213,0.5204199,"\int \frac{1}{(d+e x)^3 \left(a+b (d+e x)^2+c (d+e x)^4\right)^2} \, dx","Integrate[1/((d + e*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2),x]","\frac{\frac{\left(6 a^2 c^2-6 a b^2 c-4 a b c \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}+b^4\right) \log \left(-\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{\left(-6 a^2 c^2+6 a b^2 c-4 a b c \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}-b^4\right) \log \left(\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{a \left(-3 a b c-2 a c^2 (d+e x)^2+b^3+b^2 c (d+e x)^2\right)}{\left(4 a c-b^2\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)}-\frac{a}{(d+e x)^2}-4 b \log (d+e x)}{2 a^3 e}","\frac{b \log \left(a+b (d+e x)^2+c (d+e x)^4\right)}{2 a^3 e}-\frac{2 b \log (d+e x)}{a^3 e}-\frac{b^2-3 a c}{a^2 e \left(b^2-4 a c\right) (d+e x)^2}-\frac{\left(6 a^2 c^2-6 a b^2 c+b^4\right) \tanh ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right)}{a^3 e \left(b^2-4 a c\right)^{3/2}}+\frac{-2 a c+b^2+b c (d+e x)^2}{2 a e \left(b^2-4 a c\right) (d+e x)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)}",1,"(-(a/(d + e*x)^2) + (a*(b^3 - 3*a*b*c + b^2*c*(d + e*x)^2 - 2*a*c^2*(d + e*x)^2))/((-b^2 + 4*a*c)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) - 4*b*Log[d + e*x] + ((b^4 - 6*a*b^2*c + 6*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 4*a*b*c*Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/(b^2 - 4*a*c)^(3/2) + ((-b^4 + 6*a*b^2*c - 6*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 4*a*b*c*Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/(b^2 - 4*a*c)^(3/2))/(2*a^3*e)","A",1
629,1,384,408,2.982627,"\int \frac{1}{(d+e x)^4 \left(a+b (d+e x)^2+c (d+e x)^4\right)^2} \, dx","Integrate[1/((d + e*x)^4*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2),x]","\frac{\frac{6 (d+e x) \left(2 a^2 c^2-4 a b^2 c-3 a b c^2 (d+e x)^2+b^4+b^3 c (d+e x)^2\right)}{\left(b^2-4 a c\right) \left(a+(d+e x)^2 \left(b+c (d+e x)^2\right)\right)}+\frac{3 \sqrt{2} \sqrt{c} \left(28 a^2 c^2-29 a b^2 c-19 a b c \sqrt{b^2-4 a c}+5 b^3 \sqrt{b^2-4 a c}+5 b^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{2} \sqrt{c} \left(-28 a^2 c^2+29 a b^2 c-19 a b c \sqrt{b^2-4 a c}+5 b^3 \sqrt{b^2-4 a c}-5 b^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{4 a}{(d+e x)^3}+\frac{24 b}{d+e x}}{12 a^3 e}","\frac{b \left(5 b^2-19 a c\right)}{2 a^3 e \left(b^2-4 a c\right) (d+e x)}-\frac{5 b^2-14 a c}{6 a^2 e \left(b^2-4 a c\right) (d+e x)^3}+\frac{\sqrt{c} \left(28 a^2 c^2-29 a b^2 c+b \left(5 b^2-19 a c\right) \sqrt{b^2-4 a c}+5 b^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} a^3 e \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left(28 a^2 c^2-29 a b^2 c-b \left(5 b^2-19 a c\right) \sqrt{b^2-4 a c}+5 b^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} a^3 e \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{-2 a c+b^2+b c (d+e x)^2}{2 a e \left(b^2-4 a c\right) (d+e x)^3 \left(a+b (d+e x)^2+c (d+e x)^4\right)}",1,"((-4*a)/(d + e*x)^3 + (24*b)/(d + e*x) + (6*(d + e*x)*(b^4 - 4*a*b^2*c + 2*a^2*c^2 + b^3*c*(d + e*x)^2 - 3*a*b*c^2*(d + e*x)^2))/((b^2 - 4*a*c)*(a + (d + e*x)^2*(b + c*(d + e*x)^2))) + (3*Sqrt[2]*Sqrt[c]*(5*b^4 - 29*a*b^2*c + 28*a^2*c^2 + 5*b^3*Sqrt[b^2 - 4*a*c] - 19*a*b*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[2]*Sqrt[c]*(-5*b^4 + 29*a*b^2*c - 28*a^2*c^2 + 5*b^3*Sqrt[b^2 - 4*a*c] - 19*a*b*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(12*a^3*e)","A",1
630,1,328,341,4.4340872,"\int \frac{(d+e x)^4}{\left(a+b (d+e x)^2+c (d+e x)^4\right)^3} \, dx","Integrate[(d + e*x)^4/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3,x]","\frac{\frac{(d+e x) \left(4 a c-7 b^2-12 b c (d+e x)^2\right)}{\left(b^2-4 a c\right)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)}-\frac{2 \left(-2 a (d+e x)-b (d+e x)^3\right)}{\left(b^2-4 a c\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)^2}+\frac{3 \sqrt{2} \sqrt{c} \left(-2 b \sqrt{b^2-4 a c}+4 a c+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\left(b^2-4 a c\right)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{3 \sqrt{2} \sqrt{c} \left(2 b \sqrt{b^2-4 a c}+4 a c+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\left(b^2-4 a c\right)^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}}{8 e}","\frac{(d+e x) \left(2 a+b (d+e x)^2\right)}{4 e \left(b^2-4 a c\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)^2}-\frac{(d+e x) \left(-4 a c+7 b^2+12 b c (d+e x)^2\right)}{8 e \left(b^2-4 a c\right)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)}+\frac{3 \sqrt{c} \left(-2 b \sqrt{b^2-4 a c}+4 a c+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{4 \sqrt{2} e \left(b^2-4 a c\right)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{3 \sqrt{c} \left(2 b \sqrt{b^2-4 a c}+4 a c+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{4 \sqrt{2} e \left(b^2-4 a c\right)^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}",1,"((-2*(-2*a*(d + e*x) - b*(d + e*x)^3))/((b^2 - 4*a*c)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + ((d + e*x)*(-7*b^2 + 4*a*c - 12*b*c*(d + e*x)^2))/((b^2 - 4*a*c)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (3*Sqrt[2]*Sqrt[c]*(3*b^2 + 4*a*c - 2*b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (3*Sqrt[2]*Sqrt[c]*(3*b^2 + 4*a*c + 2*b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(8*e)","A",1
631,1,146,150,0.2060229,"\int \frac{(d+e x)^3}{\left(a+b (d+e x)^2+c (d+e x)^4\right)^3} \, dx","Integrate[(d + e*x)^3/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3,x]","\frac{\frac{\left(b^2-4 a c\right) \left(2 a+b (d+e x)^2\right)}{\left(a+(d+e x)^2 \left(b+c (d+e x)^2\right)\right)^2}-\frac{12 b c \tan ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}-\frac{3 b \left(b+2 c (d+e x)^2\right)}{a+b (d+e x)^2+c (d+e x)^4}}{4 e \left(b^2-4 a c\right)^2}","\frac{2 a+b (d+e x)^2}{4 e \left(b^2-4 a c\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)^2}-\frac{3 b \left(b+2 c (d+e x)^2\right)}{4 e \left(b^2-4 a c\right)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)}+\frac{3 b c \tanh ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right)}{e \left(b^2-4 a c\right)^{5/2}}",1,"((-3*b*(b + 2*c*(d + e*x)^2))/(a + b*(d + e*x)^2 + c*(d + e*x)^4) + ((b^2 - 4*a*c)*(2*a + b*(d + e*x)^2))/(a + (d + e*x)^2*(b + c*(d + e*x)^2))^2 - (12*b*c*ArcTan[(b + 2*c*(d + e*x)^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(4*(b^2 - 4*a*c)^2*e)","A",1
632,1,382,363,4.8611927,"\int \frac{(d+e x)^2}{\left(a+b (d+e x)^2+c (d+e x)^4\right)^3} \, dx","Integrate[(d + e*x)^2/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3,x]","\frac{-\frac{4 \left(b (d+e x)+2 c (d+e x)^3\right)}{\left(b^2-4 a c\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)^2}+\frac{2 (d+e x) \left(8 a b c+20 a c^2 (d+e x)^2+b^3+b^2 c (d+e x)^2\right)}{a \left(b^2-4 a c\right)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)}+\frac{\sqrt{2} \sqrt{c} \left(b^2 \sqrt{b^2-4 a c}+20 a c \sqrt{b^2-4 a c}-52 a b c+b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{a \left(b^2-4 a c\right)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left(b^2 \sqrt{b^2-4 a c}+20 a c \sqrt{b^2-4 a c}+52 a b c-b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{a \left(b^2-4 a c\right)^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}}{16 e}","-\frac{(d+e x) \left(b+2 c (d+e x)^2\right)}{4 e \left(b^2-4 a c\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)^2}+\frac{(d+e x) \left(c \left(20 a c+b^2\right) (d+e x)^2+b \left(8 a c+b^2\right)\right)}{8 a e \left(b^2-4 a c\right)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)}+\frac{\sqrt{c} \left(\frac{b \left(b^2-52 a c\right)}{\sqrt{b^2-4 a c}}+20 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{8 \sqrt{2} a e \left(b^2-4 a c\right)^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left(-\frac{b \left(b^2-52 a c\right)}{\sqrt{b^2-4 a c}}+20 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{8 \sqrt{2} a e \left(b^2-4 a c\right)^2 \sqrt{\sqrt{b^2-4 a c}+b}}",1,"((-4*(b*(d + e*x) + 2*c*(d + e*x)^3))/((b^2 - 4*a*c)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + (2*(d + e*x)*(b^3 + 8*a*b*c + b^2*c*(d + e*x)^2 + 20*a*c^2*(d + e*x)^2))/(a*(b^2 - 4*a*c)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (Sqrt[2]*Sqrt[c]*(b^3 - 52*a*b*c + b^2*Sqrt[b^2 - 4*a*c] + 20*a*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(-b^3 + 52*a*b*c + b^2*Sqrt[b^2 - 4*a*c] + 20*a*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(16*e)","A",1
633,1,147,152,0.1835916,"\int \frac{d+e x}{\left(a+b (d+e x)^2+c (d+e x)^4\right)^3} \, dx","Integrate[(d + e*x)/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3,x]","\frac{\frac{24 c^2 \tan ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+\frac{\left(b^2-4 a c\right) \left(-b-2 c (d+e x)^2\right)}{\left(a+b (d+e x)^2+c (d+e x)^4\right)^2}+\frac{6 c \left(b+2 c (d+e x)^2\right)}{a+b (d+e x)^2+c (d+e x)^4}}{4 e \left(b^2-4 a c\right)^2}","-\frac{6 c^2 \tanh ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right)}{e \left(b^2-4 a c\right)^{5/2}}+\frac{3 c \left(b+2 c (d+e x)^2\right)}{2 e \left(b^2-4 a c\right)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)}+\frac{-b-2 c (d+e x)^2}{4 e \left(b^2-4 a c\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)^2}",1,"(((b^2 - 4*a*c)*(-b - 2*c*(d + e*x)^2))/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2 + (6*c*(b + 2*c*(d + e*x)^2))/(a + b*(d + e*x)^2 + c*(d + e*x)^4) + (24*c^2*ArcTan[(b + 2*c*(d + e*x)^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(4*(b^2 - 4*a*c)^2*e)","A",1
634,1,463,437,6.1691567,"\int \frac{1}{\left(a+b (d+e x)^2+c (d+e x)^4\right)^3} \, dx","Integrate[(a + b*(d + e*x)^2 + c*(d + e*x)^4)^(-3),x]","\frac{28 a^2 c^2 (d+e x)-25 a b^2 c (d+e x)-24 a b c^2 (d+e x)^3+3 b^4 (d+e x)+3 b^3 c (d+e x)^3}{8 a^2 e \left(4 a c-b^2\right)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)}+\frac{3 \sqrt{c} \left(56 a^2 c^2-10 a b^2 c-8 a b c \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}+b^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{8 \sqrt{2} a^2 e \left(b^2-4 a c\right)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{c} \left(-56 a^2 c^2+10 a b^2 c-8 a b c \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}-b^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{8 \sqrt{2} a^2 e \left(b^2-4 a c\right)^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 a c (d+e x)-b^2 (d+e x)-b c (d+e x)^3}{4 a e \left(4 a c-b^2\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)^2}","\frac{\left(\frac{d}{e}+x\right) \left(3 b c e^2 \left(b^2-8 a c\right) \left(\frac{d}{e}+x\right)^2+\left(b^2-7 a c\right) \left(3 b^2-4 a c\right)\right)}{8 a^2 \left(b^2-4 a c\right)^2 \left(a+b e^2 \left(\frac{d}{e}+x\right)^2+c e^4 \left(\frac{d}{e}+x\right)^4\right)}+\frac{3 \sqrt{c} \left(56 a^2 c^2-10 a b^2 c+b \left(b^2-8 a c\right) \sqrt{b^2-4 a c}+b^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{8 \sqrt{2} a^2 e \left(b^2-4 a c\right)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{3 \sqrt{c} \left(56 a^2 c^2-10 a b^2 c-b \left(b^2-8 a c\right) \sqrt{b^2-4 a c}+b^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{8 \sqrt{2} a^2 e \left(b^2-4 a c\right)^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{\left(\frac{d}{e}+x\right) \left(-2 a c+b^2+b c e^2 \left(\frac{d}{e}+x\right)^2\right)}{4 a \left(b^2-4 a c\right) \left(a+b e^2 \left(\frac{d}{e}+x\right)^2+c e^4 \left(\frac{d}{e}+x\right)^4\right)^2}",1,"(-(b^2*(d + e*x)) + 2*a*c*(d + e*x) - b*c*(d + e*x)^3)/(4*a*(-b^2 + 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + (3*b^4*(d + e*x) - 25*a*b^2*c*(d + e*x) + 28*a^2*c^2*(d + e*x) + 3*b^3*c*(d + e*x)^3 - 24*a*b*c^2*(d + e*x)^3)/(8*a^2*(-b^2 + 4*a*c)^2*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (3*Sqrt[c]*(b^4 - 10*a*b^2*c + 56*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 8*a*b*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]*e) + (3*Sqrt[c]*(-b^4 + 10*a*b^2*c - 56*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 8*a*b*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]*e)","A",1
635,1,391,255,3.9487719,"\int \frac{1}{(d+e x) \left(a+b (d+e x)^2+c (d+e x)^4\right)^3} \, dx","Integrate[1/((d + e*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3),x]","\frac{\frac{a^2 \left(2 a c-b^2-b c (d+e x)^2\right)}{\left(4 a c-b^2\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)^2}+\frac{a \left(16 a^2 c^2-15 a b^2 c-14 a b c^2 (d+e x)^2+2 b^4+2 b^3 c (d+e x)^2\right)}{\left(b^2-4 a c\right)^2 \left(a+(d+e x)^2 \left(b+c (d+e x)^2\right)\right)}-\frac{\left(16 a^2 c^2 \sqrt{b^2-4 a c}+30 a^2 b c^2-10 a b^3 c-8 a b^2 c \sqrt{b^2-4 a c}+b^4 \sqrt{b^2-4 a c}+b^5\right) \log \left(-\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right)}{\left(b^2-4 a c\right)^{5/2}}+\frac{\left(-16 a^2 c^2 \sqrt{b^2-4 a c}+30 a^2 b c^2-10 a b^3 c+8 a b^2 c \sqrt{b^2-4 a c}-b^4 \sqrt{b^2-4 a c}+b^5\right) \log \left(\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right)}{\left(b^2-4 a c\right)^{5/2}}+4 \log (d+e x)}{4 a^3 e}","-\frac{\log \left(a+b (d+e x)^2+c (d+e x)^4\right)}{4 a^3 e}+\frac{\log (d+e x)}{a^3 e}+\frac{16 a^2 c^2+2 b c \left(b^2-7 a c\right) (d+e x)^2-15 a b^2 c+2 b^4}{4 a^2 e \left(b^2-4 a c\right)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)}+\frac{b \left(30 a^2 c^2-10 a b^2 c+b^4\right) \tanh ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right)}{2 a^3 e \left(b^2-4 a c\right)^{5/2}}+\frac{-2 a c+b^2+b c (d+e x)^2}{4 a e \left(b^2-4 a c\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)^2}",1,"((a^2*(-b^2 + 2*a*c - b*c*(d + e*x)^2))/((-b^2 + 4*a*c)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + (a*(2*b^4 - 15*a*b^2*c + 16*a^2*c^2 + 2*b^3*c*(d + e*x)^2 - 14*a*b*c^2*(d + e*x)^2))/((b^2 - 4*a*c)^2*(a + (d + e*x)^2*(b + c*(d + e*x)^2))) + 4*Log[d + e*x] - ((b^5 - 10*a*b^3*c + 30*a^2*b*c^2 + b^4*Sqrt[b^2 - 4*a*c] - 8*a*b^2*c*Sqrt[b^2 - 4*a*c] + 16*a^2*c^2*Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/(b^2 - 4*a*c)^(5/2) + ((b^5 - 10*a*b^3*c + 30*a^2*b*c^2 - b^4*Sqrt[b^2 - 4*a*c] + 8*a*b^2*c*Sqrt[b^2 - 4*a*c] - 16*a^2*c^2*Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/(b^2 - 4*a*c)^(5/2))/(4*a^3*e)","A",1
636,1,560,484,6.2374064,"\int \frac{1}{(d+e x)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)^3} \, dx","Integrate[1/((d + e*x)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3),x]","-\frac{1}{a^3 e (d+e x)}+\frac{-3 a b c (d+e x)-2 a c^2 (d+e x)^3+b^3 (d+e x)+b^2 c (d+e x)^3}{4 a^2 e \left(4 a c-b^2\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)^2}-\frac{3 \sqrt{c} \left(60 a^2 c^2 \sqrt{b^2-4 a c}+124 a^2 b c^2-47 a b^3 c-37 a b^2 c \sqrt{b^2-4 a c}+5 b^4 \sqrt{b^2-4 a c}+5 b^5\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{8 \sqrt{2} a^3 e \left(b^2-4 a c\right)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{3 \sqrt{c} \left(60 a^2 c^2 \sqrt{b^2-4 a c}-124 a^2 b c^2+47 a b^3 c-37 a b^2 c \sqrt{b^2-4 a c}+5 b^4 \sqrt{b^2-4 a c}-5 b^5\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{8 \sqrt{2} a^3 e \left(b^2-4 a c\right)^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{-84 a^2 b c^2 (d+e x)-52 a^2 c^3 (d+e x)^3+52 a b^3 c (d+e x)+47 a b^2 c^2 (d+e x)^3-7 b^5 (d+e x)-7 b^4 c (d+e x)^3}{8 a^3 e \left(4 a c-b^2\right)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)}","-\frac{3 \left(5 b^2-12 a c\right) \left(b^2-5 a c\right)}{8 a^3 e \left(b^2-4 a c\right)^2 (d+e x)}+\frac{36 a^2 c^2+b c \left(5 b^2-32 a c\right) (d+e x)^2-35 a b^2 c+5 b^4}{8 a^2 e \left(b^2-4 a c\right)^2 (d+e x) \left(a+b (d+e x)^2+c (d+e x)^4\right)}-\frac{3 \sqrt{c} \left(\frac{b \left(124 a^2 c^2-47 a b^2 c+5 b^4\right)}{\sqrt{b^2-4 a c}}+\left(5 b^2-12 a c\right) \left(b^2-5 a c\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{8 \sqrt{2} a^3 e \left(b^2-4 a c\right)^2 \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{3 \sqrt{c} \left(\left(5 b^2-12 a c\right) \left(b^2-5 a c\right)-\frac{124 a^2 b c^2-47 a b^3 c+5 b^5}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{8 \sqrt{2} a^3 e \left(b^2-4 a c\right)^2 \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{-2 a c+b^2+b c (d+e x)^2}{4 a e \left(b^2-4 a c\right) (d+e x) \left(a+b (d+e x)^2+c (d+e x)^4\right)^2}",1,"-(1/(a^3*e*(d + e*x))) + (b^3*(d + e*x) - 3*a*b*c*(d + e*x) + b^2*c*(d + e*x)^3 - 2*a*c^2*(d + e*x)^3)/(4*a^2*(-b^2 + 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + (-7*b^5*(d + e*x) + 52*a*b^3*c*(d + e*x) - 84*a^2*b*c^2*(d + e*x) - 7*b^4*c*(d + e*x)^3 + 47*a*b^2*c^2*(d + e*x)^3 - 52*a^2*c^3*(d + e*x)^3)/(8*a^3*(-b^2 + 4*a*c)^2*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) - (3*Sqrt[c]*(5*b^5 - 47*a*b^3*c + 124*a^2*b*c^2 + 5*b^4*Sqrt[b^2 - 4*a*c] - 37*a*b^2*c*Sqrt[b^2 - 4*a*c] + 60*a^2*c^2*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^3*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]*e) - (3*Sqrt[c]*(-5*b^5 + 47*a*b^3*c - 124*a^2*b*c^2 + 5*b^4*Sqrt[b^2 - 4*a*c] - 37*a*b^2*c*Sqrt[b^2 - 4*a*c] + 60*a^2*c^2*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^3*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]*e)","A",1
637,1,491,325,6.1793465,"\int \frac{1}{(d+e x)^3 \left(a+b (d+e x)^2+c (d+e x)^4\right)^3} \, dx","Integrate[1/((d + e*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3),x]","-\frac{3 b \log (d+e x)}{a^4 e}-\frac{1}{2 a^3 e (d+e x)^2}+\frac{-3 a b c-2 a c^2 (d+e x)^2+b^3+b^2 c (d+e x)^2}{4 a^2 e \left(4 a c-b^2\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)^2}+\frac{-46 a^2 b c^2-28 a^2 c^3 (d+e x)^2+29 a b^3 c+26 a b^2 c^2 (d+e x)^2-4 b^5-4 b^4 c (d+e x)^2}{4 a^3 e \left(4 a c-b^2\right)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)}+\frac{3 \left(-20 a^3 c^3+30 a^2 b^2 c^2+16 a^2 b c^2 \sqrt{b^2-4 a c}-10 a b^4 c+b^5 \sqrt{b^2-4 a c}-8 a b^3 c \sqrt{b^2-4 a c}+b^6\right) \log \left(-\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right)}{4 a^4 e \left(b^2-4 a c\right)^{5/2}}+\frac{3 \left(20 a^3 c^3-30 a^2 b^2 c^2+16 a^2 b c^2 \sqrt{b^2-4 a c}+10 a b^4 c+b^5 \sqrt{b^2-4 a c}-8 a b^3 c \sqrt{b^2-4 a c}-b^6\right) \log \left(\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right)}{4 a^4 e \left(b^2-4 a c\right)^{5/2}}","\frac{3 b \log \left(a+b (d+e x)^2+c (d+e x)^4\right)}{4 a^4 e}-\frac{3 b \log (d+e x)}{a^4 e}-\frac{3 \left(b^2-5 a c\right) \left(b^2-2 a c\right)}{2 a^3 e \left(b^2-4 a c\right)^2 (d+e x)^2}+\frac{20 a^2 c^2+3 b c \left(b^2-6 a c\right) (d+e x)^2-20 a b^2 c+3 b^4}{4 a^2 e \left(b^2-4 a c\right)^2 (d+e x)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)}-\frac{3 \left(-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right) \tanh ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right)}{2 a^4 e \left(b^2-4 a c\right)^{5/2}}+\frac{-2 a c+b^2+b c (d+e x)^2}{4 a e \left(b^2-4 a c\right) (d+e x)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)^2}",1,"-1/2*1/(a^3*e*(d + e*x)^2) + (b^3 - 3*a*b*c + b^2*c*(d + e*x)^2 - 2*a*c^2*(d + e*x)^2)/(4*a^2*(-b^2 + 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + (-4*b^5 + 29*a*b^3*c - 46*a^2*b*c^2 - 4*b^4*c*(d + e*x)^2 + 26*a*b^2*c^2*(d + e*x)^2 - 28*a^2*c^3*(d + e*x)^2)/(4*a^3*(-b^2 + 4*a*c)^2*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) - (3*b*Log[d + e*x])/(a^4*e) + (3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3 + b^5*Sqrt[b^2 - 4*a*c] - 8*a*b^3*c*Sqrt[b^2 - 4*a*c] + 16*a^2*b*c^2*Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/(4*a^4*(b^2 - 4*a*c)^(5/2)*e) + (3*(-b^6 + 10*a*b^4*c - 30*a^2*b^2*c^2 + 20*a^3*c^3 + b^5*Sqrt[b^2 - 4*a*c] - 8*a*b^3*c*Sqrt[b^2 - 4*a*c] + 16*a^2*b*c^2*Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/(4*a^4*(b^2 - 4*a*c)^(5/2)*e)","A",1
638,1,222,202,0.0490972,"\int \frac{(d f+e f x)^4}{a+b (d+e x)^2+c (d+e x)^4} \, dx","Integrate[(d*f + e*f*x)^4/(a + b*(d + e*x)^2 + c*(d + e*x)^4),x]","\frac{f^4 \left(-\frac{\sqrt{2} \left(b \sqrt{b^2-4 a c}+2 a c-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \left(b \sqrt{b^2-4 a c}-2 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+2 \sqrt{c} (d+e x)\right)}{2 c^{3/2} e}","-\frac{f^4 \left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} c^{3/2} e \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{f^4 \left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} c^{3/2} e \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{f^4 x}{c}",1,"(f^4*(2*Sqrt[c]*(d + e*x) - (Sqrt[2]*(-b^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*(b^2 - 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]])))/(2*c^(3/2)*e)","A",1
639,1,80,87,0.0440572,"\int \frac{(d f+e f x)^3}{a+b (d+e x)^2+c (d+e x)^4} \, dx","Integrate[(d*f + e*f*x)^3/(a + b*(d + e*x)^2 + c*(d + e*x)^4),x]","\frac{f^3 \left(\log \left(a+b (d+e x)^2+c (d+e x)^4\right)-\frac{2 b \tan ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}\right)}{4 c e}","\frac{b f^3 \tanh ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right)}{2 c e \sqrt{b^2-4 a c}}+\frac{f^3 \log \left(a+b (d+e x)^2+c (d+e x)^4\right)}{4 c e}",1,"(f^3*((-2*b*ArcTan[(b + 2*c*(d + e*x)^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + Log[a + b*(d + e*x)^2 + c*(d + e*x)^4]))/(4*c*e)","A",1
640,1,178,170,0.1007469,"\int \frac{(d f+e f x)^2}{a+b (d+e x)^2+c (d+e x)^4} \, dx","Integrate[(d*f + e*f*x)^2/(a + b*(d + e*x)^2 + c*(d + e*x)^4),x]","\frac{f^2 \left(\left(\sqrt{b^2-4 a c}-b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)+\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{\sqrt{b^2-4 a c}+b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)\right)}{\sqrt{2} \sqrt{c} e \sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}","\frac{f^2 \sqrt{\sqrt{b^2-4 a c}+b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} \sqrt{c} e \sqrt{b^2-4 a c}}-\frac{f^2 \sqrt{b-\sqrt{b^2-4 a c}} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} \sqrt{c} e \sqrt{b^2-4 a c}}",1,"(f^2*((-b + Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]] + Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[b + Sqrt[b^2 - 4*a*c]]*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]]))/(Sqrt[2]*Sqrt[c]*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]*e)","A",1
641,1,47,44,0.0158038,"\int \frac{d f+e f x}{a+b (d+e x)^2+c (d+e x)^4} \, dx","Integrate[(d*f + e*f*x)/(a + b*(d + e*x)^2 + c*(d + e*x)^4),x]","\frac{f \tan ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{4 a c-b^2}}\right)}{e \sqrt{4 a c-b^2}}","-\frac{f \tanh ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right)}{e \sqrt{b^2-4 a c}}",1,"(f*ArcTan[(b + 2*c*(d + e*x)^2)/Sqrt[-b^2 + 4*a*c]])/(Sqrt[-b^2 + 4*a*c]*e)","A",1
642,1,131,103,0.0688779,"\int \frac{1}{(d f+e f x) \left(a+b (d+e x)^2+c (d+e x)^4\right)} \, dx","Integrate[1/((d*f + e*f*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)),x]","\frac{4 \sqrt{b^2-4 a c} \log (d+e x)-\left(\sqrt{b^2-4 a c}+b\right) \log \left(-\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right)+\left(b-\sqrt{b^2-4 a c}\right) \log \left(\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right)}{4 a e f \sqrt{b^2-4 a c}}","\frac{b \tanh ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right)}{2 a e f \sqrt{b^2-4 a c}}-\frac{\log \left(a+b (d+e x)^2+c (d+e x)^4\right)}{4 a e f}+\frac{\log (d+e x)}{a e f}",1,"(4*Sqrt[b^2 - 4*a*c]*Log[d + e*x] - (b + Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2] + (b - Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/(4*a*Sqrt[b^2 - 4*a*c]*e*f)","A",1
643,1,209,204,0.3422285,"\int \frac{1}{(d f+e f x)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)} \, dx","Integrate[1/((d*f + e*f*x)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)),x]","-\frac{\frac{\sqrt{2} \sqrt{c} \left(\sqrt{b^2-4 a c}+b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left(\sqrt{b^2-4 a c}-b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2}{d+e x}}{2 a e f^2}","-\frac{\sqrt{c} \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} a e f^2 \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} a e f^2 \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{1}{a e f^2 (d+e x)}",1,"-1/2*(2/(d + e*x) + (Sqrt[2]*Sqrt[c]*(b + Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(-b + Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(a*e*f^2)","A",1
644,1,157,133,0.1360137,"\int \frac{1}{(d f+e f x)^3 \left(a+b (d+e x)^2+c (d+e x)^4\right)} \, dx","Integrate[1/((d*f + e*f*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)),x]","\frac{\frac{\left(b \sqrt{b^2-4 a c}-2 a c+b^2\right) \log \left(-\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right)}{\sqrt{b^2-4 a c}}+\frac{\left(b \sqrt{b^2-4 a c}+2 a c-b^2\right) \log \left(\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right)}{\sqrt{b^2-4 a c}}-\frac{2 a}{(d+e x)^2}-4 b \log (d+e x)}{4 a^2 e f^3}","-\frac{\left(b^2-2 a c\right) \tanh ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right)}{2 a^2 e f^3 \sqrt{b^2-4 a c}}+\frac{b \log \left(a+b (d+e x)^2+c (d+e x)^4\right)}{4 a^2 e f^3}-\frac{b \log (d+e x)}{a^2 e f^3}-\frac{1}{2 a e f^3 (d+e x)^2}",1,"((-2*a)/(d + e*x)^2 - 4*b*Log[d + e*x] + ((b^2 - 2*a*c + b*Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/Sqrt[b^2 - 4*a*c] + ((-b^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/Sqrt[b^2 - 4*a*c])/(4*a^2*e*f^3)","A",1
645,1,238,236,0.1031432,"\int \frac{1}{(d f+e f x)^4 \left(a+b (d+e x)^2+c (d+e x)^4\right)} \, dx","Integrate[1/((d*f + e*f*x)^4*(a + b*(d + e*x)^2 + c*(d + e*x)^4)),x]","\frac{\frac{3 \sqrt{2} \sqrt{c} \left(b \sqrt{b^2-4 a c}-2 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{2} \sqrt{c} \left(b \sqrt{b^2-4 a c}+2 a c-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{2 a}{(d+e x)^3}+\frac{6 b}{d+e x}}{6 a^2 e f^4}","\frac{\sqrt{c} \left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} a^2 e f^4 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} a^2 e f^4 \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{b}{a^2 e f^4 (d+e x)}-\frac{1}{3 a e f^4 (d+e x)^3}",1,"((-2*a)/(d + e*x)^3 + (6*b)/(d + e*x) + (3*Sqrt[2]*Sqrt[c]*(b^2 - 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[2]*Sqrt[c]*(-b^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(6*a^2*e*f^4)","A",1
646,1,266,279,0.461391,"\int \frac{(d f+e f x)^4}{\left(a+b (d+e x)^2+c (d+e x)^4\right)^2} \, dx","Integrate[(d*f + e*f*x)^4/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2,x]","\frac{f^4 \left(-\frac{2 \left(-2 a (d+e x)-b (d+e x)^3\right)}{\left(b^2-4 a c\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)}+\frac{\sqrt{2} \left(b \sqrt{b^2-4 a c}-4 a c-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left(b \sqrt{b^2-4 a c}+4 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{c} \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}\right)}{4 e}","\frac{f^4 (d+e x) \left(2 a+b (d+e x)^2\right)}{2 e \left(b^2-4 a c\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)}+\frac{f^4 \left(b-\frac{4 a c+b^2}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} \sqrt{c} e \left(b^2-4 a c\right) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{f^4 \left(b \sqrt{b^2-4 a c}+4 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} \sqrt{c} e \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}",1,"(f^4*((-2*(-2*a*(d + e*x) - b*(d + e*x)^3))/((b^2 - 4*a*c)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (Sqrt[2]*(-b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(b^2 + 4*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])))/(4*e)","A",1
647,1,103,103,0.1260188,"\int \frac{(d f+e f x)^3}{\left(a+b (d+e x)^2+c (d+e x)^4\right)^2} \, dx","Integrate[(d*f + e*f*x)^3/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2,x]","\frac{f^3 \left(\frac{2 a+b (d+e x)^2}{\left(b^2-4 a c\right) \left(a+(d+e x)^2 \left(b+c (d+e x)^2\right)\right)}-\frac{2 b \tan ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{3/2}}\right)}{2 e}","\frac{f^3 \left(2 a+b (d+e x)^2\right)}{2 e \left(b^2-4 a c\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)}-\frac{b f^3 \tanh ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right)}{e \left(b^2-4 a c\right)^{3/2}}",1,"(f^3*((2*a + b*(d + e*x)^2)/((b^2 - 4*a*c)*(a + (d + e*x)^2*(b + c*(d + e*x)^2))) - (2*b*ArcTan[(b + 2*c*(d + e*x)^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2)))/(2*e)","A",1
648,1,250,263,0.9388193,"\int \frac{(d f+e f x)^2}{\left(a+b (d+e x)^2+c (d+e x)^4\right)^2} \, dx","Integrate[(d*f + e*f*x)^2/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2,x]","-\frac{f^2 \left(\frac{b (d+e x)+2 c (d+e x)^3}{\left(b^2-4 a c\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)}+\frac{\sqrt{2} \sqrt{c} \left(\sqrt{b^2-4 a c}-2 b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left(\sqrt{b^2-4 a c}+2 b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 e}","-\frac{f^2 (d+e x) \left(b+2 c (d+e x)^2\right)}{2 e \left(b^2-4 a c\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)}+\frac{\sqrt{c} f^2 \left(2 b-\sqrt{b^2-4 a c}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} e \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} f^2 \left(\sqrt{b^2-4 a c}+2 b\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} e \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}",1,"-1/2*(f^2*((b*(d + e*x) + 2*c*(d + e*x)^3)/((b^2 - 4*a*c)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (Sqrt[2]*Sqrt[c]*(-2*b + Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(2*b + Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])))/e","A",1
649,1,99,98,0.1225037,"\int \frac{d f+e f x}{\left(a+b (d+e x)^2+c (d+e x)^4\right)^2} \, dx","Integrate[(d*f + e*f*x)/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2,x]","-\frac{f \left(\frac{4 c \tan ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+\frac{b+2 c (d+e x)^2}{a+b (d+e x)^2+c (d+e x)^4}\right)}{2 e \left(b^2-4 a c\right)}","\frac{2 c f \tanh ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right)}{e \left(b^2-4 a c\right)^{3/2}}-\frac{f \left(b+2 c (d+e x)^2\right)}{2 e \left(b^2-4 a c\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)}",1,"-1/2*(f*((b + 2*c*(d + e*x)^2)/(a + b*(d + e*x)^2 + c*(d + e*x)^4) + (4*c*ArcTan[(b + 2*c*(d + e*x)^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c]))/((b^2 - 4*a*c)*e)","A",1
650,1,238,174,0.4359377,"\int \frac{1}{(d f+e f x) \left(a+b (d+e x)^2+c (d+e x)^4\right)^2} \, dx","Integrate[1/((d*f + e*f*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2),x]","\frac{\frac{2 a \left(-2 a c+b^2+b c (d+e x)^2\right)}{\left(b^2-4 a c\right) \left(a+(d+e x)^2 \left(b+c (d+e x)^2\right)\right)}-\frac{\left(b^2 \sqrt{b^2-4 a c}-4 a c \sqrt{b^2-4 a c}-6 a b c+b^3\right) \log \left(-\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{\left(-b^2 \sqrt{b^2-4 a c}+4 a c \sqrt{b^2-4 a c}-6 a b c+b^3\right) \log \left(\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right)}{\left(b^2-4 a c\right)^{3/2}}+4 \log (d+e x)}{4 a^2 e f}","\frac{b \left(b^2-6 a c\right) \tanh ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right)}{2 a^2 e f \left(b^2-4 a c\right)^{3/2}}-\frac{\log \left(a+b (d+e x)^2+c (d+e x)^4\right)}{4 a^2 e f}+\frac{\log (d+e x)}{a^2 e f}+\frac{-2 a c+b^2+b c (d+e x)^2}{2 a e f \left(b^2-4 a c\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)}",1,"((2*a*(b^2 - 2*a*c + b*c*(d + e*x)^2))/((b^2 - 4*a*c)*(a + (d + e*x)^2*(b + c*(d + e*x)^2))) + 4*Log[d + e*x] - ((b^3 - 6*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - 4*a*c*Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/(b^2 - 4*a*c)^(3/2) + ((b^3 - 6*a*b*c - b^2*Sqrt[b^2 - 4*a*c] + 4*a*c*Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/(b^2 - 4*a*c)^(3/2))/(4*a^2*e*f)","A",1
651,1,342,360,1.4708664,"\int \frac{1}{(d f+e f x)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)^2} \, dx","Integrate[1/((d*f + e*f*x)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2),x]","\frac{\frac{2 (d+e x) \left(-3 a b c-2 a c^2 (d+e x)^2+b^3+b^2 c (d+e x)^2\right)}{\left(4 a c-b^2\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)}+\frac{\sqrt{2} \sqrt{c} \left(-3 b^2 \sqrt{b^2-4 a c}+10 a c \sqrt{b^2-4 a c}+16 a b c-3 b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left(-3 b^2 \sqrt{b^2-4 a c}+10 a c \sqrt{b^2-4 a c}-16 a b c+3 b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{4}{d+e x}}{4 a^2 e f^2}","-\frac{3 b^2-10 a c}{2 a^2 e f^2 \left(b^2-4 a c\right) (d+e x)}-\frac{\sqrt{c} \left(\left(3 b^2-10 a c\right) \sqrt{b^2-4 a c}-16 a b c+3 b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} a^2 e f^2 \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left(-\left(3 b^2-10 a c\right) \sqrt{b^2-4 a c}-16 a b c+3 b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} a^2 e f^2 \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{-2 a c+b^2+b c (d+e x)^2}{2 a e f^2 \left(b^2-4 a c\right) (d+e x) \left(a+b (d+e x)^2+c (d+e x)^4\right)}",1,"(-4/(d + e*x) + (2*(d + e*x)*(b^3 - 3*a*b*c + b^2*c*(d + e*x)^2 - 2*a*c^2*(d + e*x)^2))/((-b^2 + 4*a*c)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (Sqrt[2]*Sqrt[c]*(-3*b^3 + 16*a*b*c - 3*b^2*Sqrt[b^2 - 4*a*c] + 10*a*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(3*b^3 - 16*a*b*c - 3*b^2*Sqrt[b^2 - 4*a*c] + 10*a*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(4*a^2*e*f^2)","A",1
652,1,287,228,0.5152404,"\int \frac{1}{(d f+e f x)^3 \left(a+b (d+e x)^2+c (d+e x)^4\right)^2} \, dx","Integrate[1/((d*f + e*f*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2),x]","\frac{\frac{\left(6 a^2 c^2-6 a b^2 c-4 a b c \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}+b^4\right) \log \left(-\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{\left(-6 a^2 c^2+6 a b^2 c-4 a b c \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}-b^4\right) \log \left(\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right)}{\left(b^2-4 a c\right)^{3/2}}+\frac{a \left(-3 a b c-2 a c^2 (d+e x)^2+b^3+b^2 c (d+e x)^2\right)}{\left(4 a c-b^2\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)}-\frac{a}{(d+e x)^2}-4 b \log (d+e x)}{2 a^3 e f^3}","\frac{b \log \left(a+b (d+e x)^2+c (d+e x)^4\right)}{2 a^3 e f^3}-\frac{2 b \log (d+e x)}{a^3 e f^3}-\frac{b^2-3 a c}{a^2 e f^3 \left(b^2-4 a c\right) (d+e x)^2}-\frac{\left(6 a^2 c^2-6 a b^2 c+b^4\right) \tanh ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right)}{a^3 e f^3 \left(b^2-4 a c\right)^{3/2}}+\frac{-2 a c+b^2+b c (d+e x)^2}{2 a e f^3 \left(b^2-4 a c\right) (d+e x)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)}",1,"(-(a/(d + e*x)^2) + (a*(b^3 - 3*a*b*c + b^2*c*(d + e*x)^2 - 2*a*c^2*(d + e*x)^2))/((-b^2 + 4*a*c)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) - 4*b*Log[d + e*x] + ((b^4 - 6*a*b^2*c + 6*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 4*a*b*c*Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/(b^2 - 4*a*c)^(3/2) + ((-b^4 + 6*a*b^2*c - 6*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 4*a*b*c*Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/(b^2 - 4*a*c)^(3/2))/(2*a^3*e*f^3)","A",1
653,1,387,423,3.0246082,"\int \frac{1}{(d f+e f x)^4 \left(a+b (d+e x)^2+c (d+e x)^4\right)^2} \, dx","Integrate[1/((d*f + e*f*x)^4*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2),x]","\frac{\frac{6 (d+e x) \left(2 a^2 c^2-4 a b^2 c-3 a b c^2 (d+e x)^2+b^4+b^3 c (d+e x)^2\right)}{\left(b^2-4 a c\right) \left(a+(d+e x)^2 \left(b+c (d+e x)^2\right)\right)}+\frac{3 \sqrt{2} \sqrt{c} \left(28 a^2 c^2-29 a b^2 c-19 a b c \sqrt{b^2-4 a c}+5 b^3 \sqrt{b^2-4 a c}+5 b^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{2} \sqrt{c} \left(-28 a^2 c^2+29 a b^2 c-19 a b c \sqrt{b^2-4 a c}+5 b^3 \sqrt{b^2-4 a c}-5 b^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{4 a}{(d+e x)^3}+\frac{24 b}{d+e x}}{12 a^3 e f^4}","\frac{b \left(5 b^2-19 a c\right)}{2 a^3 e f^4 \left(b^2-4 a c\right) (d+e x)}-\frac{5 b^2-14 a c}{6 a^2 e f^4 \left(b^2-4 a c\right) (d+e x)^3}+\frac{\sqrt{c} \left(28 a^2 c^2-29 a b^2 c+b \left(5 b^2-19 a c\right) \sqrt{b^2-4 a c}+5 b^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{2 \sqrt{2} a^3 e f^4 \left(b^2-4 a c\right)^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left(28 a^2 c^2-29 a b^2 c-b \left(5 b^2-19 a c\right) \sqrt{b^2-4 a c}+5 b^4\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{2 \sqrt{2} a^3 e f^4 \left(b^2-4 a c\right)^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{-2 a c+b^2+b c (d+e x)^2}{2 a e f^4 \left(b^2-4 a c\right) (d+e x)^3 \left(a+b (d+e x)^2+c (d+e x)^4\right)}",1,"((-4*a)/(d + e*x)^3 + (24*b)/(d + e*x) + (6*(d + e*x)*(b^4 - 4*a*b^2*c + 2*a^2*c^2 + b^3*c*(d + e*x)^2 - 3*a*b*c^2*(d + e*x)^2))/((b^2 - 4*a*c)*(a + (d + e*x)^2*(b + c*(d + e*x)^2))) + (3*Sqrt[2]*Sqrt[c]*(5*b^4 - 29*a*b^2*c + 28*a^2*c^2 + 5*b^3*Sqrt[b^2 - 4*a*c] - 19*a*b*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[2]*Sqrt[c]*(-5*b^4 + 29*a*b^2*c - 28*a^2*c^2 + 5*b^3*Sqrt[b^2 - 4*a*c] - 19*a*b*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(12*a^3*e*f^4)","A",1
654,1,331,353,4.4451943,"\int \frac{(d f+e f x)^4}{\left(a+b (d+e x)^2+c (d+e x)^4\right)^3} \, dx","Integrate[(d*f + e*f*x)^4/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3,x]","\frac{f^4 \left(\frac{(d+e x) \left(4 a c-7 b^2-12 b c (d+e x)^2\right)}{\left(b^2-4 a c\right)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)}-\frac{2 \left(-2 a (d+e x)-b (d+e x)^3\right)}{\left(b^2-4 a c\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)^2}+\frac{3 \sqrt{2} \sqrt{c} \left(-2 b \sqrt{b^2-4 a c}+4 a c+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\left(b^2-4 a c\right)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{3 \sqrt{2} \sqrt{c} \left(2 b \sqrt{b^2-4 a c}+4 a c+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\left(b^2-4 a c\right)^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}\right)}{8 e}","-\frac{f^4 (d+e x) \left(-4 a c+7 b^2+12 b c (d+e x)^2\right)}{8 e \left(b^2-4 a c\right)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)}+\frac{f^4 (d+e x) \left(2 a+b (d+e x)^2\right)}{4 e \left(b^2-4 a c\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)^2}+\frac{3 \sqrt{c} f^4 \left(-2 b \sqrt{b^2-4 a c}+4 a c+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{4 \sqrt{2} e \left(b^2-4 a c\right)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{3 \sqrt{c} f^4 \left(2 b \sqrt{b^2-4 a c}+4 a c+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{4 \sqrt{2} e \left(b^2-4 a c\right)^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}",1,"(f^4*((-2*(-2*a*(d + e*x) - b*(d + e*x)^3))/((b^2 - 4*a*c)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + ((d + e*x)*(-7*b^2 + 4*a*c - 12*b*c*(d + e*x)^2))/((b^2 - 4*a*c)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (3*Sqrt[2]*Sqrt[c]*(3*b^2 + 4*a*c - 2*b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (3*Sqrt[2]*Sqrt[c]*(3*b^2 + 4*a*c + 2*b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])))/(8*e)","A",1
655,1,149,159,0.2088425,"\int \frac{(d f+e f x)^3}{\left(a+b (d+e x)^2+c (d+e x)^4\right)^3} \, dx","Integrate[(d*f + e*f*x)^3/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3,x]","\frac{f^3 \left(\frac{\left(b^2-4 a c\right) \left(2 a+b (d+e x)^2\right)}{\left(a+(d+e x)^2 \left(b+c (d+e x)^2\right)\right)^2}-\frac{12 b c \tan ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}-\frac{3 b \left(b+2 c (d+e x)^2\right)}{a+b (d+e x)^2+c (d+e x)^4}\right)}{4 e \left(b^2-4 a c\right)^2}","-\frac{3 b f^3 \left(b+2 c (d+e x)^2\right)}{4 e \left(b^2-4 a c\right)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)}+\frac{f^3 \left(2 a+b (d+e x)^2\right)}{4 e \left(b^2-4 a c\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)^2}+\frac{3 b c f^3 \tanh ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right)}{e \left(b^2-4 a c\right)^{5/2}}",1,"(f^3*((-3*b*(b + 2*c*(d + e*x)^2))/(a + b*(d + e*x)^2 + c*(d + e*x)^4) + ((b^2 - 4*a*c)*(2*a + b*(d + e*x)^2))/(a + (d + e*x)^2*(b + c*(d + e*x)^2))^2 - (12*b*c*ArcTan[(b + 2*c*(d + e*x)^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c]))/(4*(b^2 - 4*a*c)^2*e)","A",1
656,1,385,375,4.4832887,"\int \frac{(d f+e f x)^2}{\left(a+b (d+e x)^2+c (d+e x)^4\right)^3} \, dx","Integrate[(d*f + e*f*x)^2/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3,x]","\frac{f^2 \left(-\frac{4 \left(b (d+e x)+2 c (d+e x)^3\right)}{\left(b^2-4 a c\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)^2}+\frac{2 (d+e x) \left(8 a b c+20 a c^2 (d+e x)^2+b^3+b^2 c (d+e x)^2\right)}{a \left(b^2-4 a c\right)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)}+\frac{\sqrt{2} \sqrt{c} \left(b^2 \sqrt{b^2-4 a c}+20 a c \sqrt{b^2-4 a c}-52 a b c+b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{a \left(b^2-4 a c\right)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left(b^2 \sqrt{b^2-4 a c}+20 a c \sqrt{b^2-4 a c}+52 a b c-b^3\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{a \left(b^2-4 a c\right)^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}\right)}{16 e}","\frac{f^2 (d+e x) \left(c \left(20 a c+b^2\right) (d+e x)^2+b \left(8 a c+b^2\right)\right)}{8 a e \left(b^2-4 a c\right)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)}-\frac{f^2 (d+e x) \left(b+2 c (d+e x)^2\right)}{4 e \left(b^2-4 a c\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)^2}+\frac{\sqrt{c} f^2 \left(\frac{b \left(b^2-52 a c\right)}{\sqrt{b^2-4 a c}}+20 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{8 \sqrt{2} a e \left(b^2-4 a c\right)^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} f^2 \left(-\frac{b \left(b^2-52 a c\right)}{\sqrt{b^2-4 a c}}+20 a c+b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{8 \sqrt{2} a e \left(b^2-4 a c\right)^2 \sqrt{\sqrt{b^2-4 a c}+b}}",1,"(f^2*((-4*(b*(d + e*x) + 2*c*(d + e*x)^3))/((b^2 - 4*a*c)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + (2*(d + e*x)*(b^3 + 8*a*b*c + b^2*c*(d + e*x)^2 + 20*a*c^2*(d + e*x)^2))/(a*(b^2 - 4*a*c)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (Sqrt[2]*Sqrt[c]*(b^3 - 52*a*b*c + b^2*Sqrt[b^2 - 4*a*c] + 20*a*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(-b^3 + 52*a*b*c + b^2*Sqrt[b^2 - 4*a*c] + 20*a*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])))/(16*e)","A",1
657,1,148,153,0.1785904,"\int \frac{d f+e f x}{\left(a+b (d+e x)^2+c (d+e x)^4\right)^3} \, dx","Integrate[(d*f + e*f*x)/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3,x]","\frac{f \left(\frac{24 c^2 \tan ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{4 a c-b^2}}\right)}{\sqrt{4 a c-b^2}}+\frac{\left(b^2-4 a c\right) \left(-b-2 c (d+e x)^2\right)}{\left(a+b (d+e x)^2+c (d+e x)^4\right)^2}+\frac{6 c \left(b+2 c (d+e x)^2\right)}{a+b (d+e x)^2+c (d+e x)^4}\right)}{4 e \left(b^2-4 a c\right)^2}","-\frac{6 c^2 f \tanh ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right)}{e \left(b^2-4 a c\right)^{5/2}}+\frac{3 c f \left(b+2 c (d+e x)^2\right)}{2 e \left(b^2-4 a c\right)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)}-\frac{f \left(b+2 c (d+e x)^2\right)}{4 e \left(b^2-4 a c\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)^2}",1,"(f*(((b^2 - 4*a*c)*(-b - 2*c*(d + e*x)^2))/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2 + (6*c*(b + 2*c*(d + e*x)^2))/(a + b*(d + e*x)^2 + c*(d + e*x)^4) + (24*c^2*ArcTan[(b + 2*c*(d + e*x)^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c]))/(4*(b^2 - 4*a*c)^2*e)","A",1
658,1,394,270,3.9251107,"\int \frac{1}{(d f+e f x) \left(a+b (d+e x)^2+c (d+e x)^4\right)^3} \, dx","Integrate[1/((d*f + e*f*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3),x]","\frac{\frac{a^2 \left(2 a c-b^2-b c (d+e x)^2\right)}{\left(4 a c-b^2\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)^2}+\frac{a \left(16 a^2 c^2-15 a b^2 c-14 a b c^2 (d+e x)^2+2 b^4+2 b^3 c (d+e x)^2\right)}{\left(b^2-4 a c\right)^2 \left(a+(d+e x)^2 \left(b+c (d+e x)^2\right)\right)}-\frac{\left(16 a^2 c^2 \sqrt{b^2-4 a c}+30 a^2 b c^2-10 a b^3 c-8 a b^2 c \sqrt{b^2-4 a c}+b^4 \sqrt{b^2-4 a c}+b^5\right) \log \left(-\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right)}{\left(b^2-4 a c\right)^{5/2}}+\frac{\left(-16 a^2 c^2 \sqrt{b^2-4 a c}+30 a^2 b c^2-10 a b^3 c+8 a b^2 c \sqrt{b^2-4 a c}-b^4 \sqrt{b^2-4 a c}+b^5\right) \log \left(\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right)}{\left(b^2-4 a c\right)^{5/2}}+4 \log (d+e x)}{4 a^3 e f}","-\frac{\log \left(a+b (d+e x)^2+c (d+e x)^4\right)}{4 a^3 e f}+\frac{\log (d+e x)}{a^3 e f}+\frac{16 a^2 c^2+2 b c \left(b^2-7 a c\right) (d+e x)^2-15 a b^2 c+2 b^4}{4 a^2 e f \left(b^2-4 a c\right)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)}+\frac{b \left(30 a^2 c^2-10 a b^2 c+b^4\right) \tanh ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right)}{2 a^3 e f \left(b^2-4 a c\right)^{5/2}}+\frac{-2 a c+b^2+b c (d+e x)^2}{4 a e f \left(b^2-4 a c\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)^2}",1,"((a^2*(-b^2 + 2*a*c - b*c*(d + e*x)^2))/((-b^2 + 4*a*c)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + (a*(2*b^4 - 15*a*b^2*c + 16*a^2*c^2 + 2*b^3*c*(d + e*x)^2 - 14*a*b*c^2*(d + e*x)^2))/((b^2 - 4*a*c)^2*(a + (d + e*x)^2*(b + c*(d + e*x)^2))) + 4*Log[d + e*x] - ((b^5 - 10*a*b^3*c + 30*a^2*b*c^2 + b^4*Sqrt[b^2 - 4*a*c] - 8*a*b^2*c*Sqrt[b^2 - 4*a*c] + 16*a^2*c^2*Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/(b^2 - 4*a*c)^(5/2) + ((b^5 - 10*a*b^3*c + 30*a^2*b*c^2 - b^4*Sqrt[b^2 - 4*a*c] + 8*a*b^2*c*Sqrt[b^2 - 4*a*c] - 16*a^2*c^2*Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/(b^2 - 4*a*c)^(5/2))/(4*a^3*e*f)","A",1
659,1,575,499,6.2100634,"\int \frac{1}{(d f+e f x)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)^3} \, dx","Integrate[1/((d*f + e*f*x)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3),x]","-\frac{1}{a^3 e f^2 (d+e x)}+\frac{-3 a b c (d+e x)-2 a c^2 (d+e x)^3+b^3 (d+e x)+b^2 c (d+e x)^3}{4 a^2 e f^2 \left(4 a c-b^2\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)^2}-\frac{3 \sqrt{c} \left(60 a^2 c^2 \sqrt{b^2-4 a c}+124 a^2 b c^2-47 a b^3 c-37 a b^2 c \sqrt{b^2-4 a c}+5 b^4 \sqrt{b^2-4 a c}+5 b^5\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{8 \sqrt{2} a^3 e f^2 \left(b^2-4 a c\right)^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{3 \sqrt{c} \left(60 a^2 c^2 \sqrt{b^2-4 a c}-124 a^2 b c^2+47 a b^3 c-37 a b^2 c \sqrt{b^2-4 a c}+5 b^4 \sqrt{b^2-4 a c}-5 b^5\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{8 \sqrt{2} a^3 e f^2 \left(b^2-4 a c\right)^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{-84 a^2 b c^2 (d+e x)-52 a^2 c^3 (d+e x)^3+52 a b^3 c (d+e x)+47 a b^2 c^2 (d+e x)^3-7 b^5 (d+e x)-7 b^4 c (d+e x)^3}{8 a^3 e f^2 \left(4 a c-b^2\right)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)}","-\frac{3 \left(5 b^2-12 a c\right) \left(b^2-5 a c\right)}{8 a^3 e f^2 \left(b^2-4 a c\right)^2 (d+e x)}+\frac{36 a^2 c^2+b c \left(5 b^2-32 a c\right) (d+e x)^2-35 a b^2 c+5 b^4}{8 a^2 e f^2 \left(b^2-4 a c\right)^2 (d+e x) \left(a+b (d+e x)^2+c (d+e x)^4\right)}-\frac{3 \sqrt{c} \left(\frac{b \left(124 a^2 c^2-47 a b^2 c+5 b^4\right)}{\sqrt{b^2-4 a c}}+\left(5 b^2-12 a c\right) \left(b^2-5 a c\right)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{8 \sqrt{2} a^3 e f^2 \left(b^2-4 a c\right)^2 \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{3 \sqrt{c} \left(\left(5 b^2-12 a c\right) \left(b^2-5 a c\right)-\frac{124 a^2 b c^2-47 a b^3 c+5 b^5}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{8 \sqrt{2} a^3 e f^2 \left(b^2-4 a c\right)^2 \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{-2 a c+b^2+b c (d+e x)^2}{4 a e f^2 \left(b^2-4 a c\right) (d+e x) \left(a+b (d+e x)^2+c (d+e x)^4\right)^2}",1,"-(1/(a^3*e*f^2*(d + e*x))) + (b^3*(d + e*x) - 3*a*b*c*(d + e*x) + b^2*c*(d + e*x)^3 - 2*a*c^2*(d + e*x)^3)/(4*a^2*(-b^2 + 4*a*c)*e*f^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + (-7*b^5*(d + e*x) + 52*a*b^3*c*(d + e*x) - 84*a^2*b*c^2*(d + e*x) - 7*b^4*c*(d + e*x)^3 + 47*a*b^2*c^2*(d + e*x)^3 - 52*a^2*c^3*(d + e*x)^3)/(8*a^3*(-b^2 + 4*a*c)^2*e*f^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) - (3*Sqrt[c]*(5*b^5 - 47*a*b^3*c + 124*a^2*b*c^2 + 5*b^4*Sqrt[b^2 - 4*a*c] - 37*a*b^2*c*Sqrt[b^2 - 4*a*c] + 60*a^2*c^2*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^3*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]*e*f^2) - (3*Sqrt[c]*(-5*b^5 + 47*a*b^3*c - 124*a^2*b*c^2 + 5*b^4*Sqrt[b^2 - 4*a*c] - 37*a*b^2*c*Sqrt[b^2 - 4*a*c] + 60*a^2*c^2*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^3*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]*e*f^2)","A",1
660,1,509,343,6.1567102,"\int \frac{1}{(d f+e f x)^3 \left(a+b (d+e x)^2+c (d+e x)^4\right)^3} \, dx","Integrate[1/((d*f + e*f*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3),x]","-\frac{3 b \log (d+e x)}{a^4 e f^3}-\frac{1}{2 a^3 e f^3 (d+e x)^2}+\frac{-3 a b c-2 a c^2 (d+e x)^2+b^3+b^2 c (d+e x)^2}{4 a^2 e f^3 \left(4 a c-b^2\right) \left(a+b (d+e x)^2+c (d+e x)^4\right)^2}+\frac{-46 a^2 b c^2-28 a^2 c^3 (d+e x)^2+29 a b^3 c+26 a b^2 c^2 (d+e x)^2-4 b^5-4 b^4 c (d+e x)^2}{4 a^3 e f^3 \left(4 a c-b^2\right)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)}+\frac{3 \left(-20 a^3 c^3+30 a^2 b^2 c^2+16 a^2 b c^2 \sqrt{b^2-4 a c}-10 a b^4 c+b^5 \sqrt{b^2-4 a c}-8 a b^3 c \sqrt{b^2-4 a c}+b^6\right) \log \left(-\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right)}{4 a^4 e f^3 \left(b^2-4 a c\right)^{5/2}}+\frac{3 \left(20 a^3 c^3-30 a^2 b^2 c^2+16 a^2 b c^2 \sqrt{b^2-4 a c}+10 a b^4 c+b^5 \sqrt{b^2-4 a c}-8 a b^3 c \sqrt{b^2-4 a c}-b^6\right) \log \left(\sqrt{b^2-4 a c}+b+2 c (d+e x)^2\right)}{4 a^4 e f^3 \left(b^2-4 a c\right)^{5/2}}","\frac{3 b \log \left(a+b (d+e x)^2+c (d+e x)^4\right)}{4 a^4 e f^3}-\frac{3 b \log (d+e x)}{a^4 e f^3}-\frac{3 \left(b^2-5 a c\right) \left(b^2-2 a c\right)}{2 a^3 e f^3 \left(b^2-4 a c\right)^2 (d+e x)^2}+\frac{20 a^2 c^2+3 b c \left(b^2-6 a c\right) (d+e x)^2-20 a b^2 c+3 b^4}{4 a^2 e f^3 \left(b^2-4 a c\right)^2 (d+e x)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)}-\frac{3 \left(-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right) \tanh ^{-1}\left(\frac{b+2 c (d+e x)^2}{\sqrt{b^2-4 a c}}\right)}{2 a^4 e f^3 \left(b^2-4 a c\right)^{5/2}}+\frac{-2 a c+b^2+b c (d+e x)^2}{4 a e f^3 \left(b^2-4 a c\right) (d+e x)^2 \left(a+b (d+e x)^2+c (d+e x)^4\right)^2}",1,"-1/2*1/(a^3*e*f^3*(d + e*x)^2) + (b^3 - 3*a*b*c + b^2*c*(d + e*x)^2 - 2*a*c^2*(d + e*x)^2)/(4*a^2*(-b^2 + 4*a*c)*e*f^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + (-4*b^5 + 29*a*b^3*c - 46*a^2*b*c^2 - 4*b^4*c*(d + e*x)^2 + 26*a*b^2*c^2*(d + e*x)^2 - 28*a^2*c^3*(d + e*x)^2)/(4*a^3*(-b^2 + 4*a*c)^2*e*f^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) - (3*b*Log[d + e*x])/(a^4*e*f^3) + (3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3 + b^5*Sqrt[b^2 - 4*a*c] - 8*a*b^3*c*Sqrt[b^2 - 4*a*c] + 16*a^2*b*c^2*Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/(4*a^4*(b^2 - 4*a*c)^(5/2)*e*f^3) + (3*(-b^6 + 10*a*b^4*c - 30*a^2*b^2*c^2 + 20*a^3*c^3 + b^5*Sqrt[b^2 - 4*a*c] - 8*a*b^3*c*Sqrt[b^2 - 4*a*c] + 16*a^2*b*c^2*Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/(4*a^4*(b^2 - 4*a*c)^(5/2)*e*f^3)","A",1
661,0,0,340,1.1672731,"\int \frac{x}{\sqrt{a+b (d+e x)^3+c (d+e x)^6}} \, dx","Integrate[x/Sqrt[a + b*(d + e*x)^3 + c*(d + e*x)^6],x]","\int \frac{x}{\sqrt{a+b (d+e x)^3+c (d+e x)^6}} \, dx","\frac{(d+e x)^2 \sqrt{\frac{2 c (d+e x)^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c (d+e x)^3}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{2 c (d+e x)^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c (d+e x)^3}{b+\sqrt{b^2-4 a c}}\right)}{2 e^2 \sqrt{a+b (d+e x)^3+c (d+e x)^6}}-\frac{d (d+e x) \sqrt{\frac{2 c (d+e x)^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c (d+e x)^3}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{2 c (d+e x)^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c (d+e x)^3}{b+\sqrt{b^2-4 a c}}\right)}{e^2 \sqrt{a+b (d+e x)^3+c (d+e x)^6}}",1,"Integrate[x/Sqrt[a + b*(d + e*x)^3 + c*(d + e*x)^6], x]","F",-1
662,0,0,398,0.4421578,"\int \frac{x^2}{\sqrt{a+b (d+e x)^3+c (d+e x)^6}} \, dx","Integrate[x^2/Sqrt[a + b*(d + e*x)^3 + c*(d + e*x)^6],x]","\int \frac{x^2}{\sqrt{a+b (d+e x)^3+c (d+e x)^6}} \, dx","\frac{d^2 (d+e x) \sqrt{\frac{2 c (d+e x)^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c (d+e x)^3}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{2 c (d+e x)^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c (d+e x)^3}{b+\sqrt{b^2-4 a c}}\right)}{e^3 \sqrt{a+b (d+e x)^3+c (d+e x)^6}}-\frac{d (d+e x)^2 \sqrt{\frac{2 c (d+e x)^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c (d+e x)^3}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{2 c (d+e x)^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c (d+e x)^3}{b+\sqrt{b^2-4 a c}}\right)}{e^3 \sqrt{a+b (d+e x)^3+c (d+e x)^6}}+\frac{\tanh ^{-1}\left(\frac{b+2 c (d+e x)^3}{2 \sqrt{c} \sqrt{a+b (d+e x)^3+c (d+e x)^6}}\right)}{3 \sqrt{c} e^3}",1,"Integrate[x^2/Sqrt[a + b*(d + e*x)^3 + c*(d + e*x)^6], x]","F",-1
663,1,34,34,0.0112764,"\int (2+3 x)^6 \left(1+(2+3 x)^7+(2+3 x)^{14}\right) \, dx","Integrate[(2 + 3*x)^6*(1 + (2 + 3*x)^7 + (2 + 3*x)^14),x]","\frac{1}{63} (3 x+2)^{21}+\frac{1}{42} (3 x+2)^{14}+\frac{1}{21} (3 x+2)^7","\frac{1}{63} (3 x+2)^{21}+\frac{1}{42} (3 x+2)^{14}+\frac{1}{21} (3 x+2)^7",1,"(2 + 3*x)^7/21 + (2 + 3*x)^14/42 + (2 + 3*x)^21/63","A",1
664,1,188,56,0.0103044,"\int (2+3 x)^6 \left(1+(2+3 x)^7+(2+3 x)^{14}\right)^2 \, dx","Integrate[(2 + 3*x)^6*(1 + (2 + 3*x)^7 + (2 + 3*x)^14)^2,x]","\frac{16677181699666569 x^{35}}{35}+11118121133111046 x^{34}+126005372841925188 x^{33}+924039400840784712 x^{32}+4928210137817518464 x^{31}+\frac{101849676181562048256 x^{30}}{5}+67899784121041365504 x^{29}+\frac{2625458326972530284475 x^{28}}{14}+437576396725285446564 x^{27}+875152864622814086340 x^{26}+\frac{7584660010542711771792 x^{25}}{5}+2298383223254096766840 x^{24}+3064515076512846852480 x^{23}+3614565944605222108800 x^{22}+\frac{26506949038858918036881 x^{21}}{7}+3534290697929473864098 x^{20}+2945285062308448290360 x^{19}+2194577166014752240080 x^{18}+1463104032160519033200 x^{17}+872775774067455498528 x^{16}+465517091041681015296 x^{15}+221699757548270194389 x^{14}+94069263918929616324 x^{13}+35454069480572048124 x^{12}+11821487501620716192 x^{11}+\frac{17344958593049772048 x^{10}}{5}+889942562270387136 x^9+197897276851452864 x^8+37727143432895007 x^7+6077684727888102 x^6+\frac{4057390785756924 x^5}{5}+87406679578680 x^4+7299544818384 x^3+443569828128 x^2+17451466816 x","\frac{1}{105} (3 x+2)^{35}+\frac{1}{42} (3 x+2)^{28}+\frac{1}{21} (3 x+2)^{21}+\frac{1}{21} (3 x+2)^{14}+\frac{1}{21} (3 x+2)^7",1,"17451466816*x + 443569828128*x^2 + 7299544818384*x^3 + 87406679578680*x^4 + (4057390785756924*x^5)/5 + 6077684727888102*x^6 + 37727143432895007*x^7 + 197897276851452864*x^8 + 889942562270387136*x^9 + (17344958593049772048*x^10)/5 + 11821487501620716192*x^11 + 35454069480572048124*x^12 + 94069263918929616324*x^13 + 221699757548270194389*x^14 + 465517091041681015296*x^15 + 872775774067455498528*x^16 + 1463104032160519033200*x^17 + 2194577166014752240080*x^18 + 2945285062308448290360*x^19 + 3534290697929473864098*x^20 + (26506949038858918036881*x^21)/7 + 3614565944605222108800*x^22 + 3064515076512846852480*x^23 + 2298383223254096766840*x^24 + (7584660010542711771792*x^25)/5 + 875152864622814086340*x^26 + 437576396725285446564*x^27 + (2625458326972530284475*x^28)/14 + 67899784121041365504*x^29 + (101849676181562048256*x^30)/5 + 4928210137817518464*x^31 + 924039400840784712*x^32 + 126005372841925188*x^33 + 11118121133111046*x^34 + (16677181699666569*x^35)/35","B",1