1,1,17,17,0.0014347,"\int x^4 \left(a+b x^2\right) \, dx","Integrate[x^4*(a + b*x^2),x]","\frac{a x^5}{5}+\frac{b x^7}{7}","\frac{a x^5}{5}+\frac{b x^7}{7}",1,"(a*x^5)/5 + (b*x^7)/7","A",1
2,1,17,17,0.0012513,"\int x^3 \left(a+b x^2\right) \, dx","Integrate[x^3*(a + b*x^2),x]","\frac{a x^4}{4}+\frac{b x^6}{6}","\frac{a x^4}{4}+\frac{b x^6}{6}",1,"(a*x^4)/4 + (b*x^6)/6","A",1
3,1,17,17,0.0011615,"\int x^2 \left(a+b x^2\right) \, dx","Integrate[x^2*(a + b*x^2),x]","\frac{a x^3}{3}+\frac{b x^5}{5}","\frac{a x^3}{3}+\frac{b x^5}{5}",1,"(a*x^3)/3 + (b*x^5)/5","A",1
4,1,17,17,0.0011194,"\int x \left(a+b x^2\right) \, dx","Integrate[x*(a + b*x^2),x]","\frac{a x^2}{2}+\frac{b x^4}{4}","\frac{a x^2}{2}+\frac{b x^4}{4}",1,"(a*x^2)/2 + (b*x^4)/4","A",1
5,1,12,12,0.0000768,"\int \left(a+b x^2\right) \, dx","Integrate[a + b*x^2,x]","a x+\frac{b x^3}{3}","a x+\frac{b x^3}{3}",1,"a*x + (b*x^3)/3","A",1
6,1,13,13,0.0010617,"\int \frac{a+b x^2}{x} \, dx","Integrate[(a + b*x^2)/x,x]","a \log (x)+\frac{b x^2}{2}","a \log (x)+\frac{b x^2}{2}",1,"(b*x^2)/2 + a*Log[x]","A",1
7,1,10,10,0.0008547,"\int \frac{a+b x^2}{x^2} \, dx","Integrate[(a + b*x^2)/x^2,x]","b x-\frac{a}{x}","b x-\frac{a}{x}",1,"-(a/x) + b*x","A",1
8,1,13,13,0.0026461,"\int \frac{a+b x^2}{x^3} \, dx","Integrate[(a + b*x^2)/x^3,x]","b \log (x)-\frac{a}{2 x^2}","b \log (x)-\frac{a}{2 x^2}",1,"-1/2*a/x^2 + b*Log[x]","A",1
9,1,15,15,0.00192,"\int \frac{a+b x^2}{x^4} \, dx","Integrate[(a + b*x^2)/x^4,x]","-\frac{a}{3 x^3}-\frac{b}{x}","-\frac{a}{3 x^3}-\frac{b}{x}",1,"-1/3*a/x^3 - b/x","A",1
10,1,17,17,0.0019848,"\int \frac{a+b x^2}{x^5} \, dx","Integrate[(a + b*x^2)/x^5,x]","-\frac{a}{4 x^4}-\frac{b}{2 x^2}","-\frac{a}{4 x^4}-\frac{b}{2 x^2}",1,"-1/4*a/x^4 - b/(2*x^2)","A",1
11,1,17,17,0.0019492,"\int \frac{a+b x^2}{x^6} \, dx","Integrate[(a + b*x^2)/x^6,x]","-\frac{a}{5 x^5}-\frac{b}{3 x^3}","-\frac{a}{5 x^5}-\frac{b}{3 x^3}",1,"-1/5*a/x^5 - b/(3*x^3)","A",1
12,1,17,17,0.0021149,"\int \frac{a+b x^2}{x^7} \, dx","Integrate[(a + b*x^2)/x^7,x]","-\frac{a}{6 x^6}-\frac{b}{4 x^4}","-\frac{a}{6 x^6}-\frac{b}{4 x^4}",1,"-1/6*a/x^6 - b/(4*x^4)","A",1
13,1,30,30,0.0010155,"\int x^5 \left(a+b x^2\right)^2 \, dx","Integrate[x^5*(a + b*x^2)^2,x]","\frac{a^2 x^6}{6}+\frac{1}{4} a b x^8+\frac{b^2 x^{10}}{10}","\frac{a^2 x^6}{6}+\frac{1}{4} a b x^8+\frac{b^2 x^{10}}{10}",1,"(a^2*x^6)/6 + (a*b*x^8)/4 + (b^2*x^10)/10","A",1
14,1,30,30,0.0007668,"\int x^4 \left(a+b x^2\right)^2 \, dx","Integrate[x^4*(a + b*x^2)^2,x]","\frac{a^2 x^5}{5}+\frac{2}{7} a b x^7+\frac{b^2 x^9}{9}","\frac{a^2 x^5}{5}+\frac{2}{7} a b x^7+\frac{b^2 x^9}{9}",1,"(a^2*x^5)/5 + (2*a*b*x^7)/7 + (b^2*x^9)/9","A",1
15,1,30,30,0.0007825,"\int x^3 \left(a+b x^2\right)^2 \, dx","Integrate[x^3*(a + b*x^2)^2,x]","\frac{a^2 x^4}{4}+\frac{1}{3} a b x^6+\frac{b^2 x^8}{8}","\frac{a^2 x^4}{4}+\frac{1}{3} a b x^6+\frac{b^2 x^8}{8}",1,"(a^2*x^4)/4 + (a*b*x^6)/3 + (b^2*x^8)/8","A",1
16,1,30,30,0.0007546,"\int x^2 \left(a+b x^2\right)^2 \, dx","Integrate[x^2*(a + b*x^2)^2,x]","\frac{a^2 x^3}{3}+\frac{2}{5} a b x^5+\frac{b^2 x^7}{7}","\frac{a^2 x^3}{3}+\frac{2}{5} a b x^5+\frac{b^2 x^7}{7}",1,"(a^2*x^3)/3 + (2*a*b*x^5)/5 + (b^2*x^7)/7","A",1
17,1,16,16,0.0021614,"\int x \left(a+b x^2\right)^2 \, dx","Integrate[x*(a + b*x^2)^2,x]","\frac{\left(a+b x^2\right)^3}{6 b}","\frac{\left(a+b x^2\right)^3}{6 b}",1,"(a + b*x^2)^3/(6*b)","A",1
18,1,25,25,0.0009259,"\int \left(a+b x^2\right)^2 \, dx","Integrate[(a + b*x^2)^2,x]","a^2 x+\frac{2}{3} a b x^3+\frac{b^2 x^5}{5}","a^2 x+\frac{2}{3} a b x^3+\frac{b^2 x^5}{5}",1,"a^2*x + (2*a*b*x^3)/3 + (b^2*x^5)/5","A",1
19,1,23,23,0.0008556,"\int \frac{\left(a+b x^2\right)^2}{x} \, dx","Integrate[(a + b*x^2)^2/x,x]","a^2 \log (x)+a b x^2+\frac{b^2 x^4}{4}","a^2 \log (x)+a b x^2+\frac{b^2 x^4}{4}",1,"a*b*x^2 + (b^2*x^4)/4 + a^2*Log[x]","A",1
20,1,24,24,0.0007025,"\int \frac{\left(a+b x^2\right)^2}{x^2} \, dx","Integrate[(a + b*x^2)^2/x^2,x]","-\frac{a^2}{x}+2 a b x+\frac{b^2 x^3}{3}","-\frac{a^2}{x}+2 a b x+\frac{b^2 x^3}{3}",1,"-(a^2/x) + 2*a*b*x + (b^2*x^3)/3","A",1
21,1,27,27,0.0008853,"\int \frac{\left(a+b x^2\right)^2}{x^3} \, dx","Integrate[(a + b*x^2)^2/x^3,x]","-\frac{a^2}{2 x^2}+2 a b \log (x)+\frac{b^2 x^2}{2}","-\frac{a^2}{2 x^2}+2 a b \log (x)+\frac{b^2 x^2}{2}",1,"-1/2*a^2/x^2 + (b^2*x^2)/2 + 2*a*b*Log[x]","A",1
22,1,23,23,0.00079,"\int \frac{\left(a+b x^2\right)^2}{x^4} \, dx","Integrate[(a + b*x^2)^2/x^4,x]","-\frac{a^2}{3 x^3}-\frac{2 a b}{x}+b^2 x","-\frac{a^2}{3 x^3}-\frac{2 a b}{x}+b^2 x",1,"-1/3*a^2/x^3 - (2*a*b)/x + b^2*x","A",1
23,1,24,24,0.0007786,"\int \frac{\left(a+b x^2\right)^2}{x^5} \, dx","Integrate[(a + b*x^2)^2/x^5,x]","-\frac{a^2}{4 x^4}-\frac{a b}{x^2}+b^2 \log (x)","-\frac{a^2}{4 x^4}-\frac{a b}{x^2}+b^2 \log (x)",1,"-1/4*a^2/x^4 - (a*b)/x^2 + b^2*Log[x]","A",1
24,1,28,28,0.0007984,"\int \frac{\left(a+b x^2\right)^2}{x^6} \, dx","Integrate[(a + b*x^2)^2/x^6,x]","-\frac{a^2}{5 x^5}-\frac{2 a b}{3 x^3}-\frac{b^2}{x}","-\frac{a^2}{5 x^5}-\frac{2 a b}{3 x^3}-\frac{b^2}{x}",1,"-1/5*a^2/x^5 - (2*a*b)/(3*x^3) - b^2/x","A",1
25,1,30,19,0.0007854,"\int \frac{\left(a+b x^2\right)^2}{x^7} \, dx","Integrate[(a + b*x^2)^2/x^7,x]","-\frac{a^2}{6 x^6}-\frac{a b}{2 x^4}-\frac{b^2}{2 x^2}","-\frac{\left(a+b x^2\right)^3}{6 a x^6}",1,"-1/6*a^2/x^6 - (a*b)/(2*x^4) - b^2/(2*x^2)","A",1
26,1,30,30,0.0008337,"\int \frac{\left(a+b x^2\right)^2}{x^8} \, dx","Integrate[(a + b*x^2)^2/x^8,x]","-\frac{a^2}{7 x^7}-\frac{2 a b}{5 x^5}-\frac{b^2}{3 x^3}","-\frac{a^2}{7 x^7}-\frac{2 a b}{5 x^5}-\frac{b^2}{3 x^3}",1,"-1/7*a^2/x^7 - (2*a*b)/(5*x^5) - b^2/(3*x^3)","A",1
27,1,30,30,0.0008527,"\int \frac{\left(a+b x^2\right)^2}{x^9} \, dx","Integrate[(a + b*x^2)^2/x^9,x]","-\frac{a^2}{8 x^8}-\frac{a b}{3 x^6}-\frac{b^2}{4 x^4}","-\frac{a^2}{8 x^8}-\frac{a b}{3 x^6}-\frac{b^2}{4 x^4}",1,"-1/8*a^2/x^8 - (a*b)/(3*x^6) - b^2/(4*x^4)","A",1
28,1,30,30,0.0008311,"\int \frac{\left(a+b x^2\right)^2}{x^{10}} \, dx","Integrate[(a + b*x^2)^2/x^10,x]","-\frac{a^2}{9 x^9}-\frac{2 a b}{7 x^7}-\frac{b^2}{5 x^5}","-\frac{a^2}{9 x^9}-\frac{2 a b}{7 x^7}-\frac{b^2}{5 x^5}",1,"-1/9*a^2/x^9 - (2*a*b)/(7*x^7) - b^2/(5*x^5)","A",1
29,1,43,43,0.0019599,"\int x^9 \left(a+b x^2\right)^3 \, dx","Integrate[x^9*(a + b*x^2)^3,x]","\frac{a^3 x^{10}}{10}+\frac{1}{4} a^2 b x^{12}+\frac{3}{14} a b^2 x^{14}+\frac{b^3 x^{16}}{16}","\frac{a^3 x^{10}}{10}+\frac{1}{4} a^2 b x^{12}+\frac{3}{14} a b^2 x^{14}+\frac{b^3 x^{16}}{16}",1,"(a^3*x^10)/10 + (a^2*b*x^12)/4 + (3*a*b^2*x^14)/14 + (b^3*x^16)/16","A",1
30,1,43,43,0.001884,"\int x^7 \left(a+b x^2\right)^3 \, dx","Integrate[x^7*(a + b*x^2)^3,x]","\frac{a^3 x^8}{8}+\frac{3}{10} a^2 b x^{10}+\frac{1}{4} a b^2 x^{12}+\frac{b^3 x^{14}}{14}","\frac{a^3 x^8}{8}+\frac{3}{10} a^2 b x^{10}+\frac{1}{4} a b^2 x^{12}+\frac{b^3 x^{14}}{14}",1,"(a^3*x^8)/8 + (3*a^2*b*x^10)/10 + (a*b^2*x^12)/4 + (b^3*x^14)/14","A",1
31,1,43,43,0.0017512,"\int x^5 \left(a+b x^2\right)^3 \, dx","Integrate[x^5*(a + b*x^2)^3,x]","\frac{a^3 x^6}{6}+\frac{3}{8} a^2 b x^8+\frac{3}{10} a b^2 x^{10}+\frac{b^3 x^{12}}{12}","\frac{a^3 x^6}{6}+\frac{3}{8} a^2 b x^8+\frac{3}{10} a b^2 x^{10}+\frac{b^3 x^{12}}{12}",1,"(a^3*x^6)/6 + (3*a^2*b*x^8)/8 + (3*a*b^2*x^10)/10 + (b^3*x^12)/12","A",1
32,1,43,34,0.0019727,"\int x^3 \left(a+b x^2\right)^3 \, dx","Integrate[x^3*(a + b*x^2)^3,x]","\frac{a^3 x^4}{4}+\frac{1}{2} a^2 b x^6+\frac{3}{8} a b^2 x^8+\frac{b^3 x^{10}}{10}","\frac{\left(a+b x^2\right)^5}{10 b^2}-\frac{a \left(a+b x^2\right)^4}{8 b^2}",1,"(a^3*x^4)/4 + (a^2*b*x^6)/2 + (3*a*b^2*x^8)/8 + (b^3*x^10)/10","A",1
33,1,16,16,0.002001,"\int x \left(a+b x^2\right)^3 \, dx","Integrate[x*(a + b*x^2)^3,x]","\frac{\left(a+b x^2\right)^4}{8 b}","\frac{\left(a+b x^2\right)^4}{8 b}",1,"(a + b*x^2)^4/(8*b)","A",1
34,1,39,39,0.0038002,"\int \frac{\left(a+b x^2\right)^3}{x} \, dx","Integrate[(a + b*x^2)^3/x,x]","a^3 \log (x)+\frac{3}{2} a^2 b x^2+\frac{3}{4} a b^2 x^4+\frac{b^3 x^6}{6}","a^3 \log (x)+\frac{3}{2} a^2 b x^2+\frac{3}{4} a b^2 x^4+\frac{b^3 x^6}{6}",1,"(3*a^2*b*x^2)/2 + (3*a*b^2*x^4)/4 + (b^3*x^6)/6 + a^3*Log[x]","A",1
35,1,40,40,0.0067184,"\int \frac{\left(a+b x^2\right)^3}{x^3} \, dx","Integrate[(a + b*x^2)^3/x^3,x]","-\frac{a^3}{2 x^2}+3 a^2 b \log (x)+\frac{3}{2} a b^2 x^2+\frac{b^3 x^4}{4}","-\frac{a^3}{2 x^2}+3 a^2 b \log (x)+\frac{3}{2} a b^2 x^2+\frac{b^3 x^4}{4}",1,"-1/2*a^3/x^2 + (3*a*b^2*x^2)/2 + (b^3*x^4)/4 + 3*a^2*b*Log[x]","A",1
36,1,40,40,0.0045029,"\int \frac{\left(a+b x^2\right)^3}{x^5} \, dx","Integrate[(a + b*x^2)^3/x^5,x]","-\frac{a^3}{4 x^4}-\frac{3 a^2 b}{2 x^2}+3 a b^2 \log (x)+\frac{b^3 x^2}{2}","-\frac{a^3}{4 x^4}-\frac{3 a^2 b}{2 x^2}+3 a b^2 \log (x)+\frac{b^3 x^2}{2}",1,"-1/4*a^3/x^4 - (3*a^2*b)/(2*x^2) + (b^3*x^2)/2 + 3*a*b^2*Log[x]","A",1
37,1,39,39,0.0041415,"\int \frac{\left(a+b x^2\right)^3}{x^7} \, dx","Integrate[(a + b*x^2)^3/x^7,x]","-\frac{a^3}{6 x^6}-\frac{3 a^2 b}{4 x^4}-\frac{3 a b^2}{2 x^2}+b^3 \log (x)","-\frac{a^3}{6 x^6}-\frac{3 a^2 b}{4 x^4}-\frac{3 a b^2}{2 x^2}+b^3 \log (x)",1,"-1/6*a^3/x^6 - (3*a^2*b)/(4*x^4) - (3*a*b^2)/(2*x^2) + b^3*Log[x]","A",1
38,1,43,19,0.0066568,"\int \frac{\left(a+b x^2\right)^3}{x^9} \, dx","Integrate[(a + b*x^2)^3/x^9,x]","-\frac{a^3}{8 x^8}-\frac{a^2 b}{2 x^6}-\frac{3 a b^2}{4 x^4}-\frac{b^3}{2 x^2}","-\frac{\left(a+b x^2\right)^4}{8 a x^8}",1,"-1/8*a^3/x^8 - (a^2*b)/(2*x^6) - (3*a*b^2)/(4*x^4) - b^3/(2*x^2)","B",1
39,1,43,40,0.0038569,"\int \frac{\left(a+b x^2\right)^3}{x^{11}} \, dx","Integrate[(a + b*x^2)^3/x^11,x]","-\frac{a^3}{10 x^{10}}-\frac{3 a^2 b}{8 x^8}-\frac{a b^2}{2 x^6}-\frac{b^3}{4 x^4}","\frac{b \left(a+b x^2\right)^4}{40 a^2 x^8}-\frac{\left(a+b x^2\right)^4}{10 a x^{10}}",1,"-1/10*a^3/x^10 - (3*a^2*b)/(8*x^8) - (a*b^2)/(2*x^6) - b^3/(4*x^4)","A",1
40,1,43,43,0.0038469,"\int \frac{\left(a+b x^2\right)^3}{x^{13}} \, dx","Integrate[(a + b*x^2)^3/x^13,x]","-\frac{a^3}{12 x^{12}}-\frac{3 a^2 b}{10 x^{10}}-\frac{3 a b^2}{8 x^8}-\frac{b^3}{6 x^6}","-\frac{a^3}{12 x^{12}}-\frac{3 a^2 b}{10 x^{10}}-\frac{3 a b^2}{8 x^8}-\frac{b^3}{6 x^6}",1,"-1/12*a^3/x^12 - (3*a^2*b)/(10*x^10) - (3*a*b^2)/(8*x^8) - b^3/(6*x^6)","A",1
41,1,43,43,0.0067606,"\int \frac{\left(a+b x^2\right)^3}{x^{15}} \, dx","Integrate[(a + b*x^2)^3/x^15,x]","-\frac{a^3}{14 x^{14}}-\frac{a^2 b}{4 x^{12}}-\frac{3 a b^2}{10 x^{10}}-\frac{b^3}{8 x^8}","-\frac{a^3}{14 x^{14}}-\frac{a^2 b}{4 x^{12}}-\frac{3 a b^2}{10 x^{10}}-\frac{b^3}{8 x^8}",1,"-1/14*a^3/x^14 - (a^2*b)/(4*x^12) - (3*a*b^2)/(10*x^10) - b^3/(8*x^8)","A",1
42,1,43,43,0.0018533,"\int x^6 \left(a+b x^2\right)^3 \, dx","Integrate[x^6*(a + b*x^2)^3,x]","\frac{a^3 x^7}{7}+\frac{1}{3} a^2 b x^9+\frac{3}{11} a b^2 x^{11}+\frac{b^3 x^{13}}{13}","\frac{a^3 x^7}{7}+\frac{1}{3} a^2 b x^9+\frac{3}{11} a b^2 x^{11}+\frac{b^3 x^{13}}{13}",1,"(a^3*x^7)/7 + (a^2*b*x^9)/3 + (3*a*b^2*x^11)/11 + (b^3*x^13)/13","A",1
43,1,43,43,0.0018005,"\int x^4 \left(a+b x^2\right)^3 \, dx","Integrate[x^4*(a + b*x^2)^3,x]","\frac{a^3 x^5}{5}+\frac{3}{7} a^2 b x^7+\frac{1}{3} a b^2 x^9+\frac{b^3 x^{11}}{11}","\frac{a^3 x^5}{5}+\frac{3}{7} a^2 b x^7+\frac{1}{3} a b^2 x^9+\frac{b^3 x^{11}}{11}",1,"(a^3*x^5)/5 + (3*a^2*b*x^7)/7 + (a*b^2*x^9)/3 + (b^3*x^11)/11","A",1
44,1,43,43,0.0018245,"\int x^2 \left(a+b x^2\right)^3 \, dx","Integrate[x^2*(a + b*x^2)^3,x]","\frac{a^3 x^3}{3}+\frac{3}{5} a^2 b x^5+\frac{3}{7} a b^2 x^7+\frac{b^3 x^9}{9}","\frac{a^3 x^3}{3}+\frac{3}{5} a^2 b x^5+\frac{3}{7} a b^2 x^7+\frac{b^3 x^9}{9}",1,"(a^3*x^3)/3 + (3*a^2*b*x^5)/5 + (3*a*b^2*x^7)/7 + (b^3*x^9)/9","A",1
45,1,35,35,0.0009636,"\int \left(a+b x^2\right)^3 \, dx","Integrate[(a + b*x^2)^3,x]","a^3 x+a^2 b x^3+\frac{3}{5} a b^2 x^5+\frac{b^3 x^7}{7}","a^3 x+a^2 b x^3+\frac{3}{5} a b^2 x^5+\frac{b^3 x^7}{7}",1,"a^3*x + a^2*b*x^3 + (3*a*b^2*x^5)/5 + (b^3*x^7)/7","A",1
46,1,34,34,0.0035722,"\int \frac{\left(a+b x^2\right)^3}{x^2} \, dx","Integrate[(a + b*x^2)^3/x^2,x]","-\frac{a^3}{x}+3 a^2 b x+a b^2 x^3+\frac{b^3 x^5}{5}","-\frac{a^3}{x}+3 a^2 b x+a b^2 x^3+\frac{b^3 x^5}{5}",1,"-(a^3/x) + 3*a^2*b*x + a*b^2*x^3 + (b^3*x^5)/5","A",1
47,1,37,37,0.003653,"\int \frac{\left(a+b x^2\right)^3}{x^4} \, dx","Integrate[(a + b*x^2)^3/x^4,x]","-\frac{a^3}{3 x^3}-\frac{3 a^2 b}{x}+3 a b^2 x+\frac{b^3 x^3}{3}","-\frac{a^3}{3 x^3}-\frac{3 a^2 b}{x}+3 a b^2 x+\frac{b^3 x^3}{3}",1,"-1/3*a^3/x^3 - (3*a^2*b)/x + 3*a*b^2*x + (b^3*x^3)/3","A",1
48,1,34,34,0.0050488,"\int \frac{\left(a+b x^2\right)^3}{x^6} \, dx","Integrate[(a + b*x^2)^3/x^6,x]","-\frac{a^3}{5 x^5}-\frac{a^2 b}{x^3}-\frac{3 a b^2}{x}+b^3 x","-\frac{a^3}{5 x^5}-\frac{a^2 b}{x^3}-\frac{3 a b^2}{x}+b^3 x",1,"-1/5*a^3/x^5 - (a^2*b)/x^3 - (3*a*b^2)/x + b^3*x","A",1
49,1,39,39,0.0038302,"\int \frac{\left(a+b x^2\right)^3}{x^8} \, dx","Integrate[(a + b*x^2)^3/x^8,x]","-\frac{a^3}{7 x^7}-\frac{3 a^2 b}{5 x^5}-\frac{a b^2}{x^3}-\frac{b^3}{x}","-\frac{a^3}{7 x^7}-\frac{3 a^2 b}{5 x^5}-\frac{a b^2}{x^3}-\frac{b^3}{x}",1,"-1/7*a^3/x^7 - (3*a^2*b)/(5*x^5) - (a*b^2)/x^3 - b^3/x","A",1
50,1,43,43,0.0038005,"\int \frac{\left(a+b x^2\right)^3}{x^{10}} \, dx","Integrate[(a + b*x^2)^3/x^10,x]","-\frac{a^3}{9 x^9}-\frac{3 a^2 b}{7 x^7}-\frac{3 a b^2}{5 x^5}-\frac{b^3}{3 x^3}","-\frac{a^3}{9 x^9}-\frac{3 a^2 b}{7 x^7}-\frac{3 a b^2}{5 x^5}-\frac{b^3}{3 x^3}",1,"-1/9*a^3/x^9 - (3*a^2*b)/(7*x^7) - (3*a*b^2)/(5*x^5) - b^3/(3*x^3)","A",1
51,1,43,43,0.0064557,"\int \frac{\left(a+b x^2\right)^3}{x^{12}} \, dx","Integrate[(a + b*x^2)^3/x^12,x]","-\frac{a^3}{11 x^{11}}-\frac{a^2 b}{3 x^9}-\frac{3 a b^2}{7 x^7}-\frac{b^3}{5 x^5}","-\frac{a^3}{11 x^{11}}-\frac{a^2 b}{3 x^9}-\frac{3 a b^2}{7 x^7}-\frac{b^3}{5 x^5}",1,"-1/11*a^3/x^11 - (a^2*b)/(3*x^9) - (3*a*b^2)/(7*x^7) - b^3/(5*x^5)","A",1
52,1,69,69,0.0027996,"\int x^{13} \left(a+b x^2\right)^5 \, dx","Integrate[x^13*(a + b*x^2)^5,x]","\frac{a^5 x^{14}}{14}+\frac{5}{16} a^4 b x^{16}+\frac{5}{9} a^3 b^2 x^{18}+\frac{1}{2} a^2 b^3 x^{20}+\frac{5}{22} a b^4 x^{22}+\frac{b^5 x^{24}}{24}","\frac{a^5 x^{14}}{14}+\frac{5}{16} a^4 b x^{16}+\frac{5}{9} a^3 b^2 x^{18}+\frac{1}{2} a^2 b^3 x^{20}+\frac{5}{22} a b^4 x^{22}+\frac{b^5 x^{24}}{24}",1,"(a^5*x^14)/14 + (5*a^4*b*x^16)/16 + (5*a^3*b^2*x^18)/9 + (a^2*b^3*x^20)/2 + (5*a*b^4*x^22)/22 + (b^5*x^24)/24","A",1
53,1,69,69,0.0022669,"\int x^{11} \left(a+b x^2\right)^5 \, dx","Integrate[x^11*(a + b*x^2)^5,x]","\frac{a^5 x^{12}}{12}+\frac{5}{14} a^4 b x^{14}+\frac{5}{8} a^3 b^2 x^{16}+\frac{5}{9} a^2 b^3 x^{18}+\frac{1}{4} a b^4 x^{20}+\frac{b^5 x^{22}}{22}","\frac{a^5 x^{12}}{12}+\frac{5}{14} a^4 b x^{14}+\frac{5}{8} a^3 b^2 x^{16}+\frac{5}{9} a^2 b^3 x^{18}+\frac{1}{4} a b^4 x^{20}+\frac{b^5 x^{22}}{22}",1,"(a^5*x^12)/12 + (5*a^4*b*x^14)/14 + (5*a^3*b^2*x^16)/8 + (5*a^2*b^3*x^18)/9 + (a*b^4*x^20)/4 + (b^5*x^22)/22","A",1
54,1,69,69,0.0022169,"\int x^9 \left(a+b x^2\right)^5 \, dx","Integrate[x^9*(a + b*x^2)^5,x]","\frac{a^5 x^{10}}{10}+\frac{5}{12} a^4 b x^{12}+\frac{5}{7} a^3 b^2 x^{14}+\frac{5}{8} a^2 b^3 x^{16}+\frac{5}{18} a b^4 x^{18}+\frac{b^5 x^{20}}{20}","\frac{a^5 x^{10}}{10}+\frac{5}{12} a^4 b x^{12}+\frac{5}{7} a^3 b^2 x^{14}+\frac{5}{8} a^2 b^3 x^{16}+\frac{5}{18} a b^4 x^{18}+\frac{b^5 x^{20}}{20}",1,"(a^5*x^10)/10 + (5*a^4*b*x^12)/12 + (5*a^3*b^2*x^14)/7 + (5*a^2*b^3*x^16)/8 + (5*a*b^4*x^18)/18 + (b^5*x^20)/20","A",1
55,1,69,72,0.0022657,"\int x^7 \left(a+b x^2\right)^5 \, dx","Integrate[x^7*(a + b*x^2)^5,x]","\frac{a^5 x^8}{8}+\frac{1}{2} a^4 b x^{10}+\frac{5}{6} a^3 b^2 x^{12}+\frac{5}{7} a^2 b^3 x^{14}+\frac{5}{16} a b^4 x^{16}+\frac{b^5 x^{18}}{18}","-\frac{a^3 \left(a+b x^2\right)^6}{12 b^4}+\frac{3 a^2 \left(a+b x^2\right)^7}{14 b^4}+\frac{\left(a+b x^2\right)^9}{18 b^4}-\frac{3 a \left(a+b x^2\right)^8}{16 b^4}",1,"(a^5*x^8)/8 + (a^4*b*x^10)/2 + (5*a^3*b^2*x^12)/6 + (5*a^2*b^3*x^14)/7 + (5*a*b^4*x^16)/16 + (b^5*x^18)/18","A",1
56,1,66,53,0.0022391,"\int x^5 \left(a+b x^2\right)^5 \, dx","Integrate[x^5*(a + b*x^2)^5,x]","\frac{a^5 x^6}{6}+\frac{5}{8} a^4 b x^8+a^3 b^2 x^{10}+\frac{5}{6} a^2 b^3 x^{12}+\frac{5}{14} a b^4 x^{14}+\frac{b^5 x^{16}}{16}","\frac{a^2 \left(a+b x^2\right)^6}{12 b^3}+\frac{\left(a+b x^2\right)^8}{16 b^3}-\frac{a \left(a+b x^2\right)^7}{7 b^3}",1,"(a^5*x^6)/6 + (5*a^4*b*x^8)/8 + a^3*b^2*x^10 + (5*a^2*b^3*x^12)/6 + (5*a*b^4*x^14)/14 + (b^5*x^16)/16","A",1
57,1,66,34,0.0022659,"\int x^3 \left(a+b x^2\right)^5 \, dx","Integrate[x^3*(a + b*x^2)^5,x]","\frac{a^5 x^4}{4}+\frac{5}{6} a^4 b x^6+\frac{5}{4} a^3 b^2 x^8+a^2 b^3 x^{10}+\frac{5}{12} a b^4 x^{12}+\frac{b^5 x^{14}}{14}","\frac{\left(a+b x^2\right)^7}{14 b^2}-\frac{a \left(a+b x^2\right)^6}{12 b^2}",1,"(a^5*x^4)/4 + (5*a^4*b*x^6)/6 + (5*a^3*b^2*x^8)/4 + a^2*b^3*x^10 + (5*a*b^4*x^12)/12 + (b^5*x^14)/14","A",1
58,1,16,16,0.002403,"\int x \left(a+b x^2\right)^5 \, dx","Integrate[x*(a + b*x^2)^5,x]","\frac{\left(a+b x^2\right)^6}{12 b}","\frac{\left(a+b x^2\right)^6}{12 b}",1,"(a + b*x^2)^6/(12*b)","A",1
59,1,65,65,0.0041043,"\int \frac{\left(a+b x^2\right)^5}{x} \, dx","Integrate[(a + b*x^2)^5/x,x]","a^5 \log (x)+\frac{5}{2} a^4 b x^2+\frac{5}{2} a^3 b^2 x^4+\frac{5}{3} a^2 b^3 x^6+\frac{5}{8} a b^4 x^8+\frac{b^5 x^{10}}{10}","a^5 \log (x)+\frac{5}{2} a^4 b x^2+\frac{5}{2} a^3 b^2 x^4+\frac{5}{3} a^2 b^3 x^6+\frac{5}{8} a b^4 x^8+\frac{b^5 x^{10}}{10}",1,"(5*a^4*b*x^2)/2 + (5*a^3*b^2*x^4)/2 + (5*a^2*b^3*x^6)/3 + (5*a*b^4*x^8)/8 + (b^5*x^10)/10 + a^5*Log[x]","A",1
60,1,64,64,0.0045426,"\int \frac{\left(a+b x^2\right)^5}{x^3} \, dx","Integrate[(a + b*x^2)^5/x^3,x]","-\frac{a^5}{2 x^2}+5 a^4 b \log (x)+5 a^3 b^2 x^2+\frac{5}{2} a^2 b^3 x^4+\frac{5}{6} a b^4 x^6+\frac{b^5 x^8}{8}","-\frac{a^5}{2 x^2}+5 a^4 b \log (x)+5 a^3 b^2 x^2+\frac{5}{2} a^2 b^3 x^4+\frac{5}{6} a b^4 x^6+\frac{b^5 x^8}{8}",1,"-1/2*a^5/x^2 + 5*a^3*b^2*x^2 + (5*a^2*b^3*x^4)/2 + (5*a*b^4*x^6)/6 + (b^5*x^8)/8 + 5*a^4*b*Log[x]","A",1
61,1,64,64,0.0065108,"\int \frac{\left(a+b x^2\right)^5}{x^5} \, dx","Integrate[(a + b*x^2)^5/x^5,x]","-\frac{a^5}{4 x^4}-\frac{5 a^4 b}{2 x^2}+10 a^3 b^2 \log (x)+5 a^2 b^3 x^2+\frac{5}{4} a b^4 x^4+\frac{b^5 x^6}{6}","-\frac{a^5}{4 x^4}-\frac{5 a^4 b}{2 x^2}+10 a^3 b^2 \log (x)+5 a^2 b^3 x^2+\frac{5}{4} a b^4 x^4+\frac{b^5 x^6}{6}",1,"-1/4*a^5/x^4 - (5*a^4*b)/(2*x^2) + 5*a^2*b^3*x^2 + (5*a*b^4*x^4)/4 + (b^5*x^6)/6 + 10*a^3*b^2*Log[x]","A",1
62,1,64,64,0.0048211,"\int \frac{\left(a+b x^2\right)^5}{x^7} \, dx","Integrate[(a + b*x^2)^5/x^7,x]","-\frac{a^5}{6 x^6}-\frac{5 a^4 b}{4 x^4}-\frac{5 a^3 b^2}{x^2}+10 a^2 b^3 \log (x)+\frac{5}{2} a b^4 x^2+\frac{b^5 x^4}{4}","-\frac{a^5}{6 x^6}-\frac{5 a^4 b}{4 x^4}-\frac{5 a^3 b^2}{x^2}+10 a^2 b^3 \log (x)+\frac{5}{2} a b^4 x^2+\frac{b^5 x^4}{4}",1,"-1/6*a^5/x^6 - (5*a^4*b)/(4*x^4) - (5*a^3*b^2)/x^2 + (5*a*b^4*x^2)/2 + (b^5*x^4)/4 + 10*a^2*b^3*Log[x]","A",1
63,1,64,64,0.0049841,"\int \frac{\left(a+b x^2\right)^5}{x^9} \, dx","Integrate[(a + b*x^2)^5/x^9,x]","-\frac{a^5}{8 x^8}-\frac{5 a^4 b}{6 x^6}-\frac{5 a^3 b^2}{2 x^4}-\frac{5 a^2 b^3}{x^2}+5 a b^4 \log (x)+\frac{b^5 x^2}{2}","-\frac{a^5}{8 x^8}-\frac{5 a^4 b}{6 x^6}-\frac{5 a^3 b^2}{2 x^4}-\frac{5 a^2 b^3}{x^2}+5 a b^4 \log (x)+\frac{b^5 x^2}{2}",1,"-1/8*a^5/x^8 - (5*a^4*b)/(6*x^6) - (5*a^3*b^2)/(2*x^4) - (5*a^2*b^3)/x^2 + (b^5*x^2)/2 + 5*a*b^4*Log[x]","A",1
64,1,65,65,0.0048876,"\int \frac{\left(a+b x^2\right)^5}{x^{11}} \, dx","Integrate[(a + b*x^2)^5/x^11,x]","-\frac{a^5}{10 x^{10}}-\frac{5 a^4 b}{8 x^8}-\frac{5 a^3 b^2}{3 x^6}-\frac{5 a^2 b^3}{2 x^4}-\frac{5 a b^4}{2 x^2}+b^5 \log (x)","-\frac{a^5}{10 x^{10}}-\frac{5 a^4 b}{8 x^8}-\frac{5 a^3 b^2}{3 x^6}-\frac{5 a^2 b^3}{2 x^4}-\frac{5 a b^4}{2 x^2}+b^5 \log (x)",1,"-1/10*a^5/x^10 - (5*a^4*b)/(8*x^8) - (5*a^3*b^2)/(3*x^6) - (5*a^2*b^3)/(2*x^4) - (5*a*b^4)/(2*x^2) + b^5*Log[x]","A",1
65,1,69,19,0.0044913,"\int \frac{\left(a+b x^2\right)^5}{x^{13}} \, dx","Integrate[(a + b*x^2)^5/x^13,x]","-\frac{a^5}{12 x^{12}}-\frac{a^4 b}{2 x^{10}}-\frac{5 a^3 b^2}{4 x^8}-\frac{5 a^2 b^3}{3 x^6}-\frac{5 a b^4}{4 x^4}-\frac{b^5}{2 x^2}","-\frac{\left(a+b x^2\right)^6}{12 a x^{12}}",1,"-1/12*a^5/x^12 - (a^4*b)/(2*x^10) - (5*a^3*b^2)/(4*x^8) - (5*a^2*b^3)/(3*x^6) - (5*a*b^4)/(4*x^4) - b^5/(2*x^2)","B",1
66,1,67,40,0.0063381,"\int \frac{\left(a+b x^2\right)^5}{x^{15}} \, dx","Integrate[(a + b*x^2)^5/x^15,x]","-\frac{a^5}{14 x^{14}}-\frac{5 a^4 b}{12 x^{12}}-\frac{a^3 b^2}{x^{10}}-\frac{5 a^2 b^3}{4 x^8}-\frac{5 a b^4}{6 x^6}-\frac{b^5}{4 x^4}","\frac{b \left(a+b x^2\right)^6}{84 a^2 x^{12}}-\frac{\left(a+b x^2\right)^6}{14 a x^{14}}",1,"-1/14*a^5/x^14 - (5*a^4*b)/(12*x^12) - (a^3*b^2)/x^10 - (5*a^2*b^3)/(4*x^8) - (5*a*b^4)/(6*x^6) - b^5/(4*x^4)","A",1
67,1,67,62,0.0042077,"\int \frac{\left(a+b x^2\right)^5}{x^{17}} \, dx","Integrate[(a + b*x^2)^5/x^17,x]","-\frac{a^5}{16 x^{16}}-\frac{5 a^4 b}{14 x^{14}}-\frac{5 a^3 b^2}{6 x^{12}}-\frac{a^2 b^3}{x^{10}}-\frac{5 a b^4}{8 x^8}-\frac{b^5}{6 x^6}","-\frac{b^2 \left(a+b x^2\right)^6}{336 a^3 x^{12}}+\frac{b \left(a+b x^2\right)^6}{56 a^2 x^{14}}-\frac{\left(a+b x^2\right)^6}{16 a x^{16}}",1,"-1/16*a^5/x^16 - (5*a^4*b)/(14*x^14) - (5*a^3*b^2)/(6*x^12) - (a^2*b^3)/x^10 - (5*a*b^4)/(8*x^8) - b^5/(6*x^6)","A",1
68,1,69,69,0.004283,"\int \frac{\left(a+b x^2\right)^5}{x^{19}} \, dx","Integrate[(a + b*x^2)^5/x^19,x]","-\frac{a^5}{18 x^{18}}-\frac{5 a^4 b}{16 x^{16}}-\frac{5 a^3 b^2}{7 x^{14}}-\frac{5 a^2 b^3}{6 x^{12}}-\frac{a b^4}{2 x^{10}}-\frac{b^5}{8 x^8}","-\frac{a^5}{18 x^{18}}-\frac{5 a^4 b}{16 x^{16}}-\frac{5 a^3 b^2}{7 x^{14}}-\frac{5 a^2 b^3}{6 x^{12}}-\frac{a b^4}{2 x^{10}}-\frac{b^5}{8 x^8}",1,"-1/18*a^5/x^18 - (5*a^4*b)/(16*x^16) - (5*a^3*b^2)/(7*x^14) - (5*a^2*b^3)/(6*x^12) - (a*b^4)/(2*x^10) - b^5/(8*x^8)","A",1
69,1,69,69,0.0042738,"\int \frac{\left(a+b x^2\right)^5}{x^{21}} \, dx","Integrate[(a + b*x^2)^5/x^21,x]","-\frac{a^5}{20 x^{20}}-\frac{5 a^4 b}{18 x^{18}}-\frac{5 a^3 b^2}{8 x^{16}}-\frac{5 a^2 b^3}{7 x^{14}}-\frac{5 a b^4}{12 x^{12}}-\frac{b^5}{10 x^{10}}","-\frac{a^5}{20 x^{20}}-\frac{5 a^4 b}{18 x^{18}}-\frac{5 a^3 b^2}{8 x^{16}}-\frac{5 a^2 b^3}{7 x^{14}}-\frac{5 a b^4}{12 x^{12}}-\frac{b^5}{10 x^{10}}",1,"-1/20*a^5/x^20 - (5*a^4*b)/(18*x^18) - (5*a^3*b^2)/(8*x^16) - (5*a^2*b^3)/(7*x^14) - (5*a*b^4)/(12*x^12) - b^5/(10*x^10)","A",1
70,1,69,69,0.0024544,"\int x^8 \left(a+b x^2\right)^5 \, dx","Integrate[x^8*(a + b*x^2)^5,x]","\frac{a^5 x^9}{9}+\frac{5}{11} a^4 b x^{11}+\frac{10}{13} a^3 b^2 x^{13}+\frac{2}{3} a^2 b^3 x^{15}+\frac{5}{17} a b^4 x^{17}+\frac{b^5 x^{19}}{19}","\frac{a^5 x^9}{9}+\frac{5}{11} a^4 b x^{11}+\frac{10}{13} a^3 b^2 x^{13}+\frac{2}{3} a^2 b^3 x^{15}+\frac{5}{17} a b^4 x^{17}+\frac{b^5 x^{19}}{19}",1,"(a^5*x^9)/9 + (5*a^4*b*x^11)/11 + (10*a^3*b^2*x^13)/13 + (2*a^2*b^3*x^15)/3 + (5*a*b^4*x^17)/17 + (b^5*x^19)/19","A",1
71,1,69,69,0.00227,"\int x^6 \left(a+b x^2\right)^5 \, dx","Integrate[x^6*(a + b*x^2)^5,x]","\frac{a^5 x^7}{7}+\frac{5}{9} a^4 b x^9+\frac{10}{11} a^3 b^2 x^{11}+\frac{10}{13} a^2 b^3 x^{13}+\frac{1}{3} a b^4 x^{15}+\frac{b^5 x^{17}}{17}","\frac{a^5 x^7}{7}+\frac{5}{9} a^4 b x^9+\frac{10}{11} a^3 b^2 x^{11}+\frac{10}{13} a^2 b^3 x^{13}+\frac{1}{3} a b^4 x^{15}+\frac{b^5 x^{17}}{17}",1,"(a^5*x^7)/7 + (5*a^4*b*x^9)/9 + (10*a^3*b^2*x^11)/11 + (10*a^2*b^3*x^13)/13 + (a*b^4*x^15)/3 + (b^5*x^17)/17","A",1
72,1,69,69,0.0022386,"\int x^4 \left(a+b x^2\right)^5 \, dx","Integrate[x^4*(a + b*x^2)^5,x]","\frac{a^5 x^5}{5}+\frac{5}{7} a^4 b x^7+\frac{10}{9} a^3 b^2 x^9+\frac{10}{11} a^2 b^3 x^{11}+\frac{5}{13} a b^4 x^{13}+\frac{b^5 x^{15}}{15}","\frac{a^5 x^5}{5}+\frac{5}{7} a^4 b x^7+\frac{10}{9} a^3 b^2 x^9+\frac{10}{11} a^2 b^3 x^{11}+\frac{5}{13} a b^4 x^{13}+\frac{b^5 x^{15}}{15}",1,"(a^5*x^5)/5 + (5*a^4*b*x^7)/7 + (10*a^3*b^2*x^9)/9 + (10*a^2*b^3*x^11)/11 + (5*a*b^4*x^13)/13 + (b^5*x^15)/15","A",1
73,1,66,66,0.0021754,"\int x^2 \left(a+b x^2\right)^5 \, dx","Integrate[x^2*(a + b*x^2)^5,x]","\frac{a^5 x^3}{3}+a^4 b x^5+\frac{10}{7} a^3 b^2 x^7+\frac{10}{9} a^2 b^3 x^9+\frac{5}{11} a b^4 x^{11}+\frac{b^5 x^{13}}{13}","\frac{a^5 x^3}{3}+a^4 b x^5+\frac{10}{7} a^3 b^2 x^7+\frac{10}{9} a^2 b^3 x^9+\frac{5}{11} a b^4 x^{11}+\frac{b^5 x^{13}}{13}",1,"(a^5*x^3)/3 + a^4*b*x^5 + (10*a^3*b^2*x^7)/7 + (10*a^2*b^3*x^9)/9 + (5*a*b^4*x^11)/11 + (b^5*x^13)/13","A",1
74,1,62,62,0.0012403,"\int \left(a+b x^2\right)^5 \, dx","Integrate[(a + b*x^2)^5,x]","a^5 x+\frac{5}{3} a^4 b x^3+2 a^3 b^2 x^5+\frac{10}{7} a^2 b^3 x^7+\frac{5}{9} a b^4 x^9+\frac{b^5 x^{11}}{11}","a^5 x+\frac{5}{3} a^4 b x^3+2 a^3 b^2 x^5+\frac{10}{7} a^2 b^3 x^7+\frac{5}{9} a b^4 x^9+\frac{b^5 x^{11}}{11}",1,"a^5*x + (5*a^4*b*x^3)/3 + 2*a^3*b^2*x^5 + (10*a^2*b^3*x^7)/7 + (5*a*b^4*x^9)/9 + (b^5*x^11)/11","A",1
75,1,61,61,0.0038948,"\int \frac{\left(a+b x^2\right)^5}{x^2} \, dx","Integrate[(a + b*x^2)^5/x^2,x]","-\frac{a^5}{x}+5 a^4 b x+\frac{10}{3} a^3 b^2 x^3+2 a^2 b^3 x^5+\frac{5}{7} a b^4 x^7+\frac{b^5 x^9}{9}","-\frac{a^5}{x}+5 a^4 b x+\frac{10}{3} a^3 b^2 x^3+2 a^2 b^3 x^5+\frac{5}{7} a b^4 x^7+\frac{b^5 x^9}{9}",1,"-(a^5/x) + 5*a^4*b*x + (10*a^3*b^2*x^3)/3 + 2*a^2*b^3*x^5 + (5*a*b^4*x^7)/7 + (b^5*x^9)/9","A",1
76,1,60,60,0.0040489,"\int \frac{\left(a+b x^2\right)^5}{x^4} \, dx","Integrate[(a + b*x^2)^5/x^4,x]","-\frac{a^5}{3 x^3}-\frac{5 a^4 b}{x}+10 a^3 b^2 x+\frac{10}{3} a^2 b^3 x^3+a b^4 x^5+\frac{b^5 x^7}{7}","-\frac{a^5}{3 x^3}-\frac{5 a^4 b}{x}+10 a^3 b^2 x+\frac{10}{3} a^2 b^3 x^3+a b^4 x^5+\frac{b^5 x^7}{7}",1,"-1/3*a^5/x^3 - (5*a^4*b)/x + 10*a^3*b^2*x + (10*a^2*b^3*x^3)/3 + a*b^4*x^5 + (b^5*x^7)/7","A",1
77,1,63,63,0.0041003,"\int \frac{\left(a+b x^2\right)^5}{x^6} \, dx","Integrate[(a + b*x^2)^5/x^6,x]","-\frac{a^5}{5 x^5}-\frac{5 a^4 b}{3 x^3}-\frac{10 a^3 b^2}{x}+10 a^2 b^3 x+\frac{5}{3} a b^4 x^3+\frac{b^5 x^5}{5}","-\frac{a^5}{5 x^5}-\frac{5 a^4 b}{3 x^3}-\frac{10 a^3 b^2}{x}+10 a^2 b^3 x+\frac{5}{3} a b^4 x^3+\frac{b^5 x^5}{5}",1,"-1/5*a^5/x^5 - (5*a^4*b)/(3*x^3) - (10*a^3*b^2)/x + 10*a^2*b^3*x + (5*a*b^4*x^3)/3 + (b^5*x^5)/5","A",1
78,1,61,61,0.0040717,"\int \frac{\left(a+b x^2\right)^5}{x^8} \, dx","Integrate[(a + b*x^2)^5/x^8,x]","-\frac{a^5}{7 x^7}-\frac{a^4 b}{x^5}-\frac{10 a^3 b^2}{3 x^3}-\frac{10 a^2 b^3}{x}+5 a b^4 x+\frac{b^5 x^3}{3}","-\frac{a^5}{7 x^7}-\frac{a^4 b}{x^5}-\frac{10 a^3 b^2}{3 x^3}-\frac{10 a^2 b^3}{x}+5 a b^4 x+\frac{b^5 x^3}{3}",1,"-1/7*a^5/x^7 - (a^4*b)/x^5 - (10*a^3*b^2)/(3*x^3) - (10*a^2*b^3)/x + 5*a*b^4*x + (b^5*x^3)/3","A",1
79,1,60,60,0.0052041,"\int \frac{\left(a+b x^2\right)^5}{x^{10}} \, dx","Integrate[(a + b*x^2)^5/x^10,x]","-\frac{a^5}{9 x^9}-\frac{5 a^4 b}{7 x^7}-\frac{2 a^3 b^2}{x^5}-\frac{10 a^2 b^3}{3 x^3}-\frac{5 a b^4}{x}+b^5 x","-\frac{a^5}{9 x^9}-\frac{5 a^4 b}{7 x^7}-\frac{2 a^3 b^2}{x^5}-\frac{10 a^2 b^3}{3 x^3}-\frac{5 a b^4}{x}+b^5 x",1,"-1/9*a^5/x^9 - (5*a^4*b)/(7*x^7) - (2*a^3*b^2)/x^5 - (10*a^2*b^3)/(3*x^3) - (5*a*b^4)/x + b^5*x","A",1
80,1,65,65,0.0041256,"\int \frac{\left(a+b x^2\right)^5}{x^{12}} \, dx","Integrate[(a + b*x^2)^5/x^12,x]","-\frac{a^5}{11 x^{11}}-\frac{5 a^4 b}{9 x^9}-\frac{10 a^3 b^2}{7 x^7}-\frac{2 a^2 b^3}{x^5}-\frac{5 a b^4}{3 x^3}-\frac{b^5}{x}","-\frac{a^5}{11 x^{11}}-\frac{5 a^4 b}{9 x^9}-\frac{10 a^3 b^2}{7 x^7}-\frac{2 a^2 b^3}{x^5}-\frac{5 a b^4}{3 x^3}-\frac{b^5}{x}",1,"-1/11*a^5/x^11 - (5*a^4*b)/(9*x^9) - (10*a^3*b^2)/(7*x^7) - (2*a^2*b^3)/x^5 - (5*a*b^4)/(3*x^3) - b^5/x","A",1
81,1,67,67,0.0043086,"\int \frac{\left(a+b x^2\right)^5}{x^{14}} \, dx","Integrate[(a + b*x^2)^5/x^14,x]","-\frac{a^5}{13 x^{13}}-\frac{5 a^4 b}{11 x^{11}}-\frac{10 a^3 b^2}{9 x^9}-\frac{10 a^2 b^3}{7 x^7}-\frac{a b^4}{x^5}-\frac{b^5}{3 x^3}","-\frac{a^5}{13 x^{13}}-\frac{5 a^4 b}{11 x^{11}}-\frac{10 a^3 b^2}{9 x^9}-\frac{10 a^2 b^3}{7 x^7}-\frac{a b^4}{x^5}-\frac{b^5}{3 x^3}",1,"-1/13*a^5/x^13 - (5*a^4*b)/(11*x^11) - (10*a^3*b^2)/(9*x^9) - (10*a^2*b^3)/(7*x^7) - (a*b^4)/x^5 - b^5/(3*x^3)","A",1
82,1,69,69,0.0042782,"\int \frac{\left(a+b x^2\right)^5}{x^{16}} \, dx","Integrate[(a + b*x^2)^5/x^16,x]","-\frac{a^5}{15 x^{15}}-\frac{5 a^4 b}{13 x^{13}}-\frac{10 a^3 b^2}{11 x^{11}}-\frac{10 a^2 b^3}{9 x^9}-\frac{5 a b^4}{7 x^7}-\frac{b^5}{5 x^5}","-\frac{a^5}{15 x^{15}}-\frac{5 a^4 b}{13 x^{13}}-\frac{10 a^3 b^2}{11 x^{11}}-\frac{10 a^2 b^3}{9 x^9}-\frac{5 a b^4}{7 x^7}-\frac{b^5}{5 x^5}",1,"-1/15*a^5/x^15 - (5*a^4*b)/(13*x^13) - (10*a^3*b^2)/(11*x^11) - (10*a^2*b^3)/(9*x^9) - (5*a*b^4)/(7*x^7) - b^5/(5*x^5)","A",1
83,1,69,69,0.0041541,"\int \frac{\left(a+b x^2\right)^5}{x^{18}} \, dx","Integrate[(a + b*x^2)^5/x^18,x]","-\frac{a^5}{17 x^{17}}-\frac{a^4 b}{3 x^{15}}-\frac{10 a^3 b^2}{13 x^{13}}-\frac{10 a^2 b^3}{11 x^{11}}-\frac{5 a b^4}{9 x^9}-\frac{b^5}{7 x^7}","-\frac{a^5}{17 x^{17}}-\frac{a^4 b}{3 x^{15}}-\frac{10 a^3 b^2}{13 x^{13}}-\frac{10 a^2 b^3}{11 x^{11}}-\frac{5 a b^4}{9 x^9}-\frac{b^5}{7 x^7}",1,"-1/17*a^5/x^17 - (a^4*b)/(3*x^15) - (10*a^3*b^2)/(13*x^13) - (10*a^2*b^3)/(11*x^11) - (5*a*b^4)/(9*x^9) - b^5/(7*x^7)","A",1
84,1,69,69,0.0064,"\int \frac{\left(a+b x^2\right)^5}{x^{20}} \, dx","Integrate[(a + b*x^2)^5/x^20,x]","-\frac{a^5}{19 x^{19}}-\frac{5 a^4 b}{17 x^{17}}-\frac{2 a^3 b^2}{3 x^{15}}-\frac{10 a^2 b^3}{13 x^{13}}-\frac{5 a b^4}{11 x^{11}}-\frac{b^5}{9 x^9}","-\frac{a^5}{19 x^{19}}-\frac{5 a^4 b}{17 x^{17}}-\frac{2 a^3 b^2}{3 x^{15}}-\frac{10 a^2 b^3}{13 x^{13}}-\frac{5 a b^4}{11 x^{11}}-\frac{b^5}{9 x^9}",1,"-1/19*a^5/x^19 - (5*a^4*b)/(17*x^17) - (2*a^3*b^2)/(3*x^15) - (10*a^2*b^3)/(13*x^13) - (5*a*b^4)/(11*x^11) - b^5/(9*x^9)","A",1
85,1,108,129,0.0030039,"\int x^{13} \left(a+b x^2\right)^8 \, dx","Integrate[x^13*(a + b*x^2)^8,x]","\frac{a^8 x^{14}}{14}+\frac{1}{2} a^7 b x^{16}+\frac{14}{9} a^6 b^2 x^{18}+\frac{14}{5} a^5 b^3 x^{20}+\frac{35}{11} a^4 b^4 x^{22}+\frac{7}{3} a^3 b^5 x^{24}+\frac{14}{13} a^2 b^6 x^{26}+\frac{2}{7} a b^7 x^{28}+\frac{b^8 x^{30}}{30}","\frac{a^6 \left(a+b x^2\right)^9}{18 b^7}-\frac{3 a^5 \left(a+b x^2\right)^{10}}{10 b^7}+\frac{15 a^4 \left(a+b x^2\right)^{11}}{22 b^7}-\frac{5 a^3 \left(a+b x^2\right)^{12}}{6 b^7}+\frac{15 a^2 \left(a+b x^2\right)^{13}}{26 b^7}+\frac{\left(a+b x^2\right)^{15}}{30 b^7}-\frac{3 a \left(a+b x^2\right)^{14}}{14 b^7}",1,"(a^8*x^14)/14 + (a^7*b*x^16)/2 + (14*a^6*b^2*x^18)/9 + (14*a^5*b^3*x^20)/5 + (35*a^4*b^4*x^22)/11 + (7*a^3*b^5*x^24)/3 + (14*a^2*b^6*x^26)/13 + (2*a*b^7*x^28)/7 + (b^8*x^30)/30","A",1
86,1,108,110,0.0027987,"\int x^{11} \left(a+b x^2\right)^8 \, dx","Integrate[x^11*(a + b*x^2)^8,x]","\frac{a^8 x^{12}}{12}+\frac{4}{7} a^7 b x^{14}+\frac{7}{4} a^6 b^2 x^{16}+\frac{28}{9} a^5 b^3 x^{18}+\frac{7}{2} a^4 b^4 x^{20}+\frac{28}{11} a^3 b^5 x^{22}+\frac{7}{6} a^2 b^6 x^{24}+\frac{4}{13} a b^7 x^{26}+\frac{b^8 x^{28}}{28}","-\frac{a^5 \left(a+b x^2\right)^9}{18 b^6}+\frac{a^4 \left(a+b x^2\right)^{10}}{4 b^6}-\frac{5 a^3 \left(a+b x^2\right)^{11}}{11 b^6}+\frac{5 a^2 \left(a+b x^2\right)^{12}}{12 b^6}+\frac{\left(a+b x^2\right)^{14}}{28 b^6}-\frac{5 a \left(a+b x^2\right)^{13}}{26 b^6}",1,"(a^8*x^12)/12 + (4*a^7*b*x^14)/7 + (7*a^6*b^2*x^16)/4 + (28*a^5*b^3*x^18)/9 + (7*a^4*b^4*x^20)/2 + (28*a^3*b^5*x^22)/11 + (7*a^2*b^6*x^24)/6 + (4*a*b^7*x^26)/13 + (b^8*x^28)/28","A",1
87,1,106,91,0.0026433,"\int x^9 \left(a+b x^2\right)^8 \, dx","Integrate[x^9*(a + b*x^2)^8,x]","\frac{a^8 x^{10}}{10}+\frac{2}{3} a^7 b x^{12}+2 a^6 b^2 x^{14}+\frac{7}{2} a^5 b^3 x^{16}+\frac{35}{9} a^4 b^4 x^{18}+\frac{14}{5} a^3 b^5 x^{20}+\frac{14}{11} a^2 b^6 x^{22}+\frac{1}{3} a b^7 x^{24}+\frac{b^8 x^{26}}{26}","\frac{a^4 \left(a+b x^2\right)^9}{18 b^5}-\frac{a^3 \left(a+b x^2\right)^{10}}{5 b^5}+\frac{3 a^2 \left(a+b x^2\right)^{11}}{11 b^5}+\frac{\left(a+b x^2\right)^{13}}{26 b^5}-\frac{a \left(a+b x^2\right)^{12}}{6 b^5}",1,"(a^8*x^10)/10 + (2*a^7*b*x^12)/3 + 2*a^6*b^2*x^14 + (7*a^5*b^3*x^16)/2 + (35*a^4*b^4*x^18)/9 + (14*a^3*b^5*x^20)/5 + (14*a^2*b^6*x^22)/11 + (a*b^7*x^24)/3 + (b^8*x^26)/26","A",1
88,1,106,72,0.0033066,"\int x^7 \left(a+b x^2\right)^8 \, dx","Integrate[x^7*(a + b*x^2)^8,x]","\frac{a^8 x^8}{8}+\frac{4}{5} a^7 b x^{10}+\frac{7}{3} a^6 b^2 x^{12}+4 a^5 b^3 x^{14}+\frac{35}{8} a^4 b^4 x^{16}+\frac{28}{9} a^3 b^5 x^{18}+\frac{7}{5} a^2 b^6 x^{20}+\frac{4}{11} a b^7 x^{22}+\frac{b^8 x^{24}}{24}","-\frac{a^3 \left(a+b x^2\right)^9}{18 b^4}+\frac{3 a^2 \left(a+b x^2\right)^{10}}{20 b^4}+\frac{\left(a+b x^2\right)^{12}}{24 b^4}-\frac{3 a \left(a+b x^2\right)^{11}}{22 b^4}",1,"(a^8*x^8)/8 + (4*a^7*b*x^10)/5 + (7*a^6*b^2*x^12)/3 + 4*a^5*b^3*x^14 + (35*a^4*b^4*x^16)/8 + (28*a^3*b^5*x^18)/9 + (7*a^2*b^6*x^20)/5 + (4*a*b^7*x^22)/11 + (b^8*x^24)/24","A",1
89,1,103,53,0.0026501,"\int x^5 \left(a+b x^2\right)^8 \, dx","Integrate[x^5*(a + b*x^2)^8,x]","\frac{a^8 x^6}{6}+a^7 b x^8+\frac{14}{5} a^6 b^2 x^{10}+\frac{14}{3} a^5 b^3 x^{12}+5 a^4 b^4 x^{14}+\frac{7}{2} a^3 b^5 x^{16}+\frac{14}{9} a^2 b^6 x^{18}+\frac{2}{5} a b^7 x^{20}+\frac{b^8 x^{22}}{22}","\frac{a^2 \left(a+b x^2\right)^9}{18 b^3}+\frac{\left(a+b x^2\right)^{11}}{22 b^3}-\frac{a \left(a+b x^2\right)^{10}}{10 b^3}",1,"(a^8*x^6)/6 + a^7*b*x^8 + (14*a^6*b^2*x^10)/5 + (14*a^5*b^3*x^12)/3 + 5*a^4*b^4*x^14 + (7*a^3*b^5*x^16)/2 + (14*a^2*b^6*x^18)/9 + (2*a*b^7*x^20)/5 + (b^8*x^22)/22","A",1
90,1,106,34,0.0026617,"\int x^3 \left(a+b x^2\right)^8 \, dx","Integrate[x^3*(a + b*x^2)^8,x]","\frac{a^8 x^4}{4}+\frac{4}{3} a^7 b x^6+\frac{7}{2} a^6 b^2 x^8+\frac{28}{5} a^5 b^3 x^{10}+\frac{35}{6} a^4 b^4 x^{12}+4 a^3 b^5 x^{14}+\frac{7}{4} a^2 b^6 x^{16}+\frac{4}{9} a b^7 x^{18}+\frac{b^8 x^{20}}{20}","\frac{\left(a+b x^2\right)^{10}}{20 b^2}-\frac{a \left(a+b x^2\right)^9}{18 b^2}",1,"(a^8*x^4)/4 + (4*a^7*b*x^6)/3 + (7*a^6*b^2*x^8)/2 + (28*a^5*b^3*x^10)/5 + (35*a^4*b^4*x^12)/6 + 4*a^3*b^5*x^14 + (7*a^2*b^6*x^16)/4 + (4*a*b^7*x^18)/9 + (b^8*x^20)/20","B",1
91,1,16,16,0.0018874,"\int x \left(a+b x^2\right)^8 \, dx","Integrate[x*(a + b*x^2)^8,x]","\frac{\left(a+b x^2\right)^9}{18 b}","\frac{\left(a+b x^2\right)^9}{18 b}",1,"(a + b*x^2)^9/(18*b)","A",1
92,1,100,100,0.0044572,"\int \frac{\left(a+b x^2\right)^8}{x} \, dx","Integrate[(a + b*x^2)^8/x,x]","a^8 \log (x)+4 a^7 b x^2+7 a^6 b^2 x^4+\frac{28}{3} a^5 b^3 x^6+\frac{35}{4} a^4 b^4 x^8+\frac{28}{5} a^3 b^5 x^{10}+\frac{7}{3} a^2 b^6 x^{12}+\frac{4}{7} a b^7 x^{14}+\frac{b^8 x^{16}}{16}","a^8 \log (x)+4 a^7 b x^2+7 a^6 b^2 x^4+\frac{28}{3} a^5 b^3 x^6+\frac{35}{4} a^4 b^4 x^8+\frac{28}{5} a^3 b^5 x^{10}+\frac{7}{3} a^2 b^6 x^{12}+\frac{4}{7} a b^7 x^{14}+\frac{b^8 x^{16}}{16}",1,"4*a^7*b*x^2 + 7*a^6*b^2*x^4 + (28*a^5*b^3*x^6)/3 + (35*a^4*b^4*x^8)/4 + (28*a^3*b^5*x^10)/5 + (7*a^2*b^6*x^12)/3 + (4*a*b^7*x^14)/7 + (b^8*x^16)/16 + a^8*Log[x]","A",1
93,1,99,99,0.0051093,"\int \frac{\left(a+b x^2\right)^8}{x^3} \, dx","Integrate[(a + b*x^2)^8/x^3,x]","-\frac{a^8}{2 x^2}+8 a^7 b \log (x)+14 a^6 b^2 x^2+14 a^5 b^3 x^4+\frac{35}{3} a^4 b^4 x^6+7 a^3 b^5 x^8+\frac{14}{5} a^2 b^6 x^{10}+\frac{2}{3} a b^7 x^{12}+\frac{b^8 x^{14}}{14}","-\frac{a^8}{2 x^2}+8 a^7 b \log (x)+14 a^6 b^2 x^2+14 a^5 b^3 x^4+\frac{35}{3} a^4 b^4 x^6+7 a^3 b^5 x^8+\frac{14}{5} a^2 b^6 x^{10}+\frac{2}{3} a b^7 x^{12}+\frac{b^8 x^{14}}{14}",1,"-1/2*a^8/x^2 + 14*a^6*b^2*x^2 + 14*a^5*b^3*x^4 + (35*a^4*b^4*x^6)/3 + 7*a^3*b^5*x^8 + (14*a^2*b^6*x^10)/5 + (2*a*b^7*x^12)/3 + (b^8*x^14)/14 + 8*a^7*b*Log[x]","A",1
94,1,101,101,0.0051219,"\int \frac{\left(a+b x^2\right)^8}{x^5} \, dx","Integrate[(a + b*x^2)^8/x^5,x]","-\frac{a^8}{4 x^4}-\frac{4 a^7 b}{x^2}+28 a^6 b^2 \log (x)+28 a^5 b^3 x^2+\frac{35}{2} a^4 b^4 x^4+\frac{28}{3} a^3 b^5 x^6+\frac{7}{2} a^2 b^6 x^8+\frac{4}{5} a b^7 x^{10}+\frac{b^8 x^{12}}{12}","-\frac{a^8}{4 x^4}-\frac{4 a^7 b}{x^2}+28 a^6 b^2 \log (x)+28 a^5 b^3 x^2+\frac{35}{2} a^4 b^4 x^4+\frac{28}{3} a^3 b^5 x^6+\frac{7}{2} a^2 b^6 x^8+\frac{4}{5} a b^7 x^{10}+\frac{b^8 x^{12}}{12}",1,"-1/4*a^8/x^4 - (4*a^7*b)/x^2 + 28*a^5*b^3*x^2 + (35*a^4*b^4*x^4)/2 + (28*a^3*b^5*x^6)/3 + (7*a^2*b^6*x^8)/2 + (4*a*b^7*x^10)/5 + (b^8*x^12)/12 + 28*a^6*b^2*Log[x]","A",1
95,1,94,94,0.0050981,"\int \frac{\left(a+b x^2\right)^8}{x^7} \, dx","Integrate[(a + b*x^2)^8/x^7,x]","-\frac{a^8}{6 x^6}-\frac{2 a^7 b}{x^4}-\frac{14 a^6 b^2}{x^2}+56 a^5 b^3 \log (x)+35 a^4 b^4 x^2+14 a^3 b^5 x^4+\frac{14}{3} a^2 b^6 x^6+a b^7 x^8+\frac{b^8 x^{10}}{10}","-\frac{a^8}{6 x^6}-\frac{2 a^7 b}{x^4}-\frac{14 a^6 b^2}{x^2}+56 a^5 b^3 \log (x)+35 a^4 b^4 x^2+14 a^3 b^5 x^4+\frac{14}{3} a^2 b^6 x^6+a b^7 x^8+\frac{b^8 x^{10}}{10}",1,"-1/6*a^8/x^6 - (2*a^7*b)/x^4 - (14*a^6*b^2)/x^2 + 35*a^4*b^4*x^2 + 14*a^3*b^5*x^4 + (14*a^2*b^6*x^6)/3 + a*b^7*x^8 + (b^8*x^10)/10 + 56*a^5*b^3*Log[x]","A",1
96,1,97,97,0.0049277,"\int \frac{\left(a+b x^2\right)^8}{x^9} \, dx","Integrate[(a + b*x^2)^8/x^9,x]","-\frac{a^8}{8 x^8}-\frac{4 a^7 b}{3 x^6}-\frac{7 a^6 b^2}{x^4}-\frac{28 a^5 b^3}{x^2}+70 a^4 b^4 \log (x)+28 a^3 b^5 x^2+7 a^2 b^6 x^4+\frac{4}{3} a b^7 x^6+\frac{b^8 x^8}{8}","-\frac{a^8}{8 x^8}-\frac{4 a^7 b}{3 x^6}-\frac{7 a^6 b^2}{x^4}-\frac{28 a^5 b^3}{x^2}+70 a^4 b^4 \log (x)+28 a^3 b^5 x^2+7 a^2 b^6 x^4+\frac{4}{3} a b^7 x^6+\frac{b^8 x^8}{8}",1,"-1/8*a^8/x^8 - (4*a^7*b)/(3*x^6) - (7*a^6*b^2)/x^4 - (28*a^5*b^3)/x^2 + 28*a^3*b^5*x^2 + 7*a^2*b^6*x^4 + (4*a*b^7*x^6)/3 + (b^8*x^8)/8 + 70*a^4*b^4*Log[x]","A",1
97,1,95,95,0.0050289,"\int \frac{\left(a+b x^2\right)^8}{x^{11}} \, dx","Integrate[(a + b*x^2)^8/x^11,x]","-\frac{a^8}{10 x^{10}}-\frac{a^7 b}{x^8}-\frac{14 a^6 b^2}{3 x^6}-\frac{14 a^5 b^3}{x^4}-\frac{35 a^4 b^4}{x^2}+56 a^3 b^5 \log (x)+14 a^2 b^6 x^2+2 a b^7 x^4+\frac{b^8 x^6}{6}","-\frac{a^8}{10 x^{10}}-\frac{a^7 b}{x^8}-\frac{14 a^6 b^2}{3 x^6}-\frac{14 a^5 b^3}{x^4}-\frac{35 a^4 b^4}{x^2}+56 a^3 b^5 \log (x)+14 a^2 b^6 x^2+2 a b^7 x^4+\frac{b^8 x^6}{6}",1,"-1/10*a^8/x^10 - (a^7*b)/x^8 - (14*a^6*b^2)/(3*x^6) - (14*a^5*b^3)/x^4 - (35*a^4*b^4)/x^2 + 14*a^2*b^6*x^2 + 2*a*b^7*x^4 + (b^8*x^6)/6 + 56*a^3*b^5*Log[x]","A",1
98,1,101,101,0.0049857,"\int \frac{\left(a+b x^2\right)^8}{x^{13}} \, dx","Integrate[(a + b*x^2)^8/x^13,x]","-\frac{a^8}{12 x^{12}}-\frac{4 a^7 b}{5 x^{10}}-\frac{7 a^6 b^2}{2 x^8}-\frac{28 a^5 b^3}{3 x^6}-\frac{35 a^4 b^4}{2 x^4}-\frac{28 a^3 b^5}{x^2}+28 a^2 b^6 \log (x)+4 a b^7 x^2+\frac{b^8 x^4}{4}","-\frac{a^8}{12 x^{12}}-\frac{4 a^7 b}{5 x^{10}}-\frac{7 a^6 b^2}{2 x^8}-\frac{28 a^5 b^3}{3 x^6}-\frac{35 a^4 b^4}{2 x^4}-\frac{28 a^3 b^5}{x^2}+28 a^2 b^6 \log (x)+4 a b^7 x^2+\frac{b^8 x^4}{4}",1,"-1/12*a^8/x^12 - (4*a^7*b)/(5*x^10) - (7*a^6*b^2)/(2*x^8) - (28*a^5*b^3)/(3*x^6) - (35*a^4*b^4)/(2*x^4) - (28*a^3*b^5)/x^2 + 4*a*b^7*x^2 + (b^8*x^4)/4 + 28*a^2*b^6*Log[x]","A",1
99,1,99,99,0.0051244,"\int \frac{\left(a+b x^2\right)^8}{x^{15}} \, dx","Integrate[(a + b*x^2)^8/x^15,x]","-\frac{a^8}{14 x^{14}}-\frac{2 a^7 b}{3 x^{12}}-\frac{14 a^6 b^2}{5 x^{10}}-\frac{7 a^5 b^3}{x^8}-\frac{35 a^4 b^4}{3 x^6}-\frac{14 a^3 b^5}{x^4}-\frac{14 a^2 b^6}{x^2}+8 a b^7 \log (x)+\frac{b^8 x^2}{2}","-\frac{a^8}{14 x^{14}}-\frac{2 a^7 b}{3 x^{12}}-\frac{14 a^6 b^2}{5 x^{10}}-\frac{7 a^5 b^3}{x^8}-\frac{35 a^4 b^4}{3 x^6}-\frac{14 a^3 b^5}{x^4}-\frac{14 a^2 b^6}{x^2}+8 a b^7 \log (x)+\frac{b^8 x^2}{2}",1,"-1/14*a^8/x^14 - (2*a^7*b)/(3*x^12) - (14*a^6*b^2)/(5*x^10) - (7*a^5*b^3)/x^8 - (35*a^4*b^4)/(3*x^6) - (14*a^3*b^5)/x^4 - (14*a^2*b^6)/x^2 + (b^8*x^2)/2 + 8*a*b^7*Log[x]","A",1
100,1,100,100,0.0049237,"\int \frac{\left(a+b x^2\right)^8}{x^{17}} \, dx","Integrate[(a + b*x^2)^8/x^17,x]","-\frac{a^8}{16 x^{16}}-\frac{4 a^7 b}{7 x^{14}}-\frac{7 a^6 b^2}{3 x^{12}}-\frac{28 a^5 b^3}{5 x^{10}}-\frac{35 a^4 b^4}{4 x^8}-\frac{28 a^3 b^5}{3 x^6}-\frac{7 a^2 b^6}{x^4}-\frac{4 a b^7}{x^2}+b^8 \log (x)","-\frac{a^8}{16 x^{16}}-\frac{4 a^7 b}{7 x^{14}}-\frac{7 a^6 b^2}{3 x^{12}}-\frac{28 a^5 b^3}{5 x^{10}}-\frac{35 a^4 b^4}{4 x^8}-\frac{28 a^3 b^5}{3 x^6}-\frac{7 a^2 b^6}{x^4}-\frac{4 a b^7}{x^2}+b^8 \log (x)",1,"-1/16*a^8/x^16 - (4*a^7*b)/(7*x^14) - (7*a^6*b^2)/(3*x^12) - (28*a^5*b^3)/(5*x^10) - (35*a^4*b^4)/(4*x^8) - (28*a^3*b^5)/(3*x^6) - (7*a^2*b^6)/x^4 - (4*a*b^7)/x^2 + b^8*Log[x]","A",1
101,1,100,19,0.0046617,"\int \frac{\left(a+b x^2\right)^8}{x^{19}} \, dx","Integrate[(a + b*x^2)^8/x^19,x]","-\frac{a^8}{18 x^{18}}-\frac{a^7 b}{2 x^{16}}-\frac{2 a^6 b^2}{x^{14}}-\frac{14 a^5 b^3}{3 x^{12}}-\frac{7 a^4 b^4}{x^{10}}-\frac{7 a^3 b^5}{x^8}-\frac{14 a^2 b^6}{3 x^6}-\frac{2 a b^7}{x^4}-\frac{b^8}{2 x^2}","-\frac{\left(a+b x^2\right)^9}{18 a x^{18}}",1,"-1/18*a^8/x^18 - (a^7*b)/(2*x^16) - (2*a^6*b^2)/x^14 - (14*a^5*b^3)/(3*x^12) - (7*a^4*b^4)/x^10 - (7*a^3*b^5)/x^8 - (14*a^2*b^6)/(3*x^6) - (2*a*b^7)/x^4 - b^8/(2*x^2)","B",1
102,1,106,40,0.0043916,"\int \frac{\left(a+b x^2\right)^8}{x^{21}} \, dx","Integrate[(a + b*x^2)^8/x^21,x]","-\frac{a^8}{20 x^{20}}-\frac{4 a^7 b}{9 x^{18}}-\frac{7 a^6 b^2}{4 x^{16}}-\frac{4 a^5 b^3}{x^{14}}-\frac{35 a^4 b^4}{6 x^{12}}-\frac{28 a^3 b^5}{5 x^{10}}-\frac{7 a^2 b^6}{2 x^8}-\frac{4 a b^7}{3 x^6}-\frac{b^8}{4 x^4}","\frac{b \left(a+b x^2\right)^9}{180 a^2 x^{18}}-\frac{\left(a+b x^2\right)^9}{20 a x^{20}}",1,"-1/20*a^8/x^20 - (4*a^7*b)/(9*x^18) - (7*a^6*b^2)/(4*x^16) - (4*a^5*b^3)/x^14 - (35*a^4*b^4)/(6*x^12) - (28*a^3*b^5)/(5*x^10) - (7*a^2*b^6)/(2*x^8) - (4*a*b^7)/(3*x^6) - b^8/(4*x^4)","B",1
103,1,104,62,0.0044066,"\int \frac{\left(a+b x^2\right)^8}{x^{23}} \, dx","Integrate[(a + b*x^2)^8/x^23,x]","-\frac{a^8}{22 x^{22}}-\frac{2 a^7 b}{5 x^{20}}-\frac{14 a^6 b^2}{9 x^{18}}-\frac{7 a^5 b^3}{2 x^{16}}-\frac{5 a^4 b^4}{x^{14}}-\frac{14 a^3 b^5}{3 x^{12}}-\frac{14 a^2 b^6}{5 x^{10}}-\frac{a b^7}{x^8}-\frac{b^8}{6 x^6}","-\frac{b^2 \left(a+b x^2\right)^9}{990 a^3 x^{18}}+\frac{b \left(a+b x^2\right)^9}{110 a^2 x^{20}}-\frac{\left(a+b x^2\right)^9}{22 a x^{22}}",1,"-1/22*a^8/x^22 - (2*a^7*b)/(5*x^20) - (14*a^6*b^2)/(9*x^18) - (7*a^5*b^3)/(2*x^16) - (5*a^4*b^4)/x^14 - (14*a^3*b^5)/(3*x^12) - (14*a^2*b^6)/(5*x^10) - (a*b^7)/x^8 - b^8/(6*x^6)","A",1
104,1,106,84,0.0044343,"\int \frac{\left(a+b x^2\right)^8}{x^{25}} \, dx","Integrate[(a + b*x^2)^8/x^25,x]","-\frac{a^8}{24 x^{24}}-\frac{4 a^7 b}{11 x^{22}}-\frac{7 a^6 b^2}{5 x^{20}}-\frac{28 a^5 b^3}{9 x^{18}}-\frac{35 a^4 b^4}{8 x^{16}}-\frac{4 a^3 b^5}{x^{14}}-\frac{7 a^2 b^6}{3 x^{12}}-\frac{4 a b^7}{5 x^{10}}-\frac{b^8}{8 x^8}","\frac{b^3 \left(a+b x^2\right)^9}{3960 a^4 x^{18}}-\frac{b^2 \left(a+b x^2\right)^9}{440 a^3 x^{20}}+\frac{b \left(a+b x^2\right)^9}{88 a^2 x^{22}}-\frac{\left(a+b x^2\right)^9}{24 a x^{24}}",1,"-1/24*a^8/x^24 - (4*a^7*b)/(11*x^22) - (7*a^6*b^2)/(5*x^20) - (28*a^5*b^3)/(9*x^18) - (35*a^4*b^4)/(8*x^16) - (4*a^3*b^5)/x^14 - (7*a^2*b^6)/(3*x^12) - (4*a*b^7)/(5*x^10) - b^8/(8*x^8)","A",1
105,1,106,106,0.0045335,"\int \frac{\left(a+b x^2\right)^8}{x^{27}} \, dx","Integrate[(a + b*x^2)^8/x^27,x]","-\frac{a^8}{26 x^{26}}-\frac{a^7 b}{3 x^{24}}-\frac{14 a^6 b^2}{11 x^{22}}-\frac{14 a^5 b^3}{5 x^{20}}-\frac{35 a^4 b^4}{9 x^{18}}-\frac{7 a^3 b^5}{2 x^{16}}-\frac{2 a^2 b^6}{x^{14}}-\frac{2 a b^7}{3 x^{12}}-\frac{b^8}{10 x^{10}}","-\frac{b^4 \left(a+b x^2\right)^9}{12870 a^5 x^{18}}+\frac{b^3 \left(a+b x^2\right)^9}{1430 a^4 x^{20}}-\frac{b^2 \left(a+b x^2\right)^9}{286 a^3 x^{22}}+\frac{b \left(a+b x^2\right)^9}{78 a^2 x^{24}}-\frac{\left(a+b x^2\right)^9}{26 a x^{26}}",1,"-1/26*a^8/x^26 - (a^7*b)/(3*x^24) - (14*a^6*b^2)/(11*x^22) - (14*a^5*b^3)/(5*x^20) - (35*a^4*b^4)/(9*x^18) - (7*a^3*b^5)/(2*x^16) - (2*a^2*b^6)/x^14 - (2*a*b^7)/(3*x^12) - b^8/(10*x^10)","A",1
106,1,108,108,0.0044496,"\int \frac{\left(a+b x^2\right)^8}{x^{29}} \, dx","Integrate[(a + b*x^2)^8/x^29,x]","-\frac{a^8}{28 x^{28}}-\frac{4 a^7 b}{13 x^{26}}-\frac{7 a^6 b^2}{6 x^{24}}-\frac{28 a^5 b^3}{11 x^{22}}-\frac{7 a^4 b^4}{2 x^{20}}-\frac{28 a^3 b^5}{9 x^{18}}-\frac{7 a^2 b^6}{4 x^{16}}-\frac{4 a b^7}{7 x^{14}}-\frac{b^8}{12 x^{12}}","-\frac{a^8}{28 x^{28}}-\frac{4 a^7 b}{13 x^{26}}-\frac{7 a^6 b^2}{6 x^{24}}-\frac{28 a^5 b^3}{11 x^{22}}-\frac{7 a^4 b^4}{2 x^{20}}-\frac{28 a^3 b^5}{9 x^{18}}-\frac{7 a^2 b^6}{4 x^{16}}-\frac{4 a b^7}{7 x^{14}}-\frac{b^8}{12 x^{12}}",1,"-1/28*a^8/x^28 - (4*a^7*b)/(13*x^26) - (7*a^6*b^2)/(6*x^24) - (28*a^5*b^3)/(11*x^22) - (7*a^4*b^4)/(2*x^20) - (28*a^3*b^5)/(9*x^18) - (7*a^2*b^6)/(4*x^16) - (4*a*b^7)/(7*x^14) - b^8/(12*x^12)","A",1
107,1,108,108,0.0043885,"\int \frac{\left(a+b x^2\right)^8}{x^{31}} \, dx","Integrate[(a + b*x^2)^8/x^31,x]","-\frac{a^8}{30 x^{30}}-\frac{2 a^7 b}{7 x^{28}}-\frac{14 a^6 b^2}{13 x^{26}}-\frac{7 a^5 b^3}{3 x^{24}}-\frac{35 a^4 b^4}{11 x^{22}}-\frac{14 a^3 b^5}{5 x^{20}}-\frac{14 a^2 b^6}{9 x^{18}}-\frac{a b^7}{2 x^{16}}-\frac{b^8}{14 x^{14}}","-\frac{a^8}{30 x^{30}}-\frac{2 a^7 b}{7 x^{28}}-\frac{14 a^6 b^2}{13 x^{26}}-\frac{7 a^5 b^3}{3 x^{24}}-\frac{35 a^4 b^4}{11 x^{22}}-\frac{14 a^3 b^5}{5 x^{20}}-\frac{14 a^2 b^6}{9 x^{18}}-\frac{a b^7}{2 x^{16}}-\frac{b^8}{14 x^{14}}",1,"-1/30*a^8/x^30 - (2*a^7*b)/(7*x^28) - (14*a^6*b^2)/(13*x^26) - (7*a^5*b^3)/(3*x^24) - (35*a^4*b^4)/(11*x^22) - (14*a^3*b^5)/(5*x^20) - (14*a^2*b^6)/(9*x^18) - (a*b^7)/(2*x^16) - b^8/(14*x^14)","A",1
108,1,106,106,0.004464,"\int \frac{\left(a+b x^2\right)^8}{x^{33}} \, dx","Integrate[(a + b*x^2)^8/x^33,x]","-\frac{a^8}{32 x^{32}}-\frac{4 a^7 b}{15 x^{30}}-\frac{a^6 b^2}{x^{28}}-\frac{28 a^5 b^3}{13 x^{26}}-\frac{35 a^4 b^4}{12 x^{24}}-\frac{28 a^3 b^5}{11 x^{22}}-\frac{7 a^2 b^6}{5 x^{20}}-\frac{4 a b^7}{9 x^{18}}-\frac{b^8}{16 x^{16}}","-\frac{a^8}{32 x^{32}}-\frac{4 a^7 b}{15 x^{30}}-\frac{a^6 b^2}{x^{28}}-\frac{28 a^5 b^3}{13 x^{26}}-\frac{35 a^4 b^4}{12 x^{24}}-\frac{28 a^3 b^5}{11 x^{22}}-\frac{7 a^2 b^6}{5 x^{20}}-\frac{4 a b^7}{9 x^{18}}-\frac{b^8}{16 x^{16}}",1,"-1/32*a^8/x^32 - (4*a^7*b)/(15*x^30) - (a^6*b^2)/x^28 - (28*a^5*b^3)/(13*x^26) - (35*a^4*b^4)/(12*x^24) - (28*a^3*b^5)/(11*x^22) - (7*a^2*b^6)/(5*x^20) - (4*a*b^7)/(9*x^18) - b^8/(16*x^16)","A",1
109,1,108,108,0.0028811,"\int x^8 \left(a+b x^2\right)^8 \, dx","Integrate[x^8*(a + b*x^2)^8,x]","\frac{a^8 x^9}{9}+\frac{8}{11} a^7 b x^{11}+\frac{28}{13} a^6 b^2 x^{13}+\frac{56}{15} a^5 b^3 x^{15}+\frac{70}{17} a^4 b^4 x^{17}+\frac{56}{19} a^3 b^5 x^{19}+\frac{4}{3} a^2 b^6 x^{21}+\frac{8}{23} a b^7 x^{23}+\frac{b^8 x^{25}}{25}","\frac{a^8 x^9}{9}+\frac{8}{11} a^7 b x^{11}+\frac{28}{13} a^6 b^2 x^{13}+\frac{56}{15} a^5 b^3 x^{15}+\frac{70}{17} a^4 b^4 x^{17}+\frac{56}{19} a^3 b^5 x^{19}+\frac{4}{3} a^2 b^6 x^{21}+\frac{8}{23} a b^7 x^{23}+\frac{b^8 x^{25}}{25}",1,"(a^8*x^9)/9 + (8*a^7*b*x^11)/11 + (28*a^6*b^2*x^13)/13 + (56*a^5*b^3*x^15)/15 + (70*a^4*b^4*x^17)/17 + (56*a^3*b^5*x^19)/19 + (4*a^2*b^6*x^21)/3 + (8*a*b^7*x^23)/23 + (b^8*x^25)/25","A",1
110,1,108,108,0.0029335,"\int x^6 \left(a+b x^2\right)^8 \, dx","Integrate[x^6*(a + b*x^2)^8,x]","\frac{a^8 x^7}{7}+\frac{8}{9} a^7 b x^9+\frac{28}{11} a^6 b^2 x^{11}+\frac{56}{13} a^5 b^3 x^{13}+\frac{14}{3} a^4 b^4 x^{15}+\frac{56}{17} a^3 b^5 x^{17}+\frac{28}{19} a^2 b^6 x^{19}+\frac{8}{21} a b^7 x^{21}+\frac{b^8 x^{23}}{23}","\frac{a^8 x^7}{7}+\frac{8}{9} a^7 b x^9+\frac{28}{11} a^6 b^2 x^{11}+\frac{56}{13} a^5 b^3 x^{13}+\frac{14}{3} a^4 b^4 x^{15}+\frac{56}{17} a^3 b^5 x^{17}+\frac{28}{19} a^2 b^6 x^{19}+\frac{8}{21} a b^7 x^{21}+\frac{b^8 x^{23}}{23}",1,"(a^8*x^7)/7 + (8*a^7*b*x^9)/9 + (28*a^6*b^2*x^11)/11 + (56*a^5*b^3*x^13)/13 + (14*a^4*b^4*x^15)/3 + (56*a^3*b^5*x^17)/17 + (28*a^2*b^6*x^19)/19 + (8*a*b^7*x^21)/21 + (b^8*x^23)/23","A",1
111,1,108,108,0.0025538,"\int x^4 \left(a+b x^2\right)^8 \, dx","Integrate[x^4*(a + b*x^2)^8,x]","\frac{a^8 x^5}{5}+\frac{8}{7} a^7 b x^7+\frac{28}{9} a^6 b^2 x^9+\frac{56}{11} a^5 b^3 x^{11}+\frac{70}{13} a^4 b^4 x^{13}+\frac{56}{15} a^3 b^5 x^{15}+\frac{28}{17} a^2 b^6 x^{17}+\frac{8}{19} a b^7 x^{19}+\frac{b^8 x^{21}}{21}","\frac{a^8 x^5}{5}+\frac{8}{7} a^7 b x^7+\frac{28}{9} a^6 b^2 x^9+\frac{56}{11} a^5 b^3 x^{11}+\frac{70}{13} a^4 b^4 x^{13}+\frac{56}{15} a^3 b^5 x^{15}+\frac{28}{17} a^2 b^6 x^{17}+\frac{8}{19} a b^7 x^{19}+\frac{b^8 x^{21}}{21}",1,"(a^8*x^5)/5 + (8*a^7*b*x^7)/7 + (28*a^6*b^2*x^9)/9 + (56*a^5*b^3*x^11)/11 + (70*a^4*b^4*x^13)/13 + (56*a^3*b^5*x^15)/15 + (28*a^2*b^6*x^17)/17 + (8*a*b^7*x^19)/19 + (b^8*x^21)/21","A",1
112,1,106,106,0.0026074,"\int x^2 \left(a+b x^2\right)^8 \, dx","Integrate[x^2*(a + b*x^2)^8,x]","\frac{a^8 x^3}{3}+\frac{8}{5} a^7 b x^5+4 a^6 b^2 x^7+\frac{56}{9} a^5 b^3 x^9+\frac{70}{11} a^4 b^4 x^{11}+\frac{56}{13} a^3 b^5 x^{13}+\frac{28}{15} a^2 b^6 x^{15}+\frac{8}{17} a b^7 x^{17}+\frac{b^8 x^{19}}{19}","\frac{a^8 x^3}{3}+\frac{8}{5} a^7 b x^5+4 a^6 b^2 x^7+\frac{56}{9} a^5 b^3 x^9+\frac{70}{11} a^4 b^4 x^{11}+\frac{56}{13} a^3 b^5 x^{13}+\frac{28}{15} a^2 b^6 x^{15}+\frac{8}{17} a b^7 x^{17}+\frac{b^8 x^{19}}{19}",1,"(a^8*x^3)/3 + (8*a^7*b*x^5)/5 + 4*a^6*b^2*x^7 + (56*a^5*b^3*x^9)/9 + (70*a^4*b^4*x^11)/11 + (56*a^3*b^5*x^13)/13 + (28*a^2*b^6*x^15)/15 + (8*a*b^7*x^17)/17 + (b^8*x^19)/19","A",1
113,1,101,101,0.0014124,"\int \left(a+b x^2\right)^8 \, dx","Integrate[(a + b*x^2)^8,x]","a^8 x+\frac{8}{3} a^7 b x^3+\frac{28}{5} a^6 b^2 x^5+8 a^5 b^3 x^7+\frac{70}{9} a^4 b^4 x^9+\frac{56}{11} a^3 b^5 x^{11}+\frac{28}{13} a^2 b^6 x^{13}+\frac{8}{15} a b^7 x^{15}+\frac{b^8 x^{17}}{17}","a^8 x+\frac{8}{3} a^7 b x^3+\frac{28}{5} a^6 b^2 x^5+8 a^5 b^3 x^7+\frac{70}{9} a^4 b^4 x^9+\frac{56}{11} a^3 b^5 x^{11}+\frac{28}{13} a^2 b^6 x^{13}+\frac{8}{15} a b^7 x^{15}+\frac{b^8 x^{17}}{17}",1,"a^8*x + (8*a^7*b*x^3)/3 + (28*a^6*b^2*x^5)/5 + 8*a^5*b^3*x^7 + (70*a^4*b^4*x^9)/9 + (56*a^3*b^5*x^11)/11 + (28*a^2*b^6*x^13)/13 + (8*a*b^7*x^15)/15 + (b^8*x^17)/17","A",1
114,1,100,100,0.0165972,"\int \frac{\left(a+b x^2\right)^8}{x^2} \, dx","Integrate[(a + b*x^2)^8/x^2,x]","-\frac{a^8}{x}+8 a^7 b x+\frac{28}{3} a^6 b^2 x^3+\frac{56}{5} a^5 b^3 x^5+10 a^4 b^4 x^7+\frac{56}{9} a^3 b^5 x^9+\frac{28}{11} a^2 b^6 x^{11}+\frac{8}{13} a b^7 x^{13}+\frac{b^8 x^{15}}{15}","-\frac{a^8}{x}+8 a^7 b x+\frac{28}{3} a^6 b^2 x^3+\frac{56}{5} a^5 b^3 x^5+10 a^4 b^4 x^7+\frac{56}{9} a^3 b^5 x^9+\frac{28}{11} a^2 b^6 x^{11}+\frac{8}{13} a b^7 x^{13}+\frac{b^8 x^{15}}{15}",1,"-(a^8/x) + 8*a^7*b*x + (28*a^6*b^2*x^3)/3 + (56*a^5*b^3*x^5)/5 + 10*a^4*b^4*x^7 + (56*a^3*b^5*x^9)/9 + (28*a^2*b^6*x^11)/11 + (8*a*b^7*x^13)/13 + (b^8*x^15)/15","A",1
115,1,98,98,0.007861,"\int \frac{\left(a+b x^2\right)^8}{x^4} \, dx","Integrate[(a + b*x^2)^8/x^4,x]","-\frac{a^8}{3 x^3}-\frac{8 a^7 b}{x}+28 a^6 b^2 x+\frac{56}{3} a^5 b^3 x^3+14 a^4 b^4 x^5+8 a^3 b^5 x^7+\frac{28}{9} a^2 b^6 x^9+\frac{8}{11} a b^7 x^{11}+\frac{b^8 x^{13}}{13}","-\frac{a^8}{3 x^3}-\frac{8 a^7 b}{x}+28 a^6 b^2 x+\frac{56}{3} a^5 b^3 x^3+14 a^4 b^4 x^5+8 a^3 b^5 x^7+\frac{28}{9} a^2 b^6 x^9+\frac{8}{11} a b^7 x^{11}+\frac{b^8 x^{13}}{13}",1,"-1/3*a^8/x^3 - (8*a^7*b)/x + 28*a^6*b^2*x + (56*a^5*b^3*x^3)/3 + 14*a^4*b^4*x^5 + 8*a^3*b^5*x^7 + (28*a^2*b^6*x^9)/9 + (8*a*b^7*x^11)/11 + (b^8*x^13)/13","A",1
116,1,100,100,0.0120841,"\int \frac{\left(a+b x^2\right)^8}{x^6} \, dx","Integrate[(a + b*x^2)^8/x^6,x]","-\frac{a^8}{5 x^5}-\frac{8 a^7 b}{3 x^3}-\frac{28 a^6 b^2}{x}+56 a^5 b^3 x+\frac{70}{3} a^4 b^4 x^3+\frac{56}{5} a^3 b^5 x^5+4 a^2 b^6 x^7+\frac{8}{9} a b^7 x^9+\frac{b^8 x^{11}}{11}","-\frac{a^8}{5 x^5}-\frac{8 a^7 b}{3 x^3}-\frac{28 a^6 b^2}{x}+56 a^5 b^3 x+\frac{70}{3} a^4 b^4 x^3+\frac{56}{5} a^3 b^5 x^5+4 a^2 b^6 x^7+\frac{8}{9} a b^7 x^9+\frac{b^8 x^{11}}{11}",1,"-1/5*a^8/x^5 - (8*a^7*b)/(3*x^3) - (28*a^6*b^2)/x + 56*a^5*b^3*x + (70*a^4*b^4*x^3)/3 + (56*a^3*b^5*x^5)/5 + 4*a^2*b^6*x^7 + (8*a*b^7*x^9)/9 + (b^8*x^11)/11","A",1
117,1,102,102,0.0061878,"\int \frac{\left(a+b x^2\right)^8}{x^8} \, dx","Integrate[(a + b*x^2)^8/x^8,x]","-\frac{a^8}{7 x^7}-\frac{8 a^7 b}{5 x^5}-\frac{28 a^6 b^2}{3 x^3}-\frac{56 a^5 b^3}{x}+70 a^4 b^4 x+\frac{56}{3} a^3 b^5 x^3+\frac{28}{5} a^2 b^6 x^5+\frac{8}{7} a b^7 x^7+\frac{b^8 x^9}{9}","-\frac{a^8}{7 x^7}-\frac{8 a^7 b}{5 x^5}-\frac{28 a^6 b^2}{3 x^3}-\frac{56 a^5 b^3}{x}+70 a^4 b^4 x+\frac{56}{3} a^3 b^5 x^3+\frac{28}{5} a^2 b^6 x^5+\frac{8}{7} a b^7 x^7+\frac{b^8 x^9}{9}",1,"-1/7*a^8/x^7 - (8*a^7*b)/(5*x^5) - (28*a^6*b^2)/(3*x^3) - (56*a^5*b^3)/x + 70*a^4*b^4*x + (56*a^3*b^5*x^3)/3 + (28*a^2*b^6*x^5)/5 + (8*a*b^7*x^7)/7 + (b^8*x^9)/9","A",1
118,1,102,102,0.0119157,"\int \frac{\left(a+b x^2\right)^8}{x^{10}} \, dx","Integrate[(a + b*x^2)^8/x^10,x]","-\frac{a^8}{9 x^9}-\frac{8 a^7 b}{7 x^7}-\frac{28 a^6 b^2}{5 x^5}-\frac{56 a^5 b^3}{3 x^3}-\frac{70 a^4 b^4}{x}+56 a^3 b^5 x+\frac{28}{3} a^2 b^6 x^3+\frac{8}{5} a b^7 x^5+\frac{b^8 x^7}{7}","-\frac{a^8}{9 x^9}-\frac{8 a^7 b}{7 x^7}-\frac{28 a^6 b^2}{5 x^5}-\frac{56 a^5 b^3}{3 x^3}-\frac{70 a^4 b^4}{x}+56 a^3 b^5 x+\frac{28}{3} a^2 b^6 x^3+\frac{8}{5} a b^7 x^5+\frac{b^8 x^7}{7}",1,"-1/9*a^8/x^9 - (8*a^7*b)/(7*x^7) - (28*a^6*b^2)/(5*x^5) - (56*a^5*b^3)/(3*x^3) - (70*a^4*b^4)/x + 56*a^3*b^5*x + (28*a^2*b^6*x^3)/3 + (8*a*b^7*x^5)/5 + (b^8*x^7)/7","A",1
119,1,100,100,0.0137378,"\int \frac{\left(a+b x^2\right)^8}{x^{12}} \, dx","Integrate[(a + b*x^2)^8/x^12,x]","-\frac{a^8}{11 x^{11}}-\frac{8 a^7 b}{9 x^9}-\frac{4 a^6 b^2}{x^7}-\frac{56 a^5 b^3}{5 x^5}-\frac{70 a^4 b^4}{3 x^3}-\frac{56 a^3 b^5}{x}+28 a^2 b^6 x+\frac{8}{3} a b^7 x^3+\frac{b^8 x^5}{5}","-\frac{a^8}{11 x^{11}}-\frac{8 a^7 b}{9 x^9}-\frac{4 a^6 b^2}{x^7}-\frac{56 a^5 b^3}{5 x^5}-\frac{70 a^4 b^4}{3 x^3}-\frac{56 a^3 b^5}{x}+28 a^2 b^6 x+\frac{8}{3} a b^7 x^3+\frac{b^8 x^5}{5}",1,"-1/11*a^8/x^11 - (8*a^7*b)/(9*x^9) - (4*a^6*b^2)/x^7 - (56*a^5*b^3)/(5*x^5) - (70*a^4*b^4)/(3*x^3) - (56*a^3*b^5)/x + 28*a^2*b^6*x + (8*a*b^7*x^3)/3 + (b^8*x^5)/5","A",1
120,1,98,98,0.0126587,"\int \frac{\left(a+b x^2\right)^8}{x^{14}} \, dx","Integrate[(a + b*x^2)^8/x^14,x]","-\frac{a^8}{13 x^{13}}-\frac{8 a^7 b}{11 x^{11}}-\frac{28 a^6 b^2}{9 x^9}-\frac{8 a^5 b^3}{x^7}-\frac{14 a^4 b^4}{x^5}-\frac{56 a^3 b^5}{3 x^3}-\frac{28 a^2 b^6}{x}+8 a b^7 x+\frac{b^8 x^3}{3}","-\frac{a^8}{13 x^{13}}-\frac{8 a^7 b}{11 x^{11}}-\frac{28 a^6 b^2}{9 x^9}-\frac{8 a^5 b^3}{x^7}-\frac{14 a^4 b^4}{x^5}-\frac{56 a^3 b^5}{3 x^3}-\frac{28 a^2 b^6}{x}+8 a b^7 x+\frac{b^8 x^3}{3}",1,"-1/13*a^8/x^13 - (8*a^7*b)/(11*x^11) - (28*a^6*b^2)/(9*x^9) - (8*a^5*b^3)/x^7 - (14*a^4*b^4)/x^5 - (56*a^3*b^5)/(3*x^3) - (28*a^2*b^6)/x + 8*a*b^7*x + (b^8*x^3)/3","A",1
121,1,99,99,0.0057586,"\int \frac{\left(a+b x^2\right)^8}{x^{16}} \, dx","Integrate[(a + b*x^2)^8/x^16,x]","-\frac{a^8}{15 x^{15}}-\frac{8 a^7 b}{13 x^{13}}-\frac{28 a^6 b^2}{11 x^{11}}-\frac{56 a^5 b^3}{9 x^9}-\frac{10 a^4 b^4}{x^7}-\frac{56 a^3 b^5}{5 x^5}-\frac{28 a^2 b^6}{3 x^3}-\frac{8 a b^7}{x}+b^8 x","-\frac{a^8}{15 x^{15}}-\frac{8 a^7 b}{13 x^{13}}-\frac{28 a^6 b^2}{11 x^{11}}-\frac{56 a^5 b^3}{9 x^9}-\frac{10 a^4 b^4}{x^7}-\frac{56 a^3 b^5}{5 x^5}-\frac{28 a^2 b^6}{3 x^3}-\frac{8 a b^7}{x}+b^8 x",1,"-1/15*a^8/x^15 - (8*a^7*b)/(13*x^13) - (28*a^6*b^2)/(11*x^11) - (56*a^5*b^3)/(9*x^9) - (10*a^4*b^4)/x^7 - (56*a^3*b^5)/(5*x^5) - (28*a^2*b^6)/(3*x^3) - (8*a*b^7)/x + b^8*x","A",1
122,1,104,104,0.0113155,"\int \frac{\left(a+b x^2\right)^8}{x^{18}} \, dx","Integrate[(a + b*x^2)^8/x^18,x]","-\frac{a^8}{17 x^{17}}-\frac{8 a^7 b}{15 x^{15}}-\frac{28 a^6 b^2}{13 x^{13}}-\frac{56 a^5 b^3}{11 x^{11}}-\frac{70 a^4 b^4}{9 x^9}-\frac{8 a^3 b^5}{x^7}-\frac{28 a^2 b^6}{5 x^5}-\frac{8 a b^7}{3 x^3}-\frac{b^8}{x}","-\frac{a^8}{17 x^{17}}-\frac{8 a^7 b}{15 x^{15}}-\frac{28 a^6 b^2}{13 x^{13}}-\frac{56 a^5 b^3}{11 x^{11}}-\frac{70 a^4 b^4}{9 x^9}-\frac{8 a^3 b^5}{x^7}-\frac{28 a^2 b^6}{5 x^5}-\frac{8 a b^7}{3 x^3}-\frac{b^8}{x}",1,"-1/17*a^8/x^17 - (8*a^7*b)/(15*x^15) - (28*a^6*b^2)/(13*x^13) - (56*a^5*b^3)/(11*x^11) - (70*a^4*b^4)/(9*x^9) - (8*a^3*b^5)/x^7 - (28*a^2*b^6)/(5*x^5) - (8*a*b^7)/(3*x^3) - b^8/x","A",1
123,1,106,106,0.0141887,"\int \frac{\left(a+b x^2\right)^8}{x^{20}} \, dx","Integrate[(a + b*x^2)^8/x^20,x]","-\frac{a^8}{19 x^{19}}-\frac{8 a^7 b}{17 x^{17}}-\frac{28 a^6 b^2}{15 x^{15}}-\frac{56 a^5 b^3}{13 x^{13}}-\frac{70 a^4 b^4}{11 x^{11}}-\frac{56 a^3 b^5}{9 x^9}-\frac{4 a^2 b^6}{x^7}-\frac{8 a b^7}{5 x^5}-\frac{b^8}{3 x^3}","-\frac{a^8}{19 x^{19}}-\frac{8 a^7 b}{17 x^{17}}-\frac{28 a^6 b^2}{15 x^{15}}-\frac{56 a^5 b^3}{13 x^{13}}-\frac{70 a^4 b^4}{11 x^{11}}-\frac{56 a^3 b^5}{9 x^9}-\frac{4 a^2 b^6}{x^7}-\frac{8 a b^7}{5 x^5}-\frac{b^8}{3 x^3}",1,"-1/19*a^8/x^19 - (8*a^7*b)/(17*x^17) - (28*a^6*b^2)/(15*x^15) - (56*a^5*b^3)/(13*x^13) - (70*a^4*b^4)/(11*x^11) - (56*a^3*b^5)/(9*x^9) - (4*a^2*b^6)/x^7 - (8*a*b^7)/(5*x^5) - b^8/(3*x^3)","A",1
124,1,79,79,0.0058481,"\int \frac{x^{11}}{a+b x^2} \, dx","Integrate[x^11/(a + b*x^2),x]","-\frac{a^5 \log \left(a+b x^2\right)}{2 b^6}+\frac{a^4 x^2}{2 b^5}-\frac{a^3 x^4}{4 b^4}+\frac{a^2 x^6}{6 b^3}-\frac{a x^8}{8 b^2}+\frac{x^{10}}{10 b}","-\frac{a^5 \log \left(a+b x^2\right)}{2 b^6}+\frac{a^4 x^2}{2 b^5}-\frac{a^3 x^4}{4 b^4}+\frac{a^2 x^6}{6 b^3}-\frac{a x^8}{8 b^2}+\frac{x^{10}}{10 b}",1,"(a^4*x^2)/(2*b^5) - (a^3*x^4)/(4*b^4) + (a^2*x^6)/(6*b^3) - (a*x^8)/(8*b^2) + x^10/(10*b) - (a^5*Log[a + b*x^2])/(2*b^6)","A",1
125,1,81,81,0.0349973,"\int \frac{x^{10}}{a+b x^2} \, dx","Integrate[x^10/(a + b*x^2),x]","-\frac{a^{9/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{11/2}}+\frac{a^4 x}{b^5}-\frac{a^3 x^3}{3 b^4}+\frac{a^2 x^5}{5 b^3}-\frac{a x^7}{7 b^2}+\frac{x^9}{9 b}","-\frac{a^{9/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{11/2}}+\frac{a^4 x}{b^5}-\frac{a^3 x^3}{3 b^4}+\frac{a^2 x^5}{5 b^3}-\frac{a x^7}{7 b^2}+\frac{x^9}{9 b}",1,"(a^4*x)/b^5 - (a^3*x^3)/(3*b^4) + (a^2*x^5)/(5*b^3) - (a*x^7)/(7*b^2) + x^9/(9*b) - (a^(9/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(11/2)","A",1
126,1,66,66,0.0059811,"\int \frac{x^9}{a+b x^2} \, dx","Integrate[x^9/(a + b*x^2),x]","\frac{a^4 \log \left(a+b x^2\right)}{2 b^5}-\frac{a^3 x^2}{2 b^4}+\frac{a^2 x^4}{4 b^3}-\frac{a x^6}{6 b^2}+\frac{x^8}{8 b}","\frac{a^4 \log \left(a+b x^2\right)}{2 b^5}-\frac{a^3 x^2}{2 b^4}+\frac{a^2 x^4}{4 b^3}-\frac{a x^6}{6 b^2}+\frac{x^8}{8 b}",1,"-1/2*(a^3*x^2)/b^4 + (a^2*x^4)/(4*b^3) - (a*x^6)/(6*b^2) + x^8/(8*b) + (a^4*Log[a + b*x^2])/(2*b^5)","A",1
127,1,68,68,0.0263664,"\int \frac{x^8}{a+b x^2} \, dx","Integrate[x^8/(a + b*x^2),x]","\frac{a^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{9/2}}-\frac{a^3 x}{b^4}+\frac{a^2 x^3}{3 b^3}-\frac{a x^5}{5 b^2}+\frac{x^7}{7 b}","\frac{a^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{9/2}}-\frac{a^3 x}{b^4}+\frac{a^2 x^3}{3 b^3}-\frac{a x^5}{5 b^2}+\frac{x^7}{7 b}",1,"-((a^3*x)/b^4) + (a^2*x^3)/(3*b^3) - (a*x^5)/(5*b^2) + x^7/(7*b) + (a^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(9/2)","A",1
128,1,53,53,0.0054573,"\int \frac{x^7}{a+b x^2} \, dx","Integrate[x^7/(a + b*x^2),x]","-\frac{a^3 \log \left(a+b x^2\right)}{2 b^4}+\frac{a^2 x^2}{2 b^3}-\frac{a x^4}{4 b^2}+\frac{x^6}{6 b}","-\frac{a^3 \log \left(a+b x^2\right)}{2 b^4}+\frac{a^2 x^2}{2 b^3}-\frac{a x^4}{4 b^2}+\frac{x^6}{6 b}",1,"(a^2*x^2)/(2*b^3) - (a*x^4)/(4*b^2) + x^6/(6*b) - (a^3*Log[a + b*x^2])/(2*b^4)","A",1
129,1,55,55,0.0231565,"\int \frac{x^6}{a+b x^2} \, dx","Integrate[x^6/(a + b*x^2),x]","-\frac{a^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{7/2}}+\frac{a^2 x}{b^3}-\frac{a x^3}{3 b^2}+\frac{x^5}{5 b}","-\frac{a^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{7/2}}+\frac{a^2 x}{b^3}-\frac{a x^3}{3 b^2}+\frac{x^5}{5 b}",1,"(a^2*x)/b^3 - (a*x^3)/(3*b^2) + x^5/(5*b) - (a^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(7/2)","A",1
130,1,40,40,0.0050935,"\int \frac{x^5}{a+b x^2} \, dx","Integrate[x^5/(a + b*x^2),x]","\frac{a^2 \log \left(a+b x^2\right)}{2 b^3}-\frac{a x^2}{2 b^2}+\frac{x^4}{4 b}","\frac{a^2 \log \left(a+b x^2\right)}{2 b^3}-\frac{a x^2}{2 b^2}+\frac{x^4}{4 b}",1,"-1/2*(a*x^2)/b^2 + x^4/(4*b) + (a^2*Log[a + b*x^2])/(2*b^3)","A",1
131,1,42,42,0.0189587,"\int \frac{x^4}{a+b x^2} \, dx","Integrate[x^4/(a + b*x^2),x]","\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{5/2}}-\frac{a x}{b^2}+\frac{x^3}{3 b}","\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{5/2}}-\frac{a x}{b^2}+\frac{x^3}{3 b}",1,"-((a*x)/b^2) + x^3/(3*b) + (a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(5/2)","A",1
132,1,27,27,0.0042304,"\int \frac{x^3}{a+b x^2} \, dx","Integrate[x^3/(a + b*x^2),x]","\frac{x^2}{2 b}-\frac{a \log \left(a+b x^2\right)}{2 b^2}","\frac{x^2}{2 b}-\frac{a \log \left(a+b x^2\right)}{2 b^2}",1,"x^2/(2*b) - (a*Log[a + b*x^2])/(2*b^2)","A",1
133,1,31,31,0.0082957,"\int \frac{x^2}{a+b x^2} \, dx","Integrate[x^2/(a + b*x^2),x]","\frac{x}{b}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{3/2}}","\frac{x}{b}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{3/2}}",1,"x/b - (Sqrt[a]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(3/2)","A",1
134,1,15,15,0.0020854,"\int \frac{x}{a+b x^2} \, dx","Integrate[x/(a + b*x^2),x]","\frac{\log \left(a+b x^2\right)}{2 b}","\frac{\log \left(a+b x^2\right)}{2 b}",1,"Log[a + b*x^2]/(2*b)","A",1
135,1,24,24,0.0037916,"\int \frac{1}{a+b x^2} \, dx","Integrate[(a + b*x^2)^(-1),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}",1,"ArcTan[(Sqrt[b]*x)/Sqrt[a]]/(Sqrt[a]*Sqrt[b])","A",1
136,1,22,22,0.0048706,"\int \frac{1}{x \left(a+b x^2\right)} \, dx","Integrate[1/(x*(a + b*x^2)),x]","\frac{\log (x)}{a}-\frac{\log \left(a+b x^2\right)}{2 a}","\frac{\log (x)}{a}-\frac{\log \left(a+b x^2\right)}{2 a}",1,"Log[x]/a - Log[a + b*x^2]/(2*a)","A",1
137,1,34,34,0.0122908,"\int \frac{1}{x^2 \left(a+b x^2\right)} \, dx","Integrate[1/(x^2*(a + b*x^2)),x]","-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2}}-\frac{1}{a x}","-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2}}-\frac{1}{a x}",1,"-(1/(a*x)) - (Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(3/2)","A",1
138,1,35,35,0.0066288,"\int \frac{1}{x^3 \left(a+b x^2\right)} \, dx","Integrate[1/(x^3*(a + b*x^2)),x]","\frac{b \log \left(a+b x^2\right)}{2 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{2 a x^2}","\frac{b \log \left(a+b x^2\right)}{2 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{2 a x^2}",1,"-1/2*1/(a*x^2) - (b*Log[x])/a^2 + (b*Log[a + b*x^2])/(2*a^2)","A",1
139,1,43,43,0.0199263,"\int \frac{1}{x^4 \left(a+b x^2\right)} \, dx","Integrate[1/(x^4*(a + b*x^2)),x]","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{5/2}}+\frac{b}{a^2 x}-\frac{1}{3 a x^3}","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{5/2}}+\frac{b}{a^2 x}-\frac{1}{3 a x^3}",1,"-1/3*1/(a*x^3) + b/(a^2*x) + (b^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(5/2)","A",1
140,1,49,49,0.0072199,"\int \frac{1}{x^5 \left(a+b x^2\right)} \, dx","Integrate[1/(x^5*(a + b*x^2)),x]","-\frac{b^2 \log \left(a+b x^2\right)}{2 a^3}+\frac{b^2 \log (x)}{a^3}+\frac{b}{2 a^2 x^2}-\frac{1}{4 a x^4}","-\frac{b^2 \log \left(a+b x^2\right)}{2 a^3}+\frac{b^2 \log (x)}{a^3}+\frac{b}{2 a^2 x^2}-\frac{1}{4 a x^4}",1,"-1/4*1/(a*x^4) + b/(2*a^2*x^2) + (b^2*Log[x])/a^3 - (b^2*Log[a + b*x^2])/(2*a^3)","A",1
141,1,58,58,0.0228852,"\int \frac{1}{x^6 \left(a+b x^2\right)} \, dx","Integrate[1/(x^6*(a + b*x^2)),x]","-\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{7/2}}-\frac{b^2}{a^3 x}+\frac{b}{3 a^2 x^3}-\frac{1}{5 a x^5}","-\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{7/2}}-\frac{b^2}{a^3 x}+\frac{b}{3 a^2 x^3}-\frac{1}{5 a x^5}",1,"-1/5*1/(a*x^5) + b/(3*a^2*x^3) - b^2/(a^3*x) - (b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(7/2)","A",1
142,1,63,63,0.0072108,"\int \frac{1}{x^7 \left(a+b x^2\right)} \, dx","Integrate[1/(x^7*(a + b*x^2)),x]","\frac{b^3 \log \left(a+b x^2\right)}{2 a^4}-\frac{b^3 \log (x)}{a^4}-\frac{b^2}{2 a^3 x^2}+\frac{b}{4 a^2 x^4}-\frac{1}{6 a x^6}","\frac{b^3 \log \left(a+b x^2\right)}{2 a^4}-\frac{b^3 \log (x)}{a^4}-\frac{b^2}{2 a^3 x^2}+\frac{b}{4 a^2 x^4}-\frac{1}{6 a x^6}",1,"-1/6*1/(a*x^6) + b/(4*a^2*x^4) - b^2/(2*a^3*x^2) - (b^3*Log[x])/a^4 + (b^3*Log[a + b*x^2])/(2*a^4)","A",1
143,1,69,69,0.025153,"\int \frac{1}{x^8 \left(a+b x^2\right)} \, dx","Integrate[1/(x^8*(a + b*x^2)),x]","\frac{b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{9/2}}+\frac{b^3}{a^4 x}-\frac{b^2}{3 a^3 x^3}+\frac{b}{5 a^2 x^5}-\frac{1}{7 a x^7}","\frac{b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{9/2}}+\frac{b^3}{a^4 x}-\frac{b^2}{3 a^3 x^3}+\frac{b}{5 a^2 x^5}-\frac{1}{7 a x^7}",1,"-1/7*1/(a*x^7) + b/(5*a^2*x^5) - b^2/(3*a^3*x^3) + b^3/(a^4*x) + (b^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(9/2)","A",1
144,1,75,75,0.0067354,"\int \frac{1}{x^9 \left(a+b x^2\right)} \, dx","Integrate[1/(x^9*(a + b*x^2)),x]","-\frac{b^4 \log \left(a+b x^2\right)}{2 a^5}+\frac{b^4 \log (x)}{a^5}+\frac{b^3}{2 a^4 x^2}-\frac{b^2}{4 a^3 x^4}+\frac{b}{6 a^2 x^6}-\frac{1}{8 a x^8}","-\frac{b^4 \log \left(a+b x^2\right)}{2 a^5}+\frac{b^4 \log (x)}{a^5}+\frac{b^3}{2 a^4 x^2}-\frac{b^2}{4 a^3 x^4}+\frac{b}{6 a^2 x^6}-\frac{1}{8 a x^8}",1,"-1/8*1/(a*x^8) + b/(6*a^2*x^6) - b^2/(4*a^3*x^4) + b^3/(2*a^4*x^2) + (b^4*Log[x])/a^5 - (b^4*Log[a + b*x^2])/(2*a^5)","A",1
145,1,83,94,0.0318747,"\int \frac{x^{13}}{\left(a+b x^2\right)^2} \, dx","Integrate[x^13/(a + b*x^2)^2,x]","\frac{-\frac{10 a^6}{a+b x^2}-60 a^5 \log \left(a+b x^2\right)+50 a^4 b x^2-20 a^3 b^2 x^4+10 a^2 b^3 x^6-5 a b^4 x^8+2 b^5 x^{10}}{20 b^7}","-\frac{a^6}{2 b^7 \left(a+b x^2\right)}-\frac{3 a^5 \log \left(a+b x^2\right)}{b^7}+\frac{5 a^4 x^2}{2 b^6}-\frac{a^3 x^4}{b^5}+\frac{a^2 x^6}{2 b^4}-\frac{a x^8}{4 b^3}+\frac{x^{10}}{10 b^2}",1,"(50*a^4*b*x^2 - 20*a^3*b^2*x^4 + 10*a^2*b^3*x^6 - 5*a*b^4*x^8 + 2*b^5*x^10 - (10*a^6)/(a + b*x^2) - 60*a^5*Log[a + b*x^2])/(20*b^7)","A",1
146,1,93,105,0.0565105,"\int \frac{x^{12}}{\left(a+b x^2\right)^2} \, dx","Integrate[x^12/(a + b*x^2)^2,x]","\frac{x \left(\frac{315 a^5}{a+b x^2}+3150 a^4-840 a^3 b x^2+378 a^2 b^2 x^4-180 a b^3 x^6+70 b^4 x^8\right)}{630 b^6}-\frac{11 a^{9/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{13/2}}","-\frac{11 a^{9/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{13/2}}+\frac{11 a^4 x}{2 b^6}-\frac{11 a^3 x^3}{6 b^5}+\frac{11 a^2 x^5}{10 b^4}-\frac{11 a x^7}{14 b^3}-\frac{x^{11}}{2 b \left(a+b x^2\right)}+\frac{11 x^9}{18 b^2}",1,"(x*(3150*a^4 - 840*a^3*b*x^2 + 378*a^2*b^2*x^4 - 180*a*b^3*x^6 + 70*b^4*x^8 + (315*a^5)/(a + b*x^2)))/(630*b^6) - (11*a^(9/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(13/2))","A",1
147,1,72,83,0.0250124,"\int \frac{x^{11}}{\left(a+b x^2\right)^2} \, dx","Integrate[x^11/(a + b*x^2)^2,x]","\frac{\frac{12 a^5}{a+b x^2}+60 a^4 \log \left(a+b x^2\right)-48 a^3 b x^2+18 a^2 b^2 x^4-8 a b^3 x^6+3 b^4 x^8}{24 b^6}","\frac{a^5}{2 b^6 \left(a+b x^2\right)}+\frac{5 a^4 \log \left(a+b x^2\right)}{2 b^6}-\frac{2 a^3 x^2}{b^5}+\frac{3 a^2 x^4}{4 b^4}-\frac{a x^6}{3 b^3}+\frac{x^8}{8 b^2}",1,"(-48*a^3*b*x^2 + 18*a^2*b^2*x^4 - 8*a*b^3*x^6 + 3*b^4*x^8 + (12*a^5)/(a + b*x^2) + 60*a^4*Log[a + b*x^2])/(24*b^6)","A",1
148,1,82,92,0.0570922,"\int \frac{x^{10}}{\left(a+b x^2\right)^2} \, dx","Integrate[x^10/(a + b*x^2)^2,x]","\frac{9 a^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{11/2}}+\frac{x \left(-\frac{35 a^4}{a+b x^2}-280 a^3+70 a^2 b x^2-28 a b^2 x^4+10 b^3 x^6\right)}{70 b^5}","\frac{9 a^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{11/2}}-\frac{9 a^3 x}{2 b^5}+\frac{3 a^2 x^3}{2 b^4}-\frac{9 a x^5}{10 b^3}-\frac{x^9}{2 b \left(a+b x^2\right)}+\frac{9 x^7}{14 b^2}",1,"(x*(-280*a^3 + 70*a^2*b*x^2 - 28*a*b^2*x^4 + 10*b^3*x^6 - (35*a^4)/(a + b*x^2)))/(70*b^5) + (9*a^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(11/2))","A",1
149,1,60,70,0.0246855,"\int \frac{x^9}{\left(a+b x^2\right)^2} \, dx","Integrate[x^9/(a + b*x^2)^2,x]","\frac{-\frac{3 a^4}{a+b x^2}-12 a^3 \log \left(a+b x^2\right)+9 a^2 b x^2-3 a b^2 x^4+b^3 x^6}{6 b^5}","-\frac{a^4}{2 b^5 \left(a+b x^2\right)}-\frac{2 a^3 \log \left(a+b x^2\right)}{b^5}+\frac{3 a^2 x^2}{2 b^4}-\frac{a x^4}{2 b^3}+\frac{x^6}{6 b^2}",1,"(9*a^2*b*x^2 - 3*a*b^2*x^4 + b^3*x^6 - (3*a^4)/(a + b*x^2) - 12*a^3*Log[a + b*x^2])/(6*b^5)","A",1
150,1,71,79,0.0531587,"\int \frac{x^8}{\left(a+b x^2\right)^2} \, dx","Integrate[x^8/(a + b*x^2)^2,x]","\frac{x \left(\frac{15 a^3}{a+b x^2}+90 a^2-20 a b x^2+6 b^2 x^4\right)}{30 b^4}-\frac{7 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{9/2}}","-\frac{7 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{9/2}}+\frac{7 a^2 x}{2 b^4}-\frac{7 a x^3}{6 b^3}-\frac{x^7}{2 b \left(a+b x^2\right)}+\frac{7 x^5}{10 b^2}",1,"(x*(90*a^2 - 20*a*b*x^2 + 6*b^2*x^4 + (15*a^3)/(a + b*x^2)))/(30*b^4) - (7*a^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(9/2))","A",1
151,1,49,57,0.0203813,"\int \frac{x^7}{\left(a+b x^2\right)^2} \, dx","Integrate[x^7/(a + b*x^2)^2,x]","\frac{\frac{2 a^3}{a+b x^2}+6 a^2 \log \left(a+b x^2\right)-4 a b x^2+b^2 x^4}{4 b^4}","\frac{a^3}{2 b^4 \left(a+b x^2\right)}+\frac{3 a^2 \log \left(a+b x^2\right)}{2 b^4}-\frac{a x^2}{b^3}+\frac{x^4}{4 b^2}",1,"(-4*a*b*x^2 + b^2*x^4 + (2*a^3)/(a + b*x^2) + 6*a^2*Log[a + b*x^2])/(4*b^4)","A",1
152,1,60,66,0.0428295,"\int \frac{x^6}{\left(a+b x^2\right)^2} \, dx","Integrate[x^6/(a + b*x^2)^2,x]","\frac{5 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{7/2}}+\frac{x \left(-\frac{3 a^2}{a+b x^2}-12 a+2 b x^2\right)}{6 b^3}","\frac{5 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{7/2}}-\frac{5 a x}{2 b^3}-\frac{x^5}{2 b \left(a+b x^2\right)}+\frac{5 x^3}{6 b^2}",1,"(x*(-12*a + 2*b*x^2 - (3*a^2)/(a + b*x^2)))/(6*b^3) + (5*a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(7/2))","A",1
153,1,38,44,0.0230625,"\int \frac{x^5}{\left(a+b x^2\right)^2} \, dx","Integrate[x^5/(a + b*x^2)^2,x]","\frac{-\frac{a^2}{a+b x^2}-2 a \log \left(a+b x^2\right)+b x^2}{2 b^3}","-\frac{a^2}{2 b^3 \left(a+b x^2\right)}-\frac{a \log \left(a+b x^2\right)}{b^3}+\frac{x^2}{2 b^2}",1,"(b*x^2 - a^2/(a + b*x^2) - 2*a*Log[a + b*x^2])/(2*b^3)","A",1
154,1,51,55,0.0335415,"\int \frac{x^4}{\left(a+b x^2\right)^2} \, dx","Integrate[x^4/(a + b*x^2)^2,x]","-\frac{3 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{5/2}}+\frac{a x}{2 b^2 \left(a+b x^2\right)}+\frac{x}{b^2}","-\frac{3 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{5/2}}-\frac{x^3}{2 b \left(a+b x^2\right)}+\frac{3 x}{2 b^2}",1,"x/b^2 + (a*x)/(2*b^2*(a + b*x^2)) - (3*Sqrt[a]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(5/2))","A",1
155,1,27,33,0.0078095,"\int \frac{x^3}{\left(a+b x^2\right)^2} \, dx","Integrate[x^3/(a + b*x^2)^2,x]","\frac{\frac{a}{a+b x^2}+\log \left(a+b x^2\right)}{2 b^2}","\frac{a}{2 b^2 \left(a+b x^2\right)}+\frac{\log \left(a+b x^2\right)}{2 b^2}",1,"(a/(a + b*x^2) + Log[a + b*x^2])/(2*b^2)","A",1
156,1,45,45,0.0205351,"\int \frac{x^2}{\left(a+b x^2\right)^2} \, dx","Integrate[x^2/(a + b*x^2)^2,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{a} b^{3/2}}-\frac{x}{2 b \left(a+b x^2\right)}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{a} b^{3/2}}-\frac{x}{2 b \left(a+b x^2\right)}",1,"-1/2*x/(b*(a + b*x^2)) + ArcTan[(Sqrt[b]*x)/Sqrt[a]]/(2*Sqrt[a]*b^(3/2))","A",1
157,1,16,16,0.0020133,"\int \frac{x}{\left(a+b x^2\right)^2} \, dx","Integrate[x/(a + b*x^2)^2,x]","-\frac{1}{2 b \left(a+b x^2\right)}","-\frac{1}{2 b \left(a+b x^2\right)}",1,"-1/2*1/(b*(a + b*x^2))","A",1
158,1,45,45,0.0228894,"\int \frac{1}{\left(a+b x^2\right)^2} \, dx","Integrate[(a + b*x^2)^(-2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} \sqrt{b}}+\frac{x}{2 a \left(a+b x^2\right)}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} \sqrt{b}}+\frac{x}{2 a \left(a+b x^2\right)}",1,"x/(2*a*(a + b*x^2)) + ArcTan[(Sqrt[b]*x)/Sqrt[a]]/(2*a^(3/2)*Sqrt[b])","A",1
159,1,33,38,0.0145734,"\int \frac{1}{x \left(a+b x^2\right)^2} \, dx","Integrate[1/(x*(a + b*x^2)^2),x]","\frac{\frac{a}{a+b x^2}-\log \left(a+b x^2\right)+2 \log (x)}{2 a^2}","-\frac{\log \left(a+b x^2\right)}{2 a^2}+\frac{\log (x)}{a^2}+\frac{1}{2 a \left(a+b x^2\right)}",1,"(a/(a + b*x^2) + 2*Log[x] - Log[a + b*x^2])/(2*a^2)","A",1
160,1,54,57,0.0367492,"\int \frac{1}{x^2 \left(a+b x^2\right)^2} \, dx","Integrate[1/(x^2*(a + b*x^2)^2),x]","-\frac{3 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{5/2}}-\frac{b x}{2 a^2 \left(a+b x^2\right)}-\frac{1}{a^2 x}","-\frac{3 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{5/2}}-\frac{3}{2 a^2 x}+\frac{1}{2 a x \left(a+b x^2\right)}",1,"-(1/(a^2*x)) - (b*x)/(2*a^2*(a + b*x^2)) - (3*Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(5/2))","A",1
161,1,41,49,0.0397726,"\int \frac{1}{x^3 \left(a+b x^2\right)^2} \, dx","Integrate[1/(x^3*(a + b*x^2)^2),x]","-\frac{a \left(\frac{b}{a+b x^2}+\frac{1}{x^2}\right)-2 b \log \left(a+b x^2\right)+4 b \log (x)}{2 a^3}","\frac{b \log \left(a+b x^2\right)}{a^3}-\frac{2 b \log (x)}{a^3}-\frac{b}{2 a^2 \left(a+b x^2\right)}-\frac{1}{2 a^2 x^2}",1,"-1/2*(a*(x^(-2) + b/(a + b*x^2)) + 4*b*Log[x] - 2*b*Log[a + b*x^2])/a^3","A",1
162,1,67,68,0.0425109,"\int \frac{1}{x^4 \left(a+b x^2\right)^2} \, dx","Integrate[1/(x^4*(a + b*x^2)^2),x]","\frac{5 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{7/2}}+\frac{b^2 x}{2 a^3 \left(a+b x^2\right)}+\frac{2 b}{a^3 x}-\frac{1}{3 a^2 x^3}","\frac{5 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{7/2}}+\frac{5 b}{2 a^3 x}-\frac{5}{6 a^2 x^3}+\frac{1}{2 a x^3 \left(a+b x^2\right)}",1,"-1/3*1/(a^2*x^3) + (2*b)/(a^3*x) + (b^2*x)/(2*a^3*(a + b*x^2)) + (5*b^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(7/2))","A",1
163,1,57,66,0.0517562,"\int \frac{1}{x^5 \left(a+b x^2\right)^2} \, dx","Integrate[1/(x^5*(a + b*x^2)^2),x]","\frac{-6 b^2 \log \left(a+b x^2\right)+a \left(\frac{2 b^2}{a+b x^2}-\frac{a}{x^4}+\frac{4 b}{x^2}\right)+12 b^2 \log (x)}{4 a^4}","-\frac{3 b^2 \log \left(a+b x^2\right)}{2 a^4}+\frac{3 b^2 \log (x)}{a^4}+\frac{b^2}{2 a^3 \left(a+b x^2\right)}+\frac{b}{a^3 x^2}-\frac{1}{4 a^2 x^4}",1,"(a*(-(a/x^4) + (4*b)/x^2 + (2*b^2)/(a + b*x^2)) + 12*b^2*Log[x] - 6*b^2*Log[a + b*x^2])/(4*a^4)","A",1
164,1,80,81,0.0443797,"\int \frac{1}{x^6 \left(a+b x^2\right)^2} \, dx","Integrate[1/(x^6*(a + b*x^2)^2),x]","-\frac{7 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{9/2}}-\frac{b^3 x}{2 a^4 \left(a+b x^2\right)}-\frac{3 b^2}{a^4 x}+\frac{2 b}{3 a^3 x^3}-\frac{1}{5 a^2 x^5}","-\frac{7 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{9/2}}-\frac{7 b^2}{2 a^4 x}+\frac{7 b}{6 a^3 x^3}-\frac{7}{10 a^2 x^5}+\frac{1}{2 a x^5 \left(a+b x^2\right)}",1,"-1/5*1/(a^2*x^5) + (2*b)/(3*a^3*x^3) - (3*b^2)/(a^4*x) - (b^3*x)/(2*a^4*(a + b*x^2)) - (7*b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(9/2))","A",1
165,1,68,80,0.0568853,"\int \frac{1}{x^7 \left(a+b x^2\right)^2} \, dx","Integrate[1/(x^7*(a + b*x^2)^2),x]","\frac{a \left(-\frac{a^2}{x^6}-\frac{3 b^3}{a+b x^2}+\frac{3 a b}{x^4}-\frac{9 b^2}{x^2}\right)+12 b^3 \log \left(a+b x^2\right)-24 b^3 \log (x)}{6 a^5}","\frac{2 b^3 \log \left(a+b x^2\right)}{a^5}-\frac{4 b^3 \log (x)}{a^5}-\frac{b^3}{2 a^4 \left(a+b x^2\right)}-\frac{3 b^2}{2 a^4 x^2}+\frac{b}{2 a^3 x^4}-\frac{1}{6 a^2 x^6}",1,"(a*(-(a^2/x^6) + (3*a*b)/x^4 - (9*b^2)/x^2 - (3*b^3)/(a + b*x^2)) - 24*b^3*Log[x] + 12*b^3*Log[a + b*x^2])/(6*a^5)","A",1
166,1,91,94,0.0558137,"\int \frac{1}{x^8 \left(a+b x^2\right)^2} \, dx","Integrate[1/(x^8*(a + b*x^2)^2),x]","\frac{9 b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{11/2}}+\frac{b^4 x}{2 a^5 \left(a+b x^2\right)}+\frac{4 b^3}{a^5 x}-\frac{b^2}{a^4 x^3}+\frac{2 b}{5 a^3 x^5}-\frac{1}{7 a^2 x^7}","\frac{9 b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{11/2}}+\frac{9 b^3}{2 a^5 x}-\frac{3 b^2}{2 a^4 x^3}+\frac{9 b}{10 a^3 x^5}-\frac{9}{14 a^2 x^7}+\frac{1}{2 a x^7 \left(a+b x^2\right)}",1,"-1/7*1/(a^2*x^7) + (2*b)/(5*a^3*x^5) - b^2/(a^4*x^3) + (4*b^3)/(a^5*x) + (b^4*x)/(2*a^5*(a + b*x^2)) + (9*b^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(11/2))","A",1
167,1,79,93,0.0764984,"\int \frac{1}{x^9 \left(a+b x^2\right)^2} \, dx","Integrate[1/(x^9*(a + b*x^2)^2),x]","\frac{a \left(-\frac{3 a^3}{x^8}+\frac{8 a^2 b}{x^6}+12 b^3 \left(\frac{b}{a+b x^2}+\frac{4}{x^2}\right)-\frac{18 a b^2}{x^4}\right)-60 b^4 \log \left(a+b x^2\right)+120 b^4 \log (x)}{24 a^6}","-\frac{5 b^4 \log \left(a+b x^2\right)}{2 a^6}+\frac{5 b^4 \log (x)}{a^6}+\frac{b^4}{2 a^5 \left(a+b x^2\right)}+\frac{2 b^3}{a^5 x^2}-\frac{3 b^2}{4 a^4 x^4}+\frac{b}{3 a^3 x^6}-\frac{1}{8 a^2 x^8}",1,"(a*((-3*a^3)/x^8 + (8*a^2*b)/x^6 - (18*a*b^2)/x^4 + 12*b^3*(4/x^2 + b/(a + b*x^2))) + 120*b^4*Log[x] - 60*b^4*Log[a + b*x^2])/(24*a^6)","A",1
168,1,97,114,0.0304062,"\int \frac{x^{15}}{\left(a+b x^2\right)^3} \, dx","Integrate[x^15/(a + b*x^2)^3,x]","\frac{\frac{10 a^7}{\left(a+b x^2\right)^2}-\frac{140 a^6}{a+b x^2}-420 a^5 \log \left(a+b x^2\right)+300 a^4 b x^2-100 a^3 b^2 x^4+40 a^2 b^3 x^6-15 a b^4 x^8+4 b^5 x^{10}}{40 b^8}","\frac{a^7}{4 b^8 \left(a+b x^2\right)^2}-\frac{7 a^6}{2 b^8 \left(a+b x^2\right)}-\frac{21 a^5 \log \left(a+b x^2\right)}{2 b^8}+\frac{15 a^4 x^2}{2 b^7}-\frac{5 a^3 x^4}{2 b^6}+\frac{a^2 x^6}{b^5}-\frac{3 a x^8}{8 b^4}+\frac{x^{10}}{10 b^3}",1,"(300*a^4*b*x^2 - 100*a^3*b^2*x^4 + 40*a^2*b^3*x^6 - 15*a*b^4*x^8 + 4*b^5*x^10 + (10*a^7)/(a + b*x^2)^2 - (140*a^6)/(a + b*x^2) - 420*a^5*Log[a + b*x^2])/(40*b^8)","A",1
169,1,85,100,0.0278033,"\int \frac{x^{13}}{\left(a+b x^2\right)^3} \, dx","Integrate[x^13/(a + b*x^2)^3,x]","\frac{-\frac{2 a^6}{\left(a+b x^2\right)^2}+\frac{24 a^5}{a+b x^2}+60 a^4 \log \left(a+b x^2\right)-40 a^3 b x^2+12 a^2 b^2 x^4-4 a b^3 x^6+b^4 x^8}{8 b^7}","-\frac{a^6}{4 b^7 \left(a+b x^2\right)^2}+\frac{3 a^5}{b^7 \left(a+b x^2\right)}+\frac{15 a^4 \log \left(a+b x^2\right)}{2 b^7}-\frac{5 a^3 x^2}{b^6}+\frac{3 a^2 x^4}{2 b^5}-\frac{a x^6}{2 b^4}+\frac{x^8}{8 b^3}",1,"(-40*a^3*b*x^2 + 12*a^2*b^2*x^4 - 4*a*b^3*x^6 + b^4*x^8 - (2*a^6)/(a + b*x^2)^2 + (24*a^5)/(a + b*x^2) + 60*a^4*Log[a + b*x^2])/(8*b^7)","A",1
170,1,75,87,0.0253794,"\int \frac{x^{11}}{\left(a+b x^2\right)^3} \, dx","Integrate[x^11/(a + b*x^2)^3,x]","\frac{\frac{3 a^5}{\left(a+b x^2\right)^2}-\frac{30 a^4}{a+b x^2}-60 a^3 \log \left(a+b x^2\right)+36 a^2 b x^2-9 a b^2 x^4+2 b^3 x^6}{12 b^6}","\frac{a^5}{4 b^6 \left(a+b x^2\right)^2}-\frac{5 a^4}{2 b^6 \left(a+b x^2\right)}-\frac{5 a^3 \log \left(a+b x^2\right)}{b^6}+\frac{3 a^2 x^2}{b^5}-\frac{3 a x^4}{4 b^4}+\frac{x^6}{6 b^3}",1,"(36*a^2*b*x^2 - 9*a*b^2*x^4 + 2*b^3*x^6 + (3*a^5)/(a + b*x^2)^2 - (30*a^4)/(a + b*x^2) - 60*a^3*Log[a + b*x^2])/(12*b^6)","A",1
171,1,63,74,0.0213511,"\int \frac{x^9}{\left(a+b x^2\right)^3} \, dx","Integrate[x^9/(a + b*x^2)^3,x]","\frac{-\frac{a^4}{\left(a+b x^2\right)^2}+\frac{8 a^3}{a+b x^2}+12 a^2 \log \left(a+b x^2\right)-6 a b x^2+b^2 x^4}{4 b^5}","-\frac{a^4}{4 b^5 \left(a+b x^2\right)^2}+\frac{2 a^3}{b^5 \left(a+b x^2\right)}+\frac{3 a^2 \log \left(a+b x^2\right)}{b^5}-\frac{3 a x^2}{2 b^4}+\frac{x^4}{4 b^3}",1,"(-6*a*b*x^2 + b^2*x^4 - a^4/(a + b*x^2)^2 + (8*a^3)/(a + b*x^2) + 12*a^2*Log[a + b*x^2])/(4*b^5)","A",1
172,1,48,65,0.057369,"\int \frac{x^7}{\left(a+b x^2\right)^3} \, dx","Integrate[x^7/(a + b*x^2)^3,x]","-\frac{\frac{a^2 \left(5 a+6 b x^2\right)}{\left(a+b x^2\right)^2}+6 a \log \left(a+b x^2\right)-2 b x^2}{4 b^4}","\frac{a^3}{4 b^4 \left(a+b x^2\right)^2}-\frac{3 a^2}{2 b^4 \left(a+b x^2\right)}-\frac{3 a \log \left(a+b x^2\right)}{2 b^4}+\frac{x^2}{2 b^3}",1,"-1/4*(-2*b*x^2 + (a^2*(5*a + 6*b*x^2))/(a + b*x^2)^2 + 6*a*Log[a + b*x^2])/b^4","A",1
173,1,39,49,0.0183421,"\int \frac{x^5}{\left(a+b x^2\right)^3} \, dx","Integrate[x^5/(a + b*x^2)^3,x]","\frac{\frac{a \left(3 a+4 b x^2\right)}{\left(a+b x^2\right)^2}+2 \log \left(a+b x^2\right)}{4 b^3}","-\frac{a^2}{4 b^3 \left(a+b x^2\right)^2}+\frac{a}{b^3 \left(a+b x^2\right)}+\frac{\log \left(a+b x^2\right)}{2 b^3}",1,"((a*(3*a + 4*b*x^2))/(a + b*x^2)^2 + 2*Log[a + b*x^2])/(4*b^3)","A",1
174,1,24,19,0.0069617,"\int \frac{x^3}{\left(a+b x^2\right)^3} \, dx","Integrate[x^3/(a + b*x^2)^3,x]","-\frac{a+2 b x^2}{4 b^2 \left(a+b x^2\right)^2}","\frac{x^4}{4 a \left(a+b x^2\right)^2}",1,"-1/4*(a + 2*b*x^2)/(b^2*(a + b*x^2)^2)","A",1
175,1,16,16,0.0023791,"\int \frac{x}{\left(a+b x^2\right)^3} \, dx","Integrate[x/(a + b*x^2)^3,x]","-\frac{1}{4 b \left(a+b x^2\right)^2}","-\frac{1}{4 b \left(a+b x^2\right)^2}",1,"-1/4*1/(b*(a + b*x^2)^2)","A",1
176,1,43,54,0.0378623,"\int \frac{1}{x \left(a+b x^2\right)^3} \, dx","Integrate[1/(x*(a + b*x^2)^3),x]","\frac{\frac{a \left(3 a+2 b x^2\right)}{\left(a+b x^2\right)^2}-2 \log \left(a+b x^2\right)+4 \log (x)}{4 a^3}","-\frac{\log \left(a+b x^2\right)}{2 a^3}+\frac{\log (x)}{a^3}+\frac{1}{2 a^2 \left(a+b x^2\right)}+\frac{1}{4 a \left(a+b x^2\right)^2}",1,"((a*(3*a + 2*b*x^2))/(a + b*x^2)^2 + 4*Log[x] - 2*Log[a + b*x^2])/(4*a^3)","A",1
177,1,59,67,0.0594635,"\int \frac{1}{x^3 \left(a+b x^2\right)^3} \, dx","Integrate[1/(x^3*(a + b*x^2)^3),x]","-\frac{\frac{a \left(2 a^2+9 a b x^2+6 b^2 x^4\right)}{x^2 \left(a+b x^2\right)^2}-6 b \log \left(a+b x^2\right)+12 b \log (x)}{4 a^4}","\frac{3 b \log \left(a+b x^2\right)}{2 a^4}-\frac{3 b \log (x)}{a^4}-\frac{b}{a^3 \left(a+b x^2\right)}-\frac{1}{2 a^3 x^2}-\frac{b}{4 a^2 \left(a+b x^2\right)^2}",1,"-1/4*((a*(2*a^2 + 9*a*b*x^2 + 6*b^2*x^4))/(x^2*(a + b*x^2)^2) + 12*b*Log[x] - 6*b*Log[a + b*x^2])/a^4","A",1
178,1,74,86,0.0510411,"\int \frac{1}{x^5 \left(a+b x^2\right)^3} \, dx","Integrate[1/(x^5*(a + b*x^2)^3),x]","\frac{\frac{a \left(-a^3+4 a^2 b x^2+18 a b^2 x^4+12 b^3 x^6\right)}{x^4 \left(a+b x^2\right)^2}-12 b^2 \log \left(a+b x^2\right)+24 b^2 \log (x)}{4 a^5}","-\frac{3 b^2 \log \left(a+b x^2\right)}{a^5}+\frac{6 b^2 \log (x)}{a^5}+\frac{3 b^2}{2 a^4 \left(a+b x^2\right)}+\frac{3 b}{2 a^4 x^2}+\frac{b^2}{4 a^3 \left(a+b x^2\right)^2}-\frac{1}{4 a^3 x^4}",1,"((a*(-a^3 + 4*a^2*b*x^2 + 18*a*b^2*x^4 + 12*b^3*x^6))/(x^4*(a + b*x^2)^2) + 24*b^2*Log[x] - 12*b^2*Log[a + b*x^2])/(4*a^5)","A",1
179,1,85,95,0.070953,"\int \frac{1}{x^7 \left(a+b x^2\right)^3} \, dx","Integrate[1/(x^7*(a + b*x^2)^3),x]","-\frac{\frac{a \left(2 a^4-5 a^3 b x^2+20 a^2 b^2 x^4+90 a b^3 x^6+60 b^4 x^8\right)}{x^6 \left(a+b x^2\right)^2}-60 b^3 \log \left(a+b x^2\right)+120 b^3 \log (x)}{12 a^6}","\frac{5 b^3 \log \left(a+b x^2\right)}{a^6}-\frac{10 b^3 \log (x)}{a^6}-\frac{2 b^3}{a^5 \left(a+b x^2\right)}-\frac{3 b^2}{a^5 x^2}-\frac{b^3}{4 a^4 \left(a+b x^2\right)^2}+\frac{3 b}{4 a^4 x^4}-\frac{1}{6 a^3 x^6}",1,"-1/12*((a*(2*a^4 - 5*a^3*b*x^2 + 20*a^2*b^2*x^4 + 90*a*b^3*x^6 + 60*b^4*x^8))/(x^6*(a + b*x^2)^2) + 120*b^3*Log[x] - 60*b^3*Log[a + b*x^2])/a^6","A",1
180,1,96,112,0.0606669,"\int \frac{1}{x^9 \left(a+b x^2\right)^3} \, dx","Integrate[1/(x^9*(a + b*x^2)^3),x]","\frac{\frac{a \left(-a^5+2 a^4 b x^2-5 a^3 b^2 x^4+20 a^2 b^3 x^6+90 a b^4 x^8+60 b^5 x^{10}\right)}{x^8 \left(a+b x^2\right)^2}-60 b^4 \log \left(a+b x^2\right)+120 b^4 \log (x)}{8 a^7}","-\frac{15 b^4 \log \left(a+b x^2\right)}{2 a^7}+\frac{15 b^4 \log (x)}{a^7}+\frac{5 b^4}{2 a^6 \left(a+b x^2\right)}+\frac{5 b^3}{a^6 x^2}+\frac{b^4}{4 a^5 \left(a+b x^2\right)^2}-\frac{3 b^2}{2 a^5 x^4}+\frac{b}{2 a^4 x^6}-\frac{1}{8 a^3 x^8}",1,"((a*(-a^5 + 2*a^4*b*x^2 - 5*a^3*b^2*x^4 + 20*a^2*b^3*x^6 + 90*a*b^4*x^8 + 60*b^5*x^10))/(x^8*(a + b*x^2)^2) + 120*b^4*Log[x] - 60*b^4*Log[a + b*x^2])/(8*a^7)","A",1
181,1,99,111,0.0620163,"\int \frac{x^{12}}{\left(a+b x^2\right)^3} \, dx","Integrate[x^12/(a + b*x^2)^3,x]","\frac{99 a^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 b^{13/2}}-\frac{3465 a^5 x+5775 a^4 b x^3+1848 a^3 b^2 x^5-264 a^2 b^3 x^7+88 a b^4 x^9-40 b^5 x^{11}}{280 b^6 \left(a+b x^2\right)^2}","\frac{99 a^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 b^{13/2}}-\frac{99 a^3 x}{8 b^6}+\frac{33 a^2 x^3}{8 b^5}-\frac{99 a x^5}{40 b^4}-\frac{11 x^9}{8 b^2 \left(a+b x^2\right)}-\frac{x^{11}}{4 b \left(a+b x^2\right)^2}+\frac{99 x^7}{56 b^3}",1,"-1/280*(3465*a^5*x + 5775*a^4*b*x^3 + 1848*a^3*b^2*x^5 - 264*a^2*b^3*x^7 + 88*a*b^4*x^9 - 40*b^5*x^11)/(b^6*(a + b*x^2)^2) + (99*a^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(13/2))","A",1
182,1,88,98,0.0493817,"\int \frac{x^{10}}{\left(a+b x^2\right)^3} \, dx","Integrate[x^10/(a + b*x^2)^3,x]","\frac{315 a^4 x+525 a^3 b x^3+168 a^2 b^2 x^5-24 a b^3 x^7+8 b^4 x^9}{40 b^5 \left(a+b x^2\right)^2}-\frac{63 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 b^{11/2}}","-\frac{63 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 b^{11/2}}+\frac{63 a^2 x}{8 b^5}-\frac{21 a x^3}{8 b^4}-\frac{9 x^7}{8 b^2 \left(a+b x^2\right)}-\frac{x^9}{4 b \left(a+b x^2\right)^2}+\frac{63 x^5}{40 b^3}",1,"(315*a^4*x + 525*a^3*b*x^3 + 168*a^2*b^2*x^5 - 24*a*b^3*x^7 + 8*b^4*x^9)/(40*b^5*(a + b*x^2)^2) - (63*a^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(11/2))","A",1
183,1,77,85,0.0461474,"\int \frac{x^8}{\left(a+b x^2\right)^3} \, dx","Integrate[x^8/(a + b*x^2)^3,x]","\frac{35 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 b^{9/2}}-\frac{105 a^3 x+175 a^2 b x^3+56 a b^2 x^5-8 b^3 x^7}{24 b^4 \left(a+b x^2\right)^2}","\frac{35 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 b^{9/2}}-\frac{35 a x}{8 b^4}-\frac{7 x^5}{8 b^2 \left(a+b x^2\right)}-\frac{x^7}{4 b \left(a+b x^2\right)^2}+\frac{35 x^3}{24 b^3}",1,"-1/24*(105*a^3*x + 175*a^2*b*x^3 + 56*a*b^2*x^5 - 8*b^3*x^7)/(b^4*(a + b*x^2)^2) + (35*a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(9/2))","A",1
184,1,66,74,0.0489872,"\int \frac{x^6}{\left(a+b x^2\right)^3} \, dx","Integrate[x^6/(a + b*x^2)^3,x]","\frac{15 a^2 x+25 a b x^3+8 b^2 x^5}{8 b^3 \left(a+b x^2\right)^2}-\frac{15 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 b^{7/2}}","-\frac{15 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 b^{7/2}}-\frac{5 x^3}{8 b^2 \left(a+b x^2\right)}-\frac{x^5}{4 b \left(a+b x^2\right)^2}+\frac{15 x}{8 b^3}",1,"(15*a^2*x + 25*a*b*x^3 + 8*b^2*x^5)/(8*b^3*(a + b*x^2)^2) - (15*Sqrt[a]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(7/2))","A",1
185,1,55,64,0.0406775,"\int \frac{x^4}{\left(a+b x^2\right)^3} \, dx","Integrate[x^4/(a + b*x^2)^3,x]","\frac{3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 \sqrt{a} b^{5/2}}-\frac{3 a x+5 b x^3}{8 b^2 \left(a+b x^2\right)^2}","\frac{3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 \sqrt{a} b^{5/2}}-\frac{3 x}{8 b^2 \left(a+b x^2\right)}-\frac{x^3}{4 b \left(a+b x^2\right)^2}",1,"-1/8*(3*a*x + 5*b*x^3)/(b^2*(a + b*x^2)^2) + (3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*Sqrt[a]*b^(5/2))","A",1
186,1,58,65,0.0285419,"\int \frac{x^2}{\left(a+b x^2\right)^3} \, dx","Integrate[x^2/(a + b*x^2)^3,x]","\frac{\frac{\sqrt{a} \sqrt{b} x \left(b x^2-a\right)}{\left(a+b x^2\right)^2}+\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{3/2} b^{3/2}}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{3/2} b^{3/2}}+\frac{x}{8 a b \left(a+b x^2\right)}-\frac{x}{4 b \left(a+b x^2\right)^2}",1,"((Sqrt[a]*Sqrt[b]*x*(-a + b*x^2))/(a + b*x^2)^2 + ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(3/2)*b^(3/2))","A",1
187,1,55,62,0.0375293,"\int \frac{1}{\left(a+b x^2\right)^3} \, dx","Integrate[(a + b*x^2)^(-3),x]","\frac{3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} \sqrt{b}}+\frac{5 a x+3 b x^3}{8 a^2 \left(a+b x^2\right)^2}","\frac{3 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} \sqrt{b}}+\frac{3 x}{8 a^2 \left(a+b x^2\right)}+\frac{x}{4 a \left(a+b x^2\right)^2}",1,"(5*a*x + 3*b*x^3)/(8*a^2*(a + b*x^2)^2) + (3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*Sqrt[b])","A",1
188,1,68,76,0.040441,"\int \frac{1}{x^2 \left(a+b x^2\right)^3} \, dx","Integrate[1/(x^2*(a + b*x^2)^3),x]","-\frac{15 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{7/2}}-\frac{8 a^2+25 a b x^2+15 b^2 x^4}{8 a^3 x \left(a+b x^2\right)^2}","-\frac{15 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{7/2}}-\frac{15}{8 a^3 x}+\frac{5}{8 a^2 x \left(a+b x^2\right)}+\frac{1}{4 a x \left(a+b x^2\right)^2}",1,"-1/8*(8*a^2 + 25*a*b*x^2 + 15*b^2*x^4)/(a^3*x*(a + b*x^2)^2) - (15*Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(7/2))","A",1
189,1,79,87,0.0497356,"\int \frac{1}{x^4 \left(a+b x^2\right)^3} \, dx","Integrate[1/(x^4*(a + b*x^2)^3),x]","\frac{35 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{9/2}}+\frac{-8 a^3+56 a^2 b x^2+175 a b^2 x^4+105 b^3 x^6}{24 a^4 x^3 \left(a+b x^2\right)^2}","\frac{35 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{9/2}}+\frac{35 b}{8 a^4 x}-\frac{35}{24 a^3 x^3}+\frac{7}{8 a^2 x^3 \left(a+b x^2\right)}+\frac{1}{4 a x^3 \left(a+b x^2\right)^2}",1,"(-8*a^3 + 56*a^2*b*x^2 + 175*a*b^2*x^4 + 105*b^3*x^6)/(24*a^4*x^3*(a + b*x^2)^2) + (35*b^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(9/2))","A",1
190,1,90,100,0.0544194,"\int \frac{1}{x^6 \left(a+b x^2\right)^3} \, dx","Integrate[1/(x^6*(a + b*x^2)^3),x]","-\frac{63 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{11/2}}-\frac{8 a^4-24 a^3 b x^2+168 a^2 b^2 x^4+525 a b^3 x^6+315 b^4 x^8}{40 a^5 x^5 \left(a+b x^2\right)^2}","-\frac{63 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{11/2}}-\frac{63 b^2}{8 a^5 x}+\frac{21 b}{8 a^4 x^3}-\frac{63}{40 a^3 x^5}+\frac{9}{8 a^2 x^5 \left(a+b x^2\right)}+\frac{1}{4 a x^5 \left(a+b x^2\right)^2}",1,"-1/40*(8*a^4 - 24*a^3*b*x^2 + 168*a^2*b^2*x^4 + 525*a*b^3*x^6 + 315*b^4*x^8)/(a^5*x^5*(a + b*x^2)^2) - (63*b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(11/2))","A",1
191,1,101,113,0.0562545,"\int \frac{1}{x^8 \left(a+b x^2\right)^3} \, dx","Integrate[1/(x^8*(a + b*x^2)^3),x]","\frac{99 b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{13/2}}+\frac{-40 a^5+88 a^4 b x^2-264 a^3 b^2 x^4+1848 a^2 b^3 x^6+5775 a b^4 x^8+3465 b^5 x^{10}}{280 a^6 x^7 \left(a+b x^2\right)^2}","\frac{99 b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{13/2}}+\frac{99 b^3}{8 a^6 x}-\frac{33 b^2}{8 a^5 x^3}+\frac{99 b}{40 a^4 x^5}-\frac{99}{56 a^3 x^7}+\frac{11}{8 a^2 x^7 \left(a+b x^2\right)}+\frac{1}{4 a x^7 \left(a+b x^2\right)^2}",1,"(-40*a^5 + 88*a^4*b*x^2 - 264*a^3*b^2*x^4 + 1848*a^2*b^3*x^6 + 5775*a*b^4*x^8 + 3465*b^5*x^10)/(280*a^6*x^7*(a + b*x^2)^2) + (99*b^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(13/2))","A",1
192,1,169,216,0.0483411,"\int \frac{x^{25}}{\left(a+b x^2\right)^{10}} \, dx","Integrate[x^25/(a + b*x^2)^10,x]","-\frac{35201 a^{12}+289089 a^{11} b x^2+1031616 a^{10} b^2 x^4+2074464 a^9 b^3 x^6+2529576 a^8 b^4 x^8+1831032 a^7 b^5 x^{10}+638568 a^6 b^6 x^{12}-58968 a^5 b^7 x^{14}-139482 a^4 b^8 x^{16}-43218 a^3 b^9 x^{18}+27720 a^3 \left(a+b x^2\right)^9 \log \left(a+b x^2\right)-2772 a^2 b^{10} x^{20}+252 a b^{11} x^{22}-42 b^{12} x^{24}}{252 b^{13} \left(a+b x^2\right)^9}","-\frac{a^{12}}{18 b^{13} \left(a+b x^2\right)^9}+\frac{3 a^{11}}{4 b^{13} \left(a+b x^2\right)^8}-\frac{33 a^{10}}{7 b^{13} \left(a+b x^2\right)^7}+\frac{55 a^9}{3 b^{13} \left(a+b x^2\right)^6}-\frac{99 a^8}{2 b^{13} \left(a+b x^2\right)^5}+\frac{99 a^7}{b^{13} \left(a+b x^2\right)^4}-\frac{154 a^6}{b^{13} \left(a+b x^2\right)^3}+\frac{198 a^5}{b^{13} \left(a+b x^2\right)^2}-\frac{495 a^4}{2 b^{13} \left(a+b x^2\right)}-\frac{110 a^3 \log \left(a+b x^2\right)}{b^{13}}+\frac{55 a^2 x^2}{2 b^{12}}-\frac{5 a x^4}{2 b^{11}}+\frac{x^6}{6 b^{10}}",1,"-1/252*(35201*a^12 + 289089*a^11*b*x^2 + 1031616*a^10*b^2*x^4 + 2074464*a^9*b^3*x^6 + 2529576*a^8*b^4*x^8 + 1831032*a^7*b^5*x^10 + 638568*a^6*b^6*x^12 - 58968*a^5*b^7*x^14 - 139482*a^4*b^8*x^16 - 43218*a^3*b^9*x^18 - 2772*a^2*b^10*x^20 + 252*a*b^11*x^22 - 42*b^12*x^24 + 27720*a^3*(a + b*x^2)^9*Log[a + b*x^2])/(b^13*(a + b*x^2)^9)","A",1
193,1,158,205,0.0301621,"\int \frac{x^{23}}{\left(a+b x^2\right)^{10}} \, dx","Integrate[x^23/(a + b*x^2)^10,x]","\frac{42131 a^{11}+351459 a^{10} b x^2+1281096 a^9 b^2 x^4+2656584 a^8 b^3 x^6+3402756 a^7 b^4 x^8+2704212 a^6 b^5 x^{10}+1220688 a^5 b^6 x^{12}+190512 a^4 b^7 x^{14}-77112 a^3 b^8 x^{16}-36288 a^2 b^9 x^{18}+27720 a^2 \left(a+b x^2\right)^9 \log \left(a+b x^2\right)-2772 a b^{10} x^{20}+252 b^{11} x^{22}}{1008 b^{12} \left(a+b x^2\right)^9}","\frac{a^{11}}{18 b^{12} \left(a+b x^2\right)^9}-\frac{11 a^{10}}{16 b^{12} \left(a+b x^2\right)^8}+\frac{55 a^9}{14 b^{12} \left(a+b x^2\right)^7}-\frac{55 a^8}{4 b^{12} \left(a+b x^2\right)^6}+\frac{33 a^7}{b^{12} \left(a+b x^2\right)^5}-\frac{231 a^6}{4 b^{12} \left(a+b x^2\right)^4}+\frac{77 a^5}{b^{12} \left(a+b x^2\right)^3}-\frac{165 a^4}{2 b^{12} \left(a+b x^2\right)^2}+\frac{165 a^3}{2 b^{12} \left(a+b x^2\right)}+\frac{55 a^2 \log \left(a+b x^2\right)}{2 b^{12}}-\frac{5 a x^2}{b^{11}}+\frac{x^4}{4 b^{10}}",1,"(42131*a^11 + 351459*a^10*b*x^2 + 1281096*a^9*b^2*x^4 + 2656584*a^8*b^3*x^6 + 3402756*a^7*b^4*x^8 + 2704212*a^6*b^5*x^10 + 1220688*a^5*b^6*x^12 + 190512*a^4*b^7*x^14 - 77112*a^3*b^8*x^16 - 36288*a^2*b^9*x^18 - 2772*a*b^10*x^20 + 252*b^11*x^22 + 27720*a^2*(a + b*x^2)^9*Log[a + b*x^2])/(1008*b^12*(a + b*x^2)^9)","A",1
194,1,145,188,0.0388352,"\int \frac{x^{21}}{\left(a+b x^2\right)^{10}} \, dx","Integrate[x^21/(a + b*x^2)^10,x]","-\frac{4861 a^{10}+41229 a^9 b x^2+153576 a^8 b^2 x^4+328104 a^7 b^3 x^6+439236 a^6 b^4 x^8+375732 a^5 b^5 x^{10}+197568 a^4 b^6 x^{12}+54432 a^3 b^7 x^{14}+2268 a^2 b^8 x^{16}-2268 a b^9 x^{18}+2520 a \left(a+b x^2\right)^9 \log \left(a+b x^2\right)-252 b^{10} x^{20}}{504 b^{11} \left(a+b x^2\right)^9}","-\frac{a^{10}}{18 b^{11} \left(a+b x^2\right)^9}+\frac{5 a^9}{8 b^{11} \left(a+b x^2\right)^8}-\frac{45 a^8}{14 b^{11} \left(a+b x^2\right)^7}+\frac{10 a^7}{b^{11} \left(a+b x^2\right)^6}-\frac{21 a^6}{b^{11} \left(a+b x^2\right)^5}+\frac{63 a^5}{2 b^{11} \left(a+b x^2\right)^4}-\frac{35 a^4}{b^{11} \left(a+b x^2\right)^3}+\frac{30 a^3}{b^{11} \left(a+b x^2\right)^2}-\frac{45 a^2}{2 b^{11} \left(a+b x^2\right)}-\frac{5 a \log \left(a+b x^2\right)}{b^{11}}+\frac{x^2}{2 b^{10}}",1,"-1/504*(4861*a^10 + 41229*a^9*b*x^2 + 153576*a^8*b^2*x^4 + 328104*a^7*b^3*x^6 + 439236*a^6*b^4*x^8 + 375732*a^5*b^5*x^10 + 197568*a^4*b^6*x^12 + 54432*a^3*b^7*x^14 + 2268*a^2*b^8*x^16 - 2268*a*b^9*x^18 - 252*b^10*x^20 + 2520*a*(a + b*x^2)^9*Log[a + b*x^2])/(b^11*(a + b*x^2)^9)","A",1
195,1,116,179,0.0332993,"\int \frac{x^{19}}{\left(a+b x^2\right)^{10}} \, dx","Integrate[x^19/(a + b*x^2)^10,x]","\frac{\frac{a \left(7129 a^8+61641 a^7 b x^2+235224 a^6 b^2 x^4+518616 a^5 b^3 x^6+725004 a^4 b^4 x^8+661500 a^3 b^5 x^{10}+388080 a^2 b^6 x^{12}+136080 a b^7 x^{14}+22680 b^8 x^{16}\right)}{\left(a+b x^2\right)^9}+2520 \log \left(a+b x^2\right)}{5040 b^{10}}","\frac{a^9}{18 b^{10} \left(a+b x^2\right)^9}-\frac{9 a^8}{16 b^{10} \left(a+b x^2\right)^8}+\frac{18 a^7}{7 b^{10} \left(a+b x^2\right)^7}-\frac{7 a^6}{b^{10} \left(a+b x^2\right)^6}+\frac{63 a^5}{5 b^{10} \left(a+b x^2\right)^5}-\frac{63 a^4}{4 b^{10} \left(a+b x^2\right)^4}+\frac{14 a^3}{b^{10} \left(a+b x^2\right)^3}-\frac{9 a^2}{b^{10} \left(a+b x^2\right)^2}+\frac{9 a}{2 b^{10} \left(a+b x^2\right)}+\frac{\log \left(a+b x^2\right)}{2 b^{10}}",1,"((a*(7129*a^8 + 61641*a^7*b*x^2 + 235224*a^6*b^2*x^4 + 518616*a^5*b^3*x^6 + 725004*a^4*b^4*x^8 + 661500*a^3*b^5*x^10 + 388080*a^2*b^6*x^12 + 136080*a*b^7*x^14 + 22680*b^8*x^16))/(a + b*x^2)^9 + 2520*Log[a + b*x^2])/(5040*b^10)","A",1
196,1,101,19,0.0206646,"\int \frac{x^{17}}{\left(a+b x^2\right)^{10}} \, dx","Integrate[x^17/(a + b*x^2)^10,x]","-\frac{a^8+9 a^7 b x^2+36 a^6 b^2 x^4+84 a^5 b^3 x^6+126 a^4 b^4 x^8+126 a^3 b^5 x^{10}+84 a^2 b^6 x^{12}+36 a b^7 x^{14}+9 b^8 x^{16}}{18 b^9 \left(a+b x^2\right)^9}","\frac{x^{18}}{18 a \left(a+b x^2\right)^9}",1,"-1/18*(a^8 + 9*a^7*b*x^2 + 36*a^6*b^2*x^4 + 84*a^5*b^3*x^6 + 126*a^4*b^4*x^8 + 126*a^3*b^5*x^10 + 84*a^2*b^6*x^12 + 36*a*b^7*x^14 + 9*b^8*x^16)/(b^9*(a + b*x^2)^9)","B",1
197,1,90,39,0.0166388,"\int \frac{x^{15}}{\left(a+b x^2\right)^{10}} \, dx","Integrate[x^15/(a + b*x^2)^10,x]","-\frac{a^7+9 a^6 b x^2+36 a^5 b^2 x^4+84 a^4 b^3 x^6+126 a^3 b^4 x^8+126 a^2 b^5 x^{10}+84 a b^6 x^{12}+36 b^7 x^{14}}{144 b^8 \left(a+b x^2\right)^9}","\frac{x^{16}}{144 a^2 \left(a+b x^2\right)^8}+\frac{x^{16}}{18 a \left(a+b x^2\right)^9}",1,"-1/144*(a^7 + 9*a^6*b*x^2 + 36*a^5*b^2*x^4 + 84*a^4*b^3*x^6 + 126*a^3*b^4*x^8 + 126*a^2*b^5*x^10 + 84*a*b^6*x^12 + 36*b^7*x^14)/(b^8*(a + b*x^2)^9)","B",1
198,1,79,58,0.018082,"\int \frac{x^{13}}{\left(a+b x^2\right)^{10}} \, dx","Integrate[x^13/(a + b*x^2)^10,x]","-\frac{a^6+9 a^5 b x^2+36 a^4 b^2 x^4+84 a^3 b^3 x^6+126 a^2 b^4 x^8+126 a b^5 x^{10}+84 b^6 x^{12}}{504 b^7 \left(a+b x^2\right)^9}","\frac{x^{14}}{504 a^3 \left(a+b x^2\right)^7}+\frac{x^{14}}{72 a^2 \left(a+b x^2\right)^8}+\frac{x^{14}}{18 a \left(a+b x^2\right)^9}",1,"-1/504*(a^6 + 9*a^5*b*x^2 + 36*a^4*b^2*x^4 + 84*a^3*b^3*x^6 + 126*a^2*b^4*x^8 + 126*a*b^5*x^10 + 84*b^6*x^12)/(b^7*(a + b*x^2)^9)","A",1
199,1,68,77,0.0174097,"\int \frac{x^{11}}{\left(a+b x^2\right)^{10}} \, dx","Integrate[x^11/(a + b*x^2)^10,x]","-\frac{a^5+9 a^4 b x^2+36 a^3 b^2 x^4+84 a^2 b^3 x^6+126 a b^4 x^8+126 b^5 x^{10}}{1008 b^6 \left(a+b x^2\right)^9}","\frac{x^{12}}{1008 a^4 \left(a+b x^2\right)^6}+\frac{x^{12}}{168 a^3 \left(a+b x^2\right)^7}+\frac{x^{12}}{48 a^2 \left(a+b x^2\right)^8}+\frac{x^{12}}{18 a \left(a+b x^2\right)^9}",1,"-1/1008*(a^5 + 9*a^4*b*x^2 + 36*a^3*b^2*x^4 + 84*a^2*b^3*x^6 + 126*a*b^4*x^8 + 126*b^5*x^10)/(b^6*(a + b*x^2)^9)","A",1
200,1,57,91,0.0156356,"\int \frac{x^9}{\left(a+b x^2\right)^{10}} \, dx","Integrate[x^9/(a + b*x^2)^10,x]","-\frac{a^4+9 a^3 b x^2+36 a^2 b^2 x^4+84 a b^3 x^6+126 b^4 x^8}{1260 b^5 \left(a+b x^2\right)^9}","-\frac{a^4}{18 b^5 \left(a+b x^2\right)^9}+\frac{a^3}{4 b^5 \left(a+b x^2\right)^8}-\frac{3 a^2}{7 b^5 \left(a+b x^2\right)^7}+\frac{a}{3 b^5 \left(a+b x^2\right)^6}-\frac{1}{10 b^5 \left(a+b x^2\right)^5}",1,"-1/1260*(a^4 + 9*a^3*b*x^2 + 36*a^2*b^2*x^4 + 84*a*b^3*x^6 + 126*b^4*x^8)/(b^5*(a + b*x^2)^9)","A",1
201,1,46,72,0.0113244,"\int \frac{x^7}{\left(a+b x^2\right)^{10}} \, dx","Integrate[x^7/(a + b*x^2)^10,x]","-\frac{a^3+9 a^2 b x^2+36 a b^2 x^4+84 b^3 x^6}{1008 b^4 \left(a+b x^2\right)^9}","\frac{a^3}{18 b^4 \left(a+b x^2\right)^9}-\frac{3 a^2}{16 b^4 \left(a+b x^2\right)^8}+\frac{3 a}{14 b^4 \left(a+b x^2\right)^7}-\frac{1}{12 b^4 \left(a+b x^2\right)^6}",1,"-1/1008*(a^3 + 9*a^2*b*x^2 + 36*a*b^2*x^4 + 84*b^3*x^6)/(b^4*(a + b*x^2)^9)","A",1
202,1,35,53,0.0135126,"\int \frac{x^5}{\left(a+b x^2\right)^{10}} \, dx","Integrate[x^5/(a + b*x^2)^10,x]","-\frac{a^2+9 a b x^2+36 b^2 x^4}{504 b^3 \left(a+b x^2\right)^9}","-\frac{a^2}{18 b^3 \left(a+b x^2\right)^9}+\frac{a}{8 b^3 \left(a+b x^2\right)^8}-\frac{1}{14 b^3 \left(a+b x^2\right)^7}",1,"-1/504*(a^2 + 9*a*b*x^2 + 36*b^2*x^4)/(b^3*(a + b*x^2)^9)","A",1
203,1,24,34,0.0076027,"\int \frac{x^3}{\left(a+b x^2\right)^{10}} \, dx","Integrate[x^3/(a + b*x^2)^10,x]","-\frac{a+9 b x^2}{144 b^2 \left(a+b x^2\right)^9}","\frac{a}{18 b^2 \left(a+b x^2\right)^9}-\frac{1}{16 b^2 \left(a+b x^2\right)^8}",1,"-1/144*(a + 9*b*x^2)/(b^2*(a + b*x^2)^9)","A",1
204,1,16,16,0.0026277,"\int \frac{x}{\left(a+b x^2\right)^{10}} \, dx","Integrate[x/(a + b*x^2)^10,x]","-\frac{1}{18 b \left(a+b x^2\right)^9}","-\frac{1}{18 b \left(a+b x^2\right)^9}",1,"-1/18*1/(b*(a + b*x^2)^9)","A",1
205,1,120,166,0.1035872,"\int \frac{1}{x \left(a+b x^2\right)^{10}} \, dx","Integrate[1/(x*(a + b*x^2)^10),x]","\frac{\frac{a \left(7129 a^8+41481 a^7 b x^2+120564 a^6 b^2 x^4+210756 a^5 b^3 x^6+236754 a^4 b^4 x^8+173250 a^3 b^5 x^{10}+80220 a^2 b^6 x^{12}+21420 a b^7 x^{14}+2520 b^8 x^{16}\right)}{\left(a+b x^2\right)^9}-2520 \log \left(a+b x^2\right)+5040 \log (x)}{5040 a^{10}}","-\frac{\log \left(a+b x^2\right)}{2 a^{10}}+\frac{\log (x)}{a^{10}}+\frac{1}{2 a^9 \left(a+b x^2\right)}+\frac{1}{4 a^8 \left(a+b x^2\right)^2}+\frac{1}{6 a^7 \left(a+b x^2\right)^3}+\frac{1}{8 a^6 \left(a+b x^2\right)^4}+\frac{1}{10 a^5 \left(a+b x^2\right)^5}+\frac{1}{12 a^4 \left(a+b x^2\right)^6}+\frac{1}{14 a^3 \left(a+b x^2\right)^7}+\frac{1}{16 a^2 \left(a+b x^2\right)^8}+\frac{1}{18 a \left(a+b x^2\right)^9}",1,"((a*(7129*a^8 + 41481*a^7*b*x^2 + 120564*a^6*b^2*x^4 + 210756*a^5*b^3*x^6 + 236754*a^4*b^4*x^8 + 173250*a^3*b^5*x^10 + 80220*a^2*b^6*x^12 + 21420*a*b^7*x^14 + 2520*b^8*x^16))/(a + b*x^2)^9 + 5040*Log[x] - 2520*Log[a + b*x^2])/(5040*a^10)","A",1
206,1,136,184,0.1303728,"\int \frac{1}{x^3 \left(a+b x^2\right)^{10}} \, dx","Integrate[1/(x^3*(a + b*x^2)^10),x]","-\frac{\frac{a \left(252 a^9+7129 a^8 b x^2+41481 a^7 b^2 x^4+120564 a^6 b^3 x^6+210756 a^5 b^4 x^8+236754 a^4 b^5 x^{10}+173250 a^3 b^6 x^{12}+80220 a^2 b^7 x^{14}+21420 a b^8 x^{16}+2520 b^9 x^{18}\right)}{x^2 \left(a+b x^2\right)^9}-2520 b \log \left(a+b x^2\right)+5040 b \log (x)}{504 a^{11}}","\frac{5 b \log \left(a+b x^2\right)}{a^{11}}-\frac{10 b \log (x)}{a^{11}}-\frac{9 b}{2 a^{10} \left(a+b x^2\right)}-\frac{1}{2 a^{10} x^2}-\frac{2 b}{a^9 \left(a+b x^2\right)^2}-\frac{7 b}{6 a^8 \left(a+b x^2\right)^3}-\frac{3 b}{4 a^7 \left(a+b x^2\right)^4}-\frac{b}{2 a^6 \left(a+b x^2\right)^5}-\frac{b}{3 a^5 \left(a+b x^2\right)^6}-\frac{3 b}{14 a^4 \left(a+b x^2\right)^7}-\frac{b}{8 a^3 \left(a+b x^2\right)^8}-\frac{b}{18 a^2 \left(a+b x^2\right)^9}",1,"-1/504*((a*(252*a^9 + 7129*a^8*b*x^2 + 41481*a^7*b^2*x^4 + 120564*a^6*b^3*x^6 + 210756*a^5*b^4*x^8 + 236754*a^4*b^5*x^10 + 173250*a^3*b^6*x^12 + 80220*a^2*b^7*x^14 + 21420*a*b^8*x^16 + 2520*b^9*x^18))/(x^2*(a + b*x^2)^9) + 5040*b*Log[x] - 2520*b*Log[a + b*x^2])/a^11","A",1
207,1,151,217,0.1026816,"\int \frac{1}{x^5 \left(a+b x^2\right)^{10}} \, dx","Integrate[1/(x^5*(a + b*x^2)^10),x]","\frac{\frac{a \left(-252 a^{10}+2772 a^9 b x^2+78419 a^8 b^2 x^4+456291 a^7 b^3 x^6+1326204 a^6 b^4 x^8+2318316 a^5 b^5 x^{10}+2604294 a^4 b^6 x^{12}+1905750 a^3 b^7 x^{14}+882420 a^2 b^8 x^{16}+235620 a b^9 x^{18}+27720 b^{10} x^{20}\right)}{x^4 \left(a+b x^2\right)^9}-27720 b^2 \log \left(a+b x^2\right)+55440 b^2 \log (x)}{1008 a^{12}}","-\frac{55 b^2 \log \left(a+b x^2\right)}{2 a^{12}}+\frac{55 b^2 \log (x)}{a^{12}}+\frac{45 b^2}{2 a^{11} \left(a+b x^2\right)}+\frac{5 b}{a^{11} x^2}+\frac{9 b^2}{a^{10} \left(a+b x^2\right)^2}-\frac{1}{4 a^{10} x^4}+\frac{14 b^2}{3 a^9 \left(a+b x^2\right)^3}+\frac{21 b^2}{8 a^8 \left(a+b x^2\right)^4}+\frac{3 b^2}{2 a^7 \left(a+b x^2\right)^5}+\frac{5 b^2}{6 a^6 \left(a+b x^2\right)^6}+\frac{3 b^2}{7 a^5 \left(a+b x^2\right)^7}+\frac{3 b^2}{16 a^4 \left(a+b x^2\right)^8}+\frac{b^2}{18 a^3 \left(a+b x^2\right)^9}",1,"((a*(-252*a^10 + 2772*a^9*b*x^2 + 78419*a^8*b^2*x^4 + 456291*a^7*b^3*x^6 + 1326204*a^6*b^4*x^8 + 2318316*a^5*b^5*x^10 + 2604294*a^4*b^6*x^12 + 1905750*a^3*b^7*x^14 + 882420*a^2*b^8*x^16 + 235620*a*b^9*x^18 + 27720*b^10*x^20))/(x^4*(a + b*x^2)^9) + 55440*b^2*Log[x] - 27720*b^2*Log[a + b*x^2])/(1008*a^12)","A",1
208,1,162,226,0.1273794,"\int \frac{1}{x^7 \left(a+b x^2\right)^{10}} \, dx","Integrate[1/(x^7*(a + b*x^2)^10),x]","-\frac{-27720 b^3 \log \left(a+b x^2\right)+\frac{a \left(42 a^{11}-252 a^{10} b x^2+2772 a^9 b^2 x^4+78419 a^8 b^3 x^6+456291 a^7 b^4 x^8+1326204 a^6 b^5 x^{10}+2318316 a^5 b^6 x^{12}+2604294 a^4 b^7 x^{14}+1905750 a^3 b^8 x^{16}+882420 a^2 b^9 x^{18}+235620 a b^{10} x^{20}+27720 b^{11} x^{22}\right)}{x^6 \left(a+b x^2\right)^9}+55440 b^3 \log (x)}{252 a^{13}}","\frac{110 b^3 \log \left(a+b x^2\right)}{a^{13}}-\frac{220 b^3 \log (x)}{a^{13}}-\frac{165 b^3}{2 a^{12} \left(a+b x^2\right)}-\frac{55 b^2}{2 a^{12} x^2}-\frac{30 b^3}{a^{11} \left(a+b x^2\right)^2}+\frac{5 b}{2 a^{11} x^4}-\frac{14 b^3}{a^{10} \left(a+b x^2\right)^3}-\frac{1}{6 a^{10} x^6}-\frac{7 b^3}{a^9 \left(a+b x^2\right)^4}-\frac{7 b^3}{2 a^8 \left(a+b x^2\right)^5}-\frac{5 b^3}{3 a^7 \left(a+b x^2\right)^6}-\frac{5 b^3}{7 a^6 \left(a+b x^2\right)^7}-\frac{b^3}{4 a^5 \left(a+b x^2\right)^8}-\frac{b^3}{18 a^4 \left(a+b x^2\right)^9}",1,"-1/252*((a*(42*a^11 - 252*a^10*b*x^2 + 2772*a^9*b^2*x^4 + 78419*a^8*b^3*x^6 + 456291*a^7*b^4*x^8 + 1326204*a^6*b^5*x^10 + 2318316*a^5*b^6*x^12 + 2604294*a^4*b^7*x^14 + 1905750*a^3*b^8*x^16 + 882420*a^2*b^9*x^18 + 235620*a*b^10*x^20 + 27720*b^11*x^22))/(x^6*(a + b*x^2)^9) + 55440*b^3*Log[x] - 27720*b^3*Log[a + b*x^2])/a^13","A",1
209,1,166,231,0.0937845,"\int \frac{x^{24}}{\left(a+b x^2\right)^{10}} \, dx","Integrate[x^24/(a + b*x^2)^10,x]","\frac{\frac{\sqrt{b} x \left(334639305 a^{11}+2900207310 a^{10} b x^2+11110024926 a^9 b^2 x^4+24648575094 a^8 b^3 x^6+34810986496 a^7 b^4 x^8+32314857354 a^6 b^5 x^{10}+19562592546 a^5 b^6 x^{12}+7323998514 a^4 b^7 x^{14}+1469632311 a^3 b^8 x^{16}+94961664 a^2 b^9 x^{18}-4521984 a b^{10} x^{20}+589824 b^{11} x^{22}\right)}{\left(a+b x^2\right)^9}-334639305 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2949120 b^{25/2}}","-\frac{7436429 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{65536 b^{25/2}}+\frac{7436429 a^2 x}{65536 b^{12}}-\frac{7436429 a x^3}{196608 b^{11}}-\frac{1062347 x^7}{65536 b^9 \left(a+b x^2\right)}-\frac{1062347 x^9}{294912 b^8 \left(a+b x^2\right)^2}-\frac{96577 x^{11}}{73728 b^7 \left(a+b x^2\right)^3}-\frac{7429 x^{13}}{12288 b^6 \left(a+b x^2\right)^4}-\frac{7429 x^{15}}{23040 b^5 \left(a+b x^2\right)^5}-\frac{437 x^{17}}{2304 b^4 \left(a+b x^2\right)^6}-\frac{23 x^{19}}{192 b^3 \left(a+b x^2\right)^7}-\frac{23 x^{21}}{288 b^2 \left(a+b x^2\right)^8}-\frac{x^{23}}{18 b \left(a+b x^2\right)^9}+\frac{7436429 x^5}{327680 b^{10}}",1,"((Sqrt[b]*x*(334639305*a^11 + 2900207310*a^10*b*x^2 + 11110024926*a^9*b^2*x^4 + 24648575094*a^8*b^3*x^6 + 34810986496*a^7*b^4*x^8 + 32314857354*a^6*b^5*x^10 + 19562592546*a^5*b^6*x^12 + 7323998514*a^4*b^7*x^14 + 1469632311*a^3*b^8*x^16 + 94961664*a^2*b^9*x^18 - 4521984*a*b^10*x^20 + 589824*b^11*x^22))/(a + b*x^2)^9 - 334639305*a^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2949120*b^(25/2))","A",1
210,1,155,218,0.0797573,"\int \frac{x^{22}}{\left(a+b x^2\right)^{10}} \, dx","Integrate[x^22/(a + b*x^2)^10,x]","\frac{14549535 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)+\frac{\sqrt{b} x \left(-14549535 a^{10}-126095970 a^9 b x^2-483044562 a^8 b^2 x^4-1071677178 a^7 b^3 x^6-1513521152 a^6 b^4 x^8-1404993798 a^5 b^5 x^{10}-850547502 a^4 b^6 x^{12}-318434718 a^3 b^7 x^{14}-63897057 a^2 b^8 x^{16}-4128768 a b^9 x^{18}+196608 b^{10} x^{20}\right)}{\left(a+b x^2\right)^9}}{589824 b^{23/2}}","\frac{1616615 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{65536 b^{23/2}}-\frac{1616615 a x}{65536 b^{11}}-\frac{323323 x^5}{65536 b^9 \left(a+b x^2\right)}-\frac{46189 x^7}{32768 b^8 \left(a+b x^2\right)^2}-\frac{46189 x^9}{73728 b^7 \left(a+b x^2\right)^3}-\frac{4199 x^{11}}{12288 b^6 \left(a+b x^2\right)^4}-\frac{323 x^{13}}{1536 b^5 \left(a+b x^2\right)^5}-\frac{323 x^{15}}{2304 b^4 \left(a+b x^2\right)^6}-\frac{19 x^{17}}{192 b^3 \left(a+b x^2\right)^7}-\frac{7 x^{19}}{96 b^2 \left(a+b x^2\right)^8}-\frac{x^{21}}{18 b \left(a+b x^2\right)^9}+\frac{1616615 x^3}{196608 b^{10}}",1,"((Sqrt[b]*x*(-14549535*a^10 - 126095970*a^9*b*x^2 - 483044562*a^8*b^2*x^4 - 1071677178*a^7*b^3*x^6 - 1513521152*a^6*b^4*x^8 - 1404993798*a^5*b^5*x^10 - 850547502*a^4*b^6*x^12 - 318434718*a^3*b^7*x^14 - 63897057*a^2*b^8*x^16 - 4128768*a*b^9*x^18 + 196608*b^10*x^20))/(a + b*x^2)^9 + 14549535*a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(589824*b^(23/2))","A",1
211,1,144,207,0.0759631,"\int \frac{x^{20}}{\left(a+b x^2\right)^{10}} \, dx","Integrate[x^20/(a + b*x^2)^10,x]","\frac{\frac{\sqrt{b} x \left(14549535 a^9+126095970 a^8 b x^2+483044562 a^7 b^2 x^4+1071677178 a^6 b^3 x^6+1513521152 a^5 b^4 x^8+1404993798 a^4 b^5 x^{10}+850547502 a^3 b^6 x^{12}+318434718 a^2 b^7 x^{14}+63897057 a b^8 x^{16}+4128768 b^9 x^{18}\right)}{\left(a+b x^2\right)^9}-14549535 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{4128768 b^{21/2}}","-\frac{230945 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{65536 b^{21/2}}-\frac{230945 x^3}{196608 b^9 \left(a+b x^2\right)}-\frac{46189 x^5}{98304 b^8 \left(a+b x^2\right)^2}-\frac{46189 x^7}{172032 b^7 \left(a+b x^2\right)^3}-\frac{46189 x^9}{258048 b^6 \left(a+b x^2\right)^4}-\frac{4199 x^{11}}{32256 b^5 \left(a+b x^2\right)^5}-\frac{1615 x^{13}}{16128 b^4 \left(a+b x^2\right)^6}-\frac{323 x^{15}}{4032 b^3 \left(a+b x^2\right)^7}-\frac{19 x^{17}}{288 b^2 \left(a+b x^2\right)^8}-\frac{x^{19}}{18 b \left(a+b x^2\right)^9}+\frac{230945 x}{65536 b^{10}}",1,"((Sqrt[b]*x*(14549535*a^9 + 126095970*a^8*b*x^2 + 483044562*a^7*b^2*x^4 + 1071677178*a^6*b^3*x^6 + 1513521152*a^5*b^4*x^8 + 1404993798*a^4*b^5*x^10 + 850547502*a^3*b^6*x^12 + 318434718*a^2*b^7*x^14 + 63897057*a*b^8*x^16 + 4128768*b^9*x^18))/(a + b*x^2)^9 - 14549535*Sqrt[a]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(4128768*b^(21/2))","A",1
212,1,134,197,0.0780609,"\int \frac{x^{18}}{\left(a+b x^2\right)^{10}} \, dx","Integrate[x^18/(a + b*x^2)^10,x]","\frac{\frac{765765 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a}}-\frac{\sqrt{b} x \left(765765 a^8+6636630 a^7 b x^2+25423398 a^6 b^2 x^4+56404062 a^5 b^3 x^6+79659008 a^4 b^4 x^8+73947042 a^3 b^5 x^{10}+44765658 a^2 b^6 x^{12}+16759722 a b^7 x^{14}+3363003 b^8 x^{16}\right)}{\left(a+b x^2\right)^9}}{4128768 b^{19/2}}","\frac{12155 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{65536 \sqrt{a} b^{19/2}}-\frac{12155 x}{65536 b^9 \left(a+b x^2\right)}-\frac{12155 x^3}{98304 b^8 \left(a+b x^2\right)^2}-\frac{2431 x^5}{24576 b^7 \left(a+b x^2\right)^3}-\frac{2431 x^7}{28672 b^6 \left(a+b x^2\right)^4}-\frac{2431 x^9}{32256 b^5 \left(a+b x^2\right)^5}-\frac{1105 x^{11}}{16128 b^4 \left(a+b x^2\right)^6}-\frac{85 x^{13}}{1344 b^3 \left(a+b x^2\right)^7}-\frac{17 x^{15}}{288 b^2 \left(a+b x^2\right)^8}-\frac{x^{17}}{18 b \left(a+b x^2\right)^9}",1,"(-((Sqrt[b]*x*(765765*a^8 + 6636630*a^7*b*x^2 + 25423398*a^6*b^2*x^4 + 56404062*a^5*b^3*x^6 + 79659008*a^4*b^4*x^8 + 73947042*a^3*b^5*x^10 + 44765658*a^2*b^6*x^12 + 16759722*a*b^7*x^14 + 3363003*b^8*x^16))/(a + b*x^2)^9) + (765765*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[a])/(4128768*b^(19/2))","A",1
213,1,138,198,0.072623,"\int \frac{x^{16}}{\left(a+b x^2\right)^{10}} \, dx","Integrate[x^16/(a + b*x^2)^10,x]","\frac{\frac{\sqrt{a} \sqrt{b} x \left(-45045 a^8-390390 a^7 b x^2-1495494 a^6 b^2 x^4-3317886 a^5 b^3 x^6-4685824 a^4 b^4 x^8-4349826 a^3 b^5 x^{10}-2633274 a^2 b^6 x^{12}-985866 a b^7 x^{14}+45045 b^8 x^{16}\right)}{\left(a+b x^2\right)^9}+45045 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{4128768 a^{3/2} b^{17/2}}","\frac{715 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{65536 a^{3/2} b^{17/2}}+\frac{715 x}{65536 a b^8 \left(a+b x^2\right)}-\frac{715 x}{32768 b^8 \left(a+b x^2\right)^2}-\frac{715 x^3}{24576 b^7 \left(a+b x^2\right)^3}-\frac{143 x^5}{4096 b^6 \left(a+b x^2\right)^4}-\frac{143 x^7}{3584 b^5 \left(a+b x^2\right)^5}-\frac{715 x^9}{16128 b^4 \left(a+b x^2\right)^6}-\frac{65 x^{11}}{1344 b^3 \left(a+b x^2\right)^7}-\frac{5 x^{13}}{96 b^2 \left(a+b x^2\right)^8}-\frac{x^{15}}{18 b \left(a+b x^2\right)^9}",1,"((Sqrt[a]*Sqrt[b]*x*(-45045*a^8 - 390390*a^7*b*x^2 - 1495494*a^6*b^2*x^4 - 3317886*a^5*b^3*x^6 - 4685824*a^4*b^4*x^8 - 4349826*a^3*b^5*x^10 - 2633274*a^2*b^6*x^12 - 985866*a*b^7*x^14 + 45045*b^8*x^16))/(a + b*x^2)^9 + 45045*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(4128768*a^(3/2)*b^(17/2))","A",1
214,1,138,199,0.0596733,"\int \frac{x^{14}}{\left(a+b x^2\right)^{10}} \, dx","Integrate[x^14/(a + b*x^2)^10,x]","\frac{\frac{\sqrt{a} \sqrt{b} x \left(-45045 a^8-390390 a^7 b x^2-1495494 a^6 b^2 x^4-3317886 a^5 b^3 x^6-4685824 a^4 b^4 x^8-4349826 a^3 b^5 x^{10}-2633274 a^2 b^6 x^{12}+390390 a b^7 x^{14}+45045 b^8 x^{16}\right)}{\left(a+b x^2\right)^9}+45045 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{20643840 a^{5/2} b^{15/2}}","\frac{143 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{65536 a^{5/2} b^{15/2}}+\frac{143 x}{65536 a^2 b^7 \left(a+b x^2\right)}+\frac{143 x}{98304 a b^7 \left(a+b x^2\right)^2}-\frac{143 x}{24576 b^7 \left(a+b x^2\right)^3}-\frac{143 x^3}{12288 b^6 \left(a+b x^2\right)^4}-\frac{143 x^5}{7680 b^5 \left(a+b x^2\right)^5}-\frac{143 x^7}{5376 b^4 \left(a+b x^2\right)^6}-\frac{143 x^9}{4032 b^3 \left(a+b x^2\right)^7}-\frac{13 x^{11}}{288 b^2 \left(a+b x^2\right)^8}-\frac{x^{13}}{18 b \left(a+b x^2\right)^9}",1,"((Sqrt[a]*Sqrt[b]*x*(-45045*a^8 - 390390*a^7*b*x^2 - 1495494*a^6*b^2*x^4 - 3317886*a^5*b^3*x^6 - 4685824*a^4*b^4*x^8 - 4349826*a^3*b^5*x^10 - 2633274*a^2*b^6*x^12 + 390390*a*b^7*x^14 + 45045*b^8*x^16))/(a + b*x^2)^9 + 45045*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(20643840*a^(5/2)*b^(15/2))","A",1
215,1,138,200,0.0728368,"\int \frac{x^{12}}{\left(a+b x^2\right)^{10}} \, dx","Integrate[x^12/(a + b*x^2)^10,x]","\frac{\frac{\sqrt{a} \sqrt{b} x \left(-3465 a^8-30030 a^7 b x^2-115038 a^6 b^2 x^4-255222 a^5 b^3 x^6-360448 a^4 b^4 x^8-334602 a^3 b^5 x^{10}+115038 a^2 b^6 x^{12}+30030 a b^7 x^{14}+3465 b^8 x^{16}\right)}{\left(a+b x^2\right)^9}+3465 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{4128768 a^{7/2} b^{13/2}}","\frac{55 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{65536 a^{7/2} b^{13/2}}+\frac{55 x}{65536 a^3 b^6 \left(a+b x^2\right)}+\frac{55 x}{98304 a^2 b^6 \left(a+b x^2\right)^2}+\frac{11 x}{24576 a b^6 \left(a+b x^2\right)^3}-\frac{11 x}{4096 b^6 \left(a+b x^2\right)^4}-\frac{11 x^3}{1536 b^5 \left(a+b x^2\right)^5}-\frac{11 x^5}{768 b^4 \left(a+b x^2\right)^6}-\frac{11 x^7}{448 b^3 \left(a+b x^2\right)^7}-\frac{11 x^9}{288 b^2 \left(a+b x^2\right)^8}-\frac{x^{11}}{18 b \left(a+b x^2\right)^9}",1,"((Sqrt[a]*Sqrt[b]*x*(-3465*a^8 - 30030*a^7*b*x^2 - 115038*a^6*b^2*x^4 - 255222*a^5*b^3*x^6 - 360448*a^4*b^4*x^8 - 334602*a^3*b^5*x^10 + 115038*a^2*b^6*x^12 + 30030*a*b^7*x^14 + 3465*b^8*x^16))/(a + b*x^2)^9 + 3465*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(4128768*a^(7/2)*b^(13/2))","A",1
216,1,138,201,0.0715696,"\int \frac{x^{10}}{\left(a+b x^2\right)^{10}} \, dx","Integrate[x^10/(a + b*x^2)^10,x]","\frac{\frac{\sqrt{a} \sqrt{b} x \left(-315 a^8-2730 a^7 b x^2-10458 a^6 b^2 x^4-23202 a^5 b^3 x^6-32768 a^4 b^4 x^8+23202 a^3 b^5 x^{10}+10458 a^2 b^6 x^{12}+2730 a b^7 x^{14}+315 b^8 x^{16}\right)}{\left(a+b x^2\right)^9}+315 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{589824 a^{9/2} b^{11/2}}","\frac{35 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{65536 a^{9/2} b^{11/2}}+\frac{35 x}{65536 a^4 b^5 \left(a+b x^2\right)}+\frac{35 x}{98304 a^3 b^5 \left(a+b x^2\right)^2}+\frac{7 x}{24576 a^2 b^5 \left(a+b x^2\right)^3}+\frac{x}{4096 a b^5 \left(a+b x^2\right)^4}-\frac{x}{512 b^5 \left(a+b x^2\right)^5}-\frac{5 x^3}{768 b^4 \left(a+b x^2\right)^6}-\frac{x^5}{64 b^3 \left(a+b x^2\right)^7}-\frac{x^7}{32 b^2 \left(a+b x^2\right)^8}-\frac{x^9}{18 b \left(a+b x^2\right)^9}",1,"((Sqrt[a]*Sqrt[b]*x*(-315*a^8 - 2730*a^7*b*x^2 - 10458*a^6*b^2*x^4 - 23202*a^5*b^3*x^6 - 32768*a^4*b^4*x^8 + 23202*a^3*b^5*x^10 + 10458*a^2*b^6*x^12 + 2730*a*b^7*x^14 + 315*b^8*x^16))/(a + b*x^2)^9 + 315*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(589824*a^(9/2)*b^(11/2))","A",1
217,1,138,202,0.0623329,"\int \frac{x^8}{\left(a+b x^2\right)^{10}} \, dx","Integrate[x^8/(a + b*x^2)^10,x]","\frac{\frac{\sqrt{a} \sqrt{b} x \left(-315 a^8-2730 a^7 b x^2-10458 a^6 b^2 x^4-23202 a^5 b^3 x^6+32768 a^4 b^4 x^8+23202 a^3 b^5 x^{10}+10458 a^2 b^6 x^{12}+2730 a b^7 x^{14}+315 b^8 x^{16}\right)}{\left(a+b x^2\right)^9}+315 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{589824 a^{11/2} b^{9/2}}","\frac{35 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{65536 a^{11/2} b^{9/2}}+\frac{35 x}{65536 a^5 b^4 \left(a+b x^2\right)}+\frac{35 x}{98304 a^4 b^4 \left(a+b x^2\right)^2}+\frac{7 x}{24576 a^3 b^4 \left(a+b x^2\right)^3}+\frac{x}{4096 a^2 b^4 \left(a+b x^2\right)^4}+\frac{x}{4608 a b^4 \left(a+b x^2\right)^5}-\frac{5 x}{2304 b^4 \left(a+b x^2\right)^6}-\frac{5 x^3}{576 b^3 \left(a+b x^2\right)^7}-\frac{7 x^5}{288 b^2 \left(a+b x^2\right)^8}-\frac{x^7}{18 b \left(a+b x^2\right)^9}",1,"((Sqrt[a]*Sqrt[b]*x*(-315*a^8 - 2730*a^7*b*x^2 - 10458*a^6*b^2*x^4 - 23202*a^5*b^3*x^6 + 32768*a^4*b^4*x^8 + 23202*a^3*b^5*x^10 + 10458*a^2*b^6*x^12 + 2730*a*b^7*x^14 + 315*b^8*x^16))/(a + b*x^2)^9 + 315*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(589824*a^(11/2)*b^(9/2))","A",1
218,1,138,203,0.0658826,"\int \frac{x^6}{\left(a+b x^2\right)^{10}} \, dx","Integrate[x^6/(a + b*x^2)^10,x]","\frac{\frac{\sqrt{a} \sqrt{b} x \left(-3465 a^8-30030 a^7 b x^2-115038 a^6 b^2 x^4+334602 a^5 b^3 x^6+360448 a^4 b^4 x^8+255222 a^3 b^5 x^{10}+115038 a^2 b^6 x^{12}+30030 a b^7 x^{14}+3465 b^8 x^{16}\right)}{\left(a+b x^2\right)^9}+3465 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{4128768 a^{13/2} b^{7/2}}","\frac{55 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{65536 a^{13/2} b^{7/2}}+\frac{55 x}{65536 a^6 b^3 \left(a+b x^2\right)}+\frac{55 x}{98304 a^5 b^3 \left(a+b x^2\right)^2}+\frac{11 x}{24576 a^4 b^3 \left(a+b x^2\right)^3}+\frac{11 x}{28672 a^3 b^3 \left(a+b x^2\right)^4}+\frac{11 x}{32256 a^2 b^3 \left(a+b x^2\right)^5}+\frac{5 x}{16128 a b^3 \left(a+b x^2\right)^6}-\frac{5 x}{1344 b^3 \left(a+b x^2\right)^7}-\frac{5 x^3}{288 b^2 \left(a+b x^2\right)^8}-\frac{x^5}{18 b \left(a+b x^2\right)^9}",1,"((Sqrt[a]*Sqrt[b]*x*(-3465*a^8 - 30030*a^7*b*x^2 - 115038*a^6*b^2*x^4 + 334602*a^5*b^3*x^6 + 360448*a^4*b^4*x^8 + 255222*a^3*b^5*x^10 + 115038*a^2*b^6*x^12 + 30030*a*b^7*x^14 + 3465*b^8*x^16))/(a + b*x^2)^9 + 3465*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(4128768*a^(13/2)*b^(7/2))","A",1
219,1,138,204,0.063796,"\int \frac{x^4}{\left(a+b x^2\right)^{10}} \, dx","Integrate[x^4/(a + b*x^2)^10,x]","\frac{\frac{\sqrt{a} \sqrt{b} x \left(-45045 a^8-390390 a^7 b x^2+2633274 a^6 b^2 x^4+4349826 a^5 b^3 x^6+4685824 a^4 b^4 x^8+3317886 a^3 b^5 x^{10}+1495494 a^2 b^6 x^{12}+390390 a b^7 x^{14}+45045 b^8 x^{16}\right)}{\left(a+b x^2\right)^9}+45045 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{20643840 a^{15/2} b^{5/2}}","\frac{143 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{65536 a^{15/2} b^{5/2}}+\frac{143 x}{65536 a^7 b^2 \left(a+b x^2\right)}+\frac{143 x}{98304 a^6 b^2 \left(a+b x^2\right)^2}+\frac{143 x}{122880 a^5 b^2 \left(a+b x^2\right)^3}+\frac{143 x}{143360 a^4 b^2 \left(a+b x^2\right)^4}+\frac{143 x}{161280 a^3 b^2 \left(a+b x^2\right)^5}+\frac{13 x}{16128 a^2 b^2 \left(a+b x^2\right)^6}+\frac{x}{1344 a b^2 \left(a+b x^2\right)^7}-\frac{x}{96 b^2 \left(a+b x^2\right)^8}-\frac{x^3}{18 b \left(a+b x^2\right)^9}",1,"((Sqrt[a]*Sqrt[b]*x*(-45045*a^8 - 390390*a^7*b*x^2 + 2633274*a^6*b^2*x^4 + 4349826*a^5*b^3*x^6 + 4685824*a^4*b^4*x^8 + 3317886*a^3*b^5*x^10 + 1495494*a^2*b^6*x^12 + 390390*a*b^7*x^14 + 45045*b^8*x^16))/(a + b*x^2)^9 + 45045*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(20643840*a^(15/2)*b^(5/2))","A",1
220,1,138,205,0.057749,"\int \frac{x^2}{\left(a+b x^2\right)^{10}} \, dx","Integrate[x^2/(a + b*x^2)^10,x]","\frac{\frac{\sqrt{a} \sqrt{b} x \left(-45045 a^8+985866 a^7 b x^2+2633274 a^6 b^2 x^4+4349826 a^5 b^3 x^6+4685824 a^4 b^4 x^8+3317886 a^3 b^5 x^{10}+1495494 a^2 b^6 x^{12}+390390 a b^7 x^{14}+45045 b^8 x^{16}\right)}{\left(a+b x^2\right)^9}+45045 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{4128768 a^{17/2} b^{3/2}}","\frac{715 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{65536 a^{17/2} b^{3/2}}+\frac{715 x}{65536 a^8 b \left(a+b x^2\right)}+\frac{715 x}{98304 a^7 b \left(a+b x^2\right)^2}+\frac{143 x}{24576 a^6 b \left(a+b x^2\right)^3}+\frac{143 x}{28672 a^5 b \left(a+b x^2\right)^4}+\frac{143 x}{32256 a^4 b \left(a+b x^2\right)^5}+\frac{65 x}{16128 a^3 b \left(a+b x^2\right)^6}+\frac{5 x}{1344 a^2 b \left(a+b x^2\right)^7}+\frac{x}{288 a b \left(a+b x^2\right)^8}-\frac{x}{18 b \left(a+b x^2\right)^9}",1,"((Sqrt[a]*Sqrt[b]*x*(-45045*a^8 + 985866*a^7*b*x^2 + 2633274*a^6*b^2*x^4 + 4349826*a^5*b^3*x^6 + 4685824*a^4*b^4*x^8 + 3317886*a^3*b^5*x^10 + 1495494*a^2*b^6*x^12 + 390390*a*b^7*x^14 + 45045*b^8*x^16))/(a + b*x^2)^9 + 45045*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(4128768*a^(17/2)*b^(3/2))","A",1
221,1,131,181,0.1048038,"\int \frac{1}{\left(a+b x^2\right)^{10}} \, dx","Integrate[(a + b*x^2)^(-10),x]","\frac{\frac{765765 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{19/2} \sqrt{b}}+\frac{3363003 a^8 x+16759722 a^7 b x^3+44765658 a^6 b^2 x^5+73947042 a^5 b^3 x^7+79659008 a^4 b^4 x^9+56404062 a^3 b^5 x^{11}+25423398 a^2 b^6 x^{13}+6636630 a b^7 x^{15}+765765 b^8 x^{17}}{a^9 \left(a+b x^2\right)^9}}{4128768}","\frac{12155 \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{65536 a^{19/2} \sqrt{b}}+\frac{12155 x}{65536 a^9 \left(a+b x^2\right)}+\frac{12155 x}{98304 a^8 \left(a+b x^2\right)^2}+\frac{2431 x}{24576 a^7 \left(a+b x^2\right)^3}+\frac{2431 x}{28672 a^6 \left(a+b x^2\right)^4}+\frac{2431 x}{32256 a^5 \left(a+b x^2\right)^5}+\frac{1105 x}{16128 a^4 \left(a+b x^2\right)^6}+\frac{85 x}{1344 a^3 \left(a+b x^2\right)^7}+\frac{17 x}{288 a^2 \left(a+b x^2\right)^8}+\frac{x}{18 a \left(a+b x^2\right)^9}",1,"((3363003*a^8*x + 16759722*a^7*b*x^3 + 44765658*a^6*b^2*x^5 + 73947042*a^5*b^3*x^7 + 79659008*a^4*b^4*x^9 + 56404062*a^3*b^5*x^11 + 25423398*a^2*b^6*x^13 + 6636630*a*b^7*x^15 + 765765*b^8*x^17)/(a^9*(a + b*x^2)^9) + (765765*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(19/2)*Sqrt[b]))/4128768","A",1
222,1,147,209,0.0993606,"\int \frac{1}{x^2 \left(a+b x^2\right)^{10}} \, dx","Integrate[1/(x^2*(a + b*x^2)^10),x]","\frac{-\frac{\sqrt{a} \left(4128768 a^9+63897057 a^8 b x^2+318434718 a^7 b^2 x^4+850547502 a^6 b^3 x^6+1404993798 a^5 b^4 x^8+1513521152 a^4 b^5 x^{10}+1071677178 a^3 b^6 x^{12}+483044562 a^2 b^7 x^{14}+126095970 a b^8 x^{16}+14549535 b^9 x^{18}\right)}{x \left(a+b x^2\right)^9}-14549535 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{4128768 a^{21/2}}","-\frac{230945 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{65536 a^{21/2}}-\frac{230945}{65536 a^{10} x}+\frac{230945}{196608 a^9 x \left(a+b x^2\right)}+\frac{46189}{98304 a^8 x \left(a+b x^2\right)^2}+\frac{46189}{172032 a^7 x \left(a+b x^2\right)^3}+\frac{46189}{258048 a^6 x \left(a+b x^2\right)^4}+\frac{4199}{32256 a^5 x \left(a+b x^2\right)^5}+\frac{1615}{16128 a^4 x \left(a+b x^2\right)^6}+\frac{323}{4032 a^3 x \left(a+b x^2\right)^7}+\frac{19}{288 a^2 x \left(a+b x^2\right)^8}+\frac{1}{18 a x \left(a+b x^2\right)^9}",1,"(-((Sqrt[a]*(4128768*a^9 + 63897057*a^8*b*x^2 + 318434718*a^7*b^2*x^4 + 850547502*a^6*b^3*x^6 + 1404993798*a^5*b^4*x^8 + 1513521152*a^4*b^5*x^10 + 1071677178*a^3*b^6*x^12 + 483044562*a^2*b^7*x^14 + 126095970*a*b^8*x^16 + 14549535*b^9*x^18))/(x*(a + b*x^2)^9)) - 14549535*Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(4128768*a^(21/2))","A",1
223,1,157,220,0.0877388,"\int \frac{1}{x^4 \left(a+b x^2\right)^{10}} \, dx","Integrate[1/(x^4*(a + b*x^2)^10),x]","\frac{\frac{\sqrt{a} \left(-196608 a^{10}+4128768 a^9 b x^2+63897057 a^8 b^2 x^4+318434718 a^7 b^3 x^6+850547502 a^6 b^4 x^8+1404993798 a^5 b^5 x^{10}+1513521152 a^4 b^6 x^{12}+1071677178 a^3 b^7 x^{14}+483044562 a^2 b^8 x^{16}+126095970 a b^9 x^{18}+14549535 b^{10} x^{20}\right)}{x^3 \left(a+b x^2\right)^9}+14549535 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{589824 a^{23/2}}","\frac{1616615 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{65536 a^{23/2}}+\frac{1616615 b}{65536 a^{11} x}-\frac{1616615}{196608 a^{10} x^3}+\frac{323323}{65536 a^9 x^3 \left(a+b x^2\right)}+\frac{46189}{32768 a^8 x^3 \left(a+b x^2\right)^2}+\frac{46189}{73728 a^7 x^3 \left(a+b x^2\right)^3}+\frac{4199}{12288 a^6 x^3 \left(a+b x^2\right)^4}+\frac{323}{1536 a^5 x^3 \left(a+b x^2\right)^5}+\frac{323}{2304 a^4 x^3 \left(a+b x^2\right)^6}+\frac{19}{192 a^3 x^3 \left(a+b x^2\right)^7}+\frac{7}{96 a^2 x^3 \left(a+b x^2\right)^8}+\frac{1}{18 a x^3 \left(a+b x^2\right)^9}",1,"((Sqrt[a]*(-196608*a^10 + 4128768*a^9*b*x^2 + 63897057*a^8*b^2*x^4 + 318434718*a^7*b^3*x^6 + 850547502*a^6*b^4*x^8 + 1404993798*a^5*b^5*x^10 + 1513521152*a^4*b^6*x^12 + 1071677178*a^3*b^7*x^14 + 483044562*a^2*b^8*x^16 + 126095970*a*b^9*x^18 + 14549535*b^10*x^20))/(x^3*(a + b*x^2)^9) + 14549535*b^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(589824*a^(23/2))","A",1
224,1,169,233,0.0928182,"\int \frac{1}{x^6 \left(a+b x^2\right)^{10}} \, dx","Integrate[1/(x^6*(a + b*x^2)^10),x]","\frac{-334639305 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)-\frac{\sqrt{a} \left(589824 a^{11}-4521984 a^{10} b x^2+94961664 a^9 b^2 x^4+1469632311 a^8 b^3 x^6+7323998514 a^7 b^4 x^8+19562592546 a^6 b^5 x^{10}+32314857354 a^5 b^6 x^{12}+34810986496 a^4 b^7 x^{14}+24648575094 a^3 b^8 x^{16}+11110024926 a^2 b^9 x^{18}+2900207310 a b^{10} x^{20}+334639305 b^{11} x^{22}\right)}{x^5 \left(a+b x^2\right)^9}}{2949120 a^{25/2}}","-\frac{7436429 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{65536 a^{25/2}}-\frac{7436429 b^2}{65536 a^{12} x}+\frac{7436429 b}{196608 a^{11} x^3}-\frac{7436429}{327680 a^{10} x^5}+\frac{1062347}{65536 a^9 x^5 \left(a+b x^2\right)}+\frac{1062347}{294912 a^8 x^5 \left(a+b x^2\right)^2}+\frac{96577}{73728 a^7 x^5 \left(a+b x^2\right)^3}+\frac{7429}{12288 a^6 x^5 \left(a+b x^2\right)^4}+\frac{7429}{23040 a^5 x^5 \left(a+b x^2\right)^5}+\frac{437}{2304 a^4 x^5 \left(a+b x^2\right)^6}+\frac{23}{192 a^3 x^5 \left(a+b x^2\right)^7}+\frac{23}{288 a^2 x^5 \left(a+b x^2\right)^8}+\frac{1}{18 a x^5 \left(a+b x^2\right)^9}",1,"(-((Sqrt[a]*(589824*a^11 - 4521984*a^10*b*x^2 + 94961664*a^9*b^2*x^4 + 1469632311*a^8*b^3*x^6 + 7323998514*a^7*b^4*x^8 + 19562592546*a^6*b^5*x^10 + 32314857354*a^5*b^6*x^12 + 34810986496*a^4*b^7*x^14 + 24648575094*a^3*b^8*x^16 + 11110024926*a^2*b^9*x^18 + 2900207310*a*b^10*x^20 + 334639305*b^11*x^22))/(x^5*(a + b*x^2)^9)) - 334639305*b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2949120*a^(25/2))","A",1
225,1,28,28,0.0057663,"\int \frac{x^3}{a-b x^2} \, dx","Integrate[x^3/(a - b*x^2),x]","-\frac{a \log \left(a-b x^2\right)}{2 b^2}-\frac{x^2}{2 b}","-\frac{a \log \left(a-b x^2\right)}{2 b^2}-\frac{x^2}{2 b}",1,"-1/2*x^2/b - (a*Log[a - b*x^2])/(2*b^2)","A",1
226,1,31,31,0.0097161,"\int \frac{x^2}{a-b x^2} \, dx","Integrate[x^2/(a - b*x^2),x]","\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{3/2}}-\frac{x}{b}","\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{3/2}}-\frac{x}{b}",1,"-(x/b) + (Sqrt[a]*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/b^(3/2)","A",1
227,1,16,16,0.0025299,"\int \frac{x}{a-b x^2} \, dx","Integrate[x/(a - b*x^2),x]","-\frac{\log \left(a-b x^2\right)}{2 b}","-\frac{\log \left(a-b x^2\right)}{2 b}",1,"-1/2*Log[a - b*x^2]/b","A",1
228,1,24,24,0.0042142,"\int \frac{1}{a-b x^2} \, dx","Integrate[(a - b*x^2)^(-1),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}",1,"ArcTanh[(Sqrt[b]*x)/Sqrt[a]]/(Sqrt[a]*Sqrt[b])","A",1
229,1,23,23,0.0064004,"\int \frac{1}{x \left(a-b x^2\right)} \, dx","Integrate[1/(x*(a - b*x^2)),x]","\frac{\log (x)}{a}-\frac{\log \left(a-b x^2\right)}{2 a}","\frac{\log (x)}{a}-\frac{\log \left(a-b x^2\right)}{2 a}",1,"Log[x]/a - Log[a - b*x^2]/(2*a)","A",1
230,1,33,33,0.011574,"\int \frac{1}{x^2 \left(a-b x^2\right)} \, dx","Integrate[1/(x^2*(a - b*x^2)),x]","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2}}-\frac{1}{a x}","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{a^{3/2}}-\frac{1}{a x}",1,"-(1/(a*x)) + (Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/a^(3/2)","A",1
231,1,35,35,0.0081649,"\int \frac{1}{x^3 \left(a-b x^2\right)} \, dx","Integrate[1/(x^3*(a - b*x^2)),x]","-\frac{b \log \left(a-b x^2\right)}{2 a^2}+\frac{b \log (x)}{a^2}-\frac{1}{2 a x^2}","-\frac{b \log \left(a-b x^2\right)}{2 a^2}+\frac{b \log (x)}{a^2}-\frac{1}{2 a x^2}",1,"-1/2*1/(a*x^2) + (b*Log[x])/a^2 - (b*Log[a - b*x^2])/(2*a^2)","A",1
232,1,29,35,0.0130448,"\int \frac{x^3}{\left(a-b x^2\right)^2} \, dx","Integrate[x^3/(a - b*x^2)^2,x]","\frac{\frac{a}{a-b x^2}+\log \left(a-b x^2\right)}{2 b^2}","\frac{a}{2 b^2 \left(a-b x^2\right)}+\frac{\log \left(a-b x^2\right)}{2 b^2}",1,"(a/(a - b*x^2) + Log[a - b*x^2])/(2*b^2)","A",1
233,1,47,46,0.0301686,"\int \frac{x^2}{\left(a-b x^2\right)^2} \, dx","Integrate[x^2/(a - b*x^2)^2,x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{a} b^{3/2}}-\frac{x}{2 b \left(b x^2-a\right)}","\frac{x}{2 b \left(a-b x^2\right)}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 \sqrt{a} b^{3/2}}",1,"-1/2*x/(b*(-a + b*x^2)) - ArcTanh[(Sqrt[b]*x)/Sqrt[a]]/(2*Sqrt[a]*b^(3/2))","A",1
234,1,17,17,0.0022268,"\int \frac{x}{\left(a-b x^2\right)^2} \, dx","Integrate[x/(a - b*x^2)^2,x]","\frac{1}{2 b \left(a-b x^2\right)}","\frac{1}{2 b \left(a-b x^2\right)}",1,"1/(2*b*(a - b*x^2))","A",1
235,1,47,46,0.0167322,"\int \frac{1}{\left(a-b x^2\right)^2} \, dx","Integrate[(a - b*x^2)^(-2),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} \sqrt{b}}-\frac{x}{2 a \left(b x^2-a\right)}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{3/2} \sqrt{b}}+\frac{x}{2 a \left(a-b x^2\right)}",1,"-1/2*x/(a*(-a + b*x^2)) + ArcTanh[(Sqrt[b]*x)/Sqrt[a]]/(2*a^(3/2)*Sqrt[b])","A",1
236,1,35,40,0.0198132,"\int \frac{1}{x \left(a-b x^2\right)^2} \, dx","Integrate[1/(x*(a - b*x^2)^2),x]","\frac{\frac{a}{a-b x^2}-\log \left(a-b x^2\right)+2 \log (x)}{2 a^2}","-\frac{\log \left(a-b x^2\right)}{2 a^2}+\frac{\log (x)}{a^2}+\frac{1}{2 a \left(a-b x^2\right)}",1,"(a/(a - b*x^2) + 2*Log[x] - Log[a - b*x^2])/(2*a^2)","A",1
237,1,56,58,0.0362567,"\int \frac{1}{x^2 \left(a-b x^2\right)^2} \, dx","Integrate[1/(x^2*(a - b*x^2)^2),x]","\frac{3 \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{5/2}}-\frac{b x}{2 a^2 \left(b x^2-a\right)}-\frac{1}{a^2 x}","\frac{3 \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 a^{5/2}}-\frac{3}{2 a^2 x}+\frac{1}{2 a x \left(a-b x^2\right)}",1,"-(1/(a^2*x)) - (b*x)/(2*a^2*(-a + b*x^2)) + (3*Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(5/2))","A",1
238,1,44,52,0.0333013,"\int \frac{1}{x^3 \left(a-b x^2\right)^2} \, dx","Integrate[1/(x^3*(a - b*x^2)^2),x]","\frac{\frac{a b}{a-b x^2}-2 b \log \left(a-b x^2\right)-\frac{a}{x^2}+4 b \log (x)}{2 a^3}","-\frac{b \log \left(a-b x^2\right)}{a^3}+\frac{2 b \log (x)}{a^3}+\frac{b}{2 a^2 \left(a-b x^2\right)}-\frac{1}{2 a^2 x^2}",1,"(-(a/x^2) + (a*b)/(a - b*x^2) + 4*b*Log[x] - 2*b*Log[a - b*x^2])/(2*a^3)","A",1
239,1,25,20,0.0093184,"\int \frac{x^3}{\left(a-b x^2\right)^3} \, dx","Integrate[x^3/(a - b*x^2)^3,x]","-\frac{a-2 b x^2}{4 b^2 \left(a-b x^2\right)^2}","\frac{x^4}{4 a \left(a-b x^2\right)^2}",1,"-1/4*(a - 2*b*x^2)/(b^2*(a - b*x^2)^2)","A",1
240,1,56,67,0.0310098,"\int \frac{x^2}{\left(a-b x^2\right)^3} \, dx","Integrate[x^2/(a - b*x^2)^3,x]","\frac{x \left(a+b x^2\right)}{8 a b \left(a-b x^2\right)^2}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{3/2} b^{3/2}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{3/2} b^{3/2}}-\frac{x}{8 a b \left(a-b x^2\right)}+\frac{x}{4 b \left(a-b x^2\right)^2}",1,"(x*(a + b*x^2))/(8*a*b*(a - b*x^2)^2) - ArcTanh[(Sqrt[b]*x)/Sqrt[a]]/(8*a^(3/2)*b^(3/2))","A",1
241,1,17,17,0.0021888,"\int \frac{x}{\left(a-b x^2\right)^3} \, dx","Integrate[x/(a - b*x^2)^3,x]","\frac{1}{4 b \left(a-b x^2\right)^2}","\frac{1}{4 b \left(a-b x^2\right)^2}",1,"1/(4*b*(a - b*x^2)^2)","A",1
242,1,56,64,0.040779,"\int \frac{1}{\left(a-b x^2\right)^3} \, dx","Integrate[(a - b*x^2)^(-3),x]","\frac{3 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} \sqrt{b}}+\frac{5 a x-3 b x^3}{8 a^2 \left(a-b x^2\right)^2}","\frac{3 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{5/2} \sqrt{b}}+\frac{3 x}{8 a^2 \left(a-b x^2\right)}+\frac{x}{4 a \left(a-b x^2\right)^2}",1,"(5*a*x - 3*b*x^3)/(8*a^2*(a - b*x^2)^2) + (3*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*Sqrt[b])","A",1
243,1,45,57,0.0370319,"\int \frac{1}{x \left(a-b x^2\right)^3} \, dx","Integrate[1/(x*(a - b*x^2)^3),x]","\frac{\frac{a \left(3 a-2 b x^2\right)}{\left(a-b x^2\right)^2}-2 \log \left(a-b x^2\right)+4 \log (x)}{4 a^3}","-\frac{\log \left(a-b x^2\right)}{2 a^3}+\frac{\log (x)}{a^3}+\frac{1}{2 a^2 \left(a-b x^2\right)}+\frac{1}{4 a \left(a-b x^2\right)^2}",1,"((a*(3*a - 2*b*x^2))/(a - b*x^2)^2 + 4*Log[x] - 2*Log[a - b*x^2])/(4*a^3)","A",1
244,1,69,78,0.0467388,"\int \frac{1}{x^2 \left(a-b x^2\right)^3} \, dx","Integrate[1/(x^2*(a - b*x^2)^3),x]","\frac{15 \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{7/2}}+\frac{-8 a^2+25 a b x^2-15 b^2 x^4}{8 a^3 x \left(a-b x^2\right)^2}","\frac{15 \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{8 a^{7/2}}-\frac{15}{8 a^3 x}+\frac{5}{8 a^2 x \left(a-b x^2\right)}+\frac{1}{4 a x \left(a-b x^2\right)^2}",1,"(-8*a^2 + 25*a*b*x^2 - 15*b^2*x^4)/(8*a^3*x*(a - b*x^2)^2) + (15*Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(7/2))","A",1
245,1,60,69,0.0564925,"\int \frac{1}{x^3 \left(a-b x^2\right)^3} \, dx","Integrate[1/(x^3*(a - b*x^2)^3),x]","\frac{\frac{a \left(-2 a^2+9 a b x^2-6 b^2 x^4\right)}{\left(a x-b x^3\right)^2}-6 b \log \left(a-b x^2\right)+12 b \log (x)}{4 a^4}","-\frac{3 b \log \left(a-b x^2\right)}{2 a^4}+\frac{3 b \log (x)}{a^4}+\frac{b}{a^3 \left(a-b x^2\right)}-\frac{1}{2 a^3 x^2}+\frac{b}{4 a^2 \left(a-b x^2\right)^2}",1,"((a*(-2*a^2 + 9*a*b*x^2 - 6*b^2*x^4))/(a*x - b*x^3)^2 + 12*b*Log[x] - 6*b*Log[a - b*x^2])/(4*a^4)","A",1
246,1,25,36,0.0103534,"\int \frac{x^3}{\left(a-b x^2\right)^5} \, dx","Integrate[x^3/(a - b*x^2)^5,x]","-\frac{a-4 b x^2}{24 b^2 \left(a-b x^2\right)^4}","\frac{a}{8 b^2 \left(a-b x^2\right)^4}-\frac{1}{6 b^2 \left(a-b x^2\right)^3}",1,"-1/24*(a - 4*b*x^2)/(b^2*(a - b*x^2)^4)","A",1
247,1,81,109,0.0507382,"\int \frac{x^2}{\left(a-b x^2\right)^5} \, dx","Integrate[x^2/(a - b*x^2)^5,x]","\frac{15 a^3 x+73 a^2 b x^3-55 a b^2 x^5+15 b^3 x^7}{384 a^3 b \left(a-b x^2\right)^4}-\frac{5 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{128 a^{7/2} b^{3/2}}","-\frac{5 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{128 a^{7/2} b^{3/2}}-\frac{5 x}{128 a^3 b \left(a-b x^2\right)}-\frac{5 x}{192 a^2 b \left(a-b x^2\right)^2}-\frac{x}{48 a b \left(a-b x^2\right)^3}+\frac{x}{8 b \left(a-b x^2\right)^4}",1,"(15*a^3*x + 73*a^2*b*x^3 - 55*a*b^2*x^5 + 15*b^3*x^7)/(384*a^3*b*(a - b*x^2)^4) - (5*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/(128*a^(7/2)*b^(3/2))","A",1
248,1,17,17,0.0037246,"\int \frac{x}{\left(a-b x^2\right)^5} \, dx","Integrate[x/(a - b*x^2)^5,x]","\frac{1}{8 b \left(a-b x^2\right)^4}","\frac{1}{8 b \left(a-b x^2\right)^4}",1,"1/(8*b*(a - b*x^2)^4)","A",1
249,1,79,100,0.0442697,"\int \frac{1}{\left(a-b x^2\right)^5} \, dx","Integrate[(a - b*x^2)^(-5),x]","\frac{\frac{\sqrt{a} x \left(279 a^3-511 a^2 b x^2+385 a b^2 x^4-105 b^3 x^6\right)}{\left(a-b x^2\right)^4}+\frac{105 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{b}}}{384 a^{9/2}}","\frac{35 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{128 a^{9/2} \sqrt{b}}+\frac{35 x}{128 a^4 \left(a-b x^2\right)}+\frac{35 x}{192 a^3 \left(a-b x^2\right)^2}+\frac{7 x}{48 a^2 \left(a-b x^2\right)^3}+\frac{x}{8 a \left(a-b x^2\right)^4}",1,"((Sqrt[a]*x*(279*a^3 - 511*a^2*b*x^2 + 385*a*b^2*x^4 - 105*b^3*x^6))/(a - b*x^2)^4 + (105*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[b])/(384*a^(9/2))","A",1
250,1,67,91,0.0354507,"\int \frac{1}{x \left(a-b x^2\right)^5} \, dx","Integrate[1/(x*(a - b*x^2)^5),x]","\frac{\frac{a \left(25 a^3-52 a^2 b x^2+42 a b^2 x^4-12 b^3 x^6\right)}{\left(a-b x^2\right)^4}-12 \log \left(a-b x^2\right)+24 \log (x)}{24 a^5}","-\frac{\log \left(a-b x^2\right)}{2 a^5}+\frac{\log (x)}{a^5}+\frac{1}{2 a^4 \left(a-b x^2\right)}+\frac{1}{4 a^3 \left(a-b x^2\right)^2}+\frac{1}{6 a^2 \left(a-b x^2\right)^3}+\frac{1}{8 a \left(a-b x^2\right)^4}",1,"((a*(25*a^3 - 52*a^2*b*x^2 + 42*a*b^2*x^4 - 12*b^3*x^6))/(a - b*x^2)^4 + 24*Log[x] - 12*Log[a - b*x^2])/(24*a^5)","A",1
251,1,92,118,0.0551559,"\int \frac{1}{x^2 \left(a-b x^2\right)^5} \, dx","Integrate[1/(x^2*(a - b*x^2)^5),x]","\frac{\frac{\sqrt{a} \left(-128 a^4+837 a^3 b x^2-1533 a^2 b^2 x^4+1155 a b^3 x^6-315 b^4 x^8\right)}{x \left(a-b x^2\right)^4}+315 \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{128 a^{11/2}}","\frac{315 \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{128 a^{11/2}}-\frac{315}{128 a^5 x}+\frac{105}{128 a^4 x \left(a-b x^2\right)}+\frac{21}{64 a^3 x \left(a-b x^2\right)^2}+\frac{3}{16 a^2 x \left(a-b x^2\right)^3}+\frac{1}{8 a x \left(a-b x^2\right)^4}",1,"((Sqrt[a]*(-128*a^4 + 837*a^3*b*x^2 - 1533*a^2*b^2*x^4 + 1155*a*b^3*x^6 - 315*b^4*x^8))/(x*(a - b*x^2)^4) + 315*Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/(128*a^(11/2))","A",1
252,1,83,106,0.0679567,"\int \frac{1}{x^3 \left(a-b x^2\right)^5} \, dx","Integrate[1/(x^3*(a - b*x^2)^5),x]","\frac{\frac{a \left(-12 a^4+125 a^3 b x^2-260 a^2 b^2 x^4+210 a b^3 x^6-60 b^4 x^8\right)}{x^2 \left(a-b x^2\right)^4}-60 b \log \left(a-b x^2\right)+120 b \log (x)}{24 a^6}","-\frac{5 b \log \left(a-b x^2\right)}{2 a^6}+\frac{5 b \log (x)}{a^6}+\frac{2 b}{a^5 \left(a-b x^2\right)}-\frac{1}{2 a^5 x^2}+\frac{3 b}{4 a^4 \left(a-b x^2\right)^2}+\frac{b}{3 a^3 \left(a-b x^2\right)^3}+\frac{b}{8 a^2 \left(a-b x^2\right)^4}",1,"((a*(-12*a^4 + 125*a^3*b*x^2 - 260*a^2*b^2*x^4 + 210*a*b^3*x^6 - 60*b^4*x^8))/(x^2*(a - b*x^2)^4) + 120*b*Log[x] - 60*b*Log[a - b*x^2])/(24*a^6)","A",1
253,1,15,15,0.0041314,"\int \frac{1}{x \left(1+b x^2\right)} \, dx","Integrate[1/(x*(1 + b*x^2)),x]","\log (x)-\frac{1}{2} \log \left(b x^2+1\right)","\log (x)-\frac{1}{2} \log \left(b x^2+1\right)",1,"Log[x] - Log[1 + b*x^2]/2","A",1
254,1,18,18,0.004964,"\int \frac{1}{x \left(-1+b x^2\right)} \, dx","Integrate[1/(x*(-1 + b*x^2)),x]","\frac{1}{2} \log \left(1-b x^2\right)-\log (x)","\frac{1}{2} \log \left(1-b x^2\right)-\log (x)",1,"-Log[x] + Log[1 - b*x^2]/2","A",1
255,1,26,26,0.0057288,"\int \frac{1}{x^3 \left(1+b x^2\right)} \, dx","Integrate[1/(x^3*(1 + b*x^2)),x]","\frac{1}{2} b \log \left(b x^2+1\right)-b \log (x)-\frac{1}{2 x^2}","\frac{1}{2} b \log \left(b x^2+1\right)-b \log (x)-\frac{1}{2 x^2}",1,"-1/2*1/x^2 - b*Log[x] + (b*Log[1 + b*x^2])/2","A",1
256,1,27,27,0.0044962,"\int \frac{1}{x^3 \left(-1+b x^2\right)} \, dx","Integrate[1/(x^3*(-1 + b*x^2)),x]","\frac{1}{2} b \log \left(1-b x^2\right)-b \log (x)+\frac{1}{2 x^2}","\frac{1}{2} b \log \left(1-b x^2\right)-b \log (x)+\frac{1}{2 x^2}",1,"1/(2*x^2) - b*Log[x] + (b*Log[1 - b*x^2])/2","A",1
257,1,28,30,0.0109666,"\int \frac{1}{-1+a+a x^2} \, dx","Integrate[(-1 + a + a*x^2)^(-1),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{a-1}}\right)}{\sqrt{a-1} \sqrt{a}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{1-a}}\right)}{\sqrt{(1-a) a}}",1,"ArcTan[(Sqrt[a]*x)/Sqrt[-1 + a]]/(Sqrt[-1 + a]*Sqrt[a])","A",1
258,1,44,37,0.0163068,"\int \frac{1}{-c-d+(c-d) x^2} \, dx","Integrate[(-c - d + (c - d)*x^2)^(-1),x]","\frac{\tan ^{-1}\left(\frac{x \sqrt{c-d}}{\sqrt{-c-d}}\right)}{\sqrt{-c-d} \sqrt{c-d}}","-\frac{\tanh ^{-1}\left(\frac{x \sqrt{c-d}}{\sqrt{c+d}}\right)}{\sqrt{c-d} \sqrt{c+d}}",1,"ArcTan[(Sqrt[c - d]*x)/Sqrt[-c - d]]/(Sqrt[-c - d]*Sqrt[c - d])","A",1
259,1,25,28,0.0134151,"\int \frac{1}{x \left(1+b x^2\right)^2} \, dx","Integrate[1/(x*(1 + b*x^2)^2),x]","\frac{1}{2 b x^2+2}-\frac{1}{2} \log \left(b x^2+1\right)+\log (x)","\frac{1}{2 \left(b x^2+1\right)}-\frac{1}{2} \log \left(b x^2+1\right)+\log (x)",1,"(2 + 2*b*x^2)^(-1) + Log[x] - Log[1 + b*x^2]/2","A",1
260,1,26,30,0.0141585,"\int \frac{1}{x \left(-1+b x^2\right)^2} \, dx","Integrate[1/(x*(-1 + b*x^2)^2),x]","\frac{1}{2-2 b x^2}-\frac{1}{2} \log \left(1-b x^2\right)+\log (x)","\frac{1}{2 \left(1-b x^2\right)}-\frac{1}{2} \log \left(1-b x^2\right)+\log (x)",1,"(2 - 2*b*x^2)^(-1) + Log[x] - Log[1 - b*x^2]/2","A",1
261,1,36,34,0.0156384,"\int \frac{1}{a+(b-a c) x^2} \, dx","Integrate[(a + (b - a*c)*x^2)^(-1),x]","\frac{\tanh ^{-1}\left(\frac{x \sqrt{a c-b}}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{a c-b}}","\frac{\tan ^{-1}\left(\frac{x \sqrt{b-a c}}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b-a c}}",1,"ArcTanh[(Sqrt[-b + a*c]*x)/Sqrt[a]]/(Sqrt[a]*Sqrt[-b + a*c])","A",1
262,1,36,34,0.0106083,"\int \frac{1}{a-(b-a c) x^2} \, dx","Integrate[(a - (b - a*c)*x^2)^(-1),x]","\frac{\tan ^{-1}\left(\frac{x \sqrt{a c-b}}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{a c-b}}","\frac{\tanh ^{-1}\left(\frac{x \sqrt{b-a c}}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b-a c}}",1,"ArcTan[(Sqrt[-b + a*c]*x)/Sqrt[a]]/(Sqrt[a]*Sqrt[-b + a*c])","A",1
263,1,50,50,0.0215875,"\int \frac{1}{c (a-d)-(b-c) x^2} \, dx","Integrate[(c*(a - d) - (b - c)*x^2)^(-1),x]","\frac{\tan ^{-1}\left(\frac{x \sqrt{c-b}}{\sqrt{c} \sqrt{a-d}}\right)}{\sqrt{c} \sqrt{a-d} \sqrt{c-b}}","\frac{\tanh ^{-1}\left(\frac{x \sqrt{b-c}}{\sqrt{c} \sqrt{a-d}}\right)}{\sqrt{c} \sqrt{a-d} \sqrt{b-c}}",1,"ArcTan[(Sqrt[-b + c]*x)/(Sqrt[c]*Sqrt[a - d])]/(Sqrt[c]*Sqrt[-b + c]*Sqrt[a - d])","A",1
264,1,21,21,0.0050074,"\int x^{7/2} \left(a+b x^2\right) \, dx","Integrate[x^(7/2)*(a + b*x^2),x]","\frac{2}{9} a x^{9/2}+\frac{2}{13} b x^{13/2}","\frac{2}{9} a x^{9/2}+\frac{2}{13} b x^{13/2}",1,"(2*a*x^(9/2))/9 + (2*b*x^(13/2))/13","A",1
265,1,21,21,0.0047116,"\int x^{5/2} \left(a+b x^2\right) \, dx","Integrate[x^(5/2)*(a + b*x^2),x]","\frac{2}{7} a x^{7/2}+\frac{2}{11} b x^{11/2}","\frac{2}{7} a x^{7/2}+\frac{2}{11} b x^{11/2}",1,"(2*a*x^(7/2))/7 + (2*b*x^(11/2))/11","A",1
266,1,21,21,0.0048047,"\int x^{3/2} \left(a+b x^2\right) \, dx","Integrate[x^(3/2)*(a + b*x^2),x]","\frac{2}{5} a x^{5/2}+\frac{2}{9} b x^{9/2}","\frac{2}{5} a x^{5/2}+\frac{2}{9} b x^{9/2}",1,"(2*a*x^(5/2))/5 + (2*b*x^(9/2))/9","A",1
267,1,21,21,0.0046748,"\int \sqrt{x} \left(a+b x^2\right) \, dx","Integrate[Sqrt[x]*(a + b*x^2),x]","\frac{2}{3} a x^{3/2}+\frac{2}{7} b x^{7/2}","\frac{2}{3} a x^{3/2}+\frac{2}{7} b x^{7/2}",1,"(2*a*x^(3/2))/3 + (2*b*x^(7/2))/7","A",1
268,1,19,19,0.0045341,"\int \frac{a+b x^2}{\sqrt{x}} \, dx","Integrate[(a + b*x^2)/Sqrt[x],x]","2 a \sqrt{x}+\frac{2}{5} b x^{5/2}","2 a \sqrt{x}+\frac{2}{5} b x^{5/2}",1,"2*a*Sqrt[x] + (2*b*x^(5/2))/5","A",1
269,1,19,19,0.0053748,"\int \frac{a+b x^2}{x^{3/2}} \, dx","Integrate[(a + b*x^2)/x^(3/2),x]","\frac{2}{3} b x^{3/2}-\frac{2 a}{\sqrt{x}}","\frac{2}{3} b x^{3/2}-\frac{2 a}{\sqrt{x}}",1,"(-2*a)/Sqrt[x] + (2*b*x^(3/2))/3","A",1
270,1,19,19,0.0060548,"\int \frac{a+b x^2}{x^{5/2}} \, dx","Integrate[(a + b*x^2)/x^(5/2),x]","2 b \sqrt{x}-\frac{2 a}{3 x^{3/2}}","2 b \sqrt{x}-\frac{2 a}{3 x^{3/2}}",1,"(-2*a)/(3*x^(3/2)) + 2*b*Sqrt[x]","A",1
271,1,19,19,0.0056428,"\int \frac{a+b x^2}{x^{7/2}} \, dx","Integrate[(a + b*x^2)/x^(7/2),x]","-\frac{2 a}{5 x^{5/2}}-\frac{2 b}{\sqrt{x}}","-\frac{2 a}{5 x^{5/2}}-\frac{2 b}{\sqrt{x}}",1,"(-2*a)/(5*x^(5/2)) - (2*b)/Sqrt[x]","A",1
272,1,30,36,0.0085871,"\int x^{7/2} \left(a+b x^2\right)^2 \, dx","Integrate[x^(7/2)*(a + b*x^2)^2,x]","\frac{2 x^{9/2} \left(221 a^2+306 a b x^2+117 b^2 x^4\right)}{1989}","\frac{2}{9} a^2 x^{9/2}+\frac{4}{13} a b x^{13/2}+\frac{2}{17} b^2 x^{17/2}",1,"(2*x^(9/2)*(221*a^2 + 306*a*b*x^2 + 117*b^2*x^4))/1989","A",1
273,1,30,36,0.0079457,"\int x^{5/2} \left(a+b x^2\right)^2 \, dx","Integrate[x^(5/2)*(a + b*x^2)^2,x]","\frac{2 x^{7/2} \left(165 a^2+210 a b x^2+77 b^2 x^4\right)}{1155}","\frac{2}{7} a^2 x^{7/2}+\frac{4}{11} a b x^{11/2}+\frac{2}{15} b^2 x^{15/2}",1,"(2*x^(7/2)*(165*a^2 + 210*a*b*x^2 + 77*b^2*x^4))/1155","A",1
274,1,30,36,0.0079011,"\int x^{3/2} \left(a+b x^2\right)^2 \, dx","Integrate[x^(3/2)*(a + b*x^2)^2,x]","\frac{2}{585} x^{5/2} \left(117 a^2+130 a b x^2+45 b^2 x^4\right)","\frac{2}{5} a^2 x^{5/2}+\frac{4}{9} a b x^{9/2}+\frac{2}{13} b^2 x^{13/2}",1,"(2*x^(5/2)*(117*a^2 + 130*a*b*x^2 + 45*b^2*x^4))/585","A",1
275,1,30,36,0.0077059,"\int \sqrt{x} \left(a+b x^2\right)^2 \, dx","Integrate[Sqrt[x]*(a + b*x^2)^2,x]","\frac{2}{231} x^{3/2} \left(77 a^2+66 a b x^2+21 b^2 x^4\right)","\frac{2}{3} a^2 x^{3/2}+\frac{4}{7} a b x^{7/2}+\frac{2}{11} b^2 x^{11/2}",1,"(2*x^(3/2)*(77*a^2 + 66*a*b*x^2 + 21*b^2*x^4))/231","A",1
276,1,30,34,0.0079217,"\int \frac{\left(a+b x^2\right)^2}{\sqrt{x}} \, dx","Integrate[(a + b*x^2)^2/Sqrt[x],x]","\frac{2}{45} \sqrt{x} \left(45 a^2+18 a b x^2+5 b^2 x^4\right)","2 a^2 \sqrt{x}+\frac{4}{5} a b x^{5/2}+\frac{2}{9} b^2 x^{9/2}",1,"(2*Sqrt[x]*(45*a^2 + 18*a*b*x^2 + 5*b^2*x^4))/45","A",1
277,1,30,34,0.0089791,"\int \frac{\left(a+b x^2\right)^2}{x^{3/2}} \, dx","Integrate[(a + b*x^2)^2/x^(3/2),x]","\frac{2 \left(-21 a^2+14 a b x^2+3 b^2 x^4\right)}{21 \sqrt{x}}","-\frac{2 a^2}{\sqrt{x}}+\frac{4}{3} a b x^{3/2}+\frac{2}{7} b^2 x^{7/2}",1,"(2*(-21*a^2 + 14*a*b*x^2 + 3*b^2*x^4))/(21*Sqrt[x])","A",1
278,1,30,34,0.0087509,"\int \frac{\left(a+b x^2\right)^2}{x^{5/2}} \, dx","Integrate[(a + b*x^2)^2/x^(5/2),x]","\frac{2 \left(-5 a^2+30 a b x^2+3 b^2 x^4\right)}{15 x^{3/2}}","-\frac{2 a^2}{3 x^{3/2}}+4 a b \sqrt{x}+\frac{2}{5} b^2 x^{5/2}",1,"(2*(-5*a^2 + 30*a*b*x^2 + 3*b^2*x^4))/(15*x^(3/2))","A",1
279,1,30,34,0.0092405,"\int \frac{\left(a+b x^2\right)^2}{x^{7/2}} \, dx","Integrate[(a + b*x^2)^2/x^(7/2),x]","\frac{2 \left(-3 a^2-30 a b x^2+5 b^2 x^4\right)}{15 x^{5/2}}","-\frac{2 a^2}{5 x^{5/2}}-\frac{4 a b}{\sqrt{x}}+\frac{2}{3} b^2 x^{3/2}",1,"(2*(-3*a^2 - 30*a*b*x^2 + 5*b^2*x^4))/(15*x^(5/2))","A",1
280,1,41,51,0.0116745,"\int x^{7/2} \left(a+b x^2\right)^3 \, dx","Integrate[x^(7/2)*(a + b*x^2)^3,x]","\frac{2 x^{9/2} \left(1547 a^3+3213 a^2 b x^2+2457 a b^2 x^4+663 b^3 x^6\right)}{13923}","\frac{2}{9} a^3 x^{9/2}+\frac{6}{13} a^2 b x^{13/2}+\frac{6}{17} a b^2 x^{17/2}+\frac{2}{21} b^3 x^{21/2}",1,"(2*x^(9/2)*(1547*a^3 + 3213*a^2*b*x^2 + 2457*a*b^2*x^4 + 663*b^3*x^6))/13923","A",1
281,1,41,51,0.0101644,"\int x^{5/2} \left(a+b x^2\right)^3 \, dx","Integrate[x^(5/2)*(a + b*x^2)^3,x]","\frac{2 x^{7/2} \left(1045 a^3+1995 a^2 b x^2+1463 a b^2 x^4+385 b^3 x^6\right)}{7315}","\frac{2}{7} a^3 x^{7/2}+\frac{6}{11} a^2 b x^{11/2}+\frac{2}{5} a b^2 x^{15/2}+\frac{2}{19} b^3 x^{19/2}",1,"(2*x^(7/2)*(1045*a^3 + 1995*a^2*b*x^2 + 1463*a*b^2*x^4 + 385*b^3*x^6))/7315","A",1
282,1,41,51,0.0102903,"\int x^{3/2} \left(a+b x^2\right)^3 \, dx","Integrate[x^(3/2)*(a + b*x^2)^3,x]","\frac{2 x^{5/2} \left(663 a^3+1105 a^2 b x^2+765 a b^2 x^4+195 b^3 x^6\right)}{3315}","\frac{2}{5} a^3 x^{5/2}+\frac{2}{3} a^2 b x^{9/2}+\frac{6}{13} a b^2 x^{13/2}+\frac{2}{17} b^3 x^{17/2}",1,"(2*x^(5/2)*(663*a^3 + 1105*a^2*b*x^2 + 765*a*b^2*x^4 + 195*b^3*x^6))/3315","A",1
283,1,41,51,0.0101446,"\int \sqrt{x} \left(a+b x^2\right)^3 \, dx","Integrate[Sqrt[x]*(a + b*x^2)^3,x]","\frac{2 x^{3/2} \left(385 a^3+495 a^2 b x^2+315 a b^2 x^4+77 b^3 x^6\right)}{1155}","\frac{2}{3} a^3 x^{3/2}+\frac{6}{7} a^2 b x^{7/2}+\frac{6}{11} a b^2 x^{11/2}+\frac{2}{15} b^3 x^{15/2}",1,"(2*x^(3/2)*(385*a^3 + 495*a^2*b*x^2 + 315*a*b^2*x^4 + 77*b^3*x^6))/1155","A",1
284,1,41,49,0.0101948,"\int \frac{\left(a+b x^2\right)^3}{\sqrt{x}} \, dx","Integrate[(a + b*x^2)^3/Sqrt[x],x]","\frac{2}{195} \sqrt{x} \left(195 a^3+117 a^2 b x^2+65 a b^2 x^4+15 b^3 x^6\right)","2 a^3 \sqrt{x}+\frac{6}{5} a^2 b x^{5/2}+\frac{2}{3} a b^2 x^{9/2}+\frac{2}{13} b^3 x^{13/2}",1,"(2*Sqrt[x]*(195*a^3 + 117*a^2*b*x^2 + 65*a*b^2*x^4 + 15*b^3*x^6))/195","A",1
285,1,41,47,0.0112571,"\int \frac{\left(a+b x^2\right)^3}{x^{3/2}} \, dx","Integrate[(a + b*x^2)^3/x^(3/2),x]","\frac{2 \left(-77 a^3+77 a^2 b x^2+33 a b^2 x^4+7 b^3 x^6\right)}{77 \sqrt{x}}","-\frac{2 a^3}{\sqrt{x}}+2 a^2 b x^{3/2}+\frac{6}{7} a b^2 x^{7/2}+\frac{2}{11} b^3 x^{11/2}",1,"(2*(-77*a^3 + 77*a^2*b*x^2 + 33*a*b^2*x^4 + 7*b^3*x^6))/(77*Sqrt[x])","A",1
286,1,41,49,0.0116952,"\int \frac{\left(a+b x^2\right)^3}{x^{5/2}} \, dx","Integrate[(a + b*x^2)^3/x^(5/2),x]","\frac{2 \left(-15 a^3+135 a^2 b x^2+27 a b^2 x^4+5 b^3 x^6\right)}{45 x^{3/2}}","-\frac{2 a^3}{3 x^{3/2}}+6 a^2 b \sqrt{x}+\frac{6}{5} a b^2 x^{5/2}+\frac{2}{9} b^3 x^{9/2}",1,"(2*(-15*a^3 + 135*a^2*b*x^2 + 27*a*b^2*x^4 + 5*b^3*x^6))/(45*x^(3/2))","A",1
287,1,41,47,0.0111316,"\int \frac{\left(a+b x^2\right)^3}{x^{7/2}} \, dx","Integrate[(a + b*x^2)^3/x^(7/2),x]","\frac{2 \left(-7 a^3-105 a^2 b x^2+35 a b^2 x^4+5 b^3 x^6\right)}{35 x^{5/2}}","-\frac{2 a^3}{5 x^{5/2}}-\frac{6 a^2 b}{\sqrt{x}}+2 a b^2 x^{3/2}+\frac{2}{7} b^3 x^{7/2}",1,"(2*(-7*a^3 - 105*a^2*b*x^2 + 35*a*b^2*x^4 + 5*b^3*x^6))/(35*x^(5/2))","A",1
288,1,203,215,0.0682224,"\int \frac{x^{7/2}}{a+b x^2} \, dx","Integrate[x^(7/2)/(a + b*x^2),x]","\frac{-5 \sqrt{2} a^{5/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+5 \sqrt{2} a^{5/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-10 \sqrt{2} a^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)+10 \sqrt{2} a^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)-40 a \sqrt[4]{b} \sqrt{x}+8 b^{5/4} x^{5/2}}{20 b^{9/4}}","-\frac{a^{5/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} b^{9/4}}+\frac{a^{5/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} b^{9/4}}-\frac{a^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} b^{9/4}}+\frac{a^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} b^{9/4}}-\frac{2 a \sqrt{x}}{b^2}+\frac{2 x^{5/2}}{5 b}",1,"(-40*a*b^(1/4)*Sqrt[x] + 8*b^(5/4)*x^(5/2) - 10*Sqrt[2]*a^(5/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 10*Sqrt[2]*a^(5/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 5*Sqrt[2]*a^(5/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 5*Sqrt[2]*a^(5/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(20*b^(9/4))","A",1
289,1,78,204,0.0271531,"\int \frac{x^{5/2}}{a+b x^2} \, dx","Integrate[x^(5/2)/(a + b*x^2),x]","\frac{(-a)^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{-a}}\right)}{b^{7/4}}-\frac{(-a)^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{-a}}\right)}{b^{7/4}}+\frac{2 x^{3/2}}{3 b}","-\frac{a^{3/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} b^{7/4}}+\frac{a^{3/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} b^{7/4}}+\frac{a^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} b^{7/4}}-\frac{a^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} b^{7/4}}+\frac{2 x^{3/2}}{3 b}",1,"(2*x^(3/2))/(3*b) + ((-a)^(3/4)*ArcTan[(b^(1/4)*Sqrt[x])/(-a)^(1/4)])/b^(7/4) - ((-a)^(3/4)*ArcTanh[(b^(1/4)*Sqrt[x])/(-a)^(1/4)])/b^(7/4)","A",1
290,1,189,202,0.0386063,"\int \frac{x^{3/2}}{a+b x^2} \, dx","Integrate[x^(3/2)/(a + b*x^2),x]","\frac{\sqrt{2} \sqrt[4]{a} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-\sqrt{2} \sqrt[4]{a} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+2 \sqrt{2} \sqrt[4]{a} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)-2 \sqrt{2} \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)+8 \sqrt[4]{b} \sqrt{x}}{4 b^{5/4}}","\frac{\sqrt[4]{a} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} b^{5/4}}-\frac{\sqrt[4]{a} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} b^{5/4}}+\frac{\sqrt[4]{a} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} b^{5/4}}-\frac{\sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} b^{5/4}}+\frac{2 \sqrt{x}}{b}",1,"(8*b^(1/4)*Sqrt[x] + 2*Sqrt[2]*a^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 2*Sqrt[2]*a^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + Sqrt[2]*a^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - Sqrt[2]*a^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(4*b^(5/4))","A",1
291,1,54,192,0.0254738,"\int \frac{\sqrt{x}}{a+b x^2} \, dx","Integrate[Sqrt[x]/(a + b*x^2),x]","\frac{a \left(\tan ^{-1}\left(\frac{a \sqrt[4]{b} \sqrt{x}}{(-a)^{5/4}}\right)+\tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{-a}}\right)\right)}{(-a)^{5/4} b^{3/4}}","\frac{\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} \sqrt[4]{a} b^{3/4}}-\frac{\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} \sqrt[4]{a} b^{3/4}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} \sqrt[4]{a} b^{3/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} \sqrt[4]{a} b^{3/4}}",1,"(a*(ArcTan[(a*b^(1/4)*Sqrt[x])/(-a)^(5/4)] + ArcTanh[(b^(1/4)*Sqrt[x])/(-a)^(1/4)]))/((-a)^(5/4)*b^(3/4))","A",1
292,1,146,192,0.0338452,"\int \frac{1}{\sqrt{x} \left(a+b x^2\right)} \, dx","Integrate[1/(Sqrt[x]*(a + b*x^2)),x]","\frac{-\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{3/4} \sqrt[4]{b}}","-\frac{\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{3/4} \sqrt[4]{b}}+\frac{\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{3/4} \sqrt[4]{b}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} a^{3/4} \sqrt[4]{b}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} a^{3/4} \sqrt[4]{b}}",1,"(-2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(3/4)*b^(1/4))","A",1
293,1,27,202,0.0056431,"\int \frac{1}{x^{3/2} \left(a+b x^2\right)} \, dx","Integrate[1/(x^(3/2)*(a + b*x^2)),x]","-\frac{2 \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\frac{b x^2}{a}\right)}{a \sqrt{x}}","-\frac{\sqrt[4]{b} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{5/4}}+\frac{\sqrt[4]{b} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{5/4}}+\frac{\sqrt[4]{b} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} a^{5/4}}-\frac{\sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} a^{5/4}}-\frac{2}{a \sqrt{x}}",1,"(-2*Hypergeometric2F1[-1/4, 1, 3/4, -((b*x^2)/a)])/(a*Sqrt[x])","C",1
294,1,29,204,0.0065341,"\int \frac{1}{x^{5/2} \left(a+b x^2\right)} \, dx","Integrate[1/(x^(5/2)*(a + b*x^2)),x]","-\frac{2 \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\frac{b x^2}{a}\right)}{3 a x^{3/2}}","\frac{b^{3/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{7/4}}-\frac{b^{3/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{7/4}}+\frac{b^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} a^{7/4}}-\frac{b^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} a^{7/4}}-\frac{2}{3 a x^{3/2}}",1,"(-2*Hypergeometric2F1[-3/4, 1, 1/4, -((b*x^2)/a)])/(3*a*x^(3/2))","C",1
295,1,29,215,0.0062824,"\int \frac{1}{x^{7/2} \left(a+b x^2\right)} \, dx","Integrate[1/(x^(7/2)*(a + b*x^2)),x]","-\frac{2 \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\frac{b x^2}{a}\right)}{5 a x^{5/2}}","\frac{b^{5/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{9/4}}-\frac{b^{5/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{2 \sqrt{2} a^{9/4}}-\frac{b^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt{2} a^{9/4}}+\frac{b^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt{2} a^{9/4}}+\frac{2 b}{a^2 \sqrt{x}}-\frac{2}{5 a x^{5/2}}",1,"(-2*Hypergeometric2F1[-5/4, 1, -1/4, -((b*x^2)/a)])/(5*a*x^(5/2))","C",1
296,1,221,230,0.1032507,"\int \frac{x^{7/2}}{\left(a+b x^2\right)^2} \, dx","Integrate[x^(7/2)/(a + b*x^2)^2,x]","\frac{\frac{32 b^{5/4} x^{5/2}}{a+b x^2}+\frac{40 a \sqrt[4]{b} \sqrt{x}}{a+b x^2}+5 \sqrt{2} \sqrt[4]{a} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)-5 \sqrt{2} \sqrt[4]{a} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)+10 \sqrt{2} \sqrt[4]{a} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)-10 \sqrt{2} \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{16 b^{9/4}}","\frac{5 \sqrt[4]{a} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} b^{9/4}}-\frac{5 \sqrt[4]{a} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} b^{9/4}}+\frac{5 \sqrt[4]{a} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} b^{9/4}}-\frac{5 \sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} b^{9/4}}-\frac{x^{5/2}}{2 b \left(a+b x^2\right)}+\frac{5 \sqrt{x}}{2 b^2}",1,"((40*a*b^(1/4)*Sqrt[x])/(a + b*x^2) + (32*b^(5/4)*x^(5/2))/(a + b*x^2) + 10*Sqrt[2]*a^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 10*Sqrt[2]*a^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 5*Sqrt[2]*a^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - 5*Sqrt[2]*a^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(16*b^(9/4))","A",1
297,1,43,218,0.0129605,"\int \frac{x^{5/2}}{\left(a+b x^2\right)^2} \, dx","Integrate[x^(5/2)/(a + b*x^2)^2,x]","\frac{2 x^{3/2} \left(\frac{\, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-\frac{b x^2}{a}\right)}{a}-\frac{1}{a+b x^2}\right)}{b}","\frac{3 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} \sqrt[4]{a} b^{7/4}}-\frac{3 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} \sqrt[4]{a} b^{7/4}}-\frac{3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} \sqrt[4]{a} b^{7/4}}+\frac{3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} \sqrt[4]{a} b^{7/4}}-\frac{x^{3/2}}{2 b \left(a+b x^2\right)}",1,"(2*x^(3/2)*(-(a + b*x^2)^(-1) + Hypergeometric2F1[3/4, 2, 7/4, -((b*x^2)/a)]/a))/b","C",1
298,1,198,218,0.1008254,"\int \frac{x^{3/2}}{\left(a+b x^2\right)^2} \, dx","Integrate[x^(3/2)/(a + b*x^2)^2,x]","\frac{-\frac{\sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{3/4}}+\frac{\sqrt{2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{3/4}}-\frac{2 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{a^{3/4}}+\frac{2 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{a^{3/4}}-\frac{8 \sqrt[4]{b} \sqrt{x}}{a+b x^2}}{16 b^{5/4}}","-\frac{\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{3/4} b^{5/4}}+\frac{\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{3/4} b^{5/4}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} a^{3/4} b^{5/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{3/4} b^{5/4}}-\frac{\sqrt{x}}{2 b \left(a+b x^2\right)}",1,"((-8*b^(1/4)*Sqrt[x])/(a + b*x^2) - (2*Sqrt[2]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(3/4) + (2*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(3/4) - (Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(3/4) + (Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(3/4))/(16*b^(5/4))","A",1
299,1,29,218,0.0059239,"\int \frac{\sqrt{x}}{\left(a+b x^2\right)^2} \, dx","Integrate[Sqrt[x]/(a + b*x^2)^2,x]","\frac{2 x^{3/2} \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-\frac{b x^2}{a}\right)}{3 a^2}","\frac{\log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{5/4} b^{3/4}}-\frac{\log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{5/4} b^{3/4}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} a^{5/4} b^{3/4}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{5/4} b^{3/4}}+\frac{x^{3/2}}{2 a \left(a+b x^2\right)}",1,"(2*x^(3/2)*Hypergeometric2F1[3/4, 2, 7/4, -((b*x^2)/a)])/(3*a^2)","C",1
300,1,199,218,0.1021991,"\int \frac{1}{\sqrt{x} \left(a+b x^2\right)^2} \, dx","Integrate[1/(Sqrt[x]*(a + b*x^2)^2),x]","\frac{\frac{8 a^{3/4} \sqrt{x}}{a+b x^2}-\frac{3 \sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{\sqrt[4]{b}}+\frac{3 \sqrt{2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{\sqrt[4]{b}}-\frac{6 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt[4]{b}}+\frac{6 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt[4]{b}}}{16 a^{7/4}}","-\frac{3 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{7/4} \sqrt[4]{b}}+\frac{3 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{7/4} \sqrt[4]{b}}-\frac{3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} a^{7/4} \sqrt[4]{b}}+\frac{3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{7/4} \sqrt[4]{b}}+\frac{\sqrt{x}}{2 a \left(a+b x^2\right)}",1,"((8*a^(3/4)*Sqrt[x])/(a + b*x^2) - (6*Sqrt[2]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(1/4) + (6*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(1/4) - (3*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(1/4) + (3*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(1/4))/(16*a^(7/4))","A",1
301,1,27,230,0.0057793,"\int \frac{1}{x^{3/2} \left(a+b x^2\right)^2} \, dx","Integrate[1/(x^(3/2)*(a + b*x^2)^2),x]","-\frac{2 \, _2F_1\left(-\frac{1}{4},2;\frac{3}{4};-\frac{b x^2}{a}\right)}{a^2 \sqrt{x}}","-\frac{5 \sqrt[4]{b} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{9/4}}+\frac{5 \sqrt[4]{b} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{9/4}}+\frac{5 \sqrt[4]{b} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} a^{9/4}}-\frac{5 \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{9/4}}-\frac{5}{2 a^2 \sqrt{x}}+\frac{1}{2 a \sqrt{x} \left(a+b x^2\right)}",1,"(-2*Hypergeometric2F1[-1/4, 2, 3/4, -((b*x^2)/a)])/(a^2*Sqrt[x])","C",1
302,1,29,230,0.0066657,"\int \frac{1}{x^{5/2} \left(a+b x^2\right)^2} \, dx","Integrate[1/(x^(5/2)*(a + b*x^2)^2),x]","-\frac{2 \, _2F_1\left(-\frac{3}{4},2;\frac{1}{4};-\frac{b x^2}{a}\right)}{3 a^2 x^{3/2}}","\frac{7 b^{3/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{11/4}}-\frac{7 b^{3/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{11/4}}+\frac{7 b^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} a^{11/4}}-\frac{7 b^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{11/4}}-\frac{7}{6 a^2 x^{3/2}}+\frac{1}{2 a x^{3/2} \left(a+b x^2\right)}",1,"(-2*Hypergeometric2F1[-3/4, 2, 1/4, -((b*x^2)/a)])/(3*a^2*x^(3/2))","C",1
303,1,29,243,0.0065262,"\int \frac{1}{x^{7/2} \left(a+b x^2\right)^2} \, dx","Integrate[1/(x^(7/2)*(a + b*x^2)^2),x]","-\frac{2 \, _2F_1\left(-\frac{5}{4},2;-\frac{1}{4};-\frac{b x^2}{a}\right)}{5 a^2 x^{5/2}}","\frac{9 b^{5/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{13/4}}-\frac{9 b^{5/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{8 \sqrt{2} a^{13/4}}-\frac{9 b^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{4 \sqrt{2} a^{13/4}}+\frac{9 b^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{4 \sqrt{2} a^{13/4}}+\frac{9 b}{2 a^3 \sqrt{x}}-\frac{9}{10 a^2 x^{5/2}}+\frac{1}{2 a x^{5/2} \left(a+b x^2\right)}",1,"(-2*Hypergeometric2F1[-5/4, 2, -1/4, -((b*x^2)/a)])/(5*a^2*x^(5/2))","C",1
304,1,242,239,0.1128157,"\int \frac{x^{7/2}}{\left(a+b x^2\right)^3} \, dx","Integrate[x^(7/2)/(a + b*x^2)^3,x]","\frac{-\frac{15 \sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{3/4}}+\frac{15 \sqrt{2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{3/4}}-\frac{30 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{a^{3/4}}+\frac{30 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{a^{3/4}}-\frac{256 b^{5/4} x^{5/2}}{\left(a+b x^2\right)^2}+\frac{40 \sqrt[4]{b} \sqrt{x}}{a+b x^2}-\frac{160 a \sqrt[4]{b} \sqrt{x}}{\left(a+b x^2\right)^2}}{384 b^{9/4}}","-\frac{5 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{3/4} b^{9/4}}+\frac{5 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{3/4} b^{9/4}}-\frac{5 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{32 \sqrt{2} a^{3/4} b^{9/4}}+\frac{5 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{32 \sqrt{2} a^{3/4} b^{9/4}}-\frac{5 \sqrt{x}}{16 b^2 \left(a+b x^2\right)}-\frac{x^{5/2}}{4 b \left(a+b x^2\right)^2}",1,"((-160*a*b^(1/4)*Sqrt[x])/(a + b*x^2)^2 - (256*b^(5/4)*x^(5/2))/(a + b*x^2)^2 + (40*b^(1/4)*Sqrt[x])/(a + b*x^2) - (30*Sqrt[2]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(3/4) + (30*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(3/4) - (15*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(3/4) + (15*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(3/4))/(384*b^(9/4))","A",1
305,1,45,242,0.0168013,"\int \frac{x^{5/2}}{\left(a+b x^2\right)^3} \, dx","Integrate[x^(5/2)/(a + b*x^2)^3,x]","\frac{2 x^{3/2} \left(\frac{\, _2F_1\left(\frac{3}{4},3;\frac{7}{4};-\frac{b x^2}{a}\right)}{a^2}-\frac{1}{\left(a+b x^2\right)^2}\right)}{5 b}","\frac{3 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{5/4} b^{7/4}}-\frac{3 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{5/4} b^{7/4}}-\frac{3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{32 \sqrt{2} a^{5/4} b^{7/4}}+\frac{3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{32 \sqrt{2} a^{5/4} b^{7/4}}+\frac{3 x^{3/2}}{16 a b \left(a+b x^2\right)}-\frac{x^{3/2}}{4 b \left(a+b x^2\right)^2}",1,"(2*x^(3/2)*(-(a + b*x^2)^(-2) + Hypergeometric2F1[3/4, 3, 7/4, -((b*x^2)/a)]/a^2))/(5*b)","C",1
306,1,223,242,0.1089887,"\int \frac{x^{3/2}}{\left(a+b x^2\right)^3} \, dx","Integrate[x^(3/2)/(a + b*x^2)^3,x]","\frac{-\frac{3 \sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{7/4}}+\frac{3 \sqrt{2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{a^{7/4}}-\frac{6 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{a^{7/4}}+\frac{6 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{a^{7/4}}+\frac{8 \sqrt[4]{b} \sqrt{x}}{a^2+a b x^2}-\frac{32 \sqrt[4]{b} \sqrt{x}}{\left(a+b x^2\right)^2}}{128 b^{5/4}}","-\frac{3 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{7/4} b^{5/4}}+\frac{3 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{7/4} b^{5/4}}-\frac{3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{32 \sqrt{2} a^{7/4} b^{5/4}}+\frac{3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{32 \sqrt{2} a^{7/4} b^{5/4}}+\frac{\sqrt{x}}{16 a b \left(a+b x^2\right)}-\frac{\sqrt{x}}{4 b \left(a+b x^2\right)^2}",1,"((-32*b^(1/4)*Sqrt[x])/(a + b*x^2)^2 + (8*b^(1/4)*Sqrt[x])/(a^2 + a*b*x^2) - (6*Sqrt[2]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(7/4) + (6*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(7/4) - (3*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(7/4) + (3*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(7/4))/(128*b^(5/4))","A",1
307,1,29,239,0.0059665,"\int \frac{\sqrt{x}}{\left(a+b x^2\right)^3} \, dx","Integrate[Sqrt[x]/(a + b*x^2)^3,x]","\frac{2 x^{3/2} \, _2F_1\left(\frac{3}{4},3;\frac{7}{4};-\frac{b x^2}{a}\right)}{3 a^3}","\frac{5 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{9/4} b^{3/4}}-\frac{5 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{9/4} b^{3/4}}-\frac{5 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{32 \sqrt{2} a^{9/4} b^{3/4}}+\frac{5 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{32 \sqrt{2} a^{9/4} b^{3/4}}+\frac{5 x^{3/2}}{16 a^2 \left(a+b x^2\right)}+\frac{x^{3/2}}{4 a \left(a+b x^2\right)^2}",1,"(2*x^(3/2)*Hypergeometric2F1[3/4, 3, 7/4, -((b*x^2)/a)])/(3*a^3)","C",1
308,1,220,239,0.0830858,"\int \frac{1}{\sqrt{x} \left(a+b x^2\right)^3} \, dx","Integrate[1/(Sqrt[x]*(a + b*x^2)^3),x]","\frac{\frac{32 a^{7/4} \sqrt{x}}{\left(a+b x^2\right)^2}+\frac{56 a^{3/4} \sqrt{x}}{a+b x^2}-\frac{21 \sqrt{2} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{\sqrt[4]{b}}+\frac{21 \sqrt{2} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{\sqrt[4]{b}}-\frac{42 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt[4]{b}}+\frac{42 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{\sqrt[4]{b}}}{128 a^{11/4}}","-\frac{21 \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{11/4} \sqrt[4]{b}}+\frac{21 \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{11/4} \sqrt[4]{b}}-\frac{21 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{32 \sqrt{2} a^{11/4} \sqrt[4]{b}}+\frac{21 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{32 \sqrt{2} a^{11/4} \sqrt[4]{b}}+\frac{7 \sqrt{x}}{16 a^2 \left(a+b x^2\right)}+\frac{\sqrt{x}}{4 a \left(a+b x^2\right)^2}",1,"((32*a^(7/4)*Sqrt[x])/(a + b*x^2)^2 + (56*a^(3/4)*Sqrt[x])/(a + b*x^2) - (42*Sqrt[2]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(1/4) + (42*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(1/4) - (21*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(1/4) + (21*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(1/4))/(128*a^(11/4))","A",1
309,1,27,251,0.0063528,"\int \frac{1}{x^{3/2} \left(a+b x^2\right)^3} \, dx","Integrate[1/(x^(3/2)*(a + b*x^2)^3),x]","-\frac{2 \, _2F_1\left(-\frac{1}{4},3;\frac{3}{4};-\frac{b x^2}{a}\right)}{a^3 \sqrt{x}}","-\frac{45 \sqrt[4]{b} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{13/4}}+\frac{45 \sqrt[4]{b} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{13/4}}+\frac{45 \sqrt[4]{b} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{32 \sqrt{2} a^{13/4}}-\frac{45 \sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{32 \sqrt{2} a^{13/4}}-\frac{45}{16 a^3 \sqrt{x}}+\frac{9}{16 a^2 \sqrt{x} \left(a+b x^2\right)}+\frac{1}{4 a \sqrt{x} \left(a+b x^2\right)^2}",1,"(-2*Hypergeometric2F1[-1/4, 3, 3/4, -((b*x^2)/a)])/(a^3*Sqrt[x])","C",1
310,1,29,251,0.0064967,"\int \frac{1}{x^{5/2} \left(a+b x^2\right)^3} \, dx","Integrate[1/(x^(5/2)*(a + b*x^2)^3),x]","-\frac{2 \, _2F_1\left(-\frac{3}{4},3;\frac{1}{4};-\frac{b x^2}{a}\right)}{3 a^3 x^{3/2}}","\frac{77 b^{3/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{15/4}}-\frac{77 b^{3/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{15/4}}+\frac{77 b^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{32 \sqrt{2} a^{15/4}}-\frac{77 b^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{32 \sqrt{2} a^{15/4}}-\frac{77}{48 a^3 x^{3/2}}+\frac{11}{16 a^2 x^{3/2} \left(a+b x^2\right)}+\frac{1}{4 a x^{3/2} \left(a+b x^2\right)^2}",1,"(-2*Hypergeometric2F1[-3/4, 3, 1/4, -((b*x^2)/a)])/(3*a^3*x^(3/2))","C",1
311,1,29,264,0.0073116,"\int \frac{1}{x^{7/2} \left(a+b x^2\right)^3} \, dx","Integrate[1/(x^(7/2)*(a + b*x^2)^3),x]","-\frac{2 \, _2F_1\left(-\frac{5}{4},3;-\frac{1}{4};-\frac{b x^2}{a}\right)}{5 a^3 x^{5/2}}","\frac{117 b^{5/4} \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{17/4}}-\frac{117 b^{5/4} \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right)}{64 \sqrt{2} a^{17/4}}-\frac{117 b^{5/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{32 \sqrt{2} a^{17/4}}+\frac{117 b^{5/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}+1\right)}{32 \sqrt{2} a^{17/4}}+\frac{117 b}{16 a^4 \sqrt{x}}-\frac{117}{80 a^3 x^{5/2}}+\frac{13}{16 a^2 x^{5/2} \left(a+b x^2\right)}+\frac{1}{4 a x^{5/2} \left(a+b x^2\right)^2}",1,"(-2*Hypergeometric2F1[-5/4, 3, -1/4, -((b*x^2)/a)])/(5*a^3*x^(5/2))","C",1
312,1,48,58,0.0162891,"\int \frac{\sqrt{x}}{a-b x^2} \, dx","Integrate[Sqrt[x]/(a - b*x^2),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)-\tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt[4]{a} b^{3/4}}","\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt[4]{a} b^{3/4}}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a}}\right)}{\sqrt[4]{a} b^{3/4}}",1,"(-ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)] + ArcTanh[(b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(1/4)*b^(3/4))","A",1
313,1,108,108,0.0249845,"\int \frac{x^{7/2}}{1+x^2} \, dx","Integrate[x^(7/2)/(1 + x^2),x]","\frac{2 x^{5/2}}{5}-2 \sqrt{x}-\frac{\log \left(x-\sqrt{2} \sqrt{x}+1\right)}{2 \sqrt{2}}+\frac{\log \left(x+\sqrt{2} \sqrt{x}+1\right)}{2 \sqrt{2}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)}{\sqrt{2}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)}{\sqrt{2}}","\frac{2 x^{5/2}}{5}-2 \sqrt{x}-\frac{\log \left(x-\sqrt{2} \sqrt{x}+1\right)}{2 \sqrt{2}}+\frac{\log \left(x+\sqrt{2} \sqrt{x}+1\right)}{2 \sqrt{2}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)}{\sqrt{2}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)}{\sqrt{2}}",1,"-2*Sqrt[x] + (2*x^(5/2))/5 - ArcTan[1 - Sqrt[2]*Sqrt[x]]/Sqrt[2] + ArcTan[1 + Sqrt[2]*Sqrt[x]]/Sqrt[2] - Log[1 - Sqrt[2]*Sqrt[x] + x]/(2*Sqrt[2]) + Log[1 + Sqrt[2]*Sqrt[x] + x]/(2*Sqrt[2])","A",1
314,1,24,101,0.0067795,"\int \frac{x^{5/2}}{1+x^2} \, dx","Integrate[x^(5/2)/(1 + x^2),x]","-\frac{2}{3} x^{3/2} \left(\, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-x^2\right)-1\right)","\frac{2 x^{3/2}}{3}-\frac{\log \left(x-\sqrt{2} \sqrt{x}+1\right)}{2 \sqrt{2}}+\frac{\log \left(x+\sqrt{2} \sqrt{x}+1\right)}{2 \sqrt{2}}+\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)}{\sqrt{2}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)}{\sqrt{2}}",1,"(-2*x^(3/2)*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -x^2]))/3","C",1
315,1,99,99,0.0155447,"\int \frac{x^{3/2}}{1+x^2} \, dx","Integrate[x^(3/2)/(1 + x^2),x]","2 \sqrt{x}+\frac{\log \left(x-\sqrt{2} \sqrt{x}+1\right)}{2 \sqrt{2}}-\frac{\log \left(x+\sqrt{2} \sqrt{x}+1\right)}{2 \sqrt{2}}+\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)}{\sqrt{2}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)}{\sqrt{2}}","2 \sqrt{x}+\frac{\log \left(x-\sqrt{2} \sqrt{x}+1\right)}{2 \sqrt{2}}-\frac{\log \left(x+\sqrt{2} \sqrt{x}+1\right)}{2 \sqrt{2}}+\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)}{\sqrt{2}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)}{\sqrt{2}}",1,"2*Sqrt[x] + ArcTan[1 - Sqrt[2]*Sqrt[x]]/Sqrt[2] - ArcTan[1 + Sqrt[2]*Sqrt[x]]/Sqrt[2] + Log[1 - Sqrt[2]*Sqrt[x] + x]/(2*Sqrt[2]) - Log[1 + Sqrt[2]*Sqrt[x] + x]/(2*Sqrt[2])","A",1
316,1,22,92,0.0046555,"\int \frac{\sqrt{x}}{1+x^2} \, dx","Integrate[Sqrt[x]/(1 + x^2),x]","\frac{2}{3} x^{3/2} \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-x^2\right)","\frac{\log \left(x-\sqrt{2} \sqrt{x}+1\right)}{2 \sqrt{2}}-\frac{\log \left(x+\sqrt{2} \sqrt{x}+1\right)}{2 \sqrt{2}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)}{\sqrt{2}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)}{\sqrt{2}}",1,"(2*x^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -x^2])/3","C",1
317,1,76,92,0.0134631,"\int \frac{1}{\sqrt{x} \left(1+x^2\right)} \, dx","Integrate[1/(Sqrt[x]*(1 + x^2)),x]","\frac{-\log \left(x-\sqrt{2} \sqrt{x}+1\right)+\log \left(x+\sqrt{2} \sqrt{x}+1\right)-2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)+2 \tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)}{2 \sqrt{2}}","-\frac{\log \left(x-\sqrt{2} \sqrt{x}+1\right)}{2 \sqrt{2}}+\frac{\log \left(x+\sqrt{2} \sqrt{x}+1\right)}{2 \sqrt{2}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)}{\sqrt{2}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)}{\sqrt{2}}",1,"(-2*ArcTan[1 - Sqrt[2]*Sqrt[x]] + 2*ArcTan[1 + Sqrt[2]*Sqrt[x]] - Log[1 - Sqrt[2]*Sqrt[x] + x] + Log[1 + Sqrt[2]*Sqrt[x] + x])/(2*Sqrt[2])","A",1
318,1,20,99,0.0052837,"\int \frac{1}{x^{3/2} \left(1+x^2\right)} \, dx","Integrate[1/(x^(3/2)*(1 + x^2)),x]","-\frac{2 \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-x^2\right)}{\sqrt{x}}","-\frac{2}{\sqrt{x}}-\frac{\log \left(x-\sqrt{2} \sqrt{x}+1\right)}{2 \sqrt{2}}+\frac{\log \left(x+\sqrt{2} \sqrt{x}+1\right)}{2 \sqrt{2}}+\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)}{\sqrt{2}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)}{\sqrt{2}}",1,"(-2*Hypergeometric2F1[-1/4, 1, 3/4, -x^2])/Sqrt[x]","C",1
319,1,22,101,0.0056577,"\int \frac{1}{x^{5/2} \left(1+x^2\right)} \, dx","Integrate[1/(x^(5/2)*(1 + x^2)),x]","-\frac{2 \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-x^2\right)}{3 x^{3/2}}","-\frac{2}{3 x^{3/2}}+\frac{\log \left(x-\sqrt{2} \sqrt{x}+1\right)}{2 \sqrt{2}}-\frac{\log \left(x+\sqrt{2} \sqrt{x}+1\right)}{2 \sqrt{2}}+\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)}{\sqrt{2}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)}{\sqrt{2}}",1,"(-2*Hypergeometric2F1[-3/4, 1, 1/4, -x^2])/(3*x^(3/2))","C",1
320,1,22,108,0.005389,"\int \frac{1}{x^{7/2} \left(1+x^2\right)} \, dx","Integrate[1/(x^(7/2)*(1 + x^2)),x]","-\frac{2 \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-x^2\right)}{5 x^{5/2}}","-\frac{2}{5 x^{5/2}}+\frac{2}{\sqrt{x}}+\frac{\log \left(x-\sqrt{2} \sqrt{x}+1\right)}{2 \sqrt{2}}-\frac{\log \left(x+\sqrt{2} \sqrt{x}+1\right)}{2 \sqrt{2}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)}{\sqrt{2}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)}{\sqrt{2}}",1,"(-2*Hypergeometric2F1[-5/4, 1, -1/4, -x^2])/(5*x^(5/2))","C",1
321,1,121,122,0.050171,"\int \frac{x^{7/2}}{\left(1+x^2\right)^2} \, dx","Integrate[x^(7/2)/(1 + x^2)^2,x]","\frac{1}{16} \left(\frac{40 \sqrt{x}}{x^2+1}+\frac{32 x^{5/2}}{x^2+1}+5 \sqrt{2} \log \left(x-\sqrt{2} \sqrt{x}+1\right)-5 \sqrt{2} \log \left(x+\sqrt{2} \sqrt{x}+1\right)+10 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)-10 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)\right)","-\frac{x^{5/2}}{2 \left(x^2+1\right)}+\frac{5 \sqrt{x}}{2}+\frac{5 \log \left(x-\sqrt{2} \sqrt{x}+1\right)}{8 \sqrt{2}}-\frac{5 \log \left(x+\sqrt{2} \sqrt{x}+1\right)}{8 \sqrt{2}}+\frac{5 \tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)}{4 \sqrt{2}}-\frac{5 \tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)}{4 \sqrt{2}}",1,"((40*Sqrt[x])/(1 + x^2) + (32*x^(5/2))/(1 + x^2) + 10*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[x]] - 10*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[x]] + 5*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[x] + x] - 5*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[x] + x])/16","A",1
322,1,30,113,0.011336,"\int \frac{x^{5/2}}{\left(1+x^2\right)^2} \, dx","Integrate[x^(5/2)/(1 + x^2)^2,x]","2 x^{3/2} \left(\, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-x^2\right)-\frac{1}{x^2+1}\right)","-\frac{x^{3/2}}{2 \left(x^2+1\right)}+\frac{3 \log \left(x-\sqrt{2} \sqrt{x}+1\right)}{8 \sqrt{2}}-\frac{3 \log \left(x+\sqrt{2} \sqrt{x}+1\right)}{8 \sqrt{2}}-\frac{3 \tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)}{4 \sqrt{2}}+\frac{3 \tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)}{4 \sqrt{2}}",1,"2*x^(3/2)*(-(1 + x^2)^(-1) + Hypergeometric2F1[3/4, 2, 7/4, -x^2])","C",1
323,1,106,113,0.0453836,"\int \frac{x^{3/2}}{\left(1+x^2\right)^2} \, dx","Integrate[x^(3/2)/(1 + x^2)^2,x]","\frac{1}{16} \left(-\frac{8 \sqrt{x}}{x^2+1}-\sqrt{2} \log \left(x-\sqrt{2} \sqrt{x}+1\right)+\sqrt{2} \log \left(x+\sqrt{2} \sqrt{x}+1\right)-2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)+2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)\right)","-\frac{\sqrt{x}}{2 \left(x^2+1\right)}-\frac{\log \left(x-\sqrt{2} \sqrt{x}+1\right)}{8 \sqrt{2}}+\frac{\log \left(x+\sqrt{2} \sqrt{x}+1\right)}{8 \sqrt{2}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)}{4 \sqrt{2}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)}{4 \sqrt{2}}",1,"((-8*Sqrt[x])/(1 + x^2) - 2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[x]] + 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[x]] - Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[x] + x] + Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[x] + x])/16","A",1
324,1,22,113,0.0047843,"\int \frac{\sqrt{x}}{\left(1+x^2\right)^2} \, dx","Integrate[Sqrt[x]/(1 + x^2)^2,x]","\frac{2}{3} x^{3/2} \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-x^2\right)","\frac{x^{3/2}}{2 \left(x^2+1\right)}+\frac{\log \left(x-\sqrt{2} \sqrt{x}+1\right)}{8 \sqrt{2}}-\frac{\log \left(x+\sqrt{2} \sqrt{x}+1\right)}{8 \sqrt{2}}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)}{4 \sqrt{2}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)}{4 \sqrt{2}}",1,"(2*x^(3/2)*Hypergeometric2F1[3/4, 2, 7/4, -x^2])/3","C",1
325,1,107,113,0.0446689,"\int \frac{1}{\sqrt{x} \left(1+x^2\right)^2} \, dx","Integrate[1/(Sqrt[x]*(1 + x^2)^2),x]","\frac{1}{16} \left(\frac{8 \sqrt{x}}{x^2+1}-3 \sqrt{2} \log \left(x-\sqrt{2} \sqrt{x}+1\right)+3 \sqrt{2} \log \left(x+\sqrt{2} \sqrt{x}+1\right)-6 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)+6 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)\right)","\frac{\sqrt{x}}{2 \left(x^2+1\right)}-\frac{3 \log \left(x-\sqrt{2} \sqrt{x}+1\right)}{8 \sqrt{2}}+\frac{3 \log \left(x+\sqrt{2} \sqrt{x}+1\right)}{8 \sqrt{2}}-\frac{3 \tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)}{4 \sqrt{2}}+\frac{3 \tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)}{4 \sqrt{2}}",1,"((8*Sqrt[x])/(1 + x^2) - 6*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[x]] + 6*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[x]] - 3*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[x] + x] + 3*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[x] + x])/16","A",1
326,1,20,122,0.0051356,"\int \frac{1}{x^{3/2} \left(1+x^2\right)^2} \, dx","Integrate[1/(x^(3/2)*(1 + x^2)^2),x]","-\frac{2 \, _2F_1\left(-\frac{1}{4},2;\frac{3}{4};-x^2\right)}{\sqrt{x}}","\frac{1}{2 \sqrt{x} \left(x^2+1\right)}-\frac{5}{2 \sqrt{x}}-\frac{5 \log \left(x-\sqrt{2} \sqrt{x}+1\right)}{8 \sqrt{2}}+\frac{5 \log \left(x+\sqrt{2} \sqrt{x}+1\right)}{8 \sqrt{2}}+\frac{5 \tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)}{4 \sqrt{2}}-\frac{5 \tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)}{4 \sqrt{2}}",1,"(-2*Hypergeometric2F1[-1/4, 2, 3/4, -x^2])/Sqrt[x]","C",1
327,1,22,122,0.005285,"\int \frac{1}{x^{5/2} \left(1+x^2\right)^2} \, dx","Integrate[1/(x^(5/2)*(1 + x^2)^2),x]","-\frac{2 \, _2F_1\left(-\frac{3}{4},2;\frac{1}{4};-x^2\right)}{3 x^{3/2}}","-\frac{7}{6 x^{3/2}}+\frac{1}{2 x^{3/2} \left(x^2+1\right)}+\frac{7 \log \left(x-\sqrt{2} \sqrt{x}+1\right)}{8 \sqrt{2}}-\frac{7 \log \left(x+\sqrt{2} \sqrt{x}+1\right)}{8 \sqrt{2}}+\frac{7 \tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)}{4 \sqrt{2}}-\frac{7 \tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)}{4 \sqrt{2}}",1,"(-2*Hypergeometric2F1[-3/4, 2, 1/4, -x^2])/(3*x^(3/2))","C",1
328,1,22,131,0.0067672,"\int \frac{1}{x^{7/2} \left(1+x^2\right)^2} \, dx","Integrate[1/(x^(7/2)*(1 + x^2)^2),x]","-\frac{2 \, _2F_1\left(-\frac{5}{4},2;-\frac{1}{4};-x^2\right)}{5 x^{5/2}}","-\frac{9}{10 x^{5/2}}+\frac{1}{2 x^{5/2} \left(x^2+1\right)}+\frac{9}{2 \sqrt{x}}+\frac{9 \log \left(x-\sqrt{2} \sqrt{x}+1\right)}{8 \sqrt{2}}-\frac{9 \log \left(x+\sqrt{2} \sqrt{x}+1\right)}{8 \sqrt{2}}-\frac{9 \tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)}{4 \sqrt{2}}+\frac{9 \tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)}{4 \sqrt{2}}",1,"(-2*Hypergeometric2F1[-5/4, 2, -1/4, -x^2])/(5*x^(5/2))","C",1
329,1,135,129,0.046302,"\int \frac{x^{7/2}}{\left(1+x^2\right)^3} \, dx","Integrate[x^(7/2)/(1 + x^2)^3,x]","\frac{1}{384} \left(\frac{40 \sqrt{x}}{x^2+1}-\frac{160 \sqrt{x}}{\left(x^2+1\right)^2}-\frac{256 x^{5/2}}{\left(x^2+1\right)^2}-15 \sqrt{2} \log \left(x-\sqrt{2} \sqrt{x}+1\right)+15 \sqrt{2} \log \left(x+\sqrt{2} \sqrt{x}+1\right)-30 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)+30 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)\right)","-\frac{5 \sqrt{x}}{16 \left(x^2+1\right)}-\frac{x^{5/2}}{4 \left(x^2+1\right)^2}-\frac{5 \log \left(x-\sqrt{2} \sqrt{x}+1\right)}{64 \sqrt{2}}+\frac{5 \log \left(x+\sqrt{2} \sqrt{x}+1\right)}{64 \sqrt{2}}-\frac{5 \tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)}{32 \sqrt{2}}+\frac{5 \tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)}{32 \sqrt{2}}",1,"((-160*Sqrt[x])/(1 + x^2)^2 - (256*x^(5/2))/(1 + x^2)^2 + (40*Sqrt[x])/(1 + x^2) - 30*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[x]] + 30*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[x]] - 15*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[x] + x] + 15*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[x] + x])/384","A",1
330,1,32,129,0.0124671,"\int \frac{x^{5/2}}{\left(1+x^2\right)^3} \, dx","Integrate[x^(5/2)/(1 + x^2)^3,x]","\frac{2}{5} x^{3/2} \left(\, _2F_1\left(\frac{3}{4},3;\frac{7}{4};-x^2\right)-\frac{1}{\left(x^2+1\right)^2}\right)","\frac{3 x^{3/2}}{16 \left(x^2+1\right)}-\frac{x^{3/2}}{4 \left(x^2+1\right)^2}+\frac{3 \log \left(x-\sqrt{2} \sqrt{x}+1\right)}{64 \sqrt{2}}-\frac{3 \log \left(x+\sqrt{2} \sqrt{x}+1\right)}{64 \sqrt{2}}-\frac{3 \tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)}{32 \sqrt{2}}+\frac{3 \tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)}{32 \sqrt{2}}",1,"(2*x^(3/2)*(-(1 + x^2)^(-2) + Hypergeometric2F1[3/4, 3, 7/4, -x^2]))/5","C",1
331,1,121,129,0.0311933,"\int \frac{x^{3/2}}{\left(1+x^2\right)^3} \, dx","Integrate[x^(3/2)/(1 + x^2)^3,x]","\frac{1}{128} \left(\frac{8 \sqrt{x}}{x^2+1}-\frac{32 \sqrt{x}}{\left(x^2+1\right)^2}-3 \sqrt{2} \log \left(x-\sqrt{2} \sqrt{x}+1\right)+3 \sqrt{2} \log \left(x+\sqrt{2} \sqrt{x}+1\right)-6 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)+6 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)\right)","\frac{\sqrt{x}}{16 \left(x^2+1\right)}-\frac{\sqrt{x}}{4 \left(x^2+1\right)^2}-\frac{3 \log \left(x-\sqrt{2} \sqrt{x}+1\right)}{64 \sqrt{2}}+\frac{3 \log \left(x+\sqrt{2} \sqrt{x}+1\right)}{64 \sqrt{2}}-\frac{3 \tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)}{32 \sqrt{2}}+\frac{3 \tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)}{32 \sqrt{2}}",1,"((-32*Sqrt[x])/(1 + x^2)^2 + (8*Sqrt[x])/(1 + x^2) - 6*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[x]] + 6*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[x]] - 3*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[x] + x] + 3*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[x] + x])/128","A",1
332,1,22,129,0.0045533,"\int \frac{\sqrt{x}}{\left(1+x^2\right)^3} \, dx","Integrate[Sqrt[x]/(1 + x^2)^3,x]","\frac{2}{3} x^{3/2} \, _2F_1\left(\frac{3}{4},3;\frac{7}{4};-x^2\right)","\frac{5 x^{3/2}}{16 \left(x^2+1\right)}+\frac{x^{3/2}}{4 \left(x^2+1\right)^2}+\frac{5 \log \left(x-\sqrt{2} \sqrt{x}+1\right)}{64 \sqrt{2}}-\frac{5 \log \left(x+\sqrt{2} \sqrt{x}+1\right)}{64 \sqrt{2}}-\frac{5 \tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)}{32 \sqrt{2}}+\frac{5 \tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)}{32 \sqrt{2}}",1,"(2*x^(3/2)*Hypergeometric2F1[3/4, 3, 7/4, -x^2])/3","C",1
333,1,121,129,0.0331821,"\int \frac{1}{\sqrt{x} \left(1+x^2\right)^3} \, dx","Integrate[1/(Sqrt[x]*(1 + x^2)^3),x]","\frac{1}{128} \left(\frac{56 \sqrt{x}}{x^2+1}+\frac{32 \sqrt{x}}{\left(x^2+1\right)^2}-21 \sqrt{2} \log \left(x-\sqrt{2} \sqrt{x}+1\right)+21 \sqrt{2} \log \left(x+\sqrt{2} \sqrt{x}+1\right)-42 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)+42 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)\right)","\frac{7 \sqrt{x}}{16 \left(x^2+1\right)}+\frac{\sqrt{x}}{4 \left(x^2+1\right)^2}-\frac{21 \log \left(x-\sqrt{2} \sqrt{x}+1\right)}{64 \sqrt{2}}+\frac{21 \log \left(x+\sqrt{2} \sqrt{x}+1\right)}{64 \sqrt{2}}-\frac{21 \tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)}{32 \sqrt{2}}+\frac{21 \tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)}{32 \sqrt{2}}",1,"((32*Sqrt[x])/(1 + x^2)^2 + (56*Sqrt[x])/(1 + x^2) - 42*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[x]] + 42*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[x]] - 21*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[x] + x] + 21*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[x] + x])/128","A",1
334,1,20,138,0.0053231,"\int \frac{1}{x^{3/2} \left(1+x^2\right)^3} \, dx","Integrate[1/(x^(3/2)*(1 + x^2)^3),x]","-\frac{2 \, _2F_1\left(-\frac{1}{4},3;\frac{3}{4};-x^2\right)}{\sqrt{x}}","\frac{9}{16 \sqrt{x} \left(x^2+1\right)}+\frac{1}{4 \sqrt{x} \left(x^2+1\right)^2}-\frac{45}{16 \sqrt{x}}-\frac{45 \log \left(x-\sqrt{2} \sqrt{x}+1\right)}{64 \sqrt{2}}+\frac{45 \log \left(x+\sqrt{2} \sqrt{x}+1\right)}{64 \sqrt{2}}+\frac{45 \tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)}{32 \sqrt{2}}-\frac{45 \tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)}{32 \sqrt{2}}",1,"(-2*Hypergeometric2F1[-1/4, 3, 3/4, -x^2])/Sqrt[x]","C",1
335,1,22,138,0.0054345,"\int \frac{1}{x^{5/2} \left(1+x^2\right)^3} \, dx","Integrate[1/(x^(5/2)*(1 + x^2)^3),x]","-\frac{2 \, _2F_1\left(-\frac{3}{4},3;\frac{1}{4};-x^2\right)}{3 x^{3/2}}","-\frac{77}{48 x^{3/2}}+\frac{11}{16 x^{3/2} \left(x^2+1\right)}+\frac{1}{4 x^{3/2} \left(x^2+1\right)^2}+\frac{77 \log \left(x-\sqrt{2} \sqrt{x}+1\right)}{64 \sqrt{2}}-\frac{77 \log \left(x+\sqrt{2} \sqrt{x}+1\right)}{64 \sqrt{2}}+\frac{77 \tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)}{32 \sqrt{2}}-\frac{77 \tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)}{32 \sqrt{2}}",1,"(-2*Hypergeometric2F1[-3/4, 3, 1/4, -x^2])/(3*x^(3/2))","C",1
336,1,22,147,0.0056021,"\int \frac{1}{x^{7/2} \left(1+x^2\right)^3} \, dx","Integrate[1/(x^(7/2)*(1 + x^2)^3),x]","-\frac{2 \, _2F_1\left(-\frac{5}{4},3;-\frac{1}{4};-x^2\right)}{5 x^{5/2}}","-\frac{117}{80 x^{5/2}}+\frac{13}{16 x^{5/2} \left(x^2+1\right)}+\frac{1}{4 x^{5/2} \left(x^2+1\right)^2}+\frac{117}{16 \sqrt{x}}+\frac{117 \log \left(x-\sqrt{2} \sqrt{x}+1\right)}{64 \sqrt{2}}-\frac{117 \log \left(x+\sqrt{2} \sqrt{x}+1\right)}{64 \sqrt{2}}-\frac{117 \tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)}{32 \sqrt{2}}+\frac{117 \tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)}{32 \sqrt{2}}",1,"(-2*Hypergeometric2F1[-5/4, 3, -1/4, -x^2])/(5*x^(5/2))","C",1
337,1,15,15,0.0038056,"\int \frac{\sqrt{x}}{1-x^2} \, dx","Integrate[Sqrt[x]/(1 - x^2),x]","\tanh ^{-1}\left(\sqrt{x}\right)-\tan ^{-1}\left(\sqrt{x}\right)","\tanh ^{-1}\left(\sqrt{x}\right)-\tan ^{-1}\left(\sqrt{x}\right)",1,"-ArcTan[Sqrt[x]] + ArcTanh[Sqrt[x]]","A",1
338,1,22,73,0.0049064,"\int \frac{x^{2/3}}{1+x^2} \, dx","Integrate[x^(2/3)/(1 + x^2),x]","\frac{3}{5} x^{5/3} \, _2F_1\left(\frac{5}{6},1;\frac{11}{6};-x^2\right)","-\frac{1}{2} \sqrt{3} \tanh ^{-1}\left(\frac{\sqrt{3} \sqrt[3]{x}}{x^{2/3}+1}\right)-\frac{1}{2} \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{x}\right)+\frac{1}{2} \tan ^{-1}\left(2 \sqrt[3]{x}+\sqrt{3}\right)+\tan ^{-1}\left(\sqrt[3]{x}\right)",1,"(3*x^(5/3)*Hypergeometric2F1[5/6, 1, 11/6, -x^2])/5","C",1
339,1,88,97,0.0551239,"\int x^m \left(a+b x^2\right)^5 \, dx","Integrate[x^m*(a + b*x^2)^5,x]","x^{m+1} \left(\frac{a^5}{m+1}+\frac{5 a^4 b x^2}{m+3}+\frac{10 a^3 b^2 x^4}{m+5}+\frac{10 a^2 b^3 x^6}{m+7}+\frac{5 a b^4 x^8}{m+9}+\frac{b^5 x^{10}}{m+11}\right)","\frac{a^5 x^{m+1}}{m+1}+\frac{5 a^4 b x^{m+3}}{m+3}+\frac{10 a^3 b^2 x^{m+5}}{m+5}+\frac{10 a^2 b^3 x^{m+7}}{m+7}+\frac{5 a b^4 x^{m+9}}{m+9}+\frac{b^5 x^{m+11}}{m+11}",1,"x^(1 + m)*(a^5/(1 + m) + (5*a^4*b*x^2)/(3 + m) + (10*a^3*b^2*x^4)/(5 + m) + (10*a^2*b^3*x^6)/(7 + m) + (5*a*b^4*x^8)/(9 + m) + (b^5*x^10)/(11 + m))","A",1
340,1,72,79,0.035023,"\int x^m \left(a+b x^2\right)^4 \, dx","Integrate[x^m*(a + b*x^2)^4,x]","x^{m+1} \left(\frac{a^4}{m+1}+\frac{4 a^3 b x^2}{m+3}+\frac{6 a^2 b^2 x^4}{m+5}+\frac{4 a b^3 x^6}{m+7}+\frac{b^4 x^8}{m+9}\right)","\frac{a^4 x^{m+1}}{m+1}+\frac{4 a^3 b x^{m+3}}{m+3}+\frac{6 a^2 b^2 x^{m+5}}{m+5}+\frac{4 a b^3 x^{m+7}}{m+7}+\frac{b^4 x^{m+9}}{m+9}",1,"x^(1 + m)*(a^4/(1 + m) + (4*a^3*b*x^2)/(3 + m) + (6*a^2*b^2*x^4)/(5 + m) + (4*a*b^3*x^6)/(7 + m) + (b^4*x^8)/(9 + m))","A",1
341,1,56,61,0.0307725,"\int x^m \left(a+b x^2\right)^3 \, dx","Integrate[x^m*(a + b*x^2)^3,x]","x^{m+1} \left(\frac{a^3}{m+1}+\frac{3 a^2 b x^2}{m+3}+\frac{3 a b^2 x^4}{m+5}+\frac{b^3 x^6}{m+7}\right)","\frac{a^3 x^{m+1}}{m+1}+\frac{3 a^2 b x^{m+3}}{m+3}+\frac{3 a b^2 x^{m+5}}{m+5}+\frac{b^3 x^{m+7}}{m+7}",1,"x^(1 + m)*(a^3/(1 + m) + (3*a^2*b*x^2)/(3 + m) + (3*a*b^2*x^4)/(5 + m) + (b^3*x^6)/(7 + m))","A",1
342,1,40,43,0.0229903,"\int x^m \left(a+b x^2\right)^2 \, dx","Integrate[x^m*(a + b*x^2)^2,x]","x^{m+1} \left(\frac{a^2}{m+1}+\frac{2 a b x^2}{m+3}+\frac{b^2 x^4}{m+5}\right)","\frac{a^2 x^{m+1}}{m+1}+\frac{2 a b x^{m+3}}{m+3}+\frac{b^2 x^{m+5}}{m+5}",1,"x^(1 + m)*(a^2/(1 + m) + (2*a*b*x^2)/(3 + m) + (b^2*x^4)/(5 + m))","A",1
343,1,25,25,0.0137921,"\int x^m \left(a+b x^2\right) \, dx","Integrate[x^m*(a + b*x^2),x]","\frac{a x^{m+1}}{m+1}+\frac{b x^{m+3}}{m+3}","\frac{a x^{m+1}}{m+1}+\frac{b x^{m+3}}{m+3}",1,"(a*x^(1 + m))/(1 + m) + (b*x^(3 + m))/(3 + m)","A",1
344,1,41,39,0.0081768,"\int \frac{x^m}{a+b x^2} \, dx","Integrate[x^m/(a + b*x^2),x]","\frac{x^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+1}{2}+1;-\frac{b x^2}{a}\right)}{a (m+1)}","\frac{x^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a (m+1)}",1,"(x^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, 1 + (1 + m)/2, -((b*x^2)/a)])/(a*(1 + m))","A",1
345,1,41,39,0.0073252,"\int \frac{x^m}{\left(a+b x^2\right)^2} \, dx","Integrate[x^m/(a + b*x^2)^2,x]","\frac{x^{m+1} \, _2F_1\left(2,\frac{m+1}{2};\frac{m+1}{2}+1;-\frac{b x^2}{a}\right)}{a^2 (m+1)}","\frac{x^{m+1} \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a^2 (m+1)}",1,"(x^(1 + m)*Hypergeometric2F1[2, (1 + m)/2, 1 + (1 + m)/2, -((b*x^2)/a)])/(a^2*(1 + m))","A",1
346,1,41,39,0.0075158,"\int \frac{x^m}{\left(a+b x^2\right)^3} \, dx","Integrate[x^m/(a + b*x^2)^3,x]","\frac{x^{m+1} \, _2F_1\left(3,\frac{m+1}{2};\frac{m+1}{2}+1;-\frac{b x^2}{a}\right)}{a^3 (m+1)}","\frac{x^{m+1} \, _2F_1\left(3,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a^3 (m+1)}",1,"(x^(1 + m)*Hypergeometric2F1[3, (1 + m)/2, 1 + (1 + m)/2, -((b*x^2)/a)])/(a^3*(1 + m))","A",1
347,1,45,44,0.0143739,"\int \frac{(c x)^{1+m}}{a+b x^2} \, dx","Integrate[(c*x)^(1 + m)/(a + b*x^2),x]","\frac{c x^2 (c x)^m \, _2F_1\left(1,\frac{m+2}{2};\frac{m+2}{2}+1;-\frac{b x^2}{a}\right)}{a (m+2)}","\frac{(c x)^{m+2} \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-\frac{b x^2}{a}\right)}{a c (m+2)}",1,"(c*x^2*(c*x)^m*Hypergeometric2F1[1, (2 + m)/2, 1 + (2 + m)/2, -((b*x^2)/a)])/(a*(2 + m))","A",1
348,1,42,44,0.0079556,"\int \frac{(c x)^m}{a+b x^2} \, dx","Integrate[(c*x)^m/(a + b*x^2),x]","\frac{x (c x)^m \, _2F_1\left(1,\frac{m+1}{2};\frac{m+1}{2}+1;-\frac{b x^2}{a}\right)}{a (m+1)}","\frac{(c x)^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a c (m+1)}",1,"(x*(c*x)^m*Hypergeometric2F1[1, (1 + m)/2, 1 + (1 + m)/2, -((b*x^2)/a)])/(a*(1 + m))","A",1
349,1,38,38,0.0076541,"\int \frac{(c x)^{-1+m}}{a+b x^2} \, dx","Integrate[(c*x)^(-1 + m)/(a + b*x^2),x]","\frac{x (c x)^{m-1} \, _2F_1\left(1,\frac{m}{2};\frac{m}{2}+1;-\frac{b x^2}{a}\right)}{a m}","\frac{(c x)^m \, _2F_1\left(1,\frac{m}{2};\frac{m+2}{2};-\frac{b x^2}{a}\right)}{a c m}",1,"(x*(c*x)^(-1 + m)*Hypergeometric2F1[1, m/2, 1 + m/2, -((b*x^2)/a)])/(a*m)","A",1
350,1,44,47,0.0116588,"\int \frac{(c x)^{-2+m}}{a+b x^2} \, dx","Integrate[(c*x)^(-2 + m)/(a + b*x^2),x]","\frac{x (c x)^{m-2} \, _2F_1\left(1,\frac{m-1}{2};\frac{m-1}{2}+1;-\frac{b x^2}{a}\right)}{a (m-1)}","-\frac{(c x)^{m-1} \, _2F_1\left(1,\frac{m-1}{2};\frac{m+1}{2};-\frac{b x^2}{a}\right)}{a c (1-m)}",1,"(x*(c*x)^(-2 + m)*Hypergeometric2F1[1, (-1 + m)/2, 1 + (-1 + m)/2, -((b*x^2)/a)])/(a*(-1 + m))","A",1
351,1,44,45,0.0115678,"\int \frac{(c x)^{-3+m}}{a+b x^2} \, dx","Integrate[(c*x)^(-3 + m)/(a + b*x^2),x]","\frac{x (c x)^{m-3} \, _2F_1\left(1,\frac{m-2}{2};\frac{m-2}{2}+1;-\frac{b x^2}{a}\right)}{a (m-2)}","-\frac{(c x)^{m-2} \, _2F_1\left(1,\frac{m-2}{2};\frac{m}{2};-\frac{b x^2}{a}\right)}{a c (2-m)}",1,"(x*(c*x)^(-3 + m)*Hypergeometric2F1[1, (-2 + m)/2, 1 + (-2 + m)/2, -((b*x^2)/a)])/(a*(-2 + m))","A",1
352,1,38,36,0.0091106,"\int \frac{x^m}{\left(1+\frac{a x^2}{b}\right)^2} \, dx","Integrate[x^m/(1 + (a*x^2)/b)^2,x]","\frac{x^{m+1} \, _2F_1\left(2,\frac{m+1}{2};\frac{m+1}{2}+1;-\frac{a x^2}{b}\right)}{m+1}","\frac{x^{m+1} \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{a x^2}{b}\right)}{m+1}",1,"(x^(1 + m)*Hypergeometric2F1[2, (1 + m)/2, 1 + (1 + m)/2, -((a*x^2)/b)])/(1 + m)","A",1
353,1,50,80,0.0264661,"\int x^7 \sqrt{a+b x^2} \, dx","Integrate[x^7*Sqrt[a + b*x^2],x]","\frac{\left(a+b x^2\right)^{3/2} \left(-16 a^3+24 a^2 b x^2-30 a b^2 x^4+35 b^3 x^6\right)}{315 b^4}","-\frac{a^3 \left(a+b x^2\right)^{3/2}}{3 b^4}+\frac{3 a^2 \left(a+b x^2\right)^{5/2}}{5 b^4}+\frac{\left(a+b x^2\right)^{9/2}}{9 b^4}-\frac{3 a \left(a+b x^2\right)^{7/2}}{7 b^4}",1,"((a + b*x^2)^(3/2)*(-16*a^3 + 24*a^2*b*x^2 - 30*a*b^2*x^4 + 35*b^3*x^6))/(315*b^4)","A",1
354,1,39,59,0.0189525,"\int x^5 \sqrt{a+b x^2} \, dx","Integrate[x^5*Sqrt[a + b*x^2],x]","\frac{\left(a+b x^2\right)^{3/2} \left(8 a^2-12 a b x^2+15 b^2 x^4\right)}{105 b^3}","\frac{a^2 \left(a+b x^2\right)^{3/2}}{3 b^3}+\frac{\left(a+b x^2\right)^{7/2}}{7 b^3}-\frac{2 a \left(a+b x^2\right)^{5/2}}{5 b^3}",1,"((a + b*x^2)^(3/2)*(8*a^2 - 12*a*b*x^2 + 15*b^2*x^4))/(105*b^3)","A",1
355,1,28,38,0.0141072,"\int x^3 \sqrt{a+b x^2} \, dx","Integrate[x^3*Sqrt[a + b*x^2],x]","\frac{\left(a+b x^2\right)^{3/2} \left(3 b x^2-2 a\right)}{15 b^2}","\frac{\left(a+b x^2\right)^{5/2}}{5 b^2}-\frac{a \left(a+b x^2\right)^{3/2}}{3 b^2}",1,"((a + b*x^2)^(3/2)*(-2*a + 3*b*x^2))/(15*b^2)","A",1
356,1,18,18,0.0022685,"\int x \sqrt{a+b x^2} \, dx","Integrate[x*Sqrt[a + b*x^2],x]","\frac{\left(a+b x^2\right)^{3/2}}{3 b}","\frac{\left(a+b x^2\right)^{3/2}}{3 b}",1,"(a + b*x^2)^(3/2)/(3*b)","A",1
357,1,37,37,0.008077,"\int \frac{\sqrt{a+b x^2}}{x} \, dx","Integrate[Sqrt[a + b*x^2]/x,x]","\sqrt{a+b x^2}-\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)","\sqrt{a+b x^2}-\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)",1,"Sqrt[a + b*x^2] - Sqrt[a]*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]","A",1
358,1,59,47,0.032818,"\int \frac{\sqrt{a+b x^2}}{x^3} \, dx","Integrate[Sqrt[a + b*x^2]/x^3,x]","-\frac{b x^2 \sqrt{\frac{b x^2}{a}+1} \tanh ^{-1}\left(\sqrt{\frac{b x^2}{a}+1}\right)+a+b x^2}{2 x^2 \sqrt{a+b x^2}}","-\frac{\sqrt{a+b x^2}}{2 x^2}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{2 \sqrt{a}}",1,"-1/2*(a + b*x^2 + b*x^2*Sqrt[1 + (b*x^2)/a]*ArcTanh[Sqrt[1 + (b*x^2)/a]])/(x^2*Sqrt[a + b*x^2])","A",1
359,1,39,71,0.0085478,"\int \frac{\sqrt{a+b x^2}}{x^5} \, dx","Integrate[Sqrt[a + b*x^2]/x^5,x]","-\frac{b^2 \left(a+b x^2\right)^{3/2} \, _2F_1\left(\frac{3}{2},3;\frac{5}{2};\frac{b x^2}{a}+1\right)}{3 a^3}","\frac{b^2 \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{8 a^{3/2}}-\frac{b \sqrt{a+b x^2}}{8 a x^2}-\frac{\sqrt{a+b x^2}}{4 x^4}",1,"-1/3*(b^2*(a + b*x^2)^(3/2)*Hypergeometric2F1[3/2, 3, 5/2, 1 + (b*x^2)/a])/a^3","C",1
360,1,39,95,0.0083955,"\int \frac{\sqrt{a+b x^2}}{x^7} \, dx","Integrate[Sqrt[a + b*x^2]/x^7,x]","\frac{b^3 \left(a+b x^2\right)^{3/2} \, _2F_1\left(\frac{3}{2},4;\frac{5}{2};\frac{b x^2}{a}+1\right)}{3 a^4}","-\frac{b^3 \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{16 a^{5/2}}+\frac{b^2 \sqrt{a+b x^2}}{16 a^2 x^2}-\frac{\sqrt{a+b x^2}}{6 x^6}-\frac{b \sqrt{a+b x^2}}{24 a x^4}",1,"(b^3*(a + b*x^2)^(3/2)*Hypergeometric2F1[3/2, 4, 5/2, 1 + (b*x^2)/a])/(3*a^4)","C",1
361,1,77,94,0.0335569,"\int x^4 \sqrt{a+b x^2} \, dx","Integrate[x^4*Sqrt[a + b*x^2],x]","\frac{a^3 \log \left(\sqrt{b} \sqrt{a+b x^2}+b x\right)}{16 b^{5/2}}+\sqrt{a+b x^2} \left(-\frac{a^2 x}{16 b^2}+\frac{a x^3}{24 b}+\frac{x^5}{6}\right)","\frac{a^3 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 b^{5/2}}-\frac{a^2 x \sqrt{a+b x^2}}{16 b^2}+\frac{1}{6} x^5 \sqrt{a+b x^2}+\frac{a x^3 \sqrt{a+b x^2}}{24 b}",1,"Sqrt[a + b*x^2]*(-1/16*(a^2*x)/b^2 + (a*x^3)/(24*b) + x^5/6) + (a^3*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(16*b^(5/2))","A",1
362,1,64,70,0.0238099,"\int x^2 \sqrt{a+b x^2} \, dx","Integrate[x^2*Sqrt[a + b*x^2],x]","\sqrt{a+b x^2} \left(\frac{a x}{8 b}+\frac{x^3}{4}\right)-\frac{a^2 \log \left(\sqrt{b} \sqrt{a+b x^2}+b x\right)}{8 b^{3/2}}","-\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 b^{3/2}}+\frac{a x \sqrt{a+b x^2}}{8 b}+\frac{1}{4} x^3 \sqrt{a+b x^2}",1,"Sqrt[a + b*x^2]*((a*x)/(8*b) + x^3/4) - (a^2*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(8*b^(3/2))","A",1
363,1,49,46,0.0142851,"\int \sqrt{a+b x^2} \, dx","Integrate[Sqrt[a + b*x^2],x]","\frac{1}{2} x \sqrt{a+b x^2}+\frac{a \log \left(\sqrt{b} \sqrt{a+b x^2}+b x\right)}{2 \sqrt{b}}","\frac{1}{2} x \sqrt{a+b x^2}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 \sqrt{b}}",1,"(x*Sqrt[a + b*x^2])/2 + (a*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(2*Sqrt[b])","A",1
364,1,63,42,0.0719338,"\int \frac{\sqrt{a+b x^2}}{x^2} \, dx","Integrate[Sqrt[a + b*x^2]/x^2,x]","-\frac{-\sqrt{a} \sqrt{b} x \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)+a+b x^2}{x \sqrt{a+b x^2}}","\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)-\frac{\sqrt{a+b x^2}}{x}",1,"-((a + b*x^2 - Sqrt[a]*Sqrt[b]*x*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(x*Sqrt[a + b*x^2]))","A",1
365,1,21,21,0.0052437,"\int \frac{\sqrt{a+b x^2}}{x^4} \, dx","Integrate[Sqrt[a + b*x^2]/x^4,x]","-\frac{\left(a+b x^2\right)^{3/2}}{3 a x^3}","-\frac{\left(a+b x^2\right)^{3/2}}{3 a x^3}",1,"-1/3*(a + b*x^2)^(3/2)/(a*x^3)","A",1
366,1,31,44,0.008671,"\int \frac{\sqrt{a+b x^2}}{x^6} \, dx","Integrate[Sqrt[a + b*x^2]/x^6,x]","\frac{\left(a+b x^2\right)^{3/2} \left(2 b x^2-3 a\right)}{15 a^2 x^5}","\frac{2 b \left(a+b x^2\right)^{3/2}}{15 a^2 x^3}-\frac{\left(a+b x^2\right)^{3/2}}{5 a x^5}",1,"((a + b*x^2)^(3/2)*(-3*a + 2*b*x^2))/(15*a^2*x^5)","A",1
367,1,42,68,0.0097621,"\int \frac{\sqrt{a+b x^2}}{x^8} \, dx","Integrate[Sqrt[a + b*x^2]/x^8,x]","-\frac{\left(a+b x^2\right)^{3/2} \left(15 a^2-12 a b x^2+8 b^2 x^4\right)}{105 a^3 x^7}","-\frac{8 b^2 \left(a+b x^2\right)^{3/2}}{105 a^3 x^3}+\frac{4 b \left(a+b x^2\right)^{3/2}}{35 a^2 x^5}-\frac{\left(a+b x^2\right)^{3/2}}{7 a x^7}",1,"-1/105*((a + b*x^2)^(3/2)*(15*a^2 - 12*a*b*x^2 + 8*b^2*x^4))/(a^3*x^7)","A",1
368,1,53,92,0.0121279,"\int \frac{\sqrt{a+b x^2}}{x^{10}} \, dx","Integrate[Sqrt[a + b*x^2]/x^10,x]","\frac{\left(a+b x^2\right)^{3/2} \left(-35 a^3+30 a^2 b x^2-24 a b^2 x^4+16 b^3 x^6\right)}{315 a^4 x^9}","\frac{16 b^3 \left(a+b x^2\right)^{3/2}}{315 a^4 x^3}-\frac{8 b^2 \left(a+b x^2\right)^{3/2}}{105 a^3 x^5}+\frac{2 b \left(a+b x^2\right)^{3/2}}{21 a^2 x^7}-\frac{\left(a+b x^2\right)^{3/2}}{9 a x^9}",1,"((a + b*x^2)^(3/2)*(-35*a^3 + 30*a^2*b*x^2 - 24*a*b^2*x^4 + 16*b^3*x^6))/(315*a^4*x^9)","A",1
369,1,50,80,0.0257607,"\int x^7 \left(a+b x^2\right)^{3/2} \, dx","Integrate[x^7*(a + b*x^2)^(3/2),x]","\frac{\left(a+b x^2\right)^{5/2} \left(-16 a^3+40 a^2 b x^2-70 a b^2 x^4+105 b^3 x^6\right)}{1155 b^4}","-\frac{a^3 \left(a+b x^2\right)^{5/2}}{5 b^4}+\frac{3 a^2 \left(a+b x^2\right)^{7/2}}{7 b^4}+\frac{\left(a+b x^2\right)^{11/2}}{11 b^4}-\frac{a \left(a+b x^2\right)^{9/2}}{3 b^4}",1,"((a + b*x^2)^(5/2)*(-16*a^3 + 40*a^2*b*x^2 - 70*a*b^2*x^4 + 105*b^3*x^6))/(1155*b^4)","A",1
370,1,39,59,0.0185706,"\int x^5 \left(a+b x^2\right)^{3/2} \, dx","Integrate[x^5*(a + b*x^2)^(3/2),x]","\frac{\left(a+b x^2\right)^{5/2} \left(8 a^2-20 a b x^2+35 b^2 x^4\right)}{315 b^3}","\frac{a^2 \left(a+b x^2\right)^{5/2}}{5 b^3}+\frac{\left(a+b x^2\right)^{9/2}}{9 b^3}-\frac{2 a \left(a+b x^2\right)^{7/2}}{7 b^3}",1,"((a + b*x^2)^(5/2)*(8*a^2 - 20*a*b*x^2 + 35*b^2*x^4))/(315*b^3)","A",1
371,1,28,38,0.0142575,"\int x^3 \left(a+b x^2\right)^{3/2} \, dx","Integrate[x^3*(a + b*x^2)^(3/2),x]","\frac{\left(a+b x^2\right)^{5/2} \left(5 b x^2-2 a\right)}{35 b^2}","\frac{\left(a+b x^2\right)^{7/2}}{7 b^2}-\frac{a \left(a+b x^2\right)^{5/2}}{5 b^2}",1,"((a + b*x^2)^(5/2)*(-2*a + 5*b*x^2))/(35*b^2)","A",1
372,1,18,18,0.0032622,"\int x \left(a+b x^2\right)^{3/2} \, dx","Integrate[x*(a + b*x^2)^(3/2),x]","\frac{\left(a+b x^2\right)^{5/2}}{5 b}","\frac{\left(a+b x^2\right)^{5/2}}{5 b}",1,"(a + b*x^2)^(5/2)/(5*b)","A",1
373,1,50,54,0.0209019,"\int \frac{\left(a+b x^2\right)^{3/2}}{x} \, dx","Integrate[(a + b*x^2)^(3/2)/x,x]","\frac{1}{3} \sqrt{a+b x^2} \left(4 a+b x^2\right)-a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)","a^{3/2} \left(-\tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)\right)+a \sqrt{a+b x^2}+\frac{1}{3} \left(a+b x^2\right)^{3/2}",1,"(Sqrt[a + b*x^2]*(4*a + b*x^2))/3 - a^(3/2)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]","A",1
374,1,37,63,0.0089414,"\int \frac{\left(a+b x^2\right)^{3/2}}{x^3} \, dx","Integrate[(a + b*x^2)^(3/2)/x^3,x]","\frac{b \left(a+b x^2\right)^{5/2} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{b x^2}{a}+1\right)}{5 a^2}","-\frac{\left(a+b x^2\right)^{3/2}}{2 x^2}+\frac{3}{2} b \sqrt{a+b x^2}-\frac{3}{2} \sqrt{a} b \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)",1,"(b*(a + b*x^2)^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, 1 + (b*x^2)/a])/(5*a^2)","C",1
375,1,76,68,0.03726,"\int \frac{\left(a+b x^2\right)^{3/2}}{x^5} \, dx","Integrate[(a + b*x^2)^(3/2)/x^5,x]","-\frac{2 a^2+3 b^2 x^4 \sqrt{\frac{b x^2}{a}+1} \tanh ^{-1}\left(\sqrt{\frac{b x^2}{a}+1}\right)+7 a b x^2+5 b^2 x^4}{8 x^4 \sqrt{a+b x^2}}","-\frac{3 b^2 \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{8 \sqrt{a}}-\frac{3 b \sqrt{a+b x^2}}{8 x^2}-\frac{\left(a+b x^2\right)^{3/2}}{4 x^4}",1,"-1/8*(2*a^2 + 7*a*b*x^2 + 5*b^2*x^4 + 3*b^2*x^4*Sqrt[1 + (b*x^2)/a]*ArcTanh[Sqrt[1 + (b*x^2)/a]])/(x^4*Sqrt[a + b*x^2])","A",1
376,1,39,92,0.0097015,"\int \frac{\left(a+b x^2\right)^{3/2}}{x^7} \, dx","Integrate[(a + b*x^2)^(3/2)/x^7,x]","\frac{b^3 \left(a+b x^2\right)^{5/2} \, _2F_1\left(\frac{5}{2},4;\frac{7}{2};\frac{b x^2}{a}+1\right)}{5 a^4}","\frac{b^3 \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{16 a^{3/2}}-\frac{b^2 \sqrt{a+b x^2}}{16 a x^2}-\frac{\left(a+b x^2\right)^{3/2}}{6 x^6}-\frac{b \sqrt{a+b x^2}}{8 x^4}",1,"(b^3*(a + b*x^2)^(5/2)*Hypergeometric2F1[5/2, 4, 7/2, 1 + (b*x^2)/a])/(5*a^4)","C",1
377,1,39,116,0.0101064,"\int \frac{\left(a+b x^2\right)^{3/2}}{x^9} \, dx","Integrate[(a + b*x^2)^(3/2)/x^9,x]","-\frac{b^4 \left(a+b x^2\right)^{5/2} \, _2F_1\left(\frac{5}{2},5;\frac{7}{2};\frac{b x^2}{a}+1\right)}{5 a^5}","-\frac{3 b^4 \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{128 a^{5/2}}+\frac{3 b^3 \sqrt{a+b x^2}}{128 a^2 x^2}-\frac{b^2 \sqrt{a+b x^2}}{64 a x^4}-\frac{\left(a+b x^2\right)^{3/2}}{8 x^8}-\frac{b \sqrt{a+b x^2}}{16 x^6}",1,"-1/5*(b^4*(a + b*x^2)^(5/2)*Hypergeometric2F1[5/2, 5, 7/2, 1 + (b*x^2)/a])/a^5","C",1
378,1,94,115,0.1355924,"\int x^4 \left(a+b x^2\right)^{3/2} \, dx","Integrate[x^4*(a + b*x^2)^(3/2),x]","\frac{\sqrt{a+b x^2} \left(\frac{3 a^{7/2} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{\frac{b x^2}{a}+1}}+\sqrt{b} x \left(-3 a^3+2 a^2 b x^2+24 a b^2 x^4+16 b^3 x^6\right)\right)}{128 b^{5/2}}","\frac{3 a^4 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{128 b^{5/2}}-\frac{3 a^3 x \sqrt{a+b x^2}}{128 b^2}+\frac{a^2 x^3 \sqrt{a+b x^2}}{64 b}+\frac{1}{8} x^5 \left(a+b x^2\right)^{3/2}+\frac{1}{16} a x^5 \sqrt{a+b x^2}",1,"(Sqrt[a + b*x^2]*(Sqrt[b]*x*(-3*a^3 + 2*a^2*b*x^2 + 24*a*b^2*x^4 + 16*b^3*x^6) + (3*a^(7/2)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(128*b^(5/2))","A",1
379,1,83,91,0.1183043,"\int x^2 \left(a+b x^2\right)^{3/2} \, dx","Integrate[x^2*(a + b*x^2)^(3/2),x]","\frac{\sqrt{a+b x^2} \left(\sqrt{b} x \left(3 a^2+14 a b x^2+8 b^2 x^4\right)-\frac{3 a^{5/2} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{\frac{b x^2}{a}+1}}\right)}{48 b^{3/2}}","-\frac{a^3 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 b^{3/2}}+\frac{a^2 x \sqrt{a+b x^2}}{16 b}+\frac{1}{8} a x^3 \sqrt{a+b x^2}+\frac{1}{6} x^3 \left(a+b x^2\right)^{3/2}",1,"(Sqrt[a + b*x^2]*(Sqrt[b]*x*(3*a^2 + 14*a*b*x^2 + 8*b^2*x^4) - (3*a^(5/2)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(48*b^(3/2))","A",1
380,1,65,65,0.0893905,"\int \left(a+b x^2\right)^{3/2} \, dx","Integrate[(a + b*x^2)^(3/2),x]","\frac{1}{8} \sqrt{a+b x^2} \left(\frac{3 a^{3/2} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{\frac{b x^2}{a}+1}}+5 a x+2 b x^3\right)","\frac{3 a^2 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 \sqrt{b}}+\frac{3}{8} a x \sqrt{a+b x^2}+\frac{1}{4} x \left(a+b x^2\right)^{3/2}",1,"(Sqrt[a + b*x^2]*(5*a*x + 2*b*x^3 + (3*a^(3/2)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*Sqrt[1 + (b*x^2)/a])))/8","A",1
381,1,50,63,0.008353,"\int \frac{\left(a+b x^2\right)^{3/2}}{x^2} \, dx","Integrate[(a + b*x^2)^(3/2)/x^2,x]","-\frac{a \sqrt{a+b x^2} \, _2F_1\left(-\frac{3}{2},-\frac{1}{2};\frac{1}{2};-\frac{b x^2}{a}\right)}{x \sqrt{\frac{b x^2}{a}+1}}","-\frac{\left(a+b x^2\right)^{3/2}}{x}+\frac{3}{2} b x \sqrt{a+b x^2}+\frac{3}{2} a \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)",1,"-((a*Sqrt[a + b*x^2]*Hypergeometric2F1[-3/2, -1/2, 1/2, -((b*x^2)/a)])/(x*Sqrt[1 + (b*x^2)/a]))","C",1
382,1,52,61,0.0084243,"\int \frac{\left(a+b x^2\right)^{3/2}}{x^4} \, dx","Integrate[(a + b*x^2)^(3/2)/x^4,x]","-\frac{a \sqrt{a+b x^2} \, _2F_1\left(-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};-\frac{b x^2}{a}\right)}{3 x^3 \sqrt{\frac{b x^2}{a}+1}}","b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)-\frac{b \sqrt{a+b x^2}}{x}-\frac{\left(a+b x^2\right)^{3/2}}{3 x^3}",1,"-1/3*(a*Sqrt[a + b*x^2]*Hypergeometric2F1[-3/2, -3/2, -1/2, -((b*x^2)/a)])/(x^3*Sqrt[1 + (b*x^2)/a])","C",1
383,1,21,21,0.0064649,"\int \frac{\left(a+b x^2\right)^{3/2}}{x^6} \, dx","Integrate[(a + b*x^2)^(3/2)/x^6,x]","-\frac{\left(a+b x^2\right)^{5/2}}{5 a x^5}","-\frac{\left(a+b x^2\right)^{5/2}}{5 a x^5}",1,"-1/5*(a + b*x^2)^(5/2)/(a*x^5)","A",1
384,1,31,44,0.0096684,"\int \frac{\left(a+b x^2\right)^{3/2}}{x^8} \, dx","Integrate[(a + b*x^2)^(3/2)/x^8,x]","\frac{\left(a+b x^2\right)^{5/2} \left(2 b x^2-5 a\right)}{35 a^2 x^7}","\frac{2 b \left(a+b x^2\right)^{5/2}}{35 a^2 x^5}-\frac{\left(a+b x^2\right)^{5/2}}{7 a x^7}",1,"((a + b*x^2)^(5/2)*(-5*a + 2*b*x^2))/(35*a^2*x^7)","A",1
385,1,42,68,0.0107718,"\int \frac{\left(a+b x^2\right)^{3/2}}{x^{10}} \, dx","Integrate[(a + b*x^2)^(3/2)/x^10,x]","-\frac{\left(a+b x^2\right)^{5/2} \left(35 a^2-20 a b x^2+8 b^2 x^4\right)}{315 a^3 x^9}","-\frac{8 b^2 \left(a+b x^2\right)^{5/2}}{315 a^3 x^5}+\frac{4 b \left(a+b x^2\right)^{5/2}}{63 a^2 x^7}-\frac{\left(a+b x^2\right)^{5/2}}{9 a x^9}",1,"-1/315*((a + b*x^2)^(5/2)*(35*a^2 - 20*a*b*x^2 + 8*b^2*x^4))/(a^3*x^9)","A",1
386,1,53,92,0.0134846,"\int \frac{\left(a+b x^2\right)^{3/2}}{x^{12}} \, dx","Integrate[(a + b*x^2)^(3/2)/x^12,x]","\frac{\left(a+b x^2\right)^{5/2} \left(-105 a^3+70 a^2 b x^2-40 a b^2 x^4+16 b^3 x^6\right)}{1155 a^4 x^{11}}","\frac{16 b^3 \left(a+b x^2\right)^{5/2}}{1155 a^4 x^5}-\frac{8 b^2 \left(a+b x^2\right)^{5/2}}{231 a^3 x^7}+\frac{2 b \left(a+b x^2\right)^{5/2}}{33 a^2 x^9}-\frac{\left(a+b x^2\right)^{5/2}}{11 a x^{11}}",1,"((a + b*x^2)^(5/2)*(-105*a^3 + 70*a^2*b*x^2 - 40*a*b^2*x^4 + 16*b^3*x^6))/(1155*a^4*x^11)","A",1
387,1,50,80,0.0277134,"\int x^7 \left(a+b x^2\right)^{5/2} \, dx","Integrate[x^7*(a + b*x^2)^(5/2),x]","\frac{\left(a+b x^2\right)^{7/2} \left(-16 a^3+56 a^2 b x^2-126 a b^2 x^4+231 b^3 x^6\right)}{3003 b^4}","-\frac{a^3 \left(a+b x^2\right)^{7/2}}{7 b^4}+\frac{a^2 \left(a+b x^2\right)^{9/2}}{3 b^4}+\frac{\left(a+b x^2\right)^{13/2}}{13 b^4}-\frac{3 a \left(a+b x^2\right)^{11/2}}{11 b^4}",1,"((a + b*x^2)^(7/2)*(-16*a^3 + 56*a^2*b*x^2 - 126*a*b^2*x^4 + 231*b^3*x^6))/(3003*b^4)","A",1
388,1,39,59,0.0211353,"\int x^5 \left(a+b x^2\right)^{5/2} \, dx","Integrate[x^5*(a + b*x^2)^(5/2),x]","\frac{\left(a+b x^2\right)^{7/2} \left(8 a^2-28 a b x^2+63 b^2 x^4\right)}{693 b^3}","\frac{a^2 \left(a+b x^2\right)^{7/2}}{7 b^3}+\frac{\left(a+b x^2\right)^{11/2}}{11 b^3}-\frac{2 a \left(a+b x^2\right)^{9/2}}{9 b^3}",1,"((a + b*x^2)^(7/2)*(8*a^2 - 28*a*b*x^2 + 63*b^2*x^4))/(693*b^3)","A",1
389,1,28,38,0.0170444,"\int x^3 \left(a+b x^2\right)^{5/2} \, dx","Integrate[x^3*(a + b*x^2)^(5/2),x]","\frac{\left(a+b x^2\right)^{7/2} \left(7 b x^2-2 a\right)}{63 b^2}","\frac{\left(a+b x^2\right)^{9/2}}{9 b^2}-\frac{a \left(a+b x^2\right)^{7/2}}{7 b^2}",1,"((a + b*x^2)^(7/2)*(-2*a + 7*b*x^2))/(63*b^2)","A",1
390,1,18,18,0.0041119,"\int x \left(a+b x^2\right)^{5/2} \, dx","Integrate[x*(a + b*x^2)^(5/2),x]","\frac{\left(a+b x^2\right)^{7/2}}{7 b}","\frac{\left(a+b x^2\right)^{7/2}}{7 b}",1,"(a + b*x^2)^(7/2)/(7*b)","A",1
391,1,62,72,0.0269876,"\int \frac{\left(a+b x^2\right)^{5/2}}{x} \, dx","Integrate[(a + b*x^2)^(5/2)/x,x]","\frac{1}{15} \sqrt{a+b x^2} \left(23 a^2+11 a b x^2+3 b^2 x^4\right)-a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)","a^{5/2} \left(-\tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)\right)+a^2 \sqrt{a+b x^2}+\frac{1}{3} a \left(a+b x^2\right)^{3/2}+\frac{1}{5} \left(a+b x^2\right)^{5/2}",1,"(Sqrt[a + b*x^2]*(23*a^2 + 11*a*b*x^2 + 3*b^2*x^4))/15 - a^(5/2)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]","A",1
392,1,37,80,0.0097546,"\int \frac{\left(a+b x^2\right)^{5/2}}{x^3} \, dx","Integrate[(a + b*x^2)^(5/2)/x^3,x]","\frac{b \left(a+b x^2\right)^{7/2} \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};\frac{b x^2}{a}+1\right)}{7 a^2}","-\frac{5}{2} a^{3/2} b \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)-\frac{\left(a+b x^2\right)^{5/2}}{2 x^2}+\frac{5}{6} b \left(a+b x^2\right)^{3/2}+\frac{5}{2} a b \sqrt{a+b x^2}",1,"(b*(a + b*x^2)^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, 1 + (b*x^2)/a])/(7*a^2)","C",1
393,1,39,86,0.0101798,"\int \frac{\left(a+b x^2\right)^{5/2}}{x^5} \, dx","Integrate[(a + b*x^2)^(5/2)/x^5,x]","-\frac{b^2 \left(a+b x^2\right)^{7/2} \, _2F_1\left(3,\frac{7}{2};\frac{9}{2};\frac{b x^2}{a}+1\right)}{7 a^3}","\frac{15}{8} b^2 \sqrt{a+b x^2}-\frac{15}{8} \sqrt{a} b^2 \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)-\frac{5 b \left(a+b x^2\right)^{3/2}}{8 x^2}-\frac{\left(a+b x^2\right)^{5/2}}{4 x^4}",1,"-1/7*(b^2*(a + b*x^2)^(7/2)*Hypergeometric2F1[3, 7/2, 9/2, 1 + (b*x^2)/a])/a^3","C",1
394,1,87,89,0.042154,"\int \frac{\left(a+b x^2\right)^{5/2}}{x^7} \, dx","Integrate[(a + b*x^2)^(5/2)/x^7,x]","-\frac{8 a^3+34 a^2 b x^2+15 b^3 x^6 \sqrt{\frac{b x^2}{a}+1} \tanh ^{-1}\left(\sqrt{\frac{b x^2}{a}+1}\right)+59 a b^2 x^4+33 b^3 x^6}{48 x^6 \sqrt{a+b x^2}}","-\frac{5 b^3 \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{16 \sqrt{a}}-\frac{5 b^2 \sqrt{a+b x^2}}{16 x^2}-\frac{\left(a+b x^2\right)^{5/2}}{6 x^6}-\frac{5 b \left(a+b x^2\right)^{3/2}}{24 x^4}",1,"-1/48*(8*a^3 + 34*a^2*b*x^2 + 59*a*b^2*x^4 + 33*b^3*x^6 + 15*b^3*x^6*Sqrt[1 + (b*x^2)/a]*ArcTanh[Sqrt[1 + (b*x^2)/a]])/(x^6*Sqrt[a + b*x^2])","A",1
395,1,39,113,0.0107732,"\int \frac{\left(a+b x^2\right)^{5/2}}{x^9} \, dx","Integrate[(a + b*x^2)^(5/2)/x^9,x]","-\frac{b^4 \left(a+b x^2\right)^{7/2} \, _2F_1\left(\frac{7}{2},5;\frac{9}{2};\frac{b x^2}{a}+1\right)}{7 a^5}","\frac{5 b^4 \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{128 a^{3/2}}-\frac{5 b^3 \sqrt{a+b x^2}}{128 a x^2}-\frac{5 b^2 \sqrt{a+b x^2}}{64 x^4}-\frac{\left(a+b x^2\right)^{5/2}}{8 x^8}-\frac{5 b \left(a+b x^2\right)^{3/2}}{48 x^6}",1,"-1/7*(b^4*(a + b*x^2)^(7/2)*Hypergeometric2F1[7/2, 5, 9/2, 1 + (b*x^2)/a])/a^5","C",1
396,1,39,137,0.0112494,"\int \frac{\left(a+b x^2\right)^{5/2}}{x^{11}} \, dx","Integrate[(a + b*x^2)^(5/2)/x^11,x]","\frac{b^5 \left(a+b x^2\right)^{7/2} \, _2F_1\left(\frac{7}{2},6;\frac{9}{2};\frac{b x^2}{a}+1\right)}{7 a^6}","-\frac{3 b^5 \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{256 a^{5/2}}+\frac{3 b^4 \sqrt{a+b x^2}}{256 a^2 x^2}-\frac{b^3 \sqrt{a+b x^2}}{128 a x^4}-\frac{b^2 \sqrt{a+b x^2}}{32 x^6}-\frac{\left(a+b x^2\right)^{5/2}}{10 x^{10}}-\frac{b \left(a+b x^2\right)^{3/2}}{16 x^8}",1,"(b^5*(a + b*x^2)^(7/2)*Hypergeometric2F1[7/2, 6, 9/2, 1 + (b*x^2)/a])/(7*a^6)","C",1
397,1,105,136,0.1540366,"\int x^4 \left(a+b x^2\right)^{5/2} \, dx","Integrate[x^4*(a + b*x^2)^(5/2),x]","\frac{\sqrt{a+b x^2} \left(\frac{15 a^{9/2} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{\frac{b x^2}{a}+1}}+\sqrt{b} x \left(-15 a^4+10 a^3 b x^2+248 a^2 b^2 x^4+336 a b^3 x^6+128 b^4 x^8\right)\right)}{1280 b^{5/2}}","\frac{3 a^5 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{256 b^{5/2}}-\frac{3 a^4 x \sqrt{a+b x^2}}{256 b^2}+\frac{a^3 x^3 \sqrt{a+b x^2}}{128 b}+\frac{1}{32} a^2 x^5 \sqrt{a+b x^2}+\frac{1}{16} a x^5 \left(a+b x^2\right)^{3/2}+\frac{1}{10} x^5 \left(a+b x^2\right)^{5/2}",1,"(Sqrt[a + b*x^2]*(Sqrt[b]*x*(-15*a^4 + 10*a^3*b*x^2 + 248*a^2*b^2*x^4 + 336*a*b^3*x^6 + 128*b^4*x^8) + (15*a^(9/2)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(1280*b^(5/2))","A",1
398,1,94,112,0.1370103,"\int x^2 \left(a+b x^2\right)^{5/2} \, dx","Integrate[x^2*(a + b*x^2)^(5/2),x]","\frac{\sqrt{a+b x^2} \left(\sqrt{b} x \left(15 a^3+118 a^2 b x^2+136 a b^2 x^4+48 b^3 x^6\right)-\frac{15 a^{7/2} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{\frac{b x^2}{a}+1}}\right)}{384 b^{3/2}}","-\frac{5 a^4 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{128 b^{3/2}}+\frac{5 a^3 x \sqrt{a+b x^2}}{128 b}+\frac{5}{64} a^2 x^3 \sqrt{a+b x^2}+\frac{5}{48} a x^3 \left(a+b x^2\right)^{3/2}+\frac{1}{8} x^3 \left(a+b x^2\right)^{5/2}",1,"(Sqrt[a + b*x^2]*(Sqrt[b]*x*(15*a^3 + 118*a^2*b*x^2 + 136*a*b^2*x^4 + 48*b^3*x^6) - (15*a^(7/2)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(384*b^(3/2))","A",1
399,1,76,84,0.1096232,"\int \left(a+b x^2\right)^{5/2} \, dx","Integrate[(a + b*x^2)^(5/2),x]","\frac{1}{48} \sqrt{a+b x^2} \left(\frac{15 a^{5/2} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{\frac{b x^2}{a}+1}}+33 a^2 x+26 a b x^3+8 b^2 x^5\right)","\frac{5 a^3 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{16 \sqrt{b}}+\frac{5}{16} a^2 x \sqrt{a+b x^2}+\frac{5}{24} a x \left(a+b x^2\right)^{3/2}+\frac{1}{6} x \left(a+b x^2\right)^{5/2}",1,"(Sqrt[a + b*x^2]*(33*a^2*x + 26*a*b*x^3 + 8*b^2*x^5 + (15*a^(5/2)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*Sqrt[1 + (b*x^2)/a])))/48","A",1
400,1,52,83,0.0076658,"\int \frac{\left(a+b x^2\right)^{5/2}}{x^2} \, dx","Integrate[(a + b*x^2)^(5/2)/x^2,x]","-\frac{a^2 \sqrt{a+b x^2} \, _2F_1\left(-\frac{5}{2},-\frac{1}{2};\frac{1}{2};-\frac{b x^2}{a}\right)}{x \sqrt{\frac{b x^2}{a}+1}}","\frac{15}{8} a^2 \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)-\frac{\left(a+b x^2\right)^{5/2}}{x}+\frac{5}{4} b x \left(a+b x^2\right)^{3/2}+\frac{15}{8} a b x \sqrt{a+b x^2}",1,"-((a^2*Sqrt[a + b*x^2]*Hypergeometric2F1[-5/2, -1/2, 1/2, -((b*x^2)/a)])/(x*Sqrt[1 + (b*x^2)/a]))","C",1
401,1,54,86,0.0080719,"\int \frac{\left(a+b x^2\right)^{5/2}}{x^4} \, dx","Integrate[(a + b*x^2)^(5/2)/x^4,x]","-\frac{a^2 \sqrt{a+b x^2} \, _2F_1\left(-\frac{5}{2},-\frac{3}{2};-\frac{1}{2};-\frac{b x^2}{a}\right)}{3 x^3 \sqrt{\frac{b x^2}{a}+1}}","\frac{5}{2} a b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)+\frac{5}{2} b^2 x \sqrt{a+b x^2}-\frac{5 b \left(a+b x^2\right)^{3/2}}{3 x}-\frac{\left(a+b x^2\right)^{5/2}}{3 x^3}",1,"-1/3*(a^2*Sqrt[a + b*x^2]*Hypergeometric2F1[-5/2, -3/2, -1/2, -((b*x^2)/a)])/(x^3*Sqrt[1 + (b*x^2)/a])","C",1
402,1,54,82,0.0097023,"\int \frac{\left(a+b x^2\right)^{5/2}}{x^6} \, dx","Integrate[(a + b*x^2)^(5/2)/x^6,x]","-\frac{a^2 \sqrt{a+b x^2} \, _2F_1\left(-\frac{5}{2},-\frac{5}{2};-\frac{3}{2};-\frac{b x^2}{a}\right)}{5 x^5 \sqrt{\frac{b x^2}{a}+1}}","b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)-\frac{b^2 \sqrt{a+b x^2}}{x}-\frac{\left(a+b x^2\right)^{5/2}}{5 x^5}-\frac{b \left(a+b x^2\right)^{3/2}}{3 x^3}",1,"-1/5*(a^2*Sqrt[a + b*x^2]*Hypergeometric2F1[-5/2, -5/2, -3/2, -((b*x^2)/a)])/(x^5*Sqrt[1 + (b*x^2)/a])","C",1
403,1,21,21,0.0078846,"\int \frac{\left(a+b x^2\right)^{5/2}}{x^8} \, dx","Integrate[(a + b*x^2)^(5/2)/x^8,x]","-\frac{\left(a+b x^2\right)^{7/2}}{7 a x^7}","-\frac{\left(a+b x^2\right)^{7/2}}{7 a x^7}",1,"-1/7*(a + b*x^2)^(7/2)/(a*x^7)","A",1
404,1,31,44,0.0112343,"\int \frac{\left(a+b x^2\right)^{5/2}}{x^{10}} \, dx","Integrate[(a + b*x^2)^(5/2)/x^10,x]","\frac{\left(a+b x^2\right)^{7/2} \left(2 b x^2-7 a\right)}{63 a^2 x^9}","\frac{2 b \left(a+b x^2\right)^{7/2}}{63 a^2 x^7}-\frac{\left(a+b x^2\right)^{7/2}}{9 a x^9}",1,"((a + b*x^2)^(7/2)*(-7*a + 2*b*x^2))/(63*a^2*x^9)","A",1
405,1,42,68,0.0118337,"\int \frac{\left(a+b x^2\right)^{5/2}}{x^{12}} \, dx","Integrate[(a + b*x^2)^(5/2)/x^12,x]","-\frac{\left(a+b x^2\right)^{7/2} \left(63 a^2-28 a b x^2+8 b^2 x^4\right)}{693 a^3 x^{11}}","-\frac{8 b^2 \left(a+b x^2\right)^{7/2}}{693 a^3 x^7}+\frac{4 b \left(a+b x^2\right)^{7/2}}{99 a^2 x^9}-\frac{\left(a+b x^2\right)^{7/2}}{11 a x^{11}}",1,"-1/693*((a + b*x^2)^(7/2)*(63*a^2 - 28*a*b*x^2 + 8*b^2*x^4))/(a^3*x^11)","A",1
406,1,53,92,0.014691,"\int \frac{\left(a+b x^2\right)^{5/2}}{x^{14}} \, dx","Integrate[(a + b*x^2)^(5/2)/x^14,x]","\frac{\left(a+b x^2\right)^{7/2} \left(-231 a^3+126 a^2 b x^2-56 a b^2 x^4+16 b^3 x^6\right)}{3003 a^4 x^{13}}","\frac{16 b^3 \left(a+b x^2\right)^{7/2}}{3003 a^4 x^7}-\frac{8 b^2 \left(a+b x^2\right)^{7/2}}{429 a^3 x^9}+\frac{6 b \left(a+b x^2\right)^{7/2}}{143 a^2 x^{11}}-\frac{\left(a+b x^2\right)^{7/2}}{13 a x^{13}}",1,"((a + b*x^2)^(7/2)*(-231*a^3 + 126*a^2*b*x^2 - 56*a*b^2*x^4 + 16*b^3*x^6))/(3003*a^4*x^13)","A",1
407,1,64,116,0.0155451,"\int \frac{\left(a+b x^2\right)^{5/2}}{x^{16}} \, dx","Integrate[(a + b*x^2)^(5/2)/x^16,x]","-\frac{\left(a+b x^2\right)^{7/2} \left(3003 a^4-1848 a^3 b x^2+1008 a^2 b^2 x^4-448 a b^3 x^6+128 b^4 x^8\right)}{45045 a^5 x^{15}}","-\frac{128 b^4 \left(a+b x^2\right)^{7/2}}{45045 a^5 x^7}+\frac{64 b^3 \left(a+b x^2\right)^{7/2}}{6435 a^4 x^9}-\frac{16 b^2 \left(a+b x^2\right)^{7/2}}{715 a^3 x^{11}}+\frac{8 b \left(a+b x^2\right)^{7/2}}{195 a^2 x^{13}}-\frac{\left(a+b x^2\right)^{7/2}}{15 a x^{15}}",1,"-1/45045*((a + b*x^2)^(7/2)*(3003*a^4 - 1848*a^3*b*x^2 + 1008*a^2*b^2*x^4 - 448*a*b^3*x^6 + 128*b^4*x^8))/(a^5*x^15)","A",1
408,1,75,140,0.0185337,"\int \frac{\left(a+b x^2\right)^{5/2}}{x^{18}} \, dx","Integrate[(a + b*x^2)^(5/2)/x^18,x]","\frac{\left(a+b x^2\right)^{7/2} \left(-9009 a^5+6006 a^4 b x^2-3696 a^3 b^2 x^4+2016 a^2 b^3 x^6-896 a b^4 x^8+256 b^5 x^{10}\right)}{153153 a^6 x^{17}}","\frac{256 b^5 \left(a+b x^2\right)^{7/2}}{153153 a^6 x^7}-\frac{128 b^4 \left(a+b x^2\right)^{7/2}}{21879 a^5 x^9}+\frac{32 b^3 \left(a+b x^2\right)^{7/2}}{2431 a^4 x^{11}}-\frac{16 b^2 \left(a+b x^2\right)^{7/2}}{663 a^3 x^{13}}+\frac{2 b \left(a+b x^2\right)^{7/2}}{51 a^2 x^{15}}-\frac{\left(a+b x^2\right)^{7/2}}{17 a x^{17}}",1,"((a + b*x^2)^(7/2)*(-9009*a^5 + 6006*a^4*b*x^2 - 3696*a^3*b^2*x^4 + 2016*a^2*b^3*x^6 - 896*a*b^4*x^8 + 256*b^5*x^10))/(153153*a^6*x^17)","A",1
409,1,94,161,0.0584507,"\int x^{15} \left(a+b x^2\right)^{9/2} \, dx","Integrate[x^15*(a + b*x^2)^(9/2),x]","\frac{\left(a+b x^2\right)^{11/2} \left(-2048 a^7+11264 a^6 b x^2-36608 a^5 b^2 x^4+91520 a^4 b^3 x^6-194480 a^3 b^4 x^8+369512 a^2 b^5 x^{10}-646646 a b^6 x^{12}+1062347 b^7 x^{14}\right)}{26558675 b^8}","-\frac{a^7 \left(a+b x^2\right)^{11/2}}{11 b^8}+\frac{7 a^6 \left(a+b x^2\right)^{13/2}}{13 b^8}-\frac{7 a^5 \left(a+b x^2\right)^{15/2}}{5 b^8}+\frac{35 a^4 \left(a+b x^2\right)^{17/2}}{17 b^8}-\frac{35 a^3 \left(a+b x^2\right)^{19/2}}{19 b^8}+\frac{a^2 \left(a+b x^2\right)^{21/2}}{b^8}+\frac{\left(a+b x^2\right)^{25/2}}{25 b^8}-\frac{7 a \left(a+b x^2\right)^{23/2}}{23 b^8}",1,"((a + b*x^2)^(11/2)*(-2048*a^7 + 11264*a^6*b*x^2 - 36608*a^5*b^2*x^4 + 91520*a^4*b^3*x^6 - 194480*a^3*b^4*x^8 + 369512*a^2*b^5*x^10 - 646646*a*b^6*x^12 + 1062347*b^7*x^14))/(26558675*b^8)","A",1
410,1,83,140,0.0443867,"\int x^{13} \left(a+b x^2\right)^{9/2} \, dx","Integrate[x^13*(a + b*x^2)^(9/2),x]","\frac{\left(a+b x^2\right)^{11/2} \left(1024 a^6-5632 a^5 b x^2+18304 a^4 b^2 x^4-45760 a^3 b^3 x^6+97240 a^2 b^4 x^8-184756 a b^5 x^{10}+323323 b^6 x^{12}\right)}{7436429 b^7}","\frac{a^6 \left(a+b x^2\right)^{11/2}}{11 b^7}-\frac{6 a^5 \left(a+b x^2\right)^{13/2}}{13 b^7}+\frac{a^4 \left(a+b x^2\right)^{15/2}}{b^7}-\frac{20 a^3 \left(a+b x^2\right)^{17/2}}{17 b^7}+\frac{15 a^2 \left(a+b x^2\right)^{19/2}}{19 b^7}+\frac{\left(a+b x^2\right)^{23/2}}{23 b^7}-\frac{2 a \left(a+b x^2\right)^{21/2}}{7 b^7}",1,"((a + b*x^2)^(11/2)*(1024*a^6 - 5632*a^5*b*x^2 + 18304*a^4*b^2*x^4 - 45760*a^3*b^3*x^6 + 97240*a^2*b^4*x^8 - 184756*a*b^5*x^10 + 323323*b^6*x^12))/(7436429*b^7)","A",1
411,1,72,122,0.0408919,"\int x^{11} \left(a+b x^2\right)^{9/2} \, dx","Integrate[x^11*(a + b*x^2)^(9/2),x]","\frac{\left(a+b x^2\right)^{11/2} \left(-256 a^5+1408 a^4 b x^2-4576 a^3 b^2 x^4+11440 a^2 b^3 x^6-24310 a b^4 x^8+46189 b^5 x^{10}\right)}{969969 b^6}","-\frac{a^5 \left(a+b x^2\right)^{11/2}}{11 b^6}+\frac{5 a^4 \left(a+b x^2\right)^{13/2}}{13 b^6}-\frac{2 a^3 \left(a+b x^2\right)^{15/2}}{3 b^6}+\frac{10 a^2 \left(a+b x^2\right)^{17/2}}{17 b^6}+\frac{\left(a+b x^2\right)^{21/2}}{21 b^6}-\frac{5 a \left(a+b x^2\right)^{19/2}}{19 b^6}",1,"((a + b*x^2)^(11/2)*(-256*a^5 + 1408*a^4*b*x^2 - 4576*a^3*b^2*x^4 + 11440*a^2*b^3*x^6 - 24310*a*b^4*x^8 + 46189*b^5*x^10))/(969969*b^6)","A",1
412,1,61,101,0.0347924,"\int x^9 \left(a+b x^2\right)^{9/2} \, dx","Integrate[x^9*(a + b*x^2)^(9/2),x]","\frac{\left(a+b x^2\right)^{11/2} \left(128 a^4-704 a^3 b x^2+2288 a^2 b^2 x^4-5720 a b^3 x^6+12155 b^4 x^8\right)}{230945 b^5}","\frac{a^4 \left(a+b x^2\right)^{11/2}}{11 b^5}-\frac{4 a^3 \left(a+b x^2\right)^{13/2}}{13 b^5}+\frac{2 a^2 \left(a+b x^2\right)^{15/2}}{5 b^5}+\frac{\left(a+b x^2\right)^{19/2}}{19 b^5}-\frac{4 a \left(a+b x^2\right)^{17/2}}{17 b^5}",1,"((a + b*x^2)^(11/2)*(128*a^4 - 704*a^3*b*x^2 + 2288*a^2*b^2*x^4 - 5720*a*b^3*x^6 + 12155*b^4*x^8))/(230945*b^5)","A",1
413,1,50,80,0.03038,"\int x^7 \left(a+b x^2\right)^{9/2} \, dx","Integrate[x^7*(a + b*x^2)^(9/2),x]","\frac{\left(a+b x^2\right)^{11/2} \left(-16 a^3+88 a^2 b x^2-286 a b^2 x^4+715 b^3 x^6\right)}{12155 b^4}","-\frac{a^3 \left(a+b x^2\right)^{11/2}}{11 b^4}+\frac{3 a^2 \left(a+b x^2\right)^{13/2}}{13 b^4}+\frac{\left(a+b x^2\right)^{17/2}}{17 b^4}-\frac{a \left(a+b x^2\right)^{15/2}}{5 b^4}",1,"((a + b*x^2)^(11/2)*(-16*a^3 + 88*a^2*b*x^2 - 286*a*b^2*x^4 + 715*b^3*x^6))/(12155*b^4)","A",1
414,1,39,59,0.0230481,"\int x^5 \left(a+b x^2\right)^{9/2} \, dx","Integrate[x^5*(a + b*x^2)^(9/2),x]","\frac{\left(a+b x^2\right)^{11/2} \left(8 a^2-44 a b x^2+143 b^2 x^4\right)}{2145 b^3}","\frac{a^2 \left(a+b x^2\right)^{11/2}}{11 b^3}+\frac{\left(a+b x^2\right)^{15/2}}{15 b^3}-\frac{2 a \left(a+b x^2\right)^{13/2}}{13 b^3}",1,"((a + b*x^2)^(11/2)*(8*a^2 - 44*a*b*x^2 + 143*b^2*x^4))/(2145*b^3)","A",1
415,1,28,38,0.0189226,"\int x^3 \left(a+b x^2\right)^{9/2} \, dx","Integrate[x^3*(a + b*x^2)^(9/2),x]","\frac{\left(a+b x^2\right)^{11/2} \left(11 b x^2-2 a\right)}{143 b^2}","\frac{\left(a+b x^2\right)^{13/2}}{13 b^2}-\frac{a \left(a+b x^2\right)^{11/2}}{11 b^2}",1,"((a + b*x^2)^(11/2)*(-2*a + 11*b*x^2))/(143*b^2)","A",1
416,1,18,18,0.0060377,"\int x \left(a+b x^2\right)^{9/2} \, dx","Integrate[x*(a + b*x^2)^(9/2),x]","\frac{\left(a+b x^2\right)^{11/2}}{11 b}","\frac{\left(a+b x^2\right)^{11/2}}{11 b}",1,"(a + b*x^2)^(11/2)/(11*b)","A",1
417,1,84,108,0.0398256,"\int \frac{\left(a+b x^2\right)^{9/2}}{x} \, dx","Integrate[(a + b*x^2)^(9/2)/x,x]","\frac{1}{315} \sqrt{a+b x^2} \left(563 a^4+506 a^3 b x^2+408 a^2 b^2 x^4+185 a b^3 x^6+35 b^4 x^8\right)-a^{9/2} \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)","a^{9/2} \left(-\tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)\right)+a^4 \sqrt{a+b x^2}+\frac{1}{3} a^3 \left(a+b x^2\right)^{3/2}+\frac{1}{5} a^2 \left(a+b x^2\right)^{5/2}+\frac{1}{7} a \left(a+b x^2\right)^{7/2}+\frac{1}{9} \left(a+b x^2\right)^{9/2}",1,"(Sqrt[a + b*x^2]*(563*a^4 + 506*a^3*b*x^2 + 408*a^2*b^2*x^4 + 185*a*b^3*x^6 + 35*b^4*x^8))/315 - a^(9/2)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]","A",1
418,1,37,118,0.0120404,"\int \frac{\left(a+b x^2\right)^{9/2}}{x^3} \, dx","Integrate[(a + b*x^2)^(9/2)/x^3,x]","\frac{b \left(a+b x^2\right)^{11/2} \, _2F_1\left(2,\frac{11}{2};\frac{13}{2};\frac{b x^2}{a}+1\right)}{11 a^2}","-\frac{9}{2} a^{7/2} b \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)+\frac{9}{2} a^3 b \sqrt{a+b x^2}+\frac{3}{2} a^2 b \left(a+b x^2\right)^{3/2}-\frac{\left(a+b x^2\right)^{9/2}}{2 x^2}+\frac{9}{14} b \left(a+b x^2\right)^{7/2}+\frac{9}{10} a b \left(a+b x^2\right)^{5/2}",1,"(b*(a + b*x^2)^(11/2)*Hypergeometric2F1[2, 11/2, 13/2, 1 + (b*x^2)/a])/(11*a^2)","C",1
419,1,39,126,0.0129772,"\int \frac{\left(a+b x^2\right)^{9/2}}{x^5} \, dx","Integrate[(a + b*x^2)^(9/2)/x^5,x]","-\frac{b^2 \left(a+b x^2\right)^{11/2} \, _2F_1\left(3,\frac{11}{2};\frac{13}{2};\frac{b x^2}{a}+1\right)}{11 a^3}","-\frac{63}{8} a^{5/2} b^2 \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)+\frac{63}{8} a^2 b^2 \sqrt{a+b x^2}+\frac{63}{40} b^2 \left(a+b x^2\right)^{5/2}+\frac{21}{8} a b^2 \left(a+b x^2\right)^{3/2}-\frac{9 b \left(a+b x^2\right)^{7/2}}{8 x^2}-\frac{\left(a+b x^2\right)^{9/2}}{4 x^4}",1,"-1/11*(b^2*(a + b*x^2)^(11/2)*Hypergeometric2F1[3, 11/2, 13/2, 1 + (b*x^2)/a])/a^3","C",1
420,1,39,126,0.0128651,"\int \frac{\left(a+b x^2\right)^{9/2}}{x^7} \, dx","Integrate[(a + b*x^2)^(9/2)/x^7,x]","\frac{b^3 \left(a+b x^2\right)^{11/2} \, _2F_1\left(4,\frac{11}{2};\frac{13}{2};\frac{b x^2}{a}+1\right)}{11 a^4}","-\frac{105}{16} a^{3/2} b^3 \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)+\frac{35}{16} b^3 \left(a+b x^2\right)^{3/2}+\frac{105}{16} a b^3 \sqrt{a+b x^2}-\frac{21 b^2 \left(a+b x^2\right)^{5/2}}{16 x^2}-\frac{\left(a+b x^2\right)^{9/2}}{6 x^6}-\frac{3 b \left(a+b x^2\right)^{7/2}}{8 x^4}",1,"(b^3*(a + b*x^2)^(11/2)*Hypergeometric2F1[4, 11/2, 13/2, 1 + (b*x^2)/a])/(11*a^4)","C",1
421,1,39,128,0.0124373,"\int \frac{\left(a+b x^2\right)^{9/2}}{x^9} \, dx","Integrate[(a + b*x^2)^(9/2)/x^9,x]","-\frac{b^4 \left(a+b x^2\right)^{11/2} \, _2F_1\left(5,\frac{11}{2};\frac{13}{2};\frac{b x^2}{a}+1\right)}{11 a^5}","\frac{315}{128} b^4 \sqrt{a+b x^2}-\frac{315}{128} \sqrt{a} b^4 \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)-\frac{105 b^3 \left(a+b x^2\right)^{3/2}}{128 x^2}-\frac{21 b^2 \left(a+b x^2\right)^{5/2}}{64 x^4}-\frac{\left(a+b x^2\right)^{9/2}}{8 x^8}-\frac{3 b \left(a+b x^2\right)^{7/2}}{16 x^6}",1,"-1/11*(b^4*(a + b*x^2)^(11/2)*Hypergeometric2F1[5, 11/2, 13/2, 1 + (b*x^2)/a])/a^5","C",1
422,1,109,131,0.0502402,"\int \frac{\left(a+b x^2\right)^{9/2}}{x^{11}} \, dx","Integrate[(a + b*x^2)^(9/2)/x^11,x]","-\frac{128 a^5+784 a^4 b x^2+2024 a^3 b^2 x^4+2858 a^2 b^3 x^6+315 b^5 x^{10} \sqrt{\frac{b x^2}{a}+1} \tanh ^{-1}\left(\sqrt{\frac{b x^2}{a}+1}\right)+2455 a b^4 x^8+965 b^5 x^{10}}{1280 x^{10} \sqrt{a+b x^2}}","-\frac{63 b^5 \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{256 \sqrt{a}}-\frac{63 b^4 \sqrt{a+b x^2}}{256 x^2}-\frac{21 b^3 \left(a+b x^2\right)^{3/2}}{128 x^4}-\frac{21 b^2 \left(a+b x^2\right)^{5/2}}{160 x^6}-\frac{\left(a+b x^2\right)^{9/2}}{10 x^{10}}-\frac{9 b \left(a+b x^2\right)^{7/2}}{80 x^8}",1,"-1/1280*(128*a^5 + 784*a^4*b*x^2 + 2024*a^3*b^2*x^4 + 2858*a^2*b^3*x^6 + 2455*a*b^4*x^8 + 965*b^5*x^10 + 315*b^5*x^10*Sqrt[1 + (b*x^2)/a]*ArcTanh[Sqrt[1 + (b*x^2)/a]])/(x^10*Sqrt[a + b*x^2])","A",1
423,1,39,155,0.0129368,"\int \frac{\left(a+b x^2\right)^{9/2}}{x^{13}} \, dx","Integrate[(a + b*x^2)^(9/2)/x^13,x]","-\frac{b^6 \left(a+b x^2\right)^{11/2} \, _2F_1\left(\frac{11}{2},7;\frac{13}{2};\frac{b x^2}{a}+1\right)}{11 a^7}","\frac{21 b^6 \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{1024 a^{3/2}}-\frac{21 b^5 \sqrt{a+b x^2}}{1024 a x^2}-\frac{21 b^4 \sqrt{a+b x^2}}{512 x^4}-\frac{7 b^3 \left(a+b x^2\right)^{3/2}}{128 x^6}-\frac{21 b^2 \left(a+b x^2\right)^{5/2}}{320 x^8}-\frac{\left(a+b x^2\right)^{9/2}}{12 x^{12}}-\frac{3 b \left(a+b x^2\right)^{7/2}}{40 x^{10}}",1,"-1/11*(b^6*(a + b*x^2)^(11/2)*Hypergeometric2F1[11/2, 7, 13/2, 1 + (b*x^2)/a])/a^7","C",1
424,1,39,179,0.0131435,"\int \frac{\left(a+b x^2\right)^{9/2}}{x^{15}} \, dx","Integrate[(a + b*x^2)^(9/2)/x^15,x]","\frac{b^7 \left(a+b x^2\right)^{11/2} \, _2F_1\left(\frac{11}{2},8;\frac{13}{2};\frac{b x^2}{a}+1\right)}{11 a^8}","-\frac{9 b^7 \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{2048 a^{5/2}}+\frac{9 b^6 \sqrt{a+b x^2}}{2048 a^2 x^2}-\frac{3 b^5 \sqrt{a+b x^2}}{1024 a x^4}-\frac{3 b^4 \sqrt{a+b x^2}}{256 x^6}-\frac{3 b^3 \left(a+b x^2\right)^{3/2}}{128 x^8}-\frac{3 b^2 \left(a+b x^2\right)^{5/2}}{80 x^{10}}-\frac{\left(a+b x^2\right)^{9/2}}{14 x^{14}}-\frac{3 b \left(a+b x^2\right)^{7/2}}{56 x^{12}}",1,"(b^7*(a + b*x^2)^(11/2)*Hypergeometric2F1[11/2, 8, 13/2, 1 + (b*x^2)/a])/(11*a^8)","C",1
425,1,138,202,0.219422,"\int x^6 \left(a+b x^2\right)^{9/2} \, dx","Integrate[x^6*(a + b*x^2)^(9/2),x]","\frac{\sqrt{a+b x^2} \left(\sqrt{b} x \left(315 a^7-210 a^6 b x^2+168 a^5 b^2 x^4+32624 a^4 b^3 x^6+98432 a^3 b^4 x^8+119040 a^2 b^5 x^{10}+66560 a b^6 x^{12}+14336 b^7 x^{14}\right)-\frac{315 a^{15/2} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{\frac{b x^2}{a}+1}}\right)}{229376 b^{7/2}}","-\frac{45 a^8 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{32768 b^{7/2}}+\frac{45 a^7 x \sqrt{a+b x^2}}{32768 b^3}-\frac{15 a^6 x^3 \sqrt{a+b x^2}}{16384 b^2}+\frac{3 a^5 x^5 \sqrt{a+b x^2}}{4096 b}+\frac{9 a^4 x^7 \sqrt{a+b x^2}}{2048}+\frac{3}{256} a^3 x^7 \left(a+b x^2\right)^{3/2}+\frac{3}{128} a^2 x^7 \left(a+b x^2\right)^{5/2}+\frac{9}{224} a x^7 \left(a+b x^2\right)^{7/2}+\frac{1}{16} x^7 \left(a+b x^2\right)^{9/2}",1,"(Sqrt[a + b*x^2]*(Sqrt[b]*x*(315*a^7 - 210*a^6*b*x^2 + 168*a^5*b^2*x^4 + 32624*a^4*b^3*x^6 + 98432*a^3*b^4*x^8 + 119040*a^2*b^5*x^10 + 66560*a*b^6*x^12 + 14336*b^7*x^14) - (315*a^(15/2)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(229376*b^(7/2))","A",1
426,1,127,178,0.1934843,"\int x^4 \left(a+b x^2\right)^{9/2} \, dx","Integrate[x^4*(a + b*x^2)^(9/2),x]","\frac{\sqrt{a+b x^2} \left(\frac{315 a^{13/2} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{\frac{b x^2}{a}+1}}+\sqrt{b} x \left(-315 a^6+210 a^5 b x^2+14168 a^4 b^2 x^4+39056 a^3 b^3 x^6+44928 a^2 b^4 x^8+24320 a b^5 x^{10}+5120 b^6 x^{12}\right)\right)}{71680 b^{5/2}}","\frac{9 a^7 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2048 b^{5/2}}-\frac{9 a^6 x \sqrt{a+b x^2}}{2048 b^2}+\frac{3 a^5 x^3 \sqrt{a+b x^2}}{1024 b}+\frac{3}{256} a^4 x^5 \sqrt{a+b x^2}+\frac{3}{128} a^3 x^5 \left(a+b x^2\right)^{3/2}+\frac{3}{80} a^2 x^5 \left(a+b x^2\right)^{5/2}+\frac{3}{56} a x^5 \left(a+b x^2\right)^{7/2}+\frac{1}{14} x^5 \left(a+b x^2\right)^{9/2}",1,"(Sqrt[a + b*x^2]*(Sqrt[b]*x*(-315*a^6 + 210*a^5*b*x^2 + 14168*a^4*b^2*x^4 + 39056*a^3*b^3*x^6 + 44928*a^2*b^4*x^8 + 24320*a*b^5*x^10 + 5120*b^6*x^12) + (315*a^(13/2)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(71680*b^(5/2))","A",1
427,1,116,154,0.1815265,"\int x^2 \left(a+b x^2\right)^{9/2} \, dx","Integrate[x^2*(a + b*x^2)^(9/2),x]","\frac{\sqrt{a+b x^2} \left(\sqrt{b} x \left(315 a^5+4910 a^4 b x^2+11432 a^3 b^2 x^4+12144 a^2 b^3 x^6+6272 a b^4 x^8+1280 b^5 x^{10}\right)-\frac{315 a^{11/2} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{\frac{b x^2}{a}+1}}\right)}{15360 b^{3/2}}","-\frac{21 a^6 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{1024 b^{3/2}}+\frac{21 a^5 x \sqrt{a+b x^2}}{1024 b}+\frac{21}{512} a^4 x^3 \sqrt{a+b x^2}+\frac{7}{128} a^3 x^3 \left(a+b x^2\right)^{3/2}+\frac{21}{320} a^2 x^3 \left(a+b x^2\right)^{5/2}+\frac{3}{40} a x^3 \left(a+b x^2\right)^{7/2}+\frac{1}{12} x^3 \left(a+b x^2\right)^{9/2}",1,"(Sqrt[a + b*x^2]*(Sqrt[b]*x*(315*a^5 + 4910*a^4*b*x^2 + 11432*a^3*b^2*x^4 + 12144*a^2*b^3*x^6 + 6272*a*b^4*x^8 + 1280*b^5*x^10) - (315*a^(11/2)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(15360*b^(3/2))","A",1
428,1,98,122,0.1395551,"\int \left(a+b x^2\right)^{9/2} \, dx","Integrate[(a + b*x^2)^(9/2),x]","\frac{\sqrt{a+b x^2} \left(\frac{315 a^{9/2} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{\frac{b x^2}{a}+1}}+965 a^4 x+1490 a^3 b x^3+1368 a^2 b^2 x^5+656 a b^3 x^7+128 b^4 x^9\right)}{1280}","\frac{63 a^5 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{256 \sqrt{b}}+\frac{63}{256} a^4 x \sqrt{a+b x^2}+\frac{21}{128} a^3 x \left(a+b x^2\right)^{3/2}+\frac{21}{160} a^2 x \left(a+b x^2\right)^{5/2}+\frac{9}{80} a x \left(a+b x^2\right)^{7/2}+\frac{1}{10} x \left(a+b x^2\right)^{9/2}",1,"(Sqrt[a + b*x^2]*(965*a^4*x + 1490*a^3*b*x^3 + 1368*a^2*b^2*x^5 + 656*a*b^3*x^7 + 128*b^4*x^9 + (315*a^(9/2)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*Sqrt[1 + (b*x^2)/a])))/1280","A",1
429,1,52,123,0.0091294,"\int \frac{\left(a+b x^2\right)^{9/2}}{x^2} \, dx","Integrate[(a + b*x^2)^(9/2)/x^2,x]","-\frac{a^4 \sqrt{a+b x^2} \, _2F_1\left(-\frac{9}{2},-\frac{1}{2};\frac{1}{2};-\frac{b x^2}{a}\right)}{x \sqrt{\frac{b x^2}{a}+1}}","\frac{315}{128} a^4 \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)+\frac{315}{128} a^3 b x \sqrt{a+b x^2}+\frac{105}{64} a^2 b x \left(a+b x^2\right)^{3/2}-\frac{\left(a+b x^2\right)^{9/2}}{x}+\frac{9}{8} b x \left(a+b x^2\right)^{7/2}+\frac{21}{16} a b x \left(a+b x^2\right)^{5/2}",1,"-((a^4*Sqrt[a + b*x^2]*Hypergeometric2F1[-9/2, -1/2, 1/2, -((b*x^2)/a)])/(x*Sqrt[1 + (b*x^2)/a]))","C",1
430,1,54,128,0.0099362,"\int \frac{\left(a+b x^2\right)^{9/2}}{x^4} \, dx","Integrate[(a + b*x^2)^(9/2)/x^4,x]","-\frac{a^4 \sqrt{a+b x^2} \, _2F_1\left(-\frac{9}{2},-\frac{3}{2};-\frac{1}{2};-\frac{b x^2}{a}\right)}{3 x^3 \sqrt{\frac{b x^2}{a}+1}}","\frac{105}{16} a^3 b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)+\frac{105}{16} a^2 b^2 x \sqrt{a+b x^2}+\frac{7}{2} b^2 x \left(a+b x^2\right)^{5/2}+\frac{35}{8} a b^2 x \left(a+b x^2\right)^{3/2}-\frac{3 b \left(a+b x^2\right)^{7/2}}{x}-\frac{\left(a+b x^2\right)^{9/2}}{3 x^3}",1,"-1/3*(a^4*Sqrt[a + b*x^2]*Hypergeometric2F1[-9/2, -3/2, -1/2, -((b*x^2)/a)])/(x^3*Sqrt[1 + (b*x^2)/a])","C",1
431,1,54,129,0.0105925,"\int \frac{\left(a+b x^2\right)^{9/2}}{x^6} \, dx","Integrate[(a + b*x^2)^(9/2)/x^6,x]","-\frac{a^4 \sqrt{a+b x^2} \, _2F_1\left(-\frac{9}{2},-\frac{5}{2};-\frac{3}{2};-\frac{b x^2}{a}\right)}{5 x^5 \sqrt{\frac{b x^2}{a}+1}}","\frac{63}{8} a^2 b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)+\frac{21}{4} b^3 x \left(a+b x^2\right)^{3/2}+\frac{63}{8} a b^3 x \sqrt{a+b x^2}-\frac{21 b^2 \left(a+b x^2\right)^{5/2}}{5 x}-\frac{\left(a+b x^2\right)^{9/2}}{5 x^5}-\frac{3 b \left(a+b x^2\right)^{7/2}}{5 x^3}",1,"-1/5*(a^4*Sqrt[a + b*x^2]*Hypergeometric2F1[-9/2, -5/2, -3/2, -((b*x^2)/a)])/(x^5*Sqrt[1 + (b*x^2)/a])","C",1
432,1,54,126,0.0106794,"\int \frac{\left(a+b x^2\right)^{9/2}}{x^8} \, dx","Integrate[(a + b*x^2)^(9/2)/x^8,x]","-\frac{a^4 \sqrt{a+b x^2} \, _2F_1\left(-\frac{9}{2},-\frac{7}{2};-\frac{5}{2};-\frac{b x^2}{a}\right)}{7 x^7 \sqrt{\frac{b x^2}{a}+1}}","\frac{9}{2} a b^{7/2} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)+\frac{9}{2} b^4 x \sqrt{a+b x^2}-\frac{3 b^3 \left(a+b x^2\right)^{3/2}}{x}-\frac{3 b^2 \left(a+b x^2\right)^{5/2}}{5 x^3}-\frac{\left(a+b x^2\right)^{9/2}}{7 x^7}-\frac{9 b \left(a+b x^2\right)^{7/2}}{35 x^5}",1,"-1/7*(a^4*Sqrt[a + b*x^2]*Hypergeometric2F1[-9/2, -7/2, -5/2, -((b*x^2)/a)])/(x^7*Sqrt[1 + (b*x^2)/a])","C",1
433,1,54,124,0.0105421,"\int \frac{\left(a+b x^2\right)^{9/2}}{x^{10}} \, dx","Integrate[(a + b*x^2)^(9/2)/x^10,x]","-\frac{a^4 \sqrt{a+b x^2} \, _2F_1\left(-\frac{9}{2},-\frac{9}{2};-\frac{7}{2};-\frac{b x^2}{a}\right)}{9 x^9 \sqrt{\frac{b x^2}{a}+1}}","b^{9/2} \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)-\frac{b^4 \sqrt{a+b x^2}}{x}-\frac{b^3 \left(a+b x^2\right)^{3/2}}{3 x^3}-\frac{b^2 \left(a+b x^2\right)^{5/2}}{5 x^5}-\frac{\left(a+b x^2\right)^{9/2}}{9 x^9}-\frac{b \left(a+b x^2\right)^{7/2}}{7 x^7}",1,"-1/9*(a^4*Sqrt[a + b*x^2]*Hypergeometric2F1[-9/2, -9/2, -7/2, -((b*x^2)/a)])/(x^9*Sqrt[1 + (b*x^2)/a])","C",1
434,1,21,21,0.0075958,"\int \frac{\left(a+b x^2\right)^{9/2}}{x^{12}} \, dx","Integrate[(a + b*x^2)^(9/2)/x^12,x]","-\frac{\left(a+b x^2\right)^{11/2}}{11 a x^{11}}","-\frac{\left(a+b x^2\right)^{11/2}}{11 a x^{11}}",1,"-1/11*(a + b*x^2)^(11/2)/(a*x^11)","A",1
435,1,31,44,0.0137768,"\int \frac{\left(a+b x^2\right)^{9/2}}{x^{14}} \, dx","Integrate[(a + b*x^2)^(9/2)/x^14,x]","\frac{\left(a+b x^2\right)^{11/2} \left(2 b x^2-11 a\right)}{143 a^2 x^{13}}","\frac{2 b \left(a+b x^2\right)^{11/2}}{143 a^2 x^{11}}-\frac{\left(a+b x^2\right)^{11/2}}{13 a x^{13}}",1,"((a + b*x^2)^(11/2)*(-11*a + 2*b*x^2))/(143*a^2*x^13)","A",1
436,1,42,68,0.0146755,"\int \frac{\left(a+b x^2\right)^{9/2}}{x^{16}} \, dx","Integrate[(a + b*x^2)^(9/2)/x^16,x]","-\frac{\left(a+b x^2\right)^{11/2} \left(143 a^2-44 a b x^2+8 b^2 x^4\right)}{2145 a^3 x^{15}}","-\frac{8 b^2 \left(a+b x^2\right)^{11/2}}{2145 a^3 x^{11}}+\frac{4 b \left(a+b x^2\right)^{11/2}}{195 a^2 x^{13}}-\frac{\left(a+b x^2\right)^{11/2}}{15 a x^{15}}",1,"-1/2145*((a + b*x^2)^(11/2)*(143*a^2 - 44*a*b*x^2 + 8*b^2*x^4))/(a^3*x^15)","A",1
437,1,53,92,0.0173996,"\int \frac{\left(a+b x^2\right)^{9/2}}{x^{18}} \, dx","Integrate[(a + b*x^2)^(9/2)/x^18,x]","\frac{\left(a+b x^2\right)^{11/2} \left(-715 a^3+286 a^2 b x^2-88 a b^2 x^4+16 b^3 x^6\right)}{12155 a^4 x^{17}}","\frac{16 b^3 \left(a+b x^2\right)^{11/2}}{12155 a^4 x^{11}}-\frac{8 b^2 \left(a+b x^2\right)^{11/2}}{1105 a^3 x^{13}}+\frac{2 b \left(a+b x^2\right)^{11/2}}{85 a^2 x^{15}}-\frac{\left(a+b x^2\right)^{11/2}}{17 a x^{17}}",1,"((a + b*x^2)^(11/2)*(-715*a^3 + 286*a^2*b*x^2 - 88*a*b^2*x^4 + 16*b^3*x^6))/(12155*a^4*x^17)","A",1
438,1,64,116,0.0177418,"\int \frac{\left(a+b x^2\right)^{9/2}}{x^{20}} \, dx","Integrate[(a + b*x^2)^(9/2)/x^20,x]","-\frac{\left(a+b x^2\right)^{11/2} \left(12155 a^4-5720 a^3 b x^2+2288 a^2 b^2 x^4-704 a b^3 x^6+128 b^4 x^8\right)}{230945 a^5 x^{19}}","-\frac{128 b^4 \left(a+b x^2\right)^{11/2}}{230945 a^5 x^{11}}+\frac{64 b^3 \left(a+b x^2\right)^{11/2}}{20995 a^4 x^{13}}-\frac{16 b^2 \left(a+b x^2\right)^{11/2}}{1615 a^3 x^{15}}+\frac{8 b \left(a+b x^2\right)^{11/2}}{323 a^2 x^{17}}-\frac{\left(a+b x^2\right)^{11/2}}{19 a x^{19}}",1,"-1/230945*((a + b*x^2)^(11/2)*(12155*a^4 - 5720*a^3*b*x^2 + 2288*a^2*b^2*x^4 - 704*a*b^3*x^6 + 128*b^4*x^8))/(a^5*x^19)","A",1
439,1,75,140,0.020723,"\int \frac{\left(a+b x^2\right)^{9/2}}{x^{22}} \, dx","Integrate[(a + b*x^2)^(9/2)/x^22,x]","\frac{\left(a+b x^2\right)^{11/2} \left(-46189 a^5+24310 a^4 b x^2-11440 a^3 b^2 x^4+4576 a^2 b^3 x^6-1408 a b^4 x^8+256 b^5 x^{10}\right)}{969969 a^6 x^{21}}","\frac{256 b^5 \left(a+b x^2\right)^{11/2}}{969969 a^6 x^{11}}-\frac{128 b^4 \left(a+b x^2\right)^{11/2}}{88179 a^5 x^{13}}+\frac{32 b^3 \left(a+b x^2\right)^{11/2}}{6783 a^4 x^{15}}-\frac{80 b^2 \left(a+b x^2\right)^{11/2}}{6783 a^3 x^{17}}+\frac{10 b \left(a+b x^2\right)^{11/2}}{399 a^2 x^{19}}-\frac{\left(a+b x^2\right)^{11/2}}{21 a x^{21}}",1,"((a + b*x^2)^(11/2)*(-46189*a^5 + 24310*a^4*b*x^2 - 11440*a^3*b^2*x^4 + 4576*a^2*b^3*x^6 - 1408*a*b^4*x^8 + 256*b^5*x^10))/(969969*a^6*x^21)","A",1
440,1,86,164,0.0215553,"\int \frac{\left(a+b x^2\right)^{9/2}}{x^{24}} \, dx","Integrate[(a + b*x^2)^(9/2)/x^24,x]","-\frac{\left(a+b x^2\right)^{11/2} \left(323323 a^6-184756 a^5 b x^2+97240 a^4 b^2 x^4-45760 a^3 b^3 x^6+18304 a^2 b^4 x^8-5632 a b^5 x^{10}+1024 b^6 x^{12}\right)}{7436429 a^7 x^{23}}","-\frac{1024 b^6 \left(a+b x^2\right)^{11/2}}{7436429 a^7 x^{11}}+\frac{512 b^5 \left(a+b x^2\right)^{11/2}}{676039 a^6 x^{13}}-\frac{128 b^4 \left(a+b x^2\right)^{11/2}}{52003 a^5 x^{15}}+\frac{320 b^3 \left(a+b x^2\right)^{11/2}}{52003 a^4 x^{17}}-\frac{40 b^2 \left(a+b x^2\right)^{11/2}}{3059 a^3 x^{19}}+\frac{4 b \left(a+b x^2\right)^{11/2}}{161 a^2 x^{21}}-\frac{\left(a+b x^2\right)^{11/2}}{23 a x^{23}}",1,"-1/7436429*((a + b*x^2)^(11/2)*(323323*a^6 - 184756*a^5*b*x^2 + 97240*a^4*b^2*x^4 - 45760*a^3*b^3*x^6 + 18304*a^2*b^4*x^8 - 5632*a*b^5*x^10 + 1024*b^6*x^12))/(a^7*x^23)","A",1
441,1,27,46,0.011055,"\int x^5 \sqrt{9+4 x^2} \, dx","Integrate[x^5*Sqrt[9 + 4*x^2],x]","\frac{1}{280} \left(4 x^2+9\right)^{3/2} \left(10 x^4-18 x^2+27\right)","\frac{1}{448} \left(4 x^2+9\right)^{7/2}-\frac{9}{160} \left(4 x^2+9\right)^{5/2}+\frac{27}{64} \left(4 x^2+9\right)^{3/2}",1,"((9 + 4*x^2)^(3/2)*(27 - 18*x^2 + 10*x^4))/280","A",1
442,1,39,63,0.0125765,"\int x^4 \sqrt{9+4 x^2} \, dx","Integrate[x^4*Sqrt[9 + 4*x^2],x]","\frac{1}{768} x \sqrt{4 x^2+9} \left(128 x^4+72 x^2-243\right)+\frac{729}{512} \sinh ^{-1}\left(\frac{2 x}{3}\right)","-\frac{81}{256} \sqrt{4 x^2+9} x+\frac{1}{6} \sqrt{4 x^2+9} x^5+\frac{3}{32} \sqrt{4 x^2+9} x^3+\frac{729}{512} \sinh ^{-1}\left(\frac{2 x}{3}\right)",1,"(x*Sqrt[9 + 4*x^2]*(-243 + 72*x^2 + 128*x^4))/768 + (729*ArcSinh[(2*x)/3])/512","A",1
443,1,22,31,0.0072337,"\int x^3 \sqrt{9+4 x^2} \, dx","Integrate[x^3*Sqrt[9 + 4*x^2],x]","\frac{1}{40} \left(2 x^2-3\right) \left(4 x^2+9\right)^{3/2}","\frac{1}{80} \left(4 x^2+9\right)^{5/2}-\frac{3}{16} \left(4 x^2+9\right)^{3/2}",1,"((-3 + 2*x^2)*(9 + 4*x^2)^(3/2))/40","A",1
444,1,36,45,0.0153886,"\int x^2 \sqrt{9+4 x^2} \, dx","Integrate[x^2*Sqrt[9 + 4*x^2],x]","\sqrt{4 x^2+9} \left(\frac{x^3}{4}+\frac{9 x}{32}\right)-\frac{81}{64} \sinh ^{-1}\left(\frac{2 x}{3}\right)","\frac{9}{32} \sqrt{4 x^2+9} x+\frac{1}{4} \sqrt{4 x^2+9} x^3-\frac{81}{64} \sinh ^{-1}\left(\frac{2 x}{3}\right)",1,"Sqrt[9 + 4*x^2]*((9*x)/32 + x^3/4) - (81*ArcSinh[(2*x)/3])/64","A",1
445,1,15,15,0.0019112,"\int x \sqrt{9+4 x^2} \, dx","Integrate[x*Sqrt[9 + 4*x^2],x]","\frac{1}{12} \left(4 x^2+9\right)^{3/2}","\frac{1}{12} \left(4 x^2+9\right)^{3/2}",1,"(9 + 4*x^2)^(3/2)/12","A",1
446,1,27,27,0.0062216,"\int \sqrt{9+4 x^2} \, dx","Integrate[Sqrt[9 + 4*x^2],x]","\frac{1}{2} \sqrt{4 x^2+9} x+\frac{9}{4} \sinh ^{-1}\left(\frac{2 x}{3}\right)","\frac{1}{2} \sqrt{4 x^2+9} x+\frac{9}{4} \sinh ^{-1}\left(\frac{2 x}{3}\right)",1,"(x*Sqrt[9 + 4*x^2])/2 + (9*ArcSinh[(2*x)/3])/4","A",1
447,1,30,30,0.0045497,"\int \frac{\sqrt{9+4 x^2}}{x} \, dx","Integrate[Sqrt[9 + 4*x^2]/x,x]","\sqrt{4 x^2+9}-3 \tanh ^{-1}\left(\frac{1}{3} \sqrt{4 x^2+9}\right)","\sqrt{4 x^2+9}-3 \tanh ^{-1}\left(\frac{1}{3} \sqrt{4 x^2+9}\right)",1,"Sqrt[9 + 4*x^2] - 3*ArcTanh[Sqrt[9 + 4*x^2]/3]","A",1
448,1,25,25,0.0063488,"\int \frac{\sqrt{9+4 x^2}}{x^2} \, dx","Integrate[Sqrt[9 + 4*x^2]/x^2,x]","2 \sinh ^{-1}\left(\frac{2 x}{3}\right)-\frac{\sqrt{4 x^2+9}}{x}","2 \sinh ^{-1}\left(\frac{2 x}{3}\right)-\frac{\sqrt{4 x^2+9}}{x}",1,"-(Sqrt[9 + 4*x^2]/x) + 2*ArcSinh[(2*x)/3]","A",1
449,1,37,39,0.0182928,"\int \frac{\sqrt{9+4 x^2}}{x^3} \, dx","Integrate[Sqrt[9 + 4*x^2]/x^3,x]","-\frac{\sqrt{4 x^2+9}}{2 x^2}-\frac{2}{3} \tanh ^{-1}\left(\sqrt{\frac{4 x^2}{9}+1}\right)","-\frac{\sqrt{4 x^2+9}}{2 x^2}-\frac{2}{3} \tanh ^{-1}\left(\frac{1}{3} \sqrt{4 x^2+9}\right)",1,"-1/2*Sqrt[9 + 4*x^2]/x^2 - (2*ArcTanh[Sqrt[1 + (4*x^2)/9]])/3","A",1
450,1,18,18,0.0034645,"\int \frac{\sqrt{9+4 x^2}}{x^4} \, dx","Integrate[Sqrt[9 + 4*x^2]/x^4,x]","-\frac{\left(4 x^2+9\right)^{3/2}}{27 x^3}","-\frac{\left(4 x^2+9\right)^{3/2}}{27 x^3}",1,"-1/27*(9 + 4*x^2)^(3/2)/x^3","A",1
451,1,32,57,0.0054154,"\int \frac{\sqrt{9+4 x^2}}{x^5} \, dx","Integrate[Sqrt[9 + 4*x^2]/x^5,x]","-\frac{16 \left(4 x^2+9\right)^{3/2} \, _2F_1\left(\frac{3}{2},3;\frac{5}{2};\frac{4 x^2}{9}+1\right)}{2187}","-\frac{\sqrt{4 x^2+9}}{18 x^2}+\frac{2}{27} \tanh ^{-1}\left(\frac{1}{3} \sqrt{4 x^2+9}\right)-\frac{\sqrt{4 x^2+9}}{4 x^4}",1,"(-16*(9 + 4*x^2)^(3/2)*Hypergeometric2F1[3/2, 3, 5/2, 1 + (4*x^2)/9])/2187","C",1
452,1,27,46,0.0127235,"\int x^5 \sqrt{9-4 x^2} \, dx","Integrate[x^5*Sqrt[9 - 4*x^2],x]","-\frac{1}{280} \left(9-4 x^2\right)^{3/2} \left(10 x^4+18 x^2+27\right)","-\frac{1}{448} \left(9-4 x^2\right)^{7/2}+\frac{9}{160} \left(9-4 x^2\right)^{5/2}-\frac{27}{64} \left(9-4 x^2\right)^{3/2}",1,"-1/280*((9 - 4*x^2)^(3/2)*(27 + 18*x^2 + 10*x^4))","A",1
453,1,39,63,0.0125995,"\int x^4 \sqrt{9-4 x^2} \, dx","Integrate[x^4*Sqrt[9 - 4*x^2],x]","\frac{1}{768} x \sqrt{9-4 x^2} \left(128 x^4-72 x^2-243\right)+\frac{729}{512} \sin ^{-1}\left(\frac{2 x}{3}\right)","-\frac{81}{256} \sqrt{9-4 x^2} x+\frac{1}{6} \sqrt{9-4 x^2} x^5-\frac{3}{32} \sqrt{9-4 x^2} x^3+\frac{729}{512} \sin ^{-1}\left(\frac{2 x}{3}\right)",1,"(x*Sqrt[9 - 4*x^2]*(-243 - 72*x^2 + 128*x^4))/768 + (729*ArcSin[(2*x)/3])/512","A",1
454,1,22,31,0.0092442,"\int x^3 \sqrt{9-4 x^2} \, dx","Integrate[x^3*Sqrt[9 - 4*x^2],x]","-\frac{1}{40} \left(9-4 x^2\right)^{3/2} \left(2 x^2+3\right)","\frac{1}{80} \left(9-4 x^2\right)^{5/2}-\frac{3}{16} \left(9-4 x^2\right)^{3/2}",1,"-1/40*((9 - 4*x^2)^(3/2)*(3 + 2*x^2))","A",1
455,1,36,45,0.0155685,"\int x^2 \sqrt{9-4 x^2} \, dx","Integrate[x^2*Sqrt[9 - 4*x^2],x]","\sqrt{9-4 x^2} \left(\frac{x^3}{4}-\frac{9 x}{32}\right)+\frac{81}{64} \sin ^{-1}\left(\frac{2 x}{3}\right)","-\frac{9}{32} \sqrt{9-4 x^2} x+\frac{1}{4} \sqrt{9-4 x^2} x^3+\frac{81}{64} \sin ^{-1}\left(\frac{2 x}{3}\right)",1,"Sqrt[9 - 4*x^2]*((-9*x)/32 + x^3/4) + (81*ArcSin[(2*x)/3])/64","A",1
456,1,15,15,0.0017706,"\int x \sqrt{9-4 x^2} \, dx","Integrate[x*Sqrt[9 - 4*x^2],x]","-\frac{1}{12} \left(9-4 x^2\right)^{3/2}","-\frac{1}{12} \left(9-4 x^2\right)^{3/2}",1,"-1/12*(9 - 4*x^2)^(3/2)","A",1
457,1,27,27,0.0064628,"\int \sqrt{9-4 x^2} \, dx","Integrate[Sqrt[9 - 4*x^2],x]","\frac{1}{2} \sqrt{9-4 x^2} x+\frac{9}{4} \sin ^{-1}\left(\frac{2 x}{3}\right)","\frac{1}{2} \sqrt{9-4 x^2} x+\frac{9}{4} \sin ^{-1}\left(\frac{2 x}{3}\right)",1,"(x*Sqrt[9 - 4*x^2])/2 + (9*ArcSin[(2*x)/3])/4","A",1
458,1,30,30,0.0044554,"\int \frac{\sqrt{9-4 x^2}}{x} \, dx","Integrate[Sqrt[9 - 4*x^2]/x,x]","\sqrt{9-4 x^2}-3 \tanh ^{-1}\left(\frac{1}{3} \sqrt{9-4 x^2}\right)","\sqrt{9-4 x^2}-3 \tanh ^{-1}\left(\frac{1}{3} \sqrt{9-4 x^2}\right)",1,"Sqrt[9 - 4*x^2] - 3*ArcTanh[Sqrt[9 - 4*x^2]/3]","A",1
459,1,25,25,0.0068497,"\int \frac{\sqrt{9-4 x^2}}{x^2} \, dx","Integrate[Sqrt[9 - 4*x^2]/x^2,x]","-\frac{\sqrt{9-4 x^2}}{x}-2 \sin ^{-1}\left(\frac{2 x}{3}\right)","-\frac{\sqrt{9-4 x^2}}{x}-2 \sin ^{-1}\left(\frac{2 x}{3}\right)",1,"-(Sqrt[9 - 4*x^2]/x) - 2*ArcSin[(2*x)/3]","A",1
460,1,37,39,0.0265819,"\int \frac{\sqrt{9-4 x^2}}{x^3} \, dx","Integrate[Sqrt[9 - 4*x^2]/x^3,x]","\frac{2}{3} \tanh ^{-1}\left(\sqrt{1-\frac{4 x^2}{9}}\right)-\frac{\sqrt{9-4 x^2}}{2 x^2}","\frac{2}{3} \tanh ^{-1}\left(\frac{1}{3} \sqrt{9-4 x^2}\right)-\frac{\sqrt{9-4 x^2}}{2 x^2}",1,"-1/2*Sqrt[9 - 4*x^2]/x^2 + (2*ArcTanh[Sqrt[1 - (4*x^2)/9]])/3","A",1
461,1,18,18,0.0031025,"\int \frac{\sqrt{9-4 x^2}}{x^4} \, dx","Integrate[Sqrt[9 - 4*x^2]/x^4,x]","-\frac{\left(9-4 x^2\right)^{3/2}}{27 x^3}","-\frac{\left(9-4 x^2\right)^{3/2}}{27 x^3}",1,"-1/27*(9 - 4*x^2)^(3/2)/x^3","A",1
462,1,32,57,0.005572,"\int \frac{\sqrt{9-4 x^2}}{x^5} \, dx","Integrate[Sqrt[9 - 4*x^2]/x^5,x]","-\frac{16 \left(9-4 x^2\right)^{3/2} \, _2F_1\left(\frac{3}{2},3;\frac{5}{2};1-\frac{4 x^2}{9}\right)}{2187}","\frac{\sqrt{9-4 x^2}}{18 x^2}+\frac{2}{27} \tanh ^{-1}\left(\frac{1}{3} \sqrt{9-4 x^2}\right)-\frac{\sqrt{9-4 x^2}}{4 x^4}",1,"(-16*(9 - 4*x^2)^(3/2)*Hypergeometric2F1[3/2, 3, 5/2, 1 - (4*x^2)/9])/2187","C",1
463,1,27,46,0.0095423,"\int x^5 \sqrt{-9+4 x^2} \, dx","Integrate[x^5*Sqrt[-9 + 4*x^2],x]","\frac{1}{280} \left(4 x^2-9\right)^{3/2} \left(10 x^4+18 x^2+27\right)","\frac{1}{448} \left(4 x^2-9\right)^{7/2}+\frac{9}{160} \left(4 x^2-9\right)^{5/2}+\frac{27}{64} \left(4 x^2-9\right)^{3/2}",1,"((-9 + 4*x^2)^(3/2)*(27 + 18*x^2 + 10*x^4))/280","A",1
464,1,49,72,0.0118262,"\int x^4 \sqrt{-9+4 x^2} \, dx","Integrate[x^4*Sqrt[-9 + 4*x^2],x]","\frac{1}{768} x \sqrt{4 x^2-9} \left(128 x^4-72 x^2-243\right)-\frac{729}{512} \log \left(\sqrt{4 x^2-9}+2 x\right)","-\frac{81}{256} \sqrt{4 x^2-9} x-\frac{729}{512} \tanh ^{-1}\left(\frac{2 x}{\sqrt{4 x^2-9}}\right)+\frac{1}{6} \sqrt{4 x^2-9} x^5-\frac{3}{32} \sqrt{4 x^2-9} x^3",1,"(x*Sqrt[-9 + 4*x^2]*(-243 - 72*x^2 + 128*x^4))/768 - (729*Log[2*x + Sqrt[-9 + 4*x^2]])/512","A",1
465,1,22,31,0.0068872,"\int x^3 \sqrt{-9+4 x^2} \, dx","Integrate[x^3*Sqrt[-9 + 4*x^2],x]","\frac{1}{40} \left(2 x^2+3\right) \left(4 x^2-9\right)^{3/2}","\frac{1}{80} \left(4 x^2-9\right)^{5/2}+\frac{3}{16} \left(4 x^2-9\right)^{3/2}",1,"((3 + 2*x^2)*(-9 + 4*x^2)^(3/2))/40","A",1
466,1,46,54,0.0138272,"\int x^2 \sqrt{-9+4 x^2} \, dx","Integrate[x^2*Sqrt[-9 + 4*x^2],x]","\sqrt{4 x^2-9} \left(\frac{x^3}{4}-\frac{9 x}{32}\right)-\frac{81}{64} \log \left(\sqrt{4 x^2-9}+2 x\right)","-\frac{9}{32} \sqrt{4 x^2-9} x-\frac{81}{64} \tanh ^{-1}\left(\frac{2 x}{\sqrt{4 x^2-9}}\right)+\frac{1}{4} \sqrt{4 x^2-9} x^3",1,"Sqrt[-9 + 4*x^2]*((-9*x)/32 + x^3/4) - (81*Log[2*x + Sqrt[-9 + 4*x^2]])/64","A",1
467,1,15,15,0.0019934,"\int x \sqrt{-9+4 x^2} \, dx","Integrate[x*Sqrt[-9 + 4*x^2],x]","\frac{1}{12} \left(4 x^2-9\right)^{3/2}","\frac{1}{12} \left(4 x^2-9\right)^{3/2}",1,"(-9 + 4*x^2)^(3/2)/12","A",1
468,1,37,36,0.0062575,"\int \sqrt{-9+4 x^2} \, dx","Integrate[Sqrt[-9 + 4*x^2],x]","\frac{1}{2} x \sqrt{4 x^2-9}-\frac{9}{4} \log \left(\sqrt{4 x^2-9}+2 x\right)","\frac{1}{2} x \sqrt{4 x^2-9}-\frac{9}{4} \tanh ^{-1}\left(\frac{2 x}{\sqrt{4 x^2-9}}\right)",1,"(x*Sqrt[-9 + 4*x^2])/2 - (9*Log[2*x + Sqrt[-9 + 4*x^2]])/4","A",1
469,1,30,30,0.0045342,"\int \frac{\sqrt{-9+4 x^2}}{x} \, dx","Integrate[Sqrt[-9 + 4*x^2]/x,x]","\sqrt{4 x^2-9}-3 \tan ^{-1}\left(\frac{1}{3} \sqrt{4 x^2-9}\right)","\sqrt{4 x^2-9}-3 \tan ^{-1}\left(\frac{1}{3} \sqrt{4 x^2-9}\right)",1,"Sqrt[-9 + 4*x^2] - 3*ArcTan[Sqrt[-9 + 4*x^2]/3]","A",1
470,1,48,34,0.0101302,"\int \frac{\sqrt{-9+4 x^2}}{x^2} \, dx","Integrate[Sqrt[-9 + 4*x^2]/x^2,x]","-\frac{\sqrt{4 x^2-9}+\frac{2 x \sqrt{4 x^2-9} \sin ^{-1}\left(\frac{2 x}{3}\right)}{\sqrt{9-4 x^2}}}{x}","2 \tanh ^{-1}\left(\frac{2 x}{\sqrt{4 x^2-9}}\right)-\frac{\sqrt{4 x^2-9}}{x}",1,"-((Sqrt[-9 + 4*x^2] + (2*x*Sqrt[-9 + 4*x^2]*ArcSin[(2*x)/3])/Sqrt[9 - 4*x^2])/x)","A",1
471,1,55,39,0.0082574,"\int \frac{\sqrt{-9+4 x^2}}{x^3} \, dx","Integrate[Sqrt[-9 + 4*x^2]/x^3,x]","-\frac{12 x^2+4 \sqrt{9-4 x^2} x^2 \tanh ^{-1}\left(\sqrt{1-\frac{4 x^2}{9}}\right)-27}{6 x^2 \sqrt{4 x^2-9}}","\frac{2}{3} \tan ^{-1}\left(\frac{1}{3} \sqrt{4 x^2-9}\right)-\frac{\sqrt{4 x^2-9}}{2 x^2}",1,"-1/6*(-27 + 12*x^2 + 4*x^2*Sqrt[9 - 4*x^2]*ArcTanh[Sqrt[1 - (4*x^2)/9]])/(x^2*Sqrt[-9 + 4*x^2])","A",1
472,1,18,18,0.002749,"\int \frac{\sqrt{-9+4 x^2}}{x^4} \, dx","Integrate[Sqrt[-9 + 4*x^2]/x^4,x]","\frac{\left(4 x^2-9\right)^{3/2}}{27 x^3}","\frac{\left(4 x^2-9\right)^{3/2}}{27 x^3}",1,"(-9 + 4*x^2)^(3/2)/(27*x^3)","A",1
473,1,32,57,0.0053811,"\int \frac{\sqrt{-9+4 x^2}}{x^5} \, dx","Integrate[Sqrt[-9 + 4*x^2]/x^5,x]","\frac{16 \left(4 x^2-9\right)^{3/2} \, _2F_1\left(\frac{3}{2},3;\frac{5}{2};1-\frac{4 x^2}{9}\right)}{2187}","\frac{\sqrt{4 x^2-9}}{18 x^2}+\frac{2}{27} \tan ^{-1}\left(\frac{1}{3} \sqrt{4 x^2-9}\right)-\frac{\sqrt{4 x^2-9}}{4 x^4}",1,"(16*(-9 + 4*x^2)^(3/2)*Hypergeometric2F1[3/2, 3, 5/2, 1 - (4*x^2)/9])/2187","C",1
474,1,27,46,0.0141054,"\int x^5 \sqrt{-9-4 x^2} \, dx","Integrate[x^5*Sqrt[-9 - 4*x^2],x]","-\frac{1}{280} \left(-4 x^2-9\right)^{3/2} \left(10 x^4-18 x^2+27\right)","-\frac{1}{448} \left(-4 x^2-9\right)^{7/2}-\frac{9}{160} \left(-4 x^2-9\right)^{5/2}-\frac{27}{64} \left(-4 x^2-9\right)^{3/2}",1,"-1/280*((-9 - 4*x^2)^(3/2)*(27 - 18*x^2 + 10*x^4))","A",1
475,1,48,72,0.0160357,"\int x^4 \sqrt{-9-4 x^2} \, dx","Integrate[x^4*Sqrt[-9 - 4*x^2],x]","\frac{1}{768} x \sqrt{-4 x^2-9} \left(128 x^4+72 x^2-243\right)-\frac{729}{512} \tan ^{-1}\left(\frac{2 x}{\sqrt{-4 x^2-9}}\right)","-\frac{81}{256} \sqrt{-4 x^2-9} x-\frac{729}{512} \tan ^{-1}\left(\frac{2 x}{\sqrt{-4 x^2-9}}\right)+\frac{1}{6} \sqrt{-4 x^2-9} x^5+\frac{3}{32} \sqrt{-4 x^2-9} x^3",1,"(x*Sqrt[-9 - 4*x^2]*(-243 + 72*x^2 + 128*x^4))/768 - (729*ArcTan[(2*x)/Sqrt[-9 - 4*x^2]])/512","A",1
476,1,22,31,0.0094515,"\int x^3 \sqrt{-9-4 x^2} \, dx","Integrate[x^3*Sqrt[-9 - 4*x^2],x]","\frac{1}{40} \left(-4 x^2-9\right)^{3/2} \left(3-2 x^2\right)","\frac{1}{80} \left(-4 x^2-9\right)^{5/2}+\frac{3}{16} \left(-4 x^2-9\right)^{3/2}",1,"((-9 - 4*x^2)^(3/2)*(3 - 2*x^2))/40","A",1
477,1,43,54,0.0162422,"\int x^2 \sqrt{-9-4 x^2} \, dx","Integrate[x^2*Sqrt[-9 - 4*x^2],x]","\frac{1}{64} \left(2 x \sqrt{-4 x^2-9} \left(8 x^2+9\right)+81 \tan ^{-1}\left(\frac{2 x}{\sqrt{-4 x^2-9}}\right)\right)","\frac{9}{32} \sqrt{-4 x^2-9} x+\frac{81}{64} \tan ^{-1}\left(\frac{2 x}{\sqrt{-4 x^2-9}}\right)+\frac{1}{4} \sqrt{-4 x^2-9} x^3",1,"(2*x*Sqrt[-9 - 4*x^2]*(9 + 8*x^2) + 81*ArcTan[(2*x)/Sqrt[-9 - 4*x^2]])/64","A",1
478,1,15,15,0.0017897,"\int x \sqrt{-9-4 x^2} \, dx","Integrate[x*Sqrt[-9 - 4*x^2],x]","-\frac{1}{12} \left(-4 x^2-9\right)^{3/2}","-\frac{1}{12} \left(-4 x^2-9\right)^{3/2}",1,"-1/12*(-9 - 4*x^2)^(3/2)","A",1
479,1,36,36,0.0079576,"\int \sqrt{-9-4 x^2} \, dx","Integrate[Sqrt[-9 - 4*x^2],x]","\frac{1}{4} \left(2 x \sqrt{-4 x^2-9}-9 \tan ^{-1}\left(\frac{2 x}{\sqrt{-4 x^2-9}}\right)\right)","\frac{1}{2} x \sqrt{-4 x^2-9}-\frac{9}{4} \tan ^{-1}\left(\frac{2 x}{\sqrt{-4 x^2-9}}\right)",1,"(2*x*Sqrt[-9 - 4*x^2] - 9*ArcTan[(2*x)/Sqrt[-9 - 4*x^2]])/4","A",1
480,1,30,30,0.0044475,"\int \frac{\sqrt{-9-4 x^2}}{x} \, dx","Integrate[Sqrt[-9 - 4*x^2]/x,x]","\sqrt{-4 x^2-9}-3 \tan ^{-1}\left(\frac{1}{3} \sqrt{-4 x^2-9}\right)","\sqrt{-4 x^2-9}-3 \tan ^{-1}\left(\frac{1}{3} \sqrt{-4 x^2-9}\right)",1,"Sqrt[-9 - 4*x^2] - 3*ArcTan[Sqrt[-9 - 4*x^2]/3]","A",1
481,1,49,34,0.007284,"\int \frac{\sqrt{-9-4 x^2}}{x^2} \, dx","Integrate[Sqrt[-9 - 4*x^2]/x^2,x]","\frac{\sqrt{-4 x^2-9} \left(2 x \sinh ^{-1}\left(\frac{2 x}{3}\right)-\sqrt{4 x^2+9}\right)}{x \sqrt{4 x^2+9}}","-\frac{\sqrt{-4 x^2-9}}{x}-2 \tan ^{-1}\left(\frac{2 x}{\sqrt{-4 x^2-9}}\right)",1,"(Sqrt[-9 - 4*x^2]*(-Sqrt[9 + 4*x^2] + 2*x*ArcSinh[(2*x)/3]))/(x*Sqrt[9 + 4*x^2])","A",1
482,1,55,39,0.0137225,"\int \frac{\sqrt{-9-4 x^2}}{x^3} \, dx","Integrate[Sqrt[-9 - 4*x^2]/x^3,x]","\frac{12 x^2+4 \sqrt{4 x^2+9} x^2 \tanh ^{-1}\left(\sqrt{\frac{4 x^2}{9}+1}\right)+27}{6 x^2 \sqrt{-4 x^2-9}}","-\frac{\sqrt{-4 x^2-9}}{2 x^2}-\frac{2}{3} \tan ^{-1}\left(\frac{1}{3} \sqrt{-4 x^2-9}\right)",1,"(27 + 12*x^2 + 4*x^2*Sqrt[9 + 4*x^2]*ArcTanh[Sqrt[1 + (4*x^2)/9]])/(6*x^2*Sqrt[-9 - 4*x^2])","A",1
483,1,18,18,0.0026582,"\int \frac{\sqrt{-9-4 x^2}}{x^4} \, dx","Integrate[Sqrt[-9 - 4*x^2]/x^4,x]","\frac{\left(-4 x^2-9\right)^{3/2}}{27 x^3}","\frac{\left(-4 x^2-9\right)^{3/2}}{27 x^3}",1,"(-9 - 4*x^2)^(3/2)/(27*x^3)","A",1
484,1,32,57,0.005382,"\int \frac{\sqrt{-9-4 x^2}}{x^5} \, dx","Integrate[Sqrt[-9 - 4*x^2]/x^5,x]","\frac{16 \left(-4 x^2-9\right)^{3/2} \, _2F_1\left(\frac{3}{2},3;\frac{5}{2};\frac{4 x^2}{9}+1\right)}{2187}","-\frac{\sqrt{-4 x^2-9}}{18 x^2}+\frac{2}{27} \tan ^{-1}\left(\frac{1}{3} \sqrt{-4 x^2-9}\right)-\frac{\sqrt{-4 x^2-9}}{4 x^4}",1,"(16*(-9 - 4*x^2)^(3/2)*Hypergeometric2F1[3/2, 3, 5/2, 1 + (4*x^2)/9])/2187","C",1
485,1,39,56,0.0190337,"\int \frac{x^5}{\sqrt{a+b x^2}} \, dx","Integrate[x^5/Sqrt[a + b*x^2],x]","\frac{\sqrt{a+b x^2} \left(8 a^2-4 a b x^2+3 b^2 x^4\right)}{15 b^3}","\frac{a^2 \sqrt{a+b x^2}}{b^3}+\frac{\left(a+b x^2\right)^{5/2}}{5 b^3}-\frac{2 a \left(a+b x^2\right)^{3/2}}{3 b^3}",1,"(Sqrt[a + b*x^2]*(8*a^2 - 4*a*b*x^2 + 3*b^2*x^4))/(15*b^3)","A",1
486,1,62,73,0.0256736,"\int \frac{x^4}{\sqrt{a+b x^2}} \, dx","Integrate[x^4/Sqrt[a + b*x^2],x]","\frac{3 a^2 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)+\sqrt{b} x \sqrt{a+b x^2} \left(2 b x^2-3 a\right)}{8 b^{5/2}}","\frac{3 a^2 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{8 b^{5/2}}-\frac{3 a x \sqrt{a+b x^2}}{8 b^2}+\frac{x^3 \sqrt{a+b x^2}}{4 b}",1,"(Sqrt[b]*x*Sqrt[a + b*x^2]*(-3*a + 2*b*x^2) + 3*a^2*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*b^(5/2))","A",1
487,1,27,36,0.0128865,"\int \frac{x^3}{\sqrt{a+b x^2}} \, dx","Integrate[x^3/Sqrt[a + b*x^2],x]","\frac{\left(b x^2-2 a\right) \sqrt{a+b x^2}}{3 b^2}","\frac{\left(a+b x^2\right)^{3/2}}{3 b^2}-\frac{a \sqrt{a+b x^2}}{b^2}",1,"((-2*a + b*x^2)*Sqrt[a + b*x^2])/(3*b^2)","A",1
488,1,49,49,0.01831,"\int \frac{x^2}{\sqrt{a+b x^2}} \, dx","Integrate[x^2/Sqrt[a + b*x^2],x]","\frac{x \sqrt{a+b x^2}}{2 b}-\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 b^{3/2}}","\frac{x \sqrt{a+b x^2}}{2 b}-\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 b^{3/2}}",1,"(x*Sqrt[a + b*x^2])/(2*b) - (a*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(3/2))","A",1
489,1,15,15,0.0019353,"\int \frac{x}{\sqrt{a+b x^2}} \, dx","Integrate[x/Sqrt[a + b*x^2],x]","\frac{\sqrt{a+b x^2}}{b}","\frac{\sqrt{a+b x^2}}{b}",1,"Sqrt[a + b*x^2]/b","A",1
490,1,25,25,0.0043646,"\int \frac{1}{\sqrt{a+b x^2}} \, dx","Integrate[1/Sqrt[a + b*x^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{\sqrt{b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{\sqrt{b}}",1,"ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]]/Sqrt[b]","A",1
491,1,25,25,0.0050733,"\int \frac{1}{x \sqrt{a+b x^2}} \, dx","Integrate[1/(x*Sqrt[a + b*x^2]),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{\sqrt{a}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{\sqrt{a}}",1,"-(ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]/Sqrt[a])","A",1
492,1,19,19,0.0035533,"\int \frac{1}{x^2 \sqrt{a+b x^2}} \, dx","Integrate[1/(x^2*Sqrt[a + b*x^2]),x]","-\frac{\sqrt{a+b x^2}}{a x}","-\frac{\sqrt{a+b x^2}}{a x}",1,"-(Sqrt[a + b*x^2]/(a*x))","A",1
493,1,61,50,0.0534913,"\int \frac{1}{x^3 \sqrt{a+b x^2}} \, dx","Integrate[1/(x^3*Sqrt[a + b*x^2]),x]","\frac{b \sqrt{a+b x^2} \left(\frac{\tanh ^{-1}\left(\sqrt{\frac{b x^2}{a}+1}\right)}{2 \sqrt{\frac{b x^2}{a}+1}}-\frac{a}{2 b x^2}\right)}{a^2}","\frac{b \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{2 a^{3/2}}-\frac{\sqrt{a+b x^2}}{2 a x^2}",1,"(b*Sqrt[a + b*x^2]*(-1/2*a/(b*x^2) + ArcTanh[Sqrt[1 + (b*x^2)/a]]/(2*Sqrt[1 + (b*x^2)/a])))/a^2","A",1
494,1,29,44,0.0059474,"\int \frac{1}{x^4 \sqrt{a+b x^2}} \, dx","Integrate[1/(x^4*Sqrt[a + b*x^2]),x]","-\frac{\left(a-2 b x^2\right) \sqrt{a+b x^2}}{3 a^2 x^3}","\frac{2 b \sqrt{a+b x^2}}{3 a^2 x}-\frac{\sqrt{a+b x^2}}{3 a x^3}",1,"-1/3*((a - 2*b*x^2)*Sqrt[a + b*x^2])/(a^2*x^3)","A",1
495,1,37,74,0.0070721,"\int \frac{1}{x^5 \sqrt{a+b x^2}} \, dx","Integrate[1/(x^5*Sqrt[a + b*x^2]),x]","-\frac{b^2 \sqrt{a+b x^2} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};\frac{b x^2}{a}+1\right)}{a^3}","-\frac{3 b^2 \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{8 a^{5/2}}+\frac{3 b \sqrt{a+b x^2}}{8 a^2 x^2}-\frac{\sqrt{a+b x^2}}{4 a x^4}",1,"-((b^2*Sqrt[a + b*x^2]*Hypergeometric2F1[1/2, 3, 3/2, 1 + (b*x^2)/a])/a^3)","C",1
496,1,38,55,0.0165455,"\int \frac{x^5}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[x^5/(a + b*x^2)^(3/2),x]","\frac{-8 a^2-4 a b x^2+b^2 x^4}{3 b^3 \sqrt{a+b x^2}}","-\frac{a^2}{b^3 \sqrt{a+b x^2}}-\frac{2 a \sqrt{a+b x^2}}{b^3}+\frac{\left(a+b x^2\right)^{3/2}}{3 b^3}",1,"(-8*a^2 - 4*a*b*x^2 + b^2*x^4)/(3*b^3*Sqrt[a + b*x^2])","A",1
497,1,71,68,0.0335107,"\int \frac{x^4}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[x^4/(a + b*x^2)^(3/2),x]","\frac{\sqrt{b} x \left(3 a+b x^2\right)-3 a^{3/2} \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{2 b^{5/2} \sqrt{a+b x^2}}","-\frac{3 a \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 b^{5/2}}+\frac{3 x \sqrt{a+b x^2}}{2 b^2}-\frac{x^3}{b \sqrt{a+b x^2}}",1,"(Sqrt[b]*x*(3*a + b*x^2) - 3*a^(3/2)*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(5/2)*Sqrt[a + b*x^2])","A",1
498,1,24,32,0.0111564,"\int \frac{x^3}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[x^3/(a + b*x^2)^(3/2),x]","\frac{2 a+b x^2}{b^2 \sqrt{a+b x^2}}","\frac{a}{b^2 \sqrt{a+b x^2}}+\frac{\sqrt{a+b x^2}}{b^2}",1,"(2*a + b*x^2)/(b^2*Sqrt[a + b*x^2])","A",1
499,1,59,43,0.0518446,"\int \frac{x^2}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[x^2/(a + b*x^2)^(3/2),x]","\frac{\sqrt{a} \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)-\sqrt{b} x}{b^{3/2} \sqrt{a+b x^2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{b^{3/2}}-\frac{x}{b \sqrt{a+b x^2}}",1,"(-(Sqrt[b]*x) + Sqrt[a]*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(b^(3/2)*Sqrt[a + b*x^2])","A",1
500,1,16,16,0.0029394,"\int \frac{x}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[x/(a + b*x^2)^(3/2),x]","-\frac{1}{b \sqrt{a+b x^2}}","-\frac{1}{b \sqrt{a+b x^2}}",1,"-(1/(b*Sqrt[a + b*x^2]))","A",1
501,1,16,16,0.0035764,"\int \frac{1}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[(a + b*x^2)^(-3/2),x]","\frac{x}{a \sqrt{a+b x^2}}","\frac{x}{a \sqrt{a+b x^2}}",1,"x/(a*Sqrt[a + b*x^2])","A",1
502,1,33,41,0.0060894,"\int \frac{1}{x \left(a+b x^2\right)^{3/2}} \, dx","Integrate[1/(x*(a + b*x^2)^(3/2)),x]","\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b x^2}{a}+1\right)}{a \sqrt{a+b x^2}}","\frac{1}{a \sqrt{a+b x^2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{a^{3/2}}",1,"Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*x^2)/a]/(a*Sqrt[a + b*x^2])","C",1
503,1,27,38,0.0057654,"\int \frac{1}{x^2 \left(a+b x^2\right)^{3/2}} \, dx","Integrate[1/(x^2*(a + b*x^2)^(3/2)),x]","-\frac{a+2 b x^2}{a^2 x \sqrt{a+b x^2}}","-\frac{2 b x}{a^2 \sqrt{a+b x^2}}-\frac{1}{a x \sqrt{a+b x^2}}",1,"-((a + 2*b*x^2)/(a^2*x*Sqrt[a + b*x^2]))","A",1
504,1,35,69,0.0074169,"\int \frac{1}{x^3 \left(a+b x^2\right)^{3/2}} \, dx","Integrate[1/(x^3*(a + b*x^2)^(3/2)),x]","-\frac{b \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};\frac{b x^2}{a}+1\right)}{a^2 \sqrt{a+b x^2}}","\frac{3 b \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{2 a^{5/2}}-\frac{3 b}{2 a^2 \sqrt{a+b x^2}}-\frac{1}{2 a x^2 \sqrt{a+b x^2}}",1,"-((b*Hypergeometric2F1[-1/2, 2, 1/2, 1 + (b*x^2)/a])/(a^2*Sqrt[a + b*x^2]))","C",1
505,1,40,66,0.0084184,"\int \frac{1}{x^4 \left(a+b x^2\right)^{3/2}} \, dx","Integrate[1/(x^4*(a + b*x^2)^(3/2)),x]","-\frac{a^2-4 a b x^2-8 b^2 x^4}{3 a^3 x^3 \sqrt{a+b x^2}}","\frac{8 b^2 x}{3 a^3 \sqrt{a+b x^2}}+\frac{4 b}{3 a^2 x \sqrt{a+b x^2}}-\frac{1}{3 a x^3 \sqrt{a+b x^2}}",1,"-1/3*(a^2 - 4*a*b*x^2 - 8*b^2*x^4)/(a^3*x^3*Sqrt[a + b*x^2])","A",1
506,1,90,91,0.1377214,"\int \frac{x^6}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[x^6/(a + b*x^2)^(5/2),x]","\frac{\sqrt{b} x \left(15 a^2+20 a b x^2+3 b^2 x^4\right)-15 a^{3/2} \left(a+b x^2\right) \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{6 b^{7/2} \left(a+b x^2\right)^{3/2}}","-\frac{5 a \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 b^{7/2}}+\frac{5 x \sqrt{a+b x^2}}{2 b^3}-\frac{5 x^3}{3 b^2 \sqrt{a+b x^2}}-\frac{x^5}{3 b \left(a+b x^2\right)^{3/2}}",1,"(Sqrt[b]*x*(15*a^2 + 20*a*b*x^2 + 3*b^2*x^4) - 15*a^(3/2)*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(6*b^(7/2)*(a + b*x^2)^(3/2))","A",1
507,1,39,54,0.018219,"\int \frac{x^5}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[x^5/(a + b*x^2)^(5/2),x]","\frac{8 a^2+12 a b x^2+3 b^2 x^4}{3 b^3 \left(a+b x^2\right)^{3/2}}","-\frac{a^2}{3 b^3 \left(a+b x^2\right)^{3/2}}+\frac{2 a}{b^3 \sqrt{a+b x^2}}+\frac{\sqrt{a+b x^2}}{b^3}",1,"(8*a^2 + 12*a*b*x^2 + 3*b^2*x^4)/(3*b^3*(a + b*x^2)^(3/2))","A",1
508,1,80,64,0.1172748,"\int \frac{x^4}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[x^4/(a + b*x^2)^(5/2),x]","\frac{3 \sqrt{a} \left(a+b x^2\right) \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)-\sqrt{b} x \left(3 a+4 b x^2\right)}{3 b^{5/2} \left(a+b x^2\right)^{3/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{b^{5/2}}-\frac{x}{b^2 \sqrt{a+b x^2}}-\frac{x^3}{3 b \left(a+b x^2\right)^{3/2}}",1,"(-(Sqrt[b]*x*(3*a + 4*b*x^2)) + 3*Sqrt[a]*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(3*b^(5/2)*(a + b*x^2)^(3/2))","A",1
509,1,28,36,0.0118797,"\int \frac{x^3}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[x^3/(a + b*x^2)^(5/2),x]","\frac{-2 a-3 b x^2}{3 b^2 \left(a+b x^2\right)^{3/2}}","\frac{a}{3 b^2 \left(a+b x^2\right)^{3/2}}-\frac{1}{b^2 \sqrt{a+b x^2}}",1,"(-2*a - 3*b*x^2)/(3*b^2*(a + b*x^2)^(3/2))","A",1
510,1,21,21,0.0049099,"\int \frac{x^2}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[x^2/(a + b*x^2)^(5/2),x]","\frac{x^3}{3 a \left(a+b x^2\right)^{3/2}}","\frac{x^3}{3 a \left(a+b x^2\right)^{3/2}}",1,"x^3/(3*a*(a + b*x^2)^(3/2))","A",1
511,1,18,18,0.0035478,"\int \frac{x}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[x/(a + b*x^2)^(5/2),x]","-\frac{1}{3 b \left(a+b x^2\right)^{3/2}}","-\frac{1}{3 b \left(a+b x^2\right)^{3/2}}",1,"-1/3*1/(b*(a + b*x^2)^(3/2))","A",1
512,1,29,39,0.0078498,"\int \frac{1}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[(a + b*x^2)^(-5/2),x]","\frac{x \left(3 a+2 b x^2\right)}{3 a^2 \left(a+b x^2\right)^{3/2}}","\frac{2 x}{3 a^2 \sqrt{a+b x^2}}+\frac{x}{3 a \left(a+b x^2\right)^{3/2}}",1,"(x*(3*a + 2*b*x^2))/(3*a^2*(a + b*x^2)^(3/2))","A",1
513,1,36,59,0.0067768,"\int \frac{1}{x \left(a+b x^2\right)^{5/2}} \, dx","Integrate[1/(x*(a + b*x^2)^(5/2)),x]","\frac{\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b x^2}{a}+1\right)}{3 a \left(a+b x^2\right)^{3/2}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{a^{5/2}}+\frac{1}{a^2 \sqrt{a+b x^2}}+\frac{1}{3 a \left(a+b x^2\right)^{3/2}}",1,"Hypergeometric2F1[-3/2, 1, -1/2, 1 + (b*x^2)/a]/(3*a*(a + b*x^2)^(3/2))","C",1
514,1,42,60,0.0088545,"\int \frac{1}{x^2 \left(a+b x^2\right)^{5/2}} \, dx","Integrate[1/(x^2*(a + b*x^2)^(5/2)),x]","\frac{-3 a^2-12 a b x^2-8 b^2 x^4}{3 a^3 x \left(a+b x^2\right)^{3/2}}","-\frac{8 b x}{3 a^3 \sqrt{a+b x^2}}-\frac{4 b x}{3 a^2 \left(a+b x^2\right)^{3/2}}-\frac{1}{a x \left(a+b x^2\right)^{3/2}}",1,"(-3*a^2 - 12*a*b*x^2 - 8*b^2*x^4)/(3*a^3*x*(a + b*x^2)^(3/2))","A",1
515,1,37,88,0.0076301,"\int \frac{1}{x^3 \left(a+b x^2\right)^{5/2}} \, dx","Integrate[1/(x^3*(a + b*x^2)^(5/2)),x]","-\frac{b \, _2F_1\left(-\frac{3}{2},2;-\frac{1}{2};\frac{b x^2}{a}+1\right)}{3 a^2 \left(a+b x^2\right)^{3/2}}","\frac{5 b \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{2 a^{7/2}}-\frac{5 b}{2 a^3 \sqrt{a+b x^2}}-\frac{5 b}{6 a^2 \left(a+b x^2\right)^{3/2}}-\frac{1}{2 a x^2 \left(a+b x^2\right)^{3/2}}",1,"-1/3*(b*Hypergeometric2F1[-3/2, 2, -1/2, 1 + (b*x^2)/a])/(a^2*(a + b*x^2)^(3/2))","C",1
516,1,53,86,0.0105656,"\int \frac{1}{x^4 \left(a+b x^2\right)^{5/2}} \, dx","Integrate[1/(x^4*(a + b*x^2)^(5/2)),x]","\frac{-a^3+6 a^2 b x^2+24 a b^2 x^4+16 b^3 x^6}{3 a^4 x^3 \left(a+b x^2\right)^{3/2}}","\frac{16 b^2 x}{3 a^4 \sqrt{a+b x^2}}+\frac{8 b^2 x}{3 a^3 \left(a+b x^2\right)^{3/2}}+\frac{2 b}{a^2 x \left(a+b x^2\right)^{3/2}}-\frac{1}{3 a x^3 \left(a+b x^2\right)^{3/2}}",1,"(-a^3 + 6*a^2*b*x^2 + 24*a*b^2*x^4 + 16*b^3*x^6)/(3*a^4*x^3*(a + b*x^2)^(3/2))","A",1
517,1,114,131,0.1953555,"\int \frac{x^{10}}{\left(a+b x^2\right)^{9/2}} \, dx","Integrate[x^10/(a + b*x^2)^(9/2),x]","\frac{\sqrt{b} x \left(315 a^4+1050 a^3 b x^2+1218 a^2 b^2 x^4+528 a b^3 x^6+35 b^4 x^8\right)-315 a^{3/2} \left(a+b x^2\right)^3 \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{70 b^{11/2} \left(a+b x^2\right)^{7/2}}","-\frac{9 a \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{2 b^{11/2}}+\frac{9 x \sqrt{a+b x^2}}{2 b^5}-\frac{3 x^3}{b^4 \sqrt{a+b x^2}}-\frac{3 x^5}{5 b^3 \left(a+b x^2\right)^{3/2}}-\frac{9 x^7}{35 b^2 \left(a+b x^2\right)^{5/2}}-\frac{x^9}{7 b \left(a+b x^2\right)^{7/2}}",1,"(Sqrt[b]*x*(315*a^4 + 1050*a^3*b*x^2 + 1218*a^2*b^2*x^4 + 528*a*b^3*x^6 + 35*b^4*x^8) - 315*a^(3/2)*(a + b*x^2)^3*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(70*b^(11/2)*(a + b*x^2)^(7/2))","A",1
518,1,61,94,0.0312378,"\int \frac{x^9}{\left(a+b x^2\right)^{9/2}} \, dx","Integrate[x^9/(a + b*x^2)^(9/2),x]","\frac{128 a^4+448 a^3 b x^2+560 a^2 b^2 x^4+280 a b^3 x^6+35 b^4 x^8}{35 b^5 \left(a+b x^2\right)^{7/2}}","-\frac{a^4}{7 b^5 \left(a+b x^2\right)^{7/2}}+\frac{4 a^3}{5 b^5 \left(a+b x^2\right)^{5/2}}-\frac{2 a^2}{b^5 \left(a+b x^2\right)^{3/2}}+\frac{4 a}{b^5 \sqrt{a+b x^2}}+\frac{\sqrt{a+b x^2}}{b^5}",1,"(128*a^4 + 448*a^3*b*x^2 + 560*a^2*b^2*x^4 + 280*a*b^3*x^6 + 35*b^4*x^8)/(35*b^5*(a + b*x^2)^(7/2))","A",1
519,1,101,106,0.1320785,"\int \frac{x^8}{\left(a+b x^2\right)^{9/2}} \, dx","Integrate[x^8/(a + b*x^2)^(9/2),x]","\frac{\sqrt{a} \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{9/2} \sqrt{a+b x^2}}-\frac{x \left(105 a^3+350 a^2 b x^2+406 a b^2 x^4+176 b^3 x^6\right)}{105 b^4 \left(a+b x^2\right)^{7/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{b^{9/2}}-\frac{x}{b^4 \sqrt{a+b x^2}}-\frac{x^3}{3 b^3 \left(a+b x^2\right)^{3/2}}-\frac{x^5}{5 b^2 \left(a+b x^2\right)^{5/2}}-\frac{x^7}{7 b \left(a+b x^2\right)^{7/2}}",1,"-1/105*(x*(105*a^3 + 350*a^2*b*x^2 + 406*a*b^2*x^4 + 176*b^3*x^6))/(b^4*(a + b*x^2)^(7/2)) + (Sqrt[a]*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(b^(9/2)*Sqrt[a + b*x^2])","A",1
520,1,50,75,0.0249241,"\int \frac{x^7}{\left(a+b x^2\right)^{9/2}} \, dx","Integrate[x^7/(a + b*x^2)^(9/2),x]","\frac{-16 a^3-56 a^2 b x^2-70 a b^2 x^4-35 b^3 x^6}{35 b^4 \left(a+b x^2\right)^{7/2}}","\frac{a^3}{7 b^4 \left(a+b x^2\right)^{7/2}}-\frac{3 a^2}{5 b^4 \left(a+b x^2\right)^{5/2}}+\frac{a}{b^4 \left(a+b x^2\right)^{3/2}}-\frac{1}{b^4 \sqrt{a+b x^2}}",1,"(-16*a^3 - 56*a^2*b*x^2 - 70*a*b^2*x^4 - 35*b^3*x^6)/(35*b^4*(a + b*x^2)^(7/2))","A",1
521,1,21,21,0.0061662,"\int \frac{x^6}{\left(a+b x^2\right)^{9/2}} \, dx","Integrate[x^6/(a + b*x^2)^(9/2),x]","\frac{x^7}{7 a \left(a+b x^2\right)^{7/2}}","\frac{x^7}{7 a \left(a+b x^2\right)^{7/2}}",1,"x^7/(7*a*(a + b*x^2)^(7/2))","A",1
522,1,39,59,0.0189925,"\int \frac{x^5}{\left(a+b x^2\right)^{9/2}} \, dx","Integrate[x^5/(a + b*x^2)^(9/2),x]","\frac{-8 a^2-28 a b x^2-35 b^2 x^4}{105 b^3 \left(a+b x^2\right)^{7/2}}","-\frac{a^2}{7 b^3 \left(a+b x^2\right)^{7/2}}+\frac{2 a}{5 b^3 \left(a+b x^2\right)^{5/2}}-\frac{1}{3 b^3 \left(a+b x^2\right)^{3/2}}",1,"(-8*a^2 - 28*a*b*x^2 - 35*b^2*x^4)/(105*b^3*(a + b*x^2)^(7/2))","A",1
523,1,31,44,0.0124849,"\int \frac{x^4}{\left(a+b x^2\right)^{9/2}} \, dx","Integrate[x^4/(a + b*x^2)^(9/2),x]","\frac{7 a x^5+2 b x^7}{35 a^2 \left(a+b x^2\right)^{7/2}}","\frac{2 b x^7}{35 a^2 \left(a+b x^2\right)^{7/2}}+\frac{x^5}{5 a \left(a+b x^2\right)^{7/2}}",1,"(7*a*x^5 + 2*b*x^7)/(35*a^2*(a + b*x^2)^(7/2))","A",1
524,1,28,38,0.0140944,"\int \frac{x^3}{\left(a+b x^2\right)^{9/2}} \, dx","Integrate[x^3/(a + b*x^2)^(9/2),x]","\frac{-2 a-7 b x^2}{35 b^2 \left(a+b x^2\right)^{7/2}}","\frac{a}{7 b^2 \left(a+b x^2\right)^{7/2}}-\frac{1}{5 b^2 \left(a+b x^2\right)^{5/2}}",1,"(-2*a - 7*b*x^2)/(35*b^2*(a + b*x^2)^(7/2))","A",1
525,1,42,68,0.0133219,"\int \frac{x^2}{\left(a+b x^2\right)^{9/2}} \, dx","Integrate[x^2/(a + b*x^2)^(9/2),x]","\frac{x^3 \left(35 a^2+28 a b x^2+8 b^2 x^4\right)}{105 a^3 \left(a+b x^2\right)^{7/2}}","\frac{8 b^2 x^7}{105 a^3 \left(a+b x^2\right)^{7/2}}+\frac{4 b x^5}{15 a^2 \left(a+b x^2\right)^{7/2}}+\frac{x^3}{3 a \left(a+b x^2\right)^{7/2}}",1,"(x^3*(35*a^2 + 28*a*b*x^2 + 8*b^2*x^4))/(105*a^3*(a + b*x^2)^(7/2))","A",1
526,1,18,18,0.0048054,"\int \frac{x}{\left(a+b x^2\right)^{9/2}} \, dx","Integrate[x/(a + b*x^2)^(9/2),x]","-\frac{1}{7 b \left(a+b x^2\right)^{7/2}}","-\frac{1}{7 b \left(a+b x^2\right)^{7/2}}",1,"-1/7*1/(b*(a + b*x^2)^(7/2))","A",1
527,1,51,77,0.0127873,"\int \frac{1}{\left(a+b x^2\right)^{9/2}} \, dx","Integrate[(a + b*x^2)^(-9/2),x]","\frac{x \left(35 a^3+70 a^2 b x^2+56 a b^2 x^4+16 b^3 x^6\right)}{35 a^4 \left(a+b x^2\right)^{7/2}}","\frac{16 x}{35 a^4 \sqrt{a+b x^2}}+\frac{8 x}{35 a^3 \left(a+b x^2\right)^{3/2}}+\frac{6 x}{35 a^2 \left(a+b x^2\right)^{5/2}}+\frac{x}{7 a \left(a+b x^2\right)^{7/2}}",1,"(x*(35*a^3 + 70*a^2*b*x^2 + 56*a*b^2*x^4 + 16*b^3*x^6))/(35*a^4*(a + b*x^2)^(7/2))","A",1
528,1,36,95,0.006233,"\int \frac{1}{x \left(a+b x^2\right)^{9/2}} \, dx","Integrate[1/(x*(a + b*x^2)^(9/2)),x]","\frac{\, _2F_1\left(-\frac{7}{2},1;-\frac{5}{2};\frac{b x^2}{a}+1\right)}{7 a \left(a+b x^2\right)^{7/2}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{a^{9/2}}+\frac{1}{a^4 \sqrt{a+b x^2}}+\frac{1}{3 a^3 \left(a+b x^2\right)^{3/2}}+\frac{1}{5 a^2 \left(a+b x^2\right)^{5/2}}+\frac{1}{7 a \left(a+b x^2\right)^{7/2}}",1,"Hypergeometric2F1[-7/2, 1, -5/2, 1 + (b*x^2)/a]/(7*a*(a + b*x^2)^(7/2))","C",1
529,1,64,100,0.0128022,"\int \frac{1}{x^2 \left(a+b x^2\right)^{9/2}} \, dx","Integrate[1/(x^2*(a + b*x^2)^(9/2)),x]","\frac{-35 a^4-280 a^3 b x^2-560 a^2 b^2 x^4-448 a b^3 x^6-128 b^4 x^8}{35 a^5 x \left(a+b x^2\right)^{7/2}}","-\frac{128 b x}{35 a^5 \sqrt{a+b x^2}}-\frac{64 b x}{35 a^4 \left(a+b x^2\right)^{3/2}}-\frac{48 b x}{35 a^3 \left(a+b x^2\right)^{5/2}}-\frac{8 b x}{7 a^2 \left(a+b x^2\right)^{7/2}}-\frac{1}{a x \left(a+b x^2\right)^{7/2}}",1,"(-35*a^4 - 280*a^3*b*x^2 - 560*a^2*b^2*x^4 - 448*a*b^3*x^6 - 128*b^4*x^8)/(35*a^5*x*(a + b*x^2)^(7/2))","A",1
530,1,37,126,0.0093342,"\int \frac{1}{x^3 \left(a+b x^2\right)^{9/2}} \, dx","Integrate[1/(x^3*(a + b*x^2)^(9/2)),x]","-\frac{b \, _2F_1\left(-\frac{7}{2},2;-\frac{5}{2};\frac{b x^2}{a}+1\right)}{7 a^2 \left(a+b x^2\right)^{7/2}}","\frac{9 b \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{2 a^{11/2}}-\frac{9 b}{2 a^5 \sqrt{a+b x^2}}-\frac{3 b}{2 a^4 \left(a+b x^2\right)^{3/2}}-\frac{9 b}{10 a^3 \left(a+b x^2\right)^{5/2}}-\frac{9 b}{14 a^2 \left(a+b x^2\right)^{7/2}}-\frac{1}{2 a x^2 \left(a+b x^2\right)^{7/2}}",1,"-1/7*(b*Hypergeometric2F1[-7/2, 2, -5/2, 1 + (b*x^2)/a])/(a^2*(a + b*x^2)^(7/2))","C",1
531,1,75,132,0.0142964,"\int \frac{1}{x^4 \left(a+b x^2\right)^{9/2}} \, dx","Integrate[1/(x^4*(a + b*x^2)^(9/2)),x]","\frac{-7 a^5+70 a^4 b x^2+560 a^3 b^2 x^4+1120 a^2 b^3 x^6+896 a b^4 x^8+256 b^5 x^{10}}{21 a^6 x^3 \left(a+b x^2\right)^{7/2}}","\frac{256 b^2 x}{21 a^6 \sqrt{a+b x^2}}+\frac{128 b^2 x}{21 a^5 \left(a+b x^2\right)^{3/2}}+\frac{32 b^2 x}{7 a^4 \left(a+b x^2\right)^{5/2}}+\frac{80 b^2 x}{21 a^3 \left(a+b x^2\right)^{7/2}}+\frac{10 b}{3 a^2 x \left(a+b x^2\right)^{7/2}}-\frac{1}{3 a x^3 \left(a+b x^2\right)^{7/2}}",1,"(-7*a^5 + 70*a^4*b*x^2 + 560*a^3*b^2*x^4 + 1120*a^2*b^3*x^6 + 896*a*b^4*x^8 + 256*b^5*x^10)/(21*a^6*x^3*(a + b*x^2)^(7/2))","A",1
532,1,27,46,0.0088298,"\int \frac{x^5}{\sqrt{9+4 x^2}} \, dx","Integrate[x^5/Sqrt[9 + 4*x^2],x]","\frac{1}{40} \sqrt{4 x^2+9} \left(2 x^4-6 x^2+27\right)","\frac{1}{320} \left(4 x^2+9\right)^{5/2}-\frac{3}{32} \left(4 x^2+9\right)^{3/2}+\frac{81}{64} \sqrt{4 x^2+9}",1,"(Sqrt[9 + 4*x^2]*(27 - 6*x^2 + 2*x^4))/40","A",1
533,1,34,45,0.0116036,"\int \frac{x^4}{\sqrt{9+4 x^2}} \, dx","Integrate[x^4/Sqrt[9 + 4*x^2],x]","\frac{1}{256} \left(2 x \sqrt{4 x^2+9} \left(8 x^2-27\right)+243 \sinh ^{-1}\left(\frac{2 x}{3}\right)\right)","-\frac{27}{128} \sqrt{4 x^2+9} x+\frac{1}{16} \sqrt{4 x^2+9} x^3+\frac{243}{256} \sinh ^{-1}\left(\frac{2 x}{3}\right)",1,"(2*x*Sqrt[9 + 4*x^2]*(-27 + 8*x^2) + 243*ArcSinh[(2*x)/3])/256","A",1
534,1,22,31,0.0064115,"\int \frac{x^3}{\sqrt{9+4 x^2}} \, dx","Integrate[x^3/Sqrt[9 + 4*x^2],x]","\frac{1}{24} \left(2 x^2-9\right) \sqrt{4 x^2+9}","\frac{1}{48} \left(4 x^2+9\right)^{3/2}-\frac{9}{16} \sqrt{4 x^2+9}",1,"((-9 + 2*x^2)*Sqrt[9 + 4*x^2])/24","A",1
535,1,27,27,0.0053781,"\int \frac{x^2}{\sqrt{9+4 x^2}} \, dx","Integrate[x^2/Sqrt[9 + 4*x^2],x]","\frac{1}{8} x \sqrt{4 x^2+9}-\frac{9}{16} \sinh ^{-1}\left(\frac{2 x}{3}\right)","\frac{1}{8} x \sqrt{4 x^2+9}-\frac{9}{16} \sinh ^{-1}\left(\frac{2 x}{3}\right)",1,"(x*Sqrt[9 + 4*x^2])/8 - (9*ArcSinh[(2*x)/3])/16","A",1
536,1,15,15,0.0014144,"\int \frac{x}{\sqrt{9+4 x^2}} \, dx","Integrate[x/Sqrt[9 + 4*x^2],x]","\frac{1}{4} \sqrt{4 x^2+9}","\frac{1}{4} \sqrt{4 x^2+9}",1,"Sqrt[9 + 4*x^2]/4","A",1
537,1,10,10,0.0040042,"\int \frac{1}{\sqrt{9+4 x^2}} \, dx","Integrate[1/Sqrt[9 + 4*x^2],x]","\frac{1}{2} \sinh ^{-1}\left(\frac{2 x}{3}\right)","\frac{1}{2} \sinh ^{-1}\left(\frac{2 x}{3}\right)",1,"ArcSinh[(2*x)/3]/2","A",1
538,1,20,20,0.0028989,"\int \frac{1}{x \sqrt{9+4 x^2}} \, dx","Integrate[1/(x*Sqrt[9 + 4*x^2]),x]","-\frac{1}{3} \tanh ^{-1}\left(\frac{1}{3} \sqrt{4 x^2+9}\right)","-\frac{1}{3} \tanh ^{-1}\left(\frac{1}{3} \sqrt{4 x^2+9}\right)",1,"-1/3*ArcTanh[Sqrt[9 + 4*x^2]/3]","A",1
539,1,18,18,0.0025976,"\int \frac{1}{x^2 \sqrt{9+4 x^2}} \, dx","Integrate[1/(x^2*Sqrt[9 + 4*x^2]),x]","-\frac{\sqrt{4 x^2+9}}{9 x}","-\frac{\sqrt{4 x^2+9}}{9 x}",1,"-1/9*Sqrt[9 + 4*x^2]/x","A",1
540,1,37,39,0.0110456,"\int \frac{1}{x^3 \sqrt{9+4 x^2}} \, dx","Integrate[1/(x^3*Sqrt[9 + 4*x^2]),x]","\frac{1}{54} \left(4 \tanh ^{-1}\left(\sqrt{\frac{4 x^2}{9}+1}\right)-\frac{3 \sqrt{4 x^2+9}}{x^2}\right)","\frac{2}{27} \tanh ^{-1}\left(\frac{1}{3} \sqrt{4 x^2+9}\right)-\frac{\sqrt{4 x^2+9}}{18 x^2}",1,"((-3*Sqrt[9 + 4*x^2])/x^2 + 4*ArcTanh[Sqrt[1 + (4*x^2)/9]])/54","A",1
541,1,27,37,0.0045614,"\int \frac{1}{x^4 \sqrt{9+4 x^2}} \, dx","Integrate[1/(x^4*Sqrt[9 + 4*x^2]),x]","-\frac{\left(9-8 x^2\right) \sqrt{\frac{4 x^2}{9}+1}}{81 x^3}","\frac{8 \sqrt{4 x^2+9}}{243 x}-\frac{\sqrt{4 x^2+9}}{27 x^3}",1,"-1/81*((9 - 8*x^2)*Sqrt[1 + (4*x^2)/9])/x^3","A",1
542,1,32,57,0.0052025,"\int \frac{1}{x^5 \sqrt{9+4 x^2}} \, dx","Integrate[1/(x^5*Sqrt[9 + 4*x^2]),x]","-\frac{16}{729} \sqrt{4 x^2+9} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};\frac{4 x^2}{9}+1\right)","\frac{\sqrt{4 x^2+9}}{54 x^2}-\frac{2}{81} \tanh ^{-1}\left(\frac{1}{3} \sqrt{4 x^2+9}\right)-\frac{\sqrt{4 x^2+9}}{36 x^4}",1,"(-16*Sqrt[9 + 4*x^2]*Hypergeometric2F1[1/2, 3, 3/2, 1 + (4*x^2)/9])/729","C",1
543,1,27,46,0.0089028,"\int \frac{x^5}{\sqrt{9-4 x^2}} \, dx","Integrate[x^5/Sqrt[9 - 4*x^2],x]","-\frac{1}{40} \sqrt{9-4 x^2} \left(2 x^4+6 x^2+27\right)","-\frac{1}{320} \left(9-4 x^2\right)^{5/2}+\frac{3}{32} \left(9-4 x^2\right)^{3/2}-\frac{81}{64} \sqrt{9-4 x^2}",1,"-1/40*(Sqrt[9 - 4*x^2]*(27 + 6*x^2 + 2*x^4))","A",1
544,1,34,45,0.0121218,"\int \frac{x^4}{\sqrt{9-4 x^2}} \, dx","Integrate[x^4/Sqrt[9 - 4*x^2],x]","\frac{1}{256} \left(243 \sin ^{-1}\left(\frac{2 x}{3}\right)-2 x \sqrt{9-4 x^2} \left(8 x^2+27\right)\right)","-\frac{27}{128} \sqrt{9-4 x^2} x-\frac{1}{16} \sqrt{9-4 x^2} x^3+\frac{243}{256} \sin ^{-1}\left(\frac{2 x}{3}\right)",1,"(-2*x*Sqrt[9 - 4*x^2]*(27 + 8*x^2) + 243*ArcSin[(2*x)/3])/256","A",1
545,1,22,31,0.0062845,"\int \frac{x^3}{\sqrt{9-4 x^2}} \, dx","Integrate[x^3/Sqrt[9 - 4*x^2],x]","-\frac{1}{24} \sqrt{9-4 x^2} \left(2 x^2+9\right)","\frac{1}{48} \left(9-4 x^2\right)^{3/2}-\frac{9}{16} \sqrt{9-4 x^2}",1,"-1/24*(Sqrt[9 - 4*x^2]*(9 + 2*x^2))","A",1
546,1,27,27,0.0055176,"\int \frac{x^2}{\sqrt{9-4 x^2}} \, dx","Integrate[x^2/Sqrt[9 - 4*x^2],x]","\frac{9}{16} \sin ^{-1}\left(\frac{2 x}{3}\right)-\frac{1}{8} x \sqrt{9-4 x^2}","\frac{9}{16} \sin ^{-1}\left(\frac{2 x}{3}\right)-\frac{1}{8} x \sqrt{9-4 x^2}",1,"-1/8*(x*Sqrt[9 - 4*x^2]) + (9*ArcSin[(2*x)/3])/16","A",1
547,1,15,15,0.0015945,"\int \frac{x}{\sqrt{9-4 x^2}} \, dx","Integrate[x/Sqrt[9 - 4*x^2],x]","-\frac{1}{4} \sqrt{9-4 x^2}","-\frac{1}{4} \sqrt{9-4 x^2}",1,"-1/4*Sqrt[9 - 4*x^2]","A",1
548,1,10,10,0.0043956,"\int \frac{1}{\sqrt{9-4 x^2}} \, dx","Integrate[1/Sqrt[9 - 4*x^2],x]","\frac{1}{2} \sin ^{-1}\left(\frac{2 x}{3}\right)","\frac{1}{2} \sin ^{-1}\left(\frac{2 x}{3}\right)",1,"ArcSin[(2*x)/3]/2","A",1
549,1,20,20,0.0026157,"\int \frac{1}{x \sqrt{9-4 x^2}} \, dx","Integrate[1/(x*Sqrt[9 - 4*x^2]),x]","-\frac{1}{3} \tanh ^{-1}\left(\frac{1}{3} \sqrt{9-4 x^2}\right)","-\frac{1}{3} \tanh ^{-1}\left(\frac{1}{3} \sqrt{9-4 x^2}\right)",1,"-1/3*ArcTanh[Sqrt[9 - 4*x^2]/3]","A",1
550,1,18,18,0.0025963,"\int \frac{1}{x^2 \sqrt{9-4 x^2}} \, dx","Integrate[1/(x^2*Sqrt[9 - 4*x^2]),x]","-\frac{\sqrt{9-4 x^2}}{9 x}","-\frac{\sqrt{9-4 x^2}}{9 x}",1,"-1/9*Sqrt[9 - 4*x^2]/x","A",1
551,1,37,39,0.0106584,"\int \frac{1}{x^3 \sqrt{9-4 x^2}} \, dx","Integrate[1/(x^3*Sqrt[9 - 4*x^2]),x]","\frac{1}{54} \left(-\frac{3 \sqrt{9-4 x^2}}{x^2}-4 \tanh ^{-1}\left(\sqrt{1-\frac{4 x^2}{9}}\right)\right)","-\frac{\sqrt{9-4 x^2}}{18 x^2}-\frac{2}{27} \tanh ^{-1}\left(\frac{1}{3} \sqrt{9-4 x^2}\right)",1,"((-3*Sqrt[9 - 4*x^2])/x^2 - 4*ArcTanh[Sqrt[1 - (4*x^2)/9]])/54","A",1
552,1,27,37,0.0039988,"\int \frac{1}{x^4 \sqrt{9-4 x^2}} \, dx","Integrate[1/(x^4*Sqrt[9 - 4*x^2]),x]","-\frac{\sqrt{1-\frac{4 x^2}{9}} \left(8 x^2+9\right)}{81 x^3}","-\frac{8 \sqrt{9-4 x^2}}{243 x}-\frac{\sqrt{9-4 x^2}}{27 x^3}",1,"-1/81*(Sqrt[1 - (4*x^2)/9]*(9 + 8*x^2))/x^3","A",1
553,1,32,57,0.0050994,"\int \frac{1}{x^5 \sqrt{9-4 x^2}} \, dx","Integrate[1/(x^5*Sqrt[9 - 4*x^2]),x]","-\frac{16}{729} \sqrt{9-4 x^2} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};1-\frac{4 x^2}{9}\right)","-\frac{\sqrt{9-4 x^2}}{54 x^2}-\frac{2}{81} \tanh ^{-1}\left(\frac{1}{3} \sqrt{9-4 x^2}\right)-\frac{\sqrt{9-4 x^2}}{36 x^4}",1,"(-16*Sqrt[9 - 4*x^2]*Hypergeometric2F1[1/2, 3, 3/2, 1 - (4*x^2)/9])/729","C",1
554,1,27,46,0.0085116,"\int \frac{x^5}{\sqrt{-9+4 x^2}} \, dx","Integrate[x^5/Sqrt[-9 + 4*x^2],x]","\frac{1}{40} \sqrt{4 x^2-9} \left(2 x^4+6 x^2+27\right)","\frac{1}{320} \left(4 x^2-9\right)^{5/2}+\frac{3}{32} \left(4 x^2-9\right)^{3/2}+\frac{81}{64} \sqrt{4 x^2-9}",1,"(Sqrt[-9 + 4*x^2]*(27 + 6*x^2 + 2*x^4))/40","A",1
555,1,43,54,0.0131755,"\int \frac{x^4}{\sqrt{-9+4 x^2}} \, dx","Integrate[x^4/Sqrt[-9 + 4*x^2],x]","\frac{1}{256} \left(2 x \sqrt{4 x^2-9} \left(8 x^2+27\right)+243 \tanh ^{-1}\left(\frac{2 x}{\sqrt{4 x^2-9}}\right)\right)","\frac{27}{128} \sqrt{4 x^2-9} x+\frac{243}{256} \tanh ^{-1}\left(\frac{2 x}{\sqrt{4 x^2-9}}\right)+\frac{1}{16} \sqrt{4 x^2-9} x^3",1,"(2*x*Sqrt[-9 + 4*x^2]*(27 + 8*x^2) + 243*ArcTanh[(2*x)/Sqrt[-9 + 4*x^2]])/256","A",1
556,1,22,31,0.0062867,"\int \frac{x^3}{\sqrt{-9+4 x^2}} \, dx","Integrate[x^3/Sqrt[-9 + 4*x^2],x]","\frac{1}{24} \left(2 x^2+9\right) \sqrt{4 x^2-9}","\frac{1}{48} \left(4 x^2-9\right)^{3/2}+\frac{9}{16} \sqrt{4 x^2-9}",1,"((9 + 2*x^2)*Sqrt[-9 + 4*x^2])/24","A",1
557,1,36,36,0.0060333,"\int \frac{x^2}{\sqrt{-9+4 x^2}} \, dx","Integrate[x^2/Sqrt[-9 + 4*x^2],x]","\frac{1}{8} \sqrt{4 x^2-9} x+\frac{9}{16} \tanh ^{-1}\left(\frac{2 x}{\sqrt{4 x^2-9}}\right)","\frac{1}{8} \sqrt{4 x^2-9} x+\frac{9}{16} \tanh ^{-1}\left(\frac{2 x}{\sqrt{4 x^2-9}}\right)",1,"(x*Sqrt[-9 + 4*x^2])/8 + (9*ArcTanh[(2*x)/Sqrt[-9 + 4*x^2]])/16","A",1
558,1,15,15,0.001593,"\int \frac{x}{\sqrt{-9+4 x^2}} \, dx","Integrate[x/Sqrt[-9 + 4*x^2],x]","\frac{1}{4} \sqrt{4 x^2-9}","\frac{1}{4} \sqrt{4 x^2-9}",1,"Sqrt[-9 + 4*x^2]/4","A",1
559,1,43,19,0.0033944,"\int \frac{1}{\sqrt{-9+4 x^2}} \, dx","Integrate[1/Sqrt[-9 + 4*x^2],x]","\frac{1}{4} \log \left(\frac{2 x}{\sqrt{4 x^2-9}}+1\right)-\frac{1}{4} \log \left(1-\frac{2 x}{\sqrt{4 x^2-9}}\right)","\frac{1}{2} \tanh ^{-1}\left(\frac{2 x}{\sqrt{4 x^2-9}}\right)",1,"-1/4*Log[1 - (2*x)/Sqrt[-9 + 4*x^2]] + Log[1 + (2*x)/Sqrt[-9 + 4*x^2]]/4","B",1
560,1,20,20,0.002696,"\int \frac{1}{x \sqrt{-9+4 x^2}} \, dx","Integrate[1/(x*Sqrt[-9 + 4*x^2]),x]","\frac{1}{3} \tan ^{-1}\left(\frac{1}{3} \sqrt{4 x^2-9}\right)","\frac{1}{3} \tan ^{-1}\left(\frac{1}{3} \sqrt{4 x^2-9}\right)",1,"ArcTan[Sqrt[-9 + 4*x^2]/3]/3","A",1
561,1,18,18,0.0022985,"\int \frac{1}{x^2 \sqrt{-9+4 x^2}} \, dx","Integrate[1/(x^2*Sqrt[-9 + 4*x^2]),x]","\frac{\sqrt{4 x^2-9}}{9 x}","\frac{\sqrt{4 x^2-9}}{9 x}",1,"Sqrt[-9 + 4*x^2]/(9*x)","A",1
562,1,54,39,0.0214881,"\int \frac{1}{x^3 \sqrt{-9+4 x^2}} \, dx","Integrate[1/(x^3*Sqrt[-9 + 4*x^2]),x]","\frac{4}{81} \sqrt{4 x^2-9} \left(\frac{9}{8 x^2}+\frac{\tanh ^{-1}\left(\sqrt{1-\frac{4 x^2}{9}}\right)}{2 \sqrt{1-\frac{4 x^2}{9}}}\right)","\frac{\sqrt{4 x^2-9}}{18 x^2}+\frac{2}{27} \tan ^{-1}\left(\frac{1}{3} \sqrt{4 x^2-9}\right)",1,"(4*Sqrt[-9 + 4*x^2]*(9/(8*x^2) + ArcTanh[Sqrt[1 - (4*x^2)/9]]/(2*Sqrt[1 - (4*x^2)/9])))/81","A",1
563,1,25,37,0.0043459,"\int \frac{1}{x^4 \sqrt{-9+4 x^2}} \, dx","Integrate[1/(x^4*Sqrt[-9 + 4*x^2]),x]","\frac{\sqrt{4 x^2-9} \left(8 x^2+9\right)}{243 x^3}","\frac{8 \sqrt{4 x^2-9}}{243 x}+\frac{\sqrt{4 x^2-9}}{27 x^3}",1,"(Sqrt[-9 + 4*x^2]*(9 + 8*x^2))/(243*x^3)","A",1
564,1,32,57,0.0053749,"\int \frac{1}{x^5 \sqrt{-9+4 x^2}} \, dx","Integrate[1/(x^5*Sqrt[-9 + 4*x^2]),x]","\frac{16}{729} \sqrt{4 x^2-9} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};1-\frac{4 x^2}{9}\right)","\frac{\sqrt{4 x^2-9}}{54 x^2}+\frac{2}{81} \tan ^{-1}\left(\frac{1}{3} \sqrt{4 x^2-9}\right)+\frac{\sqrt{4 x^2-9}}{36 x^4}",1,"(16*Sqrt[-9 + 4*x^2]*Hypergeometric2F1[1/2, 3, 3/2, 1 - (4*x^2)/9])/729","C",1
565,1,27,46,0.0089586,"\int \frac{x^5}{\sqrt{-9-4 x^2}} \, dx","Integrate[x^5/Sqrt[-9 - 4*x^2],x]","-\frac{1}{40} \sqrt{-4 x^2-9} \left(2 x^4-6 x^2+27\right)","-\frac{1}{320} \left(-4 x^2-9\right)^{5/2}-\frac{3}{32} \left(-4 x^2-9\right)^{3/2}-\frac{81}{64} \sqrt{-4 x^2-9}",1,"-1/40*(Sqrt[-9 - 4*x^2]*(27 - 6*x^2 + 2*x^4))","A",1
566,1,43,54,0.0138114,"\int \frac{x^4}{\sqrt{-9-4 x^2}} \, dx","Integrate[x^4/Sqrt[-9 - 4*x^2],x]","\frac{1}{256} \left(2 x \sqrt{-4 x^2-9} \left(27-8 x^2\right)+243 \tan ^{-1}\left(\frac{2 x}{\sqrt{-4 x^2-9}}\right)\right)","\frac{27}{128} \sqrt{-4 x^2-9} x+\frac{243}{256} \tan ^{-1}\left(\frac{2 x}{\sqrt{-4 x^2-9}}\right)-\frac{1}{16} \sqrt{-4 x^2-9} x^3",1,"(2*x*(27 - 8*x^2)*Sqrt[-9 - 4*x^2] + 243*ArcTan[(2*x)/Sqrt[-9 - 4*x^2]])/256","A",1
567,1,22,31,0.0082578,"\int \frac{x^3}{\sqrt{-9-4 x^2}} \, dx","Integrate[x^3/Sqrt[-9 - 4*x^2],x]","\frac{1}{24} \sqrt{-4 x^2-9} \left(9-2 x^2\right)","\frac{1}{48} \left(-4 x^2-9\right)^{3/2}+\frac{9}{16} \sqrt{-4 x^2-9}",1,"(Sqrt[-9 - 4*x^2]*(9 - 2*x^2))/24","A",1
568,1,36,36,0.006429,"\int \frac{x^2}{\sqrt{-9-4 x^2}} \, dx","Integrate[x^2/Sqrt[-9 - 4*x^2],x]","-\frac{1}{8} \sqrt{-4 x^2-9} x-\frac{9}{16} \tan ^{-1}\left(\frac{2 x}{\sqrt{-4 x^2-9}}\right)","-\frac{1}{8} \sqrt{-4 x^2-9} x-\frac{9}{16} \tan ^{-1}\left(\frac{2 x}{\sqrt{-4 x^2-9}}\right)",1,"-1/8*(x*Sqrt[-9 - 4*x^2]) - (9*ArcTan[(2*x)/Sqrt[-9 - 4*x^2]])/16","A",1
569,1,15,15,0.0015697,"\int \frac{x}{\sqrt{-9-4 x^2}} \, dx","Integrate[x/Sqrt[-9 - 4*x^2],x]","-\frac{1}{4} \sqrt{-4 x^2-9}","-\frac{1}{4} \sqrt{-4 x^2-9}",1,"-1/4*Sqrt[-9 - 4*x^2]","A",1
570,1,19,19,0.0034134,"\int \frac{1}{\sqrt{-9-4 x^2}} \, dx","Integrate[1/Sqrt[-9 - 4*x^2],x]","\frac{1}{2} \tan ^{-1}\left(\frac{2 x}{\sqrt{-4 x^2-9}}\right)","\frac{1}{2} \tan ^{-1}\left(\frac{2 x}{\sqrt{-4 x^2-9}}\right)",1,"ArcTan[(2*x)/Sqrt[-9 - 4*x^2]]/2","A",1
571,1,20,20,0.0025483,"\int \frac{1}{x \sqrt{-9-4 x^2}} \, dx","Integrate[1/(x*Sqrt[-9 - 4*x^2]),x]","\frac{1}{3} \tan ^{-1}\left(\frac{1}{3} \sqrt{-4 x^2-9}\right)","\frac{1}{3} \tan ^{-1}\left(\frac{1}{3} \sqrt{-4 x^2-9}\right)",1,"ArcTan[Sqrt[-9 - 4*x^2]/3]/3","A",1
572,1,18,18,0.0024721,"\int \frac{1}{x^2 \sqrt{-9-4 x^2}} \, dx","Integrate[1/(x^2*Sqrt[-9 - 4*x^2]),x]","\frac{\sqrt{-4 x^2-9}}{9 x}","\frac{\sqrt{-4 x^2-9}}{9 x}",1,"Sqrt[-9 - 4*x^2]/(9*x)","A",1
573,1,54,39,0.0223981,"\int \frac{1}{x^3 \sqrt{-9-4 x^2}} \, dx","Integrate[1/(x^3*Sqrt[-9 - 4*x^2]),x]","-\frac{4}{81} \sqrt{-4 x^2-9} \left(\frac{\tanh ^{-1}\left(\sqrt{\frac{4 x^2}{9}+1}\right)}{2 \sqrt{\frac{4 x^2}{9}+1}}-\frac{9}{8 x^2}\right)","\frac{\sqrt{-4 x^2-9}}{18 x^2}-\frac{2}{27} \tan ^{-1}\left(\frac{1}{3} \sqrt{-4 x^2-9}\right)",1,"(-4*Sqrt[-9 - 4*x^2]*(-9/(8*x^2) + ArcTanh[Sqrt[1 + (4*x^2)/9]]/(2*Sqrt[1 + (4*x^2)/9])))/81","A",1
574,1,25,37,0.0059829,"\int \frac{1}{x^4 \sqrt{-9-4 x^2}} \, dx","Integrate[1/(x^4*Sqrt[-9 - 4*x^2]),x]","\frac{\left(9-8 x^2\right) \sqrt{-4 x^2-9}}{243 x^3}","\frac{\sqrt{-4 x^2-9}}{27 x^3}-\frac{8 \sqrt{-4 x^2-9}}{243 x}",1,"((9 - 8*x^2)*Sqrt[-9 - 4*x^2])/(243*x^3)","A",1
575,1,32,57,0.0054276,"\int \frac{1}{x^5 \sqrt{-9-4 x^2}} \, dx","Integrate[1/(x^5*Sqrt[-9 - 4*x^2]),x]","\frac{16}{729} \sqrt{-4 x^2-9} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};\frac{4 x^2}{9}+1\right)","-\frac{\sqrt{-4 x^2-9}}{54 x^2}+\frac{2}{81} \tan ^{-1}\left(\frac{1}{3} \sqrt{-4 x^2-9}\right)+\frac{\sqrt{-4 x^2-9}}{36 x^4}",1,"(16*Sqrt[-9 - 4*x^2]*Hypergeometric2F1[1/2, 3, 3/2, 1 + (4*x^2)/9])/729","C",1
576,1,17,17,0.0073899,"\int \frac{1}{\sqrt{9+b x^2}} \, dx","Integrate[1/Sqrt[9 + b*x^2],x]","\frac{\sinh ^{-1}\left(\frac{\sqrt{b} x}{3}\right)}{\sqrt{b}}","\frac{\sinh ^{-1}\left(\frac{\sqrt{b} x}{3}\right)}{\sqrt{b}}",1,"ArcSinh[(Sqrt[b]*x)/3]/Sqrt[b]","A",1
577,1,17,17,0.0078964,"\int \frac{1}{\sqrt{9-b x^2}} \, dx","Integrate[1/Sqrt[9 - b*x^2],x]","\frac{\sin ^{-1}\left(\frac{\sqrt{b} x}{3}\right)}{\sqrt{b}}","\frac{\sin ^{-1}\left(\frac{\sqrt{b} x}{3}\right)}{\sqrt{b}}",1,"ArcSin[(Sqrt[b]*x)/3]/Sqrt[b]","A",1
578,1,25,25,0.006729,"\int \frac{1}{\sqrt{-9+b x^2}} \, dx","Integrate[1/Sqrt[-9 + b*x^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2-9}}\right)}{\sqrt{b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2-9}}\right)}{\sqrt{b}}",1,"ArcTanh[(Sqrt[b]*x)/Sqrt[-9 + b*x^2]]/Sqrt[b]","A",1
579,1,26,26,0.0067688,"\int \frac{1}{\sqrt{-9-b x^2}} \, dx","Integrate[1/Sqrt[-9 - b*x^2],x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{-b x^2-9}}\right)}{\sqrt{b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{-b x^2-9}}\right)}{\sqrt{b}}",1,"ArcTan[(Sqrt[b]*x)/Sqrt[-9 - b*x^2]]/Sqrt[b]","A",1
580,1,19,19,0.0083009,"\int \frac{1}{\sqrt{\pi +b x^2}} \, dx","Integrate[1/Sqrt[Pi + b*x^2],x]","\frac{\sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{\pi }}\right)}{\sqrt{b}}","\frac{\sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{\pi }}\right)}{\sqrt{b}}",1,"ArcSinh[(Sqrt[b]*x)/Sqrt[Pi]]/Sqrt[b]","A",1
581,1,19,19,0.0078003,"\int \frac{1}{\sqrt{\pi -b x^2}} \, dx","Integrate[1/Sqrt[Pi - b*x^2],x]","\frac{\sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{\pi }}\right)}{\sqrt{b}}","\frac{\sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{\pi }}\right)}{\sqrt{b}}",1,"ArcSin[(Sqrt[b]*x)/Sqrt[Pi]]/Sqrt[b]","A",1
582,1,27,27,0.0067432,"\int \frac{1}{\sqrt{-\pi +b x^2}} \, dx","Integrate[1/Sqrt[-Pi + b*x^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2-\pi }}\right)}{\sqrt{b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2-\pi }}\right)}{\sqrt{b}}",1,"ArcTanh[(Sqrt[b]*x)/Sqrt[-Pi + b*x^2]]/Sqrt[b]","A",1
583,1,28,28,0.0060248,"\int \frac{1}{\sqrt{-\pi -b x^2}} \, dx","Integrate[1/Sqrt[-Pi - b*x^2],x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{-b x^2-\pi }}\right)}{\sqrt{b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{-b x^2-\pi }}\right)}{\sqrt{b}}",1,"ArcTan[(Sqrt[b]*x)/Sqrt[-Pi - b*x^2]]/Sqrt[b]","A",1
584,1,25,25,0.0057112,"\int \frac{1}{\sqrt{a+b x^2}} \, dx","Integrate[1/Sqrt[a + b*x^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{\sqrt{b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{\sqrt{b}}",1,"ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]]/Sqrt[b]","A",1
585,1,26,26,0.005986,"\int \frac{1}{\sqrt{a-b x^2}} \, dx","Integrate[1/Sqrt[a - b*x^2],x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a-b x^2}}\right)}{\sqrt{b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a-b x^2}}\right)}{\sqrt{b}}",1,"ArcTan[(Sqrt[b]*x)/Sqrt[a - b*x^2]]/Sqrt[b]","A",1
586,1,27,27,0.0064165,"\int \frac{1}{\sqrt{-a+b x^2}} \, dx","Integrate[1/Sqrt[-a + b*x^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2-a}}\right)}{\sqrt{b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{b x^2-a}}\right)}{\sqrt{b}}",1,"ArcTanh[(Sqrt[b]*x)/Sqrt[-a + b*x^2]]/Sqrt[b]","A",1
587,1,28,28,0.0065307,"\int \frac{1}{\sqrt{-a-b x^2}} \, dx","Integrate[1/Sqrt[-a - b*x^2],x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{-a-b x^2}}\right)}{\sqrt{b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{-a-b x^2}}\right)}{\sqrt{b}}",1,"ArcTan[(Sqrt[b]*x)/Sqrt[-a - b*x^2]]/Sqrt[b]","A",1
588,1,16,16,0.0035557,"\int \frac{1}{\sqrt{a^2-x^2}} \, dx","Integrate[1/Sqrt[a^2 - x^2],x]","\tan ^{-1}\left(\frac{x}{\sqrt{a^2-x^2}}\right)","\tan ^{-1}\left(\frac{x}{\sqrt{a^2-x^2}}\right)",1,"ArcTan[x/Sqrt[a^2 - x^2]]","A",1
589,1,103,184,0.0646714,"\int (c x)^{7/2} \sqrt{a+b x^2} \, dx","Integrate[(c*x)^(7/2)*Sqrt[a + b*x^2],x]","\frac{2 c^3 \sqrt{c x} \sqrt{a+b x^2} \left(\sqrt{\frac{b x^2}{a}+1} \left(-5 a^2+2 a b x^2+7 b^2 x^4\right)+5 a^2 \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^2}{a}\right)\right)}{77 b^2 \sqrt{\frac{b x^2}{a}+1}}","\frac{10 a^{11/4} c^{7/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{231 b^{9/4} \sqrt{a+b x^2}}-\frac{20 a^2 c^3 \sqrt{c x} \sqrt{a+b x^2}}{231 b^2}+\frac{2 (c x)^{9/2} \sqrt{a+b x^2}}{11 c}+\frac{4 a c (c x)^{5/2} \sqrt{a+b x^2}}{77 b}",1,"(2*c^3*Sqrt[c*x]*Sqrt[a + b*x^2]*(Sqrt[1 + (b*x^2)/a]*(-5*a^2 + 2*a*b*x^2 + 7*b^2*x^4) + 5*a^2*Hypergeometric2F1[-1/2, 1/4, 5/4, -((b*x^2)/a)]))/(77*b^2*Sqrt[1 + (b*x^2)/a])","C",1
590,1,85,301,0.048328,"\int (c x)^{5/2} \sqrt{a+b x^2} \, dx","Integrate[(c*x)^(5/2)*Sqrt[a + b*x^2],x]","\frac{2 c (c x)^{3/2} \sqrt{a+b x^2} \left(\left(a+b x^2\right) \sqrt{\frac{b x^2}{a}+1}-a \, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right)\right)}{9 b \sqrt{\frac{b x^2}{a}+1}}","-\frac{2 a^{9/4} c^{5/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{15 b^{7/4} \sqrt{a+b x^2}}+\frac{4 a^{9/4} c^{5/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right)|\frac{1}{2}\right)}{15 b^{7/4} \sqrt{a+b x^2}}-\frac{4 a^2 c^2 \sqrt{c x} \sqrt{a+b x^2}}{15 b^{3/2} \left(\sqrt{a}+\sqrt{b} x\right)}+\frac{2 (c x)^{7/2} \sqrt{a+b x^2}}{9 c}+\frac{4 a c (c x)^{3/2} \sqrt{a+b x^2}}{45 b}",1,"(2*c*(c*x)^(3/2)*Sqrt[a + b*x^2]*((a + b*x^2)*Sqrt[1 + (b*x^2)/a] - a*Hypergeometric2F1[-1/2, 3/4, 7/4, -((b*x^2)/a)]))/(9*b*Sqrt[1 + (b*x^2)/a])","C",1
591,1,85,153,0.0438456,"\int (c x)^{3/2} \sqrt{a+b x^2} \, dx","Integrate[(c*x)^(3/2)*Sqrt[a + b*x^2],x]","\frac{2 c \sqrt{c x} \sqrt{a+b x^2} \left(\left(a+b x^2\right) \sqrt{\frac{b x^2}{a}+1}-a \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^2}{a}\right)\right)}{7 b \sqrt{\frac{b x^2}{a}+1}}","-\frac{2 a^{7/4} c^{3/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{21 b^{5/4} \sqrt{a+b x^2}}+\frac{2 (c x)^{5/2} \sqrt{a+b x^2}}{7 c}+\frac{4 a c \sqrt{c x} \sqrt{a+b x^2}}{21 b}",1,"(2*c*Sqrt[c*x]*Sqrt[a + b*x^2]*((a + b*x^2)*Sqrt[1 + (b*x^2)/a] - a*Hypergeometric2F1[-1/2, 1/4, 5/4, -((b*x^2)/a)]))/(7*b*Sqrt[1 + (b*x^2)/a])","C",1
592,1,56,269,0.0137,"\int \sqrt{c x} \sqrt{a+b x^2} \, dx","Integrate[Sqrt[c*x]*Sqrt[a + b*x^2],x]","\frac{2 x \sqrt{c x} \sqrt{a+b x^2} \, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right)}{3 \sqrt{\frac{b x^2}{a}+1}}","\frac{2 a^{5/4} \sqrt{c} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{5 b^{3/4} \sqrt{a+b x^2}}-\frac{4 a^{5/4} \sqrt{c} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right)|\frac{1}{2}\right)}{5 b^{3/4} \sqrt{a+b x^2}}+\frac{2 (c x)^{3/2} \sqrt{a+b x^2}}{5 c}+\frac{4 a \sqrt{c x} \sqrt{a+b x^2}}{5 \sqrt{b} \left(\sqrt{a}+\sqrt{b} x\right)}",1,"(2*x*Sqrt[c*x]*Sqrt[a + b*x^2]*Hypergeometric2F1[-1/2, 3/4, 7/4, -((b*x^2)/a)])/(3*Sqrt[1 + (b*x^2)/a])","C",1
593,1,54,126,0.0134535,"\int \frac{\sqrt{a+b x^2}}{\sqrt{c x}} \, dx","Integrate[Sqrt[a + b*x^2]/Sqrt[c*x],x]","\frac{2 x \sqrt{a+b x^2} \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^2}{a}\right)}{\sqrt{c x} \sqrt{\frac{b x^2}{a}+1}}","\frac{2 a^{3/4} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{3 \sqrt[4]{b} \sqrt{c} \sqrt{a+b x^2}}+\frac{2 \sqrt{c x} \sqrt{a+b x^2}}{3 c}",1,"(2*x*Sqrt[a + b*x^2]*Hypergeometric2F1[-1/2, 1/4, 5/4, -((b*x^2)/a)])/(Sqrt[c*x]*Sqrt[1 + (b*x^2)/a])","C",1
594,1,54,263,0.0132342,"\int \frac{\sqrt{a+b x^2}}{(c x)^{3/2}} \, dx","Integrate[Sqrt[a + b*x^2]/(c*x)^(3/2),x]","-\frac{2 x \sqrt{a+b x^2} \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};-\frac{b x^2}{a}\right)}{(c x)^{3/2} \sqrt{\frac{b x^2}{a}+1}}","\frac{2 \sqrt[4]{a} \sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{c^{3/2} \sqrt{a+b x^2}}-\frac{4 \sqrt[4]{a} \sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right)|\frac{1}{2}\right)}{c^{3/2} \sqrt{a+b x^2}}+\frac{4 \sqrt{b} \sqrt{c x} \sqrt{a+b x^2}}{c^2 \left(\sqrt{a}+\sqrt{b} x\right)}-\frac{2 \sqrt{a+b x^2}}{c \sqrt{c x}}",1,"(-2*x*Sqrt[a + b*x^2]*Hypergeometric2F1[-1/2, -1/4, 3/4, -((b*x^2)/a)])/((c*x)^(3/2)*Sqrt[1 + (b*x^2)/a])","C",1
595,1,56,126,0.0135816,"\int \frac{\sqrt{a+b x^2}}{(c x)^{5/2}} \, dx","Integrate[Sqrt[a + b*x^2]/(c*x)^(5/2),x]","-\frac{2 x \sqrt{a+b x^2} \, _2F_1\left(-\frac{3}{4},-\frac{1}{2};\frac{1}{4};-\frac{b x^2}{a}\right)}{3 (c x)^{5/2} \sqrt{\frac{b x^2}{a}+1}}","\frac{2 b^{3/4} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{3 \sqrt[4]{a} c^{5/2} \sqrt{a+b x^2}}-\frac{2 \sqrt{a+b x^2}}{3 c (c x)^{3/2}}",1,"(-2*x*Sqrt[a + b*x^2]*Hypergeometric2F1[-3/4, -1/2, 1/4, -((b*x^2)/a)])/(3*(c*x)^(5/2)*Sqrt[1 + (b*x^2)/a])","C",1
596,1,56,303,0.0147191,"\int \frac{\sqrt{a+b x^2}}{(c x)^{7/2}} \, dx","Integrate[Sqrt[a + b*x^2]/(c*x)^(7/2),x]","-\frac{2 x \sqrt{a+b x^2} \, _2F_1\left(-\frac{5}{4},-\frac{1}{2};-\frac{1}{4};-\frac{b x^2}{a}\right)}{5 (c x)^{7/2} \sqrt{\frac{b x^2}{a}+1}}","\frac{2 b^{5/4} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{5 a^{3/4} c^{7/2} \sqrt{a+b x^2}}-\frac{4 b^{5/4} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right)|\frac{1}{2}\right)}{5 a^{3/4} c^{7/2} \sqrt{a+b x^2}}+\frac{4 b^{3/2} \sqrt{c x} \sqrt{a+b x^2}}{5 a c^4 \left(\sqrt{a}+\sqrt{b} x\right)}-\frac{4 b \sqrt{a+b x^2}}{5 a c^3 \sqrt{c x}}-\frac{2 \sqrt{a+b x^2}}{5 c (c x)^{5/2}}",1,"(-2*x*Sqrt[a + b*x^2]*Hypergeometric2F1[-5/4, -1/2, -1/4, -((b*x^2)/a)])/(5*(c*x)^(7/2)*Sqrt[1 + (b*x^2)/a])","C",1
597,1,102,212,0.0777655,"\int (c x)^{7/2} \left(a+b x^2\right)^{3/2} \, dx","Integrate[(c*x)^(7/2)*(a + b*x^2)^(3/2),x]","\frac{2 c^3 \sqrt{c x} \sqrt{a+b x^2} \left(5 a^3 \, _2F_1\left(-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^2}{a}\right)-\left(5 a-11 b x^2\right) \left(a+b x^2\right)^2 \sqrt{\frac{b x^2}{a}+1}\right)}{165 b^2 \sqrt{\frac{b x^2}{a}+1}}","\frac{4 a^{15/4} c^{7/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{231 b^{9/4} \sqrt{a+b x^2}}-\frac{8 a^3 c^3 \sqrt{c x} \sqrt{a+b x^2}}{231 b^2}+\frac{8 a^2 c (c x)^{5/2} \sqrt{a+b x^2}}{385 b}+\frac{2 (c x)^{9/2} \left(a+b x^2\right)^{3/2}}{15 c}+\frac{4 a (c x)^{9/2} \sqrt{a+b x^2}}{55 c}",1,"(2*c^3*Sqrt[c*x]*Sqrt[a + b*x^2]*(-((5*a - 11*b*x^2)*(a + b*x^2)^2*Sqrt[1 + (b*x^2)/a]) + 5*a^3*Hypergeometric2F1[-3/2, 1/4, 5/4, -((b*x^2)/a)]))/(165*b^2*Sqrt[1 + (b*x^2)/a])","C",1
598,1,89,329,0.0631928,"\int (c x)^{5/2} \left(a+b x^2\right)^{3/2} \, dx","Integrate[(c*x)^(5/2)*(a + b*x^2)^(3/2),x]","\frac{2 c (c x)^{3/2} \sqrt{a+b x^2} \left(\left(a+b x^2\right)^2 \sqrt{\frac{b x^2}{a}+1}-a^2 \, _2F_1\left(-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right)\right)}{13 b \sqrt{\frac{b x^2}{a}+1}}","-\frac{4 a^{13/4} c^{5/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{65 b^{7/4} \sqrt{a+b x^2}}+\frac{8 a^{13/4} c^{5/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right)|\frac{1}{2}\right)}{65 b^{7/4} \sqrt{a+b x^2}}-\frac{8 a^3 c^2 \sqrt{c x} \sqrt{a+b x^2}}{65 b^{3/2} \left(\sqrt{a}+\sqrt{b} x\right)}+\frac{8 a^2 c (c x)^{3/2} \sqrt{a+b x^2}}{195 b}+\frac{2 (c x)^{7/2} \left(a+b x^2\right)^{3/2}}{13 c}+\frac{4 a (c x)^{7/2} \sqrt{a+b x^2}}{39 c}",1,"(2*c*(c*x)^(3/2)*Sqrt[a + b*x^2]*((a + b*x^2)^2*Sqrt[1 + (b*x^2)/a] - a^2*Hypergeometric2F1[-3/2, 3/4, 7/4, -((b*x^2)/a)]))/(13*b*Sqrt[1 + (b*x^2)/a])","C",1
599,1,89,181,0.0545924,"\int (c x)^{3/2} \left(a+b x^2\right)^{3/2} \, dx","Integrate[(c*x)^(3/2)*(a + b*x^2)^(3/2),x]","\frac{2 c \sqrt{c x} \sqrt{a+b x^2} \left(\left(a+b x^2\right)^2 \sqrt{\frac{b x^2}{a}+1}-a^2 \, _2F_1\left(-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^2}{a}\right)\right)}{11 b \sqrt{\frac{b x^2}{a}+1}}","-\frac{4 a^{11/4} c^{3/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{77 b^{5/4} \sqrt{a+b x^2}}+\frac{8 a^2 c \sqrt{c x} \sqrt{a+b x^2}}{77 b}+\frac{12 a (c x)^{5/2} \sqrt{a+b x^2}}{77 c}+\frac{2 (c x)^{5/2} \left(a+b x^2\right)^{3/2}}{11 c}",1,"(2*c*Sqrt[c*x]*Sqrt[a + b*x^2]*((a + b*x^2)^2*Sqrt[1 + (b*x^2)/a] - a^2*Hypergeometric2F1[-3/2, 1/4, 5/4, -((b*x^2)/a)]))/(11*b*Sqrt[1 + (b*x^2)/a])","C",1
600,1,57,297,0.0132485,"\int \sqrt{c x} \left(a+b x^2\right)^{3/2} \, dx","Integrate[Sqrt[c*x]*(a + b*x^2)^(3/2),x]","\frac{2 a x \sqrt{c x} \sqrt{a+b x^2} \, _2F_1\left(-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right)}{3 \sqrt{\frac{b x^2}{a}+1}}","\frac{4 a^{9/4} \sqrt{c} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{15 b^{3/4} \sqrt{a+b x^2}}-\frac{8 a^{9/4} \sqrt{c} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right)|\frac{1}{2}\right)}{15 b^{3/4} \sqrt{a+b x^2}}+\frac{8 a^2 \sqrt{c x} \sqrt{a+b x^2}}{15 \sqrt{b} \left(\sqrt{a}+\sqrt{b} x\right)}+\frac{4 a (c x)^{3/2} \sqrt{a+b x^2}}{15 c}+\frac{2 (c x)^{3/2} \left(a+b x^2\right)^{3/2}}{9 c}",1,"(2*a*x*Sqrt[c*x]*Sqrt[a + b*x^2]*Hypergeometric2F1[-3/2, 3/4, 7/4, -((b*x^2)/a)])/(3*Sqrt[1 + (b*x^2)/a])","C",1
601,1,55,152,0.0118432,"\int \frac{\left(a+b x^2\right)^{3/2}}{\sqrt{c x}} \, dx","Integrate[(a + b*x^2)^(3/2)/Sqrt[c*x],x]","\frac{2 a x \sqrt{a+b x^2} \, _2F_1\left(-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^2}{a}\right)}{\sqrt{c x} \sqrt{\frac{b x^2}{a}+1}}","\frac{4 a^{7/4} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{7 \sqrt[4]{b} \sqrt{c} \sqrt{a+b x^2}}+\frac{4 a \sqrt{c x} \sqrt{a+b x^2}}{7 c}+\frac{2 \sqrt{c x} \left(a+b x^2\right)^{3/2}}{7 c}",1,"(2*a*x*Sqrt[a + b*x^2]*Hypergeometric2F1[-3/2, 1/4, 5/4, -((b*x^2)/a)])/(Sqrt[c*x]*Sqrt[1 + (b*x^2)/a])","C",1
602,1,55,296,0.0138636,"\int \frac{\left(a+b x^2\right)^{3/2}}{(c x)^{3/2}} \, dx","Integrate[(a + b*x^2)^(3/2)/(c*x)^(3/2),x]","-\frac{2 a x \sqrt{a+b x^2} \, _2F_1\left(-\frac{3}{2},-\frac{1}{4};\frac{3}{4};-\frac{b x^2}{a}\right)}{(c x)^{3/2} \sqrt{\frac{b x^2}{a}+1}}","\frac{12 a^{5/4} \sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{5 c^{3/2} \sqrt{a+b x^2}}-\frac{24 a^{5/4} \sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right)|\frac{1}{2}\right)}{5 c^{3/2} \sqrt{a+b x^2}}+\frac{12 b (c x)^{3/2} \sqrt{a+b x^2}}{5 c^3}+\frac{24 a \sqrt{b} \sqrt{c x} \sqrt{a+b x^2}}{5 c^2 \left(\sqrt{a}+\sqrt{b} x\right)}-\frac{2 \left(a+b x^2\right)^{3/2}}{c \sqrt{c x}}",1,"(-2*a*x*Sqrt[a + b*x^2]*Hypergeometric2F1[-3/2, -1/4, 3/4, -((b*x^2)/a)])/((c*x)^(3/2)*Sqrt[1 + (b*x^2)/a])","C",1
603,1,57,152,0.0135835,"\int \frac{\left(a+b x^2\right)^{3/2}}{(c x)^{5/2}} \, dx","Integrate[(a + b*x^2)^(3/2)/(c*x)^(5/2),x]","-\frac{2 a x \sqrt{a+b x^2} \, _2F_1\left(-\frac{3}{2},-\frac{3}{4};\frac{1}{4};-\frac{b x^2}{a}\right)}{3 (c x)^{5/2} \sqrt{\frac{b x^2}{a}+1}}","\frac{4 a^{3/4} b^{3/4} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{3 c^{5/2} \sqrt{a+b x^2}}+\frac{4 b \sqrt{c x} \sqrt{a+b x^2}}{3 c^3}-\frac{2 \left(a+b x^2\right)^{3/2}}{3 c (c x)^{3/2}}",1,"(-2*a*x*Sqrt[a + b*x^2]*Hypergeometric2F1[-3/2, -3/4, 1/4, -((b*x^2)/a)])/(3*(c*x)^(5/2)*Sqrt[1 + (b*x^2)/a])","C",1
604,1,57,297,0.013297,"\int \frac{\left(a+b x^2\right)^{3/2}}{(c x)^{7/2}} \, dx","Integrate[(a + b*x^2)^(3/2)/(c*x)^(7/2),x]","-\frac{2 a x \sqrt{a+b x^2} \, _2F_1\left(-\frac{3}{2},-\frac{5}{4};-\frac{1}{4};-\frac{b x^2}{a}\right)}{5 (c x)^{7/2} \sqrt{\frac{b x^2}{a}+1}}","\frac{12 \sqrt[4]{a} b^{5/4} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{5 c^{7/2} \sqrt{a+b x^2}}-\frac{24 \sqrt[4]{a} b^{5/4} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right)|\frac{1}{2}\right)}{5 c^{7/2} \sqrt{a+b x^2}}+\frac{24 b^{3/2} \sqrt{c x} \sqrt{a+b x^2}}{5 c^4 \left(\sqrt{a}+\sqrt{b} x\right)}-\frac{12 b \sqrt{a+b x^2}}{5 c^3 \sqrt{c x}}-\frac{2 \left(a+b x^2\right)^{3/2}}{5 c (c x)^{5/2}}",1,"(-2*a*x*Sqrt[a + b*x^2]*Hypergeometric2F1[-3/2, -5/4, -1/4, -((b*x^2)/a)])/(5*(c*x)^(7/2)*Sqrt[1 + (b*x^2)/a])","C",1
605,1,57,152,0.0146449,"\int \frac{\left(a+b x^2\right)^{3/2}}{(c x)^{9/2}} \, dx","Integrate[(a + b*x^2)^(3/2)/(c*x)^(9/2),x]","-\frac{2 a x \sqrt{a+b x^2} \, _2F_1\left(-\frac{7}{4},-\frac{3}{2};-\frac{3}{4};-\frac{b x^2}{a}\right)}{7 (c x)^{9/2} \sqrt{\frac{b x^2}{a}+1}}","\frac{4 b^{7/4} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{7 \sqrt[4]{a} c^{9/2} \sqrt{a+b x^2}}-\frac{4 b \sqrt{a+b x^2}}{7 c^3 (c x)^{3/2}}-\frac{2 \left(a+b x^2\right)^{3/2}}{7 c (c x)^{7/2}}",1,"(-2*a*x*Sqrt[a + b*x^2]*Hypergeometric2F1[-7/4, -3/2, -3/4, -((b*x^2)/a)])/(7*(c*x)^(9/2)*Sqrt[1 + (b*x^2)/a])","C",1
606,1,57,331,0.0140532,"\int \frac{\left(a+b x^2\right)^{3/2}}{(c x)^{11/2}} \, dx","Integrate[(a + b*x^2)^(3/2)/(c*x)^(11/2),x]","-\frac{2 a x \sqrt{a+b x^2} \, _2F_1\left(-\frac{9}{4},-\frac{3}{2};-\frac{5}{4};-\frac{b x^2}{a}\right)}{9 (c x)^{11/2} \sqrt{\frac{b x^2}{a}+1}}","\frac{4 b^{9/4} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{15 a^{3/4} c^{11/2} \sqrt{a+b x^2}}-\frac{8 b^{9/4} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right)|\frac{1}{2}\right)}{15 a^{3/4} c^{11/2} \sqrt{a+b x^2}}+\frac{8 b^{5/2} \sqrt{c x} \sqrt{a+b x^2}}{15 a c^6 \left(\sqrt{a}+\sqrt{b} x\right)}-\frac{8 b^2 \sqrt{a+b x^2}}{15 a c^5 \sqrt{c x}}-\frac{4 b \sqrt{a+b x^2}}{15 c^3 (c x)^{5/2}}-\frac{2 \left(a+b x^2\right)^{3/2}}{9 c (c x)^{9/2}}",1,"(-2*a*x*Sqrt[a + b*x^2]*Hypergeometric2F1[-9/4, -3/2, -5/4, -((b*x^2)/a)])/(9*(c*x)^(11/2)*Sqrt[1 + (b*x^2)/a])","C",1
607,1,74,128,0.0305751,"\int (c x)^{5/2} \sqrt{3 a-2 a x^2} \, dx","Integrate[(c*x)^(5/2)*Sqrt[3*a - 2*a*x^2],x]","\frac{c \sqrt{a \left(3-2 x^2\right)} (c x)^{3/2} \left(3 \sqrt{3} \, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{2 x^2}{3}\right)-\left(3-2 x^2\right)^{3/2}\right)}{9 \sqrt{3-2 x^2}}","-\frac{3 \sqrt[4]{6} a c^2 \sqrt{3-2 x^2} \sqrt{c x} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right)\right|2\right)}{5 \sqrt{x} \sqrt{3 a-2 a x^2}}+\frac{2 \sqrt{3 a-2 a x^2} (c x)^{7/2}}{9 c}-\frac{2}{15} c \sqrt{3 a-2 a x^2} (c x)^{3/2}",1,"(c*(c*x)^(3/2)*Sqrt[a*(3 - 2*x^2)]*(-(3 - 2*x^2)^(3/2) + 3*Sqrt[3]*Hypergeometric2F1[-1/2, 3/4, 7/4, (2*x^2)/3]))/(9*Sqrt[3 - 2*x^2])","C",1
608,1,74,117,0.023626,"\int (c x)^{3/2} \sqrt{3 a-2 a x^2} \, dx","Integrate[(c*x)^(3/2)*Sqrt[3*a - 2*a*x^2],x]","\frac{c \sqrt{a \left(3-2 x^2\right)} \sqrt{c x} \left(3 \sqrt{3} \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};\frac{2 x^2}{3}\right)-\left(3-2 x^2\right)^{3/2}\right)}{7 \sqrt{3-2 x^2}}","\frac{6^{3/4} a c^{3/2} \sqrt{3-2 x^2} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt[4]{\frac{2}{3}} \sqrt{c x}}{\sqrt{c}}\right),-1\right)}{7 \sqrt{a \left(3-2 x^2\right)}}+\frac{2 \sqrt{3 a-2 a x^2} (c x)^{5/2}}{7 c}-\frac{2}{7} c \sqrt{3 a-2 a x^2} \sqrt{c x}",1,"(c*Sqrt[c*x]*Sqrt[a*(3 - 2*x^2)]*(-(3 - 2*x^2)^(3/2) + 3*Sqrt[3]*Hypergeometric2F1[-1/2, 1/4, 5/4, (2*x^2)/3]))/(7*Sqrt[3 - 2*x^2])","C",1
609,1,51,99,0.0152491,"\int \sqrt{c x} \sqrt{3 a-2 a x^2} \, dx","Integrate[Sqrt[c*x]*Sqrt[3*a - 2*a*x^2],x]","\frac{2 x \sqrt{a \left(3-2 x^2\right)} \sqrt{c x} \, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{2 x^2}{3}\right)}{\sqrt{9-6 x^2}}","\frac{2 \sqrt{3 a-2 a x^2} (c x)^{3/2}}{5 c}-\frac{6 \sqrt[4]{6} a \sqrt{3-2 x^2} \sqrt{c x} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right)\right|2\right)}{5 \sqrt{x} \sqrt{3 a-2 a x^2}}",1,"(2*x*Sqrt[c*x]*Sqrt[a*(3 - 2*x^2)]*Hypergeometric2F1[-1/2, 3/4, 7/4, (2*x^2)/3])/Sqrt[9 - 6*x^2]","C",1
610,1,51,94,0.0133238,"\int \frac{\sqrt{3 a-2 a x^2}}{\sqrt{c x}} \, dx","Integrate[Sqrt[3*a - 2*a*x^2]/Sqrt[c*x],x]","\frac{2 x \sqrt{a \left(9-6 x^2\right)} \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};\frac{2 x^2}{3}\right)}{\sqrt{3-2 x^2} \sqrt{c x}}","\frac{2\ 2^{3/4} a \sqrt{3-2 x^2} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt[4]{\frac{2}{3}} \sqrt{c x}}{\sqrt{c}}\right),-1\right)}{\sqrt[4]{3} \sqrt{c} \sqrt{a \left(3-2 x^2\right)}}+\frac{2 \sqrt{3 a-2 a x^2} \sqrt{c x}}{3 c}",1,"(2*x*Sqrt[a*(9 - 6*x^2)]*Hypergeometric2F1[-1/2, 1/4, 5/4, (2*x^2)/3])/(Sqrt[c*x]*Sqrt[3 - 2*x^2])","C",1
611,1,51,98,0.0131903,"\int \frac{\sqrt{3 a-2 a x^2}}{(c x)^{3/2}} \, dx","Integrate[Sqrt[3*a - 2*a*x^2]/(c*x)^(3/2),x]","-\frac{2 x \sqrt{a \left(9-6 x^2\right)} \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\frac{2 x^2}{3}\right)}{\sqrt{3-2 x^2} (c x)^{3/2}}","\frac{4 \sqrt[4]{6} a \sqrt{3-2 x^2} \sqrt{c x} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right)\right|2\right)}{c^2 \sqrt{x} \sqrt{3 a-2 a x^2}}-\frac{2 \sqrt{3 a-2 a x^2}}{c \sqrt{c x}}",1,"(-2*x*Sqrt[a*(9 - 6*x^2)]*Hypergeometric2F1[-1/2, -1/4, 3/4, (2*x^2)/3])/((c*x)^(3/2)*Sqrt[3 - 2*x^2])","C",1
612,1,51,96,0.0132972,"\int \frac{\sqrt{3 a-2 a x^2}}{(c x)^{5/2}} \, dx","Integrate[Sqrt[3*a - 2*a*x^2]/(c*x)^(5/2),x]","-\frac{2 x \sqrt{a \left(3-2 x^2\right)} \, _2F_1\left(-\frac{3}{4},-\frac{1}{2};\frac{1}{4};\frac{2 x^2}{3}\right)}{\sqrt{9-6 x^2} (c x)^{5/2}}","-\frac{4\ 2^{3/4} a \sqrt{3-2 x^2} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt[4]{\frac{2}{3}} \sqrt{c x}}{\sqrt{c}}\right),-1\right)}{3 \sqrt[4]{3} c^{5/2} \sqrt{a \left(3-2 x^2\right)}}-\frac{2 \sqrt{3 a-2 a x^2}}{3 c (c x)^{3/2}}",1,"(-2*x*Sqrt[a*(3 - 2*x^2)]*Hypergeometric2F1[-3/4, -1/2, 1/4, (2*x^2)/3])/((c*x)^(5/2)*Sqrt[9 - 6*x^2])","C",1
613,1,87,156,0.0303055,"\int \frac{(c x)^{7/2}}{\sqrt{a+b x^2}} \, dx","Integrate[(c*x)^(7/2)/Sqrt[a + b*x^2],x]","\frac{2 c^3 \sqrt{c x} \left(5 a^2 \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^2}{a}\right)-5 a^2-2 a b x^2+3 b^2 x^4\right)}{21 b^2 \sqrt{a+b x^2}}","\frac{5 a^{7/4} c^{7/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{21 b^{9/4} \sqrt{a+b x^2}}-\frac{10 a c^3 \sqrt{c x} \sqrt{a+b x^2}}{21 b^2}+\frac{2 c (c x)^{5/2} \sqrt{a+b x^2}}{7 b}",1,"(2*c^3*Sqrt[c*x]*(-5*a^2 - 2*a*b*x^2 + 3*b^2*x^4 + 5*a^2*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^2)/a)]))/(21*b^2*Sqrt[a + b*x^2])","C",1
614,1,69,273,0.0259072,"\int \frac{(c x)^{5/2}}{\sqrt{a+b x^2}} \, dx","Integrate[(c*x)^(5/2)/Sqrt[a + b*x^2],x]","\frac{2 c (c x)^{3/2} \left(-a \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right)+a+b x^2\right)}{5 b \sqrt{a+b x^2}}","-\frac{3 a^{5/4} c^{5/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{5 b^{7/4} \sqrt{a+b x^2}}+\frac{6 a^{5/4} c^{5/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right)|\frac{1}{2}\right)}{5 b^{7/4} \sqrt{a+b x^2}}-\frac{6 a c^2 \sqrt{c x} \sqrt{a+b x^2}}{5 b^{3/2} \left(\sqrt{a}+\sqrt{b} x\right)}+\frac{2 c (c x)^{3/2} \sqrt{a+b x^2}}{5 b}",1,"(2*c*(c*x)^(3/2)*(a + b*x^2 - a*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((b*x^2)/a)]))/(5*b*Sqrt[a + b*x^2])","C",1
615,1,69,127,0.0235908,"\int \frac{(c x)^{3/2}}{\sqrt{a+b x^2}} \, dx","Integrate[(c*x)^(3/2)/Sqrt[a + b*x^2],x]","\frac{2 c \sqrt{c x} \left(-a \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^2}{a}\right)+a+b x^2\right)}{3 b \sqrt{a+b x^2}}","\frac{2 c \sqrt{c x} \sqrt{a+b x^2}}{3 b}-\frac{a^{3/4} c^{3/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{3 b^{5/4} \sqrt{a+b x^2}}",1,"(2*c*Sqrt[c*x]*(a + b*x^2 - a*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^2)/a)]))/(3*b*Sqrt[a + b*x^2])","C",1
616,1,56,236,0.0129737,"\int \frac{\sqrt{c x}}{\sqrt{a+b x^2}} \, dx","Integrate[Sqrt[c*x]/Sqrt[a + b*x^2],x]","\frac{2 x \sqrt{c x} \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right)}{3 \sqrt{a+b x^2}}","\frac{\sqrt[4]{a} \sqrt{c} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{b^{3/4} \sqrt{a+b x^2}}-\frac{2 \sqrt[4]{a} \sqrt{c} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right)|\frac{1}{2}\right)}{b^{3/4} \sqrt{a+b x^2}}+\frac{2 \sqrt{c x} \sqrt{a+b x^2}}{\sqrt{b} \left(\sqrt{a}+\sqrt{b} x\right)}",1,"(2*x*Sqrt[c*x]*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((b*x^2)/a)])/(3*Sqrt[a + b*x^2])","C",1
617,1,54,97,0.013956,"\int \frac{1}{\sqrt{c x} \sqrt{a+b x^2}} \, dx","Integrate[1/(Sqrt[c*x]*Sqrt[a + b*x^2]),x]","\frac{2 x \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^2}{a}\right)}{\sqrt{c x} \sqrt{a+b x^2}}","\frac{\left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{\sqrt[4]{a} \sqrt[4]{b} \sqrt{c} \sqrt{a+b x^2}}",1,"(2*x*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^2)/a)])/(Sqrt[c*x]*Sqrt[a + b*x^2])","C",1
618,1,54,268,0.0128309,"\int \frac{1}{(c x)^{3/2} \sqrt{a+b x^2}} \, dx","Integrate[1/((c*x)^(3/2)*Sqrt[a + b*x^2]),x]","-\frac{2 x \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{b x^2}{a}\right)}{(c x)^{3/2} \sqrt{a+b x^2}}","\frac{\sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{a^{3/4} c^{3/2} \sqrt{a+b x^2}}-\frac{2 \sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right)|\frac{1}{2}\right)}{a^{3/4} c^{3/2} \sqrt{a+b x^2}}+\frac{2 \sqrt{b} \sqrt{c x} \sqrt{a+b x^2}}{a c^2 \left(\sqrt{a}+\sqrt{b} x\right)}-\frac{2 \sqrt{a+b x^2}}{a c \sqrt{c x}}",1,"(-2*x*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[-1/4, 1/2, 3/4, -((b*x^2)/a)])/((c*x)^(3/2)*Sqrt[a + b*x^2])","C",1
619,1,56,129,0.0132968,"\int \frac{1}{(c x)^{5/2} \sqrt{a+b x^2}} \, dx","Integrate[1/((c*x)^(5/2)*Sqrt[a + b*x^2]),x]","-\frac{2 x \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{3}{4},\frac{1}{2};\frac{1}{4};-\frac{b x^2}{a}\right)}{3 (c x)^{5/2} \sqrt{a+b x^2}}","-\frac{b^{3/4} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{3 a^{5/4} c^{5/2} \sqrt{a+b x^2}}-\frac{2 \sqrt{a+b x^2}}{3 a c (c x)^{3/2}}",1,"(-2*x*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[-3/4, 1/2, 1/4, -((b*x^2)/a)])/(3*(c*x)^(5/2)*Sqrt[a + b*x^2])","C",1
620,1,56,306,0.0132062,"\int \frac{1}{(c x)^{7/2} \sqrt{a+b x^2}} \, dx","Integrate[1/((c*x)^(7/2)*Sqrt[a + b*x^2]),x]","-\frac{2 x \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{5}{4},\frac{1}{2};-\frac{1}{4};-\frac{b x^2}{a}\right)}{5 (c x)^{7/2} \sqrt{a+b x^2}}","-\frac{3 b^{5/4} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{5 a^{7/4} c^{7/2} \sqrt{a+b x^2}}+\frac{6 b^{5/4} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right)|\frac{1}{2}\right)}{5 a^{7/4} c^{7/2} \sqrt{a+b x^2}}-\frac{6 b^{3/2} \sqrt{c x} \sqrt{a+b x^2}}{5 a^2 c^4 \left(\sqrt{a}+\sqrt{b} x\right)}+\frac{6 b \sqrt{a+b x^2}}{5 a^2 c^3 \sqrt{c x}}-\frac{2 \sqrt{a+b x^2}}{5 a c (c x)^{5/2}}",1,"(-2*x*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[-5/4, 1/2, -1/4, -((b*x^2)/a)])/(5*(c*x)^(7/2)*Sqrt[a + b*x^2])","C",1
621,1,74,153,0.0312451,"\int \frac{(c x)^{7/2}}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[(c*x)^(7/2)/(a + b*x^2)^(3/2),x]","\frac{c^3 \sqrt{c x} \left(-5 a \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^2}{a}\right)+5 a+2 b x^2\right)}{3 b^2 \sqrt{a+b x^2}}","-\frac{5 a^{3/4} c^{7/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{6 b^{9/4} \sqrt{a+b x^2}}+\frac{5 c^3 \sqrt{c x} \sqrt{a+b x^2}}{3 b^2}-\frac{c (c x)^{5/2}}{b \sqrt{a+b x^2}}",1,"(c^3*Sqrt[c*x]*(5*a + 2*b*x^2 - 5*a*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^2)/a)]))/(3*b^2*Sqrt[a + b*x^2])","C",1
622,1,60,266,0.0277985,"\int \frac{(c x)^{5/2}}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[(c*x)^(5/2)/(a + b*x^2)^(3/2),x]","-\frac{2 c (c x)^{3/2} \left(\sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{b x^2}{a}\right)-1\right)}{b \sqrt{a+b x^2}}","\frac{3 \sqrt[4]{a} c^{5/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{2 b^{7/4} \sqrt{a+b x^2}}-\frac{3 \sqrt[4]{a} c^{5/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right)|\frac{1}{2}\right)}{b^{7/4} \sqrt{a+b x^2}}+\frac{3 c^2 \sqrt{c x} \sqrt{a+b x^2}}{b^{3/2} \left(\sqrt{a}+\sqrt{b} x\right)}-\frac{c (c x)^{3/2}}{b \sqrt{a+b x^2}}",1,"(-2*c*(c*x)^(3/2)*(-1 + Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[3/4, 3/2, 7/4, -((b*x^2)/a)]))/(b*Sqrt[a + b*x^2])","C",1
623,1,59,125,0.0251576,"\int \frac{(c x)^{3/2}}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[(c*x)^(3/2)/(a + b*x^2)^(3/2),x]","\frac{c \sqrt{c x} \left(\sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^2}{a}\right)-1\right)}{b \sqrt{a+b x^2}}","\frac{c^{3/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{2 \sqrt[4]{a} b^{5/4} \sqrt{a+b x^2}}-\frac{c \sqrt{c x}}{b \sqrt{a+b x^2}}",1,"(c*Sqrt[c*x]*(-1 + Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^2)/a)]))/(b*Sqrt[a + b*x^2])","C",1
624,1,59,266,0.0134377,"\int \frac{\sqrt{c x}}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[Sqrt[c*x]/(a + b*x^2)^(3/2),x]","\frac{2 x \sqrt{c x} \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{b x^2}{a}\right)}{3 a \sqrt{a+b x^2}}","-\frac{\sqrt{c} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{2 a^{3/4} b^{3/4} \sqrt{a+b x^2}}+\frac{\sqrt{c} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right)|\frac{1}{2}\right)}{a^{3/4} b^{3/4} \sqrt{a+b x^2}}+\frac{(c x)^{3/2}}{a c \sqrt{a+b x^2}}-\frac{\sqrt{c x} \sqrt{a+b x^2}}{a \sqrt{b} \left(\sqrt{a}+\sqrt{b} x\right)}",1,"(2*x*Sqrt[c*x]*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[3/4, 3/2, 7/4, -((b*x^2)/a)])/(3*a*Sqrt[a + b*x^2])","C",1
625,1,59,126,0.0211117,"\int \frac{1}{\sqrt{c x} \left(a+b x^2\right)^{3/2}} \, dx","Integrate[1/(Sqrt[c*x]*(a + b*x^2)^(3/2)),x]","\frac{x \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^2}{a}\right)+x}{a \sqrt{c x} \sqrt{a+b x^2}}","\frac{\left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{2 a^{5/4} \sqrt[4]{b} \sqrt{c} \sqrt{a+b x^2}}+\frac{\sqrt{c x}}{a c \sqrt{a+b x^2}}",1,"(x + x*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^2)/a)])/(a*Sqrt[c*x]*Sqrt[a + b*x^2])","C",1
626,1,57,296,0.0132083,"\int \frac{1}{(c x)^{3/2} \left(a+b x^2\right)^{3/2}} \, dx","Integrate[1/((c*x)^(3/2)*(a + b*x^2)^(3/2)),x]","-\frac{2 x \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{1}{4},\frac{3}{2};\frac{3}{4};-\frac{b x^2}{a}\right)}{a (c x)^{3/2} \sqrt{a+b x^2}}","\frac{3 \sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{2 a^{7/4} c^{3/2} \sqrt{a+b x^2}}-\frac{3 \sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right)|\frac{1}{2}\right)}{a^{7/4} c^{3/2} \sqrt{a+b x^2}}+\frac{3 \sqrt{b} \sqrt{c x} \sqrt{a+b x^2}}{a^2 c^2 \left(\sqrt{a}+\sqrt{b} x\right)}-\frac{3 \sqrt{a+b x^2}}{a^2 c \sqrt{c x}}+\frac{1}{a c \sqrt{c x} \sqrt{a+b x^2}}",1,"(-2*x*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[-1/4, 3/2, 3/4, -((b*x^2)/a)])/(a*(c*x)^(3/2)*Sqrt[a + b*x^2])","C",1
627,1,59,154,0.0127215,"\int \frac{1}{(c x)^{5/2} \left(a+b x^2\right)^{3/2}} \, dx","Integrate[1/((c*x)^(5/2)*(a + b*x^2)^(3/2)),x]","-\frac{2 x \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{3}{4},\frac{3}{2};\frac{1}{4};-\frac{b x^2}{a}\right)}{3 a (c x)^{5/2} \sqrt{a+b x^2}}","-\frac{5 b^{3/4} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{6 a^{9/4} c^{5/2} \sqrt{a+b x^2}}-\frac{5 \sqrt{a+b x^2}}{3 a^2 c (c x)^{3/2}}+\frac{1}{a c (c x)^{3/2} \sqrt{a+b x^2}}",1,"(-2*x*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[-3/4, 3/2, 1/4, -((b*x^2)/a)])/(3*a*(c*x)^(5/2)*Sqrt[a + b*x^2])","C",1
628,1,59,331,0.0131425,"\int \frac{1}{(c x)^{7/2} \left(a+b x^2\right)^{3/2}} \, dx","Integrate[1/((c*x)^(7/2)*(a + b*x^2)^(3/2)),x]","-\frac{2 x \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{5}{4},\frac{3}{2};-\frac{1}{4};-\frac{b x^2}{a}\right)}{5 a (c x)^{7/2} \sqrt{a+b x^2}}","-\frac{21 b^{5/4} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{10 a^{11/4} c^{7/2} \sqrt{a+b x^2}}+\frac{21 b^{5/4} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right)|\frac{1}{2}\right)}{5 a^{11/4} c^{7/2} \sqrt{a+b x^2}}-\frac{21 b^{3/2} \sqrt{c x} \sqrt{a+b x^2}}{5 a^3 c^4 \left(\sqrt{a}+\sqrt{b} x\right)}+\frac{21 b \sqrt{a+b x^2}}{5 a^3 c^3 \sqrt{c x}}-\frac{7 \sqrt{a+b x^2}}{5 a^2 c (c x)^{5/2}}+\frac{1}{a c (c x)^{5/2} \sqrt{a+b x^2}}",1,"(-2*x*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[-5/4, 3/2, -1/4, -((b*x^2)/a)])/(5*a*(c*x)^(7/2)*Sqrt[a + b*x^2])","C",1
629,1,80,155,0.0558386,"\int \frac{(c x)^{7/2}}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[(c*x)^(7/2)/(a + b*x^2)^(5/2),x]","\frac{c^3 \sqrt{c x} \left(5 \left(a+b x^2\right) \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^2}{a}\right)-5 a-7 b x^2\right)}{6 b^2 \left(a+b x^2\right)^{3/2}}","\frac{5 c^{7/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{12 \sqrt[4]{a} b^{9/4} \sqrt{a+b x^2}}-\frac{5 c^3 \sqrt{c x}}{6 b^2 \sqrt{a+b x^2}}-\frac{c (c x)^{5/2}}{3 b \left(a+b x^2\right)^{3/2}}",1,"(c^3*Sqrt[c*x]*(-5*a - 7*b*x^2 + 5*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^2)/a)]))/(6*b^2*(a + b*x^2)^(3/2))","C",1
630,1,74,304,0.0402156,"\int \frac{(c x)^{5/2}}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[(c*x)^(5/2)/(a + b*x^2)^(5/2),x]","\frac{2 c (c x)^{3/2} \left(\left(a+b x^2\right) \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{3}{4},\frac{5}{2};\frac{7}{4};-\frac{b x^2}{a}\right)-a\right)}{3 a b \left(a+b x^2\right)^{3/2}}","-\frac{c^{5/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{4 a^{3/4} b^{7/4} \sqrt{a+b x^2}}+\frac{c^{5/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right)|\frac{1}{2}\right)}{2 a^{3/4} b^{7/4} \sqrt{a+b x^2}}-\frac{c^2 \sqrt{c x} \sqrt{a+b x^2}}{2 a b^{3/2} \left(\sqrt{a}+\sqrt{b} x\right)}+\frac{c (c x)^{3/2}}{2 a b \sqrt{a+b x^2}}-\frac{c (c x)^{3/2}}{3 b \left(a+b x^2\right)^{3/2}}",1,"(2*c*(c*x)^(3/2)*(-a + (a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[3/4, 5/2, 7/4, -((b*x^2)/a)]))/(3*a*b*(a + b*x^2)^(3/2))","C",1
631,1,79,156,0.0443236,"\int \frac{(c x)^{3/2}}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[(c*x)^(3/2)/(a + b*x^2)^(5/2),x]","\frac{c \sqrt{c x} \left(\left(a+b x^2\right) \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^2}{a}\right)-a+b x^2\right)}{6 a b \left(a+b x^2\right)^{3/2}}","\frac{c^{3/2} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{12 a^{5/4} b^{5/4} \sqrt{a+b x^2}}+\frac{c \sqrt{c x}}{6 a b \sqrt{a+b x^2}}-\frac{c \sqrt{c x}}{3 b \left(a+b x^2\right)^{3/2}}",1,"(c*Sqrt[c*x]*(-a + b*x^2 + (a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^2)/a)]))/(6*a*b*(a + b*x^2)^(3/2))","C",1
632,1,59,302,0.0130762,"\int \frac{\sqrt{c x}}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[Sqrt[c*x]/(a + b*x^2)^(5/2),x]","\frac{2 x \sqrt{c x} \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{3}{4},\frac{5}{2};\frac{7}{4};-\frac{b x^2}{a}\right)}{3 a^2 \sqrt{a+b x^2}}","-\frac{\sqrt{c} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{4 a^{7/4} b^{3/4} \sqrt{a+b x^2}}+\frac{\sqrt{c} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right)|\frac{1}{2}\right)}{2 a^{7/4} b^{3/4} \sqrt{a+b x^2}}+\frac{(c x)^{3/2}}{2 a^2 c \sqrt{a+b x^2}}-\frac{\sqrt{c x} \sqrt{a+b x^2}}{2 a^2 \sqrt{b} \left(\sqrt{a}+\sqrt{b} x\right)}+\frac{(c x)^{3/2}}{3 a c \left(a+b x^2\right)^{3/2}}",1,"(2*x*Sqrt[c*x]*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[3/4, 5/2, 7/4, -((b*x^2)/a)])/(3*a^2*Sqrt[a + b*x^2])","C",1
633,1,79,157,0.0386622,"\int \frac{1}{\sqrt{c x} \left(a+b x^2\right)^{5/2}} \, dx","Integrate[1/(Sqrt[c*x]*(a + b*x^2)^(5/2)),x]","\frac{5 x \left(a+b x^2\right) \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^2}{a}\right)+7 a x+5 b x^3}{6 a^2 \sqrt{c x} \left(a+b x^2\right)^{3/2}}","\frac{5 \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{12 a^{9/4} \sqrt[4]{b} \sqrt{c} \sqrt{a+b x^2}}+\frac{5 \sqrt{c x}}{6 a^2 c \sqrt{a+b x^2}}+\frac{\sqrt{c x}}{3 a c \left(a+b x^2\right)^{3/2}}",1,"(7*a*x + 5*b*x^3 + 5*x*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^2)/a)])/(6*a^2*Sqrt[c*x]*(a + b*x^2)^(3/2))","C",1
634,1,57,333,0.0132306,"\int \frac{1}{(c x)^{3/2} \left(a+b x^2\right)^{5/2}} \, dx","Integrate[1/((c*x)^(3/2)*(a + b*x^2)^(5/2)),x]","-\frac{2 x \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{1}{4},\frac{5}{2};\frac{3}{4};-\frac{b x^2}{a}\right)}{a^2 (c x)^{3/2} \sqrt{a+b x^2}}","\frac{7 \sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{4 a^{11/4} c^{3/2} \sqrt{a+b x^2}}-\frac{7 \sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right)|\frac{1}{2}\right)}{2 a^{11/4} c^{3/2} \sqrt{a+b x^2}}+\frac{7 \sqrt{b} \sqrt{c x} \sqrt{a+b x^2}}{2 a^3 c^2 \left(\sqrt{a}+\sqrt{b} x\right)}-\frac{7 \sqrt{a+b x^2}}{2 a^3 c \sqrt{c x}}+\frac{7}{6 a^2 c \sqrt{c x} \sqrt{a+b x^2}}+\frac{1}{3 a c \sqrt{c x} \left(a+b x^2\right)^{3/2}}",1,"(-2*x*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[-1/4, 5/2, 3/4, -((b*x^2)/a)])/(a^2*(c*x)^(3/2)*Sqrt[a + b*x^2])","C",1
635,1,59,185,0.0151807,"\int \frac{1}{(c x)^{5/2} \left(a+b x^2\right)^{5/2}} \, dx","Integrate[1/((c*x)^(5/2)*(a + b*x^2)^(5/2)),x]","-\frac{2 x \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{3}{4},\frac{5}{2};\frac{1}{4};-\frac{b x^2}{a}\right)}{3 a^2 (c x)^{5/2} \sqrt{a+b x^2}}","-\frac{5 b^{3/4} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{4 a^{13/4} c^{5/2} \sqrt{a+b x^2}}-\frac{5 \sqrt{a+b x^2}}{2 a^3 c (c x)^{3/2}}+\frac{3}{2 a^2 c (c x)^{3/2} \sqrt{a+b x^2}}+\frac{1}{3 a c (c x)^{3/2} \left(a+b x^2\right)^{3/2}}",1,"(-2*x*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[-3/4, 5/2, 1/4, -((b*x^2)/a)])/(3*a^2*(c*x)^(5/2)*Sqrt[a + b*x^2])","C",1
636,1,59,362,0.0132253,"\int \frac{1}{(c x)^{7/2} \left(a+b x^2\right)^{5/2}} \, dx","Integrate[1/((c*x)^(7/2)*(a + b*x^2)^(5/2)),x]","-\frac{2 x \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{5}{4},\frac{5}{2};-\frac{1}{4};-\frac{b x^2}{a}\right)}{5 a^2 (c x)^{7/2} \sqrt{a+b x^2}}","-\frac{77 b^{5/4} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right),\frac{1}{2}\right)}{20 a^{15/4} c^{7/2} \sqrt{a+b x^2}}+\frac{77 b^{5/4} \left(\sqrt{a}+\sqrt{b} x\right) \sqrt{\frac{a+b x^2}{\left(\sqrt{a}+\sqrt{b} x\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a} \sqrt{c}}\right)|\frac{1}{2}\right)}{10 a^{15/4} c^{7/2} \sqrt{a+b x^2}}-\frac{77 b^{3/2} \sqrt{c x} \sqrt{a+b x^2}}{10 a^4 c^4 \left(\sqrt{a}+\sqrt{b} x\right)}+\frac{77 b \sqrt{a+b x^2}}{10 a^4 c^3 \sqrt{c x}}-\frac{77 \sqrt{a+b x^2}}{30 a^3 c (c x)^{5/2}}+\frac{11}{6 a^2 c (c x)^{5/2} \sqrt{a+b x^2}}+\frac{1}{3 a c (c x)^{5/2} \left(a+b x^2\right)^{3/2}}",1,"(-2*x*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[-5/4, 5/2, -1/4, -((b*x^2)/a)])/(5*a^2*(c*x)^(7/2)*Sqrt[a + b*x^2])","C",1
637,1,61,107,0.0328953,"\int \frac{(c x)^{5/2}}{\sqrt{3 a-2 a x^2}} \, dx","Integrate[(c*x)^(5/2)/Sqrt[3*a - 2*a*x^2],x]","\frac{c (c x)^{3/2} \left(\sqrt{9-6 x^2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{2 x^2}{3}\right)+2 x^2-3\right)}{5 \sqrt{a \left(3-2 x^2\right)}}","-\frac{9 \sqrt[4]{3} c^2 \sqrt{3-2 x^2} \sqrt{c x} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right)\right|2\right)}{5\ 2^{3/4} \sqrt{x} \sqrt{3 a-2 a x^2}}-\frac{c \sqrt{3 a-2 a x^2} (c x)^{3/2}}{5 a}",1,"(c*(c*x)^(3/2)*(-3 + 2*x^2 + Sqrt[9 - 6*x^2]*Hypergeometric2F1[1/2, 3/4, 7/4, (2*x^2)/3]))/(5*Sqrt[a*(3 - 2*x^2)])","C",1
638,1,61,88,0.0223362,"\int \frac{(c x)^{3/2}}{\sqrt{3 a-2 a x^2}} \, dx","Integrate[(c*x)^(3/2)/Sqrt[3*a - 2*a*x^2],x]","\frac{c \sqrt{c x} \left(\sqrt{9-6 x^2} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{2 x^2}{3}\right)+2 x^2-3\right)}{3 \sqrt{a \left(3-2 x^2\right)}}","\frac{c^{3/2} \sqrt{3-2 x^2} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt[4]{\frac{2}{3}} \sqrt{c x}}{\sqrt{c}}\right),-1\right)}{\sqrt[4]{6} \sqrt{a \left(3-2 x^2\right)}}-\frac{c \sqrt{3 a-2 a x^2} \sqrt{c x}}{3 a}",1,"(c*Sqrt[c*x]*(-3 + 2*x^2 + Sqrt[9 - 6*x^2]*Hypergeometric2F1[1/4, 1/2, 5/4, (2*x^2)/3]))/(3*Sqrt[a*(3 - 2*x^2)])","C",1
639,1,53,67,0.016197,"\int \frac{\sqrt{c x}}{\sqrt{3 a-2 a x^2}} \, dx","Integrate[Sqrt[c*x]/Sqrt[3*a - 2*a*x^2],x]","\frac{2 x \sqrt{3-2 x^2} \sqrt{c x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{2 x^2}{3}\right)}{3 \sqrt{a \left(9-6 x^2\right)}}","-\frac{\sqrt[4]{6} \sqrt{3-2 x^2} \sqrt{c x} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right)\right|2\right)}{\sqrt{x} \sqrt{3 a-2 a x^2}}",1,"(2*x*Sqrt[c*x]*Sqrt[3 - 2*x^2]*Hypergeometric2F1[1/2, 3/4, 7/4, (2*x^2)/3])/(3*Sqrt[a*(9 - 6*x^2)])","C",1
640,1,56,63,0.0173608,"\int \frac{1}{\sqrt{c x} \sqrt{3 a-2 a x^2}} \, dx","Integrate[1/(Sqrt[c*x]*Sqrt[3*a - 2*a*x^2]),x]","\frac{2 x \sqrt{3-2 x^2} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{2 x^2}{3}\right)}{\sqrt{3} \sqrt{a \left(3-2 x^2\right)} \sqrt{c x}}","\frac{2^{3/4} \sqrt{3-2 x^2} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt[4]{\frac{2}{3}} \sqrt{c x}}{\sqrt{c}}\right),-1\right)}{\sqrt[4]{3} \sqrt{c} \sqrt{a \left(3-2 x^2\right)}}",1,"(2*x*Sqrt[3 - 2*x^2]*Hypergeometric2F1[1/4, 1/2, 5/4, (2*x^2)/3])/(Sqrt[3]*Sqrt[c*x]*Sqrt[a*(3 - 2*x^2)])","C",1
641,1,51,107,0.0168042,"\int \frac{1}{(c x)^{3/2} \sqrt{3 a-2 a x^2}} \, dx","Integrate[1/((c*x)^(3/2)*Sqrt[3*a - 2*a*x^2]),x]","-\frac{2 x \sqrt{3-2 x^2} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};\frac{2 x^2}{3}\right)}{\sqrt{a \left(9-6 x^2\right)} (c x)^{3/2}}","\frac{2 \sqrt[4]{2} \sqrt{3-2 x^2} \sqrt{c x} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right)\right|2\right)}{3^{3/4} c^2 \sqrt{x} \sqrt{3 a-2 a x^2}}-\frac{2 \sqrt{3 a-2 a x^2}}{3 a c \sqrt{c x}}",1,"(-2*x*Sqrt[3 - 2*x^2]*Hypergeometric2F1[-1/4, 1/2, 3/4, (2*x^2)/3])/((c*x)^(3/2)*Sqrt[a*(9 - 6*x^2)])","C",1
642,1,53,98,0.015456,"\int \frac{1}{(c x)^{5/2} \sqrt{3 a-2 a x^2}} \, dx","Integrate[1/((c*x)^(5/2)*Sqrt[3*a - 2*a*x^2]),x]","-\frac{2 x \sqrt{3-2 x^2} \, _2F_1\left(-\frac{3}{4},\frac{1}{2};\frac{1}{4};\frac{2 x^2}{3}\right)}{3 \sqrt{a \left(9-6 x^2\right)} (c x)^{5/2}}","\frac{2\ 2^{3/4} \sqrt{3-2 x^2} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt[4]{\frac{2}{3}} \sqrt{c x}}{\sqrt{c}}\right),-1\right)}{9 \sqrt[4]{3} c^{5/2} \sqrt{a \left(3-2 x^2\right)}}-\frac{2 \sqrt{3 a-2 a x^2}}{9 a c (c x)^{3/2}}",1,"(-2*x*Sqrt[3 - 2*x^2]*Hypergeometric2F1[-3/4, 1/2, 1/4, (2*x^2)/3])/(3*(c*x)^(5/2)*Sqrt[a*(9 - 6*x^2)])","C",1
643,1,59,110,0.0247115,"\int \frac{(c x)^{5/2}}{\left(3 a-2 a x^2\right)^{3/2}} \, dx","Integrate[(c*x)^(5/2)/(3*a - 2*a*x^2)^(3/2),x]","\frac{c (c x)^{3/2} \left(\sqrt{9-6 x^2} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};\frac{2 x^2}{3}\right)-3\right)}{3 a \sqrt{a \left(3-2 x^2\right)}}","\frac{3 \sqrt[4]{3} c^2 \sqrt{3-2 x^2} \sqrt{c x} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right)\right|2\right)}{2\ 2^{3/4} a \sqrt{x} \sqrt{3 a-2 a x^2}}+\frac{c (c x)^{3/2}}{2 a \sqrt{3 a-2 a x^2}}",1,"(c*(c*x)^(3/2)*(-3 + Sqrt[9 - 6*x^2]*Hypergeometric2F1[3/4, 3/2, 7/4, (2*x^2)/3]))/(3*a*Sqrt[a*(3 - 2*x^2)])","C",1
644,1,59,94,0.0214187,"\int \frac{(c x)^{3/2}}{\left(3 a-2 a x^2\right)^{3/2}} \, dx","Integrate[(c*x)^(3/2)/(3*a - 2*a*x^2)^(3/2),x]","-\frac{c \sqrt{c x} \left(\sqrt{9-6 x^2} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{2 x^2}{3}\right)-3\right)}{6 a \sqrt{a \left(3-2 x^2\right)}}","\frac{c \sqrt{c x}}{2 a \sqrt{3 a-2 a x^2}}-\frac{c^{3/2} \sqrt{3-2 x^2} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt[4]{\frac{2}{3}} \sqrt{c x}}{\sqrt{c}}\right),-1\right)}{2 \sqrt[4]{6} a \sqrt{a \left(3-2 x^2\right)}}",1,"-1/6*(c*Sqrt[c*x]*(-3 + Sqrt[9 - 6*x^2]*Hypergeometric2F1[1/4, 1/2, 5/4, (2*x^2)/3]))/(a*Sqrt[a*(3 - 2*x^2)])","C",1
645,1,58,101,0.0169173,"\int \frac{\sqrt{c x}}{\left(3 a-2 a x^2\right)^{3/2}} \, dx","Integrate[Sqrt[c*x]/(3*a - 2*a*x^2)^(3/2),x]","\frac{2 x \left(3-2 x^2\right)^{3/2} \sqrt{c x} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};\frac{2 x^2}{3}\right)}{9 \sqrt{3} \left(a \left(3-2 x^2\right)\right)^{3/2}}","\frac{(c x)^{3/2}}{3 a c \sqrt{3 a-2 a x^2}}+\frac{\sqrt{3-2 x^2} \sqrt{c x} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right)\right|2\right)}{6^{3/4} a \sqrt{x} \sqrt{3 a-2 a x^2}}",1,"(2*x*Sqrt[c*x]*(3 - 2*x^2)^(3/2)*Hypergeometric2F1[3/4, 3/2, 7/4, (2*x^2)/3])/(9*Sqrt[3]*(a*(3 - 2*x^2))^(3/2))","C",1
646,1,59,96,0.0251687,"\int \frac{1}{\sqrt{c x} \left(3 a-2 a x^2\right)^{3/2}} \, dx","Integrate[1/(Sqrt[c*x]*(3*a - 2*a*x^2)^(3/2)),x]","\frac{x \left(\sqrt{9-6 x^2} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{2 x^2}{3}\right)+3\right)}{9 a \sqrt{a \left(3-2 x^2\right)} \sqrt{c x}}","\frac{\sqrt{3-2 x^2} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt[4]{\frac{2}{3}} \sqrt{c x}}{\sqrt{c}}\right),-1\right)}{3 \sqrt[4]{6} a \sqrt{c} \sqrt{a \left(3-2 x^2\right)}}+\frac{\sqrt{c x}}{3 a c \sqrt{3 a-2 a x^2}}",1,"(x*(3 + Sqrt[9 - 6*x^2]*Hypergeometric2F1[1/4, 1/2, 5/4, (2*x^2)/3]))/(9*a*Sqrt[c*x]*Sqrt[a*(3 - 2*x^2)])","C",1
647,1,58,140,0.0214728,"\int \frac{1}{(c x)^{3/2} \left(3 a-2 a x^2\right)^{3/2}} \, dx","Integrate[1/((c*x)^(3/2)*(3*a - 2*a*x^2)^(3/2)),x]","-\frac{2 x \left(3-2 x^2\right)^{3/2} \, _2F_1\left(-\frac{1}{4},\frac{3}{2};\frac{3}{4};\frac{2 x^2}{3}\right)}{3 \sqrt{3} \left(a \left(3-2 x^2\right)\right)^{3/2} (c x)^{3/2}}","-\frac{\sqrt{3 a-2 a x^2}}{3 a^2 c \sqrt{c x}}+\frac{\sqrt[4]{2} \sqrt{3-2 x^2} \sqrt{c x} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right)\right|2\right)}{3^{3/4} a c^2 \sqrt{x} \sqrt{3 a-2 a x^2}}+\frac{1}{3 a c \sqrt{3 a-2 a x^2} \sqrt{c x}}",1,"(-2*x*(3 - 2*x^2)^(3/2)*Hypergeometric2F1[-1/4, 3/2, 3/4, (2*x^2)/3])/(3*Sqrt[3]*(c*x)^(3/2)*(a*(3 - 2*x^2))^(3/2))","C",1
648,1,58,132,0.0194363,"\int \frac{1}{(c x)^{5/2} \left(3 a-2 a x^2\right)^{3/2}} \, dx","Integrate[1/((c*x)^(5/2)*(3*a - 2*a*x^2)^(3/2)),x]","-\frac{2 x \left(3-2 x^2\right)^{3/2} \, _2F_1\left(-\frac{3}{4},\frac{3}{2};\frac{1}{4};\frac{2 x^2}{3}\right)}{9 \sqrt{3} \left(a \left(3-2 x^2\right)\right)^{3/2} (c x)^{5/2}}","\frac{5\ 2^{3/4} \sqrt{3-2 x^2} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt[4]{\frac{2}{3}} \sqrt{c x}}{\sqrt{c}}\right),-1\right)}{27 \sqrt[4]{3} a c^{5/2} \sqrt{a \left(3-2 x^2\right)}}-\frac{5 \sqrt{3 a-2 a x^2}}{27 a^2 c (c x)^{3/2}}+\frac{1}{3 a c \sqrt{3 a-2 a x^2} (c x)^{3/2}}",1,"(-2*x*(3 - 2*x^2)^(3/2)*Hypergeometric2F1[-3/4, 3/2, 1/4, (2*x^2)/3])/(9*Sqrt[3]*(c*x)^(5/2)*(a*(3 - 2*x^2))^(3/2))","C",1
649,1,24,21,0.0068874,"\int \frac{1}{\sqrt{x} \sqrt{1-a^2 x^2}} \, dx","Integrate[1/(Sqrt[x]*Sqrt[1 - a^2*x^2]),x]","2 \sqrt{x} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};a^2 x^2\right)","\frac{2 \operatorname{EllipticF}\left(\sin ^{-1}\left(\sqrt{a} \sqrt{x}\right),-1\right)}{\sqrt{a}}",1,"2*Sqrt[x]*Hypergeometric2F1[1/4, 1/2, 5/4, a^2*x^2]","C",1
650,1,23,67,0.0061332,"\int \frac{1}{\sqrt{x} \sqrt{1+a x^2}} \, dx","Integrate[1/(Sqrt[x]*Sqrt[1 + a*x^2]),x]","2 \sqrt{x} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-a x^2\right)","\frac{\left(\sqrt{a} x+1\right) \sqrt{\frac{a x^2+1}{\left(\sqrt{a} x+1\right)^2}} \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\sqrt[4]{a} \sqrt{x}\right),\frac{1}{2}\right)}{\sqrt[4]{a} \sqrt{a x^2+1}}",1,"2*Sqrt[x]*Hypergeometric2F1[1/4, 1/2, 5/4, -(a*x^2)]","C",1
651,1,66,50,0.0217168,"\int x^m \left(a+b x^2\right)^{3/2} \, dx","Integrate[x^m*(a + b*x^2)^(3/2),x]","\frac{a x^{m+1} \sqrt{a+b x^2} \, _2F_1\left(-\frac{3}{2},\frac{m+1}{2};\frac{m+1}{2}+1;-\frac{b x^2}{a}\right)}{(m+1) \sqrt{\frac{b x^2}{a}+1}}","\frac{x^{m+1} \left(a+b x^2\right)^{5/2} \, _2F_1\left(1,\frac{m+6}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a (m+1)}",1,"(a*x^(1 + m)*Sqrt[a + b*x^2]*Hypergeometric2F1[-3/2, (1 + m)/2, 1 + (1 + m)/2, -((b*x^2)/a)])/((1 + m)*Sqrt[1 + (b*x^2)/a])","A",1
652,1,65,50,0.0135593,"\int x^m \sqrt{a+b x^2} \, dx","Integrate[x^m*Sqrt[a + b*x^2],x]","\frac{x^{m+1} \sqrt{a+b x^2} \, _2F_1\left(-\frac{1}{2},\frac{m+1}{2};\frac{m+1}{2}+1;-\frac{b x^2}{a}\right)}{(m+1) \sqrt{\frac{b x^2}{a}+1}}","\frac{x^{m+1} \left(a+b x^2\right)^{3/2} \, _2F_1\left(1,\frac{m+4}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a (m+1)}",1,"(x^(1 + m)*Sqrt[a + b*x^2]*Hypergeometric2F1[-1/2, (1 + m)/2, 1 + (1 + m)/2, -((b*x^2)/a)])/((1 + m)*Sqrt[1 + (b*x^2)/a])","A",1
653,1,65,50,0.0162607,"\int \frac{x^m}{\sqrt{a+b x^2}} \, dx","Integrate[x^m/Sqrt[a + b*x^2],x]","\frac{x^{m+1} \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+1}{2}+1;-\frac{b x^2}{a}\right)}{(m+1) \sqrt{a+b x^2}}","\frac{x^{m+1} \sqrt{a+b x^2} \, _2F_1\left(1,\frac{m+2}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a (m+1)}",1,"(x^(1 + m)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/2, (1 + m)/2, 1 + (1 + m)/2, -((b*x^2)/a)])/((1 + m)*Sqrt[a + b*x^2])","A",1
654,1,68,48,0.0178319,"\int \frac{x^m}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[x^m/(a + b*x^2)^(3/2),x]","\frac{x^{m+1} \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{3}{2},\frac{m+1}{2};\frac{m+1}{2}+1;-\frac{b x^2}{a}\right)}{a (m+1) \sqrt{a+b x^2}}","\frac{x^{m+1} \, _2F_1\left(1,\frac{m}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a (m+1) \sqrt{a+b x^2}}",1,"(x^(1 + m)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[3/2, (1 + m)/2, 1 + (1 + m)/2, -((b*x^2)/a)])/(a*(1 + m)*Sqrt[a + b*x^2])","A",1
655,1,68,50,0.0183044,"\int \frac{x^m}{\left(a+b x^2\right)^{5/2}} \, dx","Integrate[x^m/(a + b*x^2)^(5/2),x]","\frac{x^{m+1} \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{5}{2},\frac{m+1}{2};\frac{m+1}{2}+1;-\frac{b x^2}{a}\right)}{a^2 (m+1) \sqrt{a+b x^2}}","\frac{x^{m+1} \, _2F_1\left(1,\frac{m-2}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a (m+1) \left(a+b x^2\right)^{3/2}}",1,"(x^(1 + m)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[5/2, (1 + m)/2, 1 + (1 + m)/2, -((b*x^2)/a)])/(a^2*(1 + m)*Sqrt[a + b*x^2])","A",1
656,1,65,50,0.0176142,"\int \frac{x^{2+m}}{\sqrt{a+b x^2}} \, dx","Integrate[x^(2 + m)/Sqrt[a + b*x^2],x]","\frac{x^{m+3} \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+3}{2}+1;-\frac{b x^2}{a}\right)}{(m+3) \sqrt{a+b x^2}}","\frac{x^{m+3} \sqrt{a+b x^2} \, _2F_1\left(1,\frac{m+4}{2};\frac{m+5}{2};-\frac{b x^2}{a}\right)}{a (m+3)}",1,"(x^(3 + m)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/2, (3 + m)/2, 1 + (3 + m)/2, -((b*x^2)/a)])/((3 + m)*Sqrt[a + b*x^2])","A",1
657,1,65,50,0.0180808,"\int \frac{x^{1+m}}{\sqrt{a+b x^2}} \, dx","Integrate[x^(1 + m)/Sqrt[a + b*x^2],x]","\frac{x^{m+2} \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+2}{2}+1;-\frac{b x^2}{a}\right)}{(m+2) \sqrt{a+b x^2}}","\frac{x^{m+2} \sqrt{a+b x^2} \, _2F_1\left(1,\frac{m+3}{2};\frac{m+4}{2};-\frac{b x^2}{a}\right)}{a (m+2)}",1,"(x^(2 + m)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/2, (2 + m)/2, 1 + (2 + m)/2, -((b*x^2)/a)])/((2 + m)*Sqrt[a + b*x^2])","A",1
658,1,65,50,0.0142909,"\int \frac{x^m}{\sqrt{a+b x^2}} \, dx","Integrate[x^m/Sqrt[a + b*x^2],x]","\frac{x^{m+1} \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+1}{2}+1;-\frac{b x^2}{a}\right)}{(m+1) \sqrt{a+b x^2}}","\frac{x^{m+1} \sqrt{a+b x^2} \, _2F_1\left(1,\frac{m+2}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a (m+1)}",1,"(x^(1 + m)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/2, (1 + m)/2, 1 + (1 + m)/2, -((b*x^2)/a)])/((1 + m)*Sqrt[a + b*x^2])","A",1
659,1,57,46,0.0110554,"\int \frac{x^{-1+m}}{\sqrt{a+b x^2}} \, dx","Integrate[x^(-1 + m)/Sqrt[a + b*x^2],x]","\frac{x^m \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{m}{2};\frac{m}{2}+1;-\frac{b x^2}{a}\right)}{m \sqrt{a+b x^2}}","\frac{x^m \sqrt{a+b x^2} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+2}{2};-\frac{b x^2}{a}\right)}{a m}",1,"(x^m*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/2, m/2, 1 + m/2, -((b*x^2)/a)])/(m*Sqrt[a + b*x^2])","A",1
660,1,65,51,0.019473,"\int \frac{x^{-2+m}}{\sqrt{a+b x^2}} \, dx","Integrate[x^(-2 + m)/Sqrt[a + b*x^2],x]","\frac{x^{m-1} \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{m-1}{2};\frac{m-1}{2}+1;-\frac{b x^2}{a}\right)}{(m-1) \sqrt{a+b x^2}}","-\frac{x^{m-1} \sqrt{a+b x^2} \, _2F_1\left(1,\frac{m}{2};\frac{m+1}{2};-\frac{b x^2}{a}\right)}{a (1-m)}",1,"(x^(-1 + m)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/2, (-1 + m)/2, 1 + (-1 + m)/2, -((b*x^2)/a)])/((-1 + m)*Sqrt[a + b*x^2])","A",1
661,1,104,17,0.1080488,"\int \frac{x^{1+m} \left(a (2+m)+b (3+m) x^2\right)}{\sqrt{a+b x^2}} \, dx","Integrate[(x^(1 + m)*(a*(2 + m) + b*(3 + m)*x^2))/Sqrt[a + b*x^2],x]","\frac{x^{m+2} \sqrt{\frac{b x^2}{a}+1} \left(b (m+3) x^2 \, _2F_1\left(\frac{1}{2},\frac{m+4}{2};\frac{m+6}{2};-\frac{b x^2}{a}\right)+a (m+4) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};-\frac{b x^2}{a}\right)\right)}{(m+4) \sqrt{a+b x^2}}","x^{m+2} \sqrt{a+b x^2}",1,"(x^(2 + m)*Sqrt[1 + (b*x^2)/a]*(a*(4 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, -((b*x^2)/a)] + b*(3 + m)*x^2*Hypergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, -((b*x^2)/a)]))/((4 + m)*Sqrt[a + b*x^2])","C",1
662,1,104,17,0.0690578,"\int \left(\frac{a (2+m) x^{1+m}}{\sqrt{a+b x^2}}+\frac{b (3+m) x^{3+m}}{\sqrt{a+b x^2}}\right) \, dx","Integrate[(a*(2 + m)*x^(1 + m))/Sqrt[a + b*x^2] + (b*(3 + m)*x^(3 + m))/Sqrt[a + b*x^2],x]","\frac{x^{m+2} \sqrt{\frac{b x^2}{a}+1} \left(b (m+3) x^2 \, _2F_1\left(\frac{1}{2},\frac{m+4}{2};\frac{m+6}{2};-\frac{b x^2}{a}\right)+a (m+4) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};-\frac{b x^2}{a}\right)\right)}{(m+4) \sqrt{a+b x^2}}","x^{m+2} \sqrt{a+b x^2}",1,"(x^(2 + m)*Sqrt[1 + (b*x^2)/a]*(a*(4 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, -((b*x^2)/a)] + b*(3 + m)*x^2*Hypergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, -((b*x^2)/a)]))/((4 + m)*Sqrt[a + b*x^2])","C",1
663,1,103,15,0.1093934,"\int \frac{x^{-1+m} \left(a m+b (-1+m) x^2\right)}{\left(a+b x^2\right)^{3/2}} \, dx","Integrate[(x^(-1 + m)*(a*m + b*(-1 + m)*x^2))/(a + b*x^2)^(3/2),x]","\frac{x^m \sqrt{\frac{b x^2}{a}+1} \left(b (m-1) x^2 \, _2F_1\left(\frac{3}{2},\frac{m+2}{2};\frac{m+4}{2};-\frac{b x^2}{a}\right)+a (m+2) \, _2F_1\left(\frac{3}{2},\frac{m}{2};\frac{m+2}{2};-\frac{b x^2}{a}\right)\right)}{a (m+2) \sqrt{a+b x^2}}","\frac{x^m}{\sqrt{a+b x^2}}",1,"(x^m*Sqrt[1 + (b*x^2)/a]*(a*(2 + m)*Hypergeometric2F1[3/2, m/2, (2 + m)/2, -((b*x^2)/a)] + b*(-1 + m)*x^2*Hypergeometric2F1[3/2, (2 + m)/2, (4 + m)/2, -((b*x^2)/a)]))/(a*(2 + m)*Sqrt[a + b*x^2])","C",1
664,1,103,15,0.0646906,"\int \left(-\frac{b x^{1+m}}{\left(a+b x^2\right)^{3/2}}+\frac{m x^{-1+m}}{\sqrt{a+b x^2}}\right) \, dx","Integrate[-((b*x^(1 + m))/(a + b*x^2)^(3/2)) + (m*x^(-1 + m))/Sqrt[a + b*x^2],x]","\frac{x^m \sqrt{\frac{b x^2}{a}+1} \left(b (m-1) x^2 \, _2F_1\left(\frac{3}{2},\frac{m+2}{2};\frac{m+4}{2};-\frac{b x^2}{a}\right)+a (m+2) \, _2F_1\left(\frac{3}{2},\frac{m}{2};\frac{m+2}{2};-\frac{b x^2}{a}\right)\right)}{a (m+2) \sqrt{a+b x^2}}","\frac{x^m}{\sqrt{a+b x^2}}",1,"(x^m*Sqrt[1 + (b*x^2)/a]*(a*(2 + m)*Hypergeometric2F1[3/2, m/2, (2 + m)/2, -((b*x^2)/a)] + b*(-1 + m)*x^2*Hypergeometric2F1[3/2, (2 + m)/2, (4 + m)/2, -((b*x^2)/a)]))/(a*(2 + m)*Sqrt[a + b*x^2])","C",1
665,1,50,80,0.0252617,"\int x^7 \sqrt[3]{a+b x^2} \, dx","Integrate[x^7*(a + b*x^2)^(1/3),x]","\frac{3 \left(a+b x^2\right)^{4/3} \left(-81 a^3+108 a^2 b x^2-126 a b^2 x^4+140 b^3 x^6\right)}{3640 b^4}","-\frac{3 a^3 \left(a+b x^2\right)^{4/3}}{8 b^4}+\frac{9 a^2 \left(a+b x^2\right)^{7/3}}{14 b^4}+\frac{3 \left(a+b x^2\right)^{13/3}}{26 b^4}-\frac{9 a \left(a+b x^2\right)^{10/3}}{20 b^4}",1,"(3*(a + b*x^2)^(4/3)*(-81*a^3 + 108*a^2*b*x^2 - 126*a*b^2*x^4 + 140*b^3*x^6))/(3640*b^4)","A",1
666,1,39,59,0.0177322,"\int x^5 \sqrt[3]{a+b x^2} \, dx","Integrate[x^5*(a + b*x^2)^(1/3),x]","\frac{3 \left(a+b x^2\right)^{4/3} \left(9 a^2-12 a b x^2+14 b^2 x^4\right)}{280 b^3}","\frac{3 a^2 \left(a+b x^2\right)^{4/3}}{8 b^3}+\frac{3 \left(a+b x^2\right)^{10/3}}{20 b^3}-\frac{3 a \left(a+b x^2\right)^{7/3}}{7 b^3}",1,"(3*(a + b*x^2)^(4/3)*(9*a^2 - 12*a*b*x^2 + 14*b^2*x^4))/(280*b^3)","A",1
667,1,28,38,0.0134819,"\int x^3 \sqrt[3]{a+b x^2} \, dx","Integrate[x^3*(a + b*x^2)^(1/3),x]","\frac{3 \left(a+b x^2\right)^{4/3} \left(4 b x^2-3 a\right)}{56 b^2}","\frac{3 \left(a+b x^2\right)^{7/3}}{14 b^2}-\frac{3 a \left(a+b x^2\right)^{4/3}}{8 b^2}",1,"(3*(a + b*x^2)^(4/3)*(-3*a + 4*b*x^2))/(56*b^2)","A",1
668,1,18,18,0.0034336,"\int x \sqrt[3]{a+b x^2} \, dx","Integrate[x*(a + b*x^2)^(1/3),x]","\frac{3 \left(a+b x^2\right)^{4/3}}{8 b}","\frac{3 \left(a+b x^2\right)^{4/3}}{8 b}",1,"(3*(a + b*x^2)^(4/3))/(8*b)","A",1
669,1,126,101,0.0557149,"\int \frac{\sqrt[3]{a+b x^2}}{x} \, dx","Integrate[(a + b*x^2)^(1/3)/x,x]","\frac{1}{4} \left(-\sqrt[3]{a} \log \left(a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}\right)+6 \sqrt[3]{a+b x^2}+2 \sqrt[3]{a} \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)-2 \sqrt{3} \sqrt[3]{a} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b x^2}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)\right)","\frac{3}{2} \sqrt[3]{a+b x^2}+\frac{3}{4} \sqrt[3]{a} \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)-\frac{1}{2} \sqrt{3} \sqrt[3]{a} \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^2}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)-\frac{1}{2} \sqrt[3]{a} \log (x)",1,"(6*(a + b*x^2)^(1/3) - 2*Sqrt[3]*a^(1/3)*ArcTan[(1 + (2*(a + b*x^2)^(1/3))/a^(1/3))/Sqrt[3]] + 2*a^(1/3)*Log[a^(1/3) - (a + b*x^2)^(1/3)] - a^(1/3)*Log[a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3)])/4","A",1
670,1,37,107,0.0085256,"\int \frac{\sqrt[3]{a+b x^2}}{x^3} \, dx","Integrate[(a + b*x^2)^(1/3)/x^3,x]","\frac{3 b \left(a+b x^2\right)^{4/3} \, _2F_1\left(\frac{4}{3},2;\frac{7}{3};\frac{b x^2}{a}+1\right)}{8 a^2}","\frac{b \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{4 a^{2/3}}-\frac{b \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^2}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{2 \sqrt{3} a^{2/3}}-\frac{b \log (x)}{6 a^{2/3}}-\frac{\sqrt[3]{a+b x^2}}{2 x^2}",1,"(3*b*(a + b*x^2)^(4/3)*Hypergeometric2F1[4/3, 2, 7/3, 1 + (b*x^2)/a])/(8*a^2)","C",1
671,1,39,135,0.0083981,"\int \frac{\sqrt[3]{a+b x^2}}{x^5} \, dx","Integrate[(a + b*x^2)^(1/3)/x^5,x]","-\frac{3 b^2 \left(a+b x^2\right)^{4/3} \, _2F_1\left(\frac{4}{3},3;\frac{7}{3};\frac{b x^2}{a}+1\right)}{8 a^3}","-\frac{b^2 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{12 a^{5/3}}+\frac{b^2 \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^2}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{6 \sqrt{3} a^{5/3}}+\frac{b^2 \log (x)}{18 a^{5/3}}-\frac{b \sqrt[3]{a+b x^2}}{12 a x^2}-\frac{\sqrt[3]{a+b x^2}}{4 x^4}",1,"(-3*b^2*(a + b*x^2)^(4/3)*Hypergeometric2F1[4/3, 3, 7/3, 1 + (b*x^2)/a])/(8*a^3)","C",1
672,1,94,314,0.0528442,"\int x^4 \sqrt[3]{a+b x^2} \, dx","Integrate[x^4*(a + b*x^2)^(1/3),x]","\frac{3 x \sqrt[3]{a+b x^2} \left(\sqrt[3]{\frac{b x^2}{a}+1} \left(-9 a^2+2 a b x^2+11 b^2 x^4\right)+9 a^2 \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)\right)}{187 b^2 \sqrt[3]{\frac{b x^2}{a}+1}}","-\frac{54\ 3^{3/4} \sqrt{2-\sqrt{3}} a^3 \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}\right),4 \sqrt{3}-7\right)}{935 b^3 x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}-\frac{54 a^2 x \sqrt[3]{a+b x^2}}{935 b^2}+\frac{3}{17} x^5 \sqrt[3]{a+b x^2}+\frac{6 a x^3 \sqrt[3]{a+b x^2}}{187 b}",1,"(3*x*(a + b*x^2)^(1/3)*((1 + (b*x^2)/a)^(1/3)*(-9*a^2 + 2*a*b*x^2 + 11*b^2*x^4) + 9*a^2*Hypergeometric2F1[-1/3, 1/2, 3/2, -((b*x^2)/a)]))/(187*b^2*(1 + (b*x^2)/a)^(1/3))","C",1
673,1,62,290,0.0486775,"\int x^2 \sqrt[3]{a+b x^2} \, dx","Integrate[x^2*(a + b*x^2)^(1/3),x]","\frac{3 x \sqrt[3]{a+b x^2} \left(-\frac{a \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)}{\sqrt[3]{\frac{b x^2}{a}+1}}+a+b x^2\right)}{11 b}","\frac{6\ 3^{3/4} \sqrt{2-\sqrt{3}} a^2 \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}\right),4 \sqrt{3}-7\right)}{55 b^2 x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}+\frac{6 a x \sqrt[3]{a+b x^2}}{55 b}+\frac{3}{11} x^3 \sqrt[3]{a+b x^2}",1,"(3*x*(a + b*x^2)^(1/3)*(a + b*x^2 - (a*Hypergeometric2F1[-1/3, 1/2, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^(1/3)))/(11*b)","C",1
674,1,46,266,0.0054535,"\int \sqrt[3]{a+b x^2} \, dx","Integrate[(a + b*x^2)^(1/3),x]","\frac{x \sqrt[3]{a+b x^2} \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)}{\sqrt[3]{\frac{b x^2}{a}+1}}","\frac{3}{5} x \sqrt[3]{a+b x^2}-\frac{2\ 3^{3/4} \sqrt{2-\sqrt{3}} a \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}\right),4 \sqrt{3}-7\right)}{5 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}",1,"(x*(a + b*x^2)^(1/3)*Hypergeometric2F1[-1/3, 1/2, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^(1/3)","C",1
675,1,49,260,0.009122,"\int \frac{\sqrt[3]{a+b x^2}}{x^2} \, dx","Integrate[(a + b*x^2)^(1/3)/x^2,x]","-\frac{\sqrt[3]{a+b x^2} \, _2F_1\left(-\frac{1}{2},-\frac{1}{3};\frac{1}{2};-\frac{b x^2}{a}\right)}{x \sqrt[3]{\frac{b x^2}{a}+1}}","-\frac{2 \sqrt{2-\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}\right),4 \sqrt{3}-7\right)}{\sqrt[4]{3} x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}-\frac{\sqrt[3]{a+b x^2}}{x}",1,"-(((a + b*x^2)^(1/3)*Hypergeometric2F1[-1/2, -1/3, 1/2, -((b*x^2)/a)])/(x*(1 + (b*x^2)/a)^(1/3)))","C",1
676,1,51,290,0.0096349,"\int \frac{\sqrt[3]{a+b x^2}}{x^4} \, dx","Integrate[(a + b*x^2)^(1/3)/x^4,x]","-\frac{\sqrt[3]{a+b x^2} \, _2F_1\left(-\frac{3}{2},-\frac{1}{3};-\frac{1}{2};-\frac{b x^2}{a}\right)}{3 x^3 \sqrt[3]{\frac{b x^2}{a}+1}}","\frac{2 \sqrt{2-\sqrt{3}} b \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}\right),4 \sqrt{3}-7\right)}{9 \sqrt[4]{3} a x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}-\frac{2 b \sqrt[3]{a+b x^2}}{9 a x}-\frac{\sqrt[3]{a+b x^2}}{3 x^3}",1,"-1/3*((a + b*x^2)^(1/3)*Hypergeometric2F1[-3/2, -1/3, -1/2, -((b*x^2)/a)])/(x^3*(1 + (b*x^2)/a)^(1/3))","C",1
677,1,50,80,0.0256252,"\int x^7 \left(a+b x^2\right)^{2/3} \, dx","Integrate[x^7*(a + b*x^2)^(2/3),x]","\frac{3 \left(a+b x^2\right)^{5/3} \left(-81 a^3+135 a^2 b x^2-180 a b^2 x^4+220 b^3 x^6\right)}{6160 b^4}","-\frac{3 a^3 \left(a+b x^2\right)^{5/3}}{10 b^4}+\frac{9 a^2 \left(a+b x^2\right)^{8/3}}{16 b^4}+\frac{3 \left(a+b x^2\right)^{14/3}}{28 b^4}-\frac{9 a \left(a+b x^2\right)^{11/3}}{22 b^4}",1,"(3*(a + b*x^2)^(5/3)*(-81*a^3 + 135*a^2*b*x^2 - 180*a*b^2*x^4 + 220*b^3*x^6))/(6160*b^4)","A",1
678,1,39,59,0.0175759,"\int x^5 \left(a+b x^2\right)^{2/3} \, dx","Integrate[x^5*(a + b*x^2)^(2/3),x]","\frac{3 \left(a+b x^2\right)^{5/3} \left(9 a^2-15 a b x^2+20 b^2 x^4\right)}{440 b^3}","\frac{3 a^2 \left(a+b x^2\right)^{5/3}}{10 b^3}+\frac{3 \left(a+b x^2\right)^{11/3}}{22 b^3}-\frac{3 a \left(a+b x^2\right)^{8/3}}{8 b^3}",1,"(3*(a + b*x^2)^(5/3)*(9*a^2 - 15*a*b*x^2 + 20*b^2*x^4))/(440*b^3)","A",1
679,1,28,38,0.0139983,"\int x^3 \left(a+b x^2\right)^{2/3} \, dx","Integrate[x^3*(a + b*x^2)^(2/3),x]","\frac{3 \left(a+b x^2\right)^{5/3} \left(5 b x^2-3 a\right)}{80 b^2}","\frac{3 \left(a+b x^2\right)^{8/3}}{16 b^2}-\frac{3 a \left(a+b x^2\right)^{5/3}}{10 b^2}",1,"(3*(a + b*x^2)^(5/3)*(-3*a + 5*b*x^2))/(80*b^2)","A",1
680,1,18,18,0.0039031,"\int x \left(a+b x^2\right)^{2/3} \, dx","Integrate[x*(a + b*x^2)^(2/3),x]","\frac{3 \left(a+b x^2\right)^{5/3}}{10 b}","\frac{3 \left(a+b x^2\right)^{5/3}}{10 b}",1,"(3*(a + b*x^2)^(5/3))/(10*b)","A",1
681,1,93,101,0.0367783,"\int \frac{\left(a+b x^2\right)^{2/3}}{x} \, dx","Integrate[(a + b*x^2)^(2/3)/x,x]","\frac{1}{4} \left(3 \left(a^{2/3} \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)+\left(a+b x^2\right)^{2/3}\right)+2 \sqrt{3} a^{2/3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b x^2}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)-2 a^{2/3} \log (x)\right)","\frac{3}{4} a^{2/3} \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)+\frac{1}{2} \sqrt{3} a^{2/3} \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^2}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)-\frac{1}{2} a^{2/3} \log (x)+\frac{3}{4} \left(a+b x^2\right)^{2/3}",1,"(2*Sqrt[3]*a^(2/3)*ArcTan[(1 + (2*(a + b*x^2)^(1/3))/a^(1/3))/Sqrt[3]] - 2*a^(2/3)*Log[x] + 3*((a + b*x^2)^(2/3) + a^(2/3)*Log[a^(1/3) - (a + b*x^2)^(1/3)]))/4","A",1
682,1,37,104,0.0083093,"\int \frac{\left(a+b x^2\right)^{2/3}}{x^3} \, dx","Integrate[(a + b*x^2)^(2/3)/x^3,x]","\frac{3 b \left(a+b x^2\right)^{5/3} \, _2F_1\left(\frac{5}{3},2;\frac{8}{3};\frac{b x^2}{a}+1\right)}{10 a^2}","-\frac{\left(a+b x^2\right)^{2/3}}{2 x^2}+\frac{b \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{2 \sqrt[3]{a}}+\frac{b \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^2}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} \sqrt[3]{a}}-\frac{b \log (x)}{3 \sqrt[3]{a}}",1,"(3*b*(a + b*x^2)^(5/3)*Hypergeometric2F1[5/3, 2, 8/3, 1 + (b*x^2)/a])/(10*a^2)","C",1
683,1,39,135,0.0088159,"\int \frac{\left(a+b x^2\right)^{2/3}}{x^5} \, dx","Integrate[(a + b*x^2)^(2/3)/x^5,x]","-\frac{3 b^2 \left(a+b x^2\right)^{5/3} \, _2F_1\left(\frac{5}{3},3;\frac{8}{3};\frac{b x^2}{a}+1\right)}{10 a^3}","-\frac{b^2 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{12 a^{4/3}}-\frac{b^2 \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^2}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{6 \sqrt{3} a^{4/3}}+\frac{b^2 \log (x)}{18 a^{4/3}}-\frac{b \left(a+b x^2\right)^{2/3}}{6 a x^2}-\frac{\left(a+b x^2\right)^{2/3}}{4 x^4}",1,"(-3*b^2*(a + b*x^2)^(5/3)*Hypergeometric2F1[5/3, 3, 8/3, 1 + (b*x^2)/a])/(10*a^3)","C",1
684,1,94,601,0.0533337,"\int x^4 \left(a+b x^2\right)^{2/3} \, dx","Integrate[x^4*(a + b*x^2)^(2/3),x]","\frac{3 x \left(a+b x^2\right)^{2/3} \left(\left(\frac{b x^2}{a}+1\right)^{2/3} \left(-9 a^2+4 a b x^2+13 b^2 x^4\right)+9 a^2 \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)\right)}{247 b^2 \left(\frac{b x^2}{a}+1\right)^{2/3}}","-\frac{108 \sqrt{2} 3^{3/4} a^{10/3} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}\right),4 \sqrt{3}-7\right)}{1729 b^3 x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}+\frac{162 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{10/3} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b x^2+a}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b x^2+a}}\right)|-7+4 \sqrt{3}\right)}{1729 b^3 x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}-\frac{324 a^3 x}{1729 b^2 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}-\frac{108 a^2 x \left(a+b x^2\right)^{2/3}}{1729 b^2}+\frac{3}{19} x^5 \left(a+b x^2\right)^{2/3}+\frac{12 a x^3 \left(a+b x^2\right)^{2/3}}{247 b}",1,"(3*x*(a + b*x^2)^(2/3)*((1 + (b*x^2)/a)^(2/3)*(-9*a^2 + 4*a*b*x^2 + 13*b^2*x^4) + 9*a^2*Hypergeometric2F1[-2/3, 1/2, 3/2, -((b*x^2)/a)]))/(247*b^2*(1 + (b*x^2)/a)^(2/3))","C",1
685,1,62,577,0.0523656,"\int x^2 \left(a+b x^2\right)^{2/3} \, dx","Integrate[x^2*(a + b*x^2)^(2/3),x]","\frac{3 x \left(a+b x^2\right)^{2/3} \left(-\frac{a \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)}{\left(\frac{b x^2}{a}+1\right)^{2/3}}+a+b x^2\right)}{13 b}","\frac{12 \sqrt{2} 3^{3/4} a^{7/3} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}\right),4 \sqrt{3}-7\right)}{91 b^2 x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}-\frac{18 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{7/3} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b x^2+a}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b x^2+a}}\right)|-7+4 \sqrt{3}\right)}{91 b^2 x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}+\frac{36 a^2 x}{91 b \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}+\frac{12 a x \left(a+b x^2\right)^{2/3}}{91 b}+\frac{3}{13} x^3 \left(a+b x^2\right)^{2/3}",1,"(3*x*(a + b*x^2)^(2/3)*(a + b*x^2 - (a*Hypergeometric2F1[-2/3, 1/2, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^(2/3)))/(13*b)","C",1
686,1,46,550,0.0060259,"\int \left(a+b x^2\right)^{2/3} \, dx","Integrate[(a + b*x^2)^(2/3),x]","\frac{x \left(a+b x^2\right)^{2/3} \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)}{\left(\frac{b x^2}{a}+1\right)^{2/3}}","-\frac{4 \sqrt{2} 3^{3/4} a^{4/3} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}\right),4 \sqrt{3}-7\right)}{7 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}+\frac{6 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{4/3} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b x^2+a}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b x^2+a}}\right)|-7+4 \sqrt{3}\right)}{7 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}-\frac{12 a x}{7 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}+\frac{3}{7} x \left(a+b x^2\right)^{2/3}",1,"(x*(a + b*x^2)^(2/3)*Hypergeometric2F1[-2/3, 1/2, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^(2/3)","C",1
687,1,49,538,0.0106088,"\int \frac{\left(a+b x^2\right)^{2/3}}{x^2} \, dx","Integrate[(a + b*x^2)^(2/3)/x^2,x]","-\frac{\left(a+b x^2\right)^{2/3} \, _2F_1\left(-\frac{2}{3},-\frac{1}{2};\frac{1}{2};-\frac{b x^2}{a}\right)}{x \left(\frac{b x^2}{a}+1\right)^{2/3}}","-\frac{4 \sqrt{2} \sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}\right),4 \sqrt{3}-7\right)}{\sqrt[4]{3} x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}+\frac{2 \sqrt[4]{3} \sqrt{2+\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b x^2+a}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b x^2+a}}\right)|-7+4 \sqrt{3}\right)}{x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}-\frac{4 b x}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}-\frac{\left(a+b x^2\right)^{2/3}}{x}",1,"-(((a + b*x^2)^(2/3)*Hypergeometric2F1[-2/3, -1/2, 1/2, -((b*x^2)/a)])/(x*(1 + (b*x^2)/a)^(2/3)))","C",1
688,1,51,575,0.0110737,"\int \frac{\left(a+b x^2\right)^{2/3}}{x^4} \, dx","Integrate[(a + b*x^2)^(2/3)/x^4,x]","-\frac{\left(a+b x^2\right)^{2/3} \, _2F_1\left(-\frac{3}{2},-\frac{2}{3};-\frac{1}{2};-\frac{b x^2}{a}\right)}{3 x^3 \left(\frac{b x^2}{a}+1\right)^{2/3}}","-\frac{4 \sqrt{2} b \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}\right),4 \sqrt{3}-7\right)}{9 \sqrt[4]{3} a^{2/3} x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}+\frac{2 \sqrt{2+\sqrt{3}} b \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b x^2+a}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b x^2+a}}\right)|-7+4 \sqrt{3}\right)}{3\ 3^{3/4} a^{2/3} x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}-\frac{4 b^2 x}{9 a \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}-\frac{4 b \left(a+b x^2\right)^{2/3}}{9 a x}-\frac{\left(a+b x^2\right)^{2/3}}{3 x^3}",1,"-1/3*((a + b*x^2)^(2/3)*Hypergeometric2F1[-3/2, -2/3, -1/2, -((b*x^2)/a)])/(x^3*(1 + (b*x^2)/a)^(2/3))","C",1
689,1,50,80,0.0266624,"\int x^7 \left(a+b x^2\right)^{4/3} \, dx","Integrate[x^7*(a + b*x^2)^(4/3),x]","\frac{3 \left(a+b x^2\right)^{7/3} \left(-81 a^3+189 a^2 b x^2-315 a b^2 x^4+455 b^3 x^6\right)}{14560 b^4}","-\frac{3 a^3 \left(a+b x^2\right)^{7/3}}{14 b^4}+\frac{9 a^2 \left(a+b x^2\right)^{10/3}}{20 b^4}+\frac{3 \left(a+b x^2\right)^{16/3}}{32 b^4}-\frac{9 a \left(a+b x^2\right)^{13/3}}{26 b^4}",1,"(3*(a + b*x^2)^(7/3)*(-81*a^3 + 189*a^2*b*x^2 - 315*a*b^2*x^4 + 455*b^3*x^6))/(14560*b^4)","A",1
690,1,39,59,0.0192524,"\int x^5 \left(a+b x^2\right)^{4/3} \, dx","Integrate[x^5*(a + b*x^2)^(4/3),x]","\frac{3 \left(a+b x^2\right)^{7/3} \left(9 a^2-21 a b x^2+35 b^2 x^4\right)}{910 b^3}","\frac{3 a^2 \left(a+b x^2\right)^{7/3}}{14 b^3}+\frac{3 \left(a+b x^2\right)^{13/3}}{26 b^3}-\frac{3 a \left(a+b x^2\right)^{10/3}}{10 b^3}",1,"(3*(a + b*x^2)^(7/3)*(9*a^2 - 21*a*b*x^2 + 35*b^2*x^4))/(910*b^3)","A",1
691,1,28,38,0.0149453,"\int x^3 \left(a+b x^2\right)^{4/3} \, dx","Integrate[x^3*(a + b*x^2)^(4/3),x]","\frac{3 \left(a+b x^2\right)^{7/3} \left(7 b x^2-3 a\right)}{140 b^2}","\frac{3 \left(a+b x^2\right)^{10/3}}{20 b^2}-\frac{3 a \left(a+b x^2\right)^{7/3}}{14 b^2}",1,"(3*(a + b*x^2)^(7/3)*(-3*a + 7*b*x^2))/(140*b^2)","A",1
692,1,18,18,0.0048527,"\int x \left(a+b x^2\right)^{4/3} \, dx","Integrate[x*(a + b*x^2)^(4/3),x]","\frac{3 \left(a+b x^2\right)^{7/3}}{14 b}","\frac{3 \left(a+b x^2\right)^{7/3}}{14 b}",1,"(3*(a + b*x^2)^(7/3))/(14*b)","A",1
693,1,144,117,0.0450035,"\int \frac{\left(a+b x^2\right)^{4/3}}{x} \, dx","Integrate[(a + b*x^2)^(4/3)/x,x]","\frac{1}{8} \left(4 a^{4/3} \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)-2 a^{4/3} \log \left(a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}\right)-4 \sqrt{3} a^{4/3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b x^2}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)+3 b x^2 \sqrt[3]{a+b x^2}+15 a \sqrt[3]{a+b x^2}\right)","\frac{3}{4} a^{4/3} \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)-\frac{1}{2} \sqrt{3} a^{4/3} \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^2}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)-\frac{1}{2} a^{4/3} \log (x)+\frac{3}{2} a \sqrt[3]{a+b x^2}+\frac{3}{8} \left(a+b x^2\right)^{4/3}",1,"(15*a*(a + b*x^2)^(1/3) + 3*b*x^2*(a + b*x^2)^(1/3) - 4*Sqrt[3]*a^(4/3)*ArcTan[(1 + (2*(a + b*x^2)^(1/3))/a^(1/3))/Sqrt[3]] + 4*a^(4/3)*Log[a^(1/3) - (a + b*x^2)^(1/3)] - 2*a^(4/3)*Log[a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3)])/8","A",1
694,1,37,116,0.0090187,"\int \frac{\left(a+b x^2\right)^{4/3}}{x^3} \, dx","Integrate[(a + b*x^2)^(4/3)/x^3,x]","\frac{3 b \left(a+b x^2\right)^{7/3} \, _2F_1\left(2,\frac{7}{3};\frac{10}{3};\frac{b x^2}{a}+1\right)}{14 a^2}","-\frac{\left(a+b x^2\right)^{4/3}}{2 x^2}+2 b \sqrt[3]{a+b x^2}+\sqrt[3]{a} b \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)-\frac{2 \sqrt[3]{a} b \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^2}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3}}-\frac{2}{3} \sqrt[3]{a} b \log (x)",1,"(3*b*(a + b*x^2)^(7/3)*Hypergeometric2F1[2, 7/3, 10/3, 1 + (b*x^2)/a])/(14*a^2)","C",1
695,1,39,132,0.0097001,"\int \frac{\left(a+b x^2\right)^{4/3}}{x^5} \, dx","Integrate[(a + b*x^2)^(4/3)/x^5,x]","-\frac{3 b^2 \left(a+b x^2\right)^{7/3} \, _2F_1\left(\frac{7}{3},3;\frac{10}{3};\frac{b x^2}{a}+1\right)}{14 a^3}","\frac{b^2 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{6 a^{2/3}}-\frac{b^2 \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^2}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{2/3}}-\frac{b^2 \log (x)}{9 a^{2/3}}-\frac{b \sqrt[3]{a+b x^2}}{3 x^2}-\frac{\left(a+b x^2\right)^{4/3}}{4 x^4}",1,"(-3*b^2*(a + b*x^2)^(7/3)*Hypergeometric2F1[7/3, 3, 10/3, 1 + (b*x^2)/a])/(14*a^3)","C",1
696,1,79,335,0.0777618,"\int x^4 \left(a+b x^2\right)^{4/3} \, dx","Integrate[x^4*(a + b*x^2)^(4/3),x]","\frac{3 x \sqrt[3]{a+b x^2} \left(\frac{9 a^3 \, _2F_1\left(-\frac{4}{3},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)}{\sqrt[3]{\frac{b x^2}{a}+1}}-\left(9 a-17 b x^2\right) \left(a+b x^2\right)^2\right)}{391 b^2}","-\frac{432\ 3^{3/4} \sqrt{2-\sqrt{3}} a^4 \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}\right),4 \sqrt{3}-7\right)}{21505 b^3 x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}-\frac{432 a^3 x \sqrt[3]{a+b x^2}}{21505 b^2}+\frac{48 a^2 x^3 \sqrt[3]{a+b x^2}}{4301 b}+\frac{3}{23} x^5 \left(a+b x^2\right)^{4/3}+\frac{24}{391} a x^5 \sqrt[3]{a+b x^2}",1,"(3*x*(a + b*x^2)^(1/3)*(-((9*a - 17*b*x^2)*(a + b*x^2)^2) + (9*a^3*Hypergeometric2F1[-4/3, 1/2, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^(1/3)))/(391*b^2)","C",1
697,1,67,311,0.0625474,"\int x^2 \left(a+b x^2\right)^{4/3} \, dx","Integrate[x^2*(a + b*x^2)^(4/3),x]","\frac{3 x \sqrt[3]{a+b x^2} \left(\left(a+b x^2\right)^2-\frac{a^2 \, _2F_1\left(-\frac{4}{3},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)}{\sqrt[3]{\frac{b x^2}{a}+1}}\right)}{17 b}","\frac{48\ 3^{3/4} \sqrt{2-\sqrt{3}} a^3 \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}\right),4 \sqrt{3}-7\right)}{935 b^2 x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}+\frac{48 a^2 x \sqrt[3]{a+b x^2}}{935 b}+\frac{24}{187} a x^3 \sqrt[3]{a+b x^2}+\frac{3}{17} x^3 \left(a+b x^2\right)^{4/3}",1,"(3*x*(a + b*x^2)^(1/3)*((a + b*x^2)^2 - (a^2*Hypergeometric2F1[-4/3, 1/2, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^(1/3)))/(17*b)","C",1
698,1,47,285,0.0066967,"\int \left(a+b x^2\right)^{4/3} \, dx","Integrate[(a + b*x^2)^(4/3),x]","\frac{a x \sqrt[3]{a+b x^2} \, _2F_1\left(-\frac{4}{3},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)}{\sqrt[3]{\frac{b x^2}{a}+1}}","-\frac{16\ 3^{3/4} \sqrt{2-\sqrt{3}} a^2 \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}\right),4 \sqrt{3}-7\right)}{55 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}+\frac{24}{55} a x \sqrt[3]{a+b x^2}+\frac{3}{11} x \left(a+b x^2\right)^{4/3}",1,"(a*x*(a + b*x^2)^(1/3)*Hypergeometric2F1[-4/3, 1/2, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^(1/3)","C",1
699,1,50,280,0.0096051,"\int \frac{\left(a+b x^2\right)^{4/3}}{x^2} \, dx","Integrate[(a + b*x^2)^(4/3)/x^2,x]","-\frac{a \sqrt[3]{a+b x^2} \, _2F_1\left(-\frac{4}{3},-\frac{1}{2};\frac{1}{2};-\frac{b x^2}{a}\right)}{x \sqrt[3]{\frac{b x^2}{a}+1}}","-\frac{16 \sqrt{2-\sqrt{3}} a \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}\right),4 \sqrt{3}-7\right)}{5 \sqrt[4]{3} x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}-\frac{\left(a+b x^2\right)^{4/3}}{x}+\frac{8}{5} b x \sqrt[3]{a+b x^2}",1,"-((a*(a + b*x^2)^(1/3)*Hypergeometric2F1[-4/3, -1/2, 1/2, -((b*x^2)/a)])/(x*(1 + (b*x^2)/a)^(1/3)))","C",1
700,1,52,284,0.0097313,"\int \frac{\left(a+b x^2\right)^{4/3}}{x^4} \, dx","Integrate[(a + b*x^2)^(4/3)/x^4,x]","-\frac{a \sqrt[3]{a+b x^2} \, _2F_1\left(-\frac{3}{2},-\frac{4}{3};-\frac{1}{2};-\frac{b x^2}{a}\right)}{3 x^3 \sqrt[3]{\frac{b x^2}{a}+1}}","-\frac{16 \sqrt{2-\sqrt{3}} b \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}\right),4 \sqrt{3}-7\right)}{9 \sqrt[4]{3} x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}-\frac{8 b \sqrt[3]{a+b x^2}}{9 x}-\frac{\left(a+b x^2\right)^{4/3}}{3 x^3}",1,"-1/3*(a*(a + b*x^2)^(1/3)*Hypergeometric2F1[-3/2, -4/3, -1/2, -((b*x^2)/a)])/(x^3*(1 + (b*x^2)/a)^(1/3))","C",1
701,1,13,13,0.0047812,"\int x \left(-1+x^2\right)^{7/3} \, dx","Integrate[x*(-1 + x^2)^(7/3),x]","\frac{3}{20} \left(x^2-1\right)^{10/3}","\frac{3}{20} \left(x^2-1\right)^{10/3}",1,"(3*(-1 + x^2)^(10/3))/20","A",1
702,1,50,80,0.0290571,"\int \frac{x^7}{\sqrt[3]{a+b x^2}} \, dx","Integrate[x^7/(a + b*x^2)^(1/3),x]","\frac{3 \left(a+b x^2\right)^{2/3} \left(-81 a^3+54 a^2 b x^2-45 a b^2 x^4+40 b^3 x^6\right)}{880 b^4}","-\frac{3 a^3 \left(a+b x^2\right)^{2/3}}{4 b^4}+\frac{9 a^2 \left(a+b x^2\right)^{5/3}}{10 b^4}+\frac{3 \left(a+b x^2\right)^{11/3}}{22 b^4}-\frac{9 a \left(a+b x^2\right)^{8/3}}{16 b^4}",1,"(3*(a + b*x^2)^(2/3)*(-81*a^3 + 54*a^2*b*x^2 - 45*a*b^2*x^4 + 40*b^3*x^6))/(880*b^4)","A",1
703,1,39,59,0.0209584,"\int \frac{x^5}{\sqrt[3]{a+b x^2}} \, dx","Integrate[x^5/(a + b*x^2)^(1/3),x]","\frac{3 \left(a+b x^2\right)^{2/3} \left(9 a^2-6 a b x^2+5 b^2 x^4\right)}{80 b^3}","\frac{3 a^2 \left(a+b x^2\right)^{2/3}}{4 b^3}+\frac{3 \left(a+b x^2\right)^{8/3}}{16 b^3}-\frac{3 a \left(a+b x^2\right)^{5/3}}{5 b^3}",1,"(3*(a + b*x^2)^(2/3)*(9*a^2 - 6*a*b*x^2 + 5*b^2*x^4))/(80*b^3)","A",1
704,1,28,38,0.0127812,"\int \frac{x^3}{\sqrt[3]{a+b x^2}} \, dx","Integrate[x^3/(a + b*x^2)^(1/3),x]","\frac{3 \left(a+b x^2\right)^{2/3} \left(2 b x^2-3 a\right)}{20 b^2}","\frac{3 \left(a+b x^2\right)^{5/3}}{10 b^2}-\frac{3 a \left(a+b x^2\right)^{2/3}}{4 b^2}",1,"(3*(a + b*x^2)^(2/3)*(-3*a + 2*b*x^2))/(20*b^2)","A",1
705,1,18,18,0.0026754,"\int \frac{x}{\sqrt[3]{a+b x^2}} \, dx","Integrate[x/(a + b*x^2)^(1/3),x]","\frac{3 \left(a+b x^2\right)^{2/3}}{4 b}","\frac{3 \left(a+b x^2\right)^{2/3}}{4 b}",1,"(3*(a + b*x^2)^(2/3))/(4*b)","A",1
706,1,70,86,0.0302221,"\int \frac{1}{x \sqrt[3]{a+b x^2}} \, dx","Integrate[1/(x*(a + b*x^2)^(1/3)),x]","\frac{3 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b x^2}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)-2 \log (x)}{4 \sqrt[3]{a}}","\frac{3 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{4 \sqrt[3]{a}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^2}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{2 \sqrt[3]{a}}-\frac{\log (x)}{2 \sqrt[3]{a}}",1,"(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*x^2)^(1/3))/a^(1/3))/Sqrt[3]] - 2*Log[x] + 3*Log[a^(1/3) - (a + b*x^2)^(1/3)])/(4*a^(1/3))","A",1
707,1,37,110,0.0071094,"\int \frac{1}{x^3 \sqrt[3]{a+b x^2}} \, dx","Integrate[1/(x^3*(a + b*x^2)^(1/3)),x]","\frac{3 b \left(a+b x^2\right)^{2/3} \, _2F_1\left(\frac{2}{3},2;\frac{5}{3};\frac{b x^2}{a}+1\right)}{4 a^2}","-\frac{b \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{4 a^{4/3}}-\frac{b \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^2}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{2 \sqrt{3} a^{4/3}}+\frac{b \log (x)}{6 a^{4/3}}-\frac{\left(a+b x^2\right)^{2/3}}{2 a x^2}",1,"(3*b*(a + b*x^2)^(2/3)*Hypergeometric2F1[2/3, 2, 5/3, 1 + (b*x^2)/a])/(4*a^2)","C",1
708,1,39,138,0.0077372,"\int \frac{1}{x^5 \sqrt[3]{a+b x^2}} \, dx","Integrate[1/(x^5*(a + b*x^2)^(1/3)),x]","-\frac{3 b^2 \left(a+b x^2\right)^{2/3} \, _2F_1\left(\frac{2}{3},3;\frac{5}{3};\frac{b x^2}{a}+1\right)}{4 a^3}","\frac{b^2 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{6 a^{7/3}}+\frac{b^2 \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^2}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{7/3}}-\frac{b^2 \log (x)}{9 a^{7/3}}+\frac{b \left(a+b x^2\right)^{2/3}}{3 a^2 x^2}-\frac{\left(a+b x^2\right)^{2/3}}{4 a x^4}",1,"(-3*b^2*(a + b*x^2)^(2/3)*Hypergeometric2F1[2/3, 3, 5/3, 1 + (b*x^2)/a])/(4*a^3)","C",1
709,1,79,580,0.0255743,"\int \frac{x^4}{\sqrt[3]{a+b x^2}} \, dx","Integrate[x^4/(a + b*x^2)^(1/3),x]","\frac{3 \left(9 a^2 x \sqrt[3]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)-9 a^2 x-2 a b x^3+7 b^2 x^5\right)}{91 b^2 \sqrt[3]{a+b x^2}}","-\frac{27 \sqrt{2} 3^{3/4} a^{7/3} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}\right),4 \sqrt{3}-7\right)}{91 b^3 x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}+\frac{81 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{7/3} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b x^2+a}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b x^2+a}}\right)|-7+4 \sqrt{3}\right)}{182 b^3 x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}-\frac{81 a^2 x}{91 b^2 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}-\frac{27 a x \left(a+b x^2\right)^{2/3}}{91 b^2}+\frac{3 x^3 \left(a+b x^2\right)^{2/3}}{13 b}",1,"(3*(-9*a^2*x - 2*a*b*x^3 + 7*b^2*x^5 + 9*a^2*x*(1 + (b*x^2)/a)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, -((b*x^2)/a)]))/(91*b^2*(a + b*x^2)^(1/3))","C",1
710,1,62,556,0.0185978,"\int \frac{x^2}{\sqrt[3]{a+b x^2}} \, dx","Integrate[x^2/(a + b*x^2)^(1/3),x]","\frac{3 x \left(-a \sqrt[3]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)+a+b x^2\right)}{7 b \sqrt[3]{a+b x^2}}","\frac{3 \sqrt{2} 3^{3/4} a^{4/3} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}\right),4 \sqrt{3}-7\right)}{7 b^2 x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}-\frac{9 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{4/3} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b x^2+a}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b x^2+a}}\right)|-7+4 \sqrt{3}\right)}{14 b^2 x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}+\frac{9 a x}{7 b \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}+\frac{3 x \left(a+b x^2\right)^{2/3}}{7 b}",1,"(3*x*(a + b*x^2 - a*(1 + (b*x^2)/a)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, -((b*x^2)/a)]))/(7*b*(a + b*x^2)^(1/3))","C",1
711,1,46,529,0.0066935,"\int \frac{1}{\sqrt[3]{a+b x^2}} \, dx","Integrate[(a + b*x^2)^(-1/3),x]","\frac{x \sqrt[3]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)}{\sqrt[3]{a+b x^2}}","-\frac{\sqrt{2} 3^{3/4} \sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}\right),4 \sqrt{3}-7\right)}{b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}+\frac{3 \sqrt[4]{3} \sqrt{2+\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b x^2+a}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b x^2+a}}\right)|-7+4 \sqrt{3}\right)}{2 b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}-\frac{3 x}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}",1,"(x*(1 + (b*x^2)/a)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, -((b*x^2)/a)])/(a + b*x^2)^(1/3)","C",1
712,1,49,546,0.0092813,"\int \frac{1}{x^2 \sqrt[3]{a+b x^2}} \, dx","Integrate[1/(x^2*(a + b*x^2)^(1/3)),x]","-\frac{\sqrt[3]{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{1}{2},\frac{1}{3};\frac{1}{2};-\frac{b x^2}{a}\right)}{x \sqrt[3]{a+b x^2}}","-\frac{\sqrt{2} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}\right),4 \sqrt{3}-7\right)}{\sqrt[4]{3} a^{2/3} x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}+\frac{\sqrt[4]{3} \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b x^2+a}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b x^2+a}}\right)|-7+4 \sqrt{3}\right)}{2 a^{2/3} x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}-\frac{b x}{a \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}-\frac{\left(a+b x^2\right)^{2/3}}{a x}",1,"-(((1 + (b*x^2)/a)^(1/3)*Hypergeometric2F1[-1/2, 1/3, 1/2, -((b*x^2)/a)])/(x*(a + b*x^2)^(1/3)))","C",1
713,1,51,578,0.0091914,"\int \frac{1}{x^4 \sqrt[3]{a+b x^2}} \, dx","Integrate[1/(x^4*(a + b*x^2)^(1/3)),x]","-\frac{\sqrt[3]{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{3}{2},\frac{1}{3};-\frac{1}{2};-\frac{b x^2}{a}\right)}{3 x^3 \sqrt[3]{a+b x^2}}","\frac{5 \sqrt{2} b \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}\right),4 \sqrt{3}-7\right)}{9 \sqrt[4]{3} a^{5/3} x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}-\frac{5 \sqrt{2+\sqrt{3}} b \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b x^2+a}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b x^2+a}}\right)|-7+4 \sqrt{3}\right)}{6\ 3^{3/4} a^{5/3} x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}+\frac{5 b^2 x}{9 a^2 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}+\frac{5 b \left(a+b x^2\right)^{2/3}}{9 a^2 x}-\frac{\left(a+b x^2\right)^{2/3}}{3 a x^3}",1,"-1/3*((1 + (b*x^2)/a)^(1/3)*Hypergeometric2F1[-3/2, 1/3, -1/2, -((b*x^2)/a)])/(x^3*(a + b*x^2)^(1/3))","C",1
714,1,50,80,0.024003,"\int \frac{x^7}{\left(a+b x^2\right)^{2/3}} \, dx","Integrate[x^7/(a + b*x^2)^(2/3),x]","\frac{3 \sqrt[3]{a+b x^2} \left(-81 a^3+27 a^2 b x^2-18 a b^2 x^4+14 b^3 x^6\right)}{280 b^4}","-\frac{3 a^3 \sqrt[3]{a+b x^2}}{2 b^4}+\frac{9 a^2 \left(a+b x^2\right)^{4/3}}{8 b^4}+\frac{3 \left(a+b x^2\right)^{10/3}}{20 b^4}-\frac{9 a \left(a+b x^2\right)^{7/3}}{14 b^4}",1,"(3*(a + b*x^2)^(1/3)*(-81*a^3 + 27*a^2*b*x^2 - 18*a*b^2*x^4 + 14*b^3*x^6))/(280*b^4)","A",1
715,1,39,59,0.0189843,"\int \frac{x^5}{\left(a+b x^2\right)^{2/3}} \, dx","Integrate[x^5/(a + b*x^2)^(2/3),x]","\frac{3 \sqrt[3]{a+b x^2} \left(9 a^2-3 a b x^2+2 b^2 x^4\right)}{28 b^3}","\frac{3 a^2 \sqrt[3]{a+b x^2}}{2 b^3}+\frac{3 \left(a+b x^2\right)^{7/3}}{14 b^3}-\frac{3 a \left(a+b x^2\right)^{4/3}}{4 b^3}",1,"(3*(a + b*x^2)^(1/3)*(9*a^2 - 3*a*b*x^2 + 2*b^2*x^4))/(28*b^3)","A",1
716,1,27,38,0.0135958,"\int \frac{x^3}{\left(a+b x^2\right)^{2/3}} \, dx","Integrate[x^3/(a + b*x^2)^(2/3),x]","\frac{3 \left(b x^2-3 a\right) \sqrt[3]{a+b x^2}}{8 b^2}","\frac{3 \left(a+b x^2\right)^{4/3}}{8 b^2}-\frac{3 a \sqrt[3]{a+b x^2}}{2 b^2}",1,"(3*(-3*a + b*x^2)*(a + b*x^2)^(1/3))/(8*b^2)","A",1
717,1,18,18,0.0027001,"\int \frac{x}{\left(a+b x^2\right)^{2/3}} \, dx","Integrate[x/(a + b*x^2)^(2/3),x]","\frac{3 \sqrt[3]{a+b x^2}}{2 b}","\frac{3 \sqrt[3]{a+b x^2}}{2 b}",1,"(3*(a + b*x^2)^(1/3))/(2*b)","A",1
718,1,101,86,0.0312307,"\int \frac{1}{x \left(a+b x^2\right)^{2/3}} \, dx","Integrate[1/(x*(a + b*x^2)^(2/3)),x]","-\frac{\log \left(a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}\right)-2 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b x^2}}{\sqrt[3]{a}}+1}{\sqrt{3}}\right)}{4 a^{2/3}}","\frac{3 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{4 a^{2/3}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^2}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{2 a^{2/3}}-\frac{\log (x)}{2 a^{2/3}}",1,"-1/4*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*x^2)^(1/3))/a^(1/3))/Sqrt[3]] - 2*Log[a^(1/3) - (a + b*x^2)^(1/3)] + Log[a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3)])/a^(2/3)","A",1
719,1,37,107,0.0076773,"\int \frac{1}{x^3 \left(a+b x^2\right)^{2/3}} \, dx","Integrate[1/(x^3*(a + b*x^2)^(2/3)),x]","\frac{3 b \sqrt[3]{a+b x^2} \, _2F_1\left(\frac{1}{3},2;\frac{4}{3};\frac{b x^2}{a}+1\right)}{2 a^2}","-\frac{b \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{2 a^{5/3}}+\frac{b \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^2}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{5/3}}+\frac{b \log (x)}{3 a^{5/3}}-\frac{\sqrt[3]{a+b x^2}}{2 a x^2}",1,"(3*b*(a + b*x^2)^(1/3)*Hypergeometric2F1[1/3, 2, 4/3, 1 + (b*x^2)/a])/(2*a^2)","C",1
720,1,39,138,0.0075303,"\int \frac{1}{x^5 \left(a+b x^2\right)^{2/3}} \, dx","Integrate[1/(x^5*(a + b*x^2)^(2/3)),x]","-\frac{3 b^2 \sqrt[3]{a+b x^2} \, _2F_1\left(\frac{1}{3},3;\frac{4}{3};\frac{b x^2}{a}+1\right)}{2 a^3}","\frac{5 b^2 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{12 a^{8/3}}-\frac{5 b^2 \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^2}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{6 \sqrt{3} a^{8/3}}-\frac{5 b^2 \log (x)}{18 a^{8/3}}+\frac{5 b \sqrt[3]{a+b x^2}}{12 a^2 x^2}-\frac{\sqrt[3]{a+b x^2}}{4 a x^4}",1,"(-3*b^2*(a + b*x^2)^(1/3)*Hypergeometric2F1[1/3, 3, 4/3, 1 + (b*x^2)/a])/(2*a^3)","C",1
721,1,79,293,0.0268812,"\int \frac{x^4}{\left(a+b x^2\right)^{2/3}} \, dx","Integrate[x^4/(a + b*x^2)^(2/3),x]","\frac{3 \left(9 a^2 x \left(\frac{b x^2}{a}+1\right)^{2/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{3}{2};-\frac{b x^2}{a}\right)-9 a^2 x-4 a b x^3+5 b^2 x^5\right)}{55 b^2 \left(a+b x^2\right)^{2/3}}","-\frac{27\ 3^{3/4} \sqrt{2-\sqrt{3}} a^2 \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}\right),4 \sqrt{3}-7\right)}{55 b^3 x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}-\frac{27 a x \sqrt[3]{a+b x^2}}{55 b^2}+\frac{3 x^3 \sqrt[3]{a+b x^2}}{11 b}",1,"(3*(-9*a^2*x - 4*a*b*x^3 + 5*b^2*x^5 + 9*a^2*x*(1 + (b*x^2)/a)^(2/3)*Hypergeometric2F1[1/2, 2/3, 3/2, -((b*x^2)/a)]))/(55*b^2*(a + b*x^2)^(2/3))","C",1
722,1,62,269,0.0215461,"\int \frac{x^2}{\left(a+b x^2\right)^{2/3}} \, dx","Integrate[x^2/(a + b*x^2)^(2/3),x]","\frac{3 x \left(-a \left(\frac{b x^2}{a}+1\right)^{2/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{3}{2};-\frac{b x^2}{a}\right)+a+b x^2\right)}{5 b \left(a+b x^2\right)^{2/3}}","\frac{3\ 3^{3/4} \sqrt{2-\sqrt{3}} a \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}\right),4 \sqrt{3}-7\right)}{5 b^2 x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}+\frac{3 x \sqrt[3]{a+b x^2}}{5 b}",1,"(3*x*(a + b*x^2 - a*(1 + (b*x^2)/a)^(2/3)*Hypergeometric2F1[1/2, 2/3, 3/2, -((b*x^2)/a)]))/(5*b*(a + b*x^2)^(2/3))","C",1
723,1,46,246,0.0083733,"\int \frac{1}{\left(a+b x^2\right)^{2/3}} \, dx","Integrate[(a + b*x^2)^(-2/3),x]","\frac{x \left(\frac{b x^2}{a}+1\right)^{2/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{3}{2};-\frac{b x^2}{a}\right)}{\left(a+b x^2\right)^{2/3}}","-\frac{3^{3/4} \sqrt{2-\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}\right),4 \sqrt{3}-7\right)}{b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}",1,"(x*(1 + (b*x^2)/a)^(2/3)*Hypergeometric2F1[1/2, 2/3, 3/2, -((b*x^2)/a)])/(a + b*x^2)^(2/3)","C",1
724,1,49,265,0.009328,"\int \frac{1}{x^2 \left(a+b x^2\right)^{2/3}} \, dx","Integrate[1/(x^2*(a + b*x^2)^(2/3)),x]","-\frac{\left(\frac{b x^2}{a}+1\right)^{2/3} \, _2F_1\left(-\frac{1}{2},\frac{2}{3};\frac{1}{2};-\frac{b x^2}{a}\right)}{x \left(a+b x^2\right)^{2/3}}","\frac{\sqrt{2-\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}\right),4 \sqrt{3}-7\right)}{\sqrt[4]{3} a x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}-\frac{\sqrt[3]{a+b x^2}}{a x}",1,"-(((1 + (b*x^2)/a)^(2/3)*Hypergeometric2F1[-1/2, 2/3, 1/2, -((b*x^2)/a)])/(x*(a + b*x^2)^(2/3)))","C",1
725,1,51,293,0.0096873,"\int \frac{1}{x^4 \left(a+b x^2\right)^{2/3}} \, dx","Integrate[1/(x^4*(a + b*x^2)^(2/3)),x]","-\frac{\left(\frac{b x^2}{a}+1\right)^{2/3} \, _2F_1\left(-\frac{3}{2},\frac{2}{3};-\frac{1}{2};-\frac{b x^2}{a}\right)}{3 x^3 \left(a+b x^2\right)^{2/3}}","-\frac{7 \sqrt{2-\sqrt{3}} b \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}\right),4 \sqrt{3}-7\right)}{9 \sqrt[4]{3} a^2 x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}+\frac{7 b \sqrt[3]{a+b x^2}}{9 a^2 x}-\frac{\sqrt[3]{a+b x^2}}{3 a x^3}",1,"-1/3*((1 + (b*x^2)/a)^(2/3)*Hypergeometric2F1[-3/2, 2/3, -1/2, -((b*x^2)/a)])/(x^3*(a + b*x^2)^(2/3))","C",1
726,1,50,80,0.0232596,"\int \frac{x^7}{\left(a+b x^2\right)^{4/3}} \, dx","Integrate[x^7/(a + b*x^2)^(4/3),x]","\frac{3 \left(81 a^3+27 a^2 b x^2-9 a b^2 x^4+5 b^3 x^6\right)}{80 b^4 \sqrt[3]{a+b x^2}}","\frac{3 a^3}{2 b^4 \sqrt[3]{a+b x^2}}+\frac{9 a^2 \left(a+b x^2\right)^{2/3}}{4 b^4}-\frac{9 a \left(a+b x^2\right)^{5/3}}{10 b^4}+\frac{3 \left(a+b x^2\right)^{8/3}}{16 b^4}",1,"(3*(81*a^3 + 27*a^2*b*x^2 - 9*a*b^2*x^4 + 5*b^3*x^6))/(80*b^4*(a + b*x^2)^(1/3))","A",1
727,1,38,59,0.017085,"\int \frac{x^5}{\left(a+b x^2\right)^{4/3}} \, dx","Integrate[x^5/(a + b*x^2)^(4/3),x]","\frac{3 \left(-9 a^2-3 a b x^2+b^2 x^4\right)}{10 b^3 \sqrt[3]{a+b x^2}}","-\frac{3 a^2}{2 b^3 \sqrt[3]{a+b x^2}}-\frac{3 a \left(a+b x^2\right)^{2/3}}{2 b^3}+\frac{3 \left(a+b x^2\right)^{5/3}}{10 b^3}",1,"(3*(-9*a^2 - 3*a*b*x^2 + b^2*x^4))/(10*b^3*(a + b*x^2)^(1/3))","A",1
728,1,27,38,0.0122414,"\int \frac{x^3}{\left(a+b x^2\right)^{4/3}} \, dx","Integrate[x^3/(a + b*x^2)^(4/3),x]","\frac{3 \left(3 a+b x^2\right)}{4 b^2 \sqrt[3]{a+b x^2}}","\frac{3 a}{2 b^2 \sqrt[3]{a+b x^2}}+\frac{3 \left(a+b x^2\right)^{2/3}}{4 b^2}",1,"(3*(3*a + b*x^2))/(4*b^2*(a + b*x^2)^(1/3))","A",1
729,1,18,18,0.0030083,"\int \frac{x}{\left(a+b x^2\right)^{4/3}} \, dx","Integrate[x/(a + b*x^2)^(4/3),x]","-\frac{3}{2 b \sqrt[3]{a+b x^2}}","-\frac{3}{2 b \sqrt[3]{a+b x^2}}",1,"-3/(2*b*(a + b*x^2)^(1/3))","A",1
730,1,36,104,0.0068544,"\int \frac{1}{x \left(a+b x^2\right)^{4/3}} \, dx","Integrate[1/(x*(a + b*x^2)^(4/3)),x]","\frac{3 \, _2F_1\left(-\frac{1}{3},1;\frac{2}{3};\frac{b x^2}{a}+1\right)}{2 a \sqrt[3]{a+b x^2}}","\frac{3 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{4 a^{4/3}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^2}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{2 a^{4/3}}-\frac{\log (x)}{2 a^{4/3}}+\frac{3}{2 a \sqrt[3]{a+b x^2}}",1,"(3*Hypergeometric2F1[-1/3, 1, 2/3, 1 + (b*x^2)/a])/(2*a*(a + b*x^2)^(1/3))","C",1
731,1,37,123,0.0080809,"\int \frac{1}{x^3 \left(a+b x^2\right)^{4/3}} \, dx","Integrate[1/(x^3*(a + b*x^2)^(4/3)),x]","-\frac{3 b \, _2F_1\left(-\frac{1}{3},2;\frac{2}{3};\frac{b x^2}{a}+1\right)}{2 a^2 \sqrt[3]{a+b x^2}}","-\frac{b \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{a^{7/3}}-\frac{2 b \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^2}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{7/3}}+\frac{2 b \log (x)}{3 a^{7/3}}-\frac{2 b}{a^2 \sqrt[3]{a+b x^2}}-\frac{1}{2 a x^2 \sqrt[3]{a+b x^2}}",1,"(-3*b*Hypergeometric2F1[-1/3, 2, 2/3, 1 + (b*x^2)/a])/(2*a^2*(a + b*x^2)^(1/3))","C",1
732,1,39,159,0.0077616,"\int \frac{1}{x^5 \left(a+b x^2\right)^{4/3}} \, dx","Integrate[1/(x^5*(a + b*x^2)^(4/3)),x]","\frac{3 b^2 \, _2F_1\left(-\frac{1}{3},3;\frac{2}{3};\frac{b x^2}{a}+1\right)}{2 a^3 \sqrt[3]{a+b x^2}}","\frac{7 b^2 \log \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{6 a^{10/3}}+\frac{7 b^2 \tan ^{-1}\left(\frac{2 \sqrt[3]{a+b x^2}+\sqrt[3]{a}}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{10/3}}-\frac{7 b^2 \log (x)}{9 a^{10/3}}+\frac{7 b^2}{3 a^3 \sqrt[3]{a+b x^2}}+\frac{7 b}{12 a^2 x^2 \sqrt[3]{a+b x^2}}-\frac{1}{4 a x^4 \sqrt[3]{a+b x^2}}",1,"(3*b^2*Hypergeometric2F1[-1/3, 3, 2/3, 1 + (b*x^2)/a])/(2*a^3*(a + b*x^2)^(1/3))","C",1
733,1,65,577,0.0251512,"\int \frac{x^4}{\left(a+b x^2\right)^{4/3}} \, dx","Integrate[x^4/(a + b*x^2)^(4/3),x]","\frac{3 x \left(-9 a \sqrt[3]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)+9 a+2 b x^2\right)}{14 b^2 \sqrt[3]{a+b x^2}}","\frac{27\ 3^{3/4} a^{4/3} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}\right),4 \sqrt{3}-7\right)}{7 \sqrt{2} b^3 x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}-\frac{81 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^{4/3} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b x^2+a}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b x^2+a}}\right)|-7+4 \sqrt{3}\right)}{28 b^3 x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}+\frac{27 x \left(a+b x^2\right)^{2/3}}{14 b^2}+\frac{81 a x}{14 b^2 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}-\frac{3 x^3}{2 b \sqrt[3]{a+b x^2}}",1,"(3*x*(9*a + 2*b*x^2 - 9*a*(1 + (b*x^2)/a)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, -((b*x^2)/a)]))/(14*b^2*(a + b*x^2)^(1/3))","C",1
734,1,55,553,0.018545,"\int \frac{x^2}{\left(a+b x^2\right)^{4/3}} \, dx","Integrate[x^2/(a + b*x^2)^(4/3),x]","\frac{3 x \left(\sqrt[3]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)-1\right)}{2 b \sqrt[3]{a+b x^2}}","-\frac{3\ 3^{3/4} \sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}\right),4 \sqrt{3}-7\right)}{\sqrt{2} b^2 x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}+\frac{9 \sqrt[4]{3} \sqrt{2+\sqrt{3}} \sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b x^2+a}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b x^2+a}}\right)|-7+4 \sqrt{3}\right)}{4 b^2 x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}-\frac{3 x}{2 b \sqrt[3]{a+b x^2}}-\frac{9 x}{2 b \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}",1,"(3*x*(-1 + (1 + (b*x^2)/a)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, -((b*x^2)/a)]))/(2*b*(a + b*x^2)^(1/3))","C",1
735,1,58,552,0.0132662,"\int \frac{1}{\left(a+b x^2\right)^{4/3}} \, dx","Integrate[(a + b*x^2)^(-4/3),x]","\frac{3 x-x \sqrt[3]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)}{2 a \sqrt[3]{a+b x^2}}","\frac{3^{3/4} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}\right),4 \sqrt{3}-7\right)}{\sqrt{2} a^{2/3} b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}-\frac{3 \sqrt[4]{3} \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b x^2+a}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b x^2+a}}\right)|-7+4 \sqrt{3}\right)}{4 a^{2/3} b x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}+\frac{3 x}{2 a \sqrt[3]{a+b x^2}}+\frac{3 x}{2 a \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}",1,"(3*x - x*(1 + (b*x^2)/a)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, -((b*x^2)/a)])/(2*a*(a + b*x^2)^(1/3))","C",1
736,1,52,571,0.0101522,"\int \frac{1}{x^2 \left(a+b x^2\right)^{4/3}} \, dx","Integrate[1/(x^2*(a + b*x^2)^(4/3)),x]","-\frac{\sqrt[3]{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{1}{2},\frac{4}{3};\frac{1}{2};-\frac{b x^2}{a}\right)}{a x \sqrt[3]{a+b x^2}}","-\frac{5 \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}\right),4 \sqrt{3}-7\right)}{\sqrt{2} \sqrt[4]{3} a^{5/3} x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}+\frac{5 \sqrt[4]{3} \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b x^2+a}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b x^2+a}}\right)|-7+4 \sqrt{3}\right)}{4 a^{5/3} x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}-\frac{5 b x}{2 a^2 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}-\frac{5 \left(a+b x^2\right)^{2/3}}{2 a^2 x}+\frac{3}{2 a x \sqrt[3]{a+b x^2}}",1,"-(((1 + (b*x^2)/a)^(1/3)*Hypergeometric2F1[-1/2, 4/3, 1/2, -((b*x^2)/a)])/(a*x*(a + b*x^2)^(1/3)))","C",1
737,1,54,599,0.009912,"\int \frac{1}{x^4 \left(a+b x^2\right)^{4/3}} \, dx","Integrate[1/(x^4*(a + b*x^2)^(4/3)),x]","-\frac{\sqrt[3]{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{3}{2},\frac{4}{3};-\frac{1}{2};-\frac{b x^2}{a}\right)}{3 a x^3 \sqrt[3]{a+b x^2}}","\frac{55 b \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}}\right),4 \sqrt{3}-7\right)}{9 \sqrt{2} \sqrt[4]{3} a^{8/3} x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}-\frac{55 \sqrt{2+\sqrt{3}} b \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{a+b x^2}+\left(a+b x^2\right)^{2/3}}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b x^2+a}}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b x^2+a}}\right)|-7+4 \sqrt{3}\right)}{12\ 3^{3/4} a^{8/3} x \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)^2}}}+\frac{55 b^2 x}{18 a^3 \left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{a+b x^2}\right)}+\frac{55 b \left(a+b x^2\right)^{2/3}}{18 a^3 x}-\frac{11 \left(a+b x^2\right)^{2/3}}{6 a^2 x^3}+\frac{3}{2 a x^3 \sqrt[3]{a+b x^2}}",1,"-1/3*((1 + (b*x^2)/a)^(1/3)*Hypergeometric2F1[-3/2, 4/3, -1/2, -((b*x^2)/a)])/(a*x^3*(a + b*x^2)^(1/3))","C",1
738,1,102,195,0.0735659,"\int (c x)^{13/3} \sqrt[3]{a+b x^2} \, dx","Integrate[(c*x)^(13/3)*(a + b*x^2)^(1/3),x]","\frac{c^3 (c x)^{4/3} \sqrt[3]{a+b x^2} \left(\sqrt[3]{\frac{b x^2}{a}+1} \left(-5 a^2+a b x^2+6 b^2 x^4\right)+5 a^2 \, _2F_1\left(-\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^2}{a}\right)\right)}{36 b^2 \sqrt[3]{\frac{b x^2}{a}+1}}","-\frac{5 a^3 c^{13/3} \log \left(\sqrt[3]{b} (c x)^{2/3}-c^{2/3} \sqrt[3]{a+b x^2}\right)}{108 b^{8/3}}-\frac{5 a^3 c^{13/3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} (c x)^{2/3}}{c^{2/3} \sqrt[3]{a+b x^2}}+1}{\sqrt{3}}\right)}{54 \sqrt{3} b^{8/3}}-\frac{5 a^2 c^3 (c x)^{4/3} \sqrt[3]{a+b x^2}}{108 b^2}+\frac{(c x)^{16/3} \sqrt[3]{a+b x^2}}{6 c}+\frac{a c (c x)^{10/3} \sqrt[3]{a+b x^2}}{36 b}",1,"(c^3*(c*x)^(4/3)*(a + b*x^2)^(1/3)*((1 + (b*x^2)/a)^(1/3)*(-5*a^2 + a*b*x^2 + 6*b^2*x^4) + 5*a^2*Hypergeometric2F1[-1/3, 2/3, 5/3, -((b*x^2)/a)]))/(36*b^2*(1 + (b*x^2)/a)^(1/3))","C",1
739,1,85,164,0.0504829,"\int (c x)^{7/3} \sqrt[3]{a+b x^2} \, dx","Integrate[(c*x)^(7/3)*(a + b*x^2)^(1/3),x]","\frac{c (c x)^{4/3} \sqrt[3]{a+b x^2} \left(\left(a+b x^2\right) \sqrt[3]{\frac{b x^2}{a}+1}-a \, _2F_1\left(-\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^2}{a}\right)\right)}{4 b \sqrt[3]{\frac{b x^2}{a}+1}}","\frac{a^2 c^{7/3} \log \left(\sqrt[3]{b} (c x)^{2/3}-c^{2/3} \sqrt[3]{a+b x^2}\right)}{12 b^{5/3}}+\frac{a^2 c^{7/3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} (c x)^{2/3}}{c^{2/3} \sqrt[3]{a+b x^2}}+1}{\sqrt{3}}\right)}{6 \sqrt{3} b^{5/3}}+\frac{(c x)^{10/3} \sqrt[3]{a+b x^2}}{4 c}+\frac{a c (c x)^{4/3} \sqrt[3]{a+b x^2}}{12 b}",1,"(c*(c*x)^(4/3)*(a + b*x^2)^(1/3)*((a + b*x^2)*(1 + (b*x^2)/a)^(1/3) - a*Hypergeometric2F1[-1/3, 2/3, 5/3, -((b*x^2)/a)]))/(4*b*(1 + (b*x^2)/a)^(1/3))","C",1
740,1,56,133,0.0135472,"\int \sqrt[3]{c x} \sqrt[3]{a+b x^2} \, dx","Integrate[(c*x)^(1/3)*(a + b*x^2)^(1/3),x]","\frac{3 x \sqrt[3]{c x} \sqrt[3]{a+b x^2} \, _2F_1\left(-\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^2}{a}\right)}{4 \sqrt[3]{\frac{b x^2}{a}+1}}","-\frac{a \sqrt[3]{c} \log \left(\sqrt[3]{b} (c x)^{2/3}-c^{2/3} \sqrt[3]{a+b x^2}\right)}{4 b^{2/3}}-\frac{a \sqrt[3]{c} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} (c x)^{2/3}}{c^{2/3} \sqrt[3]{a+b x^2}}+1}{\sqrt{3}}\right)}{2 \sqrt{3} b^{2/3}}+\frac{(c x)^{4/3} \sqrt[3]{a+b x^2}}{2 c}",1,"(3*x*(c*x)^(1/3)*(a + b*x^2)^(1/3)*Hypergeometric2F1[-1/3, 2/3, 5/3, -((b*x^2)/a)])/(4*(1 + (b*x^2)/a)^(1/3))","C",1
741,1,56,131,0.0143019,"\int \frac{\sqrt[3]{a+b x^2}}{(c x)^{5/3}} \, dx","Integrate[(a + b*x^2)^(1/3)/(c*x)^(5/3),x]","-\frac{3 x \sqrt[3]{a+b x^2} \, _2F_1\left(-\frac{1}{3},-\frac{1}{3};\frac{2}{3};-\frac{b x^2}{a}\right)}{2 (c x)^{5/3} \sqrt[3]{\frac{b x^2}{a}+1}}","-\frac{3 \sqrt[3]{b} \log \left(\sqrt[3]{b} (c x)^{2/3}-c^{2/3} \sqrt[3]{a+b x^2}\right)}{4 c^{5/3}}-\frac{\sqrt{3} \sqrt[3]{b} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} (c x)^{2/3}}{c^{2/3} \sqrt[3]{a+b x^2}}+1}{\sqrt{3}}\right)}{2 c^{5/3}}-\frac{3 \sqrt[3]{a+b x^2}}{2 c (c x)^{2/3}}",1,"(-3*x*(a + b*x^2)^(1/3)*Hypergeometric2F1[-1/3, -1/3, 2/3, -((b*x^2)/a)])/(2*(c*x)^(5/3)*(1 + (b*x^2)/a)^(1/3))","C",1
742,1,26,28,0.0092167,"\int \frac{\sqrt[3]{a+b x^2}}{(c x)^{11/3}} \, dx","Integrate[(a + b*x^2)^(1/3)/(c*x)^(11/3),x]","-\frac{3 x \left(a+b x^2\right)^{4/3}}{8 a (c x)^{11/3}}","-\frac{3 \left(a+b x^2\right)^{4/3}}{8 a c (c x)^{8/3}}",1,"(-3*x*(a + b*x^2)^(4/3))/(8*a*(c*x)^(11/3))","A",1
743,1,41,57,0.0164285,"\int \frac{\sqrt[3]{a+b x^2}}{(c x)^{17/3}} \, dx","Integrate[(a + b*x^2)^(1/3)/(c*x)^(17/3),x]","\frac{3 \sqrt[3]{c x} \left(a+b x^2\right)^{4/3} \left(3 b x^2-4 a\right)}{56 a^2 c^6 x^5}","\frac{9 \left(a+b x^2\right)^{7/3}}{56 a^2 c (c x)^{14/3}}-\frac{3 \left(a+b x^2\right)^{4/3}}{8 a c (c x)^{14/3}}",1,"(3*(c*x)^(1/3)*(a + b*x^2)^(4/3)*(-4*a + 3*b*x^2))/(56*a^2*c^6*x^5)","A",1
744,1,52,85,0.0186104,"\int \frac{\sqrt[3]{a+b x^2}}{(c x)^{23/3}} \, dx","Integrate[(a + b*x^2)^(1/3)/(c*x)^(23/3),x]","-\frac{3 \sqrt[3]{c x} \left(a+b x^2\right)^{4/3} \left(14 a^2-12 a b x^2+9 b^2 x^4\right)}{280 a^3 c^8 x^7}","-\frac{27 \left(a+b x^2\right)^{10/3}}{280 a^3 c (c x)^{20/3}}+\frac{9 \left(a+b x^2\right)^{7/3}}{28 a^2 c (c x)^{20/3}}-\frac{3 \left(a+b x^2\right)^{4/3}}{8 a c (c x)^{20/3}}",1,"(-3*(c*x)^(1/3)*(a + b*x^2)^(4/3)*(14*a^2 - 12*a*b*x^2 + 9*b^2*x^4))/(280*a^3*c^8*x^7)","A",1
745,1,63,113,0.0175181,"\int \frac{\sqrt[3]{a+b x^2}}{(c x)^{29/3}} \, dx","Integrate[(a + b*x^2)^(1/3)/(c*x)^(29/3),x]","\frac{3 \left(a+b x^2\right)^{4/3} \left(-140 a^3+126 a^2 b x^2-108 a b^2 x^4+81 b^3 x^6\right)}{3640 a^4 c^9 x^8 (c x)^{2/3}}","\frac{243 \left(a+b x^2\right)^{13/3}}{3640 a^4 c (c x)^{26/3}}-\frac{81 \left(a+b x^2\right)^{10/3}}{280 a^3 c (c x)^{26/3}}+\frac{27 \left(a+b x^2\right)^{7/3}}{56 a^2 c (c x)^{26/3}}-\frac{3 \left(a+b x^2\right)^{4/3}}{8 a c (c x)^{26/3}}",1,"(3*(a + b*x^2)^(4/3)*(-140*a^3 + 126*a^2*b*x^2 - 108*a*b^2*x^4 + 81*b^3*x^6))/(3640*a^4*c^9*x^8*(c*x)^(2/3))","A",1
746,1,103,451,0.0614064,"\int (c x)^{10/3} \sqrt[3]{a+b x^2} \, dx","Integrate[(c*x)^(10/3)*(a + b*x^2)^(1/3),x]","\frac{c^3 \sqrt[3]{c x} \sqrt[3]{a+b x^2} \left(\sqrt[3]{\frac{b x^2}{a}+1} \left(-7 a^2+2 a b x^2+9 b^2 x^4\right)+7 a^2 \, _2F_1\left(-\frac{1}{3},\frac{1}{6};\frac{7}{6};-\frac{b x^2}{a}\right)\right)}{45 b^2 \sqrt[3]{\frac{b x^2}{a}+1}}","\frac{7 a^2 c^{7/3} \sqrt[3]{c x} \sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left(a+b x^2\right)^{2/3}}+\frac{\sqrt[3]{b} c^{2/3} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}+c^{4/3}}{\left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}} \operatorname{EllipticF}\left(\cos ^{-1}\left(\frac{c^{2/3}-\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}{c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}\right),\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{135 \sqrt[4]{3} b^2 \sqrt{-\frac{\sqrt[3]{b} (c x)^{2/3} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)}{\sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}}}-\frac{14 a^2 c^3 \sqrt[3]{c x} \sqrt[3]{a+b x^2}}{135 b^2}+\frac{(c x)^{13/3} \sqrt[3]{a+b x^2}}{5 c}+\frac{2 a c (c x)^{7/3} \sqrt[3]{a+b x^2}}{45 b}",1,"(c^3*(c*x)^(1/3)*(a + b*x^2)^(1/3)*((1 + (b*x^2)/a)^(1/3)*(-7*a^2 + 2*a*b*x^2 + 9*b^2*x^4) + 7*a^2*Hypergeometric2F1[-1/3, 1/6, 7/6, -((b*x^2)/a)]))/(45*b^2*(1 + (b*x^2)/a)^(1/3))","C",1
747,1,85,418,0.0435625,"\int (c x)^{4/3} \sqrt[3]{a+b x^2} \, dx","Integrate[(c*x)^(4/3)*(a + b*x^2)^(1/3),x]","\frac{c \sqrt[3]{c x} \sqrt[3]{a+b x^2} \left(\left(a+b x^2\right) \sqrt[3]{\frac{b x^2}{a}+1}-a \, _2F_1\left(-\frac{1}{3},\frac{1}{6};\frac{7}{6};-\frac{b x^2}{a}\right)\right)}{3 b \sqrt[3]{\frac{b x^2}{a}+1}}","-\frac{a \sqrt[3]{c} \sqrt[3]{c x} \sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left(a+b x^2\right)^{2/3}}+\frac{\sqrt[3]{b} c^{2/3} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}+c^{4/3}}{\left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}} \operatorname{EllipticF}\left(\cos ^{-1}\left(\frac{c^{2/3}-\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}{c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}\right),\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{9 \sqrt[4]{3} b \sqrt{-\frac{\sqrt[3]{b} (c x)^{2/3} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)}{\sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}}}+\frac{(c x)^{7/3} \sqrt[3]{a+b x^2}}{3 c}+\frac{2 a c \sqrt[3]{c x} \sqrt[3]{a+b x^2}}{9 b}",1,"(c*(c*x)^(1/3)*(a + b*x^2)^(1/3)*((a + b*x^2)*(1 + (b*x^2)/a)^(1/3) - a*Hypergeometric2F1[-1/3, 1/6, 7/6, -((b*x^2)/a)]))/(3*b*(1 + (b*x^2)/a)^(1/3))","C",1
748,1,54,381,0.0115151,"\int \frac{\sqrt[3]{a+b x^2}}{(c x)^{2/3}} \, dx","Integrate[(a + b*x^2)^(1/3)/(c*x)^(2/3),x]","\frac{3 x \sqrt[3]{a+b x^2} \, _2F_1\left(-\frac{1}{3},\frac{1}{6};\frac{7}{6};-\frac{b x^2}{a}\right)}{(c x)^{2/3} \sqrt[3]{\frac{b x^2}{a}+1}}","\frac{\sqrt[3]{c x} \sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left(a+b x^2\right)^{2/3}}+\frac{\sqrt[3]{b} c^{2/3} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}+c^{4/3}}{\left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}} \operatorname{EllipticF}\left(\cos ^{-1}\left(\frac{c^{2/3}-\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}{c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}\right),\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{\sqrt[4]{3} c^{5/3} \sqrt{-\frac{\sqrt[3]{b} (c x)^{2/3} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)}{\sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}}}+\frac{\sqrt[3]{c x} \sqrt[3]{a+b x^2}}{c}",1,"(3*x*(a + b*x^2)^(1/3)*Hypergeometric2F1[-1/3, 1/6, 7/6, -((b*x^2)/a)])/((c*x)^(2/3)*(1 + (b*x^2)/a)^(1/3))","C",1
749,1,56,391,0.0140815,"\int \frac{\sqrt[3]{a+b x^2}}{(c x)^{8/3}} \, dx","Integrate[(a + b*x^2)^(1/3)/(c*x)^(8/3),x]","-\frac{3 x \sqrt[3]{a+b x^2} \, _2F_1\left(-\frac{5}{6},-\frac{1}{3};\frac{1}{6};-\frac{b x^2}{a}\right)}{5 (c x)^{8/3} \sqrt[3]{\frac{b x^2}{a}+1}}","\frac{3^{3/4} b \sqrt[3]{c x} \sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left(a+b x^2\right)^{2/3}}+\frac{\sqrt[3]{b} c^{2/3} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}+c^{4/3}}{\left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}} \operatorname{EllipticF}\left(\cos ^{-1}\left(\frac{c^{2/3}-\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}{c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}\right),\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{5 a c^{11/3} \sqrt{-\frac{\sqrt[3]{b} (c x)^{2/3} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)}{\sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}}}-\frac{3 \sqrt[3]{a+b x^2}}{5 c (c x)^{5/3}}",1,"(-3*x*(a + b*x^2)^(1/3)*Hypergeometric2F1[-5/6, -1/3, 1/6, -((b*x^2)/a)])/(5*(c*x)^(8/3)*(1 + (b*x^2)/a)^(1/3))","C",1
750,1,56,422,0.0136102,"\int \frac{\sqrt[3]{a+b x^2}}{(c x)^{14/3}} \, dx","Integrate[(a + b*x^2)^(1/3)/(c*x)^(14/3),x]","-\frac{3 x \sqrt[3]{a+b x^2} \, _2F_1\left(-\frac{11}{6},-\frac{1}{3};-\frac{5}{6};-\frac{b x^2}{a}\right)}{11 (c x)^{14/3} \sqrt[3]{\frac{b x^2}{a}+1}}","-\frac{3\ 3^{3/4} b^2 \sqrt[3]{c x} \sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left(a+b x^2\right)^{2/3}}+\frac{\sqrt[3]{b} c^{2/3} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}+c^{4/3}}{\left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}} \operatorname{EllipticF}\left(\cos ^{-1}\left(\frac{c^{2/3}-\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}{c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}\right),\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{55 a^2 c^{17/3} \sqrt{-\frac{\sqrt[3]{b} (c x)^{2/3} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)}{\sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}}}-\frac{6 b \sqrt[3]{a+b x^2}}{55 a c^3 (c x)^{5/3}}-\frac{3 \sqrt[3]{a+b x^2}}{11 c (c x)^{11/3}}",1,"(-3*x*(a + b*x^2)^(1/3)*Hypergeometric2F1[-11/6, -1/3, -5/6, -((b*x^2)/a)])/(11*(c*x)^(14/3)*(1 + (b*x^2)/a)^(1/3))","C",1
751,1,56,58,0.0117085,"\int (c x)^{2/3} \sqrt[3]{a+b x^2} \, dx","Integrate[(c*x)^(2/3)*(a + b*x^2)^(1/3),x]","\frac{3 x (c x)^{2/3} \sqrt[3]{a+b x^2} \, _2F_1\left(-\frac{1}{3},\frac{5}{6};\frac{11}{6};-\frac{b x^2}{a}\right)}{5 \sqrt[3]{\frac{b x^2}{a}+1}}","\frac{3 (c x)^{5/3} \sqrt[3]{a+b x^2} \, _2F_1\left(-\frac{1}{3},\frac{5}{6};\frac{11}{6};-\frac{b x^2}{a}\right)}{5 c \sqrt[3]{\frac{b x^2}{a}+1}}",1,"(3*x*(c*x)^(2/3)*(a + b*x^2)^(1/3)*Hypergeometric2F1[-1/3, 5/6, 11/6, -((b*x^2)/a)])/(5*(1 + (b*x^2)/a)^(1/3))","A",1
752,1,56,58,0.0118936,"\int \frac{\sqrt[3]{a+b x^2}}{\sqrt[3]{c x}} \, dx","Integrate[(a + b*x^2)^(1/3)/(c*x)^(1/3),x]","\frac{3 x \sqrt[3]{a+b x^2} \, _2F_1\left(-\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^2}{a}\right)}{2 \sqrt[3]{c x} \sqrt[3]{\frac{b x^2}{a}+1}}","\frac{3 (c x)^{2/3} \sqrt[3]{a+b x^2} \, _2F_1\left(-\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^2}{a}\right)}{2 c \sqrt[3]{\frac{b x^2}{a}+1}}",1,"(3*x*(a + b*x^2)^(1/3)*Hypergeometric2F1[-1/3, 1/3, 4/3, -((b*x^2)/a)])/(2*(c*x)^(1/3)*(1 + (b*x^2)/a)^(1/3))","A",1
753,1,54,56,0.0118333,"\int \frac{\sqrt[3]{a+b x^2}}{(c x)^{4/3}} \, dx","Integrate[(a + b*x^2)^(1/3)/(c*x)^(4/3),x]","-\frac{3 x \sqrt[3]{a+b x^2} \, _2F_1\left(-\frac{1}{3},-\frac{1}{6};\frac{5}{6};-\frac{b x^2}{a}\right)}{(c x)^{4/3} \sqrt[3]{\frac{b x^2}{a}+1}}","-\frac{3 \sqrt[3]{a+b x^2} \, _2F_1\left(-\frac{1}{3},-\frac{1}{6};\frac{5}{6};-\frac{b x^2}{a}\right)}{c \sqrt[3]{c x} \sqrt[3]{\frac{b x^2}{a}+1}}",1,"(-3*x*(a + b*x^2)^(1/3)*Hypergeometric2F1[-1/3, -1/6, 5/6, -((b*x^2)/a)])/((c*x)^(4/3)*(1 + (b*x^2)/a)^(1/3))","A",1
754,1,102,223,0.0808063,"\int (c x)^{13/3} \left(a+b x^2\right)^{4/3} \, dx","Integrate[(c*x)^(13/3)*(a + b*x^2)^(4/3),x]","\frac{c^3 (c x)^{4/3} \sqrt[3]{a+b x^2} \left(5 a^3 \, _2F_1\left(-\frac{4}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^2}{a}\right)-\left(5 a-9 b x^2\right) \left(a+b x^2\right)^2 \sqrt[3]{\frac{b x^2}{a}+1}\right)}{72 b^2 \sqrt[3]{\frac{b x^2}{a}+1}}","-\frac{5 a^4 c^{13/3} \log \left(\sqrt[3]{b} (c x)^{2/3}-c^{2/3} \sqrt[3]{a+b x^2}\right)}{324 b^{8/3}}-\frac{5 a^4 c^{13/3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} (c x)^{2/3}}{c^{2/3} \sqrt[3]{a+b x^2}}+1}{\sqrt{3}}\right)}{162 \sqrt{3} b^{8/3}}-\frac{5 a^3 c^3 (c x)^{4/3} \sqrt[3]{a+b x^2}}{324 b^2}+\frac{a^2 c (c x)^{10/3} \sqrt[3]{a+b x^2}}{108 b}+\frac{(c x)^{16/3} \left(a+b x^2\right)^{4/3}}{8 c}+\frac{a (c x)^{16/3} \sqrt[3]{a+b x^2}}{18 c}",1,"(c^3*(c*x)^(4/3)*(a + b*x^2)^(1/3)*(-((5*a - 9*b*x^2)*(a + b*x^2)^2*(1 + (b*x^2)/a)^(1/3)) + 5*a^3*Hypergeometric2F1[-4/3, 2/3, 5/3, -((b*x^2)/a)]))/(72*b^2*(1 + (b*x^2)/a)^(1/3))","C",1
755,1,89,192,0.0661068,"\int (c x)^{7/3} \left(a+b x^2\right)^{4/3} \, dx","Integrate[(c*x)^(7/3)*(a + b*x^2)^(4/3),x]","\frac{c (c x)^{4/3} \sqrt[3]{a+b x^2} \left(\left(a+b x^2\right)^2 \sqrt[3]{\frac{b x^2}{a}+1}-a^2 \, _2F_1\left(-\frac{4}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^2}{a}\right)\right)}{6 b \sqrt[3]{\frac{b x^2}{a}+1}}","\frac{a^3 c^{7/3} \log \left(\sqrt[3]{b} (c x)^{2/3}-c^{2/3} \sqrt[3]{a+b x^2}\right)}{27 b^{5/3}}+\frac{2 a^3 c^{7/3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} (c x)^{2/3}}{c^{2/3} \sqrt[3]{a+b x^2}}+1}{\sqrt{3}}\right)}{27 \sqrt{3} b^{5/3}}+\frac{a^2 c (c x)^{4/3} \sqrt[3]{a+b x^2}}{27 b}+\frac{(c x)^{10/3} \left(a+b x^2\right)^{4/3}}{6 c}+\frac{a (c x)^{10/3} \sqrt[3]{a+b x^2}}{9 c}",1,"(c*(c*x)^(4/3)*(a + b*x^2)^(1/3)*((a + b*x^2)^2*(1 + (b*x^2)/a)^(1/3) - a^2*Hypergeometric2F1[-4/3, 2/3, 5/3, -((b*x^2)/a)]))/(6*b*(1 + (b*x^2)/a)^(1/3))","C",1
756,1,57,163,0.0151361,"\int \sqrt[3]{c x} \left(a+b x^2\right)^{4/3} \, dx","Integrate[(c*x)^(1/3)*(a + b*x^2)^(4/3),x]","\frac{3 a x \sqrt[3]{c x} \sqrt[3]{a+b x^2} \, _2F_1\left(-\frac{4}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^2}{a}\right)}{4 \sqrt[3]{\frac{b x^2}{a}+1}}","-\frac{a^2 \sqrt[3]{c} \log \left(\sqrt[3]{b} (c x)^{2/3}-c^{2/3} \sqrt[3]{a+b x^2}\right)}{6 b^{2/3}}-\frac{a^2 \sqrt[3]{c} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} (c x)^{2/3}}{c^{2/3} \sqrt[3]{a+b x^2}}+1}{\sqrt{3}}\right)}{3 \sqrt{3} b^{2/3}}+\frac{a (c x)^{4/3} \sqrt[3]{a+b x^2}}{3 c}+\frac{(c x)^{4/3} \left(a+b x^2\right)^{4/3}}{4 c}",1,"(3*a*x*(c*x)^(1/3)*(a + b*x^2)^(1/3)*Hypergeometric2F1[-4/3, 2/3, 5/3, -((b*x^2)/a)])/(4*(1 + (b*x^2)/a)^(1/3))","C",1
757,1,57,153,0.0119946,"\int \frac{\left(a+b x^2\right)^{4/3}}{(c x)^{5/3}} \, dx","Integrate[(a + b*x^2)^(4/3)/(c*x)^(5/3),x]","-\frac{3 a x \sqrt[3]{a+b x^2} \, _2F_1\left(-\frac{4}{3},-\frac{1}{3};\frac{2}{3};-\frac{b x^2}{a}\right)}{2 (c x)^{5/3} \sqrt[3]{\frac{b x^2}{a}+1}}","-\frac{a \sqrt[3]{b} \log \left(\sqrt[3]{b} (c x)^{2/3}-c^{2/3} \sqrt[3]{a+b x^2}\right)}{c^{5/3}}-\frac{2 a \sqrt[3]{b} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} (c x)^{2/3}}{c^{2/3} \sqrt[3]{a+b x^2}}+1}{\sqrt{3}}\right)}{\sqrt{3} c^{5/3}}+\frac{2 b (c x)^{4/3} \sqrt[3]{a+b x^2}}{c^3}-\frac{3 \left(a+b x^2\right)^{4/3}}{2 c (c x)^{2/3}}",1,"(-3*a*x*(a + b*x^2)^(1/3)*Hypergeometric2F1[-4/3, -1/3, 2/3, -((b*x^2)/a)])/(2*(c*x)^(5/3)*(1 + (b*x^2)/a)^(1/3))","C",1
758,1,57,157,0.0163044,"\int \frac{\left(a+b x^2\right)^{4/3}}{(c x)^{11/3}} \, dx","Integrate[(a + b*x^2)^(4/3)/(c*x)^(11/3),x]","-\frac{3 a x \sqrt[3]{a+b x^2} \, _2F_1\left(-\frac{4}{3},-\frac{4}{3};-\frac{1}{3};-\frac{b x^2}{a}\right)}{8 (c x)^{11/3} \sqrt[3]{\frac{b x^2}{a}+1}}","-\frac{3 b^{4/3} \log \left(\sqrt[3]{b} (c x)^{2/3}-c^{2/3} \sqrt[3]{a+b x^2}\right)}{4 c^{11/3}}-\frac{\sqrt{3} b^{4/3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} (c x)^{2/3}}{c^{2/3} \sqrt[3]{a+b x^2}}+1}{\sqrt{3}}\right)}{2 c^{11/3}}-\frac{3 b \sqrt[3]{a+b x^2}}{2 c^3 (c x)^{2/3}}-\frac{3 \left(a+b x^2\right)^{4/3}}{8 c (c x)^{8/3}}",1,"(-3*a*x*(a + b*x^2)^(1/3)*Hypergeometric2F1[-4/3, -4/3, -1/3, -((b*x^2)/a)])/(8*(c*x)^(11/3)*(1 + (b*x^2)/a)^(1/3))","C",1
759,1,26,28,0.0106164,"\int \frac{\left(a+b x^2\right)^{4/3}}{(c x)^{17/3}} \, dx","Integrate[(a + b*x^2)^(4/3)/(c*x)^(17/3),x]","-\frac{3 x \left(a+b x^2\right)^{7/3}}{14 a (c x)^{17/3}}","-\frac{3 \left(a+b x^2\right)^{7/3}}{14 a c (c x)^{14/3}}",1,"(-3*x*(a + b*x^2)^(7/3))/(14*a*(c*x)^(17/3))","A",1
760,1,41,57,0.0192515,"\int \frac{\left(a+b x^2\right)^{4/3}}{(c x)^{23/3}} \, dx","Integrate[(a + b*x^2)^(4/3)/(c*x)^(23/3),x]","\frac{3 \sqrt[3]{c x} \left(a+b x^2\right)^{7/3} \left(3 b x^2-7 a\right)}{140 a^2 c^8 x^7}","\frac{9 \left(a+b x^2\right)^{10/3}}{140 a^2 c (c x)^{20/3}}-\frac{3 \left(a+b x^2\right)^{7/3}}{14 a c (c x)^{20/3}}",1,"(3*(c*x)^(1/3)*(a + b*x^2)^(7/3)*(-7*a + 3*b*x^2))/(140*a^2*c^8*x^7)","A",1
761,1,52,85,0.0163522,"\int \frac{\left(a+b x^2\right)^{4/3}}{(c x)^{29/3}} \, dx","Integrate[(a + b*x^2)^(4/3)/(c*x)^(29/3),x]","-\frac{3 \left(a+b x^2\right)^{7/3} \left(35 a^2-21 a b x^2+9 b^2 x^4\right)}{910 a^3 c^9 x^8 (c x)^{2/3}}","-\frac{27 \left(a+b x^2\right)^{13/3}}{910 a^3 c (c x)^{26/3}}+\frac{9 \left(a+b x^2\right)^{10/3}}{70 a^2 c (c x)^{26/3}}-\frac{3 \left(a+b x^2\right)^{7/3}}{14 a c (c x)^{26/3}}",1,"(-3*(a + b*x^2)^(7/3)*(35*a^2 - 21*a*b*x^2 + 9*b^2*x^4))/(910*a^3*c^9*x^8*(c*x)^(2/3))","A",1
762,1,102,479,0.07432,"\int (c x)^{10/3} \left(a+b x^2\right)^{4/3} \, dx","Integrate[(c*x)^(10/3)*(a + b*x^2)^(4/3),x]","\frac{c^3 \sqrt[3]{c x} \sqrt[3]{a+b x^2} \left(7 a^3 \, _2F_1\left(-\frac{4}{3},\frac{1}{6};\frac{7}{6};-\frac{b x^2}{a}\right)-\left(7 a-15 b x^2\right) \left(a+b x^2\right)^2 \sqrt[3]{\frac{b x^2}{a}+1}\right)}{105 b^2 \sqrt[3]{\frac{b x^2}{a}+1}}","\frac{8 a^3 c^{7/3} \sqrt[3]{c x} \sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left(a+b x^2\right)^{2/3}}+\frac{\sqrt[3]{b} c^{2/3} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}+c^{4/3}}{\left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}} \operatorname{EllipticF}\left(\cos ^{-1}\left(\frac{c^{2/3}-\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}{c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}\right),\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{405 \sqrt[4]{3} b^2 \sqrt{-\frac{\sqrt[3]{b} (c x)^{2/3} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)}{\sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}}}-\frac{16 a^3 c^3 \sqrt[3]{c x} \sqrt[3]{a+b x^2}}{405 b^2}+\frac{16 a^2 c (c x)^{7/3} \sqrt[3]{a+b x^2}}{945 b}+\frac{(c x)^{13/3} \left(a+b x^2\right)^{4/3}}{7 c}+\frac{8 a (c x)^{13/3} \sqrt[3]{a+b x^2}}{105 c}",1,"(c^3*(c*x)^(1/3)*(a + b*x^2)^(1/3)*(-((7*a - 15*b*x^2)*(a + b*x^2)^2*(1 + (b*x^2)/a)^(1/3)) + 7*a^3*Hypergeometric2F1[-4/3, 1/6, 7/6, -((b*x^2)/a)]))/(105*b^2*(1 + (b*x^2)/a)^(1/3))","C",1
763,1,89,448,0.0505647,"\int (c x)^{4/3} \left(a+b x^2\right)^{4/3} \, dx","Integrate[(c*x)^(4/3)*(a + b*x^2)^(4/3),x]","\frac{c \sqrt[3]{c x} \sqrt[3]{a+b x^2} \left(\left(a+b x^2\right)^2 \sqrt[3]{\frac{b x^2}{a}+1}-a^2 \, _2F_1\left(-\frac{4}{3},\frac{1}{6};\frac{7}{6};-\frac{b x^2}{a}\right)\right)}{5 b \sqrt[3]{\frac{b x^2}{a}+1}}","-\frac{8 a^2 \sqrt[3]{c} \sqrt[3]{c x} \sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left(a+b x^2\right)^{2/3}}+\frac{\sqrt[3]{b} c^{2/3} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}+c^{4/3}}{\left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}} \operatorname{EllipticF}\left(\cos ^{-1}\left(\frac{c^{2/3}-\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}{c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}\right),\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{135 \sqrt[4]{3} b \sqrt{-\frac{\sqrt[3]{b} (c x)^{2/3} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)}{\sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}}}+\frac{16 a^2 c \sqrt[3]{c x} \sqrt[3]{a+b x^2}}{135 b}+\frac{(c x)^{7/3} \left(a+b x^2\right)^{4/3}}{5 c}+\frac{8 a (c x)^{7/3} \sqrt[3]{a+b x^2}}{45 c}",1,"(c*(c*x)^(1/3)*(a + b*x^2)^(1/3)*((a + b*x^2)^2*(1 + (b*x^2)/a)^(1/3) - a^2*Hypergeometric2F1[-4/3, 1/6, 7/6, -((b*x^2)/a)]))/(5*b*(1 + (b*x^2)/a)^(1/3))","C",1
764,1,55,414,0.0117488,"\int \frac{\left(a+b x^2\right)^{4/3}}{(c x)^{2/3}} \, dx","Integrate[(a + b*x^2)^(4/3)/(c*x)^(2/3),x]","\frac{3 a x \sqrt[3]{a+b x^2} \, _2F_1\left(-\frac{4}{3},\frac{1}{6};\frac{7}{6};-\frac{b x^2}{a}\right)}{(c x)^{2/3} \sqrt[3]{\frac{b x^2}{a}+1}}","\frac{8 a \sqrt[3]{c x} \sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left(a+b x^2\right)^{2/3}}+\frac{\sqrt[3]{b} c^{2/3} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}+c^{4/3}}{\left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}} \operatorname{EllipticF}\left(\cos ^{-1}\left(\frac{c^{2/3}-\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}{c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}\right),\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{9 \sqrt[4]{3} c^{5/3} \sqrt{-\frac{\sqrt[3]{b} (c x)^{2/3} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)}{\sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}}}+\frac{\sqrt[3]{c x} \left(a+b x^2\right)^{4/3}}{3 c}+\frac{8 a \sqrt[3]{c x} \sqrt[3]{a+b x^2}}{9 c}",1,"(3*a*x*(a + b*x^2)^(1/3)*Hypergeometric2F1[-4/3, 1/6, 7/6, -((b*x^2)/a)])/((c*x)^(2/3)*(1 + (b*x^2)/a)^(1/3))","C",1
765,1,57,414,0.0129562,"\int \frac{\left(a+b x^2\right)^{4/3}}{(c x)^{8/3}} \, dx","Integrate[(a + b*x^2)^(4/3)/(c*x)^(8/3),x]","-\frac{3 a x \sqrt[3]{a+b x^2} \, _2F_1\left(-\frac{4}{3},-\frac{5}{6};\frac{1}{6};-\frac{b x^2}{a}\right)}{5 (c x)^{8/3} \sqrt[3]{\frac{b x^2}{a}+1}}","\frac{8 b \sqrt[3]{c x} \sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left(a+b x^2\right)^{2/3}}+\frac{\sqrt[3]{b} c^{2/3} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}+c^{4/3}}{\left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}} \operatorname{EllipticF}\left(\cos ^{-1}\left(\frac{c^{2/3}-\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}{c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}\right),\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{5 \sqrt[4]{3} c^{11/3} \sqrt{-\frac{\sqrt[3]{b} (c x)^{2/3} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)}{\sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}}}+\frac{8 b \sqrt[3]{c x} \sqrt[3]{a+b x^2}}{5 c^3}-\frac{3 \left(a+b x^2\right)^{4/3}}{5 c (c x)^{5/3}}",1,"(-3*a*x*(a + b*x^2)^(1/3)*Hypergeometric2F1[-4/3, -5/6, 1/6, -((b*x^2)/a)])/(5*(c*x)^(8/3)*(1 + (b*x^2)/a)^(1/3))","C",1
766,1,57,419,0.0143606,"\int \frac{\left(a+b x^2\right)^{4/3}}{(c x)^{14/3}} \, dx","Integrate[(a + b*x^2)^(4/3)/(c*x)^(14/3),x]","-\frac{3 a x \sqrt[3]{a+b x^2} \, _2F_1\left(-\frac{11}{6},-\frac{4}{3};-\frac{5}{6};-\frac{b x^2}{a}\right)}{11 (c x)^{14/3} \sqrt[3]{\frac{b x^2}{a}+1}}","\frac{8\ 3^{3/4} b^2 \sqrt[3]{c x} \sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left(a+b x^2\right)^{2/3}}+\frac{\sqrt[3]{b} c^{2/3} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}+c^{4/3}}{\left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}} \operatorname{EllipticF}\left(\cos ^{-1}\left(\frac{c^{2/3}-\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}{c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}\right),\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{55 a c^{17/3} \sqrt{-\frac{\sqrt[3]{b} (c x)^{2/3} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)}{\sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}}}-\frac{24 b \sqrt[3]{a+b x^2}}{55 c^3 (c x)^{5/3}}-\frac{3 \left(a+b x^2\right)^{4/3}}{11 c (c x)^{11/3}}",1,"(-3*a*x*(a + b*x^2)^(1/3)*Hypergeometric2F1[-11/6, -4/3, -5/6, -((b*x^2)/a)])/(11*(c*x)^(14/3)*(1 + (b*x^2)/a)^(1/3))","C",1
767,1,57,450,0.0138228,"\int \frac{\left(a+b x^2\right)^{4/3}}{(c x)^{20/3}} \, dx","Integrate[(a + b*x^2)^(4/3)/(c*x)^(20/3),x]","-\frac{3 a x \sqrt[3]{a+b x^2} \, _2F_1\left(-\frac{17}{6},-\frac{4}{3};-\frac{11}{6};-\frac{b x^2}{a}\right)}{17 (c x)^{20/3} \sqrt[3]{\frac{b x^2}{a}+1}}","-\frac{24\ 3^{3/4} b^3 \sqrt[3]{c x} \sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left(a+b x^2\right)^{2/3}}+\frac{\sqrt[3]{b} c^{2/3} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}+c^{4/3}}{\left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}} \operatorname{EllipticF}\left(\cos ^{-1}\left(\frac{c^{2/3}-\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}{c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}\right),\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{935 a^2 c^{23/3} \sqrt{-\frac{\sqrt[3]{b} (c x)^{2/3} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)}{\sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}}}-\frac{48 b^2 \sqrt[3]{a+b x^2}}{935 a c^5 (c x)^{5/3}}-\frac{24 b \sqrt[3]{a+b x^2}}{187 c^3 (c x)^{11/3}}-\frac{3 \left(a+b x^2\right)^{4/3}}{17 c (c x)^{17/3}}",1,"(-3*a*x*(a + b*x^2)^(1/3)*Hypergeometric2F1[-17/6, -4/3, -11/6, -((b*x^2)/a)])/(17*(c*x)^(20/3)*(1 + (b*x^2)/a)^(1/3))","C",1
768,1,57,59,0.0113076,"\int (c x)^{2/3} \left(a+b x^2\right)^{4/3} \, dx","Integrate[(c*x)^(2/3)*(a + b*x^2)^(4/3),x]","\frac{3 a x (c x)^{2/3} \sqrt[3]{a+b x^2} \, _2F_1\left(-\frac{4}{3},\frac{5}{6};\frac{11}{6};-\frac{b x^2}{a}\right)}{5 \sqrt[3]{\frac{b x^2}{a}+1}}","\frac{3 a (c x)^{5/3} \sqrt[3]{a+b x^2} \, _2F_1\left(-\frac{4}{3},\frac{5}{6};\frac{11}{6};-\frac{b x^2}{a}\right)}{5 c \sqrt[3]{\frac{b x^2}{a}+1}}",1,"(3*a*x*(c*x)^(2/3)*(a + b*x^2)^(1/3)*Hypergeometric2F1[-4/3, 5/6, 11/6, -((b*x^2)/a)])/(5*(1 + (b*x^2)/a)^(1/3))","A",1
769,1,57,59,0.0121634,"\int \frac{\left(a+b x^2\right)^{4/3}}{\sqrt[3]{c x}} \, dx","Integrate[(a + b*x^2)^(4/3)/(c*x)^(1/3),x]","\frac{3 a x \sqrt[3]{a+b x^2} \, _2F_1\left(-\frac{4}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^2}{a}\right)}{2 \sqrt[3]{c x} \sqrt[3]{\frac{b x^2}{a}+1}}","\frac{3 a (c x)^{2/3} \sqrt[3]{a+b x^2} \, _2F_1\left(-\frac{4}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^2}{a}\right)}{2 c \sqrt[3]{\frac{b x^2}{a}+1}}",1,"(3*a*x*(a + b*x^2)^(1/3)*Hypergeometric2F1[-4/3, 1/3, 4/3, -((b*x^2)/a)])/(2*(c*x)^(1/3)*(1 + (b*x^2)/a)^(1/3))","A",1
770,1,55,57,0.0120583,"\int \frac{\left(a+b x^2\right)^{4/3}}{(c x)^{4/3}} \, dx","Integrate[(a + b*x^2)^(4/3)/(c*x)^(4/3),x]","-\frac{3 a x \sqrt[3]{a+b x^2} \, _2F_1\left(-\frac{4}{3},-\frac{1}{6};\frac{5}{6};-\frac{b x^2}{a}\right)}{(c x)^{4/3} \sqrt[3]{\frac{b x^2}{a}+1}}","-\frac{3 a \sqrt[3]{a+b x^2} \, _2F_1\left(-\frac{4}{3},-\frac{1}{6};\frac{5}{6};-\frac{b x^2}{a}\right)}{c \sqrt[3]{c x} \sqrt[3]{\frac{b x^2}{a}+1}}",1,"(-3*a*x*(a + b*x^2)^(1/3)*Hypergeometric2F1[-4/3, -1/6, 5/6, -((b*x^2)/a)])/((c*x)^(4/3)*(1 + (b*x^2)/a)^(1/3))","A",1
771,1,87,198,0.0289609,"\int \frac{(c x)^{19/3}}{\left(a+b x^2\right)^{2/3}} \, dx","Integrate[(c*x)^(19/3)/(a + b*x^2)^(2/3),x]","\frac{c^5 (c x)^{4/3} \left(-20 a^3 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{b x^2}{b x^2+a}\right)+20 a^3+8 a^2 b x^2-3 a b^2 x^4+9 b^3 x^6\right)}{54 b^3 \left(a+b x^2\right)^{2/3}}","\frac{10 a^3 c^{19/3} \log \left(\sqrt[3]{b} (c x)^{2/3}-c^{2/3} \sqrt[3]{a+b x^2}\right)}{27 b^{11/3}}+\frac{20 a^3 c^{19/3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} (c x)^{2/3}}{c^{2/3} \sqrt[3]{a+b x^2}}+1}{\sqrt{3}}\right)}{27 \sqrt{3} b^{11/3}}+\frac{10 a^2 c^5 (c x)^{4/3} \sqrt[3]{a+b x^2}}{27 b^3}-\frac{2 a c^3 (c x)^{10/3} \sqrt[3]{a+b x^2}}{9 b^2}+\frac{c (c x)^{16/3} \sqrt[3]{a+b x^2}}{6 b}",1,"(c^5*(c*x)^(4/3)*(20*a^3 + 8*a^2*b*x^2 - 3*a*b^2*x^4 + 9*b^3*x^6 - 20*a^3*Hypergeometric2F1[2/3, 1, 5/3, (b*x^2)/(a + b*x^2)]))/(54*b^3*(a + b*x^2)^(2/3))","C",1
772,1,76,167,0.0229786,"\int \frac{(c x)^{13/3}}{\left(a+b x^2\right)^{2/3}} \, dx","Integrate[(c*x)^(13/3)/(a + b*x^2)^(2/3),x]","\frac{c^3 (c x)^{4/3} \left(5 a^2 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{b x^2}{b x^2+a}\right)-5 a^2-2 a b x^2+3 b^2 x^4\right)}{12 b^2 \left(a+b x^2\right)^{2/3}}","-\frac{5 a^2 c^{13/3} \log \left(\sqrt[3]{b} (c x)^{2/3}-c^{2/3} \sqrt[3]{a+b x^2}\right)}{12 b^{8/3}}-\frac{5 a^2 c^{13/3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} (c x)^{2/3}}{c^{2/3} \sqrt[3]{a+b x^2}}+1}{\sqrt{3}}\right)}{6 \sqrt{3} b^{8/3}}-\frac{5 a c^3 (c x)^{4/3} \sqrt[3]{a+b x^2}}{12 b^2}+\frac{c (c x)^{10/3} \sqrt[3]{a+b x^2}}{4 b}",1,"(c^3*(c*x)^(4/3)*(-5*a^2 - 2*a*b*x^2 + 3*b^2*x^4 + 5*a^2*Hypergeometric2F1[2/3, 1, 5/3, (b*x^2)/(a + b*x^2)]))/(12*b^2*(a + b*x^2)^(2/3))","C",1
773,1,58,131,0.0195072,"\int \frac{(c x)^{7/3}}{\left(a+b x^2\right)^{2/3}} \, dx","Integrate[(c*x)^(7/3)/(a + b*x^2)^(2/3),x]","\frac{c (c x)^{4/3} \left(-a \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{b x^2}{b x^2+a}\right)+a+b x^2\right)}{2 b \left(a+b x^2\right)^{2/3}}","\frac{a c^{7/3} \log \left(\sqrt[3]{b} (c x)^{2/3}-c^{2/3} \sqrt[3]{a+b x^2}\right)}{2 b^{5/3}}+\frac{a c^{7/3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} (c x)^{2/3}}{c^{2/3} \sqrt[3]{a+b x^2}}+1}{\sqrt{3}}\right)}{\sqrt{3} b^{5/3}}+\frac{c (c x)^{4/3} \sqrt[3]{a+b x^2}}{2 b}",1,"(c*(c*x)^(4/3)*(a + b*x^2 - a*Hypergeometric2F1[2/3, 1, 5/3, (b*x^2)/(a + b*x^2)]))/(2*b*(a + b*x^2)^(2/3))","C",1
774,1,45,106,0.0102018,"\int \frac{\sqrt[3]{c x}}{\left(a+b x^2\right)^{2/3}} \, dx","Integrate[(c*x)^(1/3)/(a + b*x^2)^(2/3),x]","\frac{3 x \sqrt[3]{c x} \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{b x^2}{b x^2+a}\right)}{4 \left(a+b x^2\right)^{2/3}}","-\frac{3 \sqrt[3]{c} \log \left(\sqrt[3]{b} (c x)^{2/3}-c^{2/3} \sqrt[3]{a+b x^2}\right)}{4 b^{2/3}}-\frac{\sqrt{3} \sqrt[3]{c} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} (c x)^{2/3}}{c^{2/3} \sqrt[3]{a+b x^2}}+1}{\sqrt{3}}\right)}{2 b^{2/3}}",1,"(3*x*(c*x)^(1/3)*Hypergeometric2F1[2/3, 1, 5/3, (b*x^2)/(a + b*x^2)])/(4*(a + b*x^2)^(2/3))","C",1
775,1,26,28,0.0063653,"\int \frac{1}{(c x)^{5/3} \left(a+b x^2\right)^{2/3}} \, dx","Integrate[1/((c*x)^(5/3)*(a + b*x^2)^(2/3)),x]","-\frac{3 x \sqrt[3]{a+b x^2}}{2 a (c x)^{5/3}}","-\frac{3 \sqrt[3]{a+b x^2}}{2 a c (c x)^{2/3}}",1,"(-3*x*(a + b*x^2)^(1/3))/(2*a*(c*x)^(5/3))","A",1
776,1,34,57,0.01854,"\int \frac{1}{(c x)^{11/3} \left(a+b x^2\right)^{2/3}} \, dx","Integrate[1/((c*x)^(11/3)*(a + b*x^2)^(2/3)),x]","-\frac{3 x \left(a-3 b x^2\right) \sqrt[3]{a+b x^2}}{8 a^2 (c x)^{11/3}}","\frac{9 \left(a+b x^2\right)^{4/3}}{8 a^2 c (c x)^{8/3}}-\frac{3 \sqrt[3]{a+b x^2}}{2 a c (c x)^{8/3}}",1,"(-3*x*(a - 3*b*x^2)*(a + b*x^2)^(1/3))/(8*a^2*(c*x)^(11/3))","A",1
777,1,52,85,0.0253872,"\int \frac{1}{(c x)^{17/3} \left(a+b x^2\right)^{2/3}} \, dx","Integrate[1/((c*x)^(17/3)*(a + b*x^2)^(2/3)),x]","-\frac{3 \sqrt[3]{c x} \sqrt[3]{a+b x^2} \left(2 a^2-3 a b x^2+9 b^2 x^4\right)}{28 a^3 c^6 x^5}","-\frac{27 \left(a+b x^2\right)^{7/3}}{28 a^3 c (c x)^{14/3}}+\frac{9 \left(a+b x^2\right)^{4/3}}{4 a^2 c (c x)^{14/3}}-\frac{3 \sqrt[3]{a+b x^2}}{2 a c (c x)^{14/3}}",1,"(-3*(c*x)^(1/3)*(a + b*x^2)^(1/3)*(2*a^2 - 3*a*b*x^2 + 9*b^2*x^4))/(28*a^3*c^6*x^5)","A",1
778,1,63,113,0.0305896,"\int \frac{1}{(c x)^{23/3} \left(a+b x^2\right)^{2/3}} \, dx","Integrate[1/((c*x)^(23/3)*(a + b*x^2)^(2/3)),x]","\frac{3 \sqrt[3]{c x} \sqrt[3]{a+b x^2} \left(-14 a^3+18 a^2 b x^2-27 a b^2 x^4+81 b^3 x^6\right)}{280 a^4 c^8 x^7}","\frac{243 \left(a+b x^2\right)^{10/3}}{280 a^4 c (c x)^{20/3}}-\frac{81 \left(a+b x^2\right)^{7/3}}{28 a^3 c (c x)^{20/3}}+\frac{27 \left(a+b x^2\right)^{4/3}}{8 a^2 c (c x)^{20/3}}-\frac{3 \sqrt[3]{a+b x^2}}{2 a c (c x)^{20/3}}",1,"(3*(c*x)^(1/3)*(a + b*x^2)^(1/3)*(-14*a^3 + 18*a^2*b*x^2 - 27*a*b^2*x^4 + 81*b^3*x^6))/(280*a^4*c^8*x^7)","A",1
779,1,87,421,0.0312336,"\int \frac{(c x)^{10/3}}{\left(a+b x^2\right)^{2/3}} \, dx","Integrate[(c*x)^(10/3)/(a + b*x^2)^(2/3),x]","\frac{c^3 \sqrt[3]{c x} \left(7 a^2 \left(\frac{b x^2}{a}+1\right)^{2/3} \, _2F_1\left(\frac{1}{6},\frac{2}{3};\frac{7}{6};-\frac{b x^2}{a}\right)-7 a^2-4 a b x^2+3 b^2 x^4\right)}{9 b^2 \left(a+b x^2\right)^{2/3}}","\frac{7 a c^{7/3} \sqrt[3]{c x} \sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left(a+b x^2\right)^{2/3}}+\frac{\sqrt[3]{b} c^{2/3} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}+c^{4/3}}{\left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}} \operatorname{EllipticF}\left(\cos ^{-1}\left(\frac{c^{2/3}-\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}{c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}\right),\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{18 \sqrt[4]{3} b^2 \sqrt{-\frac{\sqrt[3]{b} (c x)^{2/3} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)}{\sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}}}-\frac{7 a c^3 \sqrt[3]{c x} \sqrt[3]{a+b x^2}}{9 b^2}+\frac{c (c x)^{7/3} \sqrt[3]{a+b x^2}}{3 b}",1,"(c^3*(c*x)^(1/3)*(-7*a^2 - 4*a*b*x^2 + 3*b^2*x^4 + 7*a^2*(1 + (b*x^2)/a)^(2/3)*Hypergeometric2F1[1/6, 2/3, 7/6, -((b*x^2)/a)]))/(9*b^2*(a + b*x^2)^(2/3))","C",1
780,1,66,388,0.0232538,"\int \frac{(c x)^{4/3}}{\left(a+b x^2\right)^{2/3}} \, dx","Integrate[(c*x)^(4/3)/(a + b*x^2)^(2/3),x]","\frac{c \sqrt[3]{c x} \left(-a \left(\frac{b x^2}{a}+1\right)^{2/3} \, _2F_1\left(\frac{1}{6},\frac{2}{3};\frac{7}{6};-\frac{b x^2}{a}\right)+a+b x^2\right)}{b \left(a+b x^2\right)^{2/3}}","\frac{c \sqrt[3]{c x} \sqrt[3]{a+b x^2}}{b}-\frac{\sqrt[3]{c} \sqrt[3]{c x} \sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left(a+b x^2\right)^{2/3}}+\frac{\sqrt[3]{b} c^{2/3} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}+c^{4/3}}{\left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}} \operatorname{EllipticF}\left(\cos ^{-1}\left(\frac{c^{2/3}-\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}{c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}\right),\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{2 \sqrt[4]{3} b \sqrt{-\frac{\sqrt[3]{b} (c x)^{2/3} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)}{\sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}}}",1,"(c*(c*x)^(1/3)*(a + b*x^2 - a*(1 + (b*x^2)/a)^(2/3)*Hypergeometric2F1[1/6, 2/3, 7/6, -((b*x^2)/a)]))/(b*(a + b*x^2)^(2/3))","C",1
781,1,54,364,0.0148691,"\int \frac{1}{(c x)^{2/3} \left(a+b x^2\right)^{2/3}} \, dx","Integrate[1/((c*x)^(2/3)*(a + b*x^2)^(2/3)),x]","\frac{3 x \left(\frac{b x^2}{a}+1\right)^{2/3} \, _2F_1\left(\frac{1}{6},\frac{2}{3};\frac{7}{6};-\frac{b x^2}{a}\right)}{(c x)^{2/3} \left(a+b x^2\right)^{2/3}}","\frac{3^{3/4} \sqrt[3]{c x} \sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left(a+b x^2\right)^{2/3}}+\frac{\sqrt[3]{b} c^{2/3} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}+c^{4/3}}{\left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}} \operatorname{EllipticF}\left(\cos ^{-1}\left(\frac{c^{2/3}-\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}{c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}\right),\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{2 a c^{5/3} \sqrt{-\frac{\sqrt[3]{b} (c x)^{2/3} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)}{\sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}}}",1,"(3*x*(1 + (b*x^2)/a)^(2/3)*Hypergeometric2F1[1/6, 2/3, 7/6, -((b*x^2)/a)])/((c*x)^(2/3)*(a + b*x^2)^(2/3))","C",1
782,1,56,394,0.0127873,"\int \frac{1}{(c x)^{8/3} \left(a+b x^2\right)^{2/3}} \, dx","Integrate[1/((c*x)^(8/3)*(a + b*x^2)^(2/3)),x]","-\frac{3 x \left(\frac{b x^2}{a}+1\right)^{2/3} \, _2F_1\left(-\frac{5}{6},\frac{2}{3};\frac{1}{6};-\frac{b x^2}{a}\right)}{5 (c x)^{8/3} \left(a+b x^2\right)^{2/3}}","-\frac{3\ 3^{3/4} b \sqrt[3]{c x} \sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left(a+b x^2\right)^{2/3}}+\frac{\sqrt[3]{b} c^{2/3} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}+c^{4/3}}{\left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}} \operatorname{EllipticF}\left(\cos ^{-1}\left(\frac{c^{2/3}-\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}{c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}\right),\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{10 a^2 c^{11/3} \sqrt{-\frac{\sqrt[3]{b} (c x)^{2/3} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)}{\sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}}}-\frac{3 \sqrt[3]{a+b x^2}}{5 a c (c x)^{5/3}}",1,"(-3*x*(1 + (b*x^2)/a)^(2/3)*Hypergeometric2F1[-5/6, 2/3, 1/6, -((b*x^2)/a)])/(5*(c*x)^(8/3)*(a + b*x^2)^(2/3))","C",1
783,1,56,425,0.013034,"\int \frac{1}{(c x)^{14/3} \left(a+b x^2\right)^{2/3}} \, dx","Integrate[1/((c*x)^(14/3)*(a + b*x^2)^(2/3)),x]","-\frac{3 x \left(\frac{b x^2}{a}+1\right)^{2/3} \, _2F_1\left(-\frac{11}{6},\frac{2}{3};-\frac{5}{6};-\frac{b x^2}{a}\right)}{11 (c x)^{14/3} \left(a+b x^2\right)^{2/3}}","\frac{27\ 3^{3/4} b^2 \sqrt[3]{c x} \sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left(a+b x^2\right)^{2/3}}+\frac{\sqrt[3]{b} c^{2/3} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}+c^{4/3}}{\left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}} \operatorname{EllipticF}\left(\cos ^{-1}\left(\frac{c^{2/3}-\frac{\left(1-\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}{c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}}\right),\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{110 a^3 c^{17/3} \sqrt{-\frac{\sqrt[3]{b} (c x)^{2/3} \left(c^{2/3}-\frac{\sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)}{\sqrt[3]{a+b x^2} \left(c^{2/3}-\frac{\left(1+\sqrt{3}\right) \sqrt[3]{b} (c x)^{2/3}}{\sqrt[3]{a+b x^2}}\right)^2}}}+\frac{27 b \sqrt[3]{a+b x^2}}{55 a^2 c^3 (c x)^{5/3}}-\frac{3 \sqrt[3]{a+b x^2}}{11 a c (c x)^{11/3}}",1,"(-3*x*(1 + (b*x^2)/a)^(2/3)*Hypergeometric2F1[-11/6, 2/3, -5/6, -((b*x^2)/a)])/(11*(c*x)^(14/3)*(a + b*x^2)^(2/3))","C",1
784,1,56,58,0.0132012,"\int \frac{(c x)^{2/3}}{\left(a+b x^2\right)^{2/3}} \, dx","Integrate[(c*x)^(2/3)/(a + b*x^2)^(2/3),x]","\frac{3 x (c x)^{2/3} \left(\frac{b x^2}{a}+1\right)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{5}{6};\frac{11}{6};-\frac{b x^2}{a}\right)}{5 \left(a+b x^2\right)^{2/3}}","\frac{3 (c x)^{5/3} \left(\frac{b x^2}{a}+1\right)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{5}{6};\frac{11}{6};-\frac{b x^2}{a}\right)}{5 c \left(a+b x^2\right)^{2/3}}",1,"(3*x*(c*x)^(2/3)*(1 + (b*x^2)/a)^(2/3)*Hypergeometric2F1[2/3, 5/6, 11/6, -((b*x^2)/a)])/(5*(a + b*x^2)^(2/3))","A",1
785,1,56,58,0.010928,"\int \frac{1}{\sqrt[3]{c x} \left(a+b x^2\right)^{2/3}} \, dx","Integrate[1/((c*x)^(1/3)*(a + b*x^2)^(2/3)),x]","\frac{3 x \left(\frac{b x^2}{a}+1\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^2}{a}\right)}{2 \sqrt[3]{c x} \left(a+b x^2\right)^{2/3}}","\frac{3 (c x)^{2/3} \left(\frac{b x^2}{a}+1\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^2}{a}\right)}{2 c \left(a+b x^2\right)^{2/3}}",1,"(3*x*(1 + (b*x^2)/a)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -((b*x^2)/a)])/(2*(c*x)^(1/3)*(a + b*x^2)^(2/3))","A",1
786,1,54,56,0.0116987,"\int \frac{1}{(c x)^{4/3} \left(a+b x^2\right)^{2/3}} \, dx","Integrate[1/((c*x)^(4/3)*(a + b*x^2)^(2/3)),x]","-\frac{3 x \left(\frac{b x^2}{a}+1\right)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{2}{3};\frac{5}{6};-\frac{b x^2}{a}\right)}{(c x)^{4/3} \left(a+b x^2\right)^{2/3}}","-\frac{3 \left(\frac{b x^2}{a}+1\right)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{2}{3};\frac{5}{6};-\frac{b x^2}{a}\right)}{c \sqrt[3]{c x} \left(a+b x^2\right)^{2/3}}",1,"(-3*x*(1 + (b*x^2)/a)^(2/3)*Hypergeometric2F1[-1/6, 2/3, 5/6, -((b*x^2)/a)])/((c*x)^(4/3)*(a + b*x^2)^(2/3))","A",1
787,1,93,121,0.0530303,"\int x^4 \sqrt[4]{a+b x^2} \, dx","Integrate[x^4*(a + b*x^2)^(1/4),x]","\frac{2 x \sqrt[4]{a+b x^2} \left(\sqrt[4]{\frac{b x^2}{a}+1} \left(-6 a^2+a b x^2+7 b^2 x^4\right)+6 a^2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)\right)}{77 b^2 \sqrt[4]{\frac{b x^2}{a}+1}}","\frac{8 a^{7/2} \left(\frac{b x^2}{a}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{77 b^{5/2} \left(a+b x^2\right)^{3/4}}-\frac{4 a^2 x \sqrt[4]{a+b x^2}}{77 b^2}+\frac{2}{11} x^5 \sqrt[4]{a+b x^2}+\frac{2 a x^3 \sqrt[4]{a+b x^2}}{77 b}",1,"(2*x*(a + b*x^2)^(1/4)*((1 + (b*x^2)/a)^(1/4)*(-6*a^2 + a*b*x^2 + 7*b^2*x^4) + 6*a^2*Hypergeometric2F1[-1/4, 1/2, 3/2, -((b*x^2)/a)]))/(77*b^2*(1 + (b*x^2)/a)^(1/4))","C",1
788,1,62,97,0.0491208,"\int x^2 \sqrt[4]{a+b x^2} \, dx","Integrate[x^2*(a + b*x^2)^(1/4),x]","\frac{2 x \sqrt[4]{a+b x^2} \left(-\frac{a \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)}{\sqrt[4]{\frac{b x^2}{a}+1}}+a+b x^2\right)}{7 b}","-\frac{4 a^{5/2} \left(\frac{b x^2}{a}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{21 b^{3/2} \left(a+b x^2\right)^{3/4}}+\frac{2 a x \sqrt[4]{a+b x^2}}{21 b}+\frac{2}{7} x^3 \sqrt[4]{a+b x^2}",1,"(2*x*(a + b*x^2)^(1/4)*(a + b*x^2 - (a*Hypergeometric2F1[-1/4, 1/2, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^(1/4)))/(7*b)","C",1
789,1,46,75,0.0059004,"\int \sqrt[4]{a+b x^2} \, dx","Integrate[(a + b*x^2)^(1/4),x]","\frac{x \sqrt[4]{a+b x^2} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)}{\sqrt[4]{\frac{b x^2}{a}+1}}","\frac{2 a^{3/2} \left(\frac{b x^2}{a}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{3 \sqrt{b} \left(a+b x^2\right)^{3/4}}+\frac{2}{3} x \sqrt[4]{a+b x^2}",1,"(x*(a + b*x^2)^(1/4)*Hypergeometric2F1[-1/4, 1/2, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^(1/4)","C",1
790,1,49,72,0.0090965,"\int \frac{\sqrt[4]{a+b x^2}}{x^2} \, dx","Integrate[(a + b*x^2)^(1/4)/x^2,x]","-\frac{\sqrt[4]{a+b x^2} \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{1}{2};-\frac{b x^2}{a}\right)}{x \sqrt[4]{\frac{b x^2}{a}+1}}","\frac{\sqrt{a} \sqrt{b} \left(\frac{b x^2}{a}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{\left(a+b x^2\right)^{3/4}}-\frac{\sqrt[4]{a+b x^2}}{x}",1,"-(((a + b*x^2)^(1/4)*Hypergeometric2F1[-1/2, -1/4, 1/2, -((b*x^2)/a)])/(x*(1 + (b*x^2)/a)^(1/4)))","C",1
791,1,51,99,0.00923,"\int \frac{\sqrt[4]{a+b x^2}}{x^4} \, dx","Integrate[(a + b*x^2)^(1/4)/x^4,x]","-\frac{\sqrt[4]{a+b x^2} \, _2F_1\left(-\frac{3}{2},-\frac{1}{4};-\frac{1}{2};-\frac{b x^2}{a}\right)}{3 x^3 \sqrt[4]{\frac{b x^2}{a}+1}}","-\frac{b^{3/2} \left(\frac{b x^2}{a}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{6 \sqrt{a} \left(a+b x^2\right)^{3/4}}-\frac{b \sqrt[4]{a+b x^2}}{6 a x}-\frac{\sqrt[4]{a+b x^2}}{3 x^3}",1,"-1/3*((a + b*x^2)^(1/4)*Hypergeometric2F1[-3/2, -1/4, -1/2, -((b*x^2)/a)])/(x^3*(1 + (b*x^2)/a)^(1/4))","C",1
792,1,51,123,0.0097447,"\int \frac{\sqrt[4]{a+b x^2}}{x^6} \, dx","Integrate[(a + b*x^2)^(1/4)/x^6,x]","-\frac{\sqrt[4]{a+b x^2} \, _2F_1\left(-\frac{5}{2},-\frac{1}{4};-\frac{3}{2};-\frac{b x^2}{a}\right)}{5 x^5 \sqrt[4]{\frac{b x^2}{a}+1}}","\frac{b^{5/2} \left(\frac{b x^2}{a}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{12 a^{3/2} \left(a+b x^2\right)^{3/4}}+\frac{b^2 \sqrt[4]{a+b x^2}}{12 a^2 x}-\frac{\sqrt[4]{a+b x^2}}{5 x^5}-\frac{b \sqrt[4]{a+b x^2}}{30 a x^3}",1,"-1/5*((a + b*x^2)^(1/4)*Hypergeometric2F1[-5/2, -1/4, -3/2, -((b*x^2)/a)])/(x^5*(1 + (b*x^2)/a)^(1/4))","C",1
793,1,95,126,0.0582892,"\int x^4 \sqrt[4]{a-b x^2} \, dx","Integrate[x^4*(a - b*x^2)^(1/4),x]","-\frac{2 x \sqrt[4]{a-b x^2} \left(\sqrt[4]{1-\frac{b x^2}{a}} \left(6 a^2+a b x^2-7 b^2 x^4\right)-6 a^2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)\right)}{77 b^2 \sqrt[4]{1-\frac{b x^2}{a}}}","\frac{8 a^{7/2} \left(1-\frac{b x^2}{a}\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{77 b^{5/2} \left(a-b x^2\right)^{3/4}}-\frac{4 a^2 x \sqrt[4]{a-b x^2}}{77 b^2}+\frac{2}{11} x^5 \sqrt[4]{a-b x^2}-\frac{2 a x^3 \sqrt[4]{a-b x^2}}{77 b}",1,"(-2*x*(a - b*x^2)^(1/4)*((1 - (b*x^2)/a)^(1/4)*(6*a^2 + a*b*x^2 - 7*b^2*x^4) - 6*a^2*Hypergeometric2F1[-1/4, 1/2, 3/2, (b*x^2)/a]))/(77*b^2*(1 - (b*x^2)/a)^(1/4))","C",1
794,1,64,101,0.0560746,"\int x^2 \sqrt[4]{a-b x^2} \, dx","Integrate[x^2*(a - b*x^2)^(1/4),x]","\frac{2 x \sqrt[4]{a-b x^2} \left(\frac{a \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)}{\sqrt[4]{1-\frac{b x^2}{a}}}-a+b x^2\right)}{7 b}","\frac{4 a^{5/2} \left(1-\frac{b x^2}{a}\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{21 b^{3/2} \left(a-b x^2\right)^{3/4}}-\frac{2 a x \sqrt[4]{a-b x^2}}{21 b}+\frac{2}{7} x^3 \sqrt[4]{a-b x^2}",1,"(2*x*(a - b*x^2)^(1/4)*(-a + b*x^2 + (a*Hypergeometric2F1[-1/4, 1/2, 3/2, (b*x^2)/a])/(1 - (b*x^2)/a)^(1/4)))/(7*b)","C",1
795,1,47,78,0.0070647,"\int \sqrt[4]{a-b x^2} \, dx","Integrate[(a - b*x^2)^(1/4),x]","\frac{x \sqrt[4]{a-b x^2} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)}{\sqrt[4]{1-\frac{b x^2}{a}}}","\frac{2 a^{3/2} \left(1-\frac{b x^2}{a}\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{3 \sqrt{b} \left(a-b x^2\right)^{3/4}}+\frac{2}{3} x \sqrt[4]{a-b x^2}",1,"(x*(a - b*x^2)^(1/4)*Hypergeometric2F1[-1/4, 1/2, 3/2, (b*x^2)/a])/(1 - (b*x^2)/a)^(1/4)","C",1
796,1,50,76,0.010053,"\int \frac{\sqrt[4]{a-b x^2}}{x^2} \, dx","Integrate[(a - b*x^2)^(1/4)/x^2,x]","-\frac{\sqrt[4]{a-b x^2} \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{1}{2};\frac{b x^2}{a}\right)}{x \sqrt[4]{1-\frac{b x^2}{a}}}","-\frac{\sqrt{a} \sqrt{b} \left(1-\frac{b x^2}{a}\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{\left(a-b x^2\right)^{3/4}}-\frac{\sqrt[4]{a-b x^2}}{x}",1,"-(((a - b*x^2)^(1/4)*Hypergeometric2F1[-1/2, -1/4, 1/2, (b*x^2)/a])/(x*(1 - (b*x^2)/a)^(1/4)))","C",1
797,1,52,103,0.0105753,"\int \frac{\sqrt[4]{a-b x^2}}{x^4} \, dx","Integrate[(a - b*x^2)^(1/4)/x^4,x]","-\frac{\sqrt[4]{a-b x^2} \, _2F_1\left(-\frac{3}{2},-\frac{1}{4};-\frac{1}{2};\frac{b x^2}{a}\right)}{3 x^3 \sqrt[4]{1-\frac{b x^2}{a}}}","-\frac{b^{3/2} \left(1-\frac{b x^2}{a}\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{6 \sqrt{a} \left(a-b x^2\right)^{3/4}}+\frac{b \sqrt[4]{a-b x^2}}{6 a x}-\frac{\sqrt[4]{a-b x^2}}{3 x^3}",1,"-1/3*((a - b*x^2)^(1/4)*Hypergeometric2F1[-3/2, -1/4, -1/2, (b*x^2)/a])/(x^3*(1 - (b*x^2)/a)^(1/4))","C",1
798,1,52,128,0.0105432,"\int \frac{\sqrt[4]{a-b x^2}}{x^6} \, dx","Integrate[(a - b*x^2)^(1/4)/x^6,x]","-\frac{\sqrt[4]{a-b x^2} \, _2F_1\left(-\frac{5}{2},-\frac{1}{4};-\frac{3}{2};\frac{b x^2}{a}\right)}{5 x^5 \sqrt[4]{1-\frac{b x^2}{a}}}","-\frac{b^{5/2} \left(1-\frac{b x^2}{a}\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{12 a^{3/2} \left(a-b x^2\right)^{3/4}}+\frac{b^2 \sqrt[4]{a-b x^2}}{12 a^2 x}-\frac{\sqrt[4]{a-b x^2}}{5 x^5}+\frac{b \sqrt[4]{a-b x^2}}{30 a x^3}",1,"-1/5*((a - b*x^2)^(1/4)*Hypergeometric2F1[-5/2, -1/4, -3/2, (b*x^2)/a])/(x^5*(1 - (b*x^2)/a)^(1/4))","C",1
799,1,93,143,0.0558463,"\int x^4 \left(a+b x^2\right)^{3/4} \, dx","Integrate[x^4*(a + b*x^2)^(3/4),x]","\frac{2 x \left(a+b x^2\right)^{3/4} \left(\left(\frac{b x^2}{a}+1\right)^{3/4} \left(-2 a^2+a b x^2+3 b^2 x^4\right)+2 a^2 \, _2F_1\left(-\frac{3}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)\right)}{39 b^2 \left(\frac{b x^2}{a}+1\right)^{3/4}}","-\frac{8 a^{7/2} \sqrt[4]{\frac{b x^2}{a}+1} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{65 b^{5/2} \sqrt[4]{a+b x^2}}+\frac{8 a^3 x}{65 b^2 \sqrt[4]{a+b x^2}}-\frac{4 a^2 x \left(a+b x^2\right)^{3/4}}{65 b^2}+\frac{2}{13} x^5 \left(a+b x^2\right)^{3/4}+\frac{2 a x^3 \left(a+b x^2\right)^{3/4}}{39 b}",1,"(2*x*(a + b*x^2)^(3/4)*((1 + (b*x^2)/a)^(3/4)*(-2*a^2 + a*b*x^2 + 3*b^2*x^4) + 2*a^2*Hypergeometric2F1[-3/4, 1/2, 3/2, -((b*x^2)/a)]))/(39*b^2*(1 + (b*x^2)/a)^(3/4))","C",1
800,1,62,119,0.0550046,"\int x^2 \left(a+b x^2\right)^{3/4} \, dx","Integrate[x^2*(a + b*x^2)^(3/4),x]","\frac{2 x \left(a+b x^2\right)^{3/4} \left(-\frac{a \, _2F_1\left(-\frac{3}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)}{\left(\frac{b x^2}{a}+1\right)^{3/4}}+a+b x^2\right)}{9 b}","\frac{4 a^{5/2} \sqrt[4]{\frac{b x^2}{a}+1} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{15 b^{3/2} \sqrt[4]{a+b x^2}}-\frac{4 a^2 x}{15 b \sqrt[4]{a+b x^2}}+\frac{2 a x \left(a+b x^2\right)^{3/4}}{15 b}+\frac{2}{9} x^3 \left(a+b x^2\right)^{3/4}",1,"(2*x*(a + b*x^2)^(3/4)*(a + b*x^2 - (a*Hypergeometric2F1[-3/4, 1/2, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^(3/4)))/(9*b)","C",1
801,1,46,92,0.0059636,"\int \left(a+b x^2\right)^{3/4} \, dx","Integrate[(a + b*x^2)^(3/4),x]","\frac{x \left(a+b x^2\right)^{3/4} \, _2F_1\left(-\frac{3}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)}{\left(\frac{b x^2}{a}+1\right)^{3/4}}","-\frac{6 a^{3/2} \sqrt[4]{\frac{b x^2}{a}+1} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{5 \sqrt{b} \sqrt[4]{a+b x^2}}+\frac{6 a x}{5 \sqrt[4]{a+b x^2}}+\frac{2}{5} x \left(a+b x^2\right)^{3/4}",1,"(x*(a + b*x^2)^(3/4)*Hypergeometric2F1[-3/4, 1/2, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^(3/4)","C",1
802,1,49,88,0.0092411,"\int \frac{\left(a+b x^2\right)^{3/4}}{x^2} \, dx","Integrate[(a + b*x^2)^(3/4)/x^2,x]","-\frac{\left(a+b x^2\right)^{3/4} \, _2F_1\left(-\frac{3}{4},-\frac{1}{2};\frac{1}{2};-\frac{b x^2}{a}\right)}{x \left(\frac{b x^2}{a}+1\right)^{3/4}}","\frac{3 b x}{\sqrt[4]{a+b x^2}}-\frac{\left(a+b x^2\right)^{3/4}}{x}-\frac{3 \sqrt{a} \sqrt{b} \sqrt[4]{\frac{b x^2}{a}+1} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{\sqrt[4]{a+b x^2}}",1,"-(((a + b*x^2)^(3/4)*Hypergeometric2F1[-3/4, -1/2, 1/2, -((b*x^2)/a)])/(x*(1 + (b*x^2)/a)^(3/4)))","C",1
803,1,51,121,0.0096,"\int \frac{\left(a+b x^2\right)^{3/4}}{x^4} \, dx","Integrate[(a + b*x^2)^(3/4)/x^4,x]","-\frac{\left(a+b x^2\right)^{3/4} \, _2F_1\left(-\frac{3}{2},-\frac{3}{4};-\frac{1}{2};-\frac{b x^2}{a}\right)}{3 x^3 \left(\frac{b x^2}{a}+1\right)^{3/4}}","-\frac{b^{3/2} \sqrt[4]{\frac{b x^2}{a}+1} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{2 \sqrt{a} \sqrt[4]{a+b x^2}}+\frac{b^2 x}{2 a \sqrt[4]{a+b x^2}}-\frac{b \left(a+b x^2\right)^{3/4}}{2 a x}-\frac{\left(a+b x^2\right)^{3/4}}{3 x^3}",1,"-1/3*((a + b*x^2)^(3/4)*Hypergeometric2F1[-3/2, -3/4, -1/2, -((b*x^2)/a)])/(x^3*(1 + (b*x^2)/a)^(3/4))","C",1
804,1,51,145,0.0093906,"\int \frac{\left(a+b x^2\right)^{3/4}}{x^6} \, dx","Integrate[(a + b*x^2)^(3/4)/x^6,x]","-\frac{\left(a+b x^2\right)^{3/4} \, _2F_1\left(-\frac{5}{2},-\frac{3}{4};-\frac{3}{2};-\frac{b x^2}{a}\right)}{5 x^5 \left(\frac{b x^2}{a}+1\right)^{3/4}}","\frac{3 b^{5/2} \sqrt[4]{\frac{b x^2}{a}+1} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{20 a^{3/2} \sqrt[4]{a+b x^2}}-\frac{3 b^3 x}{20 a^2 \sqrt[4]{a+b x^2}}+\frac{3 b^2 \left(a+b x^2\right)^{3/4}}{20 a^2 x}-\frac{\left(a+b x^2\right)^{3/4}}{5 x^5}-\frac{b \left(a+b x^2\right)^{3/4}}{10 a x^3}",1,"-1/5*((a + b*x^2)^(3/4)*Hypergeometric2F1[-5/2, -3/4, -3/2, -((b*x^2)/a)])/(x^5*(1 + (b*x^2)/a)^(3/4))","C",1
805,1,95,126,0.0601223,"\int x^4 \left(a-b x^2\right)^{3/4} \, dx","Integrate[x^4*(a - b*x^2)^(3/4),x]","-\frac{2 x \left(a-b x^2\right)^{3/4} \left(\left(1-\frac{b x^2}{a}\right)^{3/4} \left(2 a^2+a b x^2-3 b^2 x^4\right)-2 a^2 \, _2F_1\left(-\frac{3}{4},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)\right)}{39 b^2 \left(1-\frac{b x^2}{a}\right)^{3/4}}","\frac{8 a^{7/2} \sqrt[4]{1-\frac{b x^2}{a}} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{65 b^{5/2} \sqrt[4]{a-b x^2}}-\frac{4 a^2 x \left(a-b x^2\right)^{3/4}}{65 b^2}+\frac{2}{13} x^5 \left(a-b x^2\right)^{3/4}-\frac{2 a x^3 \left(a-b x^2\right)^{3/4}}{39 b}",1,"(-2*x*(a - b*x^2)^(3/4)*((1 - (b*x^2)/a)^(3/4)*(2*a^2 + a*b*x^2 - 3*b^2*x^4) - 2*a^2*Hypergeometric2F1[-3/4, 1/2, 3/2, (b*x^2)/a]))/(39*b^2*(1 - (b*x^2)/a)^(3/4))","C",1
806,1,64,101,0.0554756,"\int x^2 \left(a-b x^2\right)^{3/4} \, dx","Integrate[x^2*(a - b*x^2)^(3/4),x]","\frac{2 x \left(a-b x^2\right)^{3/4} \left(\frac{a \, _2F_1\left(-\frac{3}{4},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)}{\left(1-\frac{b x^2}{a}\right)^{3/4}}-a+b x^2\right)}{9 b}","\frac{4 a^{5/2} \sqrt[4]{1-\frac{b x^2}{a}} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{15 b^{3/2} \sqrt[4]{a-b x^2}}-\frac{2 a x \left(a-b x^2\right)^{3/4}}{15 b}+\frac{2}{9} x^3 \left(a-b x^2\right)^{3/4}",1,"(2*x*(a - b*x^2)^(3/4)*(-a + b*x^2 + (a*Hypergeometric2F1[-3/4, 1/2, 3/2, (b*x^2)/a])/(1 - (b*x^2)/a)^(3/4)))/(9*b)","C",1
807,1,47,78,0.0073111,"\int \left(a-b x^2\right)^{3/4} \, dx","Integrate[(a - b*x^2)^(3/4),x]","\frac{x \left(a-b x^2\right)^{3/4} \, _2F_1\left(-\frac{3}{4},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)}{\left(1-\frac{b x^2}{a}\right)^{3/4}}","\frac{6 a^{3/2} \sqrt[4]{1-\frac{b x^2}{a}} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{5 \sqrt{b} \sqrt[4]{a-b x^2}}+\frac{2}{5} x \left(a-b x^2\right)^{3/4}",1,"(x*(a - b*x^2)^(3/4)*Hypergeometric2F1[-3/4, 1/2, 3/2, (b*x^2)/a])/(1 - (b*x^2)/a)^(3/4)","C",1
808,1,50,76,0.0094935,"\int \frac{\left(a-b x^2\right)^{3/4}}{x^2} \, dx","Integrate[(a - b*x^2)^(3/4)/x^2,x]","-\frac{\left(a-b x^2\right)^{3/4} \, _2F_1\left(-\frac{3}{4},-\frac{1}{2};\frac{1}{2};\frac{b x^2}{a}\right)}{x \left(1-\frac{b x^2}{a}\right)^{3/4}}","-\frac{\left(a-b x^2\right)^{3/4}}{x}-\frac{3 \sqrt{a} \sqrt{b} \sqrt[4]{1-\frac{b x^2}{a}} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{\sqrt[4]{a-b x^2}}",1,"-(((a - b*x^2)^(3/4)*Hypergeometric2F1[-3/4, -1/2, 1/2, (b*x^2)/a])/(x*(1 - (b*x^2)/a)^(3/4)))","C",1
809,1,52,103,0.0099426,"\int \frac{\left(a-b x^2\right)^{3/4}}{x^4} \, dx","Integrate[(a - b*x^2)^(3/4)/x^4,x]","-\frac{\left(a-b x^2\right)^{3/4} \, _2F_1\left(-\frac{3}{2},-\frac{3}{4};-\frac{1}{2};\frac{b x^2}{a}\right)}{3 x^3 \left(1-\frac{b x^2}{a}\right)^{3/4}}","\frac{b^{3/2} \sqrt[4]{1-\frac{b x^2}{a}} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{2 \sqrt{a} \sqrt[4]{a-b x^2}}+\frac{b \left(a-b x^2\right)^{3/4}}{2 a x}-\frac{\left(a-b x^2\right)^{3/4}}{3 x^3}",1,"-1/3*((a - b*x^2)^(3/4)*Hypergeometric2F1[-3/2, -3/4, -1/2, (b*x^2)/a])/(x^3*(1 - (b*x^2)/a)^(3/4))","C",1
810,1,52,128,0.0099186,"\int \frac{\left(a-b x^2\right)^{3/4}}{x^6} \, dx","Integrate[(a - b*x^2)^(3/4)/x^6,x]","-\frac{\left(a-b x^2\right)^{3/4} \, _2F_1\left(-\frac{5}{2},-\frac{3}{4};-\frac{3}{2};\frac{b x^2}{a}\right)}{5 x^5 \left(1-\frac{b x^2}{a}\right)^{3/4}}","\frac{3 b^{5/2} \sqrt[4]{1-\frac{b x^2}{a}} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{20 a^{3/2} \sqrt[4]{a-b x^2}}+\frac{3 b^2 \left(a-b x^2\right)^{3/4}}{20 a^2 x}-\frac{\left(a-b x^2\right)^{3/4}}{5 x^5}+\frac{b \left(a-b x^2\right)^{3/4}}{10 a x^3}",1,"-1/5*((a - b*x^2)^(3/4)*Hypergeometric2F1[-5/2, -3/4, -3/2, (b*x^2)/a])/(x^5*(1 - (b*x^2)/a)^(3/4))","C",1
811,1,47,92,0.008336,"\int \left(a+b x^2\right)^{5/4} \, dx","Integrate[(a + b*x^2)^(5/4),x]","\frac{a x \sqrt[4]{a+b x^2} \, _2F_1\left(-\frac{5}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)}{\sqrt[4]{\frac{b x^2}{a}+1}}","\frac{10 a^{5/2} \left(\frac{b x^2}{a}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{21 \sqrt{b} \left(a+b x^2\right)^{3/4}}+\frac{10}{21} a x \sqrt[4]{a+b x^2}+\frac{2}{7} x \left(a+b x^2\right)^{5/4}",1,"(a*x*(a + b*x^2)^(1/4)*Hypergeometric2F1[-5/4, 1/2, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^(1/4)","C",1
812,1,48,96,0.0096047,"\int \left(a-b x^2\right)^{5/4} \, dx","Integrate[(a - b*x^2)^(5/4),x]","\frac{a x \sqrt[4]{a-b x^2} \, _2F_1\left(-\frac{5}{4},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)}{\sqrt[4]{1-\frac{b x^2}{a}}}","\frac{10 a^{5/2} \left(1-\frac{b x^2}{a}\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{21 \sqrt{b} \left(a-b x^2\right)^{3/4}}+\frac{10}{21} a x \sqrt[4]{a-b x^2}+\frac{2}{7} x \left(a-b x^2\right)^{5/4}",1,"(a*x*(a - b*x^2)^(1/4)*Hypergeometric2F1[-5/4, 1/2, 3/2, (b*x^2)/a])/(1 - (b*x^2)/a)^(1/4)","C",1
813,1,47,111,0.007021,"\int \left(a+b x^2\right)^{7/4} \, dx","Integrate[(a + b*x^2)^(7/4),x]","\frac{a x \left(a+b x^2\right)^{3/4} \, _2F_1\left(-\frac{7}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)}{\left(\frac{b x^2}{a}+1\right)^{3/4}}","-\frac{14 a^{5/2} \sqrt[4]{\frac{b x^2}{a}+1} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{15 \sqrt{b} \sqrt[4]{a+b x^2}}+\frac{14 a^2 x}{15 \sqrt[4]{a+b x^2}}+\frac{14}{45} a x \left(a+b x^2\right)^{3/4}+\frac{2}{9} x \left(a+b x^2\right)^{7/4}",1,"(a*x*(a + b*x^2)^(3/4)*Hypergeometric2F1[-7/4, 1/2, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^(3/4)","C",1
814,1,48,96,0.0077238,"\int \left(a-b x^2\right)^{7/4} \, dx","Integrate[(a - b*x^2)^(7/4),x]","\frac{a x \left(a-b x^2\right)^{3/4} \, _2F_1\left(-\frac{7}{4},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)}{\left(1-\frac{b x^2}{a}\right)^{3/4}}","\frac{14 a^{5/2} \sqrt[4]{1-\frac{b x^2}{a}} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{15 \sqrt{b} \sqrt[4]{a-b x^2}}+\frac{14}{45} a x \left(a-b x^2\right)^{3/4}+\frac{2}{9} x \left(a-b x^2\right)^{7/4}",1,"(a*x*(a - b*x^2)^(3/4)*Hypergeometric2F1[-7/4, 1/2, 3/2, (b*x^2)/a])/(1 - (b*x^2)/a)^(3/4)","C",1
815,1,90,146,0.0334256,"\int \frac{x^6}{\sqrt[4]{a+b x^2}} \, dx","Integrate[x^6/(a + b*x^2)^(1/4),x]","\frac{2 \left(-12 a^3 x \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)+12 a^3 x+2 a^2 b x^3-a b^2 x^5+9 b^3 x^7\right)}{117 b^3 \sqrt[4]{a+b x^2}}","\frac{16 a^{7/2} \sqrt[4]{\frac{b x^2}{a}+1} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{39 b^{7/2} \sqrt[4]{a+b x^2}}-\frac{16 a^3 x}{39 b^3 \sqrt[4]{a+b x^2}}+\frac{8 a^2 x \left(a+b x^2\right)^{3/4}}{39 b^3}-\frac{20 a x^3 \left(a+b x^2\right)^{3/4}}{117 b^2}+\frac{2 x^5 \left(a+b x^2\right)^{3/4}}{13 b}",1,"(2*(12*a^3*x + 2*a^2*b*x^3 - a*b^2*x^5 + 9*b^3*x^7 - 12*a^3*x*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, -((b*x^2)/a)]))/(117*b^3*(a + b*x^2)^(1/4))","C",1
816,1,79,122,0.0227145,"\int \frac{x^4}{\sqrt[4]{a+b x^2}} \, dx","Integrate[x^4/(a + b*x^2)^(1/4),x]","\frac{2 \left(6 a^2 x \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)-6 a^2 x-a b x^3+5 b^2 x^5\right)}{45 b^2 \sqrt[4]{a+b x^2}}","-\frac{8 a^{5/2} \sqrt[4]{\frac{b x^2}{a}+1} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{15 b^{5/2} \sqrt[4]{a+b x^2}}+\frac{8 a^2 x}{15 b^2 \sqrt[4]{a+b x^2}}-\frac{4 a x \left(a+b x^2\right)^{3/4}}{15 b^2}+\frac{2 x^3 \left(a+b x^2\right)^{3/4}}{9 b}",1,"(2*(-6*a^2*x - a*b*x^3 + 5*b^2*x^5 + 6*a^2*x*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, -((b*x^2)/a)]))/(45*b^2*(a + b*x^2)^(1/4))","C",1
817,1,62,98,0.0183433,"\int \frac{x^2}{\sqrt[4]{a+b x^2}} \, dx","Integrate[x^2/(a + b*x^2)^(1/4),x]","\frac{2 x \left(-a \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)+a+b x^2\right)}{5 b \sqrt[4]{a+b x^2}}","\frac{4 a^{3/2} \sqrt[4]{\frac{b x^2}{a}+1} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{5 b^{3/2} \sqrt[4]{a+b x^2}}-\frac{4 a x}{5 b \sqrt[4]{a+b x^2}}+\frac{2 x \left(a+b x^2\right)^{3/4}}{5 b}",1,"(2*x*(a + b*x^2 - a*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, -((b*x^2)/a)]))/(5*b*(a + b*x^2)^(1/4))","C",1
818,1,46,71,0.0067022,"\int \frac{1}{\sqrt[4]{a+b x^2}} \, dx","Integrate[(a + b*x^2)^(-1/4),x]","\frac{x \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)}{\sqrt[4]{a+b x^2}}","\frac{2 x}{\sqrt[4]{a+b x^2}}-\frac{2 \sqrt{a} \sqrt[4]{\frac{b x^2}{a}+1} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{\sqrt{b} \sqrt[4]{a+b x^2}}",1,"(x*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, -((b*x^2)/a)])/(a + b*x^2)^(1/4)","C",1
819,1,49,93,0.0089452,"\int \frac{1}{x^2 \sqrt[4]{a+b x^2}} \, dx","Integrate[1/(x^2*(a + b*x^2)^(1/4)),x]","-\frac{\sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{1}{2};-\frac{b x^2}{a}\right)}{x \sqrt[4]{a+b x^2}}","\frac{b x}{a \sqrt[4]{a+b x^2}}-\frac{\left(a+b x^2\right)^{3/4}}{a x}-\frac{\sqrt{b} \sqrt[4]{\frac{b x^2}{a}+1} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{\sqrt{a} \sqrt[4]{a+b x^2}}",1,"-(((1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[-1/2, 1/4, 1/2, -((b*x^2)/a)])/(x*(a + b*x^2)^(1/4)))","C",1
820,1,51,124,0.0094411,"\int \frac{1}{x^4 \sqrt[4]{a+b x^2}} \, dx","Integrate[1/(x^4*(a + b*x^2)^(1/4)),x]","-\frac{\sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{3}{2},\frac{1}{4};-\frac{1}{2};-\frac{b x^2}{a}\right)}{3 x^3 \sqrt[4]{a+b x^2}}","\frac{b^{3/2} \sqrt[4]{\frac{b x^2}{a}+1} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{2 a^{3/2} \sqrt[4]{a+b x^2}}-\frac{b^2 x}{2 a^2 \sqrt[4]{a+b x^2}}+\frac{b \left(a+b x^2\right)^{3/4}}{2 a^2 x}-\frac{\left(a+b x^2\right)^{3/4}}{3 a x^3}",1,"-1/3*((1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[-3/2, 1/4, -1/2, -((b*x^2)/a)])/(x^3*(a + b*x^2)^(1/4))","C",1
821,1,51,148,0.00843,"\int \frac{1}{x^6 \sqrt[4]{a+b x^2}} \, dx","Integrate[1/(x^6*(a + b*x^2)^(1/4)),x]","-\frac{\sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{5}{2},\frac{1}{4};-\frac{3}{2};-\frac{b x^2}{a}\right)}{5 x^5 \sqrt[4]{a+b x^2}}","-\frac{7 b^{5/2} \sqrt[4]{\frac{b x^2}{a}+1} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{20 a^{5/2} \sqrt[4]{a+b x^2}}+\frac{7 b^3 x}{20 a^3 \sqrt[4]{a+b x^2}}-\frac{7 b^2 \left(a+b x^2\right)^{3/4}}{20 a^3 x}+\frac{7 b \left(a+b x^2\right)^{3/4}}{30 a^2 x^3}-\frac{\left(a+b x^2\right)^{3/4}}{5 a x^5}",1,"-1/5*((1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[-5/2, 1/4, -3/2, -((b*x^2)/a)])/(x^5*(a + b*x^2)^(1/4))","C",1
822,1,89,129,0.0337042,"\int \frac{x^6}{\sqrt[4]{a-b x^2}} \, dx","Integrate[x^6/(a - b*x^2)^(1/4),x]","\frac{2 x \left(12 a^3 \sqrt[4]{1-\frac{b x^2}{a}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)-12 a^3+2 a^2 b x^2+a b^2 x^4+9 b^3 x^6\right)}{117 b^3 \sqrt[4]{a-b x^2}}","\frac{16 a^{7/2} \sqrt[4]{1-\frac{b x^2}{a}} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{39 b^{7/2} \sqrt[4]{a-b x^2}}-\frac{8 a^2 x \left(a-b x^2\right)^{3/4}}{39 b^3}-\frac{20 a x^3 \left(a-b x^2\right)^{3/4}}{117 b^2}-\frac{2 x^5 \left(a-b x^2\right)^{3/4}}{13 b}",1,"(2*x*(-12*a^3 + 2*a^2*b*x^2 + a*b^2*x^4 + 9*b^3*x^6 + 12*a^3*(1 - (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (b*x^2)/a]))/(117*b^3*(a - b*x^2)^(1/4))","C",1
823,1,79,104,0.0255802,"\int \frac{x^4}{\sqrt[4]{a-b x^2}} \, dx","Integrate[x^4/(a - b*x^2)^(1/4),x]","\frac{2 \left(6 a^2 x \sqrt[4]{1-\frac{b x^2}{a}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)-6 a^2 x+a b x^3+5 b^2 x^5\right)}{45 b^2 \sqrt[4]{a-b x^2}}","\frac{8 a^{5/2} \sqrt[4]{1-\frac{b x^2}{a}} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{15 b^{5/2} \sqrt[4]{a-b x^2}}-\frac{4 a x \left(a-b x^2\right)^{3/4}}{15 b^2}-\frac{2 x^3 \left(a-b x^2\right)^{3/4}}{9 b}",1,"(2*(-6*a^2*x + a*b*x^3 + 5*b^2*x^5 + 6*a^2*x*(1 - (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (b*x^2)/a]))/(45*b^2*(a - b*x^2)^(1/4))","C",1
824,1,64,81,0.0206195,"\int \frac{x^2}{\sqrt[4]{a-b x^2}} \, dx","Integrate[x^2/(a - b*x^2)^(1/4),x]","\frac{2 x \left(a \sqrt[4]{1-\frac{b x^2}{a}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)-a+b x^2\right)}{5 b \sqrt[4]{a-b x^2}}","\frac{4 a^{3/2} \sqrt[4]{1-\frac{b x^2}{a}} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{5 b^{3/2} \sqrt[4]{a-b x^2}}-\frac{2 x \left(a-b x^2\right)^{3/4}}{5 b}",1,"(2*x*(-a + b*x^2 + a*(1 - (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (b*x^2)/a]))/(5*b*(a - b*x^2)^(1/4))","C",1
825,1,47,58,0.0081621,"\int \frac{1}{\sqrt[4]{a-b x^2}} \, dx","Integrate[(a - b*x^2)^(-1/4),x]","\frac{x \sqrt[4]{1-\frac{b x^2}{a}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)}{\sqrt[4]{a-b x^2}}","\frac{2 \sqrt{a} \sqrt[4]{1-\frac{b x^2}{a}} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{\sqrt{b} \sqrt[4]{a-b x^2}}",1,"(x*(1 - (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (b*x^2)/a])/(a - b*x^2)^(1/4)","C",1
826,1,50,79,0.0095201,"\int \frac{1}{x^2 \sqrt[4]{a-b x^2}} \, dx","Integrate[1/(x^2*(a - b*x^2)^(1/4)),x]","-\frac{\sqrt[4]{1-\frac{b x^2}{a}} \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{1}{2};\frac{b x^2}{a}\right)}{x \sqrt[4]{a-b x^2}}","-\frac{\left(a-b x^2\right)^{3/4}}{a x}-\frac{\sqrt{b} \sqrt[4]{1-\frac{b x^2}{a}} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{\sqrt{a} \sqrt[4]{a-b x^2}}",1,"-(((1 - (b*x^2)/a)^(1/4)*Hypergeometric2F1[-1/2, 1/4, 1/2, (b*x^2)/a])/(x*(a - b*x^2)^(1/4)))","C",1
827,1,52,106,0.0103369,"\int \frac{1}{x^4 \sqrt[4]{a-b x^2}} \, dx","Integrate[1/(x^4*(a - b*x^2)^(1/4)),x]","-\frac{\sqrt[4]{1-\frac{b x^2}{a}} \, _2F_1\left(-\frac{3}{2},\frac{1}{4};-\frac{1}{2};\frac{b x^2}{a}\right)}{3 x^3 \sqrt[4]{a-b x^2}}","-\frac{b^{3/2} \sqrt[4]{1-\frac{b x^2}{a}} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{2 a^{3/2} \sqrt[4]{a-b x^2}}-\frac{b \left(a-b x^2\right)^{3/4}}{2 a^2 x}-\frac{\left(a-b x^2\right)^{3/4}}{3 a x^3}",1,"-1/3*((1 - (b*x^2)/a)^(1/4)*Hypergeometric2F1[-3/2, 1/4, -1/2, (b*x^2)/a])/(x^3*(a - b*x^2)^(1/4))","C",1
828,1,52,131,0.0102976,"\int \frac{1}{x^6 \sqrt[4]{a-b x^2}} \, dx","Integrate[1/(x^6*(a - b*x^2)^(1/4)),x]","-\frac{\sqrt[4]{1-\frac{b x^2}{a}} \, _2F_1\left(-\frac{5}{2},\frac{1}{4};-\frac{3}{2};\frac{b x^2}{a}\right)}{5 x^5 \sqrt[4]{a-b x^2}}","-\frac{7 b^{5/2} \sqrt[4]{1-\frac{b x^2}{a}} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{20 a^{5/2} \sqrt[4]{a-b x^2}}-\frac{7 b^2 \left(a-b x^2\right)^{3/4}}{20 a^3 x}-\frac{7 b \left(a-b x^2\right)^{3/4}}{30 a^2 x^3}-\frac{\left(a-b x^2\right)^{3/4}}{5 a x^5}",1,"-1/5*((1 - (b*x^2)/a)^(1/4)*Hypergeometric2F1[-5/2, 1/4, -3/2, (b*x^2)/a])/(x^5*(a - b*x^2)^(1/4))","C",1
829,1,90,124,0.0351701,"\int \frac{x^6}{\left(a+b x^2\right)^{3/4}} \, dx","Integrate[x^6/(a + b*x^2)^(3/4),x]","\frac{2 \left(-20 a^3 x \left(\frac{b x^2}{a}+1\right)^{3/4} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^2}{a}\right)+20 a^3 x+10 a^2 b x^3-3 a b^2 x^5+7 b^3 x^7\right)}{77 b^3 \left(a+b x^2\right)^{3/4}}","-\frac{80 a^{7/2} \left(\frac{b x^2}{a}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{77 b^{7/2} \left(a+b x^2\right)^{3/4}}+\frac{40 a^2 x \sqrt[4]{a+b x^2}}{77 b^3}-\frac{20 a x^3 \sqrt[4]{a+b x^2}}{77 b^2}+\frac{2 x^5 \sqrt[4]{a+b x^2}}{11 b}",1,"(2*(20*a^3*x + 10*a^2*b*x^3 - 3*a*b^2*x^5 + 7*b^3*x^7 - 20*a^3*x*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, -((b*x^2)/a)]))/(77*b^3*(a + b*x^2)^(3/4))","C",1
830,1,78,100,0.0295797,"\int \frac{x^4}{\left(a+b x^2\right)^{3/4}} \, dx","Integrate[x^4/(a + b*x^2)^(3/4),x]","\frac{2 \left(2 a^2 x \left(\frac{b x^2}{a}+1\right)^{3/4} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^2}{a}\right)-2 a^2 x-a b x^3+b^2 x^5\right)}{7 b^2 \left(a+b x^2\right)^{3/4}}","\frac{8 a^{5/2} \left(\frac{b x^2}{a}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{7 b^{5/2} \left(a+b x^2\right)^{3/4}}-\frac{4 a x \sqrt[4]{a+b x^2}}{7 b^2}+\frac{2 x^3 \sqrt[4]{a+b x^2}}{7 b}",1,"(2*(-2*a^2*x - a*b*x^3 + b^2*x^5 + 2*a^2*x*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, -((b*x^2)/a)]))/(7*b^2*(a + b*x^2)^(3/4))","C",1
831,1,62,78,0.0202917,"\int \frac{x^2}{\left(a+b x^2\right)^{3/4}} \, dx","Integrate[x^2/(a + b*x^2)^(3/4),x]","\frac{2 x \left(-a \left(\frac{b x^2}{a}+1\right)^{3/4} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^2}{a}\right)+a+b x^2\right)}{3 b \left(a+b x^2\right)^{3/4}}","\frac{2 x \sqrt[4]{a+b x^2}}{3 b}-\frac{4 a^{3/2} \left(\frac{b x^2}{a}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{3 b^{3/2} \left(a+b x^2\right)^{3/4}}",1,"(2*x*(a + b*x^2 - a*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, -((b*x^2)/a)]))/(3*b*(a + b*x^2)^(3/4))","C",1
832,1,46,56,0.0066289,"\int \frac{1}{\left(a+b x^2\right)^{3/4}} \, dx","Integrate[(a + b*x^2)^(-3/4),x]","\frac{x \left(\frac{b x^2}{a}+1\right)^{3/4} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^2}{a}\right)}{\left(a+b x^2\right)^{3/4}}","\frac{2 \sqrt{a} \left(\frac{b x^2}{a}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{\sqrt{b} \left(a+b x^2\right)^{3/4}}",1,"(x*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, -((b*x^2)/a)])/(a + b*x^2)^(3/4)","C",1
833,1,49,76,0.0102601,"\int \frac{1}{x^2 \left(a+b x^2\right)^{3/4}} \, dx","Integrate[1/(x^2*(a + b*x^2)^(3/4)),x]","-\frac{\left(\frac{b x^2}{a}+1\right)^{3/4} \, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{1}{2};-\frac{b x^2}{a}\right)}{x \left(a+b x^2\right)^{3/4}}","-\frac{\sqrt{b} \left(\frac{b x^2}{a}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{\sqrt{a} \left(a+b x^2\right)^{3/4}}-\frac{\sqrt[4]{a+b x^2}}{a x}",1,"-(((1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[-1/2, 3/4, 1/2, -((b*x^2)/a)])/(x*(a + b*x^2)^(3/4)))","C",1
834,1,51,102,0.0094615,"\int \frac{1}{x^4 \left(a+b x^2\right)^{3/4}} \, dx","Integrate[1/(x^4*(a + b*x^2)^(3/4)),x]","-\frac{\left(\frac{b x^2}{a}+1\right)^{3/4} \, _2F_1\left(-\frac{3}{2},\frac{3}{4};-\frac{1}{2};-\frac{b x^2}{a}\right)}{3 x^3 \left(a+b x^2\right)^{3/4}}","\frac{5 b^{3/2} \left(\frac{b x^2}{a}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{6 a^{3/2} \left(a+b x^2\right)^{3/4}}+\frac{5 b \sqrt[4]{a+b x^2}}{6 a^2 x}-\frac{\sqrt[4]{a+b x^2}}{3 a x^3}",1,"-1/3*((1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[-3/2, 3/4, -1/2, -((b*x^2)/a)])/(x^3*(a + b*x^2)^(3/4))","C",1
835,1,51,126,0.0096975,"\int \frac{1}{x^6 \left(a+b x^2\right)^{3/4}} \, dx","Integrate[1/(x^6*(a + b*x^2)^(3/4)),x]","-\frac{\left(\frac{b x^2}{a}+1\right)^{3/4} \, _2F_1\left(-\frac{5}{2},\frac{3}{4};-\frac{3}{2};-\frac{b x^2}{a}\right)}{5 x^5 \left(a+b x^2\right)^{3/4}}","-\frac{3 b^{5/2} \left(\frac{b x^2}{a}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{4 a^{5/2} \left(a+b x^2\right)^{3/4}}-\frac{3 b^2 \sqrt[4]{a+b x^2}}{4 a^3 x}+\frac{3 b \sqrt[4]{a+b x^2}}{10 a^2 x^3}-\frac{\sqrt[4]{a+b x^2}}{5 a x^5}",1,"-1/5*((1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[-5/2, 3/4, -3/2, -((b*x^2)/a)])/(x^5*(a + b*x^2)^(3/4))","C",1
836,1,91,129,0.0357793,"\int \frac{x^6}{\left(a-b x^2\right)^{3/4}} \, dx","Integrate[x^6/(a - b*x^2)^(3/4),x]","\frac{2 \left(20 a^3 x \left(1-\frac{b x^2}{a}\right)^{3/4} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{b x^2}{a}\right)-20 a^3 x+10 a^2 b x^3+3 a b^2 x^5+7 b^3 x^7\right)}{77 b^3 \left(a-b x^2\right)^{3/4}}","\frac{80 a^{7/2} \left(1-\frac{b x^2}{a}\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{77 b^{7/2} \left(a-b x^2\right)^{3/4}}-\frac{40 a^2 x \sqrt[4]{a-b x^2}}{77 b^3}-\frac{20 a x^3 \sqrt[4]{a-b x^2}}{77 b^2}-\frac{2 x^5 \sqrt[4]{a-b x^2}}{11 b}",1,"(2*(-20*a^3*x + 10*a^2*b*x^3 + 3*a*b^2*x^5 + 7*b^3*x^7 + 20*a^3*x*(1 - (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, (b*x^2)/a]))/(77*b^3*(a - b*x^2)^(3/4))","C",1
837,1,77,104,0.0304206,"\int \frac{x^4}{\left(a-b x^2\right)^{3/4}} \, dx","Integrate[x^4/(a - b*x^2)^(3/4),x]","\frac{2 x \left(2 a^2 \left(1-\frac{b x^2}{a}\right)^{3/4} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{b x^2}{a}\right)-2 a^2+a b x^2+b^2 x^4\right)}{7 b^2 \left(a-b x^2\right)^{3/4}}","\frac{8 a^{5/2} \left(1-\frac{b x^2}{a}\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{7 b^{5/2} \left(a-b x^2\right)^{3/4}}-\frac{4 a x \sqrt[4]{a-b x^2}}{7 b^2}-\frac{2 x^3 \sqrt[4]{a-b x^2}}{7 b}",1,"(2*x*(-2*a^2 + a*b*x^2 + b^2*x^4 + 2*a^2*(1 - (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, (b*x^2)/a]))/(7*b^2*(a - b*x^2)^(3/4))","C",1
838,1,64,81,0.0212043,"\int \frac{x^2}{\left(a-b x^2\right)^{3/4}} \, dx","Integrate[x^2/(a - b*x^2)^(3/4),x]","\frac{2 x \left(a \left(1-\frac{b x^2}{a}\right)^{3/4} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{b x^2}{a}\right)-a+b x^2\right)}{3 b \left(a-b x^2\right)^{3/4}}","\frac{4 a^{3/2} \left(1-\frac{b x^2}{a}\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{3 b^{3/2} \left(a-b x^2\right)^{3/4}}-\frac{2 x \sqrt[4]{a-b x^2}}{3 b}",1,"(2*x*(-a + b*x^2 + a*(1 - (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, (b*x^2)/a]))/(3*b*(a - b*x^2)^(3/4))","C",1
839,1,47,58,0.0078145,"\int \frac{1}{\left(a-b x^2\right)^{3/4}} \, dx","Integrate[(a - b*x^2)^(-3/4),x]","\frac{x \left(1-\frac{b x^2}{a}\right)^{3/4} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{b x^2}{a}\right)}{\left(a-b x^2\right)^{3/4}}","\frac{2 \sqrt{a} \left(1-\frac{b x^2}{a}\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{\sqrt{b} \left(a-b x^2\right)^{3/4}}",1,"(x*(1 - (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, (b*x^2)/a])/(a - b*x^2)^(3/4)","C",1
840,1,50,78,0.0096995,"\int \frac{1}{x^2 \left(a-b x^2\right)^{3/4}} \, dx","Integrate[1/(x^2*(a - b*x^2)^(3/4)),x]","-\frac{\left(1-\frac{b x^2}{a}\right)^{3/4} \, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{1}{2};\frac{b x^2}{a}\right)}{x \left(a-b x^2\right)^{3/4}}","\frac{\sqrt{b} \left(1-\frac{b x^2}{a}\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{\sqrt{a} \left(a-b x^2\right)^{3/4}}-\frac{\sqrt[4]{a-b x^2}}{a x}",1,"-(((1 - (b*x^2)/a)^(3/4)*Hypergeometric2F1[-1/2, 3/4, 1/2, (b*x^2)/a])/(x*(a - b*x^2)^(3/4)))","C",1
841,1,52,106,0.010165,"\int \frac{1}{x^4 \left(a-b x^2\right)^{3/4}} \, dx","Integrate[1/(x^4*(a - b*x^2)^(3/4)),x]","-\frac{\left(1-\frac{b x^2}{a}\right)^{3/4} \, _2F_1\left(-\frac{3}{2},\frac{3}{4};-\frac{1}{2};\frac{b x^2}{a}\right)}{3 x^3 \left(a-b x^2\right)^{3/4}}","\frac{5 b^{3/2} \left(1-\frac{b x^2}{a}\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{6 a^{3/2} \left(a-b x^2\right)^{3/4}}-\frac{5 b \sqrt[4]{a-b x^2}}{6 a^2 x}-\frac{\sqrt[4]{a-b x^2}}{3 a x^3}",1,"-1/3*((1 - (b*x^2)/a)^(3/4)*Hypergeometric2F1[-3/2, 3/4, -1/2, (b*x^2)/a])/(x^3*(a - b*x^2)^(3/4))","C",1
842,1,52,131,0.0101935,"\int \frac{1}{x^6 \left(a-b x^2\right)^{3/4}} \, dx","Integrate[1/(x^6*(a - b*x^2)^(3/4)),x]","-\frac{\left(1-\frac{b x^2}{a}\right)^{3/4} \, _2F_1\left(-\frac{5}{2},\frac{3}{4};-\frac{3}{2};\frac{b x^2}{a}\right)}{5 x^5 \left(a-b x^2\right)^{3/4}}","\frac{3 b^{5/2} \left(1-\frac{b x^2}{a}\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{4 a^{5/2} \left(a-b x^2\right)^{3/4}}-\frac{3 b^2 \sqrt[4]{a-b x^2}}{4 a^3 x}-\frac{3 b \sqrt[4]{a-b x^2}}{10 a^2 x^3}-\frac{\sqrt[4]{a-b x^2}}{5 a x^5}",1,"-1/5*((1 - (b*x^2)/a)^(3/4)*Hypergeometric2F1[-5/2, 3/4, -3/2, (b*x^2)/a])/(x^5*(a - b*x^2)^(3/4))","C",1
843,1,78,124,0.0328394,"\int \frac{x^6}{\left(a+b x^2\right)^{5/4}} \, dx","Integrate[x^6/(a + b*x^2)^(5/4),x]","\frac{2 \left(12 a^2 x \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)-12 a^2 x-2 a b x^3+b^2 x^5\right)}{9 b^3 \sqrt[4]{a+b x^2}}","-\frac{16 a^{5/2} \sqrt[4]{\frac{b x^2}{a}+1} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{3 b^{7/2} \sqrt[4]{a+b x^2}}+\frac{8 a^2 x}{3 b^3 \sqrt[4]{a+b x^2}}-\frac{4 a x^3}{9 b^2 \sqrt[4]{a+b x^2}}+\frac{2 x^5}{9 b \sqrt[4]{a+b x^2}}",1,"(2*(-12*a^2*x - 2*a*b*x^3 + b^2*x^5 + 12*a^2*x*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, -((b*x^2)/a)]))/(9*b^3*(a + b*x^2)^(1/4))","C",1
844,1,65,100,0.0213146,"\int \frac{x^4}{\left(a+b x^2\right)^{5/4}} \, dx","Integrate[x^4/(a + b*x^2)^(5/4),x]","\frac{2 \left(-6 a x \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)+6 a x+b x^3\right)}{5 b^2 \sqrt[4]{a+b x^2}}","\frac{24 a^{3/2} \sqrt[4]{\frac{b x^2}{a}+1} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{5 b^{5/2} \sqrt[4]{a+b x^2}}-\frac{12 a x}{5 b^2 \sqrt[4]{a+b x^2}}+\frac{2 x^3}{5 b \sqrt[4]{a+b x^2}}",1,"(2*(6*a*x + b*x^3 - 6*a*x*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, -((b*x^2)/a)]))/(5*b^2*(a + b*x^2)^(1/4))","C",1
845,1,53,74,0.0187663,"\int \frac{x^2}{\left(a+b x^2\right)^{5/4}} \, dx","Integrate[x^2/(a + b*x^2)^(5/4),x]","\frac{2 x \left(\sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)-1\right)}{b \sqrt[4]{a+b x^2}}","\frac{2 x}{b \sqrt[4]{a+b x^2}}-\frac{4 \sqrt{a} \sqrt[4]{\frac{b x^2}{a}+1} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{b^{3/2} \sqrt[4]{a+b x^2}}",1,"(2*x*(-1 + (1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, -((b*x^2)/a)]))/(b*(a + b*x^2)^(1/4))","C",1
846,1,55,56,0.0136691,"\int \frac{1}{\left(a+b x^2\right)^{5/4}} \, dx","Integrate[(a + b*x^2)^(-5/4),x]","\frac{2 x-x \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)}{a \sqrt[4]{a+b x^2}}","\frac{2 \sqrt[4]{\frac{b x^2}{a}+1} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{\sqrt{a} \sqrt{b} \sqrt[4]{a+b x^2}}",1,"(2*x - x*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, -((b*x^2)/a)])/(a*(a + b*x^2)^(1/4))","C",1
847,1,52,76,0.0090478,"\int \frac{1}{x^2 \left(a+b x^2\right)^{5/4}} \, dx","Integrate[1/(x^2*(a + b*x^2)^(5/4)),x]","-\frac{\sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{1}{2},\frac{5}{4};\frac{1}{2};-\frac{b x^2}{a}\right)}{a x \sqrt[4]{a+b x^2}}","-\frac{3 \sqrt{b} \sqrt[4]{\frac{b x^2}{a}+1} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{a^{3/2} \sqrt[4]{a+b x^2}}-\frac{1}{a x \sqrt[4]{a+b x^2}}",1,"-(((1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[-1/2, 5/4, 1/2, -((b*x^2)/a)])/(a*x*(a + b*x^2)^(1/4)))","C",1
848,1,54,102,0.0096761,"\int \frac{1}{x^4 \left(a+b x^2\right)^{5/4}} \, dx","Integrate[1/(x^4*(a + b*x^2)^(5/4)),x]","-\frac{\sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{3}{2},\frac{5}{4};-\frac{1}{2};-\frac{b x^2}{a}\right)}{3 a x^3 \sqrt[4]{a+b x^2}}","\frac{7 b^{3/2} \sqrt[4]{\frac{b x^2}{a}+1} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{2 a^{5/2} \sqrt[4]{a+b x^2}}+\frac{7 b}{6 a^2 x \sqrt[4]{a+b x^2}}-\frac{1}{3 a x^3 \sqrt[4]{a+b x^2}}",1,"-1/3*((1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[-3/2, 5/4, -1/2, -((b*x^2)/a)])/(a*x^3*(a + b*x^2)^(1/4))","C",1
849,1,54,126,0.0095199,"\int \frac{1}{x^6 \left(a+b x^2\right)^{5/4}} \, dx","Integrate[1/(x^6*(a + b*x^2)^(5/4)),x]","-\frac{\sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{5}{2},\frac{5}{4};-\frac{3}{2};-\frac{b x^2}{a}\right)}{5 a x^5 \sqrt[4]{a+b x^2}}","-\frac{77 b^{5/2} \sqrt[4]{\frac{b x^2}{a}+1} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{20 a^{7/2} \sqrt[4]{a+b x^2}}-\frac{77 b^2}{60 a^3 x \sqrt[4]{a+b x^2}}+\frac{11 b}{30 a^2 x^3 \sqrt[4]{a+b x^2}}-\frac{1}{5 a x^5 \sqrt[4]{a+b x^2}}",1,"-1/5*((1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[-5/2, 5/4, -3/2, -((b*x^2)/a)])/(a*x^5*(a + b*x^2)^(1/4))","C",1
850,1,78,124,0.0319548,"\int \frac{x^6}{\left(a-b x^2\right)^{5/4}} \, dx","Integrate[x^6/(a - b*x^2)^(5/4),x]","-\frac{2 x \left(12 a^2 \sqrt[4]{1-\frac{b x^2}{a}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)-12 a^2+2 a b x^2+b^2 x^4\right)}{9 b^3 \sqrt[4]{a-b x^2}}","-\frac{16 a^{5/2} \sqrt[4]{1-\frac{b x^2}{a}} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{3 b^{7/2} \sqrt[4]{a-b x^2}}+\frac{8 a x \left(a-b x^2\right)^{3/4}}{3 b^3}+\frac{20 x^3 \left(a-b x^2\right)^{3/4}}{9 b^2}+\frac{2 x^5}{b \sqrt[4]{a-b x^2}}",1,"(-2*x*(-12*a^2 + 2*a*b*x^2 + b^2*x^4 + 12*a^2*(1 - (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (b*x^2)/a]))/(9*b^3*(a - b*x^2)^(1/4))","C",1
851,1,66,101,0.022577,"\int \frac{x^4}{\left(a-b x^2\right)^{5/4}} \, dx","Integrate[x^4/(a - b*x^2)^(5/4),x]","-\frac{2 \left(6 a x \sqrt[4]{1-\frac{b x^2}{a}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)-6 a x+b x^3\right)}{5 b^2 \sqrt[4]{a-b x^2}}","-\frac{24 a^{3/2} \sqrt[4]{1-\frac{b x^2}{a}} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{5 b^{5/2} \sqrt[4]{a-b x^2}}+\frac{12 x \left(a-b x^2\right)^{3/4}}{5 b^2}+\frac{2 x^3}{b \sqrt[4]{a-b x^2}}",1,"(-2*(-6*a*x + b*x^3 + 6*a*x*(1 - (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (b*x^2)/a]))/(5*b^2*(a - b*x^2)^(1/4))","C",1
852,1,56,77,0.0177902,"\int \frac{x^2}{\left(a-b x^2\right)^{5/4}} \, dx","Integrate[x^2/(a - b*x^2)^(5/4),x]","\frac{2 x-2 x \sqrt[4]{1-\frac{b x^2}{a}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)}{b \sqrt[4]{a-b x^2}}","\frac{2 x}{b \sqrt[4]{a-b x^2}}-\frac{4 \sqrt{a} \sqrt[4]{1-\frac{b x^2}{a}} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{b^{3/2} \sqrt[4]{a-b x^2}}",1,"(2*x - 2*x*(1 - (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (b*x^2)/a])/(b*(a - b*x^2)^(1/4))","C",1
853,1,56,77,0.0154308,"\int \frac{1}{\left(a-b x^2\right)^{5/4}} \, dx","Integrate[(a - b*x^2)^(-5/4),x]","\frac{2 x-x \sqrt[4]{1-\frac{b x^2}{a}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)}{a \sqrt[4]{a-b x^2}}","\frac{2 x}{a \sqrt[4]{a-b x^2}}-\frac{2 \sqrt[4]{1-\frac{b x^2}{a}} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{\sqrt{a} \sqrt{b} \sqrt[4]{a-b x^2}}",1,"(2*x - x*(1 - (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (b*x^2)/a])/(a*(a - b*x^2)^(1/4))","C",1
854,1,53,99,0.0103783,"\int \frac{1}{x^2 \left(a-b x^2\right)^{5/4}} \, dx","Integrate[1/(x^2*(a - b*x^2)^(5/4)),x]","-\frac{\sqrt[4]{1-\frac{b x^2}{a}} \, _2F_1\left(-\frac{1}{2},\frac{5}{4};\frac{1}{2};\frac{b x^2}{a}\right)}{a x \sqrt[4]{a-b x^2}}","-\frac{3 \sqrt{b} \sqrt[4]{1-\frac{b x^2}{a}} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{a^{3/2} \sqrt[4]{a-b x^2}}-\frac{3 \left(a-b x^2\right)^{3/4}}{a^2 x}+\frac{2}{a x \sqrt[4]{a-b x^2}}",1,"-(((1 - (b*x^2)/a)^(1/4)*Hypergeometric2F1[-1/2, 5/4, 1/2, (b*x^2)/a])/(a*x*(a - b*x^2)^(1/4)))","C",1
855,1,55,126,0.0106054,"\int \frac{1}{x^4 \left(a-b x^2\right)^{5/4}} \, dx","Integrate[1/(x^4*(a - b*x^2)^(5/4)),x]","-\frac{\sqrt[4]{1-\frac{b x^2}{a}} \, _2F_1\left(-\frac{3}{2},\frac{5}{4};-\frac{1}{2};\frac{b x^2}{a}\right)}{3 a x^3 \sqrt[4]{a-b x^2}}","-\frac{7 b^{3/2} \sqrt[4]{1-\frac{b x^2}{a}} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{2 a^{5/2} \sqrt[4]{a-b x^2}}-\frac{7 b \left(a-b x^2\right)^{3/4}}{2 a^3 x}-\frac{7 \left(a-b x^2\right)^{3/4}}{3 a^2 x^3}+\frac{2}{a x^3 \sqrt[4]{a-b x^2}}",1,"-1/3*((1 - (b*x^2)/a)^(1/4)*Hypergeometric2F1[-3/2, 5/4, -1/2, (b*x^2)/a])/(a*x^3*(a - b*x^2)^(1/4))","C",1
856,1,55,151,0.0108577,"\int \frac{1}{x^6 \left(a-b x^2\right)^{5/4}} \, dx","Integrate[1/(x^6*(a - b*x^2)^(5/4)),x]","-\frac{\sqrt[4]{1-\frac{b x^2}{a}} \, _2F_1\left(-\frac{5}{2},\frac{5}{4};-\frac{3}{2};\frac{b x^2}{a}\right)}{5 a x^5 \sqrt[4]{a-b x^2}}","-\frac{77 b^{5/2} \sqrt[4]{1-\frac{b x^2}{a}} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{20 a^{7/2} \sqrt[4]{a-b x^2}}-\frac{77 b^2 \left(a-b x^2\right)^{3/4}}{20 a^4 x}-\frac{77 b \left(a-b x^2\right)^{3/4}}{30 a^3 x^3}-\frac{11 \left(a-b x^2\right)^{3/4}}{5 a^2 x^5}+\frac{2}{a x^5 \sqrt[4]{a-b x^2}}",1,"-1/5*((1 - (b*x^2)/a)^(1/4)*Hypergeometric2F1[-5/2, 5/4, -3/2, (b*x^2)/a])/(a*x^5*(a - b*x^2)^(1/4))","C",1
857,1,55,78,0.0225597,"\int \frac{1}{\left(a+b x^2\right)^{7/4}} \, dx","Integrate[(a + b*x^2)^(-7/4),x]","\frac{x \left(\left(\frac{b x^2}{a}+1\right)^{3/4} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^2}{a}\right)+2\right)}{3 a \left(a+b x^2\right)^{3/4}}","\frac{2 \left(\frac{b x^2}{a}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{3 \sqrt{a} \sqrt{b} \left(a+b x^2\right)^{3/4}}+\frac{2 x}{3 a \left(a+b x^2\right)^{3/4}}",1,"(x*(2 + (1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, -((b*x^2)/a)]))/(3*a*(a + b*x^2)^(3/4))","C",1
858,1,72,78,0.0339224,"\int \frac{1}{\left(a+b x^2\right)^{9/4}} \, dx","Integrate[(a + b*x^2)^(-9/4),x]","\frac{-3 x \left(a+b x^2\right) \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)+8 a x+6 b x^3}{5 a^2 \left(a+b x^2\right)^{5/4}}","\frac{6 \sqrt[4]{\frac{b x^2}{a}+1} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{5 a^{3/2} \sqrt{b} \sqrt[4]{a+b x^2}}+\frac{2 x}{5 a \left(a+b x^2\right)^{5/4}}",1,"(8*a*x + 6*b*x^3 - 3*x*(a + b*x^2)*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, -((b*x^2)/a)])/(5*a^2*(a + b*x^2)^(5/4))","C",1
859,1,75,97,0.0351548,"\int \frac{1}{\left(a+b x^2\right)^{11/4}} \, dx","Integrate[(a + b*x^2)^(-11/4),x]","\frac{5 x \left(a+b x^2\right) \left(\frac{b x^2}{a}+1\right)^{3/4} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^2}{a}\right)+2 x \left(8 a+5 b x^2\right)}{21 a^2 \left(a+b x^2\right)^{7/4}}","\frac{10 \left(\frac{b x^2}{a}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{21 a^{3/2} \sqrt{b} \left(a+b x^2\right)^{3/4}}+\frac{10 x}{21 a^2 \left(a+b x^2\right)^{3/4}}+\frac{2 x}{7 a \left(a+b x^2\right)^{7/4}}",1,"(2*x*(8*a + 5*b*x^2) + 5*x*(a + b*x^2)*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, -((b*x^2)/a)])/(21*a^2*(a + b*x^2)^(7/4))","C",1
860,1,56,81,0.022523,"\int \frac{1}{\left(a-b x^2\right)^{7/4}} \, dx","Integrate[(a - b*x^2)^(-7/4),x]","\frac{x \left(\left(1-\frac{b x^2}{a}\right)^{3/4} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{b x^2}{a}\right)+2\right)}{3 a \left(a-b x^2\right)^{3/4}}","\frac{2 \left(1-\frac{b x^2}{a}\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{3 \sqrt{a} \sqrt{b} \left(a-b x^2\right)^{3/4}}+\frac{2 x}{3 a \left(a-b x^2\right)^{3/4}}",1,"(x*(2 + (1 - (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, (b*x^2)/a]))/(3*a*(a - b*x^2)^(3/4))","C",1
861,1,74,101,0.0363526,"\int \frac{1}{\left(a-b x^2\right)^{9/4}} \, dx","Integrate[(a - b*x^2)^(-9/4),x]","\frac{-3 x \left(a-b x^2\right) \sqrt[4]{1-\frac{b x^2}{a}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right)+8 a x-6 b x^3}{5 a^2 \left(a-b x^2\right)^{5/4}}","-\frac{6 \sqrt[4]{1-\frac{b x^2}{a}} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{5 a^{3/2} \sqrt{b} \sqrt[4]{a-b x^2}}+\frac{6 x}{5 a^2 \sqrt[4]{a-b x^2}}+\frac{2 x}{5 a \left(a-b x^2\right)^{5/4}}",1,"(8*a*x - 6*b*x^3 - 3*x*(a - b*x^2)*(1 - (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (b*x^2)/a])/(5*a^2*(a - b*x^2)^(5/4))","C",1
862,1,77,101,0.0358217,"\int \frac{1}{\left(a-b x^2\right)^{11/4}} \, dx","Integrate[(a - b*x^2)^(-11/4),x]","\frac{5 x \left(a-b x^2\right) \left(1-\frac{b x^2}{a}\right)^{3/4} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{b x^2}{a}\right)+2 x \left(8 a-5 b x^2\right)}{21 a^2 \left(a-b x^2\right)^{7/4}}","\frac{10 \left(1-\frac{b x^2}{a}\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \sin ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{21 a^{3/2} \sqrt{b} \left(a-b x^2\right)^{3/4}}+\frac{10 x}{21 a^2 \left(a-b x^2\right)^{3/4}}+\frac{2 x}{7 a \left(a-b x^2\right)^{7/4}}",1,"(2*x*(8*a - 5*b*x^2) + 5*x*(a - b*x^2)*(1 - (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, (b*x^2)/a])/(21*a^2*(a - b*x^2)^(7/4))","C",1
863,1,54,99,0.0291791,"\int \frac{x^6}{\sqrt[4]{2+3 x^2}} \, dx","Integrate[x^6/(2 + 3*x^2)^(1/4),x]","\frac{2 x \left(\left(3 x^2+2\right)^{3/4} \left(27 x^4-20 x^2+16\right)-16\ 2^{3/4} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{3 x^2}{2}\right)\right)}{1053}","\frac{32 \left(3 x^2+2\right)^{3/4} x}{1053}-\frac{128 x}{1053 \sqrt[4]{3 x^2+2}}+\frac{2}{39} \left(3 x^2+2\right)^{3/4} x^5-\frac{40 \left(3 x^2+2\right)^{3/4} x^3}{1053}+\frac{128 \sqrt[4]{2} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{1053 \sqrt{3}}",1,"(2*x*((2 + 3*x^2)^(3/4)*(16 - 20*x^2 + 27*x^4) - 16*2^(3/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (-3*x^2)/2]))/1053","C",1
864,1,49,81,0.019005,"\int \frac{x^4}{\sqrt[4]{2+3 x^2}} \, dx","Integrate[x^4/(2 + 3*x^2)^(1/4),x]","\frac{2}{135} x \left(4\ 2^{3/4} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{3 x^2}{2}\right)+\left(3 x^2+2\right)^{3/4} \left(5 x^2-4\right)\right)","-\frac{8}{135} \left(3 x^2+2\right)^{3/4} x+\frac{32 x}{135 \sqrt[4]{3 x^2+2}}+\frac{2}{27} \left(3 x^2+2\right)^{3/4} x^3-\frac{32 \sqrt[4]{2} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{135 \sqrt{3}}",1,"(2*x*((2 + 3*x^2)^(3/4)*(-4 + 5*x^2) + 4*2^(3/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (-3*x^2)/2]))/135","C",1
865,1,41,63,0.0105775,"\int \frac{x^2}{\sqrt[4]{2+3 x^2}} \, dx","Integrate[x^2/(2 + 3*x^2)^(1/4),x]","\frac{2}{15} x \left(\left(3 x^2+2\right)^{3/4}-2^{3/4} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{3 x^2}{2}\right)\right)","\frac{2}{15} \left(3 x^2+2\right)^{3/4} x-\frac{8 x}{15 \sqrt[4]{3 x^2+2}}+\frac{8 \sqrt[4]{2} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{15 \sqrt{3}}",1,"(2*x*((2 + 3*x^2)^(3/4) - 2^(3/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (-3*x^2)/2]))/15","C",1
866,1,24,43,0.0044788,"\int \frac{1}{\sqrt[4]{2+3 x^2}} \, dx","Integrate[(2 + 3*x^2)^(-1/4),x]","\frac{x \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{3 x^2}{2}\right)}{\sqrt[4]{2}}","\frac{2 x}{\sqrt[4]{3 x^2+2}}-\frac{2 \sqrt[4]{2} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{\sqrt{3}}",1,"(x*Hypergeometric2F1[1/4, 1/2, 3/2, (-3*x^2)/2])/2^(1/4)","C",1
867,1,27,63,0.0044126,"\int \frac{1}{x^2 \sqrt[4]{2+3 x^2}} \, dx","Integrate[1/(x^2*(2 + 3*x^2)^(1/4)),x]","-\frac{\, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{1}{2};-\frac{3 x^2}{2}\right)}{\sqrt[4]{2} x}","\frac{3 x}{2 \sqrt[4]{3 x^2+2}}-\frac{\left(3 x^2+2\right)^{3/4}}{2 x}-\frac{\sqrt{3} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{2^{3/4}}",1,"-(Hypergeometric2F1[-1/2, 1/4, 1/2, (-3*x^2)/2]/(2^(1/4)*x))","C",1
868,1,29,83,0.0060518,"\int \frac{1}{x^4 \sqrt[4]{2+3 x^2}} \, dx","Integrate[1/(x^4*(2 + 3*x^2)^(1/4)),x]","-\frac{\, _2F_1\left(-\frac{3}{2},\frac{1}{4};-\frac{1}{2};-\frac{3 x^2}{2}\right)}{3 \sqrt[4]{2} x^3}","-\frac{9 x}{8 \sqrt[4]{3 x^2+2}}+\frac{3 \left(3 x^2+2\right)^{3/4}}{8 x}-\frac{\left(3 x^2+2\right)^{3/4}}{6 x^3}+\frac{3 \sqrt{3} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{4\ 2^{3/4}}",1,"-1/3*Hypergeometric2F1[-3/2, 1/4, -1/2, (-3*x^2)/2]/(2^(1/4)*x^3)","C",1
869,1,29,101,0.0055427,"\int \frac{1}{x^6 \sqrt[4]{2+3 x^2}} \, dx","Integrate[1/(x^6*(2 + 3*x^2)^(1/4)),x]","-\frac{\, _2F_1\left(-\frac{5}{2},\frac{1}{4};-\frac{3}{2};-\frac{3 x^2}{2}\right)}{5 \sqrt[4]{2} x^5}","\frac{189 x}{160 \sqrt[4]{3 x^2+2}}-\frac{63 \left(3 x^2+2\right)^{3/4}}{160 x}-\frac{\left(3 x^2+2\right)^{3/4}}{10 x^5}+\frac{7 \left(3 x^2+2\right)^{3/4}}{40 x^3}-\frac{63 \sqrt{3} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{80\ 2^{3/4}}",1,"-1/5*Hypergeometric2F1[-5/2, 1/4, -3/2, (-3*x^2)/2]/(2^(1/4)*x^5)","C",1
870,1,54,83,0.0255177,"\int \frac{x^6}{\sqrt[4]{2-3 x^2}} \, dx","Integrate[x^6/(2 - 3*x^2)^(1/4),x]","-\frac{2 x \left(\left(2-3 x^2\right)^{3/4} \left(27 x^4+20 x^2+16\right)-16\ 2^{3/4} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{3 x^2}{2}\right)\right)}{1053}","-\frac{32 \left(2-3 x^2\right)^{3/4} x}{1053}-\frac{2}{39} \left(2-3 x^2\right)^{3/4} x^5-\frac{40 \left(2-3 x^2\right)^{3/4} x^3}{1053}+\frac{128 \sqrt[4]{2} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{1053 \sqrt{3}}",1,"(-2*x*((2 - 3*x^2)^(3/4)*(16 + 20*x^2 + 27*x^4) - 16*2^(3/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (3*x^2)/2]))/1053","C",1
871,1,49,65,0.0215819,"\int \frac{x^4}{\sqrt[4]{2-3 x^2}} \, dx","Integrate[x^4/(2 - 3*x^2)^(1/4),x]","-\frac{2}{135} x \left(\left(2-3 x^2\right)^{3/4} \left(5 x^2+4\right)-4\ 2^{3/4} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{3 x^2}{2}\right)\right)","-\frac{8}{135} \left(2-3 x^2\right)^{3/4} x-\frac{2}{27} \left(2-3 x^2\right)^{3/4} x^3+\frac{32 \sqrt[4]{2} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{135 \sqrt{3}}",1,"(-2*x*((2 - 3*x^2)^(3/4)*(4 + 5*x^2) - 4*2^(3/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (3*x^2)/2]))/135","C",1
872,1,41,47,0.0098707,"\int \frac{x^2}{\sqrt[4]{2-3 x^2}} \, dx","Integrate[x^2/(2 - 3*x^2)^(1/4),x]","-\frac{2}{15} x \left(\left(2-3 x^2\right)^{3/4}-2^{3/4} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{3 x^2}{2}\right)\right)","\frac{8 \sqrt[4]{2} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{15 \sqrt{3}}-\frac{2}{15} x \left(2-3 x^2\right)^{3/4}",1,"(-2*x*((2 - 3*x^2)^(3/4) - 2^(3/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (3*x^2)/2]))/15","C",1
873,1,24,28,0.0032655,"\int \frac{1}{\sqrt[4]{2-3 x^2}} \, dx","Integrate[(2 - 3*x^2)^(-1/4),x]","\frac{x \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{3 x^2}{2}\right)}{\sqrt[4]{2}}","\frac{2 \sqrt[4]{2} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{\sqrt{3}}",1,"(x*Hypergeometric2F1[1/4, 1/2, 3/2, (3*x^2)/2])/2^(1/4)","C",1
874,1,27,47,0.0047431,"\int \frac{1}{x^2 \sqrt[4]{2-3 x^2}} \, dx","Integrate[1/(x^2*(2 - 3*x^2)^(1/4)),x]","-\frac{\, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{1}{2};\frac{3 x^2}{2}\right)}{\sqrt[4]{2} x}","-\frac{\left(2-3 x^2\right)^{3/4}}{2 x}-\frac{\sqrt{3} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{2^{3/4}}",1,"-(Hypergeometric2F1[-1/2, 1/4, 1/2, (3*x^2)/2]/(2^(1/4)*x))","C",1
875,1,29,67,0.0051024,"\int \frac{1}{x^4 \sqrt[4]{2-3 x^2}} \, dx","Integrate[1/(x^4*(2 - 3*x^2)^(1/4)),x]","-\frac{\, _2F_1\left(-\frac{3}{2},\frac{1}{4};-\frac{1}{2};\frac{3 x^2}{2}\right)}{3 \sqrt[4]{2} x^3}","-\frac{3 \left(2-3 x^2\right)^{3/4}}{8 x}-\frac{\left(2-3 x^2\right)^{3/4}}{6 x^3}-\frac{3 \sqrt{3} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{4\ 2^{3/4}}",1,"-1/3*Hypergeometric2F1[-3/2, 1/4, -1/2, (3*x^2)/2]/(2^(1/4)*x^3)","C",1
876,1,29,85,0.0048416,"\int \frac{1}{x^6 \sqrt[4]{2-3 x^2}} \, dx","Integrate[1/(x^6*(2 - 3*x^2)^(1/4)),x]","-\frac{\, _2F_1\left(-\frac{5}{2},\frac{1}{4};-\frac{3}{2};\frac{3 x^2}{2}\right)}{5 \sqrt[4]{2} x^5}","-\frac{63 \left(2-3 x^2\right)^{3/4}}{160 x}-\frac{\left(2-3 x^2\right)^{3/4}}{10 x^5}-\frac{7 \left(2-3 x^2\right)^{3/4}}{40 x^3}-\frac{63 \sqrt{3} E\left(\left.\frac{1}{2} \sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right|2\right)}{80\ 2^{3/4}}",1,"-1/5*Hypergeometric2F1[-5/2, 1/4, -3/2, (3*x^2)/2]/(2^(1/4)*x^5)","C",1
877,1,54,83,0.0206664,"\int \frac{x^6}{\left(2+3 x^2\right)^{3/4}} \, dx","Integrate[x^6/(2 + 3*x^2)^(3/4),x]","\frac{2 x \left(\sqrt[4]{3 x^2+2} \left(63 x^4-60 x^2+80\right)-80 \sqrt[4]{2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{3 x^2}{2}\right)\right)}{2079}","-\frac{320\ 2^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\sqrt{\frac{3}{2}} x\right),2\right)}{2079 \sqrt{3}}+\frac{160 \sqrt[4]{3 x^2+2} x}{2079}+\frac{2}{33} \sqrt[4]{3 x^2+2} x^5-\frac{40}{693} \sqrt[4]{3 x^2+2} x^3",1,"(2*x*((2 + 3*x^2)^(1/4)*(80 - 60*x^2 + 63*x^4) - 80*2^(1/4)*Hypergeometric2F1[1/2, 3/4, 3/2, (-3*x^2)/2]))/2079","C",1
878,1,49,65,0.0198741,"\int \frac{x^4}{\left(2+3 x^2\right)^{3/4}} \, dx","Integrate[x^4/(2 + 3*x^2)^(3/4),x]","\frac{2}{63} x \left(4 \sqrt[4]{2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{3 x^2}{2}\right)+\sqrt[4]{3 x^2+2} \left(3 x^2-4\right)\right)","\frac{16\ 2^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\sqrt{\frac{3}{2}} x\right),2\right)}{63 \sqrt{3}}-\frac{8}{63} \sqrt[4]{3 x^2+2} x+\frac{2}{21} \sqrt[4]{3 x^2+2} x^3",1,"(2*x*((-4 + 3*x^2)*(2 + 3*x^2)^(1/4) + 4*2^(1/4)*Hypergeometric2F1[1/2, 3/4, 3/2, (-3*x^2)/2]))/63","C",1
879,1,41,47,0.0094028,"\int \frac{x^2}{\left(2+3 x^2\right)^{3/4}} \, dx","Integrate[x^2/(2 + 3*x^2)^(3/4),x]","\frac{2}{9} x \left(\sqrt[4]{3 x^2+2}-\sqrt[4]{2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{3 x^2}{2}\right)\right)","\frac{2}{9} x \sqrt[4]{3 x^2+2}-\frac{4\ 2^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\sqrt{\frac{3}{2}} x\right),2\right)}{9 \sqrt{3}}",1,"(2*x*((2 + 3*x^2)^(1/4) - 2^(1/4)*Hypergeometric2F1[1/2, 3/4, 3/2, (-3*x^2)/2]))/9","C",1
880,1,24,27,0.0051826,"\int \frac{1}{\left(2+3 x^2\right)^{3/4}} \, dx","Integrate[(2 + 3*x^2)^(-3/4),x]","\frac{x \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{3 x^2}{2}\right)}{2^{3/4}}","\frac{2^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\sqrt{\frac{3}{2}} x\right),2\right)}{\sqrt{3}}",1,"(x*Hypergeometric2F1[1/2, 3/4, 3/2, (-3*x^2)/2])/2^(3/4)","C",1
881,1,27,49,0.0052468,"\int \frac{1}{x^2 \left(2+3 x^2\right)^{3/4}} \, dx","Integrate[1/(x^2*(2 + 3*x^2)^(3/4)),x]","-\frac{\, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{1}{2};-\frac{3 x^2}{2}\right)}{2^{3/4} x}","-\frac{\sqrt{3} \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\sqrt{\frac{3}{2}} x\right),2\right)}{2 \sqrt[4]{2}}-\frac{\sqrt[4]{3 x^2+2}}{2 x}",1,"-(Hypergeometric2F1[-1/2, 3/4, 1/2, (-3*x^2)/2]/(2^(3/4)*x))","C",1
882,1,29,67,0.0055197,"\int \frac{1}{x^4 \left(2+3 x^2\right)^{3/4}} \, dx","Integrate[1/(x^4*(2 + 3*x^2)^(3/4)),x]","-\frac{\, _2F_1\left(-\frac{3}{2},\frac{3}{4};-\frac{1}{2};-\frac{3 x^2}{2}\right)}{3\ 2^{3/4} x^3}","\frac{5 \sqrt{3} \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\sqrt{\frac{3}{2}} x\right),2\right)}{8 \sqrt[4]{2}}+\frac{5 \sqrt[4]{3 x^2+2}}{8 x}-\frac{\sqrt[4]{3 x^2+2}}{6 x^3}",1,"-1/3*Hypergeometric2F1[-3/2, 3/4, -1/2, (-3*x^2)/2]/(2^(3/4)*x^3)","C",1
883,1,29,85,0.0052275,"\int \frac{1}{x^6 \left(2+3 x^2\right)^{3/4}} \, dx","Integrate[1/(x^6*(2 + 3*x^2)^(3/4)),x]","-\frac{\, _2F_1\left(-\frac{5}{2},\frac{3}{4};-\frac{3}{2};-\frac{3 x^2}{2}\right)}{5\ 2^{3/4} x^5}","-\frac{27 \sqrt{3} \operatorname{EllipticF}\left(\frac{1}{2} \tan ^{-1}\left(\sqrt{\frac{3}{2}} x\right),2\right)}{32 \sqrt[4]{2}}-\frac{27 \sqrt[4]{3 x^2+2}}{32 x}-\frac{\sqrt[4]{3 x^2+2}}{10 x^5}+\frac{9 \sqrt[4]{3 x^2+2}}{40 x^3}",1,"-1/5*Hypergeometric2F1[-5/2, 3/4, -3/2, (-3*x^2)/2]/(2^(3/4)*x^5)","C",1
884,1,59,83,0.03733,"\int \frac{x^6}{\left(2-3 x^2\right)^{3/4}} \, dx","Integrate[x^6/(2 - 3*x^2)^(3/4),x]","\frac{320\ 2^{3/4} \sqrt{3} \operatorname{EllipticF}\left(\frac{1}{2} \sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right),2\right)-6 x \sqrt[4]{2-3 x^2} \left(63 x^4+60 x^2+80\right)}{6237}","\frac{320\ 2^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right),2\right)}{2079 \sqrt{3}}-\frac{160 \sqrt[4]{2-3 x^2} x}{2079}-\frac{2}{33} \sqrt[4]{2-3 x^2} x^5-\frac{40}{693} \sqrt[4]{2-3 x^2} x^3",1,"(-6*x*(2 - 3*x^2)^(1/4)*(80 + 60*x^2 + 63*x^4) + 320*2^(3/4)*Sqrt[3]*EllipticF[ArcSin[Sqrt[3/2]*x]/2, 2])/6237","A",1
885,1,54,65,0.0244981,"\int \frac{x^4}{\left(2-3 x^2\right)^{3/4}} \, dx","Integrate[x^4/(2 - 3*x^2)^(3/4),x]","-\frac{2}{189} \left(3 x \sqrt[4]{2-3 x^2} \left(3 x^2+4\right)-8\ 2^{3/4} \sqrt{3} \operatorname{EllipticF}\left(\frac{1}{2} \sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right),2\right)\right)","\frac{16\ 2^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right),2\right)}{63 \sqrt{3}}-\frac{8}{63} \sqrt[4]{2-3 x^2} x-\frac{2}{21} \sqrt[4]{2-3 x^2} x^3",1,"(-2*(3*x*(2 - 3*x^2)^(1/4)*(4 + 3*x^2) - 8*2^(3/4)*Sqrt[3]*EllipticF[ArcSin[Sqrt[3/2]*x]/2, 2]))/189","A",1
886,1,47,47,0.0151121,"\int \frac{x^2}{\left(2-3 x^2\right)^{3/4}} \, dx","Integrate[x^2/(2 - 3*x^2)^(3/4),x]","\frac{4\ 2^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right),2\right)}{9 \sqrt{3}}-\frac{2}{9} x \sqrt[4]{2-3 x^2}","\frac{4\ 2^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right),2\right)}{9 \sqrt{3}}-\frac{2}{9} x \sqrt[4]{2-3 x^2}",1,"(-2*x*(2 - 3*x^2)^(1/4))/9 + (4*2^(3/4)*EllipticF[ArcSin[Sqrt[3/2]*x]/2, 2])/(9*Sqrt[3])","A",1
887,1,27,27,0.0028653,"\int \frac{1}{\left(2-3 x^2\right)^{3/4}} \, dx","Integrate[(2 - 3*x^2)^(-3/4),x]","\frac{2^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right),2\right)}{\sqrt{3}}","\frac{2^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right),2\right)}{\sqrt{3}}",1,"(2^(3/4)*EllipticF[ArcSin[Sqrt[3/2]*x]/2, 2])/Sqrt[3]","A",1
888,1,27,49,0.0049719,"\int \frac{1}{x^2 \left(2-3 x^2\right)^{3/4}} \, dx","Integrate[1/(x^2*(2 - 3*x^2)^(3/4)),x]","-\frac{\, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{1}{2};\frac{3 x^2}{2}\right)}{2^{3/4} x}","\frac{\sqrt{3} \operatorname{EllipticF}\left(\frac{1}{2} \sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right),2\right)}{2 \sqrt[4]{2}}-\frac{\sqrt[4]{2-3 x^2}}{2 x}",1,"-(Hypergeometric2F1[-1/2, 3/4, 1/2, (3*x^2)/2]/(2^(3/4)*x))","C",1
889,1,29,67,0.0048145,"\int \frac{1}{x^4 \left(2-3 x^2\right)^{3/4}} \, dx","Integrate[1/(x^4*(2 - 3*x^2)^(3/4)),x]","-\frac{\, _2F_1\left(-\frac{3}{2},\frac{3}{4};-\frac{1}{2};\frac{3 x^2}{2}\right)}{3\ 2^{3/4} x^3}","\frac{5 \sqrt{3} \operatorname{EllipticF}\left(\frac{1}{2} \sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right),2\right)}{8 \sqrt[4]{2}}-\frac{5 \sqrt[4]{2-3 x^2}}{8 x}-\frac{\sqrt[4]{2-3 x^2}}{6 x^3}",1,"-1/3*Hypergeometric2F1[-3/2, 3/4, -1/2, (3*x^2)/2]/(2^(3/4)*x^3)","C",1
890,1,29,85,0.0053196,"\int \frac{1}{x^6 \left(2-3 x^2\right)^{3/4}} \, dx","Integrate[1/(x^6*(2 - 3*x^2)^(3/4)),x]","-\frac{\, _2F_1\left(-\frac{5}{2},\frac{3}{4};-\frac{3}{2};\frac{3 x^2}{2}\right)}{5\ 2^{3/4} x^5}","\frac{27 \sqrt{3} \operatorname{EllipticF}\left(\frac{1}{2} \sin ^{-1}\left(\sqrt{\frac{3}{2}} x\right),2\right)}{32 \sqrt[4]{2}}-\frac{27 \sqrt[4]{2-3 x^2}}{32 x}-\frac{\sqrt[4]{2-3 x^2}}{10 x^5}-\frac{9 \sqrt[4]{2-3 x^2}}{40 x^3}",1,"-1/5*Hypergeometric2F1[-5/2, 3/4, -3/2, (3*x^2)/2]/(2^(3/4)*x^5)","C",1
891,1,68,258,0.0195851,"\int \frac{x^6}{\sqrt[4]{-2+3 x^2}} \, dx","Integrate[x^6/(-2 + 3*x^2)^(1/4),x]","\frac{2 x \left(16\ 2^{3/4} \sqrt[4]{2-3 x^2} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{3 x^2}{2}\right)+81 x^6+6 x^4+8 x^2-32\right)}{1053 \sqrt[4]{3 x^2-2}}","\frac{64 \sqrt[4]{2} \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{3 x^2-2}+\sqrt{2}\right) \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{3 x^2-2}}{\sqrt[4]{2}}\right),\frac{1}{2}\right)}{1053 \sqrt{3} x}+\frac{32 \left(3 x^2-2\right)^{3/4} x}{1053}+\frac{128 \sqrt[4]{3 x^2-2} x}{1053 \left(\sqrt{3 x^2-2}+\sqrt{2}\right)}-\frac{128 \sqrt[4]{2} \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{3 x^2-2}+\sqrt{2}\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{3 x^2-2}}{\sqrt[4]{2}}\right)|\frac{1}{2}\right)}{1053 \sqrt{3} x}+\frac{2}{39} \left(3 x^2-2\right)^{3/4} x^5+\frac{40 \left(3 x^2-2\right)^{3/4} x^3}{1053}",1,"(2*x*(-32 + 8*x^2 + 6*x^4 + 81*x^6 + 16*2^(3/4)*(2 - 3*x^2)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (3*x^2)/2]))/(1053*(-2 + 3*x^2)^(1/4))","C",1
892,1,63,240,0.0171229,"\int \frac{x^4}{\sqrt[4]{-2+3 x^2}} \, dx","Integrate[x^4/(-2 + 3*x^2)^(1/4),x]","\frac{2 x \left(4\ 2^{3/4} \sqrt[4]{2-3 x^2} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{3 x^2}{2}\right)+15 x^4+2 x^2-8\right)}{135 \sqrt[4]{3 x^2-2}}","\frac{16 \sqrt[4]{2} \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{3 x^2-2}+\sqrt{2}\right) \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{3 x^2-2}}{\sqrt[4]{2}}\right),\frac{1}{2}\right)}{135 \sqrt{3} x}+\frac{8}{135} \left(3 x^2-2\right)^{3/4} x+\frac{32 \sqrt[4]{3 x^2-2} x}{135 \left(\sqrt{3 x^2-2}+\sqrt{2}\right)}-\frac{32 \sqrt[4]{2} \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{3 x^2-2}+\sqrt{2}\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{3 x^2-2}}{\sqrt[4]{2}}\right)|\frac{1}{2}\right)}{135 \sqrt{3} x}+\frac{2}{27} \left(3 x^2-2\right)^{3/4} x^3",1,"(2*x*(-8 + 2*x^2 + 15*x^4 + 4*2^(3/4)*(2 - 3*x^2)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (3*x^2)/2]))/(135*(-2 + 3*x^2)^(1/4))","C",1
893,1,57,222,0.014014,"\int \frac{x^2}{\sqrt[4]{-2+3 x^2}} \, dx","Integrate[x^2/(-2 + 3*x^2)^(1/4),x]","\frac{2 x \left(2^{3/4} \sqrt[4]{2-3 x^2} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{3 x^2}{2}\right)+3 x^2-2\right)}{15 \sqrt[4]{3 x^2-2}}","\frac{4 \sqrt[4]{2} \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{3 x^2-2}+\sqrt{2}\right) \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{3 x^2-2}}{\sqrt[4]{2}}\right),\frac{1}{2}\right)}{15 \sqrt{3} x}+\frac{2}{15} \left(3 x^2-2\right)^{3/4} x+\frac{8 \sqrt[4]{3 x^2-2} x}{15 \left(\sqrt{3 x^2-2}+\sqrt{2}\right)}-\frac{8 \sqrt[4]{2} \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{3 x^2-2}+\sqrt{2}\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{3 x^2-2}}{\sqrt[4]{2}}\right)|\frac{1}{2}\right)}{15 \sqrt{3} x}",1,"(2*x*(-2 + 3*x^2 + 2^(3/4)*(2 - 3*x^2)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (3*x^2)/2]))/(15*(-2 + 3*x^2)^(1/4))","C",1
894,1,43,199,0.0064758,"\int \frac{1}{\sqrt[4]{-2+3 x^2}} \, dx","Integrate[(-2 + 3*x^2)^(-1/4),x]","\frac{x \sqrt[4]{1-\frac{3 x^2}{2}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{3 x^2}{2}\right)}{\sqrt[4]{3 x^2-2}}","\frac{\sqrt[4]{2} \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{3 x^2-2}+\sqrt{2}\right) \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{3 x^2-2}}{\sqrt[4]{2}}\right),\frac{1}{2}\right)}{\sqrt{3} x}+\frac{2 \sqrt[4]{3 x^2-2} x}{\sqrt{3 x^2-2}+\sqrt{2}}-\frac{2 \sqrt[4]{2} \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{3 x^2-2}+\sqrt{2}\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{3 x^2-2}}{\sqrt[4]{2}}\right)|\frac{1}{2}\right)}{\sqrt{3} x}",1,"(x*(1 - (3*x^2)/2)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (3*x^2)/2])/(-2 + 3*x^2)^(1/4)","C",1
895,1,46,221,0.0063835,"\int \frac{1}{x^2 \sqrt[4]{-2+3 x^2}} \, dx","Integrate[1/(x^2*(-2 + 3*x^2)^(1/4)),x]","-\frac{\sqrt[4]{1-\frac{3 x^2}{2}} \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{1}{2};\frac{3 x^2}{2}\right)}{x \sqrt[4]{3 x^2-2}}","-\frac{\sqrt{3} \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{3 x^2-2}+\sqrt{2}\right) \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{3 x^2-2}}{\sqrt[4]{2}}\right),\frac{1}{2}\right)}{2\ 2^{3/4} x}-\frac{3 \sqrt[4]{3 x^2-2} x}{2 \left(\sqrt{3 x^2-2}+\sqrt{2}\right)}+\frac{\left(3 x^2-2\right)^{3/4}}{2 x}+\frac{\sqrt{3} \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{3 x^2-2}+\sqrt{2}\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{3 x^2-2}}{\sqrt[4]{2}}\right)|\frac{1}{2}\right)}{2^{3/4} x}",1,"-(((1 - (3*x^2)/2)^(1/4)*Hypergeometric2F1[-1/2, 1/4, 1/2, (3*x^2)/2])/(x*(-2 + 3*x^2)^(1/4)))","C",1
896,1,48,242,0.0081205,"\int \frac{1}{x^4 \sqrt[4]{-2+3 x^2}} \, dx","Integrate[1/(x^4*(-2 + 3*x^2)^(1/4)),x]","-\frac{\sqrt[4]{1-\frac{3 x^2}{2}} \, _2F_1\left(-\frac{3}{2},\frac{1}{4};-\frac{1}{2};\frac{3 x^2}{2}\right)}{3 x^3 \sqrt[4]{3 x^2-2}}","-\frac{3 \sqrt{3} \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{3 x^2-2}+\sqrt{2}\right) \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{3 x^2-2}}{\sqrt[4]{2}}\right),\frac{1}{2}\right)}{8\ 2^{3/4} x}-\frac{9 \sqrt[4]{3 x^2-2} x}{8 \left(\sqrt{3 x^2-2}+\sqrt{2}\right)}+\frac{3 \left(3 x^2-2\right)^{3/4}}{8 x}+\frac{3 \sqrt{3} \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{3 x^2-2}+\sqrt{2}\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{3 x^2-2}}{\sqrt[4]{2}}\right)|\frac{1}{2}\right)}{4\ 2^{3/4} x}+\frac{\left(3 x^2-2\right)^{3/4}}{6 x^3}",1,"-1/3*((1 - (3*x^2)/2)^(1/4)*Hypergeometric2F1[-3/2, 1/4, -1/2, (3*x^2)/2])/(x^3*(-2 + 3*x^2)^(1/4))","C",1
897,1,48,260,0.0070814,"\int \frac{1}{x^6 \sqrt[4]{-2+3 x^2}} \, dx","Integrate[1/(x^6*(-2 + 3*x^2)^(1/4)),x]","-\frac{\sqrt[4]{1-\frac{3 x^2}{2}} \, _2F_1\left(-\frac{5}{2},\frac{1}{4};-\frac{3}{2};\frac{3 x^2}{2}\right)}{5 x^5 \sqrt[4]{3 x^2-2}}","-\frac{63 \sqrt{3} \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{3 x^2-2}+\sqrt{2}\right) \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{3 x^2-2}}{\sqrt[4]{2}}\right),\frac{1}{2}\right)}{160\ 2^{3/4} x}-\frac{189 \sqrt[4]{3 x^2-2} x}{160 \left(\sqrt{3 x^2-2}+\sqrt{2}\right)}+\frac{63 \left(3 x^2-2\right)^{3/4}}{160 x}+\frac{63 \sqrt{3} \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{3 x^2-2}+\sqrt{2}\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{3 x^2-2}}{\sqrt[4]{2}}\right)|\frac{1}{2}\right)}{80\ 2^{3/4} x}+\frac{\left(3 x^2-2\right)^{3/4}}{10 x^5}+\frac{7 \left(3 x^2-2\right)^{3/4}}{40 x^3}",1,"-1/5*((1 - (3*x^2)/2)^(1/4)*Hypergeometric2F1[-5/2, 1/4, -3/2, (3*x^2)/2])/(x^5*(-2 + 3*x^2)^(1/4))","C",1
898,1,68,260,0.0200122,"\int \frac{x^6}{\sqrt[4]{-2-3 x^2}} \, dx","Integrate[x^6/(-2 - 3*x^2)^(1/4),x]","\frac{2 x \left(-16\ 2^{3/4} \sqrt[4]{3 x^2+2} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{3 x^2}{2}\right)+81 x^6-6 x^4+8 x^2+32\right)}{1053 \sqrt[4]{-3 x^2-2}}","\frac{64 \sqrt[4]{2} \sqrt{-\frac{x^2}{\left(\sqrt{-3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{-3 x^2-2}+\sqrt{2}\right) \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{-3 x^2-2}}{\sqrt[4]{2}}\right),\frac{1}{2}\right)}{1053 \sqrt{3} x}-\frac{32 \left(-3 x^2-2\right)^{3/4} x}{1053}-\frac{128 \sqrt[4]{-3 x^2-2} x}{1053 \left(\sqrt{-3 x^2-2}+\sqrt{2}\right)}-\frac{128 \sqrt[4]{2} \sqrt{-\frac{x^2}{\left(\sqrt{-3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{-3 x^2-2}+\sqrt{2}\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{-3 x^2-2}}{\sqrt[4]{2}}\right)|\frac{1}{2}\right)}{1053 \sqrt{3} x}-\frac{2}{39} \left(-3 x^2-2\right)^{3/4} x^5+\frac{40 \left(-3 x^2-2\right)^{3/4} x^3}{1053}",1,"(2*x*(32 + 8*x^2 - 6*x^4 + 81*x^6 - 16*2^(3/4)*(2 + 3*x^2)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (-3*x^2)/2]))/(1053*(-2 - 3*x^2)^(1/4))","C",1
899,1,63,242,0.0167336,"\int \frac{x^4}{\sqrt[4]{-2-3 x^2}} \, dx","Integrate[x^4/(-2 - 3*x^2)^(1/4),x]","\frac{2 x \left(4\ 2^{3/4} \sqrt[4]{3 x^2+2} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{3 x^2}{2}\right)+15 x^4-2 x^2-8\right)}{135 \sqrt[4]{-3 x^2-2}}","-\frac{16 \sqrt[4]{2} \sqrt{-\frac{x^2}{\left(\sqrt{-3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{-3 x^2-2}+\sqrt{2}\right) \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{-3 x^2-2}}{\sqrt[4]{2}}\right),\frac{1}{2}\right)}{135 \sqrt{3} x}+\frac{8}{135} \left(-3 x^2-2\right)^{3/4} x+\frac{32 \sqrt[4]{-3 x^2-2} x}{135 \left(\sqrt{-3 x^2-2}+\sqrt{2}\right)}+\frac{32 \sqrt[4]{2} \sqrt{-\frac{x^2}{\left(\sqrt{-3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{-3 x^2-2}+\sqrt{2}\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{-3 x^2-2}}{\sqrt[4]{2}}\right)|\frac{1}{2}\right)}{135 \sqrt{3} x}-\frac{2}{27} \left(-3 x^2-2\right)^{3/4} x^3",1,"(2*x*(-8 - 2*x^2 + 15*x^4 + 4*2^(3/4)*(2 + 3*x^2)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (-3*x^2)/2]))/(135*(-2 - 3*x^2)^(1/4))","C",1
900,1,58,224,0.013761,"\int \frac{x^2}{\sqrt[4]{-2-3 x^2}} \, dx","Integrate[x^2/(-2 - 3*x^2)^(1/4),x]","\frac{2 x \left(-2^{3/4} \sqrt[4]{3 x^2+2} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{3 x^2}{2}\right)+3 x^2+2\right)}{15 \sqrt[4]{-3 x^2-2}}","\frac{4 \sqrt[4]{2} \sqrt{-\frac{x^2}{\left(\sqrt{-3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{-3 x^2-2}+\sqrt{2}\right) \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{-3 x^2-2}}{\sqrt[4]{2}}\right),\frac{1}{2}\right)}{15 \sqrt{3} x}-\frac{2}{15} \left(-3 x^2-2\right)^{3/4} x-\frac{8 \sqrt[4]{-3 x^2-2} x}{15 \left(\sqrt{-3 x^2-2}+\sqrt{2}\right)}-\frac{8 \sqrt[4]{2} \sqrt{-\frac{x^2}{\left(\sqrt{-3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{-3 x^2-2}+\sqrt{2}\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{-3 x^2-2}}{\sqrt[4]{2}}\right)|\frac{1}{2}\right)}{15 \sqrt{3} x}",1,"(2*x*(2 + 3*x^2 - 2^(3/4)*(2 + 3*x^2)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (-3*x^2)/2]))/(15*(-2 - 3*x^2)^(1/4))","C",1
901,1,43,202,0.0075337,"\int \frac{1}{\sqrt[4]{-2-3 x^2}} \, dx","Integrate[(-2 - 3*x^2)^(-1/4),x]","\frac{x \sqrt[4]{\frac{3 x^2}{2}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{3 x^2}{2}\right)}{\sqrt[4]{-3 x^2-2}}","-\frac{\sqrt[4]{2} \sqrt{-\frac{x^2}{\left(\sqrt{-3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{-3 x^2-2}+\sqrt{2}\right) \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{-3 x^2-2}}{\sqrt[4]{2}}\right),\frac{1}{2}\right)}{\sqrt{3} x}+\frac{2 \sqrt[4]{-3 x^2-2} x}{\sqrt{-3 x^2-2}+\sqrt{2}}+\frac{2 \sqrt[4]{2} \sqrt{-\frac{x^2}{\left(\sqrt{-3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{-3 x^2-2}+\sqrt{2}\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{-3 x^2-2}}{\sqrt[4]{2}}\right)|\frac{1}{2}\right)}{\sqrt{3} x}",1,"(x*(1 + (3*x^2)/2)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (-3*x^2)/2])/(-2 - 3*x^2)^(1/4)","C",1
902,1,46,223,0.0076029,"\int \frac{1}{x^2 \sqrt[4]{-2-3 x^2}} \, dx","Integrate[1/(x^2*(-2 - 3*x^2)^(1/4)),x]","-\frac{\sqrt[4]{\frac{3 x^2}{2}+1} \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{1}{2};-\frac{3 x^2}{2}\right)}{x \sqrt[4]{-3 x^2-2}}","-\frac{\sqrt{3} \sqrt{-\frac{x^2}{\left(\sqrt{-3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{-3 x^2-2}+\sqrt{2}\right) \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{-3 x^2-2}}{\sqrt[4]{2}}\right),\frac{1}{2}\right)}{2\ 2^{3/4} x}+\frac{3 \sqrt[4]{-3 x^2-2} x}{2 \left(\sqrt{-3 x^2-2}+\sqrt{2}\right)}+\frac{\left(-3 x^2-2\right)^{3/4}}{2 x}+\frac{\sqrt{3} \sqrt{-\frac{x^2}{\left(\sqrt{-3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{-3 x^2-2}+\sqrt{2}\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{-3 x^2-2}}{\sqrt[4]{2}}\right)|\frac{1}{2}\right)}{2^{3/4} x}",1,"-(((1 + (3*x^2)/2)^(1/4)*Hypergeometric2F1[-1/2, 1/4, 1/2, (-3*x^2)/2])/(x*(-2 - 3*x^2)^(1/4)))","C",1
903,1,48,244,0.0079226,"\int \frac{1}{x^4 \sqrt[4]{-2-3 x^2}} \, dx","Integrate[1/(x^4*(-2 - 3*x^2)^(1/4)),x]","-\frac{\sqrt[4]{\frac{3 x^2}{2}+1} \, _2F_1\left(-\frac{3}{2},\frac{1}{4};-\frac{1}{2};-\frac{3 x^2}{2}\right)}{3 x^3 \sqrt[4]{-3 x^2-2}}","\frac{3 \sqrt{3} \sqrt{-\frac{x^2}{\left(\sqrt{-3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{-3 x^2-2}+\sqrt{2}\right) \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{-3 x^2-2}}{\sqrt[4]{2}}\right),\frac{1}{2}\right)}{8\ 2^{3/4} x}-\frac{9 \sqrt[4]{-3 x^2-2} x}{8 \left(\sqrt{-3 x^2-2}+\sqrt{2}\right)}-\frac{3 \left(-3 x^2-2\right)^{3/4}}{8 x}-\frac{3 \sqrt{3} \sqrt{-\frac{x^2}{\left(\sqrt{-3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{-3 x^2-2}+\sqrt{2}\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{-3 x^2-2}}{\sqrt[4]{2}}\right)|\frac{1}{2}\right)}{4\ 2^{3/4} x}+\frac{\left(-3 x^2-2\right)^{3/4}}{6 x^3}",1,"-1/3*((1 + (3*x^2)/2)^(1/4)*Hypergeometric2F1[-3/2, 1/4, -1/2, (-3*x^2)/2])/(x^3*(-2 - 3*x^2)^(1/4))","C",1
904,1,48,262,0.0079351,"\int \frac{1}{x^6 \sqrt[4]{-2-3 x^2}} \, dx","Integrate[1/(x^6*(-2 - 3*x^2)^(1/4)),x]","-\frac{\sqrt[4]{\frac{3 x^2}{2}+1} \, _2F_1\left(-\frac{5}{2},\frac{1}{4};-\frac{3}{2};-\frac{3 x^2}{2}\right)}{5 x^5 \sqrt[4]{-3 x^2-2}}","-\frac{63 \sqrt{3} \sqrt{-\frac{x^2}{\left(\sqrt{-3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{-3 x^2-2}+\sqrt{2}\right) \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{-3 x^2-2}}{\sqrt[4]{2}}\right),\frac{1}{2}\right)}{160\ 2^{3/4} x}+\frac{189 \sqrt[4]{-3 x^2-2} x}{160 \left(\sqrt{-3 x^2-2}+\sqrt{2}\right)}+\frac{63 \left(-3 x^2-2\right)^{3/4}}{160 x}+\frac{63 \sqrt{3} \sqrt{-\frac{x^2}{\left(\sqrt{-3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{-3 x^2-2}+\sqrt{2}\right) E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{-3 x^2-2}}{\sqrt[4]{2}}\right)|\frac{1}{2}\right)}{80\ 2^{3/4} x}+\frac{\left(-3 x^2-2\right)^{3/4}}{10 x^5}-\frac{7 \left(-3 x^2-2\right)^{3/4}}{40 x^3}",1,"-1/5*((1 + (3*x^2)/2)^(1/4)*Hypergeometric2F1[-5/2, 1/4, -3/2, (-3*x^2)/2])/(x^5*(-2 - 3*x^2)^(1/4))","C",1
905,1,68,138,0.0210909,"\int \frac{x^6}{\left(-2+3 x^2\right)^{3/4}} \, dx","Integrate[x^6/(-2 + 3*x^2)^(3/4),x]","\frac{2 x \left(80 \sqrt[4]{2} \left(2-3 x^2\right)^{3/4} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{3 x^2}{2}\right)+189 x^6+54 x^4+120 x^2-160\right)}{2079 \left(3 x^2-2\right)^{3/4}}","\frac{160\ 2^{3/4} \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{3 x^2-2}+\sqrt{2}\right) \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{3 x^2-2}}{\sqrt[4]{2}}\right),\frac{1}{2}\right)}{2079 \sqrt{3} x}+\frac{160 \sqrt[4]{3 x^2-2} x}{2079}+\frac{2}{33} \sqrt[4]{3 x^2-2} x^5+\frac{40}{693} \sqrt[4]{3 x^2-2} x^3",1,"(2*x*(-160 + 120*x^2 + 54*x^4 + 189*x^6 + 80*2^(1/4)*(2 - 3*x^2)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, (3*x^2)/2]))/(2079*(-2 + 3*x^2)^(3/4))","C",1
906,1,63,120,0.0176782,"\int \frac{x^4}{\left(-2+3 x^2\right)^{3/4}} \, dx","Integrate[x^4/(-2 + 3*x^2)^(3/4),x]","\frac{2 x \left(4 \sqrt[4]{2} \left(2-3 x^2\right)^{3/4} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{3 x^2}{2}\right)+9 x^4+6 x^2-8\right)}{63 \left(3 x^2-2\right)^{3/4}}","\frac{8\ 2^{3/4} \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{3 x^2-2}+\sqrt{2}\right) \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{3 x^2-2}}{\sqrt[4]{2}}\right),\frac{1}{2}\right)}{63 \sqrt{3} x}+\frac{8}{63} \sqrt[4]{3 x^2-2} x+\frac{2}{21} \sqrt[4]{3 x^2-2} x^3",1,"(2*x*(-8 + 6*x^2 + 9*x^4 + 4*2^(1/4)*(2 - 3*x^2)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, (3*x^2)/2]))/(63*(-2 + 3*x^2)^(3/4))","C",1
907,1,57,102,0.0151406,"\int \frac{x^2}{\left(-2+3 x^2\right)^{3/4}} \, dx","Integrate[x^2/(-2 + 3*x^2)^(3/4),x]","\frac{2 x \left(\sqrt[4]{2} \left(2-3 x^2\right)^{3/4} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{3 x^2}{2}\right)+3 x^2-2\right)}{9 \left(3 x^2-2\right)^{3/4}}","\frac{2\ 2^{3/4} \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{3 x^2-2}+\sqrt{2}\right) \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{3 x^2-2}}{\sqrt[4]{2}}\right),\frac{1}{2}\right)}{9 \sqrt{3} x}+\frac{2}{9} \sqrt[4]{3 x^2-2} x",1,"(2*x*(-2 + 3*x^2 + 2^(1/4)*(2 - 3*x^2)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, (3*x^2)/2]))/(9*(-2 + 3*x^2)^(3/4))","C",1
908,1,43,82,0.0073396,"\int \frac{1}{\left(-2+3 x^2\right)^{3/4}} \, dx","Integrate[(-2 + 3*x^2)^(-3/4),x]","\frac{x \left(1-\frac{3 x^2}{2}\right)^{3/4} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{3 x^2}{2}\right)}{\left(3 x^2-2\right)^{3/4}}","\frac{\sqrt{\frac{x^2}{\left(\sqrt{3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{3 x^2-2}+\sqrt{2}\right) \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{3 x^2-2}}{\sqrt[4]{2}}\right),\frac{1}{2}\right)}{\sqrt[4]{2} \sqrt{3} x}",1,"(x*(1 - (3*x^2)/2)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, (3*x^2)/2])/(-2 + 3*x^2)^(3/4)","C",1
909,1,46,104,0.0082396,"\int \frac{1}{x^2 \left(-2+3 x^2\right)^{3/4}} \, dx","Integrate[1/(x^2*(-2 + 3*x^2)^(3/4)),x]","-\frac{\left(1-\frac{3 x^2}{2}\right)^{3/4} \, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{1}{2};\frac{3 x^2}{2}\right)}{x \left(3 x^2-2\right)^{3/4}}","\frac{\sqrt{3} \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{3 x^2-2}+\sqrt{2}\right) \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{3 x^2-2}}{\sqrt[4]{2}}\right),\frac{1}{2}\right)}{4 \sqrt[4]{2} x}+\frac{\sqrt[4]{3 x^2-2}}{2 x}",1,"-(((1 - (3*x^2)/2)^(3/4)*Hypergeometric2F1[-1/2, 3/4, 1/2, (3*x^2)/2])/(x*(-2 + 3*x^2)^(3/4)))","C",1
910,1,48,122,0.0083991,"\int \frac{1}{x^4 \left(-2+3 x^2\right)^{3/4}} \, dx","Integrate[1/(x^4*(-2 + 3*x^2)^(3/4)),x]","-\frac{\left(1-\frac{3 x^2}{2}\right)^{3/4} \, _2F_1\left(-\frac{3}{2},\frac{3}{4};-\frac{1}{2};\frac{3 x^2}{2}\right)}{3 x^3 \left(3 x^2-2\right)^{3/4}}","\frac{5 \sqrt{3} \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{3 x^2-2}+\sqrt{2}\right) \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{3 x^2-2}}{\sqrt[4]{2}}\right),\frac{1}{2}\right)}{16 \sqrt[4]{2} x}+\frac{5 \sqrt[4]{3 x^2-2}}{8 x}+\frac{\sqrt[4]{3 x^2-2}}{6 x^3}",1,"-1/3*((1 - (3*x^2)/2)^(3/4)*Hypergeometric2F1[-3/2, 3/4, -1/2, (3*x^2)/2])/(x^3*(-2 + 3*x^2)^(3/4))","C",1
911,1,48,140,0.0078299,"\int \frac{1}{x^6 \left(-2+3 x^2\right)^{3/4}} \, dx","Integrate[1/(x^6*(-2 + 3*x^2)^(3/4)),x]","-\frac{\left(1-\frac{3 x^2}{2}\right)^{3/4} \, _2F_1\left(-\frac{5}{2},\frac{3}{4};-\frac{3}{2};\frac{3 x^2}{2}\right)}{5 x^5 \left(3 x^2-2\right)^{3/4}}","\frac{27 \sqrt{3} \sqrt{\frac{x^2}{\left(\sqrt{3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{3 x^2-2}+\sqrt{2}\right) \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{3 x^2-2}}{\sqrt[4]{2}}\right),\frac{1}{2}\right)}{64 \sqrt[4]{2} x}+\frac{27 \sqrt[4]{3 x^2-2}}{32 x}+\frac{\sqrt[4]{3 x^2-2}}{10 x^5}+\frac{9 \sqrt[4]{3 x^2-2}}{40 x^3}",1,"-1/5*((1 - (3*x^2)/2)^(3/4)*Hypergeometric2F1[-5/2, 3/4, -3/2, (3*x^2)/2])/(x^5*(-2 + 3*x^2)^(3/4))","C",1
912,1,68,139,0.0200963,"\int \frac{x^6}{\left(-2-3 x^2\right)^{3/4}} \, dx","Integrate[x^6/(-2 - 3*x^2)^(3/4),x]","\frac{2 x \left(-80 \sqrt[4]{2} \left(3 x^2+2\right)^{3/4} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{3 x^2}{2}\right)+189 x^6-54 x^4+120 x^2+160\right)}{2079 \left(-3 x^2-2\right)^{3/4}}","\frac{160\ 2^{3/4} \sqrt{-\frac{x^2}{\left(\sqrt{-3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{-3 x^2-2}+\sqrt{2}\right) \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{-3 x^2-2}}{\sqrt[4]{2}}\right),\frac{1}{2}\right)}{2079 \sqrt{3} x}-\frac{160 \sqrt[4]{-3 x^2-2} x}{2079}-\frac{2}{33} \sqrt[4]{-3 x^2-2} x^5+\frac{40}{693} \sqrt[4]{-3 x^2-2} x^3",1,"(2*x*(160 + 120*x^2 - 54*x^4 + 189*x^6 - 80*2^(1/4)*(2 + 3*x^2)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, (-3*x^2)/2]))/(2079*(-2 - 3*x^2)^(3/4))","C",1
913,1,63,121,0.0187982,"\int \frac{x^4}{\left(-2-3 x^2\right)^{3/4}} \, dx","Integrate[x^4/(-2 - 3*x^2)^(3/4),x]","\frac{2 x \left(4 \sqrt[4]{2} \left(3 x^2+2\right)^{3/4} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{3 x^2}{2}\right)+9 x^4-6 x^2-8\right)}{63 \left(-3 x^2-2\right)^{3/4}}","-\frac{8\ 2^{3/4} \sqrt{-\frac{x^2}{\left(\sqrt{-3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{-3 x^2-2}+\sqrt{2}\right) \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{-3 x^2-2}}{\sqrt[4]{2}}\right),\frac{1}{2}\right)}{63 \sqrt{3} x}+\frac{8}{63} \sqrt[4]{-3 x^2-2} x-\frac{2}{21} \sqrt[4]{-3 x^2-2} x^3",1,"(2*x*(-8 - 6*x^2 + 9*x^4 + 4*2^(1/4)*(2 + 3*x^2)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, (-3*x^2)/2]))/(63*(-2 - 3*x^2)^(3/4))","C",1
914,1,58,103,0.0138751,"\int \frac{x^2}{\left(-2-3 x^2\right)^{3/4}} \, dx","Integrate[x^2/(-2 - 3*x^2)^(3/4),x]","\frac{2 x \left(-\sqrt[4]{2} \left(3 x^2+2\right)^{3/4} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{3 x^2}{2}\right)+3 x^2+2\right)}{9 \left(-3 x^2-2\right)^{3/4}}","\frac{2\ 2^{3/4} \sqrt{-\frac{x^2}{\left(\sqrt{-3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{-3 x^2-2}+\sqrt{2}\right) \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{-3 x^2-2}}{\sqrt[4]{2}}\right),\frac{1}{2}\right)}{9 \sqrt{3} x}-\frac{2}{9} x \sqrt[4]{-3 x^2-2}",1,"(2*x*(2 + 3*x^2 - 2^(1/4)*(2 + 3*x^2)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, (-3*x^2)/2]))/(9*(-2 - 3*x^2)^(3/4))","C",1
915,1,43,84,0.0078294,"\int \frac{1}{\left(-2-3 x^2\right)^{3/4}} \, dx","Integrate[(-2 - 3*x^2)^(-3/4),x]","\frac{x \left(\frac{3 x^2}{2}+1\right)^{3/4} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{3 x^2}{2}\right)}{\left(-3 x^2-2\right)^{3/4}}","-\frac{\sqrt{-\frac{x^2}{\left(\sqrt{-3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{-3 x^2-2}+\sqrt{2}\right) \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{-3 x^2-2}}{\sqrt[4]{2}}\right),\frac{1}{2}\right)}{\sqrt[4]{2} \sqrt{3} x}",1,"(x*(1 + (3*x^2)/2)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, (-3*x^2)/2])/(-2 - 3*x^2)^(3/4)","C",1
916,1,46,105,0.007957,"\int \frac{1}{x^2 \left(-2-3 x^2\right)^{3/4}} \, dx","Integrate[1/(x^2*(-2 - 3*x^2)^(3/4)),x]","-\frac{\left(\frac{3 x^2}{2}+1\right)^{3/4} \, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{1}{2};-\frac{3 x^2}{2}\right)}{x \left(-3 x^2-2\right)^{3/4}}","\frac{\sqrt{3} \sqrt{-\frac{x^2}{\left(\sqrt{-3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{-3 x^2-2}+\sqrt{2}\right) \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{-3 x^2-2}}{\sqrt[4]{2}}\right),\frac{1}{2}\right)}{4 \sqrt[4]{2} x}+\frac{\sqrt[4]{-3 x^2-2}}{2 x}",1,"-(((1 + (3*x^2)/2)^(3/4)*Hypergeometric2F1[-1/2, 3/4, 1/2, (-3*x^2)/2])/(x*(-2 - 3*x^2)^(3/4)))","C",1
917,1,48,123,0.0088008,"\int \frac{1}{x^4 \left(-2-3 x^2\right)^{3/4}} \, dx","Integrate[1/(x^4*(-2 - 3*x^2)^(3/4)),x]","-\frac{\left(\frac{3 x^2}{2}+1\right)^{3/4} \, _2F_1\left(-\frac{3}{2},\frac{3}{4};-\frac{1}{2};-\frac{3 x^2}{2}\right)}{3 x^3 \left(-3 x^2-2\right)^{3/4}}","-\frac{5 \sqrt{3} \sqrt{-\frac{x^2}{\left(\sqrt{-3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{-3 x^2-2}+\sqrt{2}\right) \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{-3 x^2-2}}{\sqrt[4]{2}}\right),\frac{1}{2}\right)}{16 \sqrt[4]{2} x}-\frac{5 \sqrt[4]{-3 x^2-2}}{8 x}+\frac{\sqrt[4]{-3 x^2-2}}{6 x^3}",1,"-1/3*((1 + (3*x^2)/2)^(3/4)*Hypergeometric2F1[-3/2, 3/4, -1/2, (-3*x^2)/2])/(x^3*(-2 - 3*x^2)^(3/4))","C",1
918,1,48,141,0.0083137,"\int \frac{1}{x^6 \left(-2-3 x^2\right)^{3/4}} \, dx","Integrate[1/(x^6*(-2 - 3*x^2)^(3/4)),x]","-\frac{\left(\frac{3 x^2}{2}+1\right)^{3/4} \, _2F_1\left(-\frac{5}{2},\frac{3}{4};-\frac{3}{2};-\frac{3 x^2}{2}\right)}{5 x^5 \left(-3 x^2-2\right)^{3/4}}","\frac{27 \sqrt{3} \sqrt{-\frac{x^2}{\left(\sqrt{-3 x^2-2}+\sqrt{2}\right)^2}} \left(\sqrt{-3 x^2-2}+\sqrt{2}\right) \operatorname{EllipticF}\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{-3 x^2-2}}{\sqrt[4]{2}}\right),\frac{1}{2}\right)}{64 \sqrt[4]{2} x}+\frac{27 \sqrt[4]{-3 x^2-2}}{32 x}+\frac{\sqrt[4]{-3 x^2-2}}{10 x^5}-\frac{9 \sqrt[4]{-3 x^2-2}}{40 x^3}",1,"-1/5*((1 + (3*x^2)/2)^(3/4)*Hypergeometric2F1[-5/2, 3/4, -3/2, (-3*x^2)/2])/(x^5*(-2 - 3*x^2)^(3/4))","C",1
919,1,102,152,0.0675414,"\int (c x)^{7/2} \sqrt[4]{a+b x^2} \, dx","Integrate[(c*x)^(7/2)*(a + b*x^2)^(1/4),x]","\frac{c^3 \sqrt{c x} \sqrt[4]{a+b x^2} \left(\sqrt[4]{\frac{b x^2}{a}+1} \left(-5 a^2+a b x^2+6 b^2 x^4\right)+5 a^2 \, _2F_1\left(-\frac{1}{4},\frac{1}{4};\frac{5}{4};-\frac{b x^2}{a}\right)\right)}{30 b^2 \sqrt[4]{\frac{b x^2}{a}+1}}","-\frac{a^{5/2} c^2 (c x)^{3/2} \left(\frac{a}{b x^2}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{12 b^{3/2} \left(a+b x^2\right)^{3/4}}-\frac{a^2 c^3 \sqrt{c x} \sqrt[4]{a+b x^2}}{12 b^2}+\frac{(c x)^{9/2} \sqrt[4]{a+b x^2}}{5 c}+\frac{a c (c x)^{5/2} \sqrt[4]{a+b x^2}}{30 b}",1,"(c^3*Sqrt[c*x]*(a + b*x^2)^(1/4)*((1 + (b*x^2)/a)^(1/4)*(-5*a^2 + a*b*x^2 + 6*b^2*x^4) + 5*a^2*Hypergeometric2F1[-1/4, 1/4, 5/4, -((b*x^2)/a)]))/(30*b^2*(1 + (b*x^2)/a)^(1/4))","C",1
920,1,85,118,0.0447824,"\int (c x)^{3/2} \sqrt[4]{a+b x^2} \, dx","Integrate[(c*x)^(3/2)*(a + b*x^2)^(1/4),x]","\frac{c \sqrt{c x} \sqrt[4]{a+b x^2} \left(\left(a+b x^2\right) \sqrt[4]{\frac{b x^2}{a}+1}-a \, _2F_1\left(-\frac{1}{4},\frac{1}{4};\frac{5}{4};-\frac{b x^2}{a}\right)\right)}{3 b \sqrt[4]{\frac{b x^2}{a}+1}}","\frac{a^{3/2} (c x)^{3/2} \left(\frac{a}{b x^2}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{6 \sqrt{b} \left(a+b x^2\right)^{3/4}}+\frac{(c x)^{5/2} \sqrt[4]{a+b x^2}}{3 c}+\frac{a c \sqrt{c x} \sqrt[4]{a+b x^2}}{6 b}",1,"(c*Sqrt[c*x]*(a + b*x^2)^(1/4)*((a + b*x^2)*(1 + (b*x^2)/a)^(1/4) - a*Hypergeometric2F1[-1/4, 1/4, 5/4, -((b*x^2)/a)]))/(3*b*(1 + (b*x^2)/a)^(1/4))","C",1
921,1,54,89,0.0125171,"\int \frac{\sqrt[4]{a+b x^2}}{\sqrt{c x}} \, dx","Integrate[(a + b*x^2)^(1/4)/Sqrt[c*x],x]","\frac{2 x \sqrt[4]{a+b x^2} \, _2F_1\left(-\frac{1}{4},\frac{1}{4};\frac{5}{4};-\frac{b x^2}{a}\right)}{\sqrt{c x} \sqrt[4]{\frac{b x^2}{a}+1}}","\frac{\sqrt{c x} \sqrt[4]{a+b x^2}}{c}-\frac{\sqrt{a} \sqrt{b} (c x)^{3/2} \left(\frac{a}{b x^2}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{c^2 \left(a+b x^2\right)^{3/4}}",1,"(2*x*(a + b*x^2)^(1/4)*Hypergeometric2F1[-1/4, 1/4, 5/4, -((b*x^2)/a)])/(Sqrt[c*x]*(1 + (b*x^2)/a)^(1/4))","C",1
922,1,56,94,0.0130618,"\int \frac{\sqrt[4]{a+b x^2}}{(c x)^{5/2}} \, dx","Integrate[(a + b*x^2)^(1/4)/(c*x)^(5/2),x]","-\frac{2 x \sqrt[4]{a+b x^2} \, _2F_1\left(-\frac{3}{4},-\frac{1}{4};\frac{1}{4};-\frac{b x^2}{a}\right)}{3 (c x)^{5/2} \sqrt[4]{\frac{b x^2}{a}+1}}","-\frac{2 b^{3/2} (c x)^{3/2} \left(\frac{a}{b x^2}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{3 \sqrt{a} c^4 \left(a+b x^2\right)^{3/4}}-\frac{2 \sqrt[4]{a+b x^2}}{3 c (c x)^{3/2}}",1,"(-2*x*(a + b*x^2)^(1/4)*Hypergeometric2F1[-3/4, -1/4, 1/4, -((b*x^2)/a)])/(3*(c*x)^(5/2)*(1 + (b*x^2)/a)^(1/4))","C",1
923,1,56,123,0.0141945,"\int \frac{\sqrt[4]{a+b x^2}}{(c x)^{9/2}} \, dx","Integrate[(a + b*x^2)^(1/4)/(c*x)^(9/2),x]","-\frac{2 x \sqrt[4]{a+b x^2} \, _2F_1\left(-\frac{7}{4},-\frac{1}{4};-\frac{3}{4};-\frac{b x^2}{a}\right)}{7 (c x)^{9/2} \sqrt[4]{\frac{b x^2}{a}+1}}","\frac{4 b^{5/2} (c x)^{3/2} \left(\frac{a}{b x^2}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{21 a^{3/2} c^6 \left(a+b x^2\right)^{3/4}}-\frac{2 b \sqrt[4]{a+b x^2}}{21 a c^3 (c x)^{3/2}}-\frac{2 \sqrt[4]{a+b x^2}}{7 c (c x)^{7/2}}",1,"(-2*x*(a + b*x^2)^(1/4)*Hypergeometric2F1[-7/4, -1/4, -3/4, -((b*x^2)/a)])/(7*(c*x)^(9/2)*(1 + (b*x^2)/a)^(1/4))","C",1
924,1,56,154,0.0137237,"\int \frac{\sqrt[4]{a+b x^2}}{(c x)^{13/2}} \, dx","Integrate[(a + b*x^2)^(1/4)/(c*x)^(13/2),x]","-\frac{2 x \sqrt[4]{a+b x^2} \, _2F_1\left(-\frac{11}{4},-\frac{1}{4};-\frac{7}{4};-\frac{b x^2}{a}\right)}{11 (c x)^{13/2} \sqrt[4]{\frac{b x^2}{a}+1}}","-\frac{8 b^{7/2} (c x)^{3/2} \left(\frac{a}{b x^2}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{77 a^{5/2} c^8 \left(a+b x^2\right)^{3/4}}+\frac{4 b^2 \sqrt[4]{a+b x^2}}{77 a^2 c^5 (c x)^{3/2}}-\frac{2 b \sqrt[4]{a+b x^2}}{77 a c^3 (c x)^{7/2}}-\frac{2 \sqrt[4]{a+b x^2}}{11 c (c x)^{11/2}}",1,"(-2*x*(a + b*x^2)^(1/4)*Hypergeometric2F1[-11/4, -1/4, -7/4, -((b*x^2)/a)])/(11*(c*x)^(13/2)*(1 + (b*x^2)/a)^(1/4))","C",1
925,1,85,147,0.0457526,"\int (c x)^{5/2} \sqrt[4]{a+b x^2} \, dx","Integrate[(c*x)^(5/2)*(a + b*x^2)^(1/4),x]","\frac{c (c x)^{3/2} \sqrt[4]{a+b x^2} \left(\left(a+b x^2\right) \sqrt[4]{\frac{b x^2}{a}+1}-a \, _2F_1\left(-\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right)\right)}{4 b \sqrt[4]{\frac{b x^2}{a}+1}}","\frac{3 a^2 c^{5/2} \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a+b x^2}}\right)}{32 b^{7/4}}-\frac{3 a^2 c^{5/2} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a+b x^2}}\right)}{32 b^{7/4}}+\frac{(c x)^{7/2} \sqrt[4]{a+b x^2}}{4 c}+\frac{a c (c x)^{3/2} \sqrt[4]{a+b x^2}}{16 b}",1,"(c*(c*x)^(3/2)*(a + b*x^2)^(1/4)*((a + b*x^2)*(1 + (b*x^2)/a)^(1/4) - a*Hypergeometric2F1[-1/4, 3/4, 7/4, -((b*x^2)/a)]))/(4*b*(1 + (b*x^2)/a)^(1/4))","C",1
926,1,56,116,0.0120658,"\int \sqrt{c x} \sqrt[4]{a+b x^2} \, dx","Integrate[Sqrt[c*x]*(a + b*x^2)^(1/4),x]","\frac{2 x \sqrt{c x} \sqrt[4]{a+b x^2} \, _2F_1\left(-\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right)}{3 \sqrt[4]{\frac{b x^2}{a}+1}}","-\frac{a \sqrt{c} \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a+b x^2}}\right)}{4 b^{3/4}}+\frac{a \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a+b x^2}}\right)}{4 b^{3/4}}+\frac{(c x)^{3/2} \sqrt[4]{a+b x^2}}{2 c}",1,"(2*x*Sqrt[c*x]*(a + b*x^2)^(1/4)*Hypergeometric2F1[-1/4, 3/4, 7/4, -((b*x^2)/a)])/(3*(1 + (b*x^2)/a)^(1/4))","C",1
927,1,54,107,0.0127956,"\int \frac{\sqrt[4]{a+b x^2}}{(c x)^{3/2}} \, dx","Integrate[(a + b*x^2)^(1/4)/(c*x)^(3/2),x]","-\frac{2 x \sqrt[4]{a+b x^2} \, _2F_1\left(-\frac{1}{4},-\frac{1}{4};\frac{3}{4};-\frac{b x^2}{a}\right)}{(c x)^{3/2} \sqrt[4]{\frac{b x^2}{a}+1}}","-\frac{\sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a+b x^2}}\right)}{c^{3/2}}+\frac{\sqrt[4]{b} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a+b x^2}}\right)}{c^{3/2}}-\frac{2 \sqrt[4]{a+b x^2}}{c \sqrt{c x}}",1,"(-2*x*(a + b*x^2)^(1/4)*Hypergeometric2F1[-1/4, -1/4, 3/4, -((b*x^2)/a)])/((c*x)^(3/2)*(1 + (b*x^2)/a)^(1/4))","C",1
928,1,26,28,0.0081268,"\int \frac{\sqrt[4]{a+b x^2}}{(c x)^{7/2}} \, dx","Integrate[(a + b*x^2)^(1/4)/(c*x)^(7/2),x]","-\frac{2 x \left(a+b x^2\right)^{5/4}}{5 a (c x)^{7/2}}","-\frac{2 \left(a+b x^2\right)^{5/4}}{5 a c (c x)^{5/2}}",1,"(-2*x*(a + b*x^2)^(5/4))/(5*a*(c*x)^(7/2))","A",1
929,1,41,57,0.0154418,"\int \frac{\sqrt[4]{a+b x^2}}{(c x)^{11/2}} \, dx","Integrate[(a + b*x^2)^(1/4)/(c*x)^(11/2),x]","\frac{2 \sqrt{c x} \left(a+b x^2\right)^{5/4} \left(4 b x^2-5 a\right)}{45 a^2 c^6 x^5}","\frac{8 \left(a+b x^2\right)^{9/4}}{45 a^2 c (c x)^{9/2}}-\frac{2 \left(a+b x^2\right)^{5/4}}{5 a c (c x)^{9/2}}",1,"(2*Sqrt[c*x]*(a + b*x^2)^(5/4)*(-5*a + 4*b*x^2))/(45*a^2*c^6*x^5)","A",1
930,1,52,85,0.0177365,"\int \frac{\sqrt[4]{a+b x^2}}{(c x)^{15/2}} \, dx","Integrate[(a + b*x^2)^(1/4)/(c*x)^(15/2),x]","-\frac{2 \sqrt{c x} \left(a+b x^2\right)^{5/4} \left(45 a^2-40 a b x^2+32 b^2 x^4\right)}{585 a^3 c^8 x^7}","-\frac{64 \left(a+b x^2\right)^{13/4}}{585 a^3 c (c x)^{13/2}}+\frac{16 \left(a+b x^2\right)^{9/4}}{45 a^2 c (c x)^{13/2}}-\frac{2 \left(a+b x^2\right)^{5/4}}{5 a c (c x)^{13/2}}",1,"(-2*Sqrt[c*x]*(a + b*x^2)^(5/4)*(45*a^2 - 40*a*b*x^2 + 32*b^2*x^4))/(585*a^3*c^8*x^7)","A",1
931,1,63,113,0.0159223,"\int \frac{\sqrt[4]{a+b x^2}}{(c x)^{19/2}} \, dx","Integrate[(a + b*x^2)^(1/4)/(c*x)^(19/2),x]","\frac{2 \left(a+b x^2\right)^{5/4} \left(-195 a^3+180 a^2 b x^2-160 a b^2 x^4+128 b^3 x^6\right)}{3315 a^4 c^9 x^8 \sqrt{c x}}","\frac{256 \left(a+b x^2\right)^{17/4}}{3315 a^4 c (c x)^{17/2}}-\frac{64 \left(a+b x^2\right)^{13/4}}{195 a^3 c (c x)^{17/2}}+\frac{8 \left(a+b x^2\right)^{9/4}}{15 a^2 c (c x)^{17/2}}-\frac{2 \left(a+b x^2\right)^{5/4}}{5 a c (c x)^{17/2}}",1,"(2*(a + b*x^2)^(5/4)*(-195*a^3 + 180*a^2*b*x^2 - 160*a*b^2*x^4 + 128*b^3*x^6))/(3315*a^4*c^9*x^8*Sqrt[c*x])","A",1
932,1,88,122,0.0550113,"\int (c x)^{3/2} \sqrt[4]{a-b x^2} \, dx","Integrate[(c*x)^(3/2)*(a - b*x^2)^(1/4),x]","\frac{c \sqrt{c x} \sqrt[4]{a-b x^2} \left(a \, _2F_1\left(-\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{b x^2}{a}\right)+\sqrt[4]{1-\frac{b x^2}{a}} \left(b x^2-a\right)\right)}{3 b \sqrt[4]{1-\frac{b x^2}{a}}}","-\frac{a^{3/2} (c x)^{3/2} \left(1-\frac{a}{b x^2}\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \csc ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{6 \sqrt{b} \left(a-b x^2\right)^{3/4}}+\frac{(c x)^{5/2} \sqrt[4]{a-b x^2}}{3 c}-\frac{a c \sqrt{c x} \sqrt[4]{a-b x^2}}{6 b}",1,"(c*Sqrt[c*x]*(a - b*x^2)^(1/4)*((-a + b*x^2)*(1 - (b*x^2)/a)^(1/4) + a*Hypergeometric2F1[-1/4, 1/4, 5/4, (b*x^2)/a]))/(3*b*(1 - (b*x^2)/a)^(1/4))","C",1
933,1,55,92,0.0143425,"\int \frac{\sqrt[4]{a-b x^2}}{\sqrt{c x}} \, dx","Integrate[(a - b*x^2)^(1/4)/Sqrt[c*x],x]","\frac{2 x \sqrt[4]{a-b x^2} \, _2F_1\left(-\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{b x^2}{a}\right)}{\sqrt{c x} \sqrt[4]{1-\frac{b x^2}{a}}}","\frac{\sqrt{c x} \sqrt[4]{a-b x^2}}{c}-\frac{\sqrt{a} \sqrt{b} (c x)^{3/2} \left(1-\frac{a}{b x^2}\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \csc ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{c^2 \left(a-b x^2\right)^{3/4}}",1,"(2*x*(a - b*x^2)^(1/4)*Hypergeometric2F1[-1/4, 1/4, 5/4, (b*x^2)/a])/(Sqrt[c*x]*(1 - (b*x^2)/a)^(1/4))","C",1
934,1,57,97,0.0158312,"\int \frac{\sqrt[4]{a-b x^2}}{(c x)^{5/2}} \, dx","Integrate[(a - b*x^2)^(1/4)/(c*x)^(5/2),x]","-\frac{2 x \sqrt[4]{a-b x^2} \, _2F_1\left(-\frac{3}{4},-\frac{1}{4};\frac{1}{4};\frac{b x^2}{a}\right)}{3 (c x)^{5/2} \sqrt[4]{1-\frac{b x^2}{a}}}","\frac{2 b^{3/2} (c x)^{3/2} \left(1-\frac{a}{b x^2}\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \csc ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{3 \sqrt{a} c^4 \left(a-b x^2\right)^{3/4}}-\frac{2 \sqrt[4]{a-b x^2}}{3 c (c x)^{3/2}}",1,"(-2*x*(a - b*x^2)^(1/4)*Hypergeometric2F1[-3/4, -1/4, 1/4, (b*x^2)/a])/(3*(c*x)^(5/2)*(1 - (b*x^2)/a)^(1/4))","C",1
935,1,57,127,0.0150341,"\int \frac{\sqrt[4]{a-b x^2}}{(c x)^{9/2}} \, dx","Integrate[(a - b*x^2)^(1/4)/(c*x)^(9/2),x]","-\frac{2 x \sqrt[4]{a-b x^2} \, _2F_1\left(-\frac{7}{4},-\frac{1}{4};-\frac{3}{4};\frac{b x^2}{a}\right)}{7 (c x)^{9/2} \sqrt[4]{1-\frac{b x^2}{a}}}","\frac{4 b^{5/2} (c x)^{3/2} \left(1-\frac{a}{b x^2}\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \csc ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{21 a^{3/2} c^6 \left(a-b x^2\right)^{3/4}}+\frac{2 b \sqrt[4]{a-b x^2}}{21 a c^3 (c x)^{3/2}}-\frac{2 \sqrt[4]{a-b x^2}}{7 c (c x)^{7/2}}",1,"(-2*x*(a - b*x^2)^(1/4)*Hypergeometric2F1[-7/4, -1/4, -3/4, (b*x^2)/a])/(7*(c*x)^(9/2)*(1 - (b*x^2)/a)^(1/4))","C",1
936,1,57,159,0.0160292,"\int \frac{\sqrt[4]{a-b x^2}}{(c x)^{13/2}} \, dx","Integrate[(a - b*x^2)^(1/4)/(c*x)^(13/2),x]","-\frac{2 x \sqrt[4]{a-b x^2} \, _2F_1\left(-\frac{11}{4},-\frac{1}{4};-\frac{7}{4};\frac{b x^2}{a}\right)}{11 (c x)^{13/2} \sqrt[4]{1-\frac{b x^2}{a}}}","\frac{8 b^{7/2} (c x)^{3/2} \left(1-\frac{a}{b x^2}\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \csc ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{77 a^{5/2} c^8 \left(a-b x^2\right)^{3/4}}+\frac{4 b^2 \sqrt[4]{a-b x^2}}{77 a^2 c^5 (c x)^{3/2}}+\frac{2 b \sqrt[4]{a-b x^2}}{77 a c^3 (c x)^{7/2}}-\frac{2 \sqrt[4]{a-b x^2}}{11 c (c x)^{11/2}}",1,"(-2*x*(a - b*x^2)^(1/4)*Hypergeometric2F1[-11/4, -1/4, -7/4, (b*x^2)/a])/(11*(c*x)^(13/2)*(1 - (b*x^2)/a)^(1/4))","C",1
937,1,88,343,0.0496509,"\int (c x)^{5/2} \sqrt[4]{a-b x^2} \, dx","Integrate[(c*x)^(5/2)*(a - b*x^2)^(1/4),x]","\frac{c (c x)^{3/2} \sqrt[4]{a-b x^2} \left(a \, _2F_1\left(-\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{b x^2}{a}\right)+\sqrt[4]{1-\frac{b x^2}{a}} \left(b x^2-a\right)\right)}{4 b \sqrt[4]{1-\frac{b x^2}{a}}}","\frac{3 a^2 c^{5/2} \log \left(\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a-b x^2}}+\sqrt{c}\right)}{64 \sqrt{2} b^{7/4}}-\frac{3 a^2 c^{5/2} \log \left(\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}+\frac{\sqrt{2} \sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a-b x^2}}+\sqrt{c}\right)}{64 \sqrt{2} b^{7/4}}-\frac{3 a^2 c^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a-b x^2}}\right)}{32 \sqrt{2} b^{7/4}}+\frac{3 a^2 c^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a-b x^2}}+1\right)}{32 \sqrt{2} b^{7/4}}+\frac{(c x)^{7/2} \sqrt[4]{a-b x^2}}{4 c}-\frac{a c (c x)^{3/2} \sqrt[4]{a-b x^2}}{16 b}",1,"(c*(c*x)^(3/2)*(a - b*x^2)^(1/4)*((-a + b*x^2)*(1 - (b*x^2)/a)^(1/4) + a*Hypergeometric2F1[-1/4, 3/4, 7/4, (b*x^2)/a]))/(4*b*(1 - (b*x^2)/a)^(1/4))","C",1
938,1,57,307,0.01245,"\int \sqrt{c x} \sqrt[4]{a-b x^2} \, dx","Integrate[Sqrt[c*x]*(a - b*x^2)^(1/4),x]","\frac{2 x \sqrt{c x} \sqrt[4]{a-b x^2} \, _2F_1\left(-\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{b x^2}{a}\right)}{3 \sqrt[4]{1-\frac{b x^2}{a}}}","\frac{a \sqrt{c} \log \left(\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a-b x^2}}+\sqrt{c}\right)}{8 \sqrt{2} b^{3/4}}-\frac{a \sqrt{c} \log \left(\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}+\frac{\sqrt{2} \sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a-b x^2}}+\sqrt{c}\right)}{8 \sqrt{2} b^{3/4}}-\frac{a \sqrt{c} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a-b x^2}}\right)}{4 \sqrt{2} b^{3/4}}+\frac{a \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a-b x^2}}+1\right)}{4 \sqrt{2} b^{3/4}}+\frac{(c x)^{3/2} \sqrt[4]{a-b x^2}}{2 c}",1,"(2*x*Sqrt[c*x]*(a - b*x^2)^(1/4)*Hypergeometric2F1[-1/4, 3/4, 7/4, (b*x^2)/a])/(3*(1 - (b*x^2)/a)^(1/4))","C",1
939,1,55,296,0.0132543,"\int \frac{\sqrt[4]{a-b x^2}}{(c x)^{3/2}} \, dx","Integrate[(a - b*x^2)^(1/4)/(c*x)^(3/2),x]","-\frac{2 x \sqrt[4]{a-b x^2} \, _2F_1\left(-\frac{1}{4},-\frac{1}{4};\frac{3}{4};\frac{b x^2}{a}\right)}{(c x)^{3/2} \sqrt[4]{1-\frac{b x^2}{a}}}","-\frac{\sqrt[4]{b} \log \left(\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a-b x^2}}+\sqrt{c}\right)}{2 \sqrt{2} c^{3/2}}+\frac{\sqrt[4]{b} \log \left(\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}+\frac{\sqrt{2} \sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a-b x^2}}+\sqrt{c}\right)}{2 \sqrt{2} c^{3/2}}+\frac{\sqrt[4]{b} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a-b x^2}}\right)}{\sqrt{2} c^{3/2}}-\frac{\sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a-b x^2}}+1\right)}{\sqrt{2} c^{3/2}}-\frac{2 \sqrt[4]{a-b x^2}}{c \sqrt{c x}}",1,"(-2*x*(a - b*x^2)^(1/4)*Hypergeometric2F1[-1/4, -1/4, 3/4, (b*x^2)/a])/((c*x)^(3/2)*(1 - (b*x^2)/a)^(1/4))","C",1
940,1,27,29,0.0086313,"\int \frac{\sqrt[4]{a-b x^2}}{(c x)^{7/2}} \, dx","Integrate[(a - b*x^2)^(1/4)/(c*x)^(7/2),x]","-\frac{2 x \left(a-b x^2\right)^{5/4}}{5 a (c x)^{7/2}}","-\frac{2 \left(a-b x^2\right)^{5/4}}{5 a c (c x)^{5/2}}",1,"(-2*x*(a - b*x^2)^(5/4))/(5*a*(c*x)^(7/2))","A",1
941,1,42,59,0.0174797,"\int \frac{\sqrt[4]{a-b x^2}}{(c x)^{11/2}} \, dx","Integrate[(a - b*x^2)^(1/4)/(c*x)^(11/2),x]","-\frac{2 \sqrt{c x} \left(a-b x^2\right)^{5/4} \left(5 a+4 b x^2\right)}{45 a^2 c^6 x^5}","\frac{8 \left(a-b x^2\right)^{9/4}}{45 a^2 c (c x)^{9/2}}-\frac{2 \left(a-b x^2\right)^{5/4}}{5 a c (c x)^{9/2}}",1,"(-2*Sqrt[c*x]*(a - b*x^2)^(5/4)*(5*a + 4*b*x^2))/(45*a^2*c^6*x^5)","A",1
942,1,53,88,0.0185091,"\int \frac{\sqrt[4]{a-b x^2}}{(c x)^{15/2}} \, dx","Integrate[(a - b*x^2)^(1/4)/(c*x)^(15/2),x]","-\frac{2 \sqrt{c x} \left(a-b x^2\right)^{5/4} \left(45 a^2+40 a b x^2+32 b^2 x^4\right)}{585 a^3 c^8 x^7}","-\frac{64 \left(a-b x^2\right)^{13/4}}{585 a^3 c (c x)^{13/2}}+\frac{16 \left(a-b x^2\right)^{9/4}}{45 a^2 c (c x)^{13/2}}-\frac{2 \left(a-b x^2\right)^{5/4}}{5 a c (c x)^{13/2}}",1,"(-2*Sqrt[c*x]*(a - b*x^2)^(5/4)*(45*a^2 + 40*a*b*x^2 + 32*b^2*x^4))/(585*a^3*c^8*x^7)","A",1
943,1,64,117,0.0167747,"\int \frac{\sqrt[4]{a-b x^2}}{(c x)^{19/2}} \, dx","Integrate[(a - b*x^2)^(1/4)/(c*x)^(19/2),x]","-\frac{2 \left(a-b x^2\right)^{5/4} \left(195 a^3+180 a^2 b x^2+160 a b^2 x^4+128 b^3 x^6\right)}{3315 a^4 c^9 x^8 \sqrt{c x}}","\frac{256 \left(a-b x^2\right)^{17/4}}{3315 a^4 c (c x)^{17/2}}-\frac{64 \left(a-b x^2\right)^{13/4}}{195 a^3 c (c x)^{17/2}}+\frac{8 \left(a-b x^2\right)^{9/4}}{15 a^2 c (c x)^{17/2}}-\frac{2 \left(a-b x^2\right)^{5/4}}{5 a c (c x)^{17/2}}",1,"(-2*(a - b*x^2)^(5/4)*(195*a^3 + 180*a^2*b*x^2 + 160*a*b^2*x^4 + 128*b^3*x^6))/(3315*a^4*c^9*x^8*Sqrt[c*x])","A",1
944,1,97,117,0.0371366,"\int \frac{(c x)^{3/2}}{\sqrt[4]{a+b x^2}} \, dx","Integrate[(c*x)^(3/2)/(a + b*x^2)^(1/4),x]","\frac{(c x)^{3/2} \left(2 \sqrt[4]{b} \sqrt{x} \left(a+b x^2\right)^{3/4}-a \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a+b x^2}}\right)-a \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a+b x^2}}\right)\right)}{4 b^{5/4} x^{3/2}}","-\frac{a c^{3/2} \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a+b x^2}}\right)}{4 b^{5/4}}-\frac{a c^{3/2} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a+b x^2}}\right)}{4 b^{5/4}}+\frac{c \sqrt{c x} \left(a+b x^2\right)^{3/4}}{2 b}",1,"((c*x)^(3/2)*(2*b^(1/4)*Sqrt[x]*(a + b*x^2)^(3/4) - a*ArcTan[(b^(1/4)*Sqrt[x])/(a + b*x^2)^(1/4)] - a*ArcTanh[(b^(1/4)*Sqrt[x])/(a + b*x^2)^(1/4)]))/(4*b^(5/4)*x^(3/2))","A",1
945,1,65,83,0.0111526,"\int \frac{1}{\sqrt{c x} \sqrt[4]{a+b x^2}} \, dx","Integrate[1/(Sqrt[c*x]*(a + b*x^2)^(1/4)),x]","\frac{\sqrt{x} \left(\tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a+b x^2}}\right)+\tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a+b x^2}}\right)\right)}{\sqrt[4]{b} \sqrt{c x}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a+b x^2}}\right)}{\sqrt[4]{b} \sqrt{c}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a+b x^2}}\right)}{\sqrt[4]{b} \sqrt{c}}",1,"(Sqrt[x]*(ArcTan[(b^(1/4)*Sqrt[x])/(a + b*x^2)^(1/4)] + ArcTanh[(b^(1/4)*Sqrt[x])/(a + b*x^2)^(1/4)]))/(b^(1/4)*Sqrt[c*x])","A",1
946,1,26,28,0.0065312,"\int \frac{1}{(c x)^{5/2} \sqrt[4]{a+b x^2}} \, dx","Integrate[1/((c*x)^(5/2)*(a + b*x^2)^(1/4)),x]","-\frac{2 x \left(a+b x^2\right)^{3/4}}{3 a (c x)^{5/2}}","-\frac{2 \left(a+b x^2\right)^{3/4}}{3 a c (c x)^{3/2}}",1,"(-2*x*(a + b*x^2)^(3/4))/(3*a*(c*x)^(5/2))","A",1
947,1,41,57,0.0183407,"\int \frac{1}{(c x)^{9/2} \sqrt[4]{a+b x^2}} \, dx","Integrate[1/((c*x)^(9/2)*(a + b*x^2)^(1/4)),x]","\frac{2 \sqrt{c x} \left(a+b x^2\right)^{3/4} \left(4 b x^2-3 a\right)}{21 a^2 c^5 x^4}","\frac{8 \left(a+b x^2\right)^{7/4}}{21 a^2 c (c x)^{7/2}}-\frac{2 \left(a+b x^2\right)^{3/4}}{3 a c (c x)^{7/2}}",1,"(2*Sqrt[c*x]*(a + b*x^2)^(3/4)*(-3*a + 4*b*x^2))/(21*a^2*c^5*x^4)","A",1
948,1,52,85,0.024401,"\int \frac{1}{(c x)^{13/2} \sqrt[4]{a+b x^2}} \, dx","Integrate[1/((c*x)^(13/2)*(a + b*x^2)^(1/4)),x]","-\frac{2 \sqrt{c x} \left(a+b x^2\right)^{3/4} \left(21 a^2-24 a b x^2+32 b^2 x^4\right)}{231 a^3 c^7 x^6}","-\frac{64 \left(a+b x^2\right)^{11/4}}{231 a^3 c (c x)^{11/2}}+\frac{16 \left(a+b x^2\right)^{7/4}}{21 a^2 c (c x)^{11/2}}-\frac{2 \left(a+b x^2\right)^{3/4}}{3 a c (c x)^{11/2}}",1,"(-2*Sqrt[c*x]*(a + b*x^2)^(3/4)*(21*a^2 - 24*a*b*x^2 + 32*b^2*x^4))/(231*a^3*c^7*x^6)","A",1
949,1,87,156,0.0323407,"\int \frac{(c x)^{9/2}}{\sqrt[4]{a+b x^2}} \, dx","Integrate[(c*x)^(9/2)/(a + b*x^2)^(1/4),x]","\frac{c^3 (c x)^{3/2} \left(7 a^2 \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right)-7 a^2-a b x^2+6 b^2 x^4\right)}{30 b^2 \sqrt[4]{a+b x^2}}","\frac{7 a^{5/2} c^4 \sqrt{c x} \sqrt[4]{\frac{a}{b x^2}+1} E\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{20 b^{5/2} \sqrt[4]{a+b x^2}}+\frac{7 a^2 c^4 x \sqrt{c x}}{20 b^2 \sqrt[4]{a+b x^2}}-\frac{7 a c^3 (c x)^{3/2} \left(a+b x^2\right)^{3/4}}{30 b^2}+\frac{c (c x)^{7/2} \left(a+b x^2\right)^{3/4}}{5 b}",1,"(c^3*(c*x)^(3/2)*(-7*a^2 - a*b*x^2 + 6*b^2*x^4 + 7*a^2*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, -((b*x^2)/a)]))/(30*b^2*(a + b*x^2)^(1/4))","C",1
950,1,69,125,0.0254869,"\int \frac{(c x)^{5/2}}{\sqrt[4]{a+b x^2}} \, dx","Integrate[(c*x)^(5/2)/(a + b*x^2)^(1/4),x]","\frac{c (c x)^{3/2} \left(-a \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right)+a+b x^2\right)}{3 b \sqrt[4]{a+b x^2}}","-\frac{a^{3/2} c^2 \sqrt{c x} \sqrt[4]{\frac{a}{b x^2}+1} E\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{2 b^{3/2} \sqrt[4]{a+b x^2}}-\frac{a c^2 x \sqrt{c x}}{2 b \sqrt[4]{a+b x^2}}+\frac{c (c x)^{3/2} \left(a+b x^2\right)^{3/4}}{3 b}",1,"(c*(c*x)^(3/2)*(a + b*x^2 - a*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, -((b*x^2)/a)]))/(3*b*(a + b*x^2)^(1/4))","C",1
951,1,56,83,0.0118584,"\int \frac{\sqrt{c x}}{\sqrt[4]{a+b x^2}} \, dx","Integrate[Sqrt[c*x]/(a + b*x^2)^(1/4),x]","\frac{2 x \sqrt{c x} \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right)}{3 \sqrt[4]{a+b x^2}}","\frac{x \sqrt{c x}}{\sqrt[4]{a+b x^2}}+\frac{\sqrt{a} \sqrt{c x} \sqrt[4]{\frac{a}{b x^2}+1} E\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{\sqrt{b} \sqrt[4]{a+b x^2}}",1,"(2*x*Sqrt[c*x]*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, -((b*x^2)/a)])/(3*(a + b*x^2)^(1/4))","C",1
952,1,54,90,0.0126318,"\int \frac{1}{(c x)^{3/2} \sqrt[4]{a+b x^2}} \, dx","Integrate[1/((c*x)^(3/2)*(a + b*x^2)^(1/4)),x]","-\frac{2 x \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{1}{4},\frac{1}{4};\frac{3}{4};-\frac{b x^2}{a}\right)}{(c x)^{3/2} \sqrt[4]{a+b x^2}}","\frac{2 \sqrt{b} \sqrt{c x} \sqrt[4]{\frac{a}{b x^2}+1} E\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{\sqrt{a} c^2 \sqrt[4]{a+b x^2}}-\frac{2}{c \sqrt{c x} \sqrt[4]{a+b x^2}}",1,"(-2*x*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[-1/4, 1/4, 3/4, -((b*x^2)/a)])/((c*x)^(3/2)*(a + b*x^2)^(1/4))","C",1
953,1,56,126,0.0130376,"\int \frac{1}{(c x)^{7/2} \sqrt[4]{a+b x^2}} \, dx","Integrate[1/((c*x)^(7/2)*(a + b*x^2)^(1/4)),x]","-\frac{2 x \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{5}{4},\frac{1}{4};-\frac{1}{4};-\frac{b x^2}{a}\right)}{5 (c x)^{7/2} \sqrt[4]{a+b x^2}}","-\frac{4 b^{3/2} \sqrt{c x} \sqrt[4]{\frac{a}{b x^2}+1} E\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{5 a^{3/2} c^4 \sqrt[4]{a+b x^2}}+\frac{4 b}{5 a c^3 \sqrt{c x} \sqrt[4]{a+b x^2}}-\frac{2 \left(a+b x^2\right)^{3/4}}{5 a c (c x)^{5/2}}",1,"(-2*x*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[-5/4, 1/4, -1/4, -((b*x^2)/a)])/(5*(c*x)^(7/2)*(a + b*x^2)^(1/4))","C",1
954,1,56,157,0.0133985,"\int \frac{1}{(c x)^{11/2} \sqrt[4]{a+b x^2}} \, dx","Integrate[1/((c*x)^(11/2)*(a + b*x^2)^(1/4)),x]","-\frac{2 x \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{9}{4},\frac{1}{4};-\frac{5}{4};-\frac{b x^2}{a}\right)}{9 (c x)^{11/2} \sqrt[4]{a+b x^2}}","\frac{8 b^{5/2} \sqrt{c x} \sqrt[4]{\frac{a}{b x^2}+1} E\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{15 a^{5/2} c^6 \sqrt[4]{a+b x^2}}-\frac{8 b^2}{15 a^2 c^5 \sqrt{c x} \sqrt[4]{a+b x^2}}+\frac{4 b \left(a+b x^2\right)^{3/4}}{15 a^2 c^3 (c x)^{5/2}}-\frac{2 \left(a+b x^2\right)^{3/4}}{9 a c (c x)^{9/2}}",1,"(-2*x*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[-9/4, 1/4, -5/4, -((b*x^2)/a)])/(9*(c*x)^(11/2)*(a + b*x^2)^(1/4))","C",1
955,1,241,308,0.1519469,"\int \frac{(c x)^{3/2}}{\sqrt[4]{a-b x^2}} \, dx","Integrate[(c*x)^(3/2)/(a - b*x^2)^(1/4),x]","-\frac{(c x)^{3/2} \left(8 \sqrt[4]{b} \sqrt{x} \left(a-b x^2\right)^{3/4}+\sqrt{2} a \log \left(\frac{\sqrt{b} x}{\sqrt{a-b x^2}}-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a-b x^2}}+1\right)-\sqrt{2} a \log \left(\frac{\sqrt{b} x}{\sqrt{a-b x^2}}+\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a-b x^2}}+1\right)+2 \sqrt{2} a \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a-b x^2}}\right)-2 \sqrt{2} a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a-b x^2}}+1\right)\right)}{16 b^{5/4} x^{3/2}}","-\frac{a c^{3/2} \log \left(\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a-b x^2}}+\sqrt{c}\right)}{8 \sqrt{2} b^{5/4}}+\frac{a c^{3/2} \log \left(\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}+\frac{\sqrt{2} \sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a-b x^2}}+\sqrt{c}\right)}{8 \sqrt{2} b^{5/4}}-\frac{a c^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a-b x^2}}\right)}{4 \sqrt{2} b^{5/4}}+\frac{a c^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a-b x^2}}+1\right)}{4 \sqrt{2} b^{5/4}}-\frac{c \sqrt{c x} \left(a-b x^2\right)^{3/4}}{2 b}",1,"-1/16*((c*x)^(3/2)*(8*b^(1/4)*Sqrt[x]*(a - b*x^2)^(3/4) + 2*Sqrt[2]*a*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/(a - b*x^2)^(1/4)] - 2*Sqrt[2]*a*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/(a - b*x^2)^(1/4)] + Sqrt[2]*a*Log[1 + (Sqrt[b]*x)/Sqrt[a - b*x^2] - (Sqrt[2]*b^(1/4)*Sqrt[x])/(a - b*x^2)^(1/4)] - Sqrt[2]*a*Log[1 + (Sqrt[b]*x)/Sqrt[a - b*x^2] + (Sqrt[2]*b^(1/4)*Sqrt[x])/(a - b*x^2)^(1/4)]))/(b^(5/4)*x^(3/2))","A",1
956,1,197,272,0.0429027,"\int \frac{1}{\sqrt{c x} \sqrt[4]{a-b x^2}} \, dx","Integrate[1/(Sqrt[c*x]*(a - b*x^2)^(1/4)),x]","\frac{\sqrt{x} \left(-\log \left(\frac{\sqrt{b} x}{\sqrt{a-b x^2}}-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a-b x^2}}+1\right)+\log \left(\frac{\sqrt{b} x}{\sqrt{a-b x^2}}+\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a-b x^2}}+1\right)-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a-b x^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a-b x^2}}+1\right)\right)}{2 \sqrt{2} \sqrt[4]{b} \sqrt{c x}}","-\frac{\log \left(\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a-b x^2}}+\sqrt{c}\right)}{2 \sqrt{2} \sqrt[4]{b} \sqrt{c}}+\frac{\log \left(\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}+\frac{\sqrt{2} \sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a-b x^2}}+\sqrt{c}\right)}{2 \sqrt{2} \sqrt[4]{b} \sqrt{c}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a-b x^2}}\right)}{\sqrt{2} \sqrt[4]{b} \sqrt{c}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a-b x^2}}+1\right)}{\sqrt{2} \sqrt[4]{b} \sqrt{c}}",1,"(Sqrt[x]*(-2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/(a - b*x^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/(a - b*x^2)^(1/4)] - Log[1 + (Sqrt[b]*x)/Sqrt[a - b*x^2] - (Sqrt[2]*b^(1/4)*Sqrt[x])/(a - b*x^2)^(1/4)] + Log[1 + (Sqrt[b]*x)/Sqrt[a - b*x^2] + (Sqrt[2]*b^(1/4)*Sqrt[x])/(a - b*x^2)^(1/4)]))/(2*Sqrt[2]*b^(1/4)*Sqrt[c*x])","A",1
957,1,27,29,0.0066677,"\int \frac{1}{(c x)^{5/2} \sqrt[4]{a-b x^2}} \, dx","Integrate[1/((c*x)^(5/2)*(a - b*x^2)^(1/4)),x]","-\frac{2 x \left(a-b x^2\right)^{3/4}}{3 a (c x)^{5/2}}","-\frac{2 \left(a-b x^2\right)^{3/4}}{3 a c (c x)^{3/2}}",1,"(-2*x*(a - b*x^2)^(3/4))/(3*a*(c*x)^(5/2))","A",1
958,1,42,59,0.0187062,"\int \frac{1}{(c x)^{9/2} \sqrt[4]{a-b x^2}} \, dx","Integrate[1/((c*x)^(9/2)*(a - b*x^2)^(1/4)),x]","-\frac{2 \sqrt{c x} \left(a-b x^2\right)^{3/4} \left(3 a+4 b x^2\right)}{21 a^2 c^5 x^4}","\frac{8 \left(a-b x^2\right)^{7/4}}{21 a^2 c (c x)^{7/2}}-\frac{2 \left(a-b x^2\right)^{3/4}}{3 a c (c x)^{7/2}}",1,"(-2*Sqrt[c*x]*(a - b*x^2)^(3/4)*(3*a + 4*b*x^2))/(21*a^2*c^5*x^4)","A",1
959,1,53,88,0.0259591,"\int \frac{1}{(c x)^{13/2} \sqrt[4]{a-b x^2}} \, dx","Integrate[1/((c*x)^(13/2)*(a - b*x^2)^(1/4)),x]","-\frac{2 \sqrt{c x} \left(a-b x^2\right)^{3/4} \left(21 a^2+24 a b x^2+32 b^2 x^4\right)}{231 a^3 c^7 x^6}","-\frac{64 \left(a-b x^2\right)^{11/4}}{231 a^3 c (c x)^{11/2}}+\frac{16 \left(a-b x^2\right)^{7/4}}{21 a^2 c (c x)^{11/2}}-\frac{2 \left(a-b x^2\right)^{3/4}}{3 a c (c x)^{11/2}}",1,"(-2*Sqrt[c*x]*(a - b*x^2)^(3/4)*(21*a^2 + 24*a*b*x^2 + 32*b^2*x^4))/(231*a^3*c^7*x^6)","A",1
960,1,71,128,0.0291771,"\int \frac{(c x)^{5/2}}{\sqrt[4]{a-b x^2}} \, dx","Integrate[(c*x)^(5/2)/(a - b*x^2)^(1/4),x]","\frac{c (c x)^{3/2} \left(a \sqrt[4]{1-\frac{b x^2}{a}} \, _2F_1\left(\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{b x^2}{a}\right)-a+b x^2\right)}{3 b \sqrt[4]{a-b x^2}}","\frac{a^{3/2} c^2 \sqrt{c x} \sqrt[4]{1-\frac{a}{b x^2}} E\left(\left.\frac{1}{2} \csc ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{2 b^{3/2} \sqrt[4]{a-b x^2}}-\frac{a c^3 \left(a-b x^2\right)^{3/4}}{2 b^2 \sqrt{c x}}-\frac{c (c x)^{3/2} \left(a-b x^2\right)^{3/4}}{3 b}",1,"(c*(c*x)^(3/2)*(-a + b*x^2 + a*(1 - (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, (b*x^2)/a]))/(3*b*(a - b*x^2)^(1/4))","C",1
961,1,57,90,0.0137227,"\int \frac{\sqrt{c x}}{\sqrt[4]{a-b x^2}} \, dx","Integrate[Sqrt[c*x]/(a - b*x^2)^(1/4),x]","\frac{2 x \sqrt{c x} \sqrt[4]{1-\frac{b x^2}{a}} \, _2F_1\left(\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{b x^2}{a}\right)}{3 \sqrt[4]{a-b x^2}}","\frac{\sqrt{a} \sqrt{c x} \sqrt[4]{1-\frac{a}{b x^2}} E\left(\left.\frac{1}{2} \csc ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{\sqrt{b} \sqrt[4]{a-b x^2}}-\frac{c \left(a-b x^2\right)^{3/4}}{b \sqrt{c x}}",1,"(2*x*Sqrt[c*x]*(1 - (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, (b*x^2)/a])/(3*(a - b*x^2)^(1/4))","C",1
962,1,55,68,0.0135699,"\int \frac{1}{(c x)^{3/2} \sqrt[4]{a-b x^2}} \, dx","Integrate[1/((c*x)^(3/2)*(a - b*x^2)^(1/4)),x]","-\frac{2 x \sqrt[4]{1-\frac{b x^2}{a}} \, _2F_1\left(-\frac{1}{4},\frac{1}{4};\frac{3}{4};\frac{b x^2}{a}\right)}{(c x)^{3/2} \sqrt[4]{a-b x^2}}","-\frac{2 \sqrt{b} \sqrt{c x} \sqrt[4]{1-\frac{a}{b x^2}} E\left(\left.\frac{1}{2} \csc ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{\sqrt{a} c^2 \sqrt[4]{a-b x^2}}",1,"(-2*x*(1 - (b*x^2)/a)^(1/4)*Hypergeometric2F1[-1/4, 1/4, 3/4, (b*x^2)/a])/((c*x)^(3/2)*(a - b*x^2)^(1/4))","C",1
963,1,57,100,0.0138736,"\int \frac{1}{(c x)^{7/2} \sqrt[4]{a-b x^2}} \, dx","Integrate[1/((c*x)^(7/2)*(a - b*x^2)^(1/4)),x]","-\frac{2 x \sqrt[4]{1-\frac{b x^2}{a}} \, _2F_1\left(-\frac{5}{4},\frac{1}{4};-\frac{1}{4};\frac{b x^2}{a}\right)}{5 (c x)^{7/2} \sqrt[4]{a-b x^2}}","-\frac{4 b^{3/2} \sqrt{c x} \sqrt[4]{1-\frac{a}{b x^2}} E\left(\left.\frac{1}{2} \csc ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{5 a^{3/2} c^4 \sqrt[4]{a-b x^2}}-\frac{2 \left(a-b x^2\right)^{3/4}}{5 a c (c x)^{5/2}}",1,"(-2*x*(1 - (b*x^2)/a)^(1/4)*Hypergeometric2F1[-5/4, 1/4, -1/4, (b*x^2)/a])/(5*(c*x)^(7/2)*(a - b*x^2)^(1/4))","C",1
964,1,57,130,0.0138126,"\int \frac{1}{(c x)^{11/2} \sqrt[4]{a-b x^2}} \, dx","Integrate[1/((c*x)^(11/2)*(a - b*x^2)^(1/4)),x]","-\frac{2 x \sqrt[4]{1-\frac{b x^2}{a}} \, _2F_1\left(-\frac{9}{4},\frac{1}{4};-\frac{5}{4};\frac{b x^2}{a}\right)}{9 (c x)^{11/2} \sqrt[4]{a-b x^2}}","-\frac{8 b^{5/2} \sqrt{c x} \sqrt[4]{1-\frac{a}{b x^2}} E\left(\left.\frac{1}{2} \csc ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{15 a^{5/2} c^6 \sqrt[4]{a-b x^2}}-\frac{4 b \left(a-b x^2\right)^{3/4}}{15 a^2 c^3 (c x)^{5/2}}-\frac{2 \left(a-b x^2\right)^{3/4}}{9 a c (c x)^{9/2}}",1,"(-2*x*(1 - (b*x^2)/a)^(1/4)*Hypergeometric2F1[-9/4, 1/4, -5/4, (b*x^2)/a])/(9*(c*x)^(11/2)*(a - b*x^2)^(1/4))","C",1
965,1,66,86,0.0261067,"\int \frac{(c x)^{3/2}}{\left(a+b x^2\right)^{3/4}} \, dx","Integrate[(c*x)^(3/2)/(a + b*x^2)^(3/4),x]","\frac{c \sqrt{c x} \left(-a \left(\frac{b x^2}{a}+1\right)^{3/4} \, _2F_1\left(\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right)+a+b x^2\right)}{b \left(a+b x^2\right)^{3/4}}","\frac{\sqrt{a} (c x)^{3/2} \left(\frac{a}{b x^2}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{\sqrt{b} \left(a+b x^2\right)^{3/4}}+\frac{c \sqrt{c x} \sqrt[4]{a+b x^2}}{b}",1,"(c*Sqrt[c*x]*(a + b*x^2 - a*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, -((b*x^2)/a)]))/(b*(a + b*x^2)^(3/4))","C",1
966,1,54,66,0.0118714,"\int \frac{1}{\sqrt{c x} \left(a+b x^2\right)^{3/4}} \, dx","Integrate[1/(Sqrt[c*x]*(a + b*x^2)^(3/4)),x]","\frac{2 x \left(\frac{b x^2}{a}+1\right)^{3/4} \, _2F_1\left(\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right)}{\sqrt{c x} \left(a+b x^2\right)^{3/4}}","-\frac{2 \sqrt{b} (c x)^{3/2} \left(\frac{a}{b x^2}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{\sqrt{a} c^2 \left(a+b x^2\right)^{3/4}}",1,"(2*x*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, -((b*x^2)/a)])/(Sqrt[c*x]*(a + b*x^2)^(3/4))","C",1
967,1,56,97,0.0129763,"\int \frac{1}{(c x)^{5/2} \left(a+b x^2\right)^{3/4}} \, dx","Integrate[1/((c*x)^(5/2)*(a + b*x^2)^(3/4)),x]","-\frac{2 x \left(\frac{b x^2}{a}+1\right)^{3/4} \, _2F_1\left(-\frac{3}{4},\frac{3}{4};\frac{1}{4};-\frac{b x^2}{a}\right)}{3 (c x)^{5/2} \left(a+b x^2\right)^{3/4}}","\frac{4 b^{3/2} (c x)^{3/2} \left(\frac{a}{b x^2}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{3 a^{3/2} c^4 \left(a+b x^2\right)^{3/4}}-\frac{2 \sqrt[4]{a+b x^2}}{3 a c (c x)^{3/2}}",1,"(-2*x*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[-3/4, 3/4, 1/4, -((b*x^2)/a)])/(3*(c*x)^(5/2)*(a + b*x^2)^(3/4))","C",1
968,1,56,126,0.0140018,"\int \frac{1}{(c x)^{9/2} \left(a+b x^2\right)^{3/4}} \, dx","Integrate[1/((c*x)^(9/2)*(a + b*x^2)^(3/4)),x]","-\frac{2 x \left(\frac{b x^2}{a}+1\right)^{3/4} \, _2F_1\left(-\frac{7}{4},\frac{3}{4};-\frac{3}{4};-\frac{b x^2}{a}\right)}{7 (c x)^{9/2} \left(a+b x^2\right)^{3/4}}","-\frac{8 b^{5/2} (c x)^{3/2} \left(\frac{a}{b x^2}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{7 a^{5/2} c^6 \left(a+b x^2\right)^{3/4}}+\frac{4 b \sqrt[4]{a+b x^2}}{7 a^2 c^3 (c x)^{3/2}}-\frac{2 \sqrt[4]{a+b x^2}}{7 a c (c x)^{7/2}}",1,"(-2*x*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[-7/4, 3/4, -3/4, -((b*x^2)/a)])/(7*(c*x)^(9/2)*(a + b*x^2)^(3/4))","C",1
969,1,56,157,0.0143117,"\int \frac{1}{(c x)^{13/2} \left(a+b x^2\right)^{3/4}} \, dx","Integrate[1/((c*x)^(13/2)*(a + b*x^2)^(3/4)),x]","-\frac{2 x \left(\frac{b x^2}{a}+1\right)^{3/4} \, _2F_1\left(-\frac{11}{4},\frac{3}{4};-\frac{7}{4};-\frac{b x^2}{a}\right)}{11 (c x)^{13/2} \left(a+b x^2\right)^{3/4}}","\frac{80 b^{7/2} (c x)^{3/2} \left(\frac{a}{b x^2}+1\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{77 a^{7/2} c^8 \left(a+b x^2\right)^{3/4}}-\frac{40 b^2 \sqrt[4]{a+b x^2}}{77 a^3 c^5 (c x)^{3/2}}+\frac{20 b \sqrt[4]{a+b x^2}}{77 a^2 c^3 (c x)^{7/2}}-\frac{2 \sqrt[4]{a+b x^2}}{11 a c (c x)^{11/2}}",1,"(-2*x*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[-11/4, 3/4, -7/4, -((b*x^2)/a)])/(11*(c*x)^(13/2)*(a + b*x^2)^(3/4))","C",1
970,1,97,117,0.0394548,"\int \frac{(c x)^{5/2}}{\left(a+b x^2\right)^{3/4}} \, dx","Integrate[(c*x)^(5/2)/(a + b*x^2)^(3/4),x]","\frac{(c x)^{5/2} \left(2 b^{3/4} x^{3/2} \sqrt[4]{a+b x^2}+3 a \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a+b x^2}}\right)-3 a \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a+b x^2}}\right)\right)}{4 b^{7/4} x^{5/2}}","\frac{3 a c^{5/2} \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a+b x^2}}\right)}{4 b^{7/4}}-\frac{3 a c^{5/2} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a+b x^2}}\right)}{4 b^{7/4}}+\frac{c (c x)^{3/2} \sqrt[4]{a+b x^2}}{2 b}",1,"((c*x)^(5/2)*(2*b^(3/4)*x^(3/2)*(a + b*x^2)^(1/4) + 3*a*ArcTan[(b^(1/4)*Sqrt[x])/(a + b*x^2)^(1/4)] - 3*a*ArcTanh[(b^(1/4)*Sqrt[x])/(a + b*x^2)^(1/4)]))/(4*b^(7/4)*x^(5/2))","A",1
971,1,67,84,0.0139338,"\int \frac{\sqrt{c x}}{\left(a+b x^2\right)^{3/4}} \, dx","Integrate[Sqrt[c*x]/(a + b*x^2)^(3/4),x]","\frac{\sqrt{c x} \left(\tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a+b x^2}}\right)-\tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{x}}{\sqrt[4]{a+b x^2}}\right)\right)}{b^{3/4} \sqrt{x}}","\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a+b x^2}}\right)}{b^{3/4}}-\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a+b x^2}}\right)}{b^{3/4}}",1,"(Sqrt[c*x]*(-ArcTan[(b^(1/4)*Sqrt[x])/(a + b*x^2)^(1/4)] + ArcTanh[(b^(1/4)*Sqrt[x])/(a + b*x^2)^(1/4)]))/(b^(3/4)*Sqrt[x])","A",1
972,1,24,26,0.0058582,"\int \frac{1}{(c x)^{3/2} \left(a+b x^2\right)^{3/4}} \, dx","Integrate[1/((c*x)^(3/2)*(a + b*x^2)^(3/4)),x]","-\frac{2 x \sqrt[4]{a+b x^2}}{a (c x)^{3/2}}","-\frac{2 \sqrt[4]{a+b x^2}}{a c \sqrt{c x}}",1,"(-2*x*(a + b*x^2)^(1/4))/(a*(c*x)^(3/2))","A",1
973,1,34,55,0.0181863,"\int \frac{1}{(c x)^{7/2} \left(a+b x^2\right)^{3/4}} \, dx","Integrate[1/((c*x)^(7/2)*(a + b*x^2)^(3/4)),x]","-\frac{2 x \left(a-4 b x^2\right) \sqrt[4]{a+b x^2}}{5 a^2 (c x)^{7/2}}","\frac{8 \left(a+b x^2\right)^{5/4}}{5 a^2 c (c x)^{5/2}}-\frac{2 \sqrt[4]{a+b x^2}}{a c (c x)^{5/2}}",1,"(-2*x*(a - 4*b*x^2)*(a + b*x^2)^(1/4))/(5*a^2*(c*x)^(7/2))","A",1
974,1,52,83,0.0244321,"\int \frac{1}{(c x)^{11/2} \left(a+b x^2\right)^{3/4}} \, dx","Integrate[1/((c*x)^(11/2)*(a + b*x^2)^(3/4)),x]","-\frac{2 \sqrt{c x} \sqrt[4]{a+b x^2} \left(5 a^2-8 a b x^2+32 b^2 x^4\right)}{45 a^3 c^6 x^5}","-\frac{64 \left(a+b x^2\right)^{9/4}}{45 a^3 c (c x)^{9/2}}+\frac{16 \left(a+b x^2\right)^{5/4}}{5 a^2 c (c x)^{9/2}}-\frac{2 \sqrt[4]{a+b x^2}}{a c (c x)^{9/2}}",1,"(-2*Sqrt[c*x]*(a + b*x^2)^(1/4)*(5*a^2 - 8*a*b*x^2 + 32*b^2*x^4))/(45*a^3*c^6*x^5)","A",1
975,1,68,91,0.0277547,"\int \frac{(c x)^{3/2}}{\left(a-b x^2\right)^{3/4}} \, dx","Integrate[(c*x)^(3/2)/(a - b*x^2)^(3/4),x]","\frac{c \sqrt{c x} \left(a \left(1-\frac{b x^2}{a}\right)^{3/4} \, _2F_1\left(\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{b x^2}{a}\right)-a+b x^2\right)}{b \left(a-b x^2\right)^{3/4}}","-\frac{\sqrt{a} (c x)^{3/2} \left(1-\frac{a}{b x^2}\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \csc ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{\sqrt{b} \left(a-b x^2\right)^{3/4}}-\frac{c \sqrt{c x} \sqrt[4]{a-b x^2}}{b}",1,"(c*Sqrt[c*x]*(-a + b*x^2 + a*(1 - (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, (b*x^2)/a]))/(b*(a - b*x^2)^(3/4))","C",1
976,1,55,68,0.012921,"\int \frac{1}{\sqrt{c x} \left(a-b x^2\right)^{3/4}} \, dx","Integrate[1/(Sqrt[c*x]*(a - b*x^2)^(3/4)),x]","\frac{2 x \left(1-\frac{b x^2}{a}\right)^{3/4} \, _2F_1\left(\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{b x^2}{a}\right)}{\sqrt{c x} \left(a-b x^2\right)^{3/4}}","-\frac{2 \sqrt{b} (c x)^{3/2} \left(1-\frac{a}{b x^2}\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \csc ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{\sqrt{a} c^2 \left(a-b x^2\right)^{3/4}}",1,"(2*x*(1 - (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, (b*x^2)/a])/(Sqrt[c*x]*(a - b*x^2)^(3/4))","C",1
977,1,57,100,0.013721,"\int \frac{1}{(c x)^{5/2} \left(a-b x^2\right)^{3/4}} \, dx","Integrate[1/((c*x)^(5/2)*(a - b*x^2)^(3/4)),x]","-\frac{2 x \left(1-\frac{b x^2}{a}\right)^{3/4} \, _2F_1\left(-\frac{3}{4},\frac{3}{4};\frac{1}{4};\frac{b x^2}{a}\right)}{3 (c x)^{5/2} \left(a-b x^2\right)^{3/4}}","-\frac{4 b^{3/2} (c x)^{3/2} \left(1-\frac{a}{b x^2}\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \csc ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{3 a^{3/2} c^4 \left(a-b x^2\right)^{3/4}}-\frac{2 \sqrt[4]{a-b x^2}}{3 a c (c x)^{3/2}}",1,"(-2*x*(1 - (b*x^2)/a)^(3/4)*Hypergeometric2F1[-3/4, 3/4, 1/4, (b*x^2)/a])/(3*(c*x)^(5/2)*(a - b*x^2)^(3/4))","C",1
978,1,57,130,0.0132342,"\int \frac{1}{(c x)^{9/2} \left(a-b x^2\right)^{3/4}} \, dx","Integrate[1/((c*x)^(9/2)*(a - b*x^2)^(3/4)),x]","-\frac{2 x \left(1-\frac{b x^2}{a}\right)^{3/4} \, _2F_1\left(-\frac{7}{4},\frac{3}{4};-\frac{3}{4};\frac{b x^2}{a}\right)}{7 (c x)^{9/2} \left(a-b x^2\right)^{3/4}}","-\frac{8 b^{5/2} (c x)^{3/2} \left(1-\frac{a}{b x^2}\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \csc ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{7 a^{5/2} c^6 \left(a-b x^2\right)^{3/4}}-\frac{4 b \sqrt[4]{a-b x^2}}{7 a^2 c^3 (c x)^{3/2}}-\frac{2 \sqrt[4]{a-b x^2}}{7 a c (c x)^{7/2}}",1,"(-2*x*(1 - (b*x^2)/a)^(3/4)*Hypergeometric2F1[-7/4, 3/4, -3/4, (b*x^2)/a])/(7*(c*x)^(9/2)*(a - b*x^2)^(3/4))","C",1
979,1,57,162,0.0134703,"\int \frac{1}{(c x)^{13/2} \left(a-b x^2\right)^{3/4}} \, dx","Integrate[1/((c*x)^(13/2)*(a - b*x^2)^(3/4)),x]","-\frac{2 x \left(1-\frac{b x^2}{a}\right)^{3/4} \, _2F_1\left(-\frac{11}{4},\frac{3}{4};-\frac{7}{4};\frac{b x^2}{a}\right)}{11 (c x)^{13/2} \left(a-b x^2\right)^{3/4}}","-\frac{80 b^{7/2} (c x)^{3/2} \left(1-\frac{a}{b x^2}\right)^{3/4} \operatorname{EllipticF}\left(\frac{1}{2} \csc ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right),2\right)}{77 a^{7/2} c^8 \left(a-b x^2\right)^{3/4}}-\frac{40 b^2 \sqrt[4]{a-b x^2}}{77 a^3 c^5 (c x)^{3/2}}-\frac{20 b \sqrt[4]{a-b x^2}}{77 a^2 c^3 (c x)^{7/2}}-\frac{2 \sqrt[4]{a-b x^2}}{11 a c (c x)^{11/2}}",1,"(-2*x*(1 - (b*x^2)/a)^(3/4)*Hypergeometric2F1[-11/4, 3/4, -7/4, (b*x^2)/a])/(11*(c*x)^(13/2)*(a - b*x^2)^(3/4))","C",1
980,1,112,308,0.0573737,"\int \frac{(c x)^{5/2}}{\left(a-b x^2\right)^{3/4}} \, dx","Integrate[(c*x)^(5/2)/(a - b*x^2)^(3/4),x]","-\frac{(c x)^{5/2} \left(-3 a \sqrt[4]{-b} \tan ^{-1}\left(\frac{\sqrt[4]{-b} \sqrt{x}}{\sqrt[4]{a-b x^2}}\right)+3 a \sqrt[4]{-b} \tanh ^{-1}\left(\frac{\sqrt[4]{-b} \sqrt{x}}{\sqrt[4]{a-b x^2}}\right)+2 b x^{3/2} \sqrt[4]{a-b x^2}\right)}{4 b^2 x^{5/2}}","\frac{3 a c^{5/2} \log \left(\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a-b x^2}}+\sqrt{c}\right)}{8 \sqrt{2} b^{7/4}}-\frac{3 a c^{5/2} \log \left(\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}+\frac{\sqrt{2} \sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a-b x^2}}+\sqrt{c}\right)}{8 \sqrt{2} b^{7/4}}-\frac{3 a c^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a-b x^2}}\right)}{4 \sqrt{2} b^{7/4}}+\frac{3 a c^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a-b x^2}}+1\right)}{4 \sqrt{2} b^{7/4}}-\frac{c (c x)^{3/2} \sqrt[4]{a-b x^2}}{2 b}",1,"-1/4*((c*x)^(5/2)*(2*b*x^(3/2)*(a - b*x^2)^(1/4) - 3*a*(-b)^(1/4)*ArcTan[((-b)^(1/4)*Sqrt[x])/(a - b*x^2)^(1/4)] + 3*a*(-b)^(1/4)*ArcTanh[((-b)^(1/4)*Sqrt[x])/(a - b*x^2)^(1/4)]))/(b^2*x^(5/2))","A",1
981,1,75,272,0.0238709,"\int \frac{\sqrt{c x}}{\left(a-b x^2\right)^{3/4}} \, dx","Integrate[Sqrt[c*x]/(a - b*x^2)^(3/4),x]","\frac{\sqrt{c x} \left(\tanh ^{-1}\left(\frac{\sqrt[4]{-b} \sqrt{x}}{\sqrt[4]{a-b x^2}}\right)-\tan ^{-1}\left(\frac{\sqrt[4]{-b} \sqrt{x}}{\sqrt[4]{a-b x^2}}\right)\right)}{(-b)^{3/4} \sqrt{x}}","\frac{\sqrt{c} \log \left(\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a-b x^2}}+\sqrt{c}\right)}{2 \sqrt{2} b^{3/4}}-\frac{\sqrt{c} \log \left(\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}+\frac{\sqrt{2} \sqrt[4]{b} \sqrt{c x}}{\sqrt[4]{a-b x^2}}+\sqrt{c}\right)}{2 \sqrt{2} b^{3/4}}-\frac{\sqrt{c} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a-b x^2}}\right)}{\sqrt{2} b^{3/4}}+\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a-b x^2}}+1\right)}{\sqrt{2} b^{3/4}}",1,"(Sqrt[c*x]*(-ArcTan[((-b)^(1/4)*Sqrt[x])/(a - b*x^2)^(1/4)] + ArcTanh[((-b)^(1/4)*Sqrt[x])/(a - b*x^2)^(1/4)]))/((-b)^(3/4)*Sqrt[x])","A",1
982,1,25,27,0.0056344,"\int \frac{1}{(c x)^{3/2} \left(a-b x^2\right)^{3/4}} \, dx","Integrate[1/((c*x)^(3/2)*(a - b*x^2)^(3/4)),x]","-\frac{2 x \sqrt[4]{a-b x^2}}{a (c x)^{3/2}}","-\frac{2 \sqrt[4]{a-b x^2}}{a c \sqrt{c x}}",1,"(-2*x*(a - b*x^2)^(1/4))/(a*(c*x)^(3/2))","A",1
983,1,35,57,0.0184565,"\int \frac{1}{(c x)^{7/2} \left(a-b x^2\right)^{3/4}} \, dx","Integrate[1/((c*x)^(7/2)*(a - b*x^2)^(3/4)),x]","-\frac{2 x \sqrt[4]{a-b x^2} \left(a+4 b x^2\right)}{5 a^2 (c x)^{7/2}}","\frac{8 \left(a-b x^2\right)^{5/4}}{5 a^2 c (c x)^{5/2}}-\frac{2 \sqrt[4]{a-b x^2}}{a c (c x)^{5/2}}",1,"(-2*x*(a - b*x^2)^(1/4)*(a + 4*b*x^2))/(5*a^2*(c*x)^(7/2))","A",1
984,1,53,86,0.0243362,"\int \frac{1}{(c x)^{11/2} \left(a-b x^2\right)^{3/4}} \, dx","Integrate[1/((c*x)^(11/2)*(a - b*x^2)^(3/4)),x]","-\frac{2 \sqrt{c x} \sqrt[4]{a-b x^2} \left(5 a^2+8 a b x^2+32 b^2 x^4\right)}{45 a^3 c^6 x^5}","-\frac{64 \left(a-b x^2\right)^{9/4}}{45 a^3 c (c x)^{9/2}}+\frac{16 \left(a-b x^2\right)^{5/4}}{5 a^2 c (c x)^{9/2}}-\frac{2 \sqrt[4]{a-b x^2}}{a c (c x)^{9/2}}",1,"(-2*Sqrt[c*x]*(a - b*x^2)^(1/4)*(5*a^2 + 8*a*b*x^2 + 32*b^2*x^4))/(45*a^3*c^6*x^5)","A",1
985,1,63,146,0.0355438,"\int \frac{(c x)^{7/2}}{\left(a+b x^2\right)^{5/4}} \, dx","Integrate[(c*x)^(7/2)/(a + b*x^2)^(5/4),x]","\frac{c (c x)^{5/2} \left(1-\sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{5}{4},\frac{5}{4};\frac{9}{4};-\frac{b x^2}{a}\right)\right)}{2 b \sqrt[4]{a+b x^2}}","-\frac{5 a c^{7/2} \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a+b x^2}}\right)}{4 b^{9/4}}-\frac{5 a c^{7/2} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a+b x^2}}\right)}{4 b^{9/4}}+\frac{5 a c^3 \sqrt{c x}}{2 b^2 \sqrt[4]{a+b x^2}}+\frac{c (c x)^{5/2}}{2 b \sqrt[4]{a+b x^2}}",1,"(c*(c*x)^(5/2)*(1 - (1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[5/4, 5/4, 9/4, -((b*x^2)/a)]))/(2*b*(a + b*x^2)^(1/4))","C",1
986,1,59,107,0.012835,"\int \frac{(c x)^{3/2}}{\left(a+b x^2\right)^{5/4}} \, dx","Integrate[(c*x)^(3/2)/(a + b*x^2)^(5/4),x]","\frac{2 x (c x)^{3/2} \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{5}{4},\frac{5}{4};\frac{9}{4};-\frac{b x^2}{a}\right)}{5 a \sqrt[4]{a+b x^2}}","\frac{c^{3/2} \tan ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a+b x^2}}\right)}{b^{5/4}}+\frac{c^{3/2} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sqrt{c x}}{\sqrt{c} \sqrt[4]{a+b x^2}}\right)}{b^{5/4}}-\frac{2 c \sqrt{c x}}{b \sqrt[4]{a+b x^2}}",1,"(2*x*(c*x)^(3/2)*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[5/4, 5/4, 9/4, -((b*x^2)/a)])/(5*a*(a + b*x^2)^(1/4))","C",1
987,1,24,26,0.0064534,"\int \frac{1}{\sqrt{c x} \left(a+b x^2\right)^{5/4}} \, dx","Integrate[1/(Sqrt[c*x]*(a + b*x^2)^(5/4)),x]","\frac{2 x}{a \sqrt{c x} \sqrt[4]{a+b x^2}}","\frac{2 \sqrt{c x}}{a c \sqrt[4]{a+b x^2}}",1,"(2*x)/(a*Sqrt[c*x]*(a + b*x^2)^(1/4))","A",1
988,1,34,55,0.0098212,"\int \frac{1}{(c x)^{5/2} \left(a+b x^2\right)^{5/4}} \, dx","Integrate[1/((c*x)^(5/2)*(a + b*x^2)^(5/4)),x]","-\frac{2 x \left(a+4 b x^2\right)}{3 a^2 (c x)^{5/2} \sqrt[4]{a+b x^2}}","\frac{2}{a c (c x)^{3/2} \sqrt[4]{a+b x^2}}-\frac{8 \left(a+b x^2\right)^{3/4}}{3 a^2 c (c x)^{3/2}}",1,"(-2*x*(a + 4*b*x^2))/(3*a^2*(c*x)^(5/2)*(a + b*x^2)^(1/4))","A",1
989,1,47,83,0.0114839,"\int \frac{1}{(c x)^{9/2} \left(a+b x^2\right)^{5/4}} \, dx","Integrate[1/((c*x)^(9/2)*(a + b*x^2)^(5/4)),x]","-\frac{2 x \left(3 a^2-8 a b x^2-32 b^2 x^4\right)}{21 a^3 (c x)^{9/2} \sqrt[4]{a+b x^2}}","\frac{64 \left(a+b x^2\right)^{7/4}}{21 a^3 c (c x)^{7/2}}-\frac{16 \left(a+b x^2\right)^{3/4}}{3 a^2 c (c x)^{7/2}}+\frac{2}{a c (c x)^{7/2} \sqrt[4]{a+b x^2}}",1,"(-2*x*(3*a^2 - 8*a*b*x^2 - 32*b^2*x^4))/(21*a^3*(c*x)^(9/2)*(a + b*x^2)^(1/4))","A",1
990,1,58,109,0.0129443,"\int \frac{1}{(c x)^{13/2} \left(a+b x^2\right)^{5/4}} \, dx","Integrate[1/((c*x)^(13/2)*(a + b*x^2)^(5/4)),x]","-\frac{2 x \left(7 a^3-12 a^2 b x^2+32 a b^2 x^4+128 b^3 x^6\right)}{77 a^4 (c x)^{13/2} \sqrt[4]{a+b x^2}}","-\frac{256 \left(a+b x^2\right)^{11/4}}{77 a^4 c (c x)^{11/2}}+\frac{64 \left(a+b x^2\right)^{7/4}}{7 a^3 c (c x)^{11/2}}-\frac{8 \left(a+b x^2\right)^{3/4}}{a^2 c (c x)^{11/2}}+\frac{2}{a c (c x)^{11/2} \sqrt[4]{a+b x^2}}",1,"(-2*x*(7*a^3 - 12*a^2*b*x^2 + 32*a*b^2*x^4 + 128*b^3*x^6))/(77*a^4*(c*x)^(13/2)*(a + b*x^2)^(1/4))","A",1
991,1,87,155,0.0370188,"\int \frac{(c x)^{13/2}}{\left(a+b x^2\right)^{5/4}} \, dx","Integrate[(c*x)^(13/2)/(a + b*x^2)^(5/4),x]","\frac{c^5 (c x)^{3/2} \left(-77 a^2 \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{3}{4},\frac{5}{4};\frac{7}{4};-\frac{b x^2}{a}\right)+77 a^2-22 a b x^2+12 b^2 x^4\right)}{60 b^3 \sqrt[4]{a+b x^2}}","\frac{77 a^{5/2} c^6 \sqrt{c x} \sqrt[4]{\frac{a}{b x^2}+1} E\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{20 b^{7/2} \sqrt[4]{a+b x^2}}+\frac{77 a^2 c^5 (c x)^{3/2}}{60 b^3 \sqrt[4]{a+b x^2}}-\frac{11 a c^3 (c x)^{7/2}}{30 b^2 \sqrt[4]{a+b x^2}}+\frac{c (c x)^{11/2}}{5 b \sqrt[4]{a+b x^2}}",1,"(c^5*(c*x)^(3/2)*(77*a^2 - 22*a*b*x^2 + 12*b^2*x^4 - 77*a^2*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[3/4, 5/4, 7/4, -((b*x^2)/a)]))/(60*b^3*(a + b*x^2)^(1/4))","C",1
992,1,74,124,0.0323694,"\int \frac{(c x)^{9/2}}{\left(a+b x^2\right)^{5/4}} \, dx","Integrate[(c*x)^(9/2)/(a + b*x^2)^(5/4),x]","\frac{c^3 (c x)^{3/2} \left(7 a \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{3}{4},\frac{5}{4};\frac{7}{4};-\frac{b x^2}{a}\right)-7 a+2 b x^2\right)}{6 b^2 \sqrt[4]{a+b x^2}}","-\frac{7 a^{3/2} c^4 \sqrt{c x} \sqrt[4]{\frac{a}{b x^2}+1} E\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{2 b^{5/2} \sqrt[4]{a+b x^2}}-\frac{7 a c^3 (c x)^{3/2}}{6 b^2 \sqrt[4]{a+b x^2}}+\frac{c (c x)^{7/2}}{3 b \sqrt[4]{a+b x^2}}",1,"(c^3*(c*x)^(3/2)*(-7*a + 2*b*x^2 + 7*a*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[3/4, 5/4, 7/4, -((b*x^2)/a)]))/(6*b^2*(a + b*x^2)^(1/4))","C",1
993,1,60,90,0.0312531,"\int \frac{(c x)^{5/2}}{\left(a+b x^2\right)^{5/4}} \, dx","Integrate[(c*x)^(5/2)/(a + b*x^2)^(5/4),x]","\frac{c (c x)^{3/2} \left(1-\sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{3}{4},\frac{5}{4};\frac{7}{4};-\frac{b x^2}{a}\right)\right)}{b \sqrt[4]{a+b x^2}}","\frac{3 \sqrt{a} c^2 \sqrt{c x} \sqrt[4]{\frac{a}{b x^2}+1} E\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{b^{3/2} \sqrt[4]{a+b x^2}}+\frac{c (c x)^{3/2}}{b \sqrt[4]{a+b x^2}}",1,"(c*(c*x)^(3/2)*(1 - (1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[3/4, 5/4, 7/4, -((b*x^2)/a)]))/(b*(a + b*x^2)^(1/4))","C",1
994,1,59,63,0.011882,"\int \frac{\sqrt{c x}}{\left(a+b x^2\right)^{5/4}} \, dx","Integrate[Sqrt[c*x]/(a + b*x^2)^(5/4),x]","\frac{2 x \sqrt{c x} \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{3}{4},\frac{5}{4};\frac{7}{4};-\frac{b x^2}{a}\right)}{3 a \sqrt[4]{a+b x^2}}","-\frac{2 \sqrt{c x} \sqrt[4]{\frac{a}{b x^2}+1} E\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{\sqrt{a} \sqrt{b} \sqrt[4]{a+b x^2}}",1,"(2*x*Sqrt[c*x]*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[3/4, 5/4, 7/4, -((b*x^2)/a)])/(3*a*(a + b*x^2)^(1/4))","C",1
995,1,57,93,0.0120141,"\int \frac{1}{(c x)^{3/2} \left(a+b x^2\right)^{5/4}} \, dx","Integrate[1/((c*x)^(3/2)*(a + b*x^2)^(5/4)),x]","-\frac{2 x \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{1}{4},\frac{5}{4};\frac{3}{4};-\frac{b x^2}{a}\right)}{a (c x)^{3/2} \sqrt[4]{a+b x^2}}","\frac{4 \sqrt{b} \sqrt{c x} \sqrt[4]{\frac{a}{b x^2}+1} E\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{a^{3/2} c^2 \sqrt[4]{a+b x^2}}-\frac{2}{a c \sqrt{c x} \sqrt[4]{a+b x^2}}",1,"(-2*x*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[-1/4, 5/4, 3/4, -((b*x^2)/a)])/(a*(c*x)^(3/2)*(a + b*x^2)^(1/4))","C",1
996,1,59,126,0.0128429,"\int \frac{1}{(c x)^{7/2} \left(a+b x^2\right)^{5/4}} \, dx","Integrate[1/((c*x)^(7/2)*(a + b*x^2)^(5/4)),x]","-\frac{2 x \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{5}{4},\frac{5}{4};-\frac{1}{4};-\frac{b x^2}{a}\right)}{5 a (c x)^{7/2} \sqrt[4]{a+b x^2}}","-\frac{24 b^{3/2} \sqrt{c x} \sqrt[4]{\frac{a}{b x^2}+1} E\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{5 a^{5/2} c^4 \sqrt[4]{a+b x^2}}+\frac{12 b}{5 a^2 c^3 \sqrt{c x} \sqrt[4]{a+b x^2}}-\frac{2}{5 a c (c x)^{5/2} \sqrt[4]{a+b x^2}}",1,"(-2*x*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[-5/4, 5/4, -1/4, -((b*x^2)/a)])/(5*a*(c*x)^(7/2)*(a + b*x^2)^(1/4))","C",1
997,1,59,157,0.0124362,"\int \frac{1}{(c x)^{11/2} \left(a+b x^2\right)^{5/4}} \, dx","Integrate[1/((c*x)^(11/2)*(a + b*x^2)^(5/4)),x]","-\frac{2 x \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{9}{4},\frac{5}{4};-\frac{5}{4};-\frac{b x^2}{a}\right)}{9 a (c x)^{11/2} \sqrt[4]{a+b x^2}}","\frac{16 b^{5/2} \sqrt{c x} \sqrt[4]{\frac{a}{b x^2}+1} E\left(\left.\frac{1}{2} \cot ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{3 a^{7/2} c^6 \sqrt[4]{a+b x^2}}-\frac{8 b^2}{3 a^3 c^5 \sqrt{c x} \sqrt[4]{a+b x^2}}+\frac{4 b}{9 a^2 c^3 (c x)^{5/2} \sqrt[4]{a+b x^2}}-\frac{2}{9 a c (c x)^{9/2} \sqrt[4]{a+b x^2}}",1,"(-2*x*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[-9/4, 5/4, -5/4, -((b*x^2)/a)])/(9*a*(c*x)^(11/2)*(a + b*x^2)^(1/4))","C",1
998,1,56,58,0.011866,"\int \frac{(c x)^{5/4}}{\sqrt[4]{a+b x^2}} \, dx","Integrate[(c*x)^(5/4)/(a + b*x^2)^(1/4),x]","\frac{4 x (c x)^{5/4} \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{9}{8};\frac{17}{8};-\frac{b x^2}{a}\right)}{9 \sqrt[4]{a+b x^2}}","\frac{4 (c x)^{9/4} \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{9}{8};\frac{17}{8};-\frac{b x^2}{a}\right)}{9 c \sqrt[4]{a+b x^2}}",1,"(4*x*(c*x)^(5/4)*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 9/8, 17/8, -((b*x^2)/a)])/(9*(a + b*x^2)^(1/4))","A",1
999,1,56,58,0.0104056,"\int \frac{(c x)^{3/4}}{\sqrt[4]{a+b x^2}} \, dx","Integrate[(c*x)^(3/4)/(a + b*x^2)^(1/4),x]","\frac{4 x (c x)^{3/4} \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{7}{8};\frac{15}{8};-\frac{b x^2}{a}\right)}{7 \sqrt[4]{a+b x^2}}","\frac{4 (c x)^{7/4} \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{7}{8};\frac{15}{8};-\frac{b x^2}{a}\right)}{7 c \sqrt[4]{a+b x^2}}",1,"(4*x*(c*x)^(3/4)*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 7/8, 15/8, -((b*x^2)/a)])/(7*(a + b*x^2)^(1/4))","A",1
1000,1,56,58,0.01156,"\int \frac{\sqrt[4]{c x}}{\sqrt[4]{a+b x^2}} \, dx","Integrate[(c*x)^(1/4)/(a + b*x^2)^(1/4),x]","\frac{4 x \sqrt[4]{c x} \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{5}{8};\frac{13}{8};-\frac{b x^2}{a}\right)}{5 \sqrt[4]{a+b x^2}}","\frac{4 (c x)^{5/4} \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{5}{8};\frac{13}{8};-\frac{b x^2}{a}\right)}{5 c \sqrt[4]{a+b x^2}}",1,"(4*x*(c*x)^(1/4)*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 5/8, 13/8, -((b*x^2)/a)])/(5*(a + b*x^2)^(1/4))","A",1
1001,1,56,58,0.0115282,"\int \frac{1}{\sqrt[4]{c x} \sqrt[4]{a+b x^2}} \, dx","Integrate[1/((c*x)^(1/4)*(a + b*x^2)^(1/4)),x]","\frac{4 x \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{3}{8};\frac{11}{8};-\frac{b x^2}{a}\right)}{3 \sqrt[4]{c x} \sqrt[4]{a+b x^2}}","\frac{4 (c x)^{3/4} \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{3}{8};\frac{11}{8};-\frac{b x^2}{a}\right)}{3 c \sqrt[4]{a+b x^2}}",1,"(4*x*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 3/8, 11/8, -((b*x^2)/a)])/(3*(c*x)^(1/4)*(a + b*x^2)^(1/4))","A",1
1002,1,54,56,0.0105167,"\int \frac{1}{(c x)^{3/4} \sqrt[4]{a+b x^2}} \, dx","Integrate[1/((c*x)^(3/4)*(a + b*x^2)^(1/4)),x]","\frac{4 x \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{8},\frac{1}{4};\frac{9}{8};-\frac{b x^2}{a}\right)}{(c x)^{3/4} \sqrt[4]{a+b x^2}}","\frac{4 \sqrt[4]{c x} \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{8},\frac{1}{4};\frac{9}{8};-\frac{b x^2}{a}\right)}{c \sqrt[4]{a+b x^2}}",1,"(4*x*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/8, 1/4, 9/8, -((b*x^2)/a)])/((c*x)^(3/4)*(a + b*x^2)^(1/4))","A",1
1003,1,54,56,0.0107996,"\int \frac{1}{(c x)^{5/4} \sqrt[4]{a+b x^2}} \, dx","Integrate[1/((c*x)^(5/4)*(a + b*x^2)^(1/4)),x]","-\frac{4 x \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{1}{8},\frac{1}{4};\frac{7}{8};-\frac{b x^2}{a}\right)}{(c x)^{5/4} \sqrt[4]{a+b x^2}}","-\frac{4 \sqrt[4]{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{1}{8},\frac{1}{4};\frac{7}{8};-\frac{b x^2}{a}\right)}{c \sqrt[4]{c x} \sqrt[4]{a+b x^2}}",1,"(-4*x*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[-1/8, 1/4, 7/8, -((b*x^2)/a)])/((c*x)^(5/4)*(a + b*x^2)^(1/4))","A",1
1004,1,59,61,0.0128521,"\int \frac{(c x)^{5/4}}{\left(a+b x^2\right)^{7/4}} \, dx","Integrate[(c*x)^(5/4)/(a + b*x^2)^(7/4),x]","\frac{4 x (c x)^{5/4} \left(\frac{b x^2}{a}+1\right)^{3/4} \, _2F_1\left(\frac{9}{8},\frac{7}{4};\frac{17}{8};-\frac{b x^2}{a}\right)}{9 a \left(a+b x^2\right)^{3/4}}","\frac{4 (c x)^{9/4} \left(\frac{b x^2}{a}+1\right)^{3/4} \, _2F_1\left(\frac{9}{8},\frac{7}{4};\frac{17}{8};-\frac{b x^2}{a}\right)}{9 a c \left(a+b x^2\right)^{3/4}}",1,"(4*x*(c*x)^(5/4)*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[9/8, 7/4, 17/8, -((b*x^2)/a)])/(9*a*(a + b*x^2)^(3/4))","A",1
1005,1,59,61,0.0121399,"\int \frac{(c x)^{3/4}}{\left(a+b x^2\right)^{7/4}} \, dx","Integrate[(c*x)^(3/4)/(a + b*x^2)^(7/4),x]","\frac{4 x (c x)^{3/4} \left(\frac{b x^2}{a}+1\right)^{3/4} \, _2F_1\left(\frac{7}{8},\frac{7}{4};\frac{15}{8};-\frac{b x^2}{a}\right)}{7 a \left(a+b x^2\right)^{3/4}}","\frac{4 (c x)^{7/4} \left(\frac{b x^2}{a}+1\right)^{3/4} \, _2F_1\left(\frac{7}{8},\frac{7}{4};\frac{15}{8};-\frac{b x^2}{a}\right)}{7 a c \left(a+b x^2\right)^{3/4}}",1,"(4*x*(c*x)^(3/4)*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[7/8, 7/4, 15/8, -((b*x^2)/a)])/(7*a*(a + b*x^2)^(3/4))","A",1
1006,1,59,61,0.0105401,"\int \frac{\sqrt[4]{c x}}{\left(a+b x^2\right)^{7/4}} \, dx","Integrate[(c*x)^(1/4)/(a + b*x^2)^(7/4),x]","\frac{4 x \sqrt[4]{c x} \left(\frac{b x^2}{a}+1\right)^{3/4} \, _2F_1\left(\frac{5}{8},\frac{7}{4};\frac{13}{8};-\frac{b x^2}{a}\right)}{5 a \left(a+b x^2\right)^{3/4}}","\frac{4 (c x)^{5/4} \left(\frac{b x^2}{a}+1\right)^{3/4} \, _2F_1\left(\frac{5}{8},\frac{7}{4};\frac{13}{8};-\frac{b x^2}{a}\right)}{5 a c \left(a+b x^2\right)^{3/4}}",1,"(4*x*(c*x)^(1/4)*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[5/8, 7/4, 13/8, -((b*x^2)/a)])/(5*a*(a + b*x^2)^(3/4))","A",1
1007,1,59,61,0.0107112,"\int \frac{1}{\sqrt[4]{c x} \left(a+b x^2\right)^{7/4}} \, dx","Integrate[1/((c*x)^(1/4)*(a + b*x^2)^(7/4)),x]","\frac{4 x \left(\frac{b x^2}{a}+1\right)^{3/4} \, _2F_1\left(\frac{3}{8},\frac{7}{4};\frac{11}{8};-\frac{b x^2}{a}\right)}{3 a \sqrt[4]{c x} \left(a+b x^2\right)^{3/4}}","\frac{4 (c x)^{3/4} \left(\frac{b x^2}{a}+1\right)^{3/4} \, _2F_1\left(\frac{3}{8},\frac{7}{4};\frac{11}{8};-\frac{b x^2}{a}\right)}{3 a c \left(a+b x^2\right)^{3/4}}",1,"(4*x*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[3/8, 7/4, 11/8, -((b*x^2)/a)])/(3*a*(c*x)^(1/4)*(a + b*x^2)^(3/4))","A",1
1008,1,57,59,0.0115534,"\int \frac{1}{(c x)^{3/4} \left(a+b x^2\right)^{7/4}} \, dx","Integrate[1/((c*x)^(3/4)*(a + b*x^2)^(7/4)),x]","\frac{4 x \left(\frac{b x^2}{a}+1\right)^{3/4} \, _2F_1\left(\frac{1}{8},\frac{7}{4};\frac{9}{8};-\frac{b x^2}{a}\right)}{a (c x)^{3/4} \left(a+b x^2\right)^{3/4}}","\frac{4 \sqrt[4]{c x} \left(\frac{b x^2}{a}+1\right)^{3/4} \, _2F_1\left(\frac{1}{8},\frac{7}{4};\frac{9}{8};-\frac{b x^2}{a}\right)}{a c \left(a+b x^2\right)^{3/4}}",1,"(4*x*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/8, 7/4, 9/8, -((b*x^2)/a)])/(a*(c*x)^(3/4)*(a + b*x^2)^(3/4))","A",1
1009,1,57,59,0.0109453,"\int \frac{1}{(c x)^{5/4} \left(a+b x^2\right)^{7/4}} \, dx","Integrate[1/((c*x)^(5/4)*(a + b*x^2)^(7/4)),x]","-\frac{4 x \left(\frac{b x^2}{a}+1\right)^{3/4} \, _2F_1\left(-\frac{1}{8},\frac{7}{4};\frac{7}{8};-\frac{b x^2}{a}\right)}{a (c x)^{5/4} \left(a+b x^2\right)^{3/4}}","-\frac{4 \left(\frac{b x^2}{a}+1\right)^{3/4} \, _2F_1\left(-\frac{1}{8},\frac{7}{4};\frac{7}{8};-\frac{b x^2}{a}\right)}{a c \sqrt[4]{c x} \left(a+b x^2\right)^{3/4}}",1,"(-4*x*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[-1/8, 7/4, 7/8, -((b*x^2)/a)])/(a*(c*x)^(5/4)*(a + b*x^2)^(3/4))","A",1
1010,1,105,345,0.0618555,"\int x^6 \sqrt[6]{a+b x^2} \, dx","Integrate[x^6*(a + b*x^2)^(1/6),x]","\frac{3 x \sqrt[6]{a+b x^2} \left(\sqrt[6]{\frac{b x^2}{a}+1} \left(27 a^3-3 a^2 b x^2+2 a b^2 x^4+32 b^3 x^6\right)-27 a^3 \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)\right)}{704 b^3 \sqrt[6]{\frac{b x^2}{a}+1}}","-\frac{81\ 3^{3/4} \sqrt{2-\sqrt{3}} a^4 \sqrt[6]{a+b x^2} \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right),4 \sqrt{3}-7\right)}{2816 b^4 x \sqrt[3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}+\frac{81 a^3 x \sqrt[6]{a+b x^2}}{2816 b^3}-\frac{9 a^2 x^3 \sqrt[6]{a+b x^2}}{704 b^2}+\frac{3}{22} x^7 \sqrt[6]{a+b x^2}+\frac{3 a x^5 \sqrt[6]{a+b x^2}}{352 b}",1,"(3*x*(a + b*x^2)^(1/6)*((1 + (b*x^2)/a)^(1/6)*(27*a^3 - 3*a^2*b*x^2 + 2*a*b^2*x^4 + 32*b^3*x^6) - 27*a^3*Hypergeometric2F1[-1/6, 1/2, 3/2, -((b*x^2)/a)]))/(704*b^3*(1 + (b*x^2)/a)^(1/6))","C",1
1011,1,93,321,0.0490067,"\int x^4 \sqrt[6]{a+b x^2} \, dx","Integrate[x^4*(a + b*x^2)^(1/6),x]","\frac{3 x \sqrt[6]{a+b x^2} \left(\sqrt[6]{\frac{b x^2}{a}+1} \left(-9 a^2+a b x^2+10 b^2 x^4\right)+9 a^2 \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)\right)}{160 b^2 \sqrt[6]{\frac{b x^2}{a}+1}}","\frac{27\ 3^{3/4} \sqrt{2-\sqrt{3}} a^3 \sqrt[6]{a+b x^2} \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right),4 \sqrt{3}-7\right)}{640 b^3 x \sqrt[3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}-\frac{27 a^2 x \sqrt[6]{a+b x^2}}{640 b^2}+\frac{3}{16} x^5 \sqrt[6]{a+b x^2}+\frac{3 a x^3 \sqrt[6]{a+b x^2}}{160 b}",1,"(3*x*(a + b*x^2)^(1/6)*((1 + (b*x^2)/a)^(1/6)*(-9*a^2 + a*b*x^2 + 10*b^2*x^4) + 9*a^2*Hypergeometric2F1[-1/6, 1/2, 3/2, -((b*x^2)/a)]))/(160*b^2*(1 + (b*x^2)/a)^(1/6))","C",1
1012,1,62,297,0.048847,"\int x^2 \sqrt[6]{a+b x^2} \, dx","Integrate[x^2*(a + b*x^2)^(1/6),x]","\frac{3 x \sqrt[6]{a+b x^2} \left(-\frac{a \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)}{\sqrt[6]{\frac{b x^2}{a}+1}}+a+b x^2\right)}{10 b}","-\frac{3\ 3^{3/4} \sqrt{2-\sqrt{3}} a^2 \sqrt[6]{a+b x^2} \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right),4 \sqrt{3}-7\right)}{40 b^2 x \sqrt[3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}+\frac{3 a x \sqrt[6]{a+b x^2}}{40 b}+\frac{3}{10} x^3 \sqrt[6]{a+b x^2}",1,"(3*x*(a + b*x^2)^(1/6)*(a + b*x^2 - (a*Hypergeometric2F1[-1/6, 1/2, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^(1/6)))/(10*b)","C",1
1013,1,46,273,0.0071269,"\int \sqrt[6]{a+b x^2} \, dx","Integrate[(a + b*x^2)^(1/6),x]","\frac{x \sqrt[6]{a+b x^2} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)}{\sqrt[6]{\frac{b x^2}{a}+1}}","\frac{3^{3/4} \sqrt{2-\sqrt{3}} a \sqrt[6]{a+b x^2} \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right),4 \sqrt{3}-7\right)}{4 b x \sqrt[3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}+\frac{3}{4} x \sqrt[6]{a+b x^2}",1,"(x*(a + b*x^2)^(1/6)*Hypergeometric2F1[-1/6, 1/2, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^(1/6)","C",1
1014,1,49,266,0.0091088,"\int \frac{\sqrt[6]{a+b x^2}}{x^2} \, dx","Integrate[(a + b*x^2)^(1/6)/x^2,x]","-\frac{\sqrt[6]{a+b x^2} \, _2F_1\left(-\frac{1}{2},-\frac{1}{6};\frac{1}{2};-\frac{b x^2}{a}\right)}{x \sqrt[6]{\frac{b x^2}{a}+1}}","\frac{\sqrt{2-\sqrt{3}} \sqrt[6]{a+b x^2} \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right),4 \sqrt{3}-7\right)}{\sqrt[4]{3} x \sqrt[3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}-\frac{\sqrt[6]{a+b x^2}}{x}",1,"-(((a + b*x^2)^(1/6)*Hypergeometric2F1[-1/2, -1/6, 1/2, -((b*x^2)/a)])/(x*(1 + (b*x^2)/a)^(1/6)))","C",1
1015,1,51,297,0.0094782,"\int \frac{\sqrt[6]{a+b x^2}}{x^4} \, dx","Integrate[(a + b*x^2)^(1/6)/x^4,x]","-\frac{\sqrt[6]{a+b x^2} \, _2F_1\left(-\frac{3}{2},-\frac{1}{6};-\frac{1}{2};-\frac{b x^2}{a}\right)}{3 x^3 \sqrt[6]{\frac{b x^2}{a}+1}}","-\frac{2 \sqrt{2-\sqrt{3}} b \sqrt[6]{a+b x^2} \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right),4 \sqrt{3}-7\right)}{9 \sqrt[4]{3} a x \sqrt[3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}-\frac{b \sqrt[6]{a+b x^2}}{9 a x}-\frac{\sqrt[6]{a+b x^2}}{3 x^3}",1,"-1/3*((a + b*x^2)^(1/6)*Hypergeometric2F1[-3/2, -1/6, -1/2, -((b*x^2)/a)])/(x^3*(1 + (b*x^2)/a)^(1/6))","C",1
1016,1,51,323,0.0092991,"\int \frac{\sqrt[6]{a+b x^2}}{x^6} \, dx","Integrate[(a + b*x^2)^(1/6)/x^6,x]","-\frac{\sqrt[6]{a+b x^2} \, _2F_1\left(-\frac{5}{2},-\frac{1}{6};-\frac{3}{2};-\frac{b x^2}{a}\right)}{5 x^5 \sqrt[6]{\frac{b x^2}{a}+1}}","\frac{16 \sqrt{2-\sqrt{3}} b^2 \sqrt[6]{a+b x^2} \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right),4 \sqrt{3}-7\right)}{135 \sqrt[4]{3} a^2 x \sqrt[3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}+\frac{8 b^2 \sqrt[6]{a+b x^2}}{135 a^2 x}-\frac{\sqrt[6]{a+b x^2}}{5 x^5}-\frac{b \sqrt[6]{a+b x^2}}{45 a x^3}",1,"-1/5*((a + b*x^2)^(1/6)*Hypergeometric2F1[-5/2, -1/6, -3/2, -((b*x^2)/a)])/(x^5*(1 + (b*x^2)/a)^(1/6))","C",1
1017,1,51,347,0.0094894,"\int \frac{\sqrt[6]{a+b x^2}}{x^8} \, dx","Integrate[(a + b*x^2)^(1/6)/x^8,x]","-\frac{\sqrt[6]{a+b x^2} \, _2F_1\left(-\frac{7}{2},-\frac{1}{6};-\frac{5}{2};-\frac{b x^2}{a}\right)}{7 x^7 \sqrt[6]{\frac{b x^2}{a}+1}}","-\frac{32 \sqrt{2-\sqrt{3}} b^3 \sqrt[6]{a+b x^2} \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right),4 \sqrt{3}-7\right)}{405 \sqrt[4]{3} a^3 x \sqrt[3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}-\frac{16 b^3 \sqrt[6]{a+b x^2}}{405 a^3 x}+\frac{2 b^2 \sqrt[6]{a+b x^2}}{135 a^2 x^3}-\frac{\sqrt[6]{a+b x^2}}{7 x^7}-\frac{b \sqrt[6]{a+b x^2}}{105 a x^5}",1,"-1/7*((a + b*x^2)^(1/6)*Hypergeometric2F1[-7/2, -1/6, -5/2, -((b*x^2)/a)])/(x^7*(1 + (b*x^2)/a)^(1/6))","C",1
1018,1,90,659,0.0321624,"\int \frac{x^6}{\sqrt[6]{a+b x^2}} \, dx","Integrate[x^6/(a + b*x^2)^(1/6),x]","\frac{3 \left(-135 a^3 x \sqrt[6]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)+135 a^3 x+15 a^2 b x^3-8 a b^2 x^5+112 b^3 x^7\right)}{2240 b^3 \sqrt[6]{a+b x^2}}","\frac{81\ 3^{3/4} a^4 \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right),4 \sqrt{3}-7\right)}{448 \sqrt{2} b^4 x \left(\frac{a}{a+b x^2}\right)^{2/3} \sqrt[6]{a+b x^2} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}-\frac{243 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^4 \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{1792 b^4 x \left(\frac{a}{a+b x^2}\right)^{2/3} \sqrt[6]{a+b x^2} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}-\frac{243 a^4 x}{896 b^3 \left(\frac{a}{a+b x^2}\right)^{2/3} \left(a+b x^2\right)^{7/6} \left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)}-\frac{243 a^3 x}{896 b^3 \sqrt[6]{a+b x^2}}+\frac{81 a^2 x \left(a+b x^2\right)^{5/6}}{448 b^3}-\frac{9 a x^3 \left(a+b x^2\right)^{5/6}}{56 b^2}+\frac{3 x^5 \left(a+b x^2\right)^{5/6}}{20 b}",1,"(3*(135*a^3*x + 15*a^2*b*x^3 - 8*a*b^2*x^5 + 112*b^3*x^7 - 135*a^3*x*(1 + (b*x^2)/a)^(1/6)*Hypergeometric2F1[1/6, 1/2, 3/2, -((b*x^2)/a)]))/(2240*b^3*(a + b*x^2)^(1/6))","C",1
1019,1,79,635,0.0243672,"\int \frac{x^4}{\sqrt[6]{a+b x^2}} \, dx","Integrate[x^4/(a + b*x^2)^(1/6),x]","\frac{3 \left(9 a^2 x \sqrt[6]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)-9 a^2 x-a b x^3+8 b^2 x^5\right)}{112 b^2 \sqrt[6]{a+b x^2}}","-\frac{27\ 3^{3/4} a^3 \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right),4 \sqrt{3}-7\right)}{112 \sqrt{2} b^3 x \left(\frac{a}{a+b x^2}\right)^{2/3} \sqrt[6]{a+b x^2} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}+\frac{81 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^3 \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{448 b^3 x \left(\frac{a}{a+b x^2}\right)^{2/3} \sqrt[6]{a+b x^2} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}+\frac{81 a^3 x}{224 b^2 \left(\frac{a}{a+b x^2}\right)^{2/3} \left(a+b x^2\right)^{7/6} \left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)}+\frac{81 a^2 x}{224 b^2 \sqrt[6]{a+b x^2}}-\frac{27 a x \left(a+b x^2\right)^{5/6}}{112 b^2}+\frac{3 x^3 \left(a+b x^2\right)^{5/6}}{14 b}",1,"(3*(-9*a^2*x - a*b*x^3 + 8*b^2*x^5 + 9*a^2*x*(1 + (b*x^2)/a)^(1/6)*Hypergeometric2F1[1/6, 1/2, 3/2, -((b*x^2)/a)]))/(112*b^2*(a + b*x^2)^(1/6))","C",1
1020,1,62,611,0.0198471,"\int \frac{x^2}{\sqrt[6]{a+b x^2}} \, dx","Integrate[x^2/(a + b*x^2)^(1/6),x]","\frac{3 x \left(-a \sqrt[6]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)+a+b x^2\right)}{8 b \sqrt[6]{a+b x^2}}","\frac{3\ 3^{3/4} a^2 \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right),4 \sqrt{3}-7\right)}{8 \sqrt{2} b^2 x \left(\frac{a}{a+b x^2}\right)^{2/3} \sqrt[6]{a+b x^2} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}-\frac{9 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^2 \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{32 b^2 x \left(\frac{a}{a+b x^2}\right)^{2/3} \sqrt[6]{a+b x^2} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}-\frac{9 a^2 x}{16 b \left(\frac{a}{a+b x^2}\right)^{2/3} \left(a+b x^2\right)^{7/6} \left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)}-\frac{9 a x}{16 b \sqrt[6]{a+b x^2}}+\frac{3 x \left(a+b x^2\right)^{5/6}}{8 b}",1,"(3*x*(a + b*x^2 - a*(1 + (b*x^2)/a)^(1/6)*Hypergeometric2F1[1/6, 1/2, 3/2, -((b*x^2)/a)]))/(8*b*(a + b*x^2)^(1/6))","C",1
1021,1,46,577,0.0068517,"\int \frac{1}{\sqrt[6]{a+b x^2}} \, dx","Integrate[(a + b*x^2)^(-1/6),x]","\frac{x \sqrt[6]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)}{\sqrt[6]{a+b x^2}}","-\frac{3^{3/4} a \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right),4 \sqrt{3}-7\right)}{\sqrt{2} b x \left(\frac{a}{a+b x^2}\right)^{2/3} \sqrt[6]{a+b x^2} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}+\frac{3 x}{2 \sqrt[6]{a+b x^2}}+\frac{3 a x}{2 \left(\frac{a}{a+b x^2}\right)^{2/3} \left(a+b x^2\right)^{7/6} \left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)}+\frac{3 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{4 b x \left(\frac{a}{a+b x^2}\right)^{2/3} \sqrt[6]{a+b x^2} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}",1,"(x*(1 + (b*x^2)/a)^(1/6)*Hypergeometric2F1[1/6, 1/2, 3/2, -((b*x^2)/a)])/(a + b*x^2)^(1/6)","C",1
1022,1,49,586,0.009696,"\int \frac{1}{x^2 \sqrt[6]{a+b x^2}} \, dx","Integrate[1/(x^2*(a + b*x^2)^(1/6)),x]","-\frac{\sqrt[6]{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{1}{2},\frac{1}{6};\frac{1}{2};-\frac{b x^2}{a}\right)}{x \sqrt[6]{a+b x^2}}","-\frac{\sqrt{2} \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right),4 \sqrt{3}-7\right)}{\sqrt[4]{3} x \left(\frac{a}{a+b x^2}\right)^{2/3} \sqrt[6]{a+b x^2} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}+\frac{b x}{a \sqrt[6]{a+b x^2}}+\frac{b x}{\left(\frac{a}{a+b x^2}\right)^{2/3} \left(a+b x^2\right)^{7/6} \left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)}-\frac{\left(a+b x^2\right)^{5/6}}{a x}+\frac{\sqrt[4]{3} \sqrt{2+\sqrt{3}} \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{2 x \left(\frac{a}{a+b x^2}\right)^{2/3} \sqrt[6]{a+b x^2} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}",1,"-(((1 + (b*x^2)/a)^(1/6)*Hypergeometric2F1[-1/2, 1/6, 1/2, -((b*x^2)/a)])/(x*(a + b*x^2)^(1/6)))","C",1
1023,1,51,633,0.008645,"\int \frac{1}{x^4 \sqrt[6]{a+b x^2}} \, dx","Integrate[1/(x^4*(a + b*x^2)^(1/6)),x]","-\frac{\sqrt[6]{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{3}{2},\frac{1}{6};-\frac{1}{2};-\frac{b x^2}{a}\right)}{3 x^3 \sqrt[6]{a+b x^2}}","\frac{4 \sqrt{2} b \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right),4 \sqrt{3}-7\right)}{9 \sqrt[4]{3} a x \left(\frac{a}{a+b x^2}\right)^{2/3} \sqrt[6]{a+b x^2} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}-\frac{4 b^2 x}{9 a^2 \sqrt[6]{a+b x^2}}+\frac{4 b \left(a+b x^2\right)^{5/6}}{9 a^2 x}-\frac{4 b^2 x}{9 a \left(\frac{a}{a+b x^2}\right)^{2/3} \left(a+b x^2\right)^{7/6} \left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)}-\frac{2 \sqrt{2+\sqrt{3}} b \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{3\ 3^{3/4} a x \left(\frac{a}{a+b x^2}\right)^{2/3} \sqrt[6]{a+b x^2} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}-\frac{\left(a+b x^2\right)^{5/6}}{3 a x^3}",1,"-1/3*((1 + (b*x^2)/a)^(1/6)*Hypergeometric2F1[-3/2, 1/6, -1/2, -((b*x^2)/a)])/(x^3*(a + b*x^2)^(1/6))","C",1
1024,1,51,661,0.0098464,"\int \frac{1}{x^6 \sqrt[6]{a+b x^2}} \, dx","Integrate[1/(x^6*(a + b*x^2)^(1/6)),x]","-\frac{\sqrt[6]{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{5}{2},\frac{1}{6};-\frac{3}{2};-\frac{b x^2}{a}\right)}{5 x^5 \sqrt[6]{a+b x^2}}","-\frac{8 \sqrt{2} b^2 \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right),4 \sqrt{3}-7\right)}{27 \sqrt[4]{3} a^2 x \left(\frac{a}{a+b x^2}\right)^{2/3} \sqrt[6]{a+b x^2} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}+\frac{8 b^3 x}{27 a^3 \sqrt[6]{a+b x^2}}-\frac{8 b^2 \left(a+b x^2\right)^{5/6}}{27 a^3 x}+\frac{8 b^3 x}{27 a^2 \left(\frac{a}{a+b x^2}\right)^{2/3} \left(a+b x^2\right)^{7/6} \left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)}+\frac{4 \sqrt{2+\sqrt{3}} b^2 \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{9\ 3^{3/4} a^2 x \left(\frac{a}{a+b x^2}\right)^{2/3} \sqrt[6]{a+b x^2} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}+\frac{2 b \left(a+b x^2\right)^{5/6}}{9 a^2 x^3}-\frac{\left(a+b x^2\right)^{5/6}}{5 a x^5}",1,"-1/5*((1 + (b*x^2)/a)^(1/6)*Hypergeometric2F1[-5/2, 1/6, -3/2, -((b*x^2)/a)])/(x^5*(a + b*x^2)^(1/6))","C",1
1025,1,89,324,0.0330981,"\int \frac{x^6}{\left(a+b x^2\right)^{5/6}} \, dx","Integrate[x^6/(a + b*x^2)^(5/6),x]","\frac{3 x \left(-27 a^3 \left(\frac{b x^2}{a}+1\right)^{5/6} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};-\frac{b x^2}{a}\right)+27 a^3+15 a^2 b x^2-4 a b^2 x^4+8 b^3 x^6\right)}{128 b^3 \left(a+b x^2\right)^{5/6}}","-\frac{81\ 3^{3/4} \sqrt{2-\sqrt{3}} a^3 \sqrt[6]{a+b x^2} \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right),4 \sqrt{3}-7\right)}{128 b^4 x \sqrt[3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}+\frac{81 a^2 x \sqrt[6]{a+b x^2}}{128 b^3}-\frac{9 a x^3 \sqrt[6]{a+b x^2}}{32 b^2}+\frac{3 x^5 \sqrt[6]{a+b x^2}}{16 b}",1,"(3*x*(27*a^3 + 15*a^2*b*x^2 - 4*a*b^2*x^4 + 8*b^3*x^6 - 27*a^3*(1 + (b*x^2)/a)^(5/6)*Hypergeometric2F1[1/2, 5/6, 3/2, -((b*x^2)/a)]))/(128*b^3*(a + b*x^2)^(5/6))","C",1
1026,1,79,300,0.0240951,"\int \frac{x^4}{\left(a+b x^2\right)^{5/6}} \, dx","Integrate[x^4/(a + b*x^2)^(5/6),x]","\frac{3 \left(9 a^2 x \left(\frac{b x^2}{a}+1\right)^{5/6} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};-\frac{b x^2}{a}\right)-9 a^2 x-5 a b x^3+4 b^2 x^5\right)}{40 b^2 \left(a+b x^2\right)^{5/6}}","\frac{27\ 3^{3/4} \sqrt{2-\sqrt{3}} a^2 \sqrt[6]{a+b x^2} \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right),4 \sqrt{3}-7\right)}{40 b^3 x \sqrt[3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}-\frac{27 a x \sqrt[6]{a+b x^2}}{40 b^2}+\frac{3 x^3 \sqrt[6]{a+b x^2}}{10 b}",1,"(3*(-9*a^2*x - 5*a*b*x^3 + 4*b^2*x^5 + 9*a^2*x*(1 + (b*x^2)/a)^(5/6)*Hypergeometric2F1[1/2, 5/6, 3/2, -((b*x^2)/a)]))/(40*b^2*(a + b*x^2)^(5/6))","C",1
1027,1,62,276,0.0196061,"\int \frac{x^2}{\left(a+b x^2\right)^{5/6}} \, dx","Integrate[x^2/(a + b*x^2)^(5/6),x]","\frac{3 x \left(-a \left(\frac{b x^2}{a}+1\right)^{5/6} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};-\frac{b x^2}{a}\right)+a+b x^2\right)}{4 b \left(a+b x^2\right)^{5/6}}","\frac{3 x \sqrt[6]{a+b x^2}}{4 b}-\frac{3\ 3^{3/4} \sqrt{2-\sqrt{3}} a \sqrt[6]{a+b x^2} \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right),4 \sqrt{3}-7\right)}{4 b^2 x \sqrt[3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}",1,"(3*x*(a + b*x^2 - a*(1 + (b*x^2)/a)^(5/6)*Hypergeometric2F1[1/2, 5/6, 3/2, -((b*x^2)/a)]))/(4*b*(a + b*x^2)^(5/6))","C",1
1028,1,46,252,0.0076681,"\int \frac{1}{\left(a+b x^2\right)^{5/6}} \, dx","Integrate[(a + b*x^2)^(-5/6),x]","\frac{x \left(\frac{b x^2}{a}+1\right)^{5/6} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};-\frac{b x^2}{a}\right)}{\left(a+b x^2\right)^{5/6}}","\frac{3^{3/4} \sqrt{2-\sqrt{3}} \sqrt[6]{a+b x^2} \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right),4 \sqrt{3}-7\right)}{b x \sqrt[3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}",1,"(x*(1 + (b*x^2)/a)^(5/6)*Hypergeometric2F1[1/2, 5/6, 3/2, -((b*x^2)/a)])/(a + b*x^2)^(5/6)","C",1
1029,1,49,273,0.0095452,"\int \frac{1}{x^2 \left(a+b x^2\right)^{5/6}} \, dx","Integrate[1/(x^2*(a + b*x^2)^(5/6)),x]","-\frac{\left(\frac{b x^2}{a}+1\right)^{5/6} \, _2F_1\left(-\frac{1}{2},\frac{5}{6};\frac{1}{2};-\frac{b x^2}{a}\right)}{x \left(a+b x^2\right)^{5/6}}","-\frac{2 \sqrt{2-\sqrt{3}} \sqrt[6]{a+b x^2} \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right),4 \sqrt{3}-7\right)}{\sqrt[4]{3} a x \sqrt[3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}-\frac{\sqrt[6]{a+b x^2}}{a x}",1,"-(((1 + (b*x^2)/a)^(5/6)*Hypergeometric2F1[-1/2, 5/6, 1/2, -((b*x^2)/a)])/(x*(a + b*x^2)^(5/6)))","C",1
1030,1,51,300,0.0095947,"\int \frac{1}{x^4 \left(a+b x^2\right)^{5/6}} \, dx","Integrate[1/(x^4*(a + b*x^2)^(5/6)),x]","-\frac{\left(\frac{b x^2}{a}+1\right)^{5/6} \, _2F_1\left(-\frac{3}{2},\frac{5}{6};-\frac{1}{2};-\frac{b x^2}{a}\right)}{3 x^3 \left(a+b x^2\right)^{5/6}}","\frac{16 \sqrt{2-\sqrt{3}} b \sqrt[6]{a+b x^2} \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right),4 \sqrt{3}-7\right)}{9 \sqrt[4]{3} a^2 x \sqrt[3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}+\frac{8 b \sqrt[6]{a+b x^2}}{9 a^2 x}-\frac{\sqrt[6]{a+b x^2}}{3 a x^3}",1,"-1/3*((1 + (b*x^2)/a)^(5/6)*Hypergeometric2F1[-3/2, 5/6, -1/2, -((b*x^2)/a)])/(x^3*(a + b*x^2)^(5/6))","C",1
1031,1,51,326,0.0103302,"\int \frac{1}{x^6 \left(a+b x^2\right)^{5/6}} \, dx","Integrate[1/(x^6*(a + b*x^2)^(5/6)),x]","-\frac{\left(\frac{b x^2}{a}+1\right)^{5/6} \, _2F_1\left(-\frac{5}{2},\frac{5}{6};-\frac{3}{2};-\frac{b x^2}{a}\right)}{5 x^5 \left(a+b x^2\right)^{5/6}}","-\frac{224 \sqrt{2-\sqrt{3}} b^2 \sqrt[6]{a+b x^2} \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right),4 \sqrt{3}-7\right)}{135 \sqrt[4]{3} a^3 x \sqrt[3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}-\frac{112 b^2 \sqrt[6]{a+b x^2}}{135 a^3 x}+\frac{14 b \sqrt[6]{a+b x^2}}{45 a^2 x^3}-\frac{\sqrt[6]{a+b x^2}}{5 a x^5}",1,"-1/5*((1 + (b*x^2)/a)^(5/6)*Hypergeometric2F1[-5/2, 5/6, -3/2, -((b*x^2)/a)])/(x^5*(a + b*x^2)^(5/6))","C",1
1032,1,79,654,0.0237402,"\int \frac{x^6}{\left(a+b x^2\right)^{7/6}} \, dx","Integrate[x^6/(a + b*x^2)^(7/6),x]","\frac{-405 a^2 x \sqrt[6]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{3}{2};-\frac{b x^2}{a}\right)+405 a^2 x-90 a b x^3+48 b^2 x^5}{224 b^3 \sqrt[6]{a+b x^2}}","-\frac{405\ 3^{3/4} a^3 \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right),4 \sqrt{3}-7\right)}{112 \sqrt{2} b^4 x \left(\frac{a}{a+b x^2}\right)^{2/3} \sqrt[6]{a+b x^2} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}+\frac{1215 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^3 \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{448 b^4 x \left(\frac{a}{a+b x^2}\right)^{2/3} \sqrt[6]{a+b x^2} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}+\frac{1215 a^3 x}{224 b^3 \left(\frac{a}{a+b x^2}\right)^{2/3} \left(a+b x^2\right)^{7/6} \left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)}+\frac{1215 a^2 x}{224 b^3 \sqrt[6]{a+b x^2}}-\frac{405 a x \left(a+b x^2\right)^{5/6}}{112 b^3}+\frac{45 x^3 \left(a+b x^2\right)^{5/6}}{14 b^2}-\frac{3 x^5}{b \sqrt[6]{a+b x^2}}",1,"(405*a^2*x - 90*a*b*x^3 + 48*b^2*x^5 - 405*a^2*x*(1 + (b*x^2)/a)^(1/6)*Hypergeometric2F1[1/2, 7/6, 3/2, -((b*x^2)/a)])/(224*b^3*(a + b*x^2)^(1/6))","C",1
1033,1,65,630,0.0234882,"\int \frac{x^4}{\left(a+b x^2\right)^{7/6}} \, dx","Integrate[x^4/(a + b*x^2)^(7/6),x]","\frac{3 x \left(9 a \sqrt[6]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{3}{2};-\frac{b x^2}{a}\right)-9 a+2 b x^2\right)}{16 b^2 \sqrt[6]{a+b x^2}}","\frac{27\ 3^{3/4} a^2 \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right),4 \sqrt{3}-7\right)}{8 \sqrt{2} b^3 x \left(\frac{a}{a+b x^2}\right)^{2/3} \sqrt[6]{a+b x^2} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}-\frac{81 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a^2 \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{32 b^3 x \left(\frac{a}{a+b x^2}\right)^{2/3} \sqrt[6]{a+b x^2} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}-\frac{81 a^2 x}{16 b^2 \left(\frac{a}{a+b x^2}\right)^{2/3} \left(a+b x^2\right)^{7/6} \left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)}+\frac{27 x \left(a+b x^2\right)^{5/6}}{8 b^2}-\frac{81 a x}{16 b^2 \sqrt[6]{a+b x^2}}-\frac{3 x^3}{b \sqrt[6]{a+b x^2}}",1,"(3*x*(-9*a + 2*b*x^2 + 9*a*(1 + (b*x^2)/a)^(1/6)*Hypergeometric2F1[1/2, 7/6, 3/2, -((b*x^2)/a)]))/(16*b^2*(a + b*x^2)^(1/6))","C",1
1034,1,58,583,0.0162754,"\int \frac{x^2}{\left(a+b x^2\right)^{7/6}} \, dx","Integrate[x^2/(a + b*x^2)^(7/6),x]","\frac{3 x-3 x \sqrt[6]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{3}{2};-\frac{b x^2}{a}\right)}{2 b \sqrt[6]{a+b x^2}}","-\frac{3\ 3^{3/4} a \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right),4 \sqrt{3}-7\right)}{\sqrt{2} b^2 x \left(\frac{a}{a+b x^2}\right)^{2/3} \sqrt[6]{a+b x^2} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}+\frac{9 \sqrt[4]{3} \sqrt{2+\sqrt{3}} a \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{4 b^2 x \left(\frac{a}{a+b x^2}\right)^{2/3} \sqrt[6]{a+b x^2} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}+\frac{3 x}{2 b \sqrt[6]{a+b x^2}}+\frac{9 a x}{2 b \left(\frac{a}{a+b x^2}\right)^{2/3} \left(a+b x^2\right)^{7/6} \left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)}",1,"(3*x - 3*x*(1 + (b*x^2)/a)^(1/6)*Hypergeometric2F1[1/2, 7/6, 3/2, -((b*x^2)/a)])/(2*b*(a + b*x^2)^(1/6))","C",1
1035,1,49,555,0.0069795,"\int \frac{1}{\left(a+b x^2\right)^{7/6}} \, dx","Integrate[(a + b*x^2)^(-7/6),x]","\frac{x \sqrt[6]{\frac{b x^2}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{3}{2};-\frac{b x^2}{a}\right)}{a \sqrt[6]{a+b x^2}}","\frac{\sqrt{2} 3^{3/4} \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right),4 \sqrt{3}-7\right)}{b x \left(\frac{a}{a+b x^2}\right)^{2/3} \sqrt[6]{a+b x^2} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}-\frac{3 x}{\left(\frac{a}{a+b x^2}\right)^{2/3} \left(a+b x^2\right)^{7/6} \left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)}-\frac{3 \sqrt[4]{3} \sqrt{2+\sqrt{3}} \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{2 b x \left(\frac{a}{a+b x^2}\right)^{2/3} \sqrt[6]{a+b x^2} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}",1,"(x*(1 + (b*x^2)/a)^(1/6)*Hypergeometric2F1[1/2, 7/6, 3/2, -((b*x^2)/a)])/(a*(a + b*x^2)^(1/6))","C",1
1036,1,52,614,0.0091546,"\int \frac{1}{x^2 \left(a+b x^2\right)^{7/6}} \, dx","Integrate[1/(x^2*(a + b*x^2)^(7/6)),x]","-\frac{\sqrt[6]{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{1}{2},\frac{7}{6};\frac{1}{2};-\frac{b x^2}{a}\right)}{a x \sqrt[6]{a+b x^2}}","-\frac{4 \sqrt{2} \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right),4 \sqrt{3}-7\right)}{\sqrt[4]{3} a x \left(\frac{a}{a+b x^2}\right)^{2/3} \sqrt[6]{a+b x^2} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}+\frac{4 b x}{a^2 \sqrt[6]{a+b x^2}}-\frac{4 \left(a+b x^2\right)^{5/6}}{a^2 x}+\frac{4 b x}{a \left(\frac{a}{a+b x^2}\right)^{2/3} \left(a+b x^2\right)^{7/6} \left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)}+\frac{3}{a x \sqrt[6]{a+b x^2}}+\frac{2 \sqrt[4]{3} \sqrt{2+\sqrt{3}} \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{a x \left(\frac{a}{a+b x^2}\right)^{2/3} \sqrt[6]{a+b x^2} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}",1,"-(((1 + (b*x^2)/a)^(1/6)*Hypergeometric2F1[-1/2, 7/6, 1/2, -((b*x^2)/a)])/(a*x*(a + b*x^2)^(1/6)))","C",1
1037,1,54,652,0.0101676,"\int \frac{1}{x^4 \left(a+b x^2\right)^{7/6}} \, dx","Integrate[1/(x^4*(a + b*x^2)^(7/6)),x]","-\frac{\sqrt[6]{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{3}{2},\frac{7}{6};-\frac{1}{2};-\frac{b x^2}{a}\right)}{3 a x^3 \sqrt[6]{a+b x^2}}","\frac{40 \sqrt{2} b \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right),4 \sqrt{3}-7\right)}{9 \sqrt[4]{3} a^2 x \left(\frac{a}{a+b x^2}\right)^{2/3} \sqrt[6]{a+b x^2} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}-\frac{40 b^2 x}{9 a^3 \sqrt[6]{a+b x^2}}+\frac{40 b \left(a+b x^2\right)^{5/6}}{9 a^3 x}-\frac{40 b^2 x}{9 a^2 \left(\frac{a}{a+b x^2}\right)^{2/3} \left(a+b x^2\right)^{7/6} \left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)}-\frac{20 \sqrt{2+\sqrt{3}} b \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{3\ 3^{3/4} a^2 x \left(\frac{a}{a+b x^2}\right)^{2/3} \sqrt[6]{a+b x^2} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}-\frac{10 \left(a+b x^2\right)^{5/6}}{3 a^2 x^3}+\frac{3}{a x^3 \sqrt[6]{a+b x^2}}",1,"-1/3*((1 + (b*x^2)/a)^(1/6)*Hypergeometric2F1[-3/2, 7/6, -1/2, -((b*x^2)/a)])/(a*x^3*(a + b*x^2)^(1/6))","C",1
1038,1,54,680,0.0101537,"\int \frac{1}{x^6 \left(a+b x^2\right)^{7/6}} \, dx","Integrate[1/(x^6*(a + b*x^2)^(7/6)),x]","-\frac{\sqrt[6]{\frac{b x^2}{a}+1} \, _2F_1\left(-\frac{5}{2},\frac{7}{6};-\frac{3}{2};-\frac{b x^2}{a}\right)}{5 a x^5 \sqrt[6]{a+b x^2}}","-\frac{128 \sqrt{2} b^2 \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{a+b x^2}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1}\right),4 \sqrt{3}-7\right)}{27 \sqrt[4]{3} a^3 x \left(\frac{a}{a+b x^2}\right)^{2/3} \sqrt[6]{a+b x^2} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}+\frac{128 b^3 x}{27 a^4 \sqrt[6]{a+b x^2}}-\frac{128 b^2 \left(a+b x^2\right)^{5/6}}{27 a^4 x}+\frac{128 b^3 x}{27 a^3 \left(\frac{a}{a+b x^2}\right)^{2/3} \left(a+b x^2\right)^{7/6} \left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)}+\frac{64 \sqrt{2+\sqrt{3}} b^2 \left(1-\sqrt[3]{\frac{a}{a+b x^2}}\right) \sqrt{\frac{\left(\frac{a}{a+b x^2}\right)^{2/3}+\sqrt[3]{\frac{a}{a+b x^2}}+1}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt[3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{9\ 3^{3/4} a^3 x \left(\frac{a}{a+b x^2}\right)^{2/3} \sqrt[6]{a+b x^2} \sqrt{-\frac{1-\sqrt[3]{\frac{a}{a+b x^2}}}{\left(-\sqrt[3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right)^2}}}+\frac{32 b \left(a+b x^2\right)^{5/6}}{9 a^3 x^3}-\frac{16 \left(a+b x^2\right)^{5/6}}{5 a^2 x^5}+\frac{3}{a x^5 \sqrt[6]{a+b x^2}}",1,"-1/5*((1 + (b*x^2)/a)^(1/6)*Hypergeometric2F1[-5/2, 7/6, -3/2, -((b*x^2)/a)])/(a*x^5*(a + b*x^2)^(1/6))","C",1
1039,1,95,100,0.0549396,"\int x^7 \left(a+b x^2\right)^p \, dx","Integrate[x^7*(a + b*x^2)^p,x]","\frac{1}{2} \left(-\frac{a^3 \left(a+b x^2\right)^{p+1}}{b^4 (p+1)}+\frac{3 a^2 \left(a+b x^2\right)^{p+2}}{b^4 (p+2)}-\frac{3 a \left(a+b x^2\right)^{p+3}}{b^4 (p+3)}+\frac{\left(a+b x^2\right)^{p+4}}{b^4 (p+4)}\right)","-\frac{a^3 \left(a+b x^2\right)^{p+1}}{2 b^4 (p+1)}+\frac{3 a^2 \left(a+b x^2\right)^{p+2}}{2 b^4 (p+2)}-\frac{3 a \left(a+b x^2\right)^{p+3}}{2 b^4 (p+3)}+\frac{\left(a+b x^2\right)^{p+4}}{2 b^4 (p+4)}",1,"(-((a^3*(a + b*x^2)^(1 + p))/(b^4*(1 + p))) + (3*a^2*(a + b*x^2)^(2 + p))/(b^4*(2 + p)) - (3*a*(a + b*x^2)^(3 + p))/(b^4*(3 + p)) + (a + b*x^2)^(4 + p)/(b^4*(4 + p)))/2","A",1
1040,1,64,72,0.0308703,"\int x^5 \left(a+b x^2\right)^p \, dx","Integrate[x^5*(a + b*x^2)^p,x]","\frac{\left(a+b x^2\right)^{p+1} \left(2 a^2-2 a b (p+1) x^2+b^2 \left(p^2+3 p+2\right) x^4\right)}{2 b^3 (p+1) (p+2) (p+3)}","\frac{a^2 \left(a+b x^2\right)^{p+1}}{2 b^3 (p+1)}-\frac{a \left(a+b x^2\right)^{p+2}}{b^3 (p+2)}+\frac{\left(a+b x^2\right)^{p+3}}{2 b^3 (p+3)}",1,"((a + b*x^2)^(1 + p)*(2*a^2 - 2*a*b*(1 + p)*x^2 + b^2*(2 + 3*p + p^2)*x^4))/(2*b^3*(1 + p)*(2 + p)*(3 + p))","A",1
1041,1,40,48,0.0188153,"\int x^3 \left(a+b x^2\right)^p \, dx","Integrate[x^3*(a + b*x^2)^p,x]","\frac{\left(a+b x^2\right)^{p+1} \left(b (p+1) x^2-a\right)}{2 b^2 (p+1) (p+2)}","\frac{\left(a+b x^2\right)^{p+2}}{2 b^2 (p+2)}-\frac{a \left(a+b x^2\right)^{p+1}}{2 b^2 (p+1)}",1,"((a + b*x^2)^(1 + p)*(-a + b*(1 + p)*x^2))/(2*b^2*(1 + p)*(2 + p))","A",1
1042,1,22,23,0.0028958,"\int x \left(a+b x^2\right)^p \, dx","Integrate[x*(a + b*x^2)^p,x]","\frac{\left(a+b x^2\right)^{p+1}}{2 b p+2 b}","\frac{\left(a+b x^2\right)^{p+1}}{2 b (p+1)}",1,"(a + b*x^2)^(1 + p)/(2*b + 2*b*p)","A",1
1043,1,41,41,0.0113796,"\int \frac{\left(a+b x^2\right)^p}{x} \, dx","Integrate[(a + b*x^2)^p/x,x]","-\frac{\left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b x^2}{a}+1\right)}{2 a (p+1)}","-\frac{\left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b x^2}{a}+1\right)}{2 a (p+1)}",1,"-1/2*((a + b*x^2)^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*x^2)/a])/(a*(1 + p))","A",1
1044,1,42,42,0.0075315,"\int \frac{\left(a+b x^2\right)^p}{x^3} \, dx","Integrate[(a + b*x^2)^p/x^3,x]","\frac{b \left(a+b x^2\right)^{p+1} \, _2F_1\left(2,p+1;p+2;\frac{b x^2}{a}+1\right)}{2 a^2 (p+1)}","\frac{b \left(a+b x^2\right)^{p+1} \, _2F_1\left(2,p+1;p+2;\frac{b x^2}{a}+1\right)}{2 a^2 (p+1)}",1,"(b*(a + b*x^2)^(1 + p)*Hypergeometric2F1[2, 1 + p, 2 + p, 1 + (b*x^2)/a])/(2*a^2*(1 + p))","A",1
1045,1,49,40,0.0076497,"\int x^6 \left(a+b x^2\right)^p \, dx","Integrate[x^6*(a + b*x^2)^p,x]","\frac{1}{7} x^7 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{7}{2},-p;\frac{9}{2};-\frac{b x^2}{a}\right)","\frac{x^7 \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+\frac{9}{2};\frac{9}{2};-\frac{b x^2}{a}\right)}{7 a}",1,"(x^7*(a + b*x^2)^p*Hypergeometric2F1[7/2, -p, 9/2, -((b*x^2)/a)])/(7*(1 + (b*x^2)/a)^p)","A",1
1046,1,49,40,0.0059028,"\int x^4 \left(a+b x^2\right)^p \, dx","Integrate[x^4*(a + b*x^2)^p,x]","\frac{1}{5} x^5 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right)","\frac{x^5 \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+\frac{7}{2};\frac{7}{2};-\frac{b x^2}{a}\right)}{5 a}",1,"(x^5*(a + b*x^2)^p*Hypergeometric2F1[5/2, -p, 7/2, -((b*x^2)/a)])/(5*(1 + (b*x^2)/a)^p)","A",1
1047,1,49,40,0.0060996,"\int x^2 \left(a+b x^2\right)^p \, dx","Integrate[x^2*(a + b*x^2)^p,x]","\frac{1}{3} x^3 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right)","\frac{x^3 \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+\frac{5}{2};\frac{5}{2};-\frac{b x^2}{a}\right)}{3 a}",1,"(x^3*(a + b*x^2)^p*Hypergeometric2F1[3/2, -p, 5/2, -((b*x^2)/a)])/(3*(1 + (b*x^2)/a)^p)","A",1
1048,1,44,35,0.0046524,"\int \left(a+b x^2\right)^p \, dx","Integrate[(a + b*x^2)^p,x]","x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)","\frac{x \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+\frac{3}{2};\frac{3}{2};-\frac{b x^2}{a}\right)}{a}",1,"(x*(a + b*x^2)^p*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p","A",1
1049,1,47,38,0.0057381,"\int \frac{\left(a+b x^2\right)^p}{x^2} \, dx","Integrate[(a + b*x^2)^p/x^2,x]","-\frac{\left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b x^2}{a}\right)}{x}","-\frac{\left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+\frac{1}{2};\frac{1}{2};-\frac{b x^2}{a}\right)}{a x}",1,"-(((a + b*x^2)^p*Hypergeometric2F1[-1/2, -p, 1/2, -((b*x^2)/a)])/(x*(1 + (b*x^2)/a)^p))","A",1
1050,1,51,42,0.0078043,"\int x^{7/2} \left(a+b x^2\right)^p \, dx","Integrate[x^(7/2)*(a + b*x^2)^p,x]","\frac{2}{9} x^{9/2} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{9}{4},-p;\frac{13}{4};-\frac{b x^2}{a}\right)","\frac{2 x^{9/2} \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+\frac{13}{4};\frac{13}{4};-\frac{b x^2}{a}\right)}{9 a}",1,"(2*x^(9/2)*(a + b*x^2)^p*Hypergeometric2F1[9/4, -p, 13/4, -((b*x^2)/a)])/(9*(1 + (b*x^2)/a)^p)","A",1
1051,1,51,42,0.0075039,"\int x^{5/2} \left(a+b x^2\right)^p \, dx","Integrate[x^(5/2)*(a + b*x^2)^p,x]","\frac{2}{7} x^{7/2} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{7}{4},-p;\frac{11}{4};-\frac{b x^2}{a}\right)","\frac{2 x^{7/2} \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+\frac{11}{4};\frac{11}{4};-\frac{b x^2}{a}\right)}{7 a}",1,"(2*x^(7/2)*(a + b*x^2)^p*Hypergeometric2F1[7/4, -p, 11/4, -((b*x^2)/a)])/(7*(1 + (b*x^2)/a)^p)","A",1
1052,1,51,42,0.0083618,"\int x^{3/2} \left(a+b x^2\right)^p \, dx","Integrate[x^(3/2)*(a + b*x^2)^p,x]","\frac{2}{5} x^{5/2} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{5}{4},-p;\frac{9}{4};-\frac{b x^2}{a}\right)","\frac{2 x^{5/2} \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+\frac{9}{4};\frac{9}{4};-\frac{b x^2}{a}\right)}{5 a}",1,"(2*x^(5/2)*(a + b*x^2)^p*Hypergeometric2F1[5/4, -p, 9/4, -((b*x^2)/a)])/(5*(1 + (b*x^2)/a)^p)","A",1
1053,1,51,42,0.0076474,"\int \sqrt{x} \left(a+b x^2\right)^p \, dx","Integrate[Sqrt[x]*(a + b*x^2)^p,x]","\frac{2}{3} x^{3/2} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{3}{4},-p;\frac{7}{4};-\frac{b x^2}{a}\right)","\frac{2 x^{3/2} \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+\frac{7}{4};\frac{7}{4};-\frac{b x^2}{a}\right)}{3 a}",1,"(2*x^(3/2)*(a + b*x^2)^p*Hypergeometric2F1[3/4, -p, 7/4, -((b*x^2)/a)])/(3*(1 + (b*x^2)/a)^p)","A",1
1054,1,49,40,0.0066145,"\int \frac{\left(a+b x^2\right)^p}{\sqrt{x}} \, dx","Integrate[(a + b*x^2)^p/Sqrt[x],x]","2 \sqrt{x} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{4},-p;\frac{5}{4};-\frac{b x^2}{a}\right)","\frac{2 \sqrt{x} \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+\frac{5}{4};\frac{5}{4};-\frac{b x^2}{a}\right)}{a}",1,"(2*Sqrt[x]*(a + b*x^2)^p*Hypergeometric2F1[1/4, -p, 5/4, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p","A",1
1055,1,49,40,0.0072875,"\int \frac{\left(a+b x^2\right)^p}{x^{3/2}} \, dx","Integrate[(a + b*x^2)^p/x^(3/2),x]","-\frac{2 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{4},-p;\frac{3}{4};-\frac{b x^2}{a}\right)}{\sqrt{x}}","-\frac{2 \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+\frac{3}{4};\frac{3}{4};-\frac{b x^2}{a}\right)}{a \sqrt{x}}",1,"(-2*(a + b*x^2)^p*Hypergeometric2F1[-1/4, -p, 3/4, -((b*x^2)/a)])/(Sqrt[x]*(1 + (b*x^2)/a)^p)","A",1
1056,1,51,42,0.007778,"\int \frac{\left(a+b x^2\right)^p}{x^{5/2}} \, dx","Integrate[(a + b*x^2)^p/x^(5/2),x]","-\frac{2 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(-\frac{3}{4},-p;\frac{1}{4};-\frac{b x^2}{a}\right)}{3 x^{3/2}}","-\frac{2 \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+\frac{1}{4};\frac{1}{4};-\frac{b x^2}{a}\right)}{3 a x^{3/2}}",1,"(-2*(a + b*x^2)^p*Hypergeometric2F1[-3/4, -p, 1/4, -((b*x^2)/a)])/(3*x^(3/2)*(1 + (b*x^2)/a)^p)","A",1
1057,1,51,42,0.0076106,"\int \frac{\left(a+b x^2\right)^p}{x^{7/2}} \, dx","Integrate[(a + b*x^2)^p/x^(7/2),x]","-\frac{2 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(-\frac{5}{4},-p;-\frac{1}{4};-\frac{b x^2}{a}\right)}{5 x^{5/2}}","-\frac{2 \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p-\frac{1}{4};-\frac{1}{4};-\frac{b x^2}{a}\right)}{5 a x^{5/2}}",1,"(-2*(a + b*x^2)^p*Hypergeometric2F1[-5/4, -p, -1/4, -((b*x^2)/a)])/(5*x^(5/2)*(1 + (b*x^2)/a)^p)","A",1
1058,1,63,53,0.0108628,"\int x^m \left(a+b x^2\right)^p \, dx","Integrate[x^m*(a + b*x^2)^p,x]","\frac{x^{m+1} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+1}{2},-p;\frac{m+1}{2}+1;-\frac{b x^2}{a}\right)}{m+1}","\frac{x^{m+1} \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,\frac{1}{2} (m+2 p+3);\frac{m+3}{2};-\frac{b x^2}{a}\right)}{a (m+1)}",1,"(x^(1 + m)*(a + b*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p, 1 + (1 + m)/2, -((b*x^2)/a)])/((1 + m)*(1 + (b*x^2)/a)^p)","A",1
1059,1,64,66,0.0112459,"\int (c x)^m \left(a+b x^2\right)^p \, dx","Integrate[(c*x)^m*(a + b*x^2)^p,x]","\frac{x (c x)^m \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+1}{2},-p;\frac{m+1}{2}+1;-\frac{b x^2}{a}\right)}{m+1}","\frac{(c x)^{m+1} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{b x^2}{a}\right)}{c (m+1)}",1,"(x*(c*x)^m*(a + b*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p, 1 + (1 + m)/2, -((b*x^2)/a)])/((1 + m)*(1 + (b*x^2)/a)^p)","A",1
1060,1,66,53,0.0153399,"\int x^{-8-2 p} \left(a+b x^2\right)^p \, dx","Integrate[x^(-8 - 2*p)*(a + b*x^2)^p,x]","-\frac{x^{-2 p-7} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(-p-\frac{7}{2},-p;-p-\frac{5}{2};-\frac{b x^2}{a}\right)}{2 p+7}","-\frac{x^{-2 p-7} \left(a+b x^2\right)^{p+1} \, _2F_1\left(-\frac{5}{2},1;\frac{1}{2} (-2 p-5);-\frac{b x^2}{a}\right)}{a (2 p+7)}",1,"-((x^(-7 - 2*p)*(a + b*x^2)^p*Hypergeometric2F1[-7/2 - p, -p, -5/2 - p, -((b*x^2)/a)])/((7 + 2*p)*(1 + (b*x^2)/a)^p))","A",1
1061,1,62,105,0.0142587,"\int x^{-7-2 p} \left(a+b x^2\right)^p \, dx","Integrate[x^(-7 - 2*p)*(a + b*x^2)^p,x]","-\frac{x^{-2 (p+3)} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(-p-3,-p;-p-2;-\frac{b x^2}{a}\right)}{2 (p+3)}","-\frac{b^2 x^{-2 (p+1)} \left(a+b x^2\right)^{p+1}}{a^3 (p+1) (p+2) (p+3)}+\frac{b x^{-2 (p+2)} \left(a+b x^2\right)^{p+1}}{a^2 (p+2) (p+3)}-\frac{x^{-2 (p+3)} \left(a+b x^2\right)^{p+1}}{2 a (p+3)}",1,"-1/2*((a + b*x^2)^p*Hypergeometric2F1[-3 - p, -p, -2 - p, -((b*x^2)/a)])/((3 + p)*x^(2*(3 + p))*(1 + (b*x^2)/a)^p)","C",1
1062,1,66,53,0.0161338,"\int x^{-6-2 p} \left(a+b x^2\right)^p \, dx","Integrate[x^(-6 - 2*p)*(a + b*x^2)^p,x]","-\frac{x^{-2 p-5} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(-p-\frac{5}{2},-p;-p-\frac{3}{2};-\frac{b x^2}{a}\right)}{2 p+5}","-\frac{x^{-2 p-5} \left(a+b x^2\right)^{p+1} \, _2F_1\left(-\frac{3}{2},1;\frac{1}{2} (-2 p-3);-\frac{b x^2}{a}\right)}{a (2 p+5)}",1,"-((x^(-5 - 2*p)*(a + b*x^2)^p*Hypergeometric2F1[-5/2 - p, -p, -3/2 - p, -((b*x^2)/a)])/((5 + 2*p)*(1 + (b*x^2)/a)^p))","A",1
1063,1,62,67,0.0151139,"\int x^{-5-2 p} \left(a+b x^2\right)^p \, dx","Integrate[x^(-5 - 2*p)*(a + b*x^2)^p,x]","-\frac{x^{-2 (p+2)} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(-p-2,-p;-p-1;-\frac{b x^2}{a}\right)}{2 (p+2)}","\frac{b x^{-2 (p+1)} \left(a+b x^2\right)^{p+1}}{2 a^2 (p+1) (p+2)}-\frac{x^{-2 (p+2)} \left(a+b x^2\right)^{p+1}}{2 a (p+2)}",1,"-1/2*((a + b*x^2)^p*Hypergeometric2F1[-2 - p, -p, -1 - p, -((b*x^2)/a)])/((2 + p)*x^(2*(2 + p))*(1 + (b*x^2)/a)^p)","C",1
1064,1,66,53,0.0153425,"\int x^{-4-2 p} \left(a+b x^2\right)^p \, dx","Integrate[x^(-4 - 2*p)*(a + b*x^2)^p,x]","-\frac{x^{-2 p-3} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(-p-\frac{3}{2},-p;-p-\frac{1}{2};-\frac{b x^2}{a}\right)}{2 p+3}","-\frac{x^{-2 p-3} \left(a+b x^2\right)^{p+1} \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2} (-2 p-1);-\frac{b x^2}{a}\right)}{a (2 p+3)}",1,"-((x^(-3 - 2*p)*(a + b*x^2)^p*Hypergeometric2F1[-3/2 - p, -p, -1/2 - p, -((b*x^2)/a)])/((3 + 2*p)*(1 + (b*x^2)/a)^p))","A",1
1065,1,29,30,0.0119542,"\int x^{-3-2 p} \left(a+b x^2\right)^p \, dx","Integrate[x^(-3 - 2*p)*(a + b*x^2)^p,x]","\frac{x^{-2 p-2} \left(a+b x^2\right)^{p+1}}{a (-2 p-2)}","-\frac{x^{-2 (p+1)} \left(a+b x^2\right)^{p+1}}{2 a (p+1)}",1,"(x^(-2 - 2*p)*(a + b*x^2)^(1 + p))/(a*(-2 - 2*p))","A",1
1066,1,66,53,0.0152142,"\int x^{-2-2 p} \left(a+b x^2\right)^p \, dx","Integrate[x^(-2 - 2*p)*(a + b*x^2)^p,x]","-\frac{x^{-2 p-1} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(-p-\frac{1}{2},-p;\frac{1}{2}-p;-\frac{b x^2}{a}\right)}{2 p+1}","-\frac{x^{-2 p-1} \left(a+b x^2\right)^{p+1} \, _2F_1\left(\frac{1}{2},1;\frac{1}{2} (1-2 p);-\frac{b x^2}{a}\right)}{a (2 p+1)}",1,"-((x^(-1 - 2*p)*(a + b*x^2)^p*Hypergeometric2F1[-1/2 - p, -p, 1/2 - p, -((b*x^2)/a)])/((1 + 2*p)*(1 + (b*x^2)/a)^p))","A",1
1067,1,56,43,0.0073384,"\int x^{-1-2 p} \left(a+b x^2\right)^p \, dx","Integrate[x^(-1 - 2*p)*(a + b*x^2)^p,x]","-\frac{x^{-2 p} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(-p,-p;1-p;-\frac{b x^2}{a}\right)}{2 p}","-\frac{x^{-2 p} \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,1;1-p;-\frac{b x^2}{a}\right)}{2 a p}",1,"-1/2*((a + b*x^2)^p*Hypergeometric2F1[-p, -p, 1 - p, -((b*x^2)/a)])/(p*x^(2*p)*(1 + (b*x^2)/a)^p)","A",1
1068,1,65,52,0.0172082,"\int x^{-2 p} \left(a+b x^2\right)^p \, dx","Integrate[(a + b*x^2)^p/x^(2*p),x]","\frac{x^{1-2 p} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2}-p,-p;\frac{3}{2}-p;-\frac{b x^2}{a}\right)}{1-2 p}","\frac{x^{1-2 p} \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,\frac{3}{2};\frac{1}{2} (3-2 p);-\frac{b x^2}{a}\right)}{a (1-2 p)}",1,"(x^(1 - 2*p)*(a + b*x^2)^p*Hypergeometric2F1[1/2 - p, -p, 3/2 - p, -((b*x^2)/a)])/((1 - 2*p)*(1 + (b*x^2)/a)^p)","A",1
1069,1,61,49,0.0174841,"\int x^{1-2 p} \left(a+b x^2\right)^p \, dx","Integrate[x^(1 - 2*p)*(a + b*x^2)^p,x]","\frac{x^{2-2 p} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(1-p,-p;2-p;-\frac{b x^2}{a}\right)}{2-2 p}","\frac{x^{2-2 p} \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,2;2-p;-\frac{b x^2}{a}\right)}{2 a (1-p)}",1,"(x^(2 - 2*p)*(a + b*x^2)^p*Hypergeometric2F1[1 - p, -p, 2 - p, -((b*x^2)/a)])/((2 - 2*p)*(1 + (b*x^2)/a)^p)","A",1
1070,1,65,52,0.0182889,"\int x^{2-2 p} \left(a+b x^2\right)^p \, dx","Integrate[x^(2 - 2*p)*(a + b*x^2)^p,x]","\frac{x^{3-2 p} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{3}{2}-p,-p;\frac{5}{2}-p;-\frac{b x^2}{a}\right)}{3-2 p}","\frac{x^{3-2 p} \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,\frac{5}{2};\frac{1}{2} (5-2 p);-\frac{b x^2}{a}\right)}{a (3-2 p)}",1,"(x^(3 - 2*p)*(a + b*x^2)^p*Hypergeometric2F1[3/2 - p, -p, 5/2 - p, -((b*x^2)/a)])/((3 - 2*p)*(1 + (b*x^2)/a)^p)","A",1
1071,1,61,49,0.0172697,"\int x^{3-2 p} \left(a+b x^2\right)^p \, dx","Integrate[x^(3 - 2*p)*(a + b*x^2)^p,x]","\frac{x^{4-2 p} \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(2-p,-p;3-p;-\frac{b x^2}{a}\right)}{4-2 p}","\frac{x^{4-2 p} \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,3;3-p;-\frac{b x^2}{a}\right)}{2 a (2-p)}",1,"(x^(4 - 2*p)*(a + b*x^2)^p*Hypergeometric2F1[2 - p, -p, 3 - p, -((b*x^2)/a)])/((4 - 2*p)*(1 + (b*x^2)/a)^p)","A",1